Theoretical description of light emission in the presence of nanoscale resonators: from classical scattering to photon states entanglement and statistics

EUSKAL HERRIKO UNIBERTSITATEA THE UNIVERSITY OF THE BASQUE COUNTRY Department of Electricity and Electronics CAMPUS OF INTERNATIONAL EXCELLENCE Theoretical description of light emission in the presence of nanoscale resonators: from classical scattering to photon states entanglement and statistics Thesis by Álvaro Nodar Villa Supervised by Prof. Javier Aizpurua Iriazabal and Dr. Rubén Esteban Llorente Donostia-San Sebastián, February 2023 (cc)2023 ALVARO NODAR VILLA (cc by-sa 4.0) ACKNOWLEDGMENTS Quisiera dar las gracias / I would like to express my gratitude: A Javier y a Rubén, por haberme dado la oportunidad de realizar esta tesis y por haberme dirigido durante todos estos años. Ha sido un viaje largo y ha habido etapas duras, pero posiblemente sin esos retos no estaría tan seguro ni tan orgulloso como lo estoy ahora de cada una de las páginas que siguen en esta tesis. Gracias por ofrecerme vuestro apoyo y vuestra paciencia. He crecido mucho bajo vuestra tutela. A Gabriel, quien también me ha supervisado durante una larga etapa de esta tesis. Gracias, Gabriel, por remangarte y echarte las cuentas conmigo. Gracias por tu dedicación, me alegra mucho haber podido contar contigo durante esta etapa. To Mike and Mikołaj, for welcoming me with open arms in Sydney. The project on the correlations really motivated me through the last stages of my thesis. I want to especially thank Mikołaj for showing me a different way of doing science. Thank you for being so humble and supportive. The last time we went to a pub, you invited me. I hope to have many chances of repaying this debt. A mi gente del segundo piso (en orden alfabético, evitemos favoritismos): Adrián, Alberto, Antton, Bruno, Carlitos, Iker, Jon, Jonathan, Mario, Martín, Mikel, Miriam, Roberto, Txemikel. ¿Qué decir? Hemos compartido el día a día durante años, ya sabéis que os considero muy buenos amigos y que sois una parte importante de mi vida. Un cachito de esta tesis es de cada uno de vosotros. Ya me invitaréis a un café en el Etna, ¿no? I would like to thank the support I received from all the friends I have made along the way in the Donosti scientific community. To the old gang: Andrea, Berni, Borinaga, Donaldi, Fede, Moritz, Thomas, Tomáš... thank you for showing me the ropes! To the new gang: Alex, Auguste, Carmela, Cris (Sanz), Edurne, Fer, Gabriele, Jorge (JOT), Jorge (Melillo), Josa, Joseba, Josu, Maria, Marina, Mathias, Matteo, Miguel Angel, Mikel, Paul, Raulillo, Stefan, Unai, Xabi... esta tesis no habría sido posible sin vuestro apoyo y amistad, gracias por estar a mi lado en esta etapa. A mis padres y a toda mi familia por todo el apoyo incondicional que me habéis brindado durante mi larga etapa de estudiante. Sé que estaréis tan orgullosos de esta tesis como lo estoy yo hoy. iii A Cris, durante estos años he sido (y sigo siendo) muy feliz a tu lado. Estoy muy agradecido por haber compartido cada día de esta etapa contigo. Gracias por apoyarme en los momentos duros y celebrar conmigo cuando las cosas salían bien. Te quiero mucho. Por último, y mucho más importante que el resto, quiero agradecer el apoyo económico brindado por el proyecto PID 2019-107432GB-I00P del Ministerio de Ciencia e Innovación y de la Agencia Estatal de Investigación, así como por el proyecto IT 1526-22 del Gobierno Vasco. Ha sido una inversión muy acertada, os los aseguro. iv RESUMEN Al pasar un dedo por el borde de una copa, su cuerpo vibra creando un sonido. Esto ocurre porque la copa soporta una resonancia mecánica, que se excita al pasar el dedo con una velocidad y presión específicas. Algo similar ocurre con la luz y las nanoestructuras fotónicas. Al iluminar algunas nanoestructuras con una longitud de onda o frecuencia (color) específicas, podemos aumentar la intensidad del campo dispersado por la nanoestructura. Por ejemplo, las nanoestructuras metálicas soportan resonancias plasmónicas que consisten en una oscilación colectiva de su densidad de carga electrónica y que pueden ser excitadas con luz a frecuencias ópticas. La excitación resonante de las corrientes de polarización en el interior de nanoestructuras dieléctricas, es otro ejemplo de cómo la luz puede excitar resonancias ópticas en otros materiales. Cada tipo de nanoestructura ofrece ciertas ventajas para controlar la luz en la nanoescala. Por ejemplo, las nanoestructuras metálicas pueden confinar la luz en volúmenes mucho más pequeños, mientras que las nanoestructuras dieléctricas tienen pérdidas por absorción mucho menores. Gran parte de este tipo de fenómenos se pueden explicar dentro del marco de la teoría de electromagnetismo clásico propuesta por James Clerk Maxwell en 1865 [1]. Desde entonces, se han desarrollado modelos analíticos (y semianalíticos) así cómo métodos numéricos que permiten analizar la respuesta de nanoestructuras fotónicas en condiciones muy generales. Por ejemplo, en esta tesis empleamos la llamada teoría de Mie para obtener una solución semianalítica a las ecuaciones de Maxwell de los campos dispersados por nanopartículas esféricas bajo diversas iluminaciones [2]. Las predicciones que se obtienen resolviendo la respuesta electromagnética clásica de nanoestructuras han sido de gran utilidad para el desarrollo de diversas aplicaciones fotónicas basadas en el aumento y la localización del campo electromagnético. Dentro de estas aplicaciones, cabe destacar las técnicas de espectroscopía aumentada por superficies [3 – 5], las técnicas de microscopía con resolución submolecular [6 – 8], los tratamientos de cáncer [9,10], ó las mejoras en la captación y almacenamiento de energía solar [11,12]. Por otra parte, los avances recientes en la fabricación y caracterización de nanoestructuras fotónicas han permitido alcanzar un nivel de precisión lo suficientemente alto como para demostrar diversos efectos cuánticos, lo cual ha generado un creciente interés en el campo de la nanofotónica cuántica durante las últimas décadas [13,14]. La nanofotónica cuántica permite describir fenómenos muy variados. Por v ejemplo, el campo electromagnético aumentado que genera una nanoestructura fotónica puede dar lugar a una fuerte interacción óptica con un emisor cuántico (por ejemplo, un punto cuántico o una molécula) situado en su entorno. Si esta interacción es muy fuerte se pueden dar fenómenos no lineales en la respuesta del sistema emisor-nanoestructura que solo pueden ser descritos con un formalismo cuántico. Al mismo tiempo, la interacción de nanoestructuras con estados de luz cuánticos es particularmente interesante por su potencial aplicación en diversas tecnologías de información cuántica. Los estados cuánticos de luz son muy resistentes a perder la información codificada en ellos al mantener su propagación, constituyéndose en excelentes candidatos para la transmisión de información cuántica. Sin embargo, las posibilidades de modificar la información codificada en estados de luz cuánticos se ven limitadas debido a la débil interacción de la luz con la materia. Una solución prometedora a este reto es aprovechar la interacción amplificada entre la luz y materia que se da al excitar las resonancias ópticas de las nanoestructuras. Esta tesis está dedicada a estudiar la interacción entre estados de luz clásicos y cuánticos con nanoestructuras fotónicas aisladas o interaccionando con emisores cuánticos. A continuación se presenta una discusión resumida de los contenidos principales de cada capítulo: En el capítulo 1 revisamos algunos aspectos de los fundamentos del electromagnetismo clásico que emplearemos en esta tesis. En concreto mostramos el tratamiento teórico para entender la respuesta a la luz de una nanopartícula esférica, una nanoestructura canónica en el campo de la nanofotónica que nos permite ilustrar diversos fenómenos de la interacción de la luz con nanoestructuras fotónicas resonantes. Para el estudio de este sistema consideramos varias descripciones. Primero introducimos la aproximación cuasiestática, válida para describir la respuesta de nanopartículas con un diámetro mucho más pequeño que la longitud de onda de la luz incidente. La aproximación cuasiestática nos permite explicar de una manera intuitiva los aspectos principales de la física de estos sistemas. Por ejemplo, describimos cómo la luz puede excitar las resonancias plasmónicas. También discutimos cómo se puede mejorar este modelo aproximado para incluir la corrección radiativa que tiene en cuenta las pérdidas de la nanoestructura al dispersar la luz incidente. Estos modelos aproximados son utilizados en el capítulo 3para analizar los diferentes mecanismos físicos que intervienen en la interacción entre una nanoestructura y un emisor cuántico. Junto con estas descripciones aproximadas, en el capítulo 1también introducimos formalmente la teoría semianalitica de Mie. En concreto, la solución de Mie permite expresar los campos dispersados por una nanopartícula esférica como una suma de distintas contribuciones, cada una de ellas correspondiendose con la excitación de diferentes modos resonantes de la nanopartícula. A lo largo de esta tesis utilizamos el modelo de Mie para analizar la interacción de la luz con nanopartículas esféricas. Finalmente, en el capítulo 1describimos las propiedades del momento angular de la luz (espín, helicidad, momento angular orbital y momento angular total), y revisamos la interacción entre haces de luz con propiedades de momento angular bien vi definidas y nanopartículas esféricas. Para ello extendemos la solución semianalitica de la teoría de Mie para describir la dispersión de este tipo de haces. Este tipo de problemas de dispersión son muy interesantes en el campo de la nanofotónica, ya que el momento angular de la luz introduce nuevos grados de libertad que pueden ser controlados con nanoestructuras y que presentan un gran potencial en diversas aplicaciones tecnológicas, por ejemplo, para aumentar la información codificada en un haz de luz [15–18] ó para detectar propiedades de substancias químicas (como la quiralidad de las moléculas) [19,20]. En el capítulo 2 revisamos algunos de los fundamentos de la nanofotónica cuántica que emplearemos en el resto de esta tesis. En particular, nos centramos en dos problemas canónicos. Primero estudiamos cómo se puede tratar la transformación de la luz incidente con un divisor de haz. Esta transformación es la base formal de nuestro estudio presentado en el capitulo 5. De manera más general, el divisor de haz es el elemento principal en muchos interferómetros, por ejemplo, el interferómetro Hanbury-Brown-Twiss (HBT) que empleamos en el capítulo 4. El interferómetro HBT permite la caracterización de la estadística del número de fotones emitidos por una fuente, y en el capitulo 2discutimos detalladamente cuál es la base de su funcionamiento y cómo dicha caracterización nos permite clasificar diferentes tipos de luz dependiendo de si los fotones tienden a llegar individualmente o en grupos. El segundo problema estudiado en este capítulo es la descripción cuántica de la interacción entre una cavidad óptica (tal cómo una nanoestructura resonante) y un emisor cuántico basada en el formalismo de electrodinámica cuántica de cavidades (cavity-quantum electrodynamics, en inglés). En concreto presentamos la formulación del Hamiltoniano que describe esta interacción según el modelo cuántico de Rabi (QRM, quantum Rabi model en inglés), el cuál es válido para cualquier valor de la energía de interacción entre la cavidad y el emisor. También introducimos el formalismo de la ecuación maestra (master equation en inglés) que describe la dinámica de un sistema cuántico interaccionando con su entorno, y que nos permite introducir las pérdidas de la cavidad y la iluminación incoherente del emisor. El Hamiltoniano del QRM y la ecuación maestra son las principales herramientas que utilizamos en el capítulo 4para describir la emisión de sistemas formados por un emisor cuántico interaccionando con una nanoestructura. El capítulo 3 está dedicado al estudio de la asimetría en las resonancias Fano que emergen en el espectro de extinción de sistemas formados por una nanoestructura metálica interaccionando débilmente con un emisor cuántico. Este tipo de sistemas emisor-nanoestructura han sido estudiados extensivamente en el contexto de las técnicas de espectroscopía de campo aumentado (surfaceenhanced spectroscopy en inglés), dónde el campo aumentado generado al excitar las resonancias plasmónicas de la nanoestructura metálica se usa para mejorar la señal espectral de una molécula en la cercanía de la nanoestructura, lo que permite detectar y caracterizar cantidades muy pequeñas de moléculas. Cuándo la fuerza de acoplamiento entre el emisor y la nanoestructura es débil, el espectro del sistema emisor-nanoestructura se caracteriza por la aparición de una resonancia tipo Fano, la cual es originada por la interferencia entre una resonancia vii espectralmente estrecha correspondiente al emisor y otra mucho más ancha (que se comporta como un continuo de modos) correspondiente a la nanoestructura metálica. Las resonancias de tipo Fano se identifican por la aparición de un cambio de la señal abrupto en una región espectral estrecha, dando lugar a lo que se denomina un perfil Fano. La forma de este perfil varía con la naturaleza de la interacción emisor-nanoestructura, y puede presentar varios grados de asimetría, pero un modelo sencillo predice que la resonancia Fano es perfectamente simétrica si el sistema es resonante, es decir, cuándo la resonancia del excitón del emisor y de la resonancia óptica de la nanoestructura tienen la misma frecuencia central. Sin embargo, trabajos experimentales recientes han demostrado que la resonancia Fano puede ser asimétrica incluso en sistemas resonantes [8]. Para entender mejor esta observación experimental, analizamos en detalle el origen de la asimetría en las resonancias de tipo Fano en sistemas emisornanoestructura resonantes. Para ello empleamos simulaciones numéricas de la respuesta óptica del sistema híbrido en tres tipos de nanoestructuras diferentes (una nanopartícula esférica de plata, una nanopartícula esférica de oro, y un dímero compuesto por dos nanopartículas esféricas de oro, todas ellas iluminadas por una onda plana), así como una serie de modelos analíticos basados en sistemas de dos osciladores armónicos acoplados. De esta manera podemos identificar cinco efectos que producen la asimetría en la señal Fano: (i) la fase que adquieren los campos inducidos por la nanoestructura al propagarse hasta el emisor cuántico (y viceversa), (ii) la excitación directa del emisor cuántico por la luz incidente sobre el sistema, así como la emisión directa del emisor cuántico al detector, (iii) las pérdidas radiativas de la nanoestructura, (iv) la contribución de los electrones de valencia a la constante dieléctrica de la nanoestructura metálica, y (v) la contribución de las resonancias de orden alto que soporta la nanoestructura. Los dos primeros efectos (la fase de propagación y la excitación y emisión directa) son claves para explicar el origen de la asimetría en todos los sistemas considerados. El impacto de la contribución de los electrones de valencia depende del material considerado. Por ejemplo, encontramos que esta contribución afecta fuertemente a la asimetría en las nanoestructuras de oro, mientras que en las nanoestructuras de plata es mucho menor. Por otra parte, la contribución de las resonancias de orden alto es pequeña en los sistemas considerados, pero puede adquirir más importancia en otro tipo de sistemas e iluminaciones, cómo la iluminación por la corriente túnel de un microscopio de efecto túnel (scanning tunneling microscope, en inglés). Por último, es necesario considerar las pérdidas radiativas de la nanoestructura para obtener una correcta descripción de la interacción emisor-nanoestructura, y por tanto para capturar correctamente la influencia de todos los otros efectos en la asimetría. En el capítulo 4 seguimos considerando un sistema formado por un emisor cuántico interaccionando con una nanoestructura metálica en condiciones resonantes. Sin embargo, a diferencia del capítulo anterior, en el capítulo 4consideramos un amplio rango de energías de acoplamiento entre la nanoestructura y el emisor, pasando del régimen de acoplamiento débil (dónde el intercambio de excitaciones viii entre la nanoestructura y el emisor es más lento que la disipación de la energía incidente por el sistema híbrido), al régimen de acoplamiento fuerte (dónde el intercambio de excitaciones es más rápido que la disipación, de forma que aparecen nuevos estados híbridos), y por último al régimen de acoplamiento ultrafuerte (caracterizado por la aparición de fenómenos no lineales asociados con términos que no conservan el número de excitaciones). Además, en este capítulo consideramos situaciones donde la iluminación es muy intensa, y para describir correctamente la respuesta del emisor consideramos que este actúa como un sistema de dos niveles (two level system, en inglés), lo cual puede introducir fenómenos no lineales adicionales (por ejemplo, el denominado bloqueo de fotones, photon blockade, en inglés). En este capítulo estudiamos las correlaciones de intensidad de la luz emitida por este sistema híbrido bajo una excitación incoherente del emisor, lo cual requiere ir más allá de la descripción clásica. Para ello utilizamos dos modelos cuánticos diferentes. Primero introducimos una formulación del QRM desarrollada recientemente que es válida para cualquier régimen de acoplamiento. En segundo lugar consideramos el modelo Jaynes-Cummings (JCM, Jaynes-Cummings model, en inglés), el cual puede derivarse a partir del QRM trás aplicar la aproximación de onda rotante (RWA, rotating wave approximation, en inglés) que desprecia los términos que no conservan el número de excitaciones en el Hamiltoniano del QRM. Esta aproximación no es válida para describir el régimen de acoplamiento ultrafuerte, donde los términos que no conservan el número de excitaciones se vuelven más importantes, sin embargo ha sido utilizada con éxito para simplificar el análisis de sistemas acoplados débilmente [21,22]. Al comparar las correlaciones de intensidad calculadas con el QRM y con el JCM observamos que, en el régimen de acoplamiento ultrafuerte, el QRM predice una emisión amontonada (bunched, en inglés), mientras que el JCM predice una emisión anti-amontonada (antibunched, en inglés). Sorprendentemente, bajo iluminaciones débiles, esta diferencia no solo se da en el régimen de acoplamiento ultrafuerte, sino que también ocurre en el régimen fuerte y débil, dónde se esperaría que el QRM y el JCM coincidieran. A continuación, analizamos en detalle la influencia de cada autoestado del sistema en la emisión, y encontramos que la emisión amontonada en el QRM se puede atribuir al decaimiento de un sólo autoestado, el polariton |3−⟩R (correspondiente al quinto estado excitado del sistema bajo energías de acoplamiento pequeñas). Debido a la presencia de términos que no conservan el número de excitaciones en el Hamiltoniano del QRM, el estado |3−⟩R puede ser excitado de manera directa desde el estado base (ground state, en inglés), así cómo decaer emitiendo múltiples fotones de manera simultánea, lo que produce la emisión amontonada del sistema. Por el contrario, en el marco del JCM el estado análogo |3−⟩ sólo puede excitarse de manera secuencial, mediante tres procesos de absorción, un mecanismo mucho menos eficiente para intensidades y fuerzas de acoplamiento suficientemente bajas. Recalcamos que esta diferencia entre el JCM y el QRM se puede extender al régimen de acoplamiento débil, donde normalmente se espera que los resultados del JCM y el QRM coincidan, lo que indica que los términos que no conservan el número ix Introduction emitter (QE, e.g. a quantum dot or a molecule) located in the proximity of the nanoparticle. If this interaction is very strong, a variety of nonlinear phenomena can occur in the response of the QE-nanostructure system that can be better described through a quantum framework. At the same time, the interaction of nanostructures with quantum states of light is particularly interesting for various quantum information technologies. Quantum states of light are very resilient to the loss of their information through propagation, establishing themselves as excellent candidates for the transmission of quantum information. However, the possibilities of controlling the information encoded in quantum light states are limited by the weak interaction of light with matter. A promising solution to this challenge is to exploit the amplified interaction between light and matter that occurs upon the excitation of optical resonances in nanostructures. This thesis is devoted to studying the interaction between classical and quantum states of light with photonic nanostructures and QE-nanostructure systems. In the following, we introduce a summary of the main contents of each chapter. In chapter 1 we summarize some of the fundamentals of classical electromagnetism that we use in this thesis. In particular, we review the response to light of a spherical nanoparticle, a canonical nanostructure in the field of nanophotonics that allows us to illustrate various phenomena of the interaction of light with resonant photonic nanostructures. We use different approaches to describe the optical response. First, we introduce the quasi-static approximation, valid for describing the response of nanoparticles with a size much smaller than the incident wavelength of light. The quasi-static approximation allows for explaining, in an intuitive way, the main aspects of light-matter interaction. For example, we describe how light can excite the plasmonic resonances in metallic nanoparticles. We also discuss how this approximated model can be improved to include the radiative correction that takes into account the increase in the losses of the nanostructure due to the scattering of light. We use these approximated models in chapter 3 to analyze the different aspects of the interaction between a QE and a metallic nanostructure. Along with these approximate descriptions, in chapter 1we formally introduce the semianalytical Mie theory that describes the exact solution of Maxwell’s equations to the field scattered by a spherical nanoparticle. Specifically, Mie’s solution allows for expressing the field scattered by a nanoparticle as a sum of different contributions, each corresponding to the excitation of different resonant modes of the nanoparticle. Throughout this thesis, we use Mie theory to analyze the interaction of light with spherical nanoparticles. Finally, in chapter 1we describe the angular momentum properties of light (spin, helicity, orbital angular momentum, and total angular momentum), and review the interaction between light beams with well-defined angular momentum properties and nanoparticles. For this purpose, we extend the semi-analytic solution of Mie theory to describe the scattering of such beams by a spherical nanoparticle. Such scattering problem is very interesting in the field of nanophotonics, since the angular momentum of light introduces new degrees of freedom that can be controlled with nanostructures and have great potential in a variety of applications, for example, 2 Introduction to increase the information encoded in a light beam [15 – 18] or to detect additional properties of chemical substances (such as the chirality of molecules) [19,20]. In chapter 2 we review the fundamentals of quantum nanophotonics that we use in this thesis. In particular, we focus on two canonical problems. First, we describe the transformation of a state of light by a beam splitter. This transformation is the formal basis of the study presented in chapter 5. Moreover, the beam splitter is the main element in many interferometers, for example, in the Hanbury-Brown-Twiss (HBT) interferometer that we use in chapter 4. The HBT interferometer enables the characterization of the statistics of the number of photons emitted by a light source. In chapter 2, we discuss in detail what is the basis of the HBT operation and how it allows us to characterize different types of light depending on whether the photons are emitted mostly separately (antibunched) or in groups (bunched). The second problem studied in this chapter is the quantum description within cavity Quantum Electrodynamics (cavity-QED) of the interaction between an optical cavity (such as a resonant nanostructure) and a QE. In particular, we present the formulation of the Hamiltonian that describes this interaction according to the quantum Rabi model (QRM), which is valid for any value of the interaction strength between the cavity and the QE. We also introduce the master equation formalism that describes the dynamics of a quantum system interacting with its environment and allows us to describe the losses and the incoherent illumination of the system. The QRM Hamiltonian and the master equation are the two pillars we use in chapter 4to describe the light emission from systems consisting of a QE interacting with a nanostructure. Chapter 3 is devoted to the study of the asymmetry in the Fano resonances that emerges in the extinction spectrum of systems formed by a metallic nanostructure interacting weakly with a QE. Such QE-nanostructure systems have been extensively studied in the context of enhanced field spectroscopy techniques, where the enhanced field generated by exciting the plasmon resonances of the metallic nanostructure is used to increase the spectral signal of the molecule. These techniques allows for the detection and characterization of very small quantities of molecules. When the coupling strength between the QE and the nanostructure is weak, the extinction spectrum of the hybrid QE-nanostructure system is characterized by the appearance of a Fano-type resonance, caused by the interference between a spectrally narrow resonance corresponding to the QE and a much wider one (which behaves as a continuum of modes) corresponding to the metallic nanostructure. Fano resonances are identified by the appearance of a sharp spectral feature, which is the so-called Fano lineshapes. These lineshapes depend on the nature of the QE-nanostructure interaction and can exhibit different degrees of asymmetry. In particular, a simple model indicates that a Fano resonance is perfectly symmetric if the system is resonant, i.e. when the central frequency of the (excitonic) resonance of the QE is tuned to the central frequency of the optical resonance of the nanostructure. However, recent experimental work has shown that the Fano resonance can be asymmetric even in resonant systems. To better understand this experimental observation, we analyze in detail the origin of the asymmetry in the Fano resonances of zero-detuned QE-nanostructure 3 Introduction systems. For this purpose, we analyze the optical response in three different types of nanostructures (a spherical silver nanoparticle, a spherical gold nanoparticle, and a dimer composed of two spherical gold nanoparticles, all illuminated by a plane wave) using numerical simulations and a series of analytical models based on coupled harmonic oscillators. In this way, we identify the different physical mechanisms that originate the Fano asymmetry in zero-detuned QE-nanostructure systems. In chapter 4 we also consider a QE interacting with a nanostructure under resonant conditions. However, unlike the previous chapter, in chapter 4we consider a wide range of coupling strengths between the nanostructure and the QE, ranging from the weak coupling regime (where the exchange of excitations between the nanostructure and the QE is slower than the dissipation of the incident energy by the system), to the strong coupling regime (where the exchange of excitations is faster than the dissipation so that new hybrid states appear), and finally to the ultra-strong coupling regime (the coupling strength becomes so large that nonlinear phenomena associated with terms that do not conserve the number of excitations becomes possible). Furthermore, in this chapter, we take into account that an (excitonic) transition of a QE acts as a two-level-system, which introduces additional nonlinear phenomena under strong illumination (for example, the so-called photon blockade). We study the intensity correlations of the light emitted by this hybrid QEnanostructure system under incoherent excitation of the QE. This study requires going beyond the classical description. In particular, we use two different quantum models. First, we introduce a recently-developed formulation of the QRM that is valid for any coupling regime. Second, we consider the Jaynes-Cummings model (JCM), which can be derived from the QRM after applying the rotating wave approximation (RWA) that neglects the terms that do not preserve the number of excitations in the Hamiltonian of the QRM. This approximation is known to fail in the ultra-strong coupling regime, where the role of the terms that do not conserve the number of excitations becomes more important, but it has been successfully used to simplify the analysis of many systems in the weak and strong coupling regime [21,22]. We compare the intensity correlations calculated with the QRM and the JCM, and observe that, in the ultra-strong coupling regime, the QRM predicts a bunched emission while the JCM predicts an antibunched emission. Surprisingly, under weak illuminations, this difference does not only occurs in the ultra-strong coupling regime but also in the strong and weak regimes, where the QRM and JCM are expected to agree. In this chapter, we analyze in detail the origin of the bunched emission in the QRM and the deviation between the QRM and the JCM, i.e. the breakdown of the RWA. In chapters 3and 4we focus on the response of nanostructures illuminated with classical beams. In contrast, in chapter 5 we consider a nanostructure illuminated by a quantum state of light. Specifically, we study the response of rotationally symmetric nanostructures illuminated by a quantum state composed of two entangled photons with information encoded in their angular momentum 4 Introduction properties. On the one hand, such states are particularly interesting for quantum information processing applications, since the angular momentum of light is a property that is not limited to two values like the spin of a trapped ion qubit. The orbital angular momentum of light opens an (in principle) infinite Hilbert space to encode information. On the other hand, rotationally symmetric nanoparticles preserve the total angular momentum of the state after scattering, thus offering the possibility of manipulating the quantum state in a controlled manner. This control in the manipulation of a quantum state of light is very interesting for processing its quantum information, but requires that the quantum purity of the incident state is respected to a high degree. We develop a general theoretical formalism to model the scattering process in these systems, which is based on the transformation of a quantum state by a lossy beam splitter. Using this formalism we calculate the output state scattered by the nanostructure and find that the purity of the incident state can be lost in the interaction with the nanostructure. We then develop a semi-analytical model based on treating the quasi-monochromatic input and output modes that allows us to identify the physical mechanism that causes the loss of purity. In summary, the research presented in this thesis advances our understanding of fundamental physical aspects of the interaction of both classical light and quantum states of light with nanostructures. 5 Chapter 1 CLASSICAL DESCRIPTION OF THE INTERACTION BETWEEN LIGHT AND MATTER AT THE NANOSCALE The main topic of this thesis is the interaction between light and matter at the nanoscale. In this chapter, we introduce the theoretical framework that we use to study the interaction between classical states of light and a nanostructure. Nanostructures can be used to control light at the nanoscale, allowing for concentrating incident electromagnetic fields in very small regions [3,23 – 26], change the polarization properties of incident light [27 – 29], or generate light of a frequency different than that of the illumination [30 – 32]. Most of these effects can be described within the theory of classical electromagnetism, where light-matter interaction can be captured Maxwell’s equations, a set of differential equations that describe how electromagnetic fields evolve in time and space in a particular dielectric configuration. By applying the appropriate boundary conditions, Maxwell’s equations can be solved to obtain the response of an arbitrary nanostructure under specific illumination. In section 1.1 of this chapter, we review the formulation of Maxwell’s equations. In sections 1.2 and 1.3 we review the analytical and semi-analytical solutions of Maxwell’s equations for a canonical nanostructure: a spherical nanoparticle. The analytical or semi-analytical solutions allow us to discuss the general properties of the optical response. Specifically, in section 1.2 we treat the nanoparticle as a polarizable object that behaves as an electric point-like dipole, a commonly used approximation in nanophotonics. We also discuss how a similar approach can be used to describe the optical response of molecules and other quantum emitters (QEs). On the other hand, in section 1.3 we introduce a semi-analytical solution to Maxwell’s equations (obtained without any approximation) for the 7 Chapter 1. Fundamentals of classical nanophotonics fields scattered by a spherical nanoparticle. We evaluate these frameworks in a canonical configuration, the scattering of linearly polarized light by a metallic spherical nanoparticle. Finally, in section 1.4 we expand the formalism introduced in section 1.3 to describe the scattering of a spherical nanoparticle illuminated by complex beams of light, in particular, beams of light with well-defined angular momentum properties, as those used in chapter 5. 1.1 Maxwell’s equations In 1865 the Scottish mathematician James Clerk Maxwell published “A Dynamical Theory of the Electromagnetic Field” [1], a paper containing the original formulation of his famous equations showing the interrelationship between electric fields E ( r, t ) and magnetic fields B ( r, t )in a dielectric medium ( E ( r, t )and B ( r, t )are evaluated at a position r and time t ). Nineteen years later, in 1884, the English mathematician Oliver Heaviside used his developments in vectorial and complex number calculus to reformulate Maxwell’s equations into the form that has been known ever since [33], ∇×E(r, t) = −∂B(r, t) ∂t , ∇×H(r, t) = ∂D(r, t) ∂t +Jext(r, t), ∇·D(r, t) = ρext(r, t), ∇·B(r, t)=0,(1.1) where Jext is the external current density, and ρext is the external charge density. The electric field displacement is D ( r, t ) = ε0E ( r, t ) + PD ( r, t ), where PD is the polarization field of the medium and ε0 is the electric permittivity in a vacuum. Similarly, the magnetizing field is H ( r, t ) = B ( r, t ) /µ0−MB ( r, t ), with MB the magnetization field of the medium and µ0 the magnetic permeability in a vacuum. In this thesis, we assume linear light-matter interaction, with PD ( r, t ) ∝E ( r, t ) and MB ( r, t ) ∝B ( r, t )and we assume that the materials we are treating are isotropic. Thus, we introduce the constitutive relationship [34 – 36] of electric field displacement and the magnetizing field, D(r, t) = ε0εE(r, t),(1.2) and B(r, t) = µ0µH(r, t),(1.3) where ε and µ are the relative dielectric and relative magnetic permittivity of the medium, respectively. For convenience, in this thesis, we treat the fields in the frequency domain, with E ( r, ω ) = ´dtE ( r, t ) eiωt/ (2 π )and B ( r, ω ) = ´dtB ( r, t ) eiωt/ (2 π ), respectively, where ω is the (angular) frequency of light. Maxwell’s equations (1.1) can then be 8 1.1. Maxwell’s equations written as: ∇×E(r, ω) = iωB(r, ω), ∇×H(r, ω) = Jext(r, ω) + iωD(r, ω), ∇·D(r, ω) = ρext(r, ω), ∇·B(r, ω) = 0.(1.4) Furthermore, for all the systems studied in this thesis, we consider the case where there are no external currents or charges [37], i.e., Jext = 0 and ρext = 0, so that Eq. (1.4) simplifies to, ∇×E(r, ω) = iωB(r, ω), ∇×B(r, ω) = iω ε0ε µ0µE(r, ω), ∇·D(r, ω)=0, ∇·B(r, ω)=0,(1.5) where we have also accounted for the constitutive relationship in Eqs. (1.2) and (1.3). In 1901 the German mathematician Heinrich Weber reformulated Eq. (1.5) in vacuum (ε= 1 and µ= 1) as [37] i∂F±(r, t) ∂t =c0∇×F±(r, t), ∇·F±(r, t)=0,(1.6) where c0= 1/√ε0µ0is the speed of light in vacuum and F±(r, t) = E(r, t)±ic0B(r, t),(1.7) are the Riemann–Silberstein vectors named after the German mathematician Bernhard Riemann, who inspired Heinrich Weber to publish Eq. (1.6) , and after the Polish-American physicist Ludwik Silberstein, who also published this equation independently of Weber’s work in 1907 [38,39]. Equation (1.6) can be easily written in the frequency domain, ∇× kF±(r, ω) = ±F±(r, ω),(1.8) where F±(r, ω) = E(r, ω)±ic0B(r, ω),(1.9) and k=ω/c0is the wavevector of light. In this thesis, we mostly use the standard formulation in Eq. (1.5) that considers the electric and magnetic fields E and B . However, we also discuss the advantages of the Riemann–Silberstein formalism (Eq. (1.8) ) when treating light 9 Chapter 1. Fundamentals of classical nanophotonics k E0 R x z εout εin Figure 1.1: Scheme of the problem studied in subsections 1.2 and 1.3. A x -polarized plane wave that propagates along the z -axis interacts with a spherical nanoparticle of radius R . We have explicitly indicated the relative dielectric permittivity inside and outside the nanoparticle, εin and εout, respectively. with well-defined angular momentum in section 1.4. 1.2 Electromagnetic response of very small objects Throughout this thesis, we often calculate the electromagnetic response of very small objects, such as a small nanoparticle or a quantum emitter (QE, e.g., a molecule, a quantum dot, or a nitrogen-vacancy center in diamond). This section discusses a very common approach to solving this problem: considering that the small object is excited (or polarized) by the illumination and behaves as a point-like electric dipole [35]. During this section we focus on describing the separate response of a nanoparticle and a QE. In chapter 3, we use the same framework introduced here to describe the response of a nanoparticle interacting with a QE. This section is structured as follows: we first introduce, in subsection 1.2.1, the fields induced (or scattered) by a spherical nanoparticle under plane-wave illumination treated within the quasistatic approximation. Next, in subsection 1.2.2, we briefly formulate the response beyond this quasistatic approximation. In subsection 1.2.3, we use the framework developed in the previous subsections to illustrate the far-field spectral response of the nanoparticle. Finally, in subsection 1.2.4 we use a similar formalism to treat the response of a QE. 1.2.1 Response of a very small spherical nanoparticle within the quasistatic approximation Near-field response We first review the solution of the near fields induced (scattered) by a very small spherical nanoparticle of radius R much smaller than the wavelength of light λ . The material of the nanoparticle has a relative dielectric permittivity εin and the 10 1.2. Electromagnetic response of very small objects medium outside the nanoparticle has a relative dielectric permittivity εout . We consider the particle to be illuminated by an incident plane wave of amplitude E0 , polarized along the x -axis, and propagating along the z -axis (Fig. 1.1). First, we consider that the particle is very small compared to the wavelength, and thus we can assume the electric field to be constant along the space. This is the so-called electrostatic approximation, within which the electromagnetic fields fulfill ∇×E(r, ω)=0, ∇×H(r, ω)=0, ∇·D(r, ω)=0, ∇·B(r, ω)=0.(1.10) in the frequency domain, where we have already accounted for the absence of external currents or charges, i.e.,Jext = 0 and ρext = 0. To solve E in Eq. (1.10) , we notice first that the rotational of the electric field is zero. Thus, we can write Eas the gradient of a scalar function [35,36,40], E(r, ω) = −∇VE(r, ω)(1.11) where the scalar function, VE , is the electric or electrostatic potential. Using the standard constitutive relations [34 – 36] of the displacement vector ( D ( r, ω ) ∝ E ( r, ω )) in Eq. (1.2) , and ∇·D ( r, ω )=0(in Eq. (1.10) ) we find that VE must satisfy Laplace equation, ∇2VE(r, ω)=0.(1.12) The fields outside the nanoparticle include the incident plane wave and the fields scattered by the nanoparticle. However, we consider that the fields induced by the nanoparticle decay with the distance, and thus, far from the particle Eout reduces to the incident plane wave, lim r→∞ Eout(r, ω) = E0ux,(1.13) with E0ux the electric field of the incident plane wave ( ux is the unity vector along the x -axis). The boundary conditions at the surface of the spherical nanoparticle, R, are given by [35,36], n×[Eout(R, ω)−Ein(R, ω)] = 0,(1.14) and n·[Dout(R, ω)−Din(R, ω)] = 0.(1.15) The solution of Eqs. (1.12)-(1.15) is [36], Ein(r, ω) = 4πε0 3εout(ω) εout(ω)+2εin(ω)E0uz,(1.16) 11 Chapter 1. Fundamentals of classical nanophotonics λ = 350 nm R = 70 nm z x Figure 1.5: Far-field directivity of a R = 70 nm radius silver nanoparticle illuminated by a x -polarized plane wave propagating along the z -direction with λ = 350 nm. The results are obtained within the quasistatic approximation (using Eqs. (1.20) , (1.35) , and (1.36) ). The directivity is shown for different θs-angles along the xz-plane. towards λ≈380 nm. Last, for the largest radius considered, R = 70 nm, the maximum enhancement factor spectra obtained with the quasistatic approximation (Fig. 1.2b) is almost an order of magnitude larger than the result in Fig. 1.4a. The main cause for this effect is the radiative correction, which according to Eqs. (1.29) and (1.36) scales with R3 . Similar to the R = 30 nm case, the radiative correction not only causes a decrease in the maximum enhancement, but it broadens the enhancement peak, and induces a red-shift in the response. Without the radiative correction, the peak of the R = 70 nm nanoparticle was centered at λ≈ 350 nm, and with the radiative correction, it is centered at λ≈ 450 nm. We show in section 1.3 that the radiative-corrected model captures the main changes of the lowest-energy resonance with increasing radius but becomes inaccurate as R increases and does not capture some phenomena as the excitation of higher order resonances. Figure 1.4b shows the enhancement factor distribution for a silver spherical nanoparticle of radius R = 70 nm illuminated at λ = 350 nm (we only evaluate the scattered fields outside the nanoparticle). The enhancement is significantly reduced within the radiative corrected model, but the region of stronger fields is still localized near the nanoparticle in a very similar way to the quasistatic solution (Fig. 1.2a). Far-field emission Next, we evaluate the fields scattered by the nanoparticle in the far-field region ( r≫λ ) within the radiative-corrected model. Figure 1.5 shows the far-field directivity for a R = 70 nm silver spherical nanoparticle illuminated at λ = 350 nm by the same x -polarized, z -propagating plane wave. The directivity is defined 18 1.2. Electromagnetic response of very small objects as [42] Da(ϕs, θs) = 4π ‚|EF F sca (rs, ϕs, θs, ω)|2dΩ|EF F sca (rs, ϕs, θs, ω)|2,(1.37) with rs≫λ (in the far-field). Da describes the emission of the fields in a ( θs, φs ) direction (in this case, we only show the dependence on θs because the emission is φs -symmetric). The directivity pattern in Fig. 1.5 shows two lobes oriented along the z -axis, where the far-field emission of the nanoparticle is maximized. Interestingly, these lobes are oriented along the direction of propagation of the incident plane wave, contrary to the near-fields generated by the nanoparticle, which are concentrated along the x-axis (polarization of the incident plane wave). 1.2.3 Optical cross-sections of a small particle The radiative corrected model introduced in this section allows us to obtain the far-field response of a spherical nanoparticle excited by a linearly polarized plane wave of amplitude E0 . The far-field spectral response of nanostructures is usually characterized by three different quantities: the absorption cross-section σabs , the scattering cross-section σsca , and the extinction cross-section σext . The absorption cross-section can be related to the power dissipated by the nanoparticle, Pabs , as Pabs ( ω ) = σabs ( ω ) I0 , where I0 is the intensity of the incident light. On the other hand, the scattering cross section relates to the power of the light elastically scattered in all directions by the nanoparticle as Psca ( ω ) = σsca ( ω ) I0 . Finally, the extinction cross-section corresponds to the sum of the absorption and scattering cross-sections, σext ( ω ) = σsca ( ω )+ σabs ( ω ), and describes the total power extincted: Pext ( ω ) = σext ( ω ) I0 . In this section we introduce all the optical cross sections for a particle in a vacuum, and the extension to other non-dissipating and linear media is straightforward. The power scattered by a nanoparticle can be obtained according to Poynting’s theorem [35,36,43] as, Psca(ω) = ¨A ¯ Ssca(r, ω)·nAdA,(1.38) where A is an arbitrary closed area surrounding the dipole, nA is the normal vector to this surface at each surface point, and ¯ Ssca(r, ω) = 1 2Re{Esca(r, ω)×Hsca(r, ω)∗}(1.39) is the time-averaged Poynting vector of the scattered electromagnetic fields. Using σsca ( ω ) = Psca ( ω ) /I0 , the expressions of Esca and Hsca in Eqs. (1.20) - (1.25) , and pa(ω) = αRC a(ω)E0, we obtain σsca(ω) = k4 6πε0|αRC a(ω)|2.(1.40) 19 Chapter 1. Fundamentals of classical nanophotonics The extinction cross-section, σext , can be directly obtained by using the optical theorem for a system driven by a linearly x -polarized plane wave propagating along z . The optical theorem relates σext with the x -component of the field scattered by the nanoparticle at some point zd≫λ(far-field) along the z-axis [36,40,44], σext(ω)=2λIm zd Esca(zd, ω)·ux E0(zd, ω)·ux,(1.41) where ux is the unity vector along the x -axis. By using Eqs. (1.20) , (1.35) , (1.36) , and (1.41) we obtain, σext(ω) = k ε0 Im{αRC a(ω)}.(1.42) Note that we have written σsca and σext in Eqs. (1.40) and (1.42) in terms of the radiative-corrected polarizability, but an equivalent expression can be found for the quasistatic approximated model, by considering the quasistatic polarizability αagiven in Eq. (1.29) instead of αRC a. Last, the absorption cross-section can be obtained from the extinction crosssection as σabs(ω) = σext(ω)−σsca(ω).(1.43) Figure 1.6 shows the evaluation of the absorption (red lines), extinction (black lines), and scattering (blue lines) cross-sections spectra for silver spherical nanoparticles of different radius. All cross-sections spectra are normalized to πR2 , the area of the geometrical cross-section of the spherical nanoparticle ( σ ( ω ) / ( πR2 )is also called the efficiency factor. Figures 1.6(a), (c), and (e) are obtained considering the radiative corrected polarizability, and Figs. 1.6 (b), (d), and (f) are obtained using the quasistatic polarizability, i.e., substituting αRC a (Eq. (1.36) ) by αa (Eq. (1.29)) in Eqs. (1.40) and (1.42). Similarly to the near-field enhancement spectra in Figs. 1.2b and 1.4a, the extinction cross-section spectrum of the nanoparticles is dominated by a single peak corresponding to the excitation of a dipolar plasmonic resonance of the nanoparticle. This peak redshifts and broadens for increasing radius, similarly to the near-field enhancement spectra (Fig. 1.4a). For the smallest radius considered, R = 5 nm, in Fig. 1.6(a), the total extinction cross-section (black line) is mostly dominated by the absorption (red line), and the scattering cross-section (blue line) is almost negligible. The weak contribution from the scattering can be understood from the quasistatic expressions. In this approximation, the polarizability of the nanoparticle follows Eq. (1.29) , and thus, scales as ∝R3 . As a consequence, σsca in Eq. (1.40) scales as ∝R6 and σext in Eq. (1.42) as ∝R3 . For small values of R , σext is thus much larger than σsca ( R3≫R6 ). On the other hand, σsca grows much faster than σabs with increasing R (even if the R6 scaling is not valid once the radiative correction becomes large). In fact, for the intermediate R = 30 nm radius (Fig. 1.6c), the values of the scattering (blue line) become comparable to the absorption (red line) cross-section. Further, the extinction cross-section spectrum for the R = 70 nm case in Fig. 1.6e is dominated 20 1.2. Electromagnetic response of very small objects abs ext sca abs ext sca abs ext sca Quasi-static R = 30 nm Radiative-corrected R = 30 nm R = 70 nmR = 70 nm R = 5 nm (b) R = 5 nm (a) (d)(c) (f)(e) abs ext sca abs ext sca abs ext sca Figure 1.6: Absorption (red lines), extinction (black lines), and scattering (blue lines) cross section spectra for silver spherical nanoparticles with different radius. The cross-section spectra are calculated using the model presented in section 1.2. Figures (a), (c), and (e) are obtained considering the radiative correction of the polarizability of the nanoparticle, and figures (b), (d), and (f) ignore this radiative correction. Figures (a)-(b), (c)-(d), and (e)-(f) show the cross-section spectra for a silver spherical nanoparticle of radius R = 5,30, and 70 nm, respectively. The cross-sections are normalized to πR2 , the area of a circle with the same radius of the spherical nanoparticle. The dielectric permittivity of silver used for these calculations is obtained from reference [41]. by the scattering contribution. By comparing these results with the ones obtained with the quasistatic approximation, we find that there is a very good agreement for the R = 5 nm nanoparticle (compare Figs. 1.6 a and b). However, for the larger R = 30 nm and R = 70 nm the quasistatic approximation breaks, similar to the results for the enhancement spectra in Figs. 1.2b and 1.4a. Most notably, the quasitatic results for R = 30 nm and R = 70 nm (in Figs. 1.6 d and f, respectively) show unphysical negative values of the absorption cross-section spectra, which violates 21 Chapter 1. Fundamentals of classical nanophotonics the conservation of energy. 1.2.4 Response of a quantum emitter In the previous sections we have discussed the response of a very small spherical nanoparticle, but a similar approach can also be valid to describe the electromagnetic response of a quantum emitter (QE), such as a molecule, a quantum dot, or a nitrogen-vacancy center in a diamond. The electromagnetic response of the QE can also be described as an oscillating electric point-like dipole, pe . In this situation, pe represents the effect of exciting the transition between the two lowest energetic level systems of the QE (the ground state, and the first excited state), energetically separated by ℏωσi . If an external field Eext drives the QE with a frequency close to ωσ , the first excited state becomes populated (or excited). Then, after an average relaxation time τr , the excitation stored in the excited state decays to the ground state by emitting a photon with average frequency ωσ . Thus, the response of the QE can be modeled by treating its electric dipole moment as a damped harmonic oscillator with damping rate γ, ∂2 ∂t2pe(t) + γ∂ ∂tpe(t) + ω2 σpe(t) = AeEext(t),(1.44) where Ae= 2ωσf2 0/ℏ, and f0is the oscillator strength [35]. Equation (1.44) can be rewritten in the frequency domain (considering that pe(t)∝e−iωt, see discussion of Eq. (1.4)) as pe(ω) = αe(ω)Eext(ω)(1.45) with αe(ω) = Ae ω2 σ−ω2−iγω ,(1.46) the polarizability of the QE. The decay rate γ of the QE in this equation can be decomposed as γ = γ0 + γintr , which account for the radiative or spontaneous decay rate of the QE, γ0 , (due to the emission of photons) and for intrinsic non-radiative losses, γintr (due to the decay of the QE without emitting photons). γ0 can be obtained from a quantum electrodynamic description of the coupling of the QE with the vacuum fields [35], resulting in γ0=ω3 σf2 0 3ℏπc3 0 (1.47) i To approximate the response of the QE as the response of an electric point-like dipole we are also considering four additional conditions satisfied in common set-ups: (i) The energy difference between the higher-energetic states (beyond the first excited state) and the ground state is much larger than the energy difference between the ground state and the first excited state. (ii) Light driving the QE has a similar frequency or the two-level transition ωσ , and thus, it does not drive higher-energetic states. (iii) Light driving the QE is not confined in effective volumes smaller than the size of the QE. (iv) The intensity of the field driving the QE is small enough to avoid non-linear effects. 22 1.3. Full electromagnetic response of spherical nanoparticles Interestingly, this value can also be obtained by considering a QE without losses and applying the radiative correction discussed in subsection 1.2.2. The polarizability of a lossless QE corresponds to considering γ= 0 in Eq. (1.46), α0 e(ω) = Ae ω2 σ−ω2.(1.48) Using the radiative-correction formula (Eq. (1.36)) we obtain, α0−RC e(ω) = =α0 e(ω)1 1−iω3 6πε0c3 0 α0 e(ω)=Ae ω2 σ−ω2−iω3 6πε0c3 0 Ae≈Ae ω2 σ−ω2−iω ω3 σf2 0 3ℏπc3 0 ,(1.49) where in the last step we have approximated ω3≈ωω2 σ , valid when the response of the QE is negligible far from resonance. From Eq. (1.46) and the last identity in Eq. (1.49) we can identify γ0 = ω3 σf2 0/ (3 ℏπc3 0 ), which is the same value as in Eq. (1.47). The equations describing the cross-section spectra of the small spherical nanoparticle can also be extended to the QE, where Eqs. (1.40) , (1.42) , and (1.43) are evaluated with the use of αeinstead of αRC a. 1.3 Full electromagnetic response of spherical nanoparticles In the previous section we have introduced a description of the response of small nanoparticles by treating them as electric dipoles. Next, we describe Mie’s or LorenzMie theory, a semi-analytical solution of Maxwell’s equations of the electromagnetic field scattered by an arbitrarily large spherical particle, which is used in chapters 3 and 5. As we discuss in this section, this complete solution of Maxwell’s equations reveals that, although nanoparticles can indeed be treated as electric dipoles, larger nanoparticles cannot. In particular, we show that large nanoparticles have a complex behavior that can be expressed as the sum of different electric and magnetic multipoles, where each multipole corresponds to a different field distribution. In subsection 1.3.1, we first review the Mie’s formalism. Then, in subsection 1.3.2 we use this formalism to calculate the optical response in a canonical scenario as studied in this thesis: the fields scattered by a spherical nanoparticle when illuminated by a linearly polarized plane wave. Further, Mie’s formalism is the starting point to describe the optical response of nanoparticles under the excitation by more complex beams in the last section of this chapter. 23 Chapter 1. Fundamentals of classical nanophotonics 1.3.1 Mie theory formulation In this subsection, we review the formulation of Mie theory as given by chapter 4 of reference [40], which is very similar to the one followed by Gustav Mie in his original paper written in 1908 [2]. This formulation depart from Helmholtz equations (or electromagnetic wave equation). The Helmholtz equations are a reformulation of Maxwell’s equations in the absence of external currents and charges ( Jext = 0 and ρext = 0 in Eq. (1.1) ). By using the constitutive relations in Eqs. (1.2) and (1.3) in Maxwell’s equations (1.1) , after some algebraic manipulation, we obtain [35,36,40] 1 c2 0 ∂2E(r, t) ∂t2=∇2E(r, t), 1 c2 0 ∂2B(r, t) ∂t2=∇2B(r, t).(1.50) In the linear regime, we can write these equations in the frequency domain using E ( r, ω ) = ´dtE ( r, t ) eiωt/ (2 π )and B ( r, ω ) = ´dtB ( r, t ) eiωt/ (2 π ), which leads to, −iω2εµ c2 0 E(r, ω) = ∇2E(r, ω), −iω2εµ c2 0 B(r, ω) = ∇2B(r, ω),(1.51) where ε and µ are the relative dielectric and magnetic permittivity of the medium, respectively. Equation (1.51) is satisfied by the following family of equations, so-called multipoles: Mb,n,m,e(r, ω) = m sin(θs)sin(−mφs)Pm n(cos(θs))B(b) n(ζ)uθs− −cos(mφs)dPm n(cos(θs)) dθs B(b) n(ζ)uφs, Mb,n,m,o(r, ω) = m sin(θs)cos(−mφs)Pm n(cos(θs))B(b) n(ζ)uθs− −sin(mφs)dPm n(cos(θs)) dθs B(b) n(ζ)uφs, (1.52a) 24 1.3. Full electromagnetic response of spherical nanoparticles Nb,n,m,e(r, ω) = B(b) n(ζ) ζcos(mφs)n(n+ 1)Pm n(cos(θs))urs+ +cos(mφs)dPm n(cos(θs)) dθs 1 ζ d[ζB(b) n(ζ)] dζ uθs+ +msin(−mφs)Pm n(cos(θs)) sin(θs) 1 ζ d[ζB(b) n(ζ)] dζ uφs, Nb,n,m,o(r, ω) = B(b) n(ζ) ζsin(mφs)n(n+ 1)Pm n(cos(θs))urs+ +sin(mφs)dPm n(cos(θs)) dθs 1 ζ d[ζB(b) n(ζ)] dζ uθs+ +mcos(−mφs)Pm n(cos(θs)) sin(θs) 1 ζ d[ζB(b) n(ζ)] dζ uφs,(1.52b) where Pm n are the associated Legendre functions of the ( m, n )-order, ζ = √εµk0rs is the optical distance (with k0 the wave number in vacuum and rs the radial spherical coordinate), φs and θs are the azimuthal and polar spherical coordinates ii , respectively. urs , uφs , and uθs are the unity vectors in spherical coordinates (corresponding to the radial, azimuthal, and polar directions, respectively). The subindex n indicates the order of the multipole, and the subindex m satisfies m∈ [ −n, n ]. The e and o labels indicate that the functions are even or odd with respect to φs , respectively. For convenience, we substitute the e and o labels of Eqs. (1.52a) and (1.52b) with a generic σ∈ {o, e} , so, for the rest of this thesis, we refer to Mb,n,m,e and Mb,n,m,o as Mb,n,m,σ , and to Nb,n,m,e , and, Nb,n,m,o as Nb,n,m,σ. Last, the label subindex b in Eqs. (1.52a) and (1.52b) indicates that the multipoles depend on the spherical Bessel function of the b -kind (and n -order), B(b) n ( ζ ), where we only need to consider b = 1 , 3(for the scenarios studied in this thesis). On the one hand, we use the b = 1, first kind functions, to describe incident beams and the field inside a nanoparticle because only the first kind functions, B(1) n ( ζ ) ≡jn ( ζ ), are finite at the origin. On the other hand, only the b = 3 third kind functions (equivalent to the first order Hankel functions, B(3) n ( ζ ) ≡h(I) n ( ζ )), behave asymptotically as an outgoing spherical wave, lim ζ→∞ B(3) n(ζ)=(−i)n+1 eiζ ζ.(1.53) Thus, they are the only appropriate functions to describe the expected behavior of light scattered by a nanoparticle (in contrast with the asymptotical behavior of b= 1,2,and 4functions, which become non-physical for large distances [40,45]). ii As in standard notation, θs is the angle with respect to the z -axis of the Cartesian coordinates. 25 Chapter 1. Fundamentals of classical nanophotonics (b)(a) (d)(c) x z y x z y x z y x z y Figure 1.7: Direction of the tangential components of M1,n,1,o and N1,n,1,e (in Eqs. (1.52a) and (1.52b) ) with n = 1 , 2over a 3D spherical cap with kr = 10, φs∈ [0 , π ], and θs∈ [0 , π ], following the plotting convention in reference [2] (the perspective of the figure shows the xz -plane in the background, see axis labels for reference). All the arrows have the same length in the 3D space, and they are colored only to help the 3D visualization. The dots represent the position at which the functions become zero. Electric and magnetic multipoles Figure 1.7a-d reproduces some of the original results from Mie [2] by evaluating the tangential components of M1,n,1,o and N1,n,1,e with n = 1 , 2over a 3D spherical cap with ζ = 10, φs∈ [0 , π ], and θs∈ [0 , π ](the perspective of the figure shows the xz -plane). We can indeed observe how the N1,1,1,e and N1,2,1,e functions in Figs. 1.7a and 1.7c, respectively, show a vectorial pattern of fields going from one or two points (“divergent” poles) to the same number of “convergent” poles (indicated with grey spots in the figure). These vectorial patterns are analogous to those obtained if we placed positive charges in the divergent and negative charges in the convergent points, forming an electric dipole in the case of N1,1,1,e (Fig. 1.7a) and an electric quadrupole for N1,2,1,e (Fig. 1.7c). As a consequence, the N -functions are often called electric or electric-type multipoles, as already pointed out by Mie in 1908 [2]. Similarly, the M1,1,1,o and M1,2,1,o functions in Figs. 1.7b and 1.7d, respectively, show a vectorial pattern of fields that surround the poles (grey spots 26 1.3. Full electromagnetic response of spherical nanoparticles in the figure), which resembles the electric field produced by a magnetic dipole and a magnetic quadrupole, respectively. Hence, the M -functions are often called magnetic or magnetic-type multipoles. Although the fields generated by a single N -function do indeed correspond to the fields generated by an electric multipole, it is convenient to point out that the representation of the N -functions followed by Mie and reproduced in Figs. 1.7a and 1.7c can be misleading [40]. In Figs. 1.7b and 1.7d, it might seem that there are charges placed at the poles, but there are not; the indicated poles (grey spots in the figure) correspond to a zero of the transverse component of the N -functions, not to a charge existing at the pole. In these positions the vectorial fields become purely radial, which cannot be shown in this representation. Despite this caveat, the identification of the vectorial patterns of the N -functions with the fields generated by different electric multiples (e.g., an electric dipole or an electric quadrupole) is quite useful to analyze the response of a nanoparticle. Expansion of fields in the electric and magnetic multipoles The multipolar functions in Eqs. (1.52a) and (1.52b) , Mb,n,m,σ and Nb,n,m,σ (with σ∈ {e, o} ) form an orthogonal basis themselves, i.e., ´d Ω Xb,n,m,σ ·X′ n′,m′,σ′∝ δK X,X′δK b,b′δK n,n′δK m,m′δK σ,σ′ with d Ωthe solid angle differential, δK the Kronecker delta, and X∈ {M,N} (we direct the reader to the full orthogonality relations of the multipoles in chapter 7 of reference [46]). Thus, any arbitrary electric (or magnetic) field Earb can be expanded onto the functions in Eqs. (1.52a) and (1.52b) as: Earb(r, ω) = n X m=−n ∞ X n=0 X σ=e,o CM b,n,m,σMb,n,m,σ(r, ω) + CN n,m,σNb,n,m,σ(r, ω) (1.54) where CM b,n,m,σ and CN b,n,m,σ are the normalized projections of Earb onto the basis {Mb,n,m,σ,Nb,n,m,σ}: CM b,n,m,σ =´dΩEarb(r, ω)·Mb,n,m,σ(r, ω) ´dΩ|Mb,n,m,σ(r, ω)|2, CN n,m,σ =´dΩEarb(r, ω)·Nb,n,m,σ(r, ω) ´dΩ|Nb,n,m,σ(r, ω)|2,(1.55) Similarly, the same method can be applied to decompose the magnetic field on the same basis. Thus, we can use this formalism to describe any arbitrary electric and magnetic field. 1.3.2 Fields scattered by a spherical nanoparticle under plane wave illumination 27 Chapter 1. Fundamentals of classical nanophotonics Electric dipole Magnetic dipole Magnetic quadrupole ext Figure 1.10: Extinction cross section spectrum of a R = 230 nm silicon spherical nanoparticle obtained with Mie theory and accounting for the contribution of all multipoles. The dielectric permittivity of silicon used for these calculations is obtained from reference [48]. nanoparticle contribute significantly to the cross-section spectra. 1.4 Angular momentum of light So far, we have focused on describing the interaction between a linearly polarized plane wave and a spherical nanoparticle. However, it is also interesting to explore the interaction between a spherical nanoparticle with more sophisticated light beams, such as light beams with well-defined angular momentum properties, as these type of beams will be used in chapter 5. The angular momentum properties introduce new degrees of freedom that can be exploited in different applications, such as sensing [19,28,54] or information processing [15–18]. There are four properties associated with the angular momentum of light, namely the spin angular momentum, the orbital angular momentum, the helicity, and the total angular momentum. In the simple case of a weakly-focused beam, so-called a paraxial beam, the spin value is directly given by its circular polarization; for example, a plane wave with circularly left polarization has spin s = +1. On the other hand, the orbital angular momentum is connected with changes in the spatial distribution of the phase on an individual wavefront (how many times the phase changes from 0to 2 π when moving along a circular trajectory around the axis of propagation). In subsection 1.4.1, we discuss the spin and orbital angular momentum in more detail for weakly focused beams with a wavefront propagating in a single defined direction (paraxial beams). On the other hand, we also consider in this section the angular momentum properties of non-paraxial beams, for example, with a spherical wavefront. For these type of fields, the spin and orbital angular momentum of light are ill-defined [55]. Thus, in that case, we introduce two auxiliary properties: the total angular momentum and the helicity. The helicity is an intrinsic property of the photons [56], defined as the spin projected in the direction of propagation, and it determines the torque that light can exert on matter [57 – 59]. On the other hand, in simple paraxial 34 1.4. Angular momentum of light beams, the total angular momentum is the sum of the spin and the orbital angular momentum. The total angular momentum of light determines the total angular momentum that can be transferred to matter, for example, to drive circular motion of nanoparticles [58 – 60]. In subsection 1.4.2, we adapt Mie theory formulation in section 1.3.2 to formally describe the helicity and total angular momentum of non-paraxial beams. Finally, in subsection 1.4.3, we derive the expressions for fields scattered by a spherical nanoparticle when illuminated by a beam with well-defined angular momentum focused by a lens, a typical experimental setup that we consider in chapter 5. 1.4.1 Paraxial beams with non-zero orbital angular momentum and spin Laguerre-Gauss beams Let us first consider the simplified case of paraxial beams to introduce the angular momentum properties of light. A paraxial light beam is an electromagnetic wave that propagates towards a fixed direction and that has a relatively small divergence or expansion while propagating (for example, the beam emitted by a laser). Formally, the defining characteristic of a paraxial beam is that one component of the wavevector, k= (kx, ky, kz), dominates the rest; for example, a z -propagating paraxial beam satisfies kz≫ ( kx + ky ). If we consider a paraxial beam propagating in free space along the z-direction with an electric field, E(r, ω) = E0(r)ei(kz−ωt)vE,(1.71) where E0 ( r )is the spatial distribution that varies with z only slowly (compared with the wavelength), and vE is the polarization vector, perpendicular to z [35,36]. We can approximate the Helmholtz equation (Eq. (1.51) ) in the paraxial regime as ∇2 ⊥E0(r)+2ik ∂E0(r) ∂z = 0.(1.72) Here, we have approximated ∂2E0 ( r ) /∂z2≈ −k2E0 ( r ) + 2 ik∂E0 ( r ) /∂zeikz (i.e., we neglect ∂2E0 ( r ) /∂z2 over the rest of the contributions due to the slow spatial variation of the fields along z ), where ∇2 ⊥ is the Laplacian in the coordinates perpendicular to the z –axis. The standard solution of E0 ( r )in Eq. (1.72) obtained in cylindrical coordinates (radial rc , polar φc , and axial zc coordinates) is [61,62], E0(r)≡LGl q(rc, φc, zc) = s2q! π(q+|l|)! w0 w(zc)rc√2 w(zc)|l| exp−r2 c w(zc)2 L|l| q2r2 c w(zc)2exp−ik r2 c 2RC(zc)exp(ilφc) exp(iψG(zc)),(1.73) 35 Chapter 1. Fundamentals of classical nanophotonics where LGl q are the Laguerre-Gauss beams of ( l, q )–order, ψG ( zc ) = arctan ( zc/zR )is the Gouy phase, RC ( zc ) = ( z2 c + z2 E ) /zc is the position-dependent radius of curvature of the beam, and zR = w2 0k/ 2is the Rayleigh range. w ( zc ) = w0p1+(zc/zR)2 in Eq. (1.73) is the position-dependent waist of the beam, which shows a minimum waist w0 at the focal plane zc = 0. In Eq. (1.73) , Lq |l| are the generalized Laguerre polynomials of (|l|, q)–order. Orbital angular momentum The field in Eqs. (1.71) and (1.73) is said to have well-defined orbital angular momentum. This means that LGl q is an eigenfunction of a particular projection of the orbital angular momentum operator, L = −i ( r×∇ ). In our case, LGl q is an eigenfunction of Lz, the projection along z, the direction of propagation, Lz=L·uz=−i∂ ∂φc .(1.74) The eigenvalue of LGl qwith respect to Lzis l, LzLGl q(r) = lLGl q(r),(1.75) which is determined by the exp(ilφc) term in Eq. (1.73) ,i.e., the rotation of the phase around the z axis directly results in the value of the orbital angular momentum carried by the beam. For illustration, we show in Fig. 1.11 the spatial distribution of the amplitude and phase of the LG0 0 , LG1 0 , and LG2 0 Laguerre-Gauss beams in the zc = 0 plane. The phase of the LGl 0 beams cycles l -times from −π to π as φc varies over [ −π, π ]. For example, the phase of LG1 0 in the figure changes (with φc ) from −π to π one time, while the phase of LG2 0 goes from −π to π twice. On the other hand, the LG0 0 beam (left column in the figure) has a constant phase, which shows its l= 0 orbital angular momentum. Spin angular momentum In free space or in a homogeneous medium, Maxwell’s equations impose that any electromagnetic wave has a polarization transverse to its propagation [63]. Hence, vEin Eq. (1.71) must have a form (in Cartesian coordinates) as: vE=  vx vy 0z ,(1.76) i.e., a zero contribution in the direction of propagation, which in this case is z . The x - and y -components, vx and vy , respectively can have any arbitrary complex value. However, specific values of vx and vy make the vE vector an eigenvector of the spin operator projected in the direction of propagation, Sz . The spin operator 36 1.4. Angular momentum of light max min π — π AmplitudePhase Figure 1.11: Representation of different Laguerre-Gauss fields in the zc = 0 focal plane. The three columns in the figure correspond from left to right to LG0 0 , LG1 0 , and LG2 0 beams. In the upper row, we plot the amplitude of the fields and in the row below we plot the phase of the fields. These beams are obtained for a z-propagating beam. in the Cartesian coordinates is [64] ←→ S=  0 0 0 0 0 −i 0i0 ux+  0 0 i 0 0 0 −i0 0 uy+  0−i0 i0 0 000 uz.(1.77) The eigenvectors of Sz=←→ S·uzare, v+=1 √2  1x iy 0z ,(1.78) and v−=1 √2  1x −iy 0z ,(1.79) These eigenvectors (normalized in Eqs. (1.78) and (1.79) ) correspond to circularly left polarized light and circularly right polarized light, respectively. v+ has a spin s = +1 eigenvalue ( Szv+ = + v+ ), and v− has a spin s = − 1eigenvalue (Szv−=−v−). As an example of a paraxial beam without a well-defined spin, we can consider a beam in Eq. (1.71) with a linear polarization vector vx=  1x 0y 0z .(1.80) The spin associated with this vector is ill-defined because vx is not an eigenvalue 37 Chapter 1. Fundamentals of classical nanophotonics of Sz . In fact, vx can be written as the sum of two different contributions with opposite spin, vx∝v++v−. Total angular momentum From the discussion above, a paraxial beam with field ELG(r, ω) = LGq l(r)ei(kz−ωt)v±(1.81) has a well-defined spin and orbital angular momentum. Thus if we define the total angular momentum as ←→ J=←→ S+L,(1.82) ELG is an eigenfunction of Jz=←→ J·uzwith eigenvalue m=l+s. (1.83) 1.4.2 Beyond the paraxial approximation Helicity and total angular momentum We consider a field that is non-paraxial, such as, a strongly focused beam or the field scattered by a nanoparticle. In this case, the orbital and the spin angular momentum become ill-defined because the polarization and the direction of propagation become position-dependent, and the orbital and spin angular momentum values are defined as the eigenvalues with respect to the orbital and spin angular momentum operators projected onto the direction of propagation. To avoid these issues, we can use two alternative quantities to describe the angular momentum properties of the field. On the one hand, it is convenient to work directly with the total angular momentum [55], as the eigenvalues from Jz are well defined for some solutions of Maxwell’s equations, such as the Mie’s multipoles (see below). On the other hand, we can address the polarization properties of light by defining the helicity as the spin projected in the direction of propagation. Formally, the helicity operator is ←→ Λ=←→ S·uk=∇× k,(1.84) where in the last identity, we have used Eq. (1.77) and the fact that uk , the unity vector along the direction of propagation can be found using the operator ←→ uk = −i∇/k [36,64]. Notably, the eigenvectors of these operators are the RiemannSilberstein vectors as defined in Eq. (1.8) . Thus, any electromagnetic field that can be written as a F+or F−Riemann-Silberstein vectors has a well-defined helicity. Multipoles with well-defined total angular momentum We can redefine the standard Mie theory multipoles described in section 1.3 to obtain multipoles with a well-defined total angular momentum operator. According 38 1.4. Angular momentum of light to reference [65], these multipoles are A(M) b,n,m(r, ω) = =i(−1)m (sign(m))mpn(n+ 1)s2n+ 1 4π (n−|m|)! (n+|m|)!(Mb,n,|m|,e +isign(m)Mb,n,|m|,e) (1.85) A(E) b,n,m(r, ω) = =(−1)m (sign(m))mpn(n+ 1)s2n+ 1 4π (n−|m|)! (n+|m|)!(Nb,n,|m|,e +isign(m)Nb,n,|m|,o) (1.86) where sign ( m ) = 1 for m≥ 0and sign ( m ) = − 1for m < 0. A(E) b,n,m and A(M) b,n,m are a combination of electric ( Nb,n,m,σ ) and magnetic ( Mb,n,|m|,σ ) multipoles, respectively and they, hence, maintain their respective electric or magnetic character. Because of the orthogonal relationships between Mb,n,|m|,σ and Nb,n,|m|,σ , A(M) b,n,m are A(E) b,n,m also orthogonal, i.e., ˚drdΩA(X) b,n,m(r, ω)A(X′) b′,n′,m′(r, ω)=0,(1.87) for b = b′ , n = n′ , m = m′ , or X = X′ , with X = M or E . The new A(M) b,n,m and A(E) b,n,m electric multipoles are eigenvalues of Jz , the z -component of the total angular momentum operator, JzA(M) b,n,m(r, ω) = mA(M) b,n,m(r, ω),(1.88) and JzA(E) b,n,m(r, ω) = mA(E) b,n,m(r, ω).(1.89) Further analysis of the angular momentum properties of A(M) b,n,m and A(E) b,n,m can be found in references [65,66]. Multipoles with well-defined helicity The A(M) b,n,m and A(E) b,n,m multipoles have well-defined total angular momentum, but not well-defined helicity. Next, we define a set of combinations of A(M) b,n,m and A(E) b,n,m that results in a new basis of multipoles with well-defined helicity. To find these combinations, we use the expressions of the Riemann-Silberstein vectors, which are the eigenfunctions of the helicity operator (Eq. (1.8) ), and are combinations of electric and magnetic fields (Eq. (1.7) ). Thus, we first write the 39 Chapter 1. Fundamentals of classical nanophotonics electric and magnetic fields using the A(M) b,n,m and A(E) b,n,m multipolesiv, E(r, ω) = X n,m [iC(E) n,mA(E) b,n,m(r, ω) + C(M) n,m A(M) b,n,m(r, ω)].(1.90) By using this expression of the electric field and applying Eqs. (1.63) , (1.85) and (1.86) on Maxwell’s equations (Eq. (1.5) ), we can obtain the expression of the magnetic field, B(r, ω) = 1 c0X n,m [C(M) n,m A(E) b,n,m(r, ω)−iC(E) n,mA(E) b,n,m(r, ω)].(1.91) Using these expressions of E and B we can write the Riemann-Silberstein vectors in Eq. (1.9) as, F±(r, ω) = X n,m (C(E) l,m +C(M) l,m )[A(M) b,n,m(r, ω)±iA(E) b,n,m(r, ω)].(1.92) F± are eigenfunctions of the helicity operator with eigenvalue Λ. It can be proved [65–67] that each term in the sum of Eq. (1.92), A(Λ) b,n,m(r, ω) = A(M) b,n,m(r, ω)+ΛiA(E) b,n,m(r, ω),(1.93) are also eigenfunctions of the helicity operator, ←→ Λ , with the same eigenvalue Λ = ± 1,i.e. A(+) b,n,m satisfies ←→ Λ A(+1) b,n,m ( r, ω ) = A(+1) b,n,m ( r, ω ), and A(−1) b,n,m satisfies ←→ Λ A(−1) b,n,m ( r, ω ) = −A(−1) b,n,m ( r, ω ). Further, the A(Λ) b,n,m multipoles are also the z - components of total angular momentum, Jz = ←→ J·uz . Thus, A(Λ) b,n,m constitutes the new multipoles we are looking for, which allows for decomposing any electromagnetic field on a basis well-suited to directly address the helicity and total angular momentum properties. 1.4.3 Scattering of a beam with well-defined angular momentum by a spherical nanoparticle In chapter 5we study the scattering of a focused light beam with well-defined angular momentum properties by a spherical nanoparticle. In this section, we show that the Mie’s formalism described in section 1.4.2 allows for tackling this scattering process. We show in Fig. 1.12 a scheme of this scattering problem, an incident circularly-polarized Laguerre-Gauss beam focused by a high-numerical-aperture lens at the center of a spherical nanoparticle. Before focusing, the Laguerre-Gauss beam is a paraxial beam and thus has a well-defined value of the spin and the orbital angular momentum (see subsection 1.4.1). For example, we can consider iv Note that for convenience, in equations (1.90) and (1.91) we chose to add a π/ 2phase (i.e., aifactor) between the electric and magnetic multipoles. 40 1.4. Angular momentum of light Incident circularly polarized LG beam High NA lens Focusing Nanoparticle (a) Collimation (b) Figure 1.12: Scheme of the scattering process studied in section 1.4.3. (a) Scheme of the illumination: an incident circularly polarized Laguerre-Gauss (LG) beam is focused on the center of a small spherical nanoparticle by using a high numerical aperture lens. In the scheme we plot the amplitude distribution of a LG0 l beam with l = 1 and s = − 1(corresponding to circular right-polarization). These values result in a total angular momentum m= 0. (b) Scheme of the collimation process: the light back-scattered by the nanoparticle is collected and collimated by the same lens used for focusing. In the focusing process the field (in (a)) incident on the aperture of the lens (vertical arrow in the shaded blue area) is mapped onto a spherical surface of radius f (dashed line) and then rotated as described in section 1.4.3. The collimation (in (b)) corresponds to the inverse process, so that the field is evaluated in the same spherical surface, rotated, and then mapped onto the aperture of the lens. a circularly right-polarized beam with a spatial distribution following a LG0 1 (see Eq. (1.73) ) Laguerre-Gauss, resulting in s = − 1, l = +1, and m = l + s = 0. We consider that the beam propagates in the z -direction so that ←→ Λ = Sz (Eq. (1.84) ), and thus s=Λ=−1. The scattered beam in the backward direction (i.e., opposite to the propagation of the incident beam) is collimated with the same lens used to focus the incident beam. After the collimation we separate the field into two contributions, one with the same helicity as the incident beam, and another contribution with opposite helicity. The collimated beams are paraxial, and thus, we can also address the s , l , of the scattered fields. We next describe the equations that we use to implement all these steps of the scattering process. The incident Laguerre-Gauss beam follows Eq. (1.81) . This beam is focused by the lens at the plane zc = 0. The first step is to write the focused field as an expansion of multipoles with well-defined helicity A(Λ) b,n,m of different order n . The multipoles A(Λ) b,n,m are defined in Eq. (1.93) . In particular, we use multipoles with b= 1 (A(Λ) 1,n,m) (see section 1.3). The focused electric field is: Efoc(r, ω) = ∞ X n=0 √2Cn(ω)A(Λ) 1,n,m(r, ω),(1.94) where, ω is the angular frequency of the light, Λis the helicity of the incident beam 41 Chapter 1. Fundamentals of classical nanophotonics (corresponding to its circular polarization, see subsection 1.4.1), and we only need to sum over multipoles with m equal to the angular momentum of the beam. For convenience, the r coordinates in Eq. (1.94) are chosen to be centered at the focal point of the lens (not to be confused with the rc , ϕc, and θc coordinates of the LG beam in Eq. (1.73)). The coefficients Cnare, Cn=i(n−1)k√2π√2n+ 1ˆθmax 0 dn m,Λ(θ)sin(θ)LGl q(fsin(θ),0,0)pcos(θ)fe−ikf dθ, (1.95) where dn m,Λ is the small Wigner d -function [68] and θmax is the maximal half-angle of the lens of numerical aperture NA = nsin ( θmax )( n is the refractive index of the medium after the lens). These coefficients are derived in reference [67] using the aplanatic lens model [35] (see scheme on Fig. 1.12a). In brief, the modeling of the focusing process can be separated into three steps. First, the incident electric field in the aperture of the lens is mapped onto a reference surface with coordinates r = f , φs∈ [0 , 2 π ], and θ∈ [ π−θmax, π ](i.e. a spherical cap situated at the focal distance, f , from the nanoparticle). Second, the mapped (vectorial) electric field is rotated such that from each point of the reference surface emerges a plane wave that propagates toward the focal point. This rotation results in the dn m,Λ function in Eq. (1.95) . Third, we obtain the field at the focal point as the sum of all these plane waves. The sum of these plane waves leads to the integration in Eq. (1.95). We next calculate the fields scattered by the spherical nanoparticle under the illumination of the strongly focused Laguerre-Gauss beams. Equation (1.61) describes the response of a spherical nanoparticle under plane wave illumination. The extension of this solution to our case is [67], Esca LG(r, ω) = ∞ X n=0 Cn(ω)Vn(ω)A(Λ) 3,n,m(r, ω) + Wn(ω)A(−Λ) 3,n,m(r, ω),(1.96) where Vn and Wn correspond to combinations of the an and bn coefficients (in Eqs. (1.62a) and (1.62b), respectively), Vn(ω) = −an(ω) + bn(ω) 2,(1.97) Wn=an(ω)−bn(ω) 2.(1.98) Note that due to the angular momentum properties of the A(Λ) b,n,m multipoles (see Eqs. (1.88) , (1.89) , and (1.93) ) the scattered field in Eq. (1.96) preserves the total angular momentum of the incoming and focused beam. This preservation is a consequence of the rotational symmetry of the nanoparticle and will be exploited in chapter 5. The backscattered field in Eq. (1.96) is collimated through the same lens that focuses the incident beam. To model this collimation using the aplanatic lens model, we follow the inverse process of the focusing. The backscattered field is evaluated at the same spherical reference surface as for the focus (dashed line in Fig. 1.12b). We then perform the inverse rotation compared to the focusing process so that the Poynting vectors of the scattered field become perpendicular to the aperture of the 42 1.4. Angular momentum of light lens at all points. Finally, we map the rotated field of the reference aperture onto the surface of the lens. This collimation process corresponds mathematically to: ECol LG (rc, φc, zc= 0, ω) = ˆ R(f, φs, θs)·Esca LG(f, φs, θs, ω)cos(θs)−1,(1.99) where ECol LG is the collimated field, ( rc , φc , zc ) are the cylindrical coordiantes in the aperture of the lens (with zc = 0 and rc = fsin ( θs )), and ˆ R ( f, φs, θs )is the position-dependent Euler rotation matrix: ˆ R(f, φs, θs) =   sin(φs)−cos(φs) 0 cos(φs) sin(φs) 0 0 0 1 ·   1 0 0 0 cos(θs)−sin(θs) 0 sin(θs) cos(θs) ·  sin(φs) cos(φs) 0 −cos(φs) sin(φs) 0 0 0 1 .(1.100) Last, the cos ( θs ) −1 factor in Eq. (1.99) accounts for the differences between the differential area at the reference spherical surface, dAS , and the differential area at the aperture of the lens, dAL ( dAS = dAL/cos ( θs ), see chapter 3 of reference [35]). 43 Chapter 2. Fundamentals of quantum nanophotonics Figure 2.1: Sketch of the classical and quantum transformation of light induced by a standard beam splitter. In the classical description, the input Ei 1 and Ei 2 modes are transformed onto the output Eo 1 and Eo 2 modes. In the quantum formalism, the input quantum ˆai 1 and ˆai 2 operators are transformed onto the output ˆao 1and ˆao 2operators. splitter. We then extend this formalism to describe the transformation by a lossy beam splitter, i.e., a beam splitter that can dissipate the incident photons and thus making it lose their corresponding energy. In this thesis we use the quantum transformation of beam splitters to describe the process occurring in a HanburyBrown Twiss interferometer in section 2.4, and to study the interaction between quantum states of light and a nanostructure in chapter 5. Classical transformation by a beam splitter We first consider the standard description of the classical transformation of light by a standard (lossless) beam splitter. This transformation can be understood as a change of basis between two input modes at the beam splitter, Ei 1 ( ω )and Ei 2 ( ω ), and the two output modes of the beam splitter Eo 1 ( ω )and Eo 2 ( ω ). The input and output channels with the corresponding fields are shown in the scheme of Fig. 2.1. The input Ei 1 ( ω )and Ei 2 ( ω )and output Eo 1 ( ω )and Eo 2 ( ω )modes are related by a unitary transformation [75], Eo 1(ω) = t1(ω)Ei 1(ω) + r1(ω)Ei 2(ω), Eo 2(ω) = r2(ω)Ei 1(ω) + t2(ω)Ei 2(ω),(2.24) where t1 and t2 are the transmittance coefficients, and r1 and r2 are the reflectance coefficients of the beam splitter [76]. For a lossy beam splitter, t1 , t2 , r1 , and r2 can take any arbitrary value as long the energy of the fields at the output of the beam splitter is lower than the value of the energy of the fields at the input of the beam splitter, i.e.,|Eo 1(ω)|2+|Eo 2(ω)|2<|Ei 1(ω)|2+|Ei 2(ω)|2. 50 2.3. Quantum transformations by a lossless and a lossy beam splitter On the other hand, for a lossless beam splitter, the energy of the input fields is the same as the output fields, i.e., |Eo 1 ( ω ) |2 + |Eo 2 ( ω ) |2 = |Ei 1 ( ω ) |2 + |Ei 2 ( ω ) |2 . The conservation of energy imposes the transmittance t -coefficients and the reflectance r-coefficients to satisfy [75,76] |t1(ω)|2+|r1(ω)|2=|t2(ω)|2+|r2(ω)|2= 1,(2.25) and t1(ω)r2(ω)∗+r1(ω)t2(ω)∗= 0.(2.26) These constraints on t1 , t2 , r1 , and r2 allow us to write Eq. (2.24) in terms of a single transmittance, and a single reflectance coefficient [75,77,78], Eo 1(ω) = t(ω)Ei 1(ω) + r(ω)Ei 2(ω), Eo 2(ω) = r(ω)Ei 1(ω) + t(ω)Ei 2(ω),(2.27) where tand rmust satisfy |r(ω)|2+|t(ω)|2= 1 and r(ω)t(ω) = −(r(ω)t(ω))∗. Quantum transformation by an energy-conserving beam splitter In quantum optics, the beam splitter transformation also corresponds to a change of basis between the output and the input states. Importantly, in quantum optics, this change of basis is defined by the annihilation operators and not directly by the quantum states [70,77,78]. Following the procedure of quantization of electromagnetic fields used in section 2.2, we introduce a set of ˆai 1 ( ω )and ˆai 2 ( ω ) annihilation operators that define the orthogonal input modes of the beam splitter and another set of ˆao 1 ( ω )and ˆao 2 ( ω )annihilation operators that define the orthogonal output modes (Fig. 2.1). The relationship between the input and output quantum operators of a lossless beam splitter is analogous to the classical transformation introduced in Eq. (2.27) [70,77,78], ˆao 1(ω) = t(ω)ˆai 1(ω) + r(ω)ˆai 2(ω), ˆao 2(ω) = r(ω)ˆai 1(ω) + t(ω)ˆai 2(ω),(2.28) where t and r are the same transmittance and reflectance coefficients as in the classical description. The close connection between the classical and quantum transformation is due to the fact that Maxwell’s equations determine the evolution of electromagnetic modes both in the classical and quantum regimes [64]. Quantum transformation by a lossy beam splitter We next consider the quantum transformation of a lossy beam splitter. In this case, the output state of the lossy beam splitter does not conserve the energy of the incident quantum state, which means that the output state can have the same or fewer photons than the incident state. We model the “dissipation” modes (also called “ancilla” modes [79]) by introducing the Langevin ˆ L1 and ˆ L2 operators associated 51 Chapter 2. Fundamentals of quantum nanophotonics with losses due, for example, to fluctuating currents in the beam splitter [78]. These operators are added directly to the quantum transformation in Eq. (2.28) , leading to the quantum transformation of a lossy beam splitter [78], ˆao 1(ω) = t1(ω)ˆai 1(ω) + r1(ω)ˆai 2(ω) + ˆ L1(ω), ˆao 2(ω) = r2(ω)ˆai 1(ω) + t2(ω)ˆai 2(ω) + ˆ L2(ω).(2.29) In the lossy beam splitter, there is no imposition for t1 = t2 or r1 = r2 . However, t1 , t2 , r1 , and r2 still correspond to the classical values of the transmittance and reflectance coefficients in Eq. (2.24) . On the other hand, the Langevin operators in Eq. (2.29) must satisfy three requirements [78,80,81]: (i) their expected value must vanish, ⟨ˆ L1(ω)⟩=⟨ˆ L† 1(ω)⟩=⟨ˆ L2(ω)⟩=⟨ˆ L† 2(ω)⟩= 0,(2.30) (ii) the Langevin operators must commute with the input operators (e.g., [ ˆ L1 ( ω ) ,ˆai 1 ( ω )] = 0), as the input fields and noise sources inside the beam splitter must be independent, and (iii) the Langevin operators do not change the canonical commutation relationships between the bosonic operators in Eq. (2.21) . Additional properties of Langevin operators and their explicit form for a one-dimensional beam splitter (a very thin one-layer beam splitter) can be found in references [78,80,81]. 2.4 The Hanbury-Brown and Twiss interferometer The Hanbury-Brown and Twiss (HBT) interferometer is a simple device typically composed of a beam splitter and two detectors. The HBT has played a crucial role in the history of quantum optics because it enables to characterize the different states of light and it has shown how photons composing a state of light can interfere between them [69,70,82,83]. In this section, we briefly review the measurements that can be performed with an HBT interferometer and how they relate to the statistics of the number of photons emitted by a light source. We first derive a mathematical formalism that describes the transformations produced in the HBT measurements (subsection 2.4.1). We then illustrate how to interpret the results from the HBT interferometry by applying this formalism to analyze the emission from a variety of canonical light sources (subsections 2.4.2-2.4.5). 2.4.1 General description of the Hanbury-Brown and Twiss interferometer Figure 2.2 shows the schematics of a standard HBT interferometer: light incident along one input path of a beam splitter is divided into two paths. At the end of both paths, a detector, D1 and D2 , is located for the horizontal and vertical paths, respectively. The HBT measures intensity correlations of light: the detection of 52 2.4. The Hanbury-Brown and Twiss interferometer Figure 2.2: Sketch of a standard HBT interferometer. The input light, which can be a classical or quantum state of light, is splitted into two output paths by a beam splitter. In this section, we choose a “50-50” beam splitter, where light is reflected or transmitted with equal probability. We indicate this transformation with the input ˆai 1 and output ˆao 1 and ˆao 2 operators introduced in Fig. 2.1. Light leaving the beam splitter is converted into an electric current at two photodetectors D1 and D2 , each of them at the end of the output paths. Both detectors are placed at the same distance from the beam splitter. The electric signal of both detectors is manipulated in an additional device to obtain the correlation measurement, g(2)(0). light simultaneously at both detectors normalized to the individual detection at each detector. The resulting measurement of the HBT is the intensity correlation, g(2) ( τ ), where τ indicates the time delay between light traveling from the beamsplitter to each detector. This thesis considers only the zero-delayed case with τ = 0, corresponding to a situation where both detectors are set at the same distance from the beam splitter. In a typical experiment based on HBT interferometry, g(2) (0) is obtained by taking the average of the intensity measured at each detector ( ⟨ˆ I1⟩ and ⟨ˆ I2⟩ at D1 and D2 , respectively) and the average of the multiplication of the intensity at both detectors (⟨ˆ I1ˆ I2⟩). The resulting intensities correlation corresponds to g(2)(0)(ω1, ω2) = ⟨ˆ I1(ω1)ˆ I2(ω2)⟩ ⟨ˆ I1(ω1)⟩⟨ˆ I2(ω2)⟩,(2.31) where ω1 and ω2 are the frequencies of detection of each detector (they correspond, for example, to the central frequency of a filter placed at the aperture of each detector) [83]. In this thesis we consider that the detectors are color-blind, i.e., all photons are detected independently of their frequency. Thus we write the “color-blind” intensity correlations as g(2)(0) = ⟨ˆ I1ˆ I2⟩ ⟨ˆ I1⟩⟨ˆ I2⟩,(2.32) where we are tracing out over the frequency degree of freedom. The intensity operators are proportional to the corresponding number of photons operator: ˆ I1∝ˆao 1†ˆao 1 and ˆ I2∝ˆao 2†ˆao 2 . Next, we consider that the beam splitter is a lossless, frequency independent, “50-50” beam splitter, which corresponds to setting t ( ω ) →t = 1 /√2 and r ( ω ) →r = i/√2 [77,78]. Assuming monochromatic 53 Chapter 2. Fundamentals of quantum nanophotonics illumination, using the beam splitter transformation introduced in Eq. (2.28) , and considering that there is no input state on the vertical path, we can write Eq. (2.32) in terms of the input operators as g(2)(0) = ⟨ˆai 1†ˆai 1†ˆai 1ˆai 1⟩ ⟨ˆai 1†ˆai 1⟩2=⟨ˆa†ˆa†ˆaˆa⟩ ⟨ˆa†ˆa⟩2,(2.33) where in the last equality, we have simplified our notation, ˆai 1≡ˆa . It can be shown (see, for example, chapter 5 in reference [70]) that g(2) (0) can be directly connected with the statistics of emission from a light source g(2)(0) = 1 + ⟨(∆ˆn)2⟩−⟨ˆn⟩ ⟨ˆn⟩2,(2.34) where ⟨ˆn⟩ is the mean number of photons emitted by the source in a time interval. The time interval that defines the number of photons in each “packet” of photons emitted by the source is called coherence time, τC , and it is inversely related to the natural line width of the spectral lines of the source. In our theoretical description, we are considering that the detectors in the HBT detect the number of photons arriving in time windows of τCvii [70]. ⟨(∆ˆn)2⟩ = |⟨ˆn2⟩−⟨ˆn⟩2| in Eq. (2.34) are the fluctuations in the average number of emitted photons. In the following subsections (subsections 2.4.2 to 2.4.5), we discuss the relationship between g(2) (0) and the statistical nature of the emission from four canonical light sources: a coherent source, a thermal source, a quantum single-photon source, and a quantum realistic two-photon source. 2.4.2 Statistics of light emitted by a coherent source Coherent sources of light, such as a laser, emit photons with the same frequency and same wavefronts. The statistics of the number of photons emitted from a coherent light source follows a characteristic Poisson’s distribution with fluctuations ⟨(∆ˆn)2⟩ = ⟨ˆn⟩ [70]. Thus, using Eq. (2.34) , we find that the HBT interferometer results in g(2)(0) = 1.(2.35) To illustrate this result, we analyze in Fig. 2.3 the statistics and intensity correlations of a coherent light source with an average emission of ⟨ˆn⟩ = 5 photons. Figure 2.3a shows the distribution of the number of photons emitted by this source (a Poisson distribution with average ⟨ˆn⟩ = 5 and fluctuations ⟨(∆ˆn)2⟩ = 5) at a time interval. Figure 2.3b shows a sketch illustrating the response of the HBT interferometer to this input source. In this sketch, the packets of photons arrive at the beam splitter at relatively regular intervals and are split towards the two detectors. Note that the beam splitter divides the incident states into a superposition of states with different vii For larger time windows, counting the photons in each packet can result in a statistical Poisson distribution, and at shorter time windows, the information in the photon intensity correlations can be lost [77,83]. 54 2.4. The Hanbury-Brown and Twiss interferometer Coherent source 0 1 2 3 4 5 6 7 n P(n) 8 4 2 1 4 6 9 25 2 3 3 3 4 3 18 6 5 3 4 6 2 4 3 6 2 2 2 3 6 8 4 9 2 4 68 2 2 4 12 3.1 2.4 7.4 Number of photons Average Measurement number 1 2 3 4 5 6 7 8 9 10 (a) (b) (c) Figure 2.3: Example of the measurement of the intensity correlations g(2) (0) of a coherent source. (a) Poissonian distribution with mean occupation number ⟨n⟩ = 5 representing the statistics of the number of photons emitted by a coherent source. The graph shows the probability distribution evaluated up to n = 8 photons. (b) Sketch of the response of the HBT interferometer under coherent illumination, where each orange circle represents an individual photon. In the figure, we represent one of the possible outcomes from the beam splitter. (c) Simulated measurement by a HBT interferometer. In the first ten rows we show (from left to right columns): the measurement number, the intensity measured by the detector in the horizontal path, ¯ I1 , the intensity measured by the detector in the vertical path, ¯ I2 , the multiplication of the intensity measured by both detectors, ¯ I1¯ I2 , and the number of photons emitted by the coherent source. These values are obtained with random number generators based on the number of photons statistically emitted by the coherent source and on the response of the beam splitter (see discussion in the text). In the last row of the table, we show the average value of ¯ I1 , ¯ I2 , and ¯ I1¯ I2 . Below the table we show the value of g(2)(0) obtained from the average values. 55 Chapter 2. Fundamentals of quantum nanophotonics numbers of photons at each output branch of the beam splitter. For example, in the figure, an incident state with N = 5 photons is split into a superposition of all possible |n1, n2⟩ states with n1 + n2 = N = 5, where n1 and n2 indicate the number of photons at each output path of the beam splitter. When measured, this superposition of states collapses into a single state; for example, in the figure, we chose it to be the |3,2⟩state. In Fig. 2.3c, we show a table that simulates the behavior of the HBT interferometer. The first ten rows correspond to ten simulated measurements under coherent illumination (measurement number in the first column). For this table and the following examples in Figs. 2.4-2.6, we first use a random number generator to obtain N , the number of photons emitted by the source in a given time interval (corresponding to each individual simulated measurement). This random number generator follows the statistic distribution of the source, i.e., a Poisson distribution in the case of Fig. 2.3. Next, we obtain the intensity of an individual measurement by each detector, ¯ I1 and ¯ I2 , shown in the second and third columns, respectively (the line over a variable indicates simulated values for a single measurement). We assign ¯ I1 to ¯n1 , a randomly generated number of photons that arrive at the D1 detector. To obtain ¯n1 we take into account that, using the beam splitter transformation in Eq. (2.28) , the output state at each detector of the HBT is |Ψo HBT⟩ = [( ˆa† 1 + iˆa† 2 ) /√2 ] N|0⟩ , with |0⟩ being the vacuum state (with zero photons). We can then obtain the probability of measuring n1 photons in the D1 detector as |⟨Ψo HBT|n1, n2⟩|2 , where the |n1, n2⟩ describe having n1 and n2 photons at the horizontal and vertical output paths of the beam splitter, respectively. The value of ¯n1 results from using a random number generator following the statistical distribution given by the |⟨Ψo HBT|n1, n2⟩|2 probabilities. Then we assign ¯ I2 to ¯n2 = N−¯n1 . Using these values of ¯ I1 and ¯ I2 we show the resulting ¯ I1¯ I2 product in the fourth column. Finally, in the last row, we give the mean values ⟨¯ I1⟩ , ⟨¯ I2⟩ , and ⟨¯ I1¯ I2⟩ , which corresponds to the average of the individual simulated measurements, ¯ I1 , ¯ I2 , and ¯ I1¯ I2 , respectively, in the rows above. The result of the HBT interferometer corresponds to g(2)(0) = ⟨¯ I1¯ I2⟩ ⟨¯ I1⟩⟨¯ I2⟩.(2.36) For the table in Fig. 2.3 we obtain g(2)(0) ≈ 0 . 99 (indicated at the bottom of the figure), which is very close to the theoretical value g(2) (0) = 1 expected from Eq. (2.35) (larger sampling results in g(2)(0) ≈ 1, not shown here). This simple example illustrates not only the performance of a standard HBT but also how the intensity correlations measured by an HBT interferometer are directly connected to the statistics of the emission of a source. 2.4.3 Statistics of light emitted by a thermal source We next consider an HBT interferometer under thermal illumination, such as light emitted by a gas discharge lamp. The number of photons emitted by a thermal 56 2.4. The Hanbury-Brown and Twiss interferometer 10 25 0 2 1 1 1 3 2 0 0 6 0 0 1 0 0 1 0 0 33 1 1 0 0 0 0 0 0 6 13 47 7 3 42 13 1.6 1.8 6.0 Number of photons Average Measurement number 1 2 3 4 5 6 7 8 9 10 Thermal source (bunching) 01 2 3 4 5 6 7 n P(n) 8 (a) (b) (c) Figure 2.4: Example of the measurement of the intensity correlations g(2) (0) of a thermal source. (a) Bose-Einstein distribution with mean occupation number ⟨n⟩ = 5 representing the statistics of the number of photons emitted by a coherent source. The graph shows the probability distribution evaluated up to n = 8 photons. (b) Sketch of the response of the HBT interferometer under coherent illumination, where each red circle represents an individual photon. In the figure, we represent one of the possible outcomes from the beam splitter. (c) Simulated measurement by a HBT interferometer. In the first ten rows we show (from left to right columns): the measurement number, the intensity measured by the detector in the horizontal path, ¯ I1 , the intensity measured by the detector in the vertical path, ¯ I2 , the multiplication of the intensity measured by both detectors, ¯ I1¯ I2 , and the number of photons emitted by the thermal source. These values are obtained with random number generators based on the number of photons statistically emitted by the thermal source and on the response of the beam splitter (see discussion in the text). In the last row of the table, we show the average value of ¯ I1 , ¯ I2 , and ¯ I1¯ I2 . Below the table we show the value of g(2)(0) obtained from the average values. 57 Chapter 2. Fundamentals of quantum nanophotonics light source follows a Bose-Einstein, positive-definite, half-normal distribution (a normal distribution, P ( n ), but only defined for positive values, n≥ 0) [70]. Figure 2.4a shows the evaluation of such distribution with a ⟨(∆ˆn)2⟩ = 5 variance. A half-normal distribution satisfies ⟨(∆ˆn)2⟩ = ⟨ˆn⟩2 + ⟨ˆn⟩ [70], and, g(2) (0) in Eq. (2.34) becomes, g(2)(0) = 2.(2.37) Figure 2.4b shows a sketch of the emission of a thermal source onto an HBT interferometer. The sketch emphasizes how this source often emits photon packages containing a significant number of photons, with a relatively long period of time between these groups with no photon emission. This behavior contrasts with the more regular emission of a coherent source (Fig. 2.3b). Statistically, if the incident photon package to the beam splitter contains many photons, the output detected state would contain a similar number of photons at each detector. Figure 2.4c shows a simulation of the HBT interferometer measurements for a thermal source. The values shown in the figure are obtained using the same methodology presented when discussing Fig. 2.3c, except for the use of the BoseEinstein statistics for the photons emitted by the thermal source (Fig. 2.4a). Consistent with the previous discussion, the number of photons emitted by the thermal source in a time interval (last column of Fig. 2.4c) is very irregular, with some intervals containing many photons and others very few or none. This type of behavior, where photons are emitted in “bunches” (packets with large numbers of photons), corresponds to a bunched emission and results in g(2) (0) > 1. viii . The last row of the table shows the average value of the intensities measured at each detector of the HBT after ten simulated measurements in the rows above. Using these average values in Eq. (2.36) we obtain g(2)(0) ≈ 2 . 08 (indicated at the bottom of the figure), a value very close to the theoretical g(2)(0) = 2 result in Eq. (2.37) . 2.4.4 Statistics of light emitted by a single photon source We study in this section the intensity correlations and statistics of light showing a very different behavior as compared to the bunched thermal emission: the emission from a single photon source. In Fig. 2.5a, we show the distribution of the number of photons emitted by a realistic single-photon source, which can emit one photon at each time interval, but can also emit zero photons. As illustrated by the sketch in Fig. 2.5b, the output state from the beam splitter can only be detected in a single output path. Thus, the intensity measured by one of the detectors is zero, so that ⟨ˆ Ihˆ Iv⟩= 0, and according to Eq. (2.32), in this case, g(2)(0) = 0.(2.38) The table in Fig. 2.5c shows the results of a simulated measurement (performed as in previous sections) for a single photon source. In the last row of the table, we viii More technically, “bunching” occurs when g(2) ( τ ) < g(2) (0). In any case, the statistical behavior of the emission of photons in bunches results in g(2) (0) > 1, and for the rest of this thesis, we refer to g(2)(0) >1as “bunching”. 58 2.4. The Hanbury-Brown and Twiss interferometer 00 01 0 1 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 1 00 0 0 0 0 0.3 0.1 0 Number of photons Average Measurement number 1 2 3 4 5 6 7 8 9 10 Single photon source (antibunching) 0 1 2 3 4 5 6 7 n P(n) 8 (a) (b) (c) Figure 2.5: Example of the measurement of the intensity correlations g(2) (0) of a single photon source. (a) Statistic distribution of the number of photons emitted by a single photon source. (b) Sketch of the response of the HBT interferometer under single photon illumination, where each blue circle represents an individual photon. In the figure, we represent one of the possible outcomes from the beam splitter. (c) Simulated measurement by a HBT interferometer. In the first ten rows we show (from left to right columns): the measurement number, the intensity measured by the detector in the horizontal path, ¯ I1 , the intensity measured by the detector in the vertical path, ¯ I2 , the multiplication of the intensity measured by both detectors, ¯ I1¯ I2 , and the number of photons emitted by the single-photon source. These values are obtained with random number generators based on the number of photons statistically emitted by the single photon source and on the response of the beam splitter (see discussion in the text). In the last row of the table, we show the average value of ¯ I1 , ¯ I2 , and ¯ I1¯ I2 . Below the table we show the value of g(2) (0) obtained from the average values. 59 Chapter 2. Fundamentals of quantum nanophotonics where ˆ H(0) C=ℏωcˆc†ˆc(2.53) is the Hamiltonian of the cavity unperturbed by the emitter, and ˆ Hσ=(ˆp−qˆ A)2 m+ˆ V , (2.54) corresponds, a priori, to the Hamiltonian of the emitter unperturbed by the cavity, where the second term ( ˆ V ) is the potential energy experienced by the charges of the emitter, and the first term (( ˆp−qˆ A ) 2/m ) is equivalent to the kinetic energy of the emitter (see discussion of the ( p−qA )term in Eq. (2.46) ). This means that in this Hamiltonian, derived in the Coulomb-gauge, ˆ HC , we can not directly identify an independent term describing the interaction between the emitter and the cavity (the cavity-emitter interaction term is hidden in the energy of the “unperturbed” emitter). So far, we have introduced the creation and annihilation operators of the cavity, and the next natural step in the quantization scheme requires introducing the analogous “creation” and “annihilation” operators of the emitter. The standard procedure to introduce these operators requires truncating the Hilbert space of the emitter, and, in order to do so, ˆ Hσ (Eq. (2.54) ) should describe the energy of the emitter unperturbed by the presence of the cavity. However, the ˆ Hσ Hamiltonian includes the action of the cavity with terms ∝ˆ A . The latter ∝ˆ A terms appears from the description of the classical momentum p in Eq. (2.43) , and it does not allow us to derive further the quantization of ˆ Hσ [84]. Moreover, if we directly truncate the ˆ V term in ˆ Hσ (Eq. (2.54) ), ˆ V will also become dependent on the field of the cavity. Before entering into details on how to avoid this issue, let us consider what would happen if we directly truncate the Hilbert space of the emitter. The truncation of the Hilbert space of the emitter is a very common approximation in which the response of an emitter, such as a molecule or a quantum dot, is described by only considering the transition between two electronic levels with the lowest energy of the emitter. After reducing the response of the emitter to only its two lowest electronic levels, the emitter is referred to as a “two-level-system”. The two lowest energetic states of the unperturbed emitter are termed as the ground |g⟩ state and the excited |e⟩ state, and the energy structure of the emitter can be approximated by these two states if two main conditions are met: (i) that the two states are well separated from the higher energy levels, and (ii) that the energy difference ∆ ECT S = ℏ|ωc−ωσ| given by the transition defined by these two levels, ℏωσ , and the energy of the electromagnetic mode of the cavity, ℏωc , is much smaller than the energy difference between ℏωc and the energy of the other electric transitions of the emitter. In this thesis, we always consider that these two conditions are met. Then, the Hamiltonian of the “unperturbed” emitter in Eq. (2.52) is effectively approximated as ˆ Hσ = ℏωσ ( |e⟩⟨e|−|g⟩⟨g| ) / 2, implying that the energy of the unperturbed TLS becomes ℏωσ/ 2if excited or −ℏωσ/ 2otherwise. It is straightforward to foresee that replacing this expression of ˆ Hσ in Eq. (2.52) completely neglects the cavity-TLS interaction, and thus, it is not 66 2.5. The Hamiltonian of cavity-QED systems correct. On the other hand, references [84,97] point out another common effective approach to describe the “unperturbed” Hamiltonian of the emitter, which consists in substituting ˆp2 m+ˆ V→ℏωσ 2(|e⟩⟨e|−|g⟩⟨g|)(2.55) in ˆ Hσ . However, this approximation introduces two crucial sources of error. First, the ˆp operator in Eq. (2.55) depends on the action of the cavity via a qˆ A term (Eq. (2.43) ). Second, in the Coulomb gauge, the Hilbert space truncation of the emitter (onto the |g⟩ and |e⟩ states) causes the potential ˆ V to gain dependence also on the field ˆ A (see the beginning of the next subsection). Thus, nor the kinetic ˆp2 term, nor the potential ˆ V term in Eq. (2.55) are suited to describe the energy of the unperturbed emitter. In the following subsections, we briefly review a solution to successfully truncate the Hilbert space of the emitter and obtain a correct expression of the QRM Hamiltonian in the Coulomb gauge. 2.5.2 Truncation of the Hilbert space of the emitter, loss of locality, and quantum Rabi Hamiltonian in the dipole gauge Loss of locality of the emitter potential Next we briefly review the general discussion in reference [84] to explain how the potential of the unperturbed emitter becomes dependent on ˆ A due to the truncation of the Hilbert space of the emitter in the Coulomb gauge. After Eq. (2.48) we discussed that the potential operator ˆ V could be written in the distance basis in terms of the function V ( dσ, d′ σ ). We expect that V ( dσ, d′ σ ) = V ( dσ ) δ ( dσ−d′ σ )with V ( dσ )the value of the classical potential that we introduced in the beginning of the previous subsection (Eq. (2.41) ). This is because we initially consider that V ( dσ, d′ σ )is a local potential in the quantum context xiv . However, as we show next, when truncating the response of the QE to only its two lowest energy levels, the locality of this potential can be lost. We first consider a complete basis of the eigenstates of the emitter, |nσ⟩ ∈ {|g⟩,|e⟩,|e2⟩,|e3⟩, ...},(2.56) where |g⟩ is the ground state, |e⟩ is the first excited state, and |e2⟩,|e3⟩, ... are the higher excited states. Then we write V(dσ, d′ σ)in Eq. (2.48) in this |nσ⟩basis, V(dσ, d′ σ) = V(dσ)δ(dσ−d′ σ) = ∞ X nσ=n′ σ V(dσ)⟨dσ|nσ⟩⟨d′ σ|n′ σ⟩,(2.57) where in a realistic emitter, the ⟨dσ|nσ⟩ and ⟨d′ σ|n′ σ⟩ terms correspond to smooth xiv Note that here we discuss locality and non-locality in a quantum context, where quantum non-locality describes the instantaneous propagation of correlations between entangled systems, or in Albert Einstein’s words “spooky action-at-a-distance” [98–100]. 67 Chapter 2. Fundamentals of quantum nanophotonics wave functions representing the spatial distribution of the eigenstates of the emitter (note that the dσ distance is a continuous variable). Next, we truncate the sum on the |nσ⟩states to the first two eigenstates, |nσ⟩ ∈ {|g⟩,|e⟩}, and thus, V(dσ, d′ σ)≈V(dσ)(⟨dσ|g⟩⟨d′ σ|e⟩+⟨dσ|e⟩⟨d′ σ|g⟩).(2.58) The two terms added inside the parenthesis in Eq. (2.58) cannot result in δ ( dσ−d′ σ ), because the Dirac delta is the sum of all the elements of the basis; in other words, the Dirac delta cannot simply be factorized as the product of two smooth wavefunctions. Thus, Eq. (2.58) results in a loss of locality of the potential. Furthermore, any non-local potential V ( dσ, d′ σ )can be expressed as a momentum-dependent potential, V ( dσ, d′ σ ) →V ( dσ, p )[84,101 – 103], and this dependence on the momentum is a significant issue in the Coulomb gauge because, according to Eq. (2.43) , p depends on the potential vector A of the electromagnetic cavity mode. As we discussed above, if the potential operator ˆ V has a dependence on the field of the cavity, it is not suited to describe the potential energy of the unperturbed emitter. The method to avoid introducing the dependence of ˆ V on ˆ A consists in changing from the Coulomb to the dipole gauge. In the dipole gauge, p only describes the properties of the unperturbed emitter, allowing one to truncate the Hamiltonian of the unperturbed emitter. After this change, it becomes possible to apply a transformation onto the Hamiltonian of the emitter in the dipole gauge and return to the Coulomb gauge. From the Coulomb gauge Lagrangian to the dipole gauge Hamiltonian The Lagrangian introduced in Eq. (2.42) is written in the Coulomb gauge. To change from the Coulomb to the dipole gauge without affecting the equations of motion of the system, we must perform a transformation of the type LD = L + dGL/dt , where GL is a function of the position variables ( dσ ), the field variables ( A and Π), and time ( t ). In particular, the GL function that describes the transformation from the Coulomb gauge to the dipole gauge is GL = −qdσA , resulting in LD=L−q(˙ dσA+dσ˙ A) = 1 4m˙ d2 σ+V(dσ) + 1 2ε0VEff ˙ A2−1 2ε0VEffω2 cA2−qdσ˙ A. (2.59) This change of gauge affects the choice of the canonical momenta from p and Πin Eqs. (2.43) and (2.44) to pD=∂L ∂˙ dσ =1 2m˙ dσ,(2.60) ΠD=∂L ∂˙ A=ε0VEff ˙ A−qdσ.(2.61) Importantly, the new canonical momentum pD in the dipole gauge does not depend on any property of the external field. Using Eqs. (2.59) - (2.61) we can obtain the 68 2.5. The Hamiltonian of cavity-QED systems classical Hamiltonian in the dipole gauge: HD-Class. =p2 D m+V(dσ) + (ΠD+qdσ)2 2ε0VEff +1 2ε0VEffω2 cA. (2.62) Next we use the correspondence principle and convert pD ,Π D , dσ , and A in Eq. (2.62) into quantum operators, ˆ H(N) D=ˆp2 D m+ˆ V+(ˆ ΠD+qˆ dσ)2 2ε0VEff +1 2ε0VEffω2 cˆ A. (2.63) In a similar manner to the Coulomb gauge (Eqs. (2.49) and (2.50) ) we can introduce the second-quantization creation ˆc† and annihilation ˆc operators associated to the electromagnetic mode of the cavity, ˆ A=rℏ 2ε0VEffωc (ˆc+ ˆc†),(2.64) and ˆ ΠD=−irℏε0VEffωc 2(ˆc−ˆc†).(2.65) By substituting these expressions on ˆ HDwe obtain, ˆ H(N) D=ˆ H(D) σ+ℏωcˆc†ˆc+(qˆ dσ)2 2εVEff −iqˆ dσ√ℏωc √2ε0VEff (ˆc−ˆc†) + ℏωc 2,(2.66) where we have defined H(D) σ = ˆp2 D/m + ˆ V , as the Hamiltonian of the unperturbed emitter. Next, we truncate the Hilbert space of the emitter. We introduce this truncation on the Hamiltonian of the system ˆ HD in four steps [84,97]: (i) We first consider the complete basis of the eigenstates of the emitter in Eq. (2.56) , and truncate it to the first two states, the ground |g⟩ and the excited |e⟩ state. (ii) We introduce the annihilation (or lowering) operator ˆσ=|g⟩⟨e|,(2.67) and the creation (or rising) operator ˆσ†=|e⟩⟨g|.(2.68) ˆσ describes the decay from the excited |e⟩ state to the ground |g⟩ state of the emitter, and ˆσ† describes the excitation from |g⟩ to |e⟩ . (iii) We write the Hamiltonian of the unperturbed emitter, H(D) σ , as the energy difference between the excited and ground state, ˆ H(D) σ=ℏωσ 2ˆσz,(2.69) 69 Chapter 2. Fundamentals of quantum nanophotonics where ˆσz= [ˆσ†,ˆσ]. (iv) We write ˆ dσand ˆpDin terms of ˆσand ˆσ†, ˆ dσ=rℏ 2mωσ (ˆσ+ ˆσ†),(2.70) ˆpD=−irℏmωσ 2(ˆσ−ˆσ†).(2.71) Then, the Hamiltonian in Eq. (2.66) can be written as, ˆ H(N) D=ℏωσ 2ˆσz+ℏωcˆc†ˆc−iℏgrωc ωσ (ˆc−ˆc†)(ˆσ+ ˆσ†) + ℏq2 4mωσε0VEff +ℏωc 2,(2.72) with [84] g=q 2√mε0VEff .(2.73) After re-normalizing (neglecting the constant terms in the Hamiltonian), we find the well-known formula of the Rabi Hamiltonian in the dipole gauge, ˆ HD=ˆ H(N) D−ℏq2 4mωσε0VEff +ℏωc 2=ℏωσ 2ˆσz+ℏωcˆc†ˆc−iℏgrωc ωσ (ˆc−ˆc†)(ˆσ+ˆσ†). (2.74) 2.5.3 From the dipole gauge to the Coulomb gauge So far we have shown the derivation of the QRM Hamiltonian in the dipole gauge. Our next and final goal is to obtain this Hamiltonian in the Coulomb gauge. In Eq. (2.59) we introduced the change of gauge as a transformation on the Lagrangian operator. However, we can also directly change between the dipole and coulomb gauges by performing a unitary transformation on the Hamiltonian of the system [84,85,87,92,104], ˆ HC=ˆ Uˆ HDˆ U†+i˙ ˆ Uˆ U†,(2.75) with ˆ U= exp(i(−ˆ GL)),(2.76) being a unitary matrix, and ˆ GL = −qˆ dσˆ A corresponds to applying the quantum correspondence principle to the same GL function that we use to transform the Lagrangian from the Coulomb gauge to the dipole gauge. Using the expressions of ˆ Aand ˆ dσin Eqs. (2.64) and (2.70), ˆ Uresults in ˆ U= expig √ωσωc (ˆσ+ ˆσ†)(ˆc+ ˆc†).(2.77) 70 2.6. Quantum dynamics in open quantum systems: the quantum master equation Note that the ˆ Utransformation does not depend explicitly on time, and thus the i˙ ˆ Uˆ U†term in Eq. (2.75) vanish, and we obtain ˆ HC=ℏωcˆc†ˆc+ˆ Uℏωσ 2ˆσzˆσˆ U†(2.78) For simplicity, we focus on the resonant case with ω0≡ωσ = ωc , where Eq. (2.78) becomes ˆ HC=ℏω0ˆc†ˆc+ℏω0 2ˆσzcos[2η(ˆc+ ˆc†)] + ˆσysin[2η(ˆc+ ˆc†)],(2.79) with ˆσy = i ( ˆσ†−ˆσ ), and η = g/ω0 . This is the expression of the QRM Hamiltonian in the Coulomb gauge. From this derivation, it might seem easier to work in the dipole gauge, where the truncation of the Hilbert space of the basis of the emitter is straightforward, and hence, the derivation of the Hamiltonian is direct. However, we note that working in the dipole gauge also presents some disadvantages. For instance, whereas in the Coulomb gauge the operator of the electric field generated by the cavity is simply proportional to ˆc , in the dipole gauge it becomes proportional to ˆc + iη ( ˆσ + ˆσ† )[85,87]. This change in the electric field operator affects, for example, the operators describing the emission and the dissipation of the system, which is very inconvenient for analyzing the dynamics in cavity-QED systems. 2.5.4 Jaynes-Cummings model Hamiltonian Many cavity-QED setups show coupling strengths that are much smaller than the natural frequency of the cavity, i.e., g≪ω0 . In this case, the Hamiltonian in Eq. (2.79) can be simplified in the limit of very small coupling η = g/ω0→ 0. In this limit, we can expand the sine and cosine functions to the first order, and we obtain ˆ HC≈ℏω0ˆc†ˆc+ℏω0 2ˆσz+ℏgˆσy(ˆc+ ˆc†).(2.80) Further, we can perform the rotating wave approximation (RWA), which consists in neglecting the double rotating, no-number conserving terms, i.e., ˆσˆc and ˆσ†ˆc† . This approximation results in the widely-used Jaynes-Cummings Hamiltonian [21,105]: ˆ HJC =ℏω0ˆc†ˆc+ℏω0 2ˆσz+iℏg(ˆσ†ˆc−ˆc†ˆσ).(2.81) 2.6 Quantum dynamics in open quantum systems: the quantum master equation Every quantum system interacts with its environment, for example with phonons at a specific temperature or simply with vacuum fluctuations. Often we know very 71 Chapter 2. Fundamentals of quantum nanophotonics little about this environment, and we approximate it as a Markovian reservoir (a Markovian reservoir has no memory of its previous interactions with the quantum system, nor is it able to create coherences due to its interaction with the system). In this section, we focus on describing how the interaction of a quantum system with its environment affects its dynamics, i.e., the time evolution of the state of the quantum system. For that purpose, we introduce the quantum master equation formalism, which allows us to describe the dynamics of the system without giving a complete description of the environment (we just approximate the environment as a Markovian reservoir). In the following subsections 2.6.1 and 2.6.2, we review the critical steps in deriving the master equation (for a complete and pedagogical derivation of the master equation, including a detailed discussion of the approximations involved, references [72,73,106] are good options). 2.6.1 From the Von Neumann equation to the Markovian master equation In quantum mechanics, there are three pictures to understand the time evolution of a system: the Schrödinger, the Interaction, and the Heisenberg pictures. The studies presented in this thesis (chapters 5and 4) are given in the Schrödinger picture, where the states describe the evolution of a system. In the Schrödinger picture, the evolution of a state (described by its density matrix, ˆρ ) follows the Von Neumann equation [72], d dt ˆρ(t) = −i ℏ[ˆ H,ˆρ(t)],(2.82) being ˆ H the Hamiltonian describing the energy of the quantum system, the energy of the environment, and the system-environment interaction. Our aim in this section is to operate the Von Neumann equation (Eq. (2.82) ) to obtain the socalled quantum master equation, an expression that focuses only on the evolution of the state of the system, described by the density matrix ˆρS , and the interaction between the system and the environment is conveniently simplified. ˆρS is contained in the total density matrix ˆρ (in Eq. (2.82) ), which describes both the state of the system (independently of the environment) and of the state of the environment (independently of the system). We can extract ˆρSfrom ˆρas, ˆρS(t) = TrR{ˆρ(t)},(2.83) where TrR denotes the trace over the Hilbert space of the environment (or reservoir), [72–74,106]. Before starting the derivation of the master equation, we need to introduce ˆ H (in Eq. (2.82) ), which is the Hamiltonian of the system and the environment. This Hamiltonian can be split into three terms, ˆ H=ˆ HS+ˆ HR+ˆ HSR,(2.84) 72 2.6. Quantum dynamics in open quantum systems: the quantum master equation where ˆ HS describes the energy of the system that we are interested in studying. For convenience, our derivation of the master equation is carried out on the basis of the |ν⟩eigenstates of ˆ HS, such that ˆ HS=X ν ℏων|ν⟩⟨ν|,(2.85) where ℏων is the eigenvalue of the |ν⟩ eigenstate. ˆ HR in Eq. (2.84) describes the energy of the environment of the system, and ˆ HSR describes the interaction energy between the system and the environment. The most general form of the ˆ HSR Hamiltonian describing the interaction between the environment and the system is [72], ˆ HSR =ℏX α ˆ Sα⊗ˆ Rα,(2.86) where ˆ Sα = ˆ S† α and ˆ Rα = ˆ R† α are hermitian operators of the system and of the environment, respectively. The subindex α runsover the different components of the Hamiltonian of the system. For instance, in the cavity-QED systems studied in this thesis, we consider two different α terms, one for the interaction between the cavity with the environment, and another term accounting for the interaction between the TLS and the environment. We now proceed to operate the Von Neumann equation (Eq. (2.82) ) to obtain the master equation. For this derivation, it is most convenient to operate Eq. (2.82) in the interaction picture. In the interaction picture, both the states and the Hamiltonian evolve in time, and the Von Neumann equation in the interaction picture reads as [72], d dt ˆρ(i)(t) = −i ℏ[ˆ H(i)(t),ˆρ(i)(t)].(2.87) Along this section we use the label “( i )” to indicate that the Hamiltonian and the density matrix of the system, ˆ H(i) and ˆρ(i) ( t ), respectively, are written in the interaction picture. ˆ H(i) and ˆρ(i) ( t )are connected with the operators in the Schrödinger picture by a unitary transformation [72,73,106]: ˆ H(i)(t) = ˆ U† (s−i)(t)ˆ HSR ˆ U(s−i)(t),(2.88) and ˆρ(i)(t) = ˆ U† (s−i)(t)ˆρˆ U(s−i)(t).(2.89) with the unitary transformation being ˆ U(s−i)(t) = exp[i(ˆ HS+ˆ HR)t/ℏ].(2.90) We next re-express Eq. (2.87) by integrating it, ˆρ(i)(t) = ˆρ(i)(0) −i ℏˆt 0 [ˆ H(i)(t′),ˆρ(i)(t′)]dt′,(2.91) 73 Chapter 2. Fundamentals of quantum nanophotonics and then substituting ˆρ(i) ( t )in Eq. (2.91) back into the right-hand side of Eq. (2.87), d dt ˆρ(i)(t) = −i ℏ[ˆ H(i)(t),ˆρ(i)(0)] −1 ℏ2ˆ H(i)(t),ˆt 0 [ˆ H(i)(t′),ˆρ(i)(t′)]dt′.(2.92) As stated above, we are interested only in the time evolution of the state of the system, which we can extract from Eq. (2.92) using the partial trace over the Hilbert space of the environment (Eq. (2.83)), d dt ˆρ(i) S(t) = −TrRi ℏ[ˆ H(i)(t),ˆρ(i)(0)] + 1 ℏ2ˆ H(i)(t),ˆt 0 [ˆ H(i)(t′),ˆρ(i)(t′)]dt′. (2.93) To operate Eq. (2.93) we need to introduce our first approximation in this derivation: we assume that the environment corresponds to a Markovian reservoir, where we assume that the dissipation of the correlations of the reservoir is much faster than any variation of the system. Mathematically this implies that [72,73,106]: ⟨ˆ R(i) α(t)⟩=Tr{ˆ R(i) α(t)ˆρ(i)(0)} ≈ 0,(2.94) for all α, where ˆ R(i) α(t) = ˆ U† (s−i)(t)ˆ Rαˆ U(s−i)(t)(2.95) are the operators of the reservoir in the interaction picture. This is the main approximation that we are going to use for the derivation of the master equation. It can be proven that the assumption in Eq. (2.94) implies [72], TrR{[ˆ H(i)(t),ˆρ(i)(0)]} ≈ 0.(2.96) Thus, Eq. (2.93) simplifies to d dt ˆρ(i) S(t)≈ −TrR1 ℏ2ˆ H(i)(t),ˆt 0 [ˆ H(i)(t′),ˆρ(i)(t′)]dt′.(2.97) Next, we introduce two further approximations in Eq. (2.97) : we first assume that the state of the system at a time t′ does not depend on the history of the interaction with the reservoir at previous times t′, and thus, [ˆ H(i)(t′),ˆρ(i)(t′)] ≈[ˆ H(i)(t′),ˆρ(i)(t)].(2.98) This approximation simplifies Eq. (2.97) to d dt ˆρ(i) S(t)≈ −TrR1 ℏ2ˆ H(i)(t),ˆt 0 [ˆ H(i)(t′),ˆρ(i)(t)]dt′.(2.99) Note that in this last equation ˆρ(i) depends on t instead of t′ as in Eq. (2.97) . Second, we introduce the Born approximation, which states that for a weak interaction 74 2.6. Quantum dynamics in open quantum systems: the quantum master equation between the system and the reservoir, we can factorize the total density matrix ˆρ(i)(t)≈ˆρ(i) S(t)⊗ˆρ(i) R,(2.100) where the state of the reservoir, described by the density matrix ˆρ(i) R , is considered to be constant in time because we assume that the time scales at which coherences of the reservoir are dissipated are much faster than the time scales at which the system varies, and thus, ˆρR is not affected by the dynamics of the system (same approximation as for Eq. (2.94) ). Using the Born approximation (Eq. (2.100) ) in Eq. (2.99), we can arrive at the so-called Redfield equation [107], d dt ˆρ(i) S(t) = −TrR1 ℏ2ˆ H(i)(t),ˆt 0 [ˆ H(i)(t′),ˆρ(i) S(t)⊗ˆρ(i) R]dt′.(2.101) If we again assume that the reservoir correlations disappear very fast in comparison with the evolution of the system, we can consider that the [ ˆ H(i) ( t′ ) ,ˆρS ( t ) (i)⊗ˆρ(i) R ] term in Eq. (2.101) is only determined by the evaluation of t′ close to t , and thus, we can approximate the lower limit of the integral in time by −∞ [72,73,106]. This results in the Born-Markov or Markovian master equation, d dt ˆρ(i) S(t) = −TrR1 ℏ2ˆ H(i)(t),ˆ∞ 0 [ˆ H(i)(t−s),ˆρ(i) S(t)⊗ˆρ(i) R]ds,(2.102) where we have substituted t′→t−sfor convenience. 2.6.2 From the Markovian master equation to the Lindbladian master equation The integral in Eq. (2.102) does not ensure that ˆρ(i) S is a positive-definite matrix, which is a requisite for any quantum density matrix. To ensure that ˆρ(i) S is positive-definite, we need to perform one last approximation, the rotating wave approximation on the interaction between the system and the reservoir (RRWA xv ). To better explain the RRWA, let us first find the expression of ˆ H(i) , the Hamiltonian in the interaction picture. We first introduced ˆ H(i) in Eq. (2.88) , where it is shown to explicitly depend on ˆ HSR , the Hamiltonian describing the system-reservoir interaction in the Scrhrödinger picture. ˆ HSR is written directly in terms of the ˆ Sα operators of the system and the ˆ Rα operators of the reservoir (Eq. (2.86) ). In the interaction picture, ˆ H(i) can also be directly expressed in terms of the operators of the system and the reservoir [72], ˆ H(i)(t) = X α ˆ S(i) α(t)⊗ˆ R(i) α(t),(2.103) xv Do not confuse with the RWA of the Jaynes-Cummings model introduced in subsection 2.5.4. 75 Chapter 3. Fano asymmetry in zero–detuned exciton–plasmon systems from the QE to the nanoantenna, i.e., Gqs x,x ( re ) ≡Gqs x,x ( re,ra ) = Gqs x,x ( ra,re )(see Eq. (1.27) ). αa and αe in Eqs. (3.1a) and (3.1b) are the polarizability of the nanoantenna and of the QE, respectively. Within a Drude model description of the permittivity of the metal, and using a Drude-Lorentz model for the optical response of the QE, the αeand αapolarizabilities become: αe(ω) = Ae ω2 0−ω2−iγ0ω,(3.2a) αa(ω) = Aa ω2 0−ω2−iκω ,(3.2b) where the strength of the coupling between the two dipoles is determined by the polarizability amplitude of the QE, Ae , and that of the plasmonic nanoantenna, Aa . The frequency ωR = ωp/√3 is the frequency of the dipolar plasmon resonance in the nanoantenna (with ωp the Drude plasma frequency). Here we chose ωR to match ω0 ( ωR = ω0 ), the resonant excitation frequency of the QE. Last, γ0 and κ are the spontaneous decay rate of the QE and the plasmonic intrinsic decay rate, respectively. Throughout this chapter we consider that the dipole moment of the excitonic transition is f0 = 0 . 05 e· nm ( e is the electron charge), which sets the polarizability amplitude of the QE Ae = 2 ω0f2 0/ℏ and the spontaneous decay rate γ0 = ω3 0f2 0/ (3 πε0ℏc3 0 )(see section 1.2.4). We do not consider other intrinsic molecular losses beyond γ0 . Substituting Eqs. (3.2a) and (3.2b) into Eqs. (3.1a) and (3.1b) one obtains: (ω2 0−ω2)pe(ω)−iωγ0pe(ω) = AeGqs x,x(re)pa(ω),(3.3a) (ω2 0−ω2)pa(ω)−iωκpa(ω) = AaGqs x,x(re)pe(ω) + AaE0.(3.3b) Here, the polarizability amplitude of the nanoantenna determines how efficiently the system is excited (via the term AaE0 ). Eqs. (3.1a) - (3.3b) and the expression used to obtain Aaare discussed in more detail in section 3.4.2. We note that Eqs. (3.3a) and (3.3b) are very similar to those obtained with phenomenological models that assume that the QE and the plasmonic nanoparticle can be treated as two coupled harmonic oscillators. This coupled-harmonicoscillators model has been frequently used to describe the interaction between quantum emitters and nanoantennas [8,125,126]. We show in Fig. 3.1b the extinction cross-section spectrum of the hybrid QE-nanoantenna system obtained using Eqs. (3.3a) and (3.3b) for different values of the separation distance d between the QE and the surface of the nanoantenna (the value of d affects the quasistatic Green’s function, Gqs x,x , and, thus, the QEnanoantenna interaction). The extinction cross section σext is normalized to the corresponding value of the bare nanoantenna σ(0) ext and it is obtained assuming that the direct emission of the QE is negligible ( pa≫pe ), which is a typical situation 82 3.2. Fano asymmetry under resonant conditions in plasmonic systems. In this case, the optical theorem (Eq. (1.41)) [40] gives σsimp ext (ω) = 2π λε0 Im{pa(ω)/E0},(3.4) where the super index “ simp ” emphasizes that this is a simplified expression that only considers the emission from the nanoantenna. The normalized extinction cross-section in Fig. 3.1b features an almost constant background and the emergence of a spectrally narrow dip at the resonant wavelength λ0 for all the separation distances d considered. The background corresponds to the very broad plasmonic response and the dip to the Fano feature. Importantly, the Fano dip obtained within this simple dipole-dipole interaction model (Eqs. (3.3a)-(3.4)) is always perfectly symmetric. We next obtain the exact electromagnetic response of the hybrid system, which we calculate from the optical theorem (Eq. (1.41) ) by obtaining the total scattered field within the rigorous Mie theory formalism introduced in section 1.3 (see also references [40,45,127]). In particular, we use Mie theory to calculate the response of the spherical nanoantenna illuminated by the incoming plane wave (section 1.3.2) and by the induced dipole moment of the QE (section 1.2.4). For all Mie theory calculations shown in this chapter, we have used an expansion of 60 multipoles, which we verified that ensures convergence. The details of the calculation of the extinction cross-section are given in section 3.3.1. Figure 3.1c shows the resulting extinction cross section obtained for the same system and distances d as in Fig. 3.1b. These exact calculations exhibit again a broad background, due to the response of the bare plasmonic nanoantenna, and a narrow Fano feature caused by the coupling between the QE exciton and the plasmonic resonance of the nanoantenna. However, the Fano lineshapes show clear differences compared to those obtained with the simple dipole-dipole interaction model (Eqs. (3.3a) - (3.4) and Fig. 3.1b). Overall, the Fano features are broader for the exact calculations than for the simple dipole model. Further, the exact calculation also results in a shift of the Fano features, induced by the photonic lamb shift, not included in the simple model. The shift and larger broadening are clearer for small nanoantenna-QE separation distances ( d < 20 nm) and are mainly a consequence of the coupling between the QE exciton and the higher-order modes of the nanoantenna [35,95,96,128 – 136]. Crucially, the Fano feature obtained within the exact calculations is not necessarily a perfectly symmetric dip, but it can take different lineshapes. This shape evolves from a broad and almost symmetric dip at short separation distances ( d < 15 nm) towards a narrow and almost symmetric peak at large separation distances ( d > 60 nm). In the range between these two extremes, the Fano feature becomes clearly asymmetric. Thus, we have shown that the prediction of a symmetric Fano dip obtained with a simple dipole-dipole interaction model can strongly differ from the results of the exact calculations, where significantly asymmetric lineshapes emerge. We emphasize that this asymmetry is not due to plasmon-exciton detuning as the resonance condition of zero detuning is preserved in all cases. In the following, we 83 Chapter 3. Fano asymmetry in zero–detuned exciton–plasmon systems analyze in detail the different physical mechanisms that lead to asymmetric Fano lineshapes for QE-nanoantenna systems under resonant conditions. 3.3 Fano lineshape 3.3.1 Analytical derivation of the Fano lineshape in the extinction cross-section First, we show that the extinction cross-section spectrum σext of a QE interacting with a nanoantenna follows the modified Fano lineshape discussed in references [118,119], σext(ω) σ(0) ext ≈(Ω(ω) + q)2+B Ω(ω)2+ 1 ,(3.5) where q is the Fano asymmetry factor (the key parameter that we analyze in this chapter), Bis the zero-dip parameter, and Ω(ω) = ω′ 0 2−ω2 ωγ′, ω′ 0=ω0+ ∆ω, (3.6) with ∆ ω the Lamb shift. γ′ is the enhanced decay rate of the QE in the presence of the nanoantenna, which can be written in terms of the Purcell Factor PFas γ′= (PF+ 1)γ0+γNR i(3.7) where γNR i are other intrinsic (non-radiative) losses of the QE (in this chapter we consider γNR i = 0). q , B ,∆ ω , and PF are the main parameters defining the Fano lineshape. In the first part of this subsection, we focus on deriving simple analytical expressions to calculate them in an arbitrary QE-nanoantenna system. In the second part of this subsection, we evaluate them for a QE interacting with a silver spherical nanoparticle and discuss in more depth their physical origin. To obtain the extinction cross-section of the QE–nanoantenna system using the optical theorem [40], evaluated for our system and illumination σext(ω) = 4π k2Re(−ikzd)e−ikzdEFF x(rd, ω) E0,(3.8) where EFF x is the x -component of the scattered electric field that the QEnanoantenna system induces on a point-like detector placed in the far-field region, at the (Cartesian) coordinates rd = (0 x, 0 y, ( zd ) z ). σext is independent of the chosen value of zd since EFF x ( rd, ω ) ∝eikzd/zd [40]. We note that throughout this derivation, it is only necessary to calculate the x -component of all of the considered electric fields because of the geometry considered [35,40]. Thus, to avoid repetition, we do not always state explicitly in the discussion below that we are referring to the x -component of the fields, or the ( x, x )-component of the Green’s functions, 84 3.3. Fano lineshape but we indicate this by an x(or (x, x)) subindex. EFF xcan be expressed as the sum of two contributions, EFF x(rd, ω) = EA x(rd, ω) + EE-Tot x(rd, ω),(3.9) where EA x is the field directly scattered by the nanoantenna in the absence of the QE and EE-Tot x is the total electric field that the QE induces at the detector considering the presence of the nanoantenna. By defining the far-field enhancement factor at the position of the detector as KFF ( rd, ω ) = EA x ( rd, ω ) /E0 we can write EA xas EA x(ω) = KFF(ω)E0.(3.10) On the other hand, EE-Tot x can be written using the Green’s function formalism as EE-Tot x(rd, ω) = GFF x,x(rd,re, ω)pe(ω),(3.11) where pe is the induced dipole moment of the QE (oriented along the x -axis) and GFF x,x is the Green’s function that describes the emission of the QE towards the detector in the presence of the nanoantenna. We decompose GFF x,x as a sum of two contributions GFF x,x(rd,re, ω) = GFF 0x,x (rd,re, ω) + SFF x,x(rd,re, ω),(3.12) where GFF 0x,x (already introduced in Eq. (1.24) ) is the vacuum Green’s function that describes the field EE x that the QE induces in the detector when no nanoantenna is present, EE x(rd, ω) = GFF 0x,x(rd,re, ω)pe(ω),(3.13) where we can approximate re≈ 0in the evaluation of GFF 0x,x (i.e., the effect of this change becomes negligible small in the detector situated in the far-field), so that GFF 0x,x (rd,re, ω)≈GFF 0x,x (rd,0, ω) = k2 4πε0zd eikzd,(3.14) On the other hand, SFF x,x is the dyadic function that describes the electric fields induced by the QE on the detector via the nanoantenna EEA x [137] (i.e. the electric fields that the nanoantenna scatters towards the detector when it is illuminated only by the QE), EEA x(rd, ω) = SFF x,x(rd,re, ω)pe(ω).(3.15) For the exact results obtained with Mie theory, we calculate SFF x,x using Mie theory as discussed in reference [127]. To evaluate Eqs. (3.11) , (3.13) , and (3.15) we need first to obtain the value of the induced dipole moment pe . pe is given by (see 85 Chapter 3. Fano asymmetry in zero–detuned exciton–plasmon systems section 1.2.4), pe(ω) = αe(ω)ENF x(re, ω),(3.16) where αeis given in Eq. (3.2a). ENF xis the field that excites the QE, ENF x(re, ω) = E0+EAE x(re, ω) + EEAE x(re, ω),(3.17) which we write as the sum of the field E0 due to the direct illumination of the plane wave, and two additional contributions induced by the presence of the nanoantenna, EAE x and EEAE x . EAE x corresponds to the electric field that the nanoantenna induces at the QE position in the absence of the QE due to the illumination by the incident plane wave, i.e. EAE x depends only on the response of the isolated nanoantenna. If we use a typical definition of the near-field enhancement factor at the position of the QE, K(re, ω) = EAE x(re, ω)/E0, we can write EAE x(re, ω) = K(re, ω)E0.(3.18) On the other hand, EEAE x is the electric field that the QE induces at its own position via the nanoantenna, that is, the field scattered by the nanoantenna at the QE position when the only source is the QE. We write this contribution as a function of pe by using again the Green’s function formalism. For doing so, we introduce the dyadic function SNF x,x that describes the emission of the QE onto itself via the nanoantenna [137], EEAE x(re, ω) = SNF x,x(re, ω)pe(ω).(3.19) Using Eqs. (3.17), (3.18), and (3.19) we find the expression for pe, pe(ω) = Ae ω′ 0(ω)2−ω2−iγ′(ω)ω[K(re, ω) + 1]E0,(3.20) with ω′ 0(ω)2=ω2 0−AeRe{SNF x,x(re, ω)}, γ′(ω) = γ0+γNR i+1 ωAeIm{SNF x,x(re, ω)}, (3.21) where γNR i is the non-radiative decay rate of the QE (in this chapter, we consider γNR i = 0). We observe that the Ae/[ω′ 0(ω)2−ω2−iγ′(ω)ω] prefactor in Eq. (3.20) follows a similar expression to the original αe polarizability of the QE (Eq. (3.2a) ) but for the central frequency, which is shifted from ω0 to ω′ 0 in Eq. (3.21) (see Eq. (3.6) ) and for the decay rate of the QE, which is augmented from γ0 to γ′ in Eq. (3.21) (see Eq. (3.7)). By using Eqs. (3.10), (3.11), and (3.20) we can write EFF xas 86 3.3. Fano lineshape EFF x(rd, ω) = =nKFF(rd, ω)+[GFF 0x,x(rd,re, ω) + SFF x,x(rd,re, ω)] | {z } GFF x,x(rd,re,ω) Ae[K(re, ω) + 1] ω′ 0(ω)2−ω2−iγ′(ω)ωoE0, (3.22) and by substituting Eq. (3.22) into Eq. (3.8) we obtain σext(ω) = =4π k2Re(−ikzd)e−ikzdKFF(rd, ω) + GFF x,x(rd,re, ω)Ae(K(re, ω) + 1) ω′ 0(ω)2−ω2−iγ′(ω)ω. (3.23) We normalize this expression by the extinction cross-section of the bare nanoantenna σ(0) ext (that can be calculated using the optical theorem as σ(0) ext ( ω ) = (4 π/k2 ) Re{ ( −ikzd ) e−ikzdKFF ( rd, ω ) } ), and obtain the normalized extinction crosssection of the hybrid system as: σext(ω) σ(0) ext(ω)= =Ren1 + 1 k2 4πσ(0) ext(ω)(−ikzd)e−ikzdGFF x,x(rd,re, ω)Ae(K(re, ω) + 1) ω′ 0(ω)2−ω2−iγ′(ω)ωo. (3.24) Next, we define Ω(ω) = ω′ 0(ω)2−ω2 γ′(ω)ω,(3.25) q(ω) = 1 2 1 k2 4πσ(0) ext(ω) Ae γ′(ω)ωRe(−ikzd)e−ikzdGFF x,x(rd,re, ω)(K(re, ω) + 1), (3.26) B(ω) = = 1 −q(ω)2−1 k2 4πσ(0) ext(ω) Ae γ′(ω)ωIm(−ikzd)e−ikzdGFF x,x(rd,re, ω)(K(re, ω) + 1), (3.27) 87 Chapter 3. Fano asymmetry in zero–detuned exciton–plasmon systems and, taking into account that Ae , γ0 , ω , ω0 , and σ(0) ext ( ω )are real numbers, we can simplify Eq. (3.24) as σext(ω) σ(0) ext(ω)=1 + 2q(ω)Ω Ω2+ 1 +−1 + q(ω)2+B(ω) Ω2+ 1 =[Ω + q(ω)]2+B(ω)2 Ω2+ 1 .(3.28) For generality, we have kept in this derivation the explicit frequency dependence of the parameters σ(0) ext ( ω ), γ′ ( ω ), ω′ 0(ω)2 , q ( ω ), and B ( ω ), so that Eq. (3.28) is exact. If we consider that the spectral width of the Fano feature (determined by γ′ ) is much smaller than the spectral width of the plasmon resonance of the nanoantenna we can evaluate all these parameters at the excitonic resonant frequency ω0, σ(0) ext ≈σ(0) ext(ω0), γ′≈γ′(ω0), ω′ 0 2≈ω′ 0(ω0)2, q ≈q(ω0), B ≈B(ω0), (3.29) and approximatexvi Eq. (3.28) as the modified Fano lineshape [8,118,119], σext(ω) σ(0) ext ≈(Ω(ω) + q)2+B2 Ω(ω)2+ 1 ,(3.30) with Ω( ω ) = ( ω′ 0 2−ω2 ) / ( ωγ′ ). To ensure that the approximation of Eq. (3.29) is justified, we consider in this chapter a dipolar oscillator strength f0 of the QE sufficiently small so that the Fano features remain very narrow [138,139] (in our calculations we use f0= 0.05e·nm, ebeing the electron charge). Next we summarize the expressions of the parameters present in Eq. (3.30), σ(0) ext =4π k2 0 Re(−ik0zd)e−ik0zdKFF(rd, ω0),(3.31a) ∆ω=ω′ 0−ω0=qω0−AeRe{SNF x,x(re, ω0)}−ω0,(3.31b) PF=γ′−γNR i γ0−1 = 1 ωAeIm{SNF x,x(re, ω0)},(3.31c) q=1 2 1 k2 0 4πσ(0) ext Ae γ′ω0 Re(−ik0zd)e−ik0zdGFF x,x(rd,re, ω0)(K(re, ω0) + 1),(3.31d) B= 1 −q2−1 k2 0 4πσ(0) ext Ae γ′ω0 Im(−ik0zd)e−ik0zdGFF x,x(rd,re, ω0)(K(re, ω0) + 1), (3.31e) where k0 = ω0c0 is the wavevector at the resonant frequency ω0 . We have verified that Eqs. (3.30) - (3.31e) describe all the spectra that we present in this chapter very accurately (i.e. the approximation given in Eq. (3.29) is valid). We note that all the parameters in Eqs. (3.31a) - (3.31e) can be obtained from standard classical xvi In the systems studied in this chapter ω0≈ω′ 0 , and thus, evaluating the parameters σ(0) ext , γ′,q, and Bparameters in Eq. (3.29) at ω0or ω′ 0gives very similar results. 88 3.3. Fano lineshape electromagnetic calculations. Finally, we simplify Eq. (3.31d) using the reciprocity theorem [140 – 142]. According to this theorem, near-field enhancement factor K ( re, ω ) = EA x ( re, ω ) /E0 is connected with the fields that the QE induces at the detector directly ( EE x ) and via the nanoantenna (EEA x) by the following equation, K(re, ω) = EEA x(re, ω) EE x(re, ω).(3.32) Using Eqs. (3.13), (3.15), and (3.32) we write the reciprocity theorem as SFF x,x(rd,re, ω) GFF 0x,x(rd,re, ω)=K(re, ω).(3.33) In principle, the reciprocity theorem [140 – 142] relates the field enhancement induced by a plane-wave with the emission of the QE in the backward direction (i.e. in the negative z -direction for the considered systems), while GFF 0x,x and SFF x,x in Eq. (3.31d) describe the emission of the QE in the forward direction (i.e. in the positive z -direction). However, the symmetry of our system implies that GFF 0x,x and SFF x,x are identical in the forward and backward direction, and Eqs. (3.32) and (3.33) are valid. By using Eq. (3.12) we can write Eq. (3.33) as GFF x,x(rd,re, ω) = GFF 0x,x(rd,re, ω)(K(re, ω) + 1).(3.34) Last, using Eqs. (3.14), (3.31d), and (3.34) we obtain q=Ae 2σ(0) extγ′c0ε0 (Im{K(re, ω0)}+Re{K(re, ω0)}Im{K(re, ω0)}),(3.35) 3.3.2 Evaluation of the Fano lineshape In the subsection above, we have shown that the extinction cross-section of a QE-nanoantenna system excited by a plane wave can be described by a modified Fano lineshape [8,118,119] (assuming that the spectral width of the QE emission is much smaller than the width of the spectral response of the nanoantenna). For convenience, we repeat here the formula of the modified Fano lineshape in Eq. (3.30) σext(ω) σ(0) ext ≈(Ω(ω) + q)2+B Ω(ω)2+ 1 , where Ω(ω)=(ω′ 0 2−ω2)/(ωγ′)(Eq. (3.25)). The modified Fano lineshape in Eq. (3.30) depends on three parameters, q , Ω( ω ), and B (calculated from Eqs. (3.31a) - (3.31e) and (3.35) ). As introduced above, q is the total asymmetry factor (also called Fano-parameter) that captures the asymmetry of the Fano lineshape of the extinction cross-section spectrum, and it is the main focus of this chapter. Ωis a normalized frequency given in Eq. (3.6) . 89 Chapter 3. Fano asymmetry in zero–detuned exciton–plasmon systems 1.0 1.1 1.2 1.3 -0.8 -0.4 0 0.4 0.8 (nm) 1.0 1.2 1.4 1.6 ext/(0) ext ext/(0) ext 3 4 4 0.4 0.6 0.8 1.0 ext/(0) ext 2 0.5 1.0 1.5 ext/(0) ext 1 (a) R = 50 nm d = 7 nm R = 20 nm d = 50 nm R = 50 nm d = 100 nm R = 70 nm d = 100 nm 2 1 3 4 (c) 2 1 3 4 (b) 2 1 3 4 (d) 2 1 3 (e) RR Figure 3.2: (a-d) Contour plots of the parameters defining the Fano lineshape, obtained within Mie theory. (a) Purcell Factor, PF , (b) Lamb Shift, ∆ ω , (c) contrast, C , and (d) total asymmetry factor, q , as a function of the distance, d (the minimum distance in the panels is d = 2 nm), from the QE to the surface of a silver spherical antenna with different radius, R . The resonance of the QE is chosen to match the frequency of resonance of the nanoantenna for all sizes of particles, R . (e) Normalized extinction spectra of the hybrid system are evaluated at points marked as 1,2,3, and 4in panels (a)-(d). The values of R and d for each point are indicated in the labels of panel (e). ∆ ω is the photonic Lamb Shift (Eq. (3.6) ) that corresponds to a slight shift in the resonant frequency from ω0 to ω′ 0 , and γ′ = ( PF + 1) γ0 are the enhanced losses of the QE (Eq. (3.7) ), which describes the broadening of the Fano dip. Both effects can be clearly observed in the spectra of Fig. 3.1c. The expression of γ′ assumes no intrinsic losses beyond γ0 , but it can be modified in a straightforward manner to include additional intrinsic losses. Last, B in Eq. (3.30) is the zero-dip parameter (Eq. (3.31e) ) that can be related to the factor q and the contrast C = p2(B+ 1)q2+ (B−1)2+q4 . Here, we define the contrast C of the Fano feature as the difference between the maximum and the minimum of the Fano feature in the normalized extinction cross-section spectrum (see inset in panel 2 of Fig. 3.2e). Thus, changes on the Fano spectral lineshape can be understood by analyzing the parameters, q , C , PF , and ∆ ω (the last two determining Ω). Note that these parameters can be obtained from the classical Green’s function and the field enhancement of the plasmonic antenna at the position of the emitter according to Eqs. (3.31a) - (3.31e) . In Fig. 3.2a-d, we systematically study the dependence of these parameters with the radius of the silver spherical nanoparticle, R , and the distance between the antenna and the emitter, d . All values are obtained from exact Mie theory calculations using the experimental values of the silver permittivity [41] and assuming resonant conditions; i.e., for each radius, we find the frequency of the dipolar plasmonic resonance of the antenna (lowest-energy peak in the extinction 90 3.3. Fano lineshape cross-section spectrum of the bare nanoantenna), and we modify the energy of the QE transition accordingly. To illustrate the resulting Fano lineshape described by the parameters in Figs. 3.2a-b, we also show, in Fig. 3.2e, the extinction cross section spectrum for four different points indicated in Figs. 3.2a-d (the values of R and dfor each point are given in the labels of Fig. 3.2e). For all the radii considered, the Purcell Factor, PF , (Fig. 3.2a), and the photonic Lamb shift ∆ ω (Fig. 3.2b) strongly increase when the QE approaches the antenna, as shown in previous studies [35,95,96,128 – 133]. This increase is due to the more efficient coupling of the QE with the plasmonic modes of the nanoantenna, particularly with high-order modes. On the other hand, the contrast, C , shows a more complex dependence with the radius, R , and the distance, d (Fig. 3.2c). We can distinguish three different distance regimes in this figure. For short distances ( d≲ 10 nm), the QE couples very efficiently to the higher-order modes of the spherical nanoparticle, and the resulting quenching of the emission [134 – 136] leads to the disappearance of the Fano dip (small contrast). For intermediate distances (compared to the radius, i.e. 10 nm ≲d≲ 3 R ), the quenching becomes less significant, and the Fano feature emerges with a reasonably big contrast (1 ≳C≳ 0 . 5, purple-reddish region in Fig. 3.2c). In this regime of distances, the contrast is smaller for spheres with R≳ 40 nm, which is mainly a consequence of two destructive interference effects, the first between the excitation of the QE by the illumination plane–wave and by the antenna-induced near fields, and the second between the light emitted by the QE directly and via the nanoantenna. Last, as the separation distance is made significantly larger than the radius ( d≳ 3 R ), the QE progressively decouples from the nanoantenna, and the extinction cross-section of the whole system evolves toward the superposition of the peak of the extinction cross-section of the QE in a vacuum on top of the broad background spectrum of the bare spherical nanoparticle. We can then express the extinction cross– section of the whole system at very long separation distances as ( σ(0) ext + σQE ext ) /σ(0) ext , where σQE ext is the extinction cross-section of the QE in a vacuum. As we have considered that the QE only has radiative losses due to the spontaneous decay, σQE ext is larger than the extinction cross-section of the bare spherical nanoparticle σ(0) ext [36] ( σQE ext = 6 π ( ω0/c0 ) 2> σ(0) ext ), and the contrast becomes higher than one, C > 1. Last, Fig. 3.2d shows the relatively complex dependence of the total asymmetry factor q with radius R and distance d . The Fano asymmetry is small ( |q|< 0 . 2) for d≲ 10 nm (corresponding to a symmetric dip) and for large distances, d≳ 150 nm, (corresponding to an almost symmetric peak). At intermediate distances (10 nm ≲d≲ 150 nm) the asymmetry is significantly larger, with a maximum value at a distance d∼ 50 nm, which is strongly dependent on the radius. We also find that at large distances ( d≳ 150 nm) the asymmetry can take negative values and show a damped oscillatory behavior of q with d (the dependence of q for a larger range of distance is studied in section 3.4.3). Understanding this complex behavior is the main objective of this chapter and it is analyzed in detail next. 91 Chapter 3. Fano asymmetry in zero–detuned exciton–plasmon systems We note that, once the radiative correction is introduced, the maximum of the extinction cross-section σ(0) ext of the bare nanoantenna dipolar resonance red-shifts spectrally with increasing size and is found at a slightly shorter wavelength than the corresponding maximum of the near–field enhancement |K| [144,145]. When not stated otherwise, we consider by default that the resonance frequency of the QE, ω0 , matches the frequency at which σ(0) ext is maximum. Below, in section 3.4.3, we show that the effect of setting ω0to the maximum of |K|is weak. We show in Fig. 3.4b (second column of the figure) the resulting total asymmetry factor q (and its contributions, qE and qR ) obtained after substituting αMD a by αRC a in the simplest dipole-dipole interaction model (3.44a) - (3.44c) . This change mainly affects the value of K , which acquires a larger real part at resonance as compared to the previous model (the phase of K deviates further from π/ 2) and, thus, q = qR increases (Eq. (3.37) ), with qE remaining equal to zero. In particular, qR becomes larger with increasing R , as the effect of the radiative correction increases with the size of the nanoantenna. qR also increases for shorter d due to the stronger enhancement |K| . However, the asymmetry remains small (hardly noticeable in Fig. 3.4b, with max ( qR ) ≈ 0 . 14). Last, we note that this low value of q may lead to think that the radiative correction is of little importance in the description of the total asymmetry factor. However, we emphasize in the last model presented in this section that, once we go beyond the quasistatic approximation, it is critical to consider a correct description of the radiative correction. Direct excitation and emission of the QE In the next step we introduce the direct excitation of the QE by the plane wave and the direct emission of the QE to the far field (third column, Fig. 3.4c). These two effects are introduced at the same time because, due to reciprocity [140 – 142], their contribution to the asymmetry is identical (demonstration in appendix A). After all these changes the response of the system is given by the following modified equations: pa(ω) = αRC a(ω)(E0+Gqs x,x(re)pe(ω)),(3.45a) pe(ω) = αe(ω)(1 + K(re, ω))E0,(3.45b) K(re, ω)E0=Gqs x,x(re)pa(ω),(3.45c) γ′=1 + ImAeαRC a(ω) γ0ω0 (Gqs x,x(re, ω))2γ0,(3.45d) σext(ω) = 2π λε0 Impa(ω) + pe(ω) E0.(3.45e) The direct excitation of the QE by the incident plane wave of amplitude E0 is described by the term αeE0 in Eq. (3.45b) . Similarly, the direct contribution from the QE to the extinction cross section is given by the term ∝Im{pe} in Eq. (3.45e) . In this scenario, the qE contribution to the asymmetry is no longer zero, and the full expression of the total asymmetry factor, q = qE + qR , needs to be 98 3.4. Dissection of the asymmetry considered (Eq. (3.36) ). On the other hand, qR remains unchanged as compared to the previous model. As shown in Fig. 3.4c the resulting qE dominates the total asymmetry factor q , and follows similar trends with distance as those described when discussing the results of the exact calculation in Fig. 3.3. qE is small at long separation distances ( d > 100 nm) because the QE and the nanoantenna start to behave independently, and also at short distances ( d≲ 20 nm) because the direct excitation of the QE is very small compared to the excitation via the nanoantenna. qE is thus maximum at intermediate distances within this model (20 nm ≲d≲ 80 nm) where the excitation of the QE via the nanoantenna is of the same order of magnitude than the direct excitation by the incident plane wave. The distance that maximizes qE follows a linear dependence with increasing radius R . More precisely, the maxima are found for an approximately constant distance between the QE and the center of the nanoantenna (d+R). This behavior occurs because in this description the near fields are evaluated using the quasistatic Green’s function, which only depends on ( d + R ) 3 . Further, despite the similar behavior of qE obtained with this model, and that obtained with the rigorous calculation (compare Figs. 3.4c and 3.3b), some differences still remain. In particular the latter decays more slowly with distance, it changes its sign as the distance increases, and the maximum of qE is found at a similar distance d for all radii. Moreover, the current model, given by Eqs. (3.45b) - (3.45e) , is clearly insufficient to reproduce the exact qR contribution (compare Figs. 3.4c and 3.3a). Full retarded Green’s function In order to further approach the exact response of the interacting system, we replace the quasistatic near-field Green’s function in Eqs. (3.45a) - (3.45e) with the full expression of the Green’s function Gx,x in Eq. (1.21) [35]. Gqs x,x ( re ) = 1 / [(2 π ) ε0 ( R + d ) 3 ](Eq. (3.43) ) is always a real number but Gx,x is complex, with a phase that changes with distance d largely due to the retardation phase associated with the propagation of the fields. Furthermore, Gx,x decays more slowly than Gqs x,x with dbecause it includes terms decaying as 1/(R+d)and i/(R+d)2 (corresponding to the farand intermediate-field contributions, respectively). These changes directly affect the phase and the modulus of the enhancement factor ( K ( re, ω ) = Gx,x ( re, ω ) pa ( ω ) /E0 ) and thus both qE and qR (Eq. (3.36) ), as shown in Fig. 3.4d (fourth column). We first observe that the distance-dependence of the amplitude and phase of K induces the change of sign of qE for d≈ 150 nm (change from red to blue color), and also the overall slower decay of its absolute value ( |qE| ) with d discussed above. We show in section 3.4.3 that qE oscillates for larger separation distances. The maxima values of qE are larger than those in the previous model, mainly due to the farand intermediate field contributions. Further, we obtain clearly larger values of |qR| than in the previous model, with values of up to |qR| ≈ 0 . 45, as compared to |qR|≲ 0 . 14 in Fig. 3.4c. According to Eq. (3.37) we can directly relate these high values of qR to changes of phase 99 Chapter 3. Fano asymmetry in zero–detuned exciton–plasmon systems of the field enhancement, φA . When the full Green’s function Gx,x is used, φA can considerably differ from π/ 2for moderate and large d , which explains the relatively large values of |qR| . |qR| is maximum for ( R + d ) ≈ 100 nm and it decays for larger distances because the field enhancement becomes very small and, thus, |qR|∝|K(re, ω)|2/γ′(Eq. (3.37)) progressively approaches zero. The asymmetry contributions qE and qR take similar absolute values of opposite signs at intermediate distances (20 nm < d < 100 nm). As a consequence, the total asymmetry q = qE + qR partially cancels in this regime, specially for radius 25 nm ≲R≲ 70 nm. Thus, q presents a saddle point centered at R≈ 40 nm and d≈ 80 nm. The qualitative dependence of q with radius and distance within this model is in good agreement with the rigorous Mie theory results (Fig. 3.2d). We thus conclude that this improved model contains the fundamental elements to capture the main features of the behavior of the total asymmetry factor. Experimental permittivity We can further increase the agreement with the results obtained with the Mie theory calculations by using the same experimental values εExp a of the permittivity of silver used in that Mie calculations (instead of the modified Drude model). The polarizability of the nanoantenna then becomes αRC-Exp a ( ω ) = αExp a ( ω )[1 / (1 − iαExp a ( ω ) k3/ (6 πε0 ))] with αExp a ( ω ) = 4 πR3 ( εExp a ( ω ) − 1) / ( εExp a ( ω )+2). Figure 3.4e (fifth column) shows the asymmetry contributions calculated with this assumption. The changes as compared with the previous model (Fig. 3.4d) are relatively small and are mostly found for small spheres ( R < 25 nm), where we find an increase of |qE| and a decrease of |qR| . Indeed, smaller spheres resonate at shorter wavelengths, for which the contribution of the d -electrons to the experimental permittivity significantly modifies the plasmonic response. The changes on the asymmetry due to the influence of the d -electrons can be larger in other materials, such as gold. For example, we show in section 3.4.3 that including this effect is crucial to accurately describe the asymmetry factor for a QE interacting with a gold nanoantenna. Improved description of the radiative correction Last we introduce a more accurate description of the radiative–corrected polarizability following reference [47]: αIRC a(ω) = αExp a(ω) 1−3 5ζ2εExp a(ω)−2 εExp a(ω)+2 −iαExp a(ω)(2π/λ)3 6πε0−3ζ4 350 (εExp a(ω))2−24εExp a(ω)+16 εExp a(ω)+2 (3.46) with ζ= 2πR/λ. We implement this improvement to the dipole-dipole interaction model, which can be summarized in a set of equations as: pa(ω) = αIRC a(ω)(E0+Gx,x(re, ω)pe(ω)),(3.47a) 100 3.4. Dissection of the asymmetry pe(ω) = αe(ω)(E0+K(re, ω)),(3.47b) K(re, ω)E0=Gx,x(re, ω)pa(ω),(3.47c) γ′=1 + ImAeαIRC a(ω) γ0ω0 (Gx,x(re, ω))2γ0,(3.47d) σext(ω) = 2π λε0 Impa(ω) + pe(ω) E0.(3.47e) For ease of reference, we summarize all the aspects that are included in Eqs. (3.47a) - (3.47e) but not in Eqs. (3.44a) - (3.44e) (the latter corresponding to the simplest model considered in this section): (i) the direct excitation and emission of the QE are included in Eqs. (3.47b) and (3.47e) , respectively, (ii) the propagation of the fields beyond the quasistatic approximation is included by the full Green’s function in Eqs. (3.47a) , (3.47c) , and (3.47d) , and (iii) we use a modified version of the polarizability of the spherical nanoparticle αIRC a in Eqs. (3.47a) and (3.47d) that incorporates the effect of the radiation damping of the nanoantenna and considers the influence of d-electrons on the permittivity of the material. Figure 3.4f (sixth column) shows that by introducing the improved radiative correction (Eq. (3.46) ) the values of the asymmetry change very little, with the largest changes occurring for R > 50 nm (as compared to the results of the previous model in Fig. 3.4e). In particular the maxima of |qR| and qE for R > 50 nm have been displaced in Fig. 3.4f towards slightly larger distances d . The reason for this displacement is that the new radiative correction redshifts the resonant wavelength for large particles, which changes the ratio between the QE-nanoantenna distance and the wavelength, (R+d)/λ (affecting the full Green’s function Gx,x). The resulting values of the total asymmetry factor q and the qR and qE contributions that are obtained within this improved model (Fig. 3.4f) are in very good agreement with the exact results shown in Figs. 3.2d and 3.3a-b for the radius and distances considered. The main difference occurs in the shortest range of distances, d < 10 nm. For such distances the Mie theory calculation results in a very large increase of the decay rate γ′ due to the coupling with the high-order modes of the plasmonic response [134], which is not included in the dipole-dipole description analyzed here. The large increase of γ′ strongly reduces the asymmetry by increasing the denominator in Eq. (3.36) . However, this decrease is hard to appreciate in the figures, as the value of q predicted by the most refined dipole-dipole interaction model (Eqs. (3.47b) - (3.47e) ) is already small for these short distances. Equations (3.47b) - (3.47e) allow for a simple quantitative description of the total asymmetry factor that enables to identify the different effects that influence the value of q . However, it is instructive to further analyze the importance of the radiative correction. In the discussion above, the introduction of the simpler radiative correction only led to a very small change of the asymmetry (compare Fig. 3.4a and Fig. 3.4b), but this effect is small only when considering very simple dipole-dipole interaction models. If the radiative correction is neglected in the final expressions (Eqs. (3.47b) - (3.47e) ) we obtain a completely inaccurate response 101 Chapter 3. Fano asymmetry in zero–detuned exciton–plasmon systems for the total asymmetry factor (and its contributions). This can be observed by comparing Fig. 3.4g (seventh column) with Fig. 3.4f, where the only difference between the two is that Fig. 3.4g ignores the radiative correction. We have verified that including the radiative correction is necessary for all the models that use the full Green’s function. We have thus shown, in this subsection, how each approximation in the dipoledipole interaction affects the description of the Fano asymmetry for zero-detuning. This has allowed us to associate the different aspects of the asymmetry with relevant physical effects. 3.4.3 The Fano asymmetry factor in additional scenarios In sections 3.2,3.3, and 3.4 we study the asymmetry factor of the Fano feature present in the extinction cross-section spectrum of a QE placed at a distance d∈ [2 , 200] nm from the surface of a single spherical nanoparticle. In particular, we studied the situation where the plasmonic mode of the nanoantenna and the exciton of the QE are resonant are the same wavelength λ0 , with λ0 the wavelength that maximizes the extinction cross-section of the bare nanoantenna (in the spectral region where the dipolar plasmonic resonance dominates the response). However, some of the dependencies of the Fano asymmetry predicted (such as the oscillatory behavior of q with the separation distance d or the influence of the d -electrons) are hard to appreciate in the situations considered so far. Therefore, we explore in this subsection three additional clarifying scenarios. First, we consider a larger range of distances d∈ [2 , 1200] nm. Second, we change the material of the spherical nanoparticle from silver to gold. Last, to assess the robustness of our results, we analyze the effect of tuning the resonant wavelength of the QE to match the maximum value of the near-field enhancement factor |K| associated with the dipolar mode instead of the maximum of the cross-section. The analysis presented here follows the same procedure as in subsection 3.4.2, i.e. we calculate the asymmetry factor q and its contributions qR and qE (Eq. (3.36) ) with a series of simple models based on the dipole-dipole approximation. The simplest model always predicts an almost negligible asymmetry factor, and we progressively incorporate different physical effects that increase the accuracy of the description. Columns a-f of Figs. 3.5,3.7, and 3.6 correspond to the same models as those developed for Fig. 3.4a-f. We note that all the models in this subsection do not include the quenching effect due to the coupling with higher order plasmonic modes [134] that strongly decrease qfor d < 10 nm. Longer range of distances So far we have focused on the behavior of the asymmetry for a range of distances d∈ [2 , 200] nm for the different dipole-dipole interaction models. We show in Fig. 3.5a-f the values of the asymmetry factor q and its contributions qR and qE calculated for a larger range of distances, d∈ [2 , 1200] nm. Figure 3.5f shows the values of q , qR , and qE using the most accurate dipole-dipole interaction model 102 3.4. Dissection of the asymmetry No RC RC IRCRC RC (a) (b) (c) (d) (e) (f) RC R R R R R R Figure 3.5: Values of the asymmetry factor calculated for a QE in the proximity of a spherical silver nanoparticle of different radius R . The system is identical as in Fig. 3.4, except that the asymmetry is calculated for a larger range of distances between the QE and the surface of the nanoparticle d∈ [2 , 1200] nm. (a)-(f) Asymmetry factor q (panels in the third row) and its contributions qE (first row) and qR (second row) calculated as a function of R and d . Each column corresponds to a different model, as indicated by the labels at the bottom, following the same scheme as used in Fig. 3.4a-f (see caption of that figure for further details). using the improved description of the radiative correction (Eqs. (3.47a) - (3.47e) ). Consistently with the discussion in subsection 3.4.2, as we increase d we can observe a clear oscillatory behavior (superimposed to a general tendency to decrease) of q , qR , and qE . The oscillatory behavior causes changes of the sign of these three factors. Comparing the results for the different models in Fig. 3.5 we find that the oscillations appear when we include the full Green’s function Gx,x in our model (compare Figs. 3.5c and d). Last, we note that Fig. 3.5a (corresponding to the simplest dipole-dipole interaction model) also shows a small but clearly non-zero value of asymmetry, q = 0 for d≲ 100 nm, which was harder to appreciate in Fig. 3.4a because of the chosen color scheme. Gold spherical nanoparticle Figure 3.6a-f shows the analysis of the asymmetry factor q of the Fano feature in the extinction cross-section spectrum and its contributions qR and qE when the single spherical nanoparticle is made of gold. The QE resonant frequency is again set to the value that maximizes the extinction cross-section spectrum near the dipolar resonance. Figure 3.6f, corresponding to the most precise dipole-dipole interaction model consider, shows that q is relatively big for a large range of 103 Chapter 3. Fano asymmetry in zero–detuned exciton–plasmon systems No RC RC IRCRC RC (a) (b) (c) (d) (e) (f) RC R R R R R R Figure 3.6: Values of the asymmetry factor calculated for a QE in the proximity of a spherical gold nanoparticle of different radius R of the nanoparticle and different distances d between the QE and the surface of the nanoparticle. (a)-(f) Asymmetry factor q (panels in the third row) and its contributions qE (first row) and qR (second row) calculated as a function of R and d . Each column corresponds to a different model, as indicated by the labels at the bottom, following the same scheme as used in Fig. 3.4a-f (see caption of that figure for further details). distances, q > 0 . 25 for d≲ 200 nm (whereas in the silver case q > 0 . 25 for 20 nm ≲d≲ 100 nm, see Fig. 3.4d and 3.4f). Further, the dependence of q with d for the gold nanoparticles shows a broad single maximum and does not change its sign in the range of distances considered. This is in contrast with the silver results, where there is a change of sign of q following an oscillatory pattern. Additionally, for the gold spherical nanoparticle (Fig. 3.6f) |qE| is overall much larger than |qR| , so that it dominates the dependence of the total asymmetry factor q with R and d , while in the silver nanoparticles |qE| and |qR| are of the same order of magnitude and both contributions strongly influence the values of q . Last, Fig. 3.6f shows that the qR contributions takes large positive values ( q > 0 . 25) for d≲ 50 nm when considering gold as the plasmonic material, whereas for silver, qR is overall negative for the same range of distances (see Fig. 3.4f). The differences between the calculations for silver and gold nanoparticles are mostly due to the contribution of the d -electrons of the material to the dielectric permittivity, which is much larger for gold. This can be confirmed by looking at Fig. 3.6d, which was obtained using the modified Drude model for gold. Specifically, we use εMD a=ε∞−ω2 p ω(ω+iκ),(3.48) 104 3.4. Dissection of the asymmetry IRC No RC RC RC RC (a) (b) (c) (d) (e) (f) RC R R R R R R Figure 3.7: Values of the asymmetry factor calculated for a QE in the proximity of a spherical silver nanoparticle of different radius R of the nanoparticle and different distances d between the QE and the surface of the nanoparticle. The asymmetry factor is calculated by tuning the QE resonance to match the frequency at which the nanoantenna near-field enhancement is maximized due to the dipolar resonance for each particular set of R and d (in contrast, in Fig. 3.4 the QE resonance was matched to the extinction maximum of the plasmonic dipolar resonance). (a)-(f) Asymmetry factor q (panels in the third row) and its contributions qE (first row) and qR (second row) calculated as a function of R and d . Each column corresponds to a different model, as indicated by the labels at the bottom, following the same scheme as used in Fig. 3.4a-f (see caption of that figure for further details). with ε∞ = 9, ℏωp = 9 . 07eV, and ℏκ = 71 meV for gold. These values are obtained from fitting the experimental data [40] for large ω . The dependence of the asymmetry factor with R and d shown in Fig. 3.6d (modified Drude model) are very similar to the results for the silver nanosphere (Fig. 3.4f). QE tuned to the frequency of the maximum field enhancement All the results presented in this chapter except for Fig. 3.7 are obtained considering that the resonant frequency of the QE matches the maximum of the extinction cross-section of the bare nanoantenna. In Fig. 3.7a-f we show the analysis of the Fano asymmetry in the extinction cross-section spectrum for the case where the resonant frequency of the QE matches the maximum of the field enhancement induced by the nanoantenna at the position of the QE (in both cases we consider the maximum that is mostly determined by the dipolar plasmonic mode). Overall, the values of |q| , |qR| , and |qE| are slightly higher in Fig. 3.7a-f than in Fig. 3.4a-f. However, these differences are small and the trends of the dependence of q , qR , and qE with the distance d and the radius of the nanoparticle R are the 105 Chapter 3. Fano asymmetry in zero–detuned exciton–plasmon systems same in Fig. 3.7a-f and in Fig. 3.4a-f. 3.5 Fano resonance in dimers In the previous sections we have analyzed in detail the asymmetry of the Fano lineshape that is revealed in the extinction cross section spectrum of a QE placed near a spherical metallic nanoparticle (Fig. 3.1a), chosen as an example of a canonical nanoantenna. To demonstrate that a similar analysis can be applied to more general nanostructures, we consider next the Fano asymmetry for a QE situated in a junction between two spherical gold nanoparticles (a dimer nanoantenna). This dimer configuration has been intensely studied because it induces a much larger near–field enhancement than the single spherical nanoparticle, as sought, for example, in surface-enhanced spectroscopy [25,26,146–151]. We show in Fig. 3.8a a scheme of the dimer configuration. The system is driven by an incident plane wave of amplitude E0 that propagates along the z -axis, and polarized along the x -axis parallel to the orientation of the point-like dipole that represents the QE and to the axis of symmetry of the two spherical nanoparticles. We consider gold instead of silver nanoparticles in this section. Despite having larger absorption losses, gold is widely used in surface-enhanced spectroscopy because it does not oxidize and it is more handleable in experiments. The permittivity of gold is taken from reference [41], the two spherical nanoparticles have a radius of R = 40 nm, and we vary their surface-to-surface distance 2 d . The emitter is placed in the middle of the gap between the two nanoparticles (at distance d from the surface of each of them), and its properties are the same as in the previous sections (strength f0 = 0 . 05 e· nm, intrinsic decay rate corresponding to the spontaneous radiative decay, and resonance frequency tuned as a function of d to always match the dipolar resonance of the nanoantenna [152], as given by the maximum of the extinction cross-section), i.e., we keep the condition of zero-detuning in all cases analyzed and shown here. Fig. 3.8b shows the extinction cross-section spectrum of this hybrid system calculated for different values of d , as obtained from Eq. (3.8) . In this section, the value of all the necessary input electromagnetic parameters (such as the near-field enhancement and the self-interaction Green’s function) are obtained from the solution of Maxwell’s equations under plane-wave of dipolar illumination as given by the Matlab package MNPBEM17 [153 – 155] (the details of these calculations are given in appendix C). A clear Fano feature is observed in all spectra, showing a qualitatively similar dependence with distance as the results of the single spherical nanoparticle (Fig. 3.1c). In both situations the Fano lineshape obtained at small distances d≲ 10 nm corresponds to a broadened and almost symmetric dip, while at much larger separation distances, d≳ 200 nm, we observe an almost symmetric narrow peak. Thus, q≈ 0in these two situations. For values of d between these two extremes, the Fano spectrum shows various degrees of asymmetry. Despite these qualitative similarities, the results obtained for the dimer nanoantenna (Fig. 3.8b) and a single silver nanoparticle (Fig. 3.1c) show some 106 3.5. Fano resonance in dimers Figure 3.8: Characterization of the Fano asymmetry in the extinction cross–section of a QE coupled to a metallic dimer obtained at zero detuning (resonant conditions). (a) Scheme of the dimer nanostructure. A QE with dipole momentum polarized along the x -axis is placed between two gold spherical nanoparticles of radius R = 40 nm at a distance d from the surface of each of them (the separation between the center of the two nanoparticles is 2( d + R )). The dimer axis is parallel to the x -axis. The system is illuminated by a plane wave propagating along the z -axis and with the electric field polarized along the x -axis. (b) Normalized extinction cross-section spectra σext/σ(0) ext of the coupled emitter-dimer nanoantenna system. The spectra are vertically displaced by 1 . 5for clarity. Each Fano lineshape is evaluated for different values of d that range from d = 2 . 5nm to d = 250 nm (see labels in the figure). The spectra are grouped in three separate panels, each of them plotted over a different spectral range, ∆ λ . (c) Dependence with distance d of the Fano total asymmetry factor q (blue line) together with its contributions qE (orange line) and qR (green line). For each separation distance d of the calculations in (b) and (c) we have set the resonance of the QE to match the frequency of resonance of the nanoantenna. clear quantitative differences. For instance, the Purcell factor PF and the photonic Lamb shift ∆ ω experienced by the emitter, which describe the broadening and shift of the Fano feature, respectively, are much larger in the case of the dimer due to the stronger field confinement [25,146,151] ( PF≈ 1 . 4 × 10 4 and ∆ ω/γ ≈ 3 . 2 × 10 5 for the dimer and d = 2 . 5nm, to be compared with PF≈ 4 . 3 × 10 2 and ∆ ω/γ ≈ 4 . 4 × 10 3 for the single silver spherical nanoparticle system of the same radius R and distance d ). We also observe that there is a clear asymmetry for a larger range of distances in the dimer as compared to the single silver nanoantenna of the same radius (compare the three panels in Fig. 3.8b and Fig. 3.1c) To study the Fano asymmetry of the dimer system in more detail, we show in Fig. 3.8c the dependence with d of the total asymmetry factor q (blue line) and its two components qE (orange line) and qR (green line), as obtained from Eq. (3.36) (with q = qE + qR ). For separation distances d≳ 30 nm the total asymmetry factor q is mainly influenced by the qE contribution, i.e. it is mostly due to the direct excitation and emission of the QE. In a similar way as for the single spherical nanoparticle, qE is larger at intermediate distances (20 nm ≲d≲ 200 nm for the 107 Chapter 4. Unbounded strong bunching and breakdown of the RWA in the QRM seven elements to characterize the dynamics of the system in the master equation introduced in section 2.6 (Eq. (2.120) ): (i) the Hamiltonian describing the energy of the CTS, (ii) the operators describing the losses of the cavity, (iii) the decay rate of the cavity, (iv) the operators describing the losses of the TLS, (v) the decay rate of the TLS, (vi) the operators describing the incoherent excitation of the TLS, and (vii) the pumping rate of the TLS. In subsections 4.2.1 and 4.2.2 we introduce these seven elements in the complete QRM and in the approximated JCM, respectively. 4.2.1 Quantum Rabi model Interaction and dynamics in the quantum Rabi model In section 2.5, we derived the QRM Hamiltonian ˆ HC describing the energy in the CTS for any arbitrary coupling strength η = g/ω0 between the cavity and the TLS. For convenience, we repeat here the formula of ˆ HC, (Eq. (2.79)), ˆ HC=ℏω0ˆc†ˆc+ℏω0 2ˆσzcos[2η(ˆc+ ˆc†)] + ˆσysin[2η(ˆc+ ˆc†)], where, again, ˆc is the annihilation operator of the cavity mode, ω0 is the resonant frequency of the cavity and of the TLS, ˆσz = [ σ†, σ ], and ˆσy = i ( ˆσ†−ˆσ ), being ˆσ the lowering operator of the TLS (see Eq. (2.67) ). Note that the ˆ HC Quantum Rabi Hamiltonian does not conserve the excitation number, but ˆ HC conserves the parity of excitations. As mentioned above, the Hamiltonian is the first element that we need to address the dynamics of the state of our system, described by the time evolution of the density matrix of the CTS, ˆρ . Next, we characterize the dissipation of the cavity. To do so, we use the dressed operators formalism introduced in subsection 2.6.2, where the dressed operators appeared in the Lindblad terms of the master equation (Eqs. (2.117) and (2.120) ). We use this same formalism applied to the ˆ Sˆc = i ( ˆc†−ˆc ) operators describing the interaction between the cavity and the environment [85,86]. As in the derivation of the master equation in section 2.6.2, it is convenient to decompose ˆ Sˆc in the basis of the eigenstates of the QRM Hamiltonian (see also Eq. (2.106)), ˆ Sˆc= ˆx(0) ˆc+X ω [ˆxˆc(ω) + ˆx† ˆc(ω)],(4.1) where ˆx(0) ˆc=X µ|µ⟩R R⟨µ|i(ˆc†−ˆc)|µ⟩R R⟨µ|,(4.2) and ˆxˆc(ω) = |ν⟩R R⟨ν|i(ˆc†−ˆc)|µ⟩R R⟨µ|,(4.3) where ω = ωµ−ων> 0, and the kets |ν⟩R and |µ⟩R are the eigenvectors of the QRM Hamiltonian, and ℏων>ℏωµ are their respective eigenvalues. We plot ∆ E = ℏ ( ων−ωG )(being ℏωG the eigenenergy of the CTS ground state) as a 114 4.2. Cavity-two-level-system Hamiltonian models (b) JCMQRM (a) R R R R RR R Figure 4.2: Eigenvalues of the CTS obtained as function of the coupling parameter η = g/ω0 within (a) the JCM and (b) the QRM. For each |ν⟩ (for the JCM) or |ν⟩R (for the QRM) eigenstate, we show ∆ E : the difference between its eigenenergy ( ℏων ) and the ground state energy (ℏωG). function of η in Fig. 4.2a. The notation for QRM eigenstates used throughout this chapter, |ν⟩R∈ {|0⟩R,|1−⟩R,|1+⟩R,|2−⟩R,|2+⟩R, . . . } , is chosen to recall the JCM polaritons ( |n±⟩ = ( |n, g⟩±|n−1, e⟩ ) /√2 ), as the two match in the limit of vanishing coupling g . In Fig. 4.2a, we can observe how the eigenvalues of |n+⟩R ( |n−⟩R ) increase (decrease) linearly with η for η≲ 0 . 1. This same behavior is followed by the JCM polaritons ( |n±⟩ ). However, whereas the JCM polaritons keep increasing or decreasing linearly with η for η≳ 0 . 1, the eigenvalues of the QRM show a more complex dependence with η , presenting crossings and anticrossings with the eigenvalues of other eigenstates. Note that throughout this chapter we keep the labeling of the eigenstates even after the eigenvalues crossing points (see color code in the figure). Each term on the right hand side of Eq. (4.1) represents a different process in the interaction of the system with its environment; ˆxˆc describes the dissipation (the cavity gives energy to the environment), ˆx† ˆc describes the incoherent pumping (the environment gives energy to the cavity), and ˆx(0) ˆc describes the so-called dephasing (there is no direct energy transfer between the cavity and the environment) [21,72,86]. In the systems studied in this thesis, we are interested in the dissipation of the cavity, described by the ˆxˆc terms. These operators are then used to build the Dˆxˆc(ω) dissipation Lindblad superoperators according to Eq. (2.117) , where each dissipation superoperator has associated a frequency-dependent dissipation rate, γˆc in the final master equation (Eq. (2.120) ). In general, γˆc can be strongly dependent on the energy ℏω of the transitions between the eigenstates. However, in this thesis, we consider simple dissipation mechanisms, where γˆc becomes constant, i.e., γˆc ( ω ) →κ , with κ being the classical plasmonic losses introduced in section 1.2, when the cavity is a plasmonic resonator. In the plasmonic CTS considered in this chapter, the cavity is characterized by resonant frequency ω0(set to ℏω0= 1 eV), and a low quality factor Q = ω0/κ , defined by the dissipation rate κ , and set to Q = 20 (thus κ = 50 meV). Thus, we only need a single Lindblad term to describe the dissipation of the cavity, κDˆxˆc[ˆρ(t)] = κ 22ˆxˆcˆρ(t)ˆx† ˆc−{ˆx† ˆcˆxˆc,ˆρ(t)}(4.4) 115 Chapter 4. Unbounded strong bunching and breakdown of the RWA in the QRM with ˆxˆc=X ωµ>ων|ν⟩R R⟨ν|i(ˆc†−ˆc)|µ⟩R R⟨µ|.(4.5) Next, we use this same approach to address the dissipation and incoherent pumping of the TLS. We first consider the operator ˆ Sˆσ = ( ˆσ† + ˆσ )describing the interaction between the TLS and the environment [85,86]. Then we decompose ˆ Sˆσ in the basis of the QRM eigenvalues: ˆ Sˆσ= ˆx(0) ˆσ+X ω [ˆxˆσ(ω) + ˆx† ˆσ(ω)],(4.6) where, ˆx(0) ˆσ=X µ|µ⟩R R⟨µ|(ˆσ†+ ˆσ)|µ⟩R R⟨µ|,(4.7) and ˆxˆσ(ω) = |ν⟩R R⟨ν|(ˆσ†+ ˆσ)|µ⟩R R⟨µ|,(4.8) with ω = ωµ−ων> 0(as in Eq. (4.3) ). ˆxˆσ and ˆx† ˆσ in Eq. (4.6) are used in Lindblad superoperators to address the dissipation, and the incoherent pumping, respectively [85,86]. For the CTS consider in this chapter, we consider frequencyindependent dissipation and pumping rates of the TLS, so we use a single Lindblad term to describe the losses of the TLS, γDˆxˆσ[ˆρ(t)] = γ 22ˆxˆσˆρ(t)ˆx† ˆσ−{ˆx† ˆσˆxˆσ,ˆρ(t)},(4.9) and a single Lindblad term to describe the incoherent pumping of the TLS, ΓDˆx† ˆσ [ˆρ(t)] = Γ 22ˆx† ˆσˆρ(t)ˆxˆσ−{ˆxˆσˆx† ˆσ,ˆρ(t)}.(4.10) Here γ and Γare the decay and incoherent pumping rates of the TLS, respectively, and ˆxˆσ=X ωµ>ων|ν⟩R R⟨ν|(ˆσ†+ ˆσ)|µ⟩R R⟨µ|.(4.11) In particular, we chose a decay rate of the TLS γ/ω0 = 10 −3 , negligible compared to the decay rate of the cavity κ . Thus, we can establish the upper limit for the WC regime as η=g/ω0≲κ/(2ω0)=0.025. We now have all the elements to describe the dynamical evolution of ˆρ as follows from the master equation in Eq. (2.120), d dt ˆρ(t) = −i ℏ[ˆ HC,ˆρ(t)] + κDˆxˆc[ˆρ(t)] + γDˆxˆσ[ˆρ(t)] + ΓDˆx† ˆσ [ˆρ(t)].(4.12) 116 4.2. Cavity-two-level-system Hamiltonian models Correlations in the quantum Rabi model To characterize the statistical properties of the emission of the CTS, we study the intensity correlations of the emitted photons, g(2) ( τ = 0), as measured by an HBT interferometer, introduced in section 2.4. The measurement of a HBT interferometer corresponds to, g(2)(τ) = ⟨I1(t+τ)I2(t)⟩ ⟨I1(t+τ)⟩⟨I2(t)⟩,(4.13) where I1 ( t + τ )and I2 ( t )are the photocurrents registered by the two detectors of the HBT interferometer, and τ is the time delay between the detection events (see Fig. 4.1a). For zero time delay ( τ = 0) and sufficiently large t (so the system reaches the steady state), this quantity is related to the statistics of photons inside the cavity as [21,85]: g(2)(0) = ⟨ˆx† ˆcˆx† ˆcˆxˆcˆxˆc⟩ss ⟨ˆx† ˆcˆxˆc⟩2 ss .(4.14) Crucially, within the QRM, g(2) (0) depends on the ˆxˆc dressed operators introduced in Eq. (4.5) , ensuring Gauge-invariance. On the other hand, ⟨ˆ O⟩ss in the expression of g(2) (0) denotes the expectation value of operator ˆ O in the steady state ( ss ). Then, by using Eq. (4.12) we can obtain the steady state of the system via the density operator ˆρss such that ∂tˆρss = 0. This framework is used to calculate the dependence of g(2) (0) (Eq. (4.14) ) on the coupling parameter η = g/ω0 , plotted in Fig. 4.1b as solid orange and blue lines, for the case of strong (Γ /γ = 10), and weak (Γ /γ = 10 −6 ) pumping, respectively. All calculations of the evolution of the steady state and expectation values in the system that we obtain in this chapter have been carried out using the Python package QuTiP [177,178]. We have considered in all of our QuTiP calculations an expansion of the Fock states of the cavity up to Nc = 10, which we verified ensures convergence. 4.2.2 Jaynes-Cummings model In section 2.5.4 we derived the JCM Hamiltonian by taking two approximations. First, we expand the interaction term in the QRM Hamiltonian as ˆσzcos 2η(ˆc+ ˆc†)+ ˆσysin 2η(ˆc+ ˆc†)= ˆσz+ 2ηˆσy(ˆc+ ˆc†) + O(η2),(4.15) and drop the nonlinear terms in η . Next, we introduce the rotating wave approximation (RWA) by removing the so-called non-number-conserving terms ˆσˆc + ˆσ†ˆc† , to obtain the JCM Hamiltonian in Eq. (2.81) that we reproduce here for convenience, ˆ HJC =ℏω0ˆc†ˆc+ℏω0 2ˆσz+iℏg(ˆσ†ˆc−ˆc†ˆσ). 117 Chapter 4. Unbounded strong bunching and breakdown of the RWA in the QRM Further, the JCM introduces additional approximations regarding the emission, dissipation, and absorption of the system. Dynamics in the Jaynes Cummings model In the JCM, the interaction between the system and the environment is simplified by treating the dissipation of the cavity and of the TLS as separate elements. Let us first explore the dissipation of the cavity in the absence of the TLS. The Hamiltonian of the single cavity mode is (see Eq. (2.53)), ˆ HCavity =ℏω0ˆc†ˆc. (4.16) The eigenstates of these Hamiltonians are the Fock number states |n⟩ (see section 2.2), with eigenvalues ℏnω0 , ( n≥ 0any integer number). The dissipation of the cavity is described using the same dressed operators formalism introduced in the section above (Eqs. (4.1) - (4.5) ). However, the dressed operators are built using the |n⟩ eigenstates of the Hamiltonian of the bare cavity. Thus, the ˆxˆc dressed operators in Eq. (4.5) now result in ˆxˆc→X ωm>ωn|n⟩⟨n|i(ˆc†−ˆc)|m⟩⟨m|=−iX n √n|n−1⟩⟨n|=−iˆc, (4.17) where, in the last identity, we have simply used the definition of the ˆc annihilation operators in Eq. (2.18) . Hence, the Lindblad dissipation term of the cavity becomes, κDˆxˆc[ˆρ(JC)(t)] →κD(−iˆc)[ˆρ(JC)(t)] = κDˆc[ˆρ(JC)(t)] = =κ 22ˆcˆρ(JC)(t)ˆc†−{ˆc†ˆc, ˆρ(JC)(t)}.(4.18) Note that here we use the label “( JC )” to differentiate the density matrix describing the state in the QRM and that in the JCM. Similarly, the dissipation of the TLS is described in the absence of the cavity. The Hamiltonian of the bare TLS is (see Eq. (2.69)), ˆ HTLS =ℏω0 2ˆσz,(4.19) with eigenstates |g⟩ and |e⟩ , and eigenvalues −ω0/ 2for |g⟩ and ω0/ 2for |e⟩ . As for the case of the cavity, the dissipation and incoherent excitation of the TLS is described using the same dressed operator formalism introduced for the QRM (Eqs. (4.6) - (4.11) ). However, the dressed operators are built using only the decomposition on the |g⟩ and |e⟩ eigenstates. Thus, the ˆxˆσ dressed operators in Eq. (4.11) become, ˆxˆσ→X ωe>ωg|g⟩⟨g|(ˆσ†+ ˆσ)|e⟩⟨e|= ˆσ. (4.20) Substituting this expression of ˆxˆσ into the dissipation and incoherent pumping 118 4.2. Cavity-two-level-system Hamiltonian models terms in Eqs. (4.9) and (4.10), we obtain γDˆxˆσ[ˆρ(JC)(t)] →γDˆσ[ˆρ(JC)(t)] = γ 22ˆσˆρ(JC)(t)ˆσ†−{ˆσ†ˆσ, ˆρ(JC)(t)},(4.21) for the TLS dissipation term, and ΓDˆx† ˆσ [ˆρ(JC)(t)] →ΓDˆσ†[ˆρ(JC)(t)] = Γ 22ˆσ†ˆρ(JC)(t)ˆσ−{ˆσˆσ†,ˆρ(JC)(t)},(4.22) for the TLS incoherent pumping term. In summary, we can write the master equation with all the contributions to the system dynamics as introduced here, as: d dt ˆρJC(t) = −i ℏ[ˆ HJC,ˆρ(JC)(t)] + κDˆc[ˆρ(JC)(t)] + γDˆσ[ˆρ(JC)(t)] + ΓDˆσ†[ˆρ(JC)(t)]. (4.23) Intensity correlations in the Jaynes Cummings model As for the QRM (in Eq. (4.14) ), the correlations arising in the JCM are also related to the statistics of photons inside of the cavity and, can be expressed in terms of the dressed cavity operators [21]. However, the operator that describes the photons of the cavity corresponding to the dressed operators in Eq. (4.17) reduce to ˆxˆc→ −iˆc, and thus, in the JCM the intensity correlations become g(2) JC (0) = ⟨ˆc†ˆc†ˆcˆc⟩ss ⟨ˆc†ˆc⟩2 ss .(4.24) The expected values in Eq. (4.24) are obtained with the steady-state ˆρ(JC) ss , which we calculate from the solution of ∂tˆρ(JC) ss = 0 in the standard master equation introduced in Eq. (4.23). Figure 4.1b (dashed lines) shows the intensity correlations g(2) JC (0) obtained applying Eq. (4.24) in the JCM for strong (dashed orange line) and weak (dashed blue line) pumping. In the former case, the JCM correctly reproduces the results of the exact QRM below the USC threshold η≲ 0 . 1, but completely fails for larger η , where the g(2) JC (0) obtained with the JCM saturates, g(2) JC (0) ≲ 2 / 3, while, in the exact QRM, g(2) (0) keeps increasing strongly with η . Furthermore, for the weak incoherent pumping (blue lines), we find significant differences between the JCM and QRM even in the WC regime (the JCM and QRM differ for η≳ 5 × 10 −3 ). This unexpected breakdown of the JCM for small η is discussed in detail in section 4.4. 119 Chapter 4. Unbounded strong bunching and breakdown of the RWA in the QRM Analytical limit of the correlations in the Jaynes-Cummings Hamiltonian The JCM has been extensively used to describe the properties of weakly-coupled CTSs [21]. In this chapter, it establishes a point of comparison to identify new features that can arise in the QRM. Further, we show next that the simplifications introduced by the JCM allow us to obtain an approximate analytical expression of g(2) (0) in the weak-illumination case, where the incoherent pumping of the TLS has a rate much lower than the TLS losses, Γ≪γ. As a first step, we identify the minimum set of operators for which the master equations form an almost closed system: v= ˆc†ˆc, ˆσ†ˆσ, ˆc†ˆσ, ˆcˆσ†,ˆc†ˆcˆσ†ˆσ, ˆc†ˆcˆcˆσ†,ˆc†ˆc†ˆcˆσ, ˆc†ˆc†ˆcˆcT . The equations of motion for the expectation values of these operators can be approximately expressed as d dt ⟨v⟩=M⟨v⟩+b,(4.25) with b= (0,Γ,0,0,0,0,0,0)T, and M=             −κ0−g−g0 0 0 0 0−γ g g 0 0 0 0 g−g−M10−g0 0 0 g−g0−M1−g0 0 0 Γ 0 0 0 −(γ+κ)g g 0 0 0 0 0 −2g−M20g 0 0 0 0 −2g0−M2g 0 0 0 0 0 −2g−2g−2κ             ,(4.26) where M1 = (Γ + κ + γ ) / 2and M2 = (Γ + γ + 3 κ ) / 2. To derive Eqs. (4.25) - (4.26) we truncated the set of equations by considering that the pumping rate of the system, Γ, is very small. Thus, in the steady-state, the TLS is mostly in the ground state, and we approximate [179] (i) ⟨ˆσˆσ†⟩ ≈ 1, (ii) ⟨ˆc†ˆc†ˆcˆcˆσˆσ†⟩≈⟨ˆc†ˆc†ˆcˆc⟩ , and (iii) ⟨ˆc†ˆc†ˆcˆcˆσ†ˆσ⟩ ≈ 0. In the steady state ∂t⟨v⟩ss = 0, and we can derive closed expressions for ⟨ˆc†ˆc⟩ss and ⟨ˆc†ˆc†ˆcˆc⟩ss. ⟨ˆc†ˆc⟩ss ≈ −4g2Γ(4g2γ+ 12g2κ+ Γγκ +γ2κ+ Γκ2+ 4γκ2+ 3κ3)× ×16g4Γγ−16g4γ2−64g4γκ + 4g2Γ2γκ −4g2Γγ2κ−8g2γ3κ−48g4κ2− −8g2Γγκ2−36g2γ2κ2−Γ2γ2κ2−2Γγ3κ2−γ4κ2−4g2Γκ3−40g2γκ3− −Γ2γκ3−6Γγ2κ3−5γ3κ3−12g2κ4−4Γγκ4−7γ2κ4−3γκ5−1,(4.27) 120 4.3. Origin of the bunching in ultrastrongly coupled systems ⟨ˆc†ˆc†ˆcˆc⟩ss ≈32g4Γ2−16g4Γγ+ 16g4γ2+ 64g4γκ −4g2Γ2γκ + 4g2Γγ2κ+ + 8g2γ3κ+ 48g4κ2+ 8g2Γγκ2+ 36g2γ2κ2+ Γ2γ2κ2+ 2Γγ3κ2+γ4κ2+ + 4g2Γκ3+ 40g2γκ3+ Γ2γκ3+ 6Γγ2κ3+ 5γ3κ3+ +12g2κ4+ 4Γγκ4+ 7γ2κ4+ 3γκ5−1.(4.28) In particular, we are interested only in the values of ⟨ˆc†ˆc⟩ss and ⟨ˆc†ˆc†ˆcˆc⟩ss in the two opposite limits of g≪ Γ(vanishing coupling), and g≫κ (beyond the SC regime). We obtain an expression that is valid in both limits by retaining the terms dependent on the leading powers of the two free parameters, Γand g , in Eqs. (4.27) and (4.28): ⟨ˆc†ˆc⟩ss ≈4g2(4g2+κ2)Γ κ3(4g2+γκ)= Γ C C+ 1 4g2+κ2 κ3,(4.29) where C= 4g2/(κγ)is the cooperativity, and ⟨ˆc†ˆc†ˆcˆc⟩ss ≈32g4Γ2 3κ2(16g4+γκ3).(4.30) The intensity correlations are found, in the two limits of interest, as g(2) JC (0) κ≫γ≫Γ≫g −−−−−−−→ 2 3 γ κ,(4.31) g(2) JC (0) g≫κ≫γ≫Γ −−−−−−−→ 2 3.(4.32) Note that g(2) JC (0) is derived here by neglecting the direct emission from the TLS, which is an invalid approximation for very small g in equation (4.31) (unless the emission of the TLS is filtered-out). If direct emission of the TLS is included (4.31) should be modified, as it is clearest in the limit of g = 0. In this case, the emission of the system is only given by the TLS, which emits one photon at a time resulting in g(2) JC = 0. Importantly, both limits in Eqs. (4.31) and (4.32) are antibunched, i.e., for all coupling strengths, we expect the JCM to result in an antibunching signal. This is confirmed by the results in Fig. 4.1b (dashed blue line), which shows the intensity correlations g(2) JC (0) obtained applying Eq. (4.24) for the JCM for weak incoherent pumping. The results of this figure also validates the limits in Eqs. (4.31) and (4.32) , with g(2) (0) = 1 . 33 × 10 −2 = (2 / 3)( γ/κ )for η = 10 −3 and g(2) (0) = 2 / 3for η= 1. 121 Chapter 4. Unbounded strong bunching and breakdown of the RWA in the QRM exact calculation diagonal approx. R R Figure 4.3: Intensity correlations as a function of the coupling parameter η = g/ω0 obtained within several approximations: exact values of g(2) (0) (solid blue line); the diagonal approximation truncated to the states {|0⟩R,|1−⟩R,|1+⟩R,|2−⟩R,|2+⟩R,|3−⟩R} in Eq. (4.33) (solid orange line); the diagonal approximation considering only the R⟨3−| ˆxˆcˆxˆc|1−⟩Rterm in the numerator of g(2) (0) in Eq. (4.37) ( |3−⟩R→ |1−⟩R , solid green line). The calculations shown in this figure are obtained within the QRM for Γ/γ = 10−3. 4.3 Origin of the bunching in ultrastrongly coupled systems In this section, we demonstrate that the strong bunching identified in the USC regime in Fig. 4.1b can be related to the characteristics (decay pathways and population) of the single |3−⟩R eigenstate. To justify the focus on that particular eigenstate of the QRM Hamiltonian, we plot in Fig. 4.3 the exact values of g(2) (0) (solid blue line) obtained for an intermediate pumping Γ /γ = 10 −3 together with approximated results. We start by approximating the steady-state density matrix as being diagonal in the basis of the |ν⟩Reigenstates of the QRM Hamiltonian, ˆρss ≈X ν Rν|ν⟩R R⟨ν|,(4.33) where Rν is the population of the |ν⟩R eigenstate. Thus, we can write the expected value of any operator ˆ Oin the steady state as: ⟨ˆ O⟩ss =Tr{ˆ Oˆρss} ≈ Tr (ˆ O X ν Rν|ν⟩R R⟨ν|!)=X ν RνR⟨ν|ˆ O|ν⟩R.(4.34) We then apply this formula to the expected value of ⟨(ˆx† ˆc)n(ˆxˆc)n⟩ss ( n = 1 and n= 2 for the numerator and denominator of g(2)(0), respectively), resulting in: ⟨(ˆx† ˆc)n(ˆxˆc)n⟩ss ≈X ν RνR⟨ν|(ˆx† ˆc)n(ˆxˆc)n|ν⟩R=X ν RνR⟨ν|(ˆx† ˆc)nˆ I(ˆxˆc)n|ν⟩R, (4.35) 122 4.3. Origin of the bunching in ultrastrongly coupled systems R0 R1R1 + R2R2 + R3 Figure 4.4: Populations of the polaritonic states Rν (see labels) of a CTS calculated with the QRM (solid lines) and JCM (dashed lines) as a function of the coupling parameter η = g/ω0 , for the intermediate pumping Γ/γ = 10−3. where we have included in the last step the identity matrix ˆ I . Because the eigenstates of the QRM Hamiltonian are orthonormal we can write the identity matrix as ˆ I=Pµ|µ⟩R R⟨µ|, and thus: X ν RνR⟨ν|(ˆx† ˆc)nˆ I(ˆxˆc)n|ν⟩R= =X µ,ν RνR⟨ν|(ˆx† ˆc)n|µ⟩R R⟨µ|(ˆxˆc)n|ν⟩R=X µ,ν Rν|R⟨µ|(ˆxˆc)n|ν⟩R|2,(4.36) where we have used the property ⟨b|ˆ O†|a⟩ = ( ⟨a|ˆ O|b⟩ ) ∗ . Applying Eq. (4.36) to n = 1 (denominator of g(2) (0) in Eq. (4.14) ) and n = 2 (numerator of g(2) (0)), results in g(2)(0) ≈Pν,µ Rν|R⟨µ|ˆxˆcˆxˆc|ν⟩R|2 Pν,µ Rν|R⟨µ|ˆxˆc|ν⟩R|22,(4.37) The intensity correlations calculated by truncating the double sum in the numerator up to |3−⟩R are shown with the solid orange line in Fig. 4.3 — this approximation gives a very good agreement with the exact calculation for η≳ 2 . 5 × 10 −2 ; as we have numerically verified, the deviation observed in the WC regime η≲ 2 . 5 × 10 −2 is not due to the truncation of the basis, but rather due to the effect of the off-diagonal terms of ˆρss. We can further approximate g(2) (0) by limiting the double sum in Eq. (4.37) over νand µto the |ν⟩=|3−⟩Rand |µ⟩=|1−⟩Rterm: g(2)(0) ≈R3−|R⟨1−|ˆxˆcˆxˆc|3−⟩R|2 Pν,µ Rν|R⟨µ|ˆxˆc|ν⟩R|22.(4.38) This approximation explores the role of the correlated two-photon emission from |3−⟩R to the |1−⟩R state. We show in Fig. 4.3 (green line) the resulting g(2) (0) 123 Chapter 4. Unbounded strong bunching and breakdown of the RWA in the QRM in both considered pumping rates considered, but these differences would be likely difficult to identify in experimental settings. On the other hand, as shown in Figs. 4.7 and 4.8b, the g(2) (0) obtained for Γ /γ = 10 −3 and Γ /γ = 10 −6 show strong qualitative differences for η≳ 2 . 5 × 10 −2 due to the direct excitation pathway of the state |3−⟩R introduced in section 4.3. The lack of sensitivity of S(1) ( ω )to this direct excitation pathway is due to the fact that the emission from the lower |1−⟩R and |1+⟩R states predominantly govern the emission spectra for η≲ 0 . 3. The excitation and emission from these eigenstates are not impacted by the direct excitation mechanism discussed above, and do not lead to the breakdown of the RWA. We thus conclude that the characterization of the correlations is a more powerful tool than measuring the one-photon emission spectra for the identification of phenomena caused by the non-number-conserving terms of the QRM Hamiltonian for coupling below the traditional USC threshold η≈0.1. 4.5 Conclusions In this chapter, we analyze the statistics of the emission from a generic quantum system comprising an incoherently driven two-level emitter interacting with a cavity. We identify the emergence of unbounded bunching as the system approaches the USC regime. By expressing the dynamics of the system in the basis of the polaritonic eigenstates of the quantum Rabi Hamiltonian, we can attribute the bunching to the singular behavior of the individual eigenstate |3−⟩R , which (i) decays through a correlated two-photon emission, and (ii) is very strongly populated by a new, direct excitation mechanism from the ground state. We show that intensity correlations g(2) (0) are a much more sensitive tool for observing the phenomena induced by the non-number-conserving terms in the QRM, than the one-photon emission spectra. Indeed, we find that the intensity correlations can identify a breakdown of the rotating wave approximation far below the conventional limit of the USC, with the exact limit determined by the rate of incoherent pumping. These findings call for experimental verification, and further theoretical studies, to verify the robustness of the identified excitation and emission mechanisms. Our model can be extended to account for more complex decay dynamics and energy structure of the quantum emitter, involving dark excitonic states, or pure dephasing, as well as the interaction with a structured reservoir. 130 Chapter 5 PRESERVATION AND DESTRUCTION OF THE PURITY OF TWO-PHOTON STATES IN THE INTERACTION WITH A NANOSCATTERER 5.1 Introduction Classical light beams can carry well-defined angular momentum, as described in chapter 1. Similarly, quantum states of light (introduced in chapter 2) can also carry well-defined angular momentum. This aspect opens up a variety of potential benefits for quantum technologies. For instance, the angular momentum of quantum states of light can be used to encode information [15 – 18]. This is particularly useful for quantum information applications as quantum states of light are very resilient to lose their entanglement and purity during propagation [180 – 182]. Thus, the states of light with well-defined angular momentum are ideal candidates as quantum information carriers. On the other hand, photons do not interact strongly with material particles and structures, limiting the possibilities of processing photonic quantum information [183,184]. Several techniques are being developed in order to enhance photon interactions, such as the use of quantum optomechanical interaction, optical metamaterials, high-density gases, slow-light materials and several others [185 – 189]. Engineering nanophotonic nanostructures for quantum information processing offers the possibility to, not only enhancing light-matter interactions, but also manipulating light in devices with a footprint of the order of the wavelength. While the use of nanostructures and the study of their optical resonances to enhance 131 Chapter 5. Loss of purity in the scattering of two-photon entangled states the classical interaction of light and matter has a long tradition [49,111,190], a formal study of the effect of the interaction of quantum states of light with such nanostructures is still needed. In this chapter, we provide a framework to study the interaction between quantum states of light with well defined angular momentum and a nanostructure. Our approach is general, but we focus on an experimentally relevant situation: the scattering of two-photon states of light by a rotationally symmetric nanostructure. Crucially, the rotational symmetry imposes the conservation of the total angular momentum m = l + s , where l and s represent the orbital and spin angular momentum, respectively. Thus, these nanostructures allow for the manipulation of states of light with a fixed and well-defined total angular momentum in a controlled manner [20,191]. 5.2 Input and output states The theoretical framework used to describe the quantum scattering process is based on an input/output general formalism [78,79,191,192]. We consider that the input and output states of the system are quantum states of light composed by two entangled photons, where the two photons have total angular momentum m = 0, and the information is encoded in their helicity Λ(defined as the projection of the spin operator ←→ S on the direction of propagation, see Eq. (1.84) ), which takes Λ = +1 or Λ = − 1values (see section 1.4) [191]. We consider a basis of four input two-photon modes that completely describes any input monochromatic two-photon states: |ψi ±(ω1, ω2)⟩=1 2nˆa† i(ω1)ˆa† i(ω2)±ˆ b† i(ω1)ˆ b† i(ω2)o|0⟩,(5.1a) |χi ±(ω1, ω2)⟩=1 2nˆa† i(ω1)ˆ b† i(ω2)±ˆ b† i(ω1)ˆa† i(ω2)o|0⟩,(5.1b) where |0⟩ is the vacuum state, and ω1 and ω2 are the frequencies of the two-photons. The basis of the two-photon output monochromatic modes also has four elements, |ψo ±(ω1, ω2)⟩ and |χo ±(ω1, ω2)⟩ , whcih follow Eqs. (5.1a) and (5.1b) , respectively, but the input “i” labels are substituted by the output “o” labels, |ψo ±(ω1, ω2)⟩=1 2nˆa† o(ω1)ˆa† o(ω2)±ˆ b† o(ω1)ˆ b† o(ω2)o|0⟩,(5.2a) |χo ±(ω1, ω2)⟩=1 2nˆa† o(ω1)ˆ b† o(ω2)±ˆ b† o(ω1)ˆa† o(ω2)o|0⟩.(5.2b) The modes of light are described by the input(output) ˆa† i(o) ( ω )and ˆ b† i(o) ( ω )operators that indicate the creation of an input(output) photon with helicity Λ = +1 or Λ = − 1, respectively. The properties of the creation and annihilation operators describing the quantization of light were discussed in section 2.2. Notably, the ˆa† i(o) ( ω )and ˆ b† i(o) ( ω )bosonic operators satisfy the canonical commutation relations (Eq. (2.21)) and operate on a single frequency ω. 132 5.2. Input and output states Recent experiments have measured a degradation of input quantum states (quantified below by means of the loss of purity) after scattering off a nanostructure [191]. In particular, in these (and other similar) experiments, the incident photon pairs are generated in a superposition of states of different frequencies, typically by a standard spontaneous parametric down-conversion (SPDC) process. In this chapter, we analyze the mechanism by which the scattering of these non-monochromatic states can result in a loss of purity. With this purpose, we first focus on the scattering of states that are frequency superpositions of the monochromatic twophoton state |ψi +(ω1, ω2)⟩ (in section 5.3.3 we study the scattering of the rest of the elements in the basis), |Ψi +⟩=¨dω1dω2ϕ(ω1, ω2)|ψi +(ω1, ω2)⟩.(5.3) where ϕ ( ω1, ω2 )is the two-photon spectral function that we approximate as the product of two Gaussian functions both centered at the central frequency ωin (or central wavelength λin = 2 πc/ωin ) and variance σ = 3 THz (this value is chosen to be similar to the one used in recent experiments [193,194]), ϕ(ω1, ω2) = 1 σ√πexp−(ω1−ωin)2 2σ2exp−(ω2−ωin)2 2σ2.(5.4) Note that this expression of ϕ ( ω1, ω2 )implies that the two photons are indistinguishable, since ϕ(ω1, ω2) = ϕ(ω2, ω1). Experimentally-accesible density matrix In order to approach a realistic experimental characterization of quantum states, we consider that the scattered states are measured through a post-selection of twophoton states, where the scattered states with less than two photons are ignored. We also consider in the following that the detectors are “blind” to the frequency degree of freedom. Then, the input and output quantum states are best described with the experimentally-accessible post-selected density matrix, ˆϱ , resulting from tracing out the frequency degree of freedom. Thus, the elements ⟨ξ|ˆϱ|ξ′⟩ of the density matrix corresponds to the results of standard quantum state tomography measurements [74,195], ⟨ξ|ˆϱi(o)|ξ′⟩=K¨dω1dω2⟨ξ(ω1, ω2)|Ψi(o) +⟩⟨Ψi(o) +|ξ′(ω1, ω2)⟩,(5.5) where K is a normalization constant that ensures Tr{ˆϱi(o)} = 1. |ξ(ω1, ω2)⟩ and |ξ′(ω1, ω2)⟩ can be any of the |ψi(o) ±(ω1, ω2)⟩ and |χi(o) ±(ω1, ω2)⟩ states given in Eqs. (5.1a) and (5.1b) , respectively. For example, Fig. 5.1 shows the density matrix ˆϱi of the incident state |Ψi +⟩ (Eq. (5.3) ) calculated using Eq. (5.5) . ˆϱi is characterized by a single non-zero element corresponding to ⟨ψ+|ˆϱi|ψ+⟩ , and there is no contribution from the other elements of the basis (Eqs. (5.1a) and 133 Chapter 5. Loss of purity in the scattering of two-photon entangled states ψ+χ+ψ−χ− ψ+ χ+ ψ− χ− 0.00 0.25 0.50 0.75 1.00 Re{ i} ψ+χ+ψ−χ− ψ+ χ+ ψ− χ− 0.00 0.25 0.50 0.75 Im{ i} Input state Figure 5.1: Real (left) and imaginary (right) components of ˆϱi , the density matrix associated to the input state |Ψi +⟩ . The two-photon spectral function is centered at ωin = 17 . 5 × 10 14 rad/s and has a variance of σ= 3 THz. Li= 0 indicates that the input state is pure. (5.1b) ). The loss of purity of such a quantum state can then be quantified using Li(o) = 1 −Tr{ ( ˆϱi(o) ) 2} (introduced in Eq. (2.3) ), where Li(o) = 0 indicates a pure state and Li(o)> 0a mixed state. The input density matrix ˆϱi in Fig. 5.1 satisfies Li= 1 −Tr{(ˆϱi)2}= 0, which confirms that ˆϱiis a pure state [74]. 5.3 Quantum transformation We next discuss how to analyze the loss of purity due to the scattering of m = 0 photons by a rotationally symmetric nanostructure. As discussed in section 1.4.3, rotationally symmetric structures conserve the total angular momentum of the incident light. However, the conservation of the total angular momentum does not imply the conservation of the vectorial degree of freedom of light, which is determined by the helicity Λ. Since the states of light studied in this chapter are determined by m and Λ(see section 1.4.2), the input electromagnetic modes with m = 0 and Λ = +1 (or Λ = − 1) can only be scattered into two different output electromagnetic modes with m = 0 (due to m conservation) and Λ = +1 or Λ = − 1. Further, we consider that photons can be lost or dissipated in the scattering process. This situation where two input electromagnetic modes are either lost or transformed into two other output electromagnetic modes is analogous to the situation produced in a lossy beam splitter [78,79,192]. Thus, we can directly adapt the transformation of lossy beam splitters that was introduced in section 2.3 (Eq. (2.29) ) to our system, resulting in the following equations connecting the output and input annihilation operators: ˆao(ω) = α+1(ω)ˆai(ω) + β+1(ω)ˆ bi(ω) + ˆ L+1(ω), ˆ bo(ω) = α−1(ω)ˆ bi(ω) + β−1(ω)ˆai(ω) + ˆ L−1(ω),(5.6) where ˆ L+1 and ˆ L−1 are the Langevin operators accounting for the losses in the scattering process (see section 2.3) [78,80,81], and α+1 , α−1 , β+1 , and β−1 are the helicity-splitting coefficients that are fully described in the next subsection. 134 5.3. Quantum transformation 5.3.1 Helicity-splitting coefficients Equation (5.6) describes the scattering of quantum states of light, but the α+1 , α−1 , β+1 , and β−1 coefficients can be calculated from the classical response of the system as obtained from Maxwell’s equations because Maxwell’s equations determine how the electromagnetic modes get transformed, both in the classical and quantum regimes [64,196]. Thus, to obtain the helicity-splitting coefficients, we consider a classical scattering problem where incident light beam with m = 0 is scattered by the nanostructure. The scattered light is separated by its helicity contributions, and each helicity contribution is detected separately. For example, in this chapter, we consider that the detection is done by coupling each helicity contribution of the scattered field to a single-mode fiber connected to a detector. In particular, we consider two classical input beams with a helicity of either Λ = +1 or Λ = − 1and total angular momentum m = 0. The electric fields for these input beams are represented by Ei +1 and Ei −1 , respectively. We then calculate the scattered fields, Esca +1 ( Esca −1 ), when the nanostructure is illuminated by the input beam Ei +1 ( Ei −1 ). The helicity-splitting coefficients are determined by projecting the scattered field into two classical output beams, Eo +1 and Eo −1 , which also have an angular momentum of m = 0 and a helicity of Λ = +1 and Λ = − 1, respectively: α+1(ω) = ¨A dA[Eo +1(r, ω)]∗·Esca +1 (r, ω), α−1(ω) = ¨A dA[Eo +1(r, ω)]∗·Esca −1(r, ω), β+1(ω) = ¨A dA[Eo −1(r, ω)]∗·Esca +1 (r, ω), β−1(ω) = ¨A dA[Eo −1(r, ω)]∗·Esca −1(r, ω). (5.7) This operation corresponds to calculating the coupling between the scattered fields and a single-mode fiber connected to the detector, and Ais the area of the fiber. In this chapter, we consider a simple detection scheme (discussed in section 5.4) where α+1 ( ω ) = α−1 ( ω ) = α ( ω )and β+1 ( ω ) = β−1 ( ω ) = β ( ω ). In this case, Eq. (5.6) simplifies to ˆao(ω) = α(ω)ˆai(ω) + β(ω)ˆ bi(ω) + ˆ L+1(ω), ˆ bo(ω) = α(ω)ˆ bi(ω) + β(ω)ˆai(ω) + ˆ L−1(ω).(5.8) 5.3.2 Output |Ψo +⟩state By considering the simplified transformation in Eq. (5.8) we can obtain a the output state |Ψo +⟩ from the projection of the input state, |Ψi +⟩ , on all the two-photon 135 Chapter 5. Loss of purity in the scattering of two-photon entangled states nanostructure Rotationally symmetric state Input state Output Figure 5.2: Scheme of the scattering process. The |Ψi +⟩ input state is scattered as a superposition of |ψo +⟩and |χo +⟩with amplitudes given by Cψϕand Cχϕ, respectively states of the output basis: |Ψo +⟩=h|ψo +(ω3, ω4)⟩⟨ψo +(ω3, ω4)|+ +|ψo −(ω3, ω4)⟩⟨ψo −(ω3, ω4)|+|χo +(ω3, ω4)⟩⟨χo +(ω3, ω4)|+ +|χo −(ω3, ω4)⟩⟨χo −(ω3, ω4)|i·¨dω1dω2ϕ(ω1, ω2)|ψi +(ω1, ω2)⟩.(5.9) To evaluate Eq. (5.9) , we first substitute the expressions of the ˆao ( ω )and ˆ bo ( ω )operators in Eq. (5.8) into the expressions of ⟨ψo +(ω1, ω2)| , ⟨ψo −(ω1, ω2)| , ⟨χo +(ω1, ω2)| , and ⟨χo −(ω1, ω2)| in Eqs. (5.2a) and (5.2b) ,i.e., we express the output basis bra states in terms of the input operators. Then we perform the projection of each output bra state onto the |ψi +(ω1, ω2)⟩ state. After some algebraic manipulation, Eq. (5.9) becomes: |Ψo +⟩=¨dω1dω2ϕ(ω1, ω2)Cψ(ω1, ω2)|ψo +(ω1, ω2)⟩+Cχ(ω1, ω2)|χo +(ω1, ω2)⟩, (5.10) with Cψ(ω1, ω2)and Cχ(ω1, ω2)defined as: Cψ(ω1, ω2) = α(ω1)α(ω2) + β(ω1)β(ω2),(5.11a) Cχ(ω1, ω2) = α(ω1)β(ω2) + β(ω1)α(ω2).(5.11b) Equation (5.10) shows that, in general, the output state is a superposition of two different entangled photon modes, |ψo +(ω1, ω2)⟩ and |χo +(ω1, ω2)⟩ . Each output mode has a different amplitude, Cψ ( ω1, ω2 ) ϕ ( ω1, ω2 )for |ψo +(ω1, ω2)⟩ and Cχ ( ω1, ω2 ) ϕ ( ω1, ω2 )for |χo +(ω1, ω2)⟩ , as we show in the scheme of figure 5.2. The Cψ ( ω1, ω2 )and Cχ ( ω1, ω2 )coefficients in Eqs. (5.11a) and (5.11b) may have a strong frequency dependence due to rapid spectral changes of the α and β coefficients, greatly affecting the purity of the output state. To illustrate the formalism, we artificially set β ( ω )=0 . 2and α ( ω ) = ωLγ/ [2( ω2 L−ω2 + iγω )] a Lorentzian function that mimics the resonant behavior of a nanostructure. Figure 136 5.3. Quantum transformation ψ+χ+ψ−χ− ψ+ χ+ ψ− χ− 0.2 0.0 0.2 0.4 0.6 Re{ o} ψ+χ+ψ−χ− ψ+ χ+ ψ− χ− 0.2 0.0 0.2 0.4 0.6 Im{ o} Output state Figure 5.3: Real (left) and imaginary (right) components of ˆϱo , the post-selected density matrix of the output state |Ψo +⟩ that results from the scattering of the incident input state |Ψi +⟩ in Fig. 5.1. For the calculation, we chose that the helicity splitting coefficient β ( ω )=0 . 2, and α ( ω )is a Lorentzian function with ωL = 17 . 5 × 10 14 rad/s and γ = 1 THz. Lo≈ 0 . 4 > 0indicates that the output state is not pure 5.3 shows the output density matrix ˆϱo calculated using Eqs. (5.5) and (5.10) - (5.11b) , for these α and β helicity-splitting coefficients. The ˆϱo obtained for the output state represents a partially coherent superposition of |ψo +(ω1, ω2)⟩ and |χo +(ω1, ω2)⟩ (as indicated by Eq. (5.10) ). The purity of this state is Lo≈ 0 . 4 > 0 (i.e., the output state is mixed). Thus, this simple example demonstrates that the purity of the incident quantum state can be lost in the interaction with a nanostructure. Origin of the loss of purity under the quasi–monochromatic approximation To identify the origin of this loss of purity, we consider that the input two-photon spectral function, ϕ ( ω1, ω2 )is quasi-monochromatic (i.e., its spectral variance σ is significantly smaller than the central frequency of the pulse, ωin ). Under this approximation, we first expand the α and β coefficients to first order around the central frequency of the two-photon spectral function ωin: α(ω)≈A1 + A′ A∆ω,(5.12) β(ω)≈B1 + B′ B∆ω,(5.13) with A = α ( ωin ), B = β ( ωin ), A′ = dα ( ω ) /dω|ωin , B′ = dβ ( ω ) /dω|ωin , and ∆ω=ω−ωin. Using Eqs. (5.12) and (5.13) we can write Eq. (5.11a) and (5.11b) as: Cψ(ω1, ω2)≈(A2+B2)[1 + (∆ω1+ ∆ω2)(Fψ+iτψ)],(5.14) Cχ(ω1, ω2)≈2AB[1 + (∆ω1+ ∆ω2)(Fχ+iτχ)],(5.15) 137 Chapter 5. Loss of purity in the scattering of two-photon entangled states with Fψ=1 |A|4+|B|4+ 2|A|2|B|2cos(2δ)|A|′ |A||A|4+|B|′ |B||B|4+ |A|2|B|2cos(2δ)|A|′ A+|B|′ B+ sin(2δ)(arg{A}′−arg{B}′),(5.16) τψ=1 |A|4+|B|4+ 2|A|2|B|2cos(2δ)arg{A}′|A|4+ arg{B}′|B|4+ |A|2|B|2cos(2δ)(arg{A}′+ arg{B}′) + sin(2δ)|B|′ |B|−|A|′ |A|,(5.17) Fχ=1 2|A|′ |A|+|B|′ |B|,(5.18) τχ=1 2(arg{A}′+ arg{B}′).(5.19) In Eqs. (5.16) - (5.19) we have introduced arg{A}′ = darg{α ( ω ) }/dω|ωin , arg{B}′ = darg{β ( ω ) }/dω|ωin , δ = arg{B} − arg{A} , |A|′ = d|α ( ω ) |/dω|ωin , and |B|′ = d|β ( ω ) |/dω|ωin . We further make the approximation 1 + x ∆ ω≈ex∆ω in Eqs. (5.14) and (5.15), which gives Cψ(ω1, ω2)≈(A2+B2)exp[(∆ω1+ ∆ω2)(Fψ+iτψ)],(5.20) Cχ(ω1, ω2)≈2AB exp[(∆ω1+ ∆ω2)(Fχ+iτχ)].(5.21) Then, the ϕ ( ω1, ω2 ) Cψ ( ω1, ω2 )and ϕ ( ω1, ω2 ) Cχ ( ω1, ω2 )functions (Eq. (5.10) ) result in: ϕ(ω1, ω2)Cψ(ω1, ω2)≈ ≈Aψexp−[ω1−(ωin +σ2FΨ)]2 2σ2+iω1τψexp−[ω2−(ωin +σ2FΨ)]2 2σ2+iω2τψ, (5.22) ϕ(ω1, ω2)Cχ(ω1, ω2)≈ ≈Aχexp−[ω1−(ωin +σ2Fχ)]2 2σ2+iω1τχexp−[ω2−(ωin +σ2Fχ)]2 2σ2+iω2τχ. (5.23) Note that AΨ , FΨ , τΨ , Aχ , Fχ , and τχ depend only on the classical response of the system evaluated at the central frequency of the incident pulse, ωin. Equations (5.22) and (5.23) show that the output two-photon modes can be represented as two distinct two-photon pulses. Each two-photon pulse can be factorized as two Gaussian pulses, one for each photon. However, the pulses 138 5.3. Quantum transformation associated with the |ψo +(ω1, ω2)⟩ and |χo +(ω1, ω2)⟩ states have different properties. The pulse in Eq. (5.22) has an amplitude Aψ , a common central frequency for both photons ( ωin + σ2Fψ ), and a central time τψ (identified from the iω1τψ and iω2τψ terms in Eq. (5.22) ). Similarly, the pulse in Eq. (5.23) has an amplitude Aχ , a central frequency ( ωin + σ2Fχ ), and a central time τχ . This implies that after the interaction with the nanostructure, the resulting quantum state is a superposition of two different quantum states with different time-frequency properties. As we show next, the loss of purity in the output state can be attributed to the time delay and frequency shift between the output pulses. Analytical expression of the loss of purity Using Eqs. (5.5) , (5.10) , (5.22) , and (5.23) we obtain an analytical expression for the purity of the output state in the quasi-monochromatic approximation, Lo=2|AΨ|2|Aχ|2eσ2(F2 χ+F2 Ψ) (|AΨ|2e2σ2F2 Ψ+|Aχ|2e2σ2F2 χ)2[eσ2∆F2−e−σ2∆τ2],(5.24) with ∆ F = FΨ−Fχ and ∆ τ = τΨ−τχ . This equation indicates that the output state is pure under any of the following conditions: • If σ = 0, corresponding to a purely monochromatic incident state. In this case, the output state is pure and consists of a superposition of two different states with amplitudes AΨand Aχ(Eqs. (5.10)-(5.23)). • If β ( ωin ) = 0, corresponding to the condition of helicity preservation [27]. In this case, the output state becomes a frequency superposition of only |ψo +(ω1, ω2)⟩states. • If α ( ωin ) = 0, corresponding to the condition of total conversion of helicity [27]. In this case, the output state also becomes a frequency superposition of only |ψo +(ω1, ω2)⟩states. • If α ( ωin ) = ±β ( ωin ), corresponding to the situation where the incident light only excites either magnetic or electric modes of the nanostructure [27,29,197]. In this case, the output state is a superposition of output |ψo +(ω1, ω2)⟩ and |χo +(ω1, ω2)⟩ states with the same amplitude, and the time delays and frequency shifts between the output pulses become zero (∆ τ = 0 and ∆ F = 0). This situation is discussed in more detail at the end of section 5.4.2. • If α ( ω )and β ( ω )are almost constant in a spectral range given by σ . From Eqs. (5.16)-(5.19) we find that this case leads to ∆F≈0and ∆τ≈0. From the conditions above, we can expect a substantial loss of purity if an incident non-monochromatic pulse ( σ = 0) excites both electric and magnetic resonances of the nanostructure (thus, α ( ωin )  = ±β ( ωin )  = 0), while α ( ω )and β(ω)change abruptly near the illumination frequency. 139