Dicke superradiance in atomic arrays: impact of dimensionality

Bachelor Thesis Department of Physics Dicke superradiance in atomic arrays: impact of dimensionality Eric Sierra Garzo Supervisor: Ana Asenjo-Garcia Co-supervisor: Stuart J. Masson Tutor UPC: Jordi Boronat Medico In partial fulfillment of the requirements for the: Bachelor’s Degree in Mathematics Bachelor’s Degree in Engineering Physics September 2021 Facultat de Matemàtiques i Estadística Abstract Many-body effects are of increasing interest in quantum optics. This work provides an examination of one of these effects, known as Dicke superradiance, where an ensemble of atoms decay in a coherent manner radiating a high intensity pulse of light to the surroundings. We show that Dicke superradiance survives in extended periodic structures forming one, two and three dimensional arrays in free space. These ideas have been discussed in the literature for several decades, but concrete advances have been minimal due to the fact that evolution occurs in a Hilbert space that grows exponentially in size with atom number. Focusing on the statistics of the emitted photons, we are able to bypass the exponential complexity and analytically characterize the superradiant behavior of large arrays and study the effect of dimensionality and other geometrical properties. Keywords: quantum optics, Dicke superradiance, superradiant burst, atomic arrays, collective decay, coherent emission. MSC2020: 81V80 Resumen Los efectos colectivos son un ´ambito de creciente inter´es dentro del estudio de la interacci´on luz-materia a nivel cu´antico. Este trabajo se centra en uno de estos efectos conocido como la superradiancia de Dicke, en el que un conjunto de ´atomos emite coherentemente produciento un pulso de luz de alta intensidad. Este trabajo demuestra que este efecto sobrevive en estructuras peri´odicas en una, dos y tres dimensiones. Esto ha sido estudiado durante decadas pero con muy pocos avances debido a que el espacio de Hilbert del sistema crece exponencialmente con el n´umero de atomos. A trav´es de la estad´ıstica de los fotones emitidos por el sistema, solucionamos el problema de la complejidad exponencial y obtenemos una condici´on anal´ıtica que caracteriza la superradiancia y que empleamos para estudiar el rol de la dimensionalidad y otras propiedades geom´etricas. Palabras clave: ´optica cu´antica, superradiancia de Dicke, destello superradiante, colecci´on de ´atomos, decaimiento colectivo, emisi´on coherente. MSC2020: 81V80 Resum Els efectes colectius s´on un `ambit de creixent inter`es en l’estudi de les interaccions llum-mat`eria a nivell qu`antic. Aquest treball es centra en un d’aquests efectes conegut com la superradi`ancia de Dicke, en la qual un conjunt d’`atoms emet coherentment produint un pols de llum d’alta intensitat. Es demostra que aquest efecte sobreviu en estructures peri`odiques d’una, dues i tres dimensions. Aquest fet s’ha estudiat durant d`ecades per`o amb pocs aven¸cos degut a que l’espai de Hilbert del sistema creix de forma exponencial. A trav´es de l’estad´ıstica dels fotons emesos pel sistema, som capa¸cos de solucionar el problema de la complexitat exponencial i obtenir una condici´o anal´ıtica que caracteritza la superradi`ancia i que ens serveix per estudiar com la dimensionalitat i altres propietats geom`etriques afecten a la pres`encia d’aquesta. Paraules clau: `optica qu`antica, superradi`ancia de Dicke, flaix superradiant, colecci´o d’`atoms, deca¨ıment col.lectiu, emissi´o coherent. MSC2020: 81V80 Contents 1 Introduction 2 2 State of the art 5 2.1 Twolevelatoms ........................... 5 2.2 Superradiance and the Dicke model . . . . . . . . . . . . . . . . . 7 2.3 Themasterequation......................... 8 2.4 Spin model for multiple atoms . . . . . . . . . . . . . . . . . . . 10 3 Photon statistics 14 3.1 Jumpoperators............................ 14 3.2 Second order correlation function . . . . . . . . . . . . . . . . . . 15 3.3 Enhancement due to subsequent photon emission . . . . . . . . . 19 3.4 Infinitearrays............................. 20 4 The algorithm for finite systems 23 5 One dimensional arrays 26 5.1 Theinfinitechain........................... 26 5.2 Finitearrays ............................. 34 6 Two dimensional arrays 38 6.1 Infinite squared array . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2 Finite squared arrays . . . . . . . . . . . . . . . . . . . . . . . . . 42 6.3 Non-squarearrays .......................... 46 7 Three dimensional arrays 49 8 Conclusions 52 9 Future work 54 1 Chapter 1 Introduction Controlled and deterministic interactions between photons and atoms have been an important field of investigation for years, with a wide range of applications from quantum sensing to quantum information processing. The interaction between single photons is extremely inefficient and so a common approach has been to focus on the interaction between photons and atomic ensembles [1]. This has enabled the study of collective effects and experimental applications in many fields and regimes. A fundamental limitation in most of the cases arises from spontaneous emission into the environment, leading to an energy decay of the system. Historically, in this context it was typically assumed that spontaneous emission of atoms occurs independently at the same rate that a single atom does. However, in the 1950s Dicke showed that collective effects changed the exponential decay to an initial increase in the photon emission rate known as “superradiance burst” or “superflourescence” [2–5]. When atoms are virtually placed at the same spatial location, they become phase locked during their decay, which translates into a collective behavior of the ensemble. This behavior opens the door to a variety of interesting collective properties and possible applications. For example, a deep study on this effect leads to the creation of a “superradiant laser” [6]. Although Dicke showed the existence of a superradiant burst, his scenario ignores coherent interactions between atoms, which become relevant in realistic regimes where atoms are not at the same spatial location. If the interatomic distance is significant, these interactions have been predicted to wash out superradiant decay [4, 5]. However, in some special situations a superradiant burst has been observed. In dense thermal gases, a short interparticle distance enhances photon emission via collective effects [7–9] while in cavity setups, the confinement of the field emulate the single spatial location condition [10, 11]. Recent experiments in ordered atomic arrays (in optical lattices and tweezer arrays) demonstrated the existence of a two-dimensional atomic mirror [12] and collective frequency shifts in a 1D atomic array [13]. Theoretical investiga- 2 tions of collective behavior in ordered arrays have also predicted the presence of extremely dark states [14, 15], directional emission [6, 16] and topological physics [17]. These predictions and observations show that ordered arrays are an exciting new platform for quantum optics ideal for studying collective effects. Here, we focus on the problem of superradiant decay in ordered atomic 1D, 2D and 3D arrays (see Fig. 1.1) and how it depends on the interatomic distance and geometry of the array. In the limit of zero interatomic distance, the problem was already solved by Dicke: a collective decay with a superradiant burst in the photon emission rate that scales as the square of the number of atoms. In the limit of infinite separation, atoms decay independently with an exponential rate as interactions between atoms disappear (see Fig. 1.2). But the intermediate regime remains mostly unsolved. This work will address this by looking at photon statistics from ordered arrays with a finite interatomic distance. Recent work in this way is limited to small number of atoms [18, 19] or just a few excitations [20] because the Hilbert space grows as 2N, where Nis the number of atoms. However, a new algorithmic approach considering only the initial statistics of the emitted field, combined with the spatial periodicity of atomic arrays, allows us to look at the photon emission rate for over 106atoms. Figure 1.1: Representation of the atomic systems. From left to right, 1D, 2D and 3D atomic arrays. dindicates the interatomic separation. This work will show that the transition from the Dicke scenario to independent decay is smooth. Therefore, the superradiant burst is shown to be conserved for low interatomic distances. With this, we define a critical distance dcritical to be the threshold for superradiance and see how this distance depends on the array, as seen in Fig. 1.2. Focusing on photon statistics, an analitic condition for superradiance is found. We derive this condition for the emission to be superradiant in terms of the decay rates of the system with a method that works for all atomic distributions and polarizations. This makes it possible to look at how the geometry of the array and the number of atoms impacts the critical distance. We find that symmetry impacts it in a crystallographic way but not in a fundamental way, i.e., it slightly changes depending on the unit cell. The case will be the same for different polarizations. However, number of atoms and dimensionality do have a big impact on critical distance. Increasing the number of atoms makes dcritical become larger. For 1D, distance increases 3 d d 0 photon emission rate inter-atomic distance, d d<dcritical d>dcritical dcritical superradiant burst exponential decay dcritical Figure 1.2: Dicke scenario shows a burst in the photon emission rate with a peak at tmax, which corresponds to the zero distance scenario (d→0). When atoms are far from each other (d→ ∞), the decay becomes exponential. Therefore, for extended arrays a crossover distance dcritical arises as the upper limit for a superradiant burst to be possible. This figure has been taken from [21]. up to a threshold that can be found analytically. For 2D and 3D, the critical distance has no threshold, it keeps increasing as the number of atoms does. This means that, from a theoretical point of view, and given an interatomic distance, the essence of superradiant decay does not disappear only by the effect of propagation of the electromagnetic field if the number of atoms is high enough. Note that this work includes part of the work done in the paper by Masson and Asenjo-Garcia [21]. Therefore, an important part presented in this work is also presented there and is by no means a copy or an attempt to supplant the work done in the paper. In sections 3.4, 4, 5, 6 and 7, we present new and original results that will soon be submitted for publication. 4 Chapter 2 State of the art In this work we treat with quantum interactions between light and an ensemble of atoms. To study our system properly we need to characterize the electromagnetic field, atoms and the interaction between them. To describe the interaction between atoms and the radiation field we need several things. First, we consider the atoms as two-level systems. This approximation is accurate when the transition energy between a ground state and an excited state of the atom is considerably different than that of other transition energies or when the transition can be isolated by choice of field polarization. Then, we describe the electromagnetic field as a quantum system. We then trace out the field via an approach based on the classical Green’s tensor of any dielectric media. Finally, the interaction is treated as a system-reservoir problem. We consider the ensemble of atoms (system) surrounded by the electromagnetic field (reservoir) and the way they interact. This leads us to what is known as the master equation, an extension of the Schr¨odinger equation, with the advantage that it allows for dissipation into the reservoir. This whole perspective of the problem is known as the spin model, commonly used in quantum optics to study light-matter interactions. The model is valid for any linear, isotropic, dielectric media, which in our case is set to be free space. 2.1 Two level atoms In an ensemble of atoms, a loss of energy may be due to spontaneous emission of photons or inelastic collisions between atoms. Therefore, in an ordered, fixed array of atoms, the only possible way to lose energy is via spontaneous emission. We consider the atoms to be two-level systems, and because our dissipative scenario, we will treat them as quantum dipoles. 5 for the atomic internal degrees of freedom, we can extend Eqs. (2.14a) and (2.14b) to Natoms as [15,19,24,26,27] ˙ρA=−i ~[H, ρA] + L[ρA] (2.18a) L[ρA] = N X i,j=1 Γij 22σj geρσi eg −σi egσj geρ−ρσi egσj ge.(2.18b) with the corresponding Hamiltonian H=~ N X i=1 ω0σi ee +~ N X i,j=1 Jijσi egσj ge.(2.19) Coherent and dissipative interactions between atoms iand jare given by [33,34] Jij =−µ0ω2 0 ~℘∗·Re{G(ri,rj, ω0)}·℘,(2.20a) Γij =2µ0ω2 0 ~℘∗·Im{G(ri,rj, ω0)}·℘.(2.20b) The Green’s tensor can be directly calculated in free space. Assuming the atoms are tightly trapped, so that we can treat atomic positions as fixed points rather than quantum variables, and using the fact that G(ri,rj, ω0) = G(rij, ω0) with rij =ri−rj, the solution of Eq. (2.16) for a dipole is [15] G(r, ω0) = eik0r 4πk2 0r3(k2 0r2+ik0r−1)1+ (−k2 0r2−3ik0r+ 3)r⊗r r2,(2.21) where r=|r|and k0= 2π/λ0=ω0/c is the wave number corresponding to the atomic transition energy. In the case of a single atom, taking a limit one can reproduce Einstein’s A coefficient as the vacuum emission rate Γii = Γ0=ω3 0|℘|2 3π~ε0c3.(2.22) The change on notation due to the Natoms system is defined so that σi eg = |eiihgi|is the atomic coherence operator between the ground and excited state of atom i. This leads us to the coherent evolution Hamiltonian H=~ N X i=1 ω0σi ee +~ N X i,j=1 Jij −iΓij 2σi egσj ge,(2.23) with the first term being the initial system energy and the second being the effective Hamiltonian Heff. The interaction between atoms now is easily describable. Note that the coupling matrix has been divided into Jand Γwith elements Jij and Γij. If we 12 look at how does this coupling matrix affects the evolution due to the effective Hamiltonian we can see that the evolution of any state is [22] |φ(t)i=X ν,τ cν,τ (0)e−i2Jνte−Γνt|ϕν,τ (0)i,(2.24) where Jνand Γνare the corresponding eigenvalues for Jand Γ.Jaccounts only for coherent evolution while Γencodes dissipation. Therefore, the key element in this work is how the decay matrix depends on the atomic positions. 13 Chapter 3 Photon statistics 3.1 Jump operators Equations (2.18) and (2.19) fully describe the system, but the size of the Hilbert space is exponentially large. Diagonalizing the decay matrix Γgives us a set of Njump operators {Oν}with corresponding decay rates {Γν}. These operators and decay rates are the eigenstates and eigenvalues of Γwith elements Γij [35]. Then, the Lindbladian of the master equation (2.18b) can be written as L[ρA] = N X ν=1 Γν 22OνρO† ν−O† νOνρ−ρO† νOν.(3.1) Now we have Ncollective operators governed by Ndecay rates, which allows us to compute the evolution using quantum trajectories [23]. These operators can be either superradiant or subradiant depending on the array, i.e., their corresponding decay rate is greater than the single-atom emission rate (Γν>Γ0) or smaller than it (Γν<Γ0). These decay rates represent how frequently the corresponding jump (from the excited to the ground state) occurs. Therefore, superradiant decay means that the jumps occur more often than in the single atom photon emission. Jump operators describe how the atoms decay. They happen stochastically, with operators acting on all atoms with phases that depend on the atomic position. When a photon is emitted by the array, the number of excitations is lowered by one. Therefore, the action of a jump operator is equivalent to the action of a superposition of atomic lowering operators such that Oν= N X i=1 αν,iσi ge,(3.2) with αν,i representing some spatial profile. 14 1 0.5 0 0 2.521.50.5 1 (a) 1.5 ‘superradiant burst’ 2normalized photon emission rate most radiant least radiant photons emitted 3 2 1 6 5 4 7 8 most radiant least radiant most radiant least radiant 20 40 60 80 100 0.3 0.2 no burst burst Figure 3.1: Decay of an initially fully excited 1D chain with N= 8 atoms and the stochastic action of jump operators. The picture shows the action of these operators for different interatomic distances. Operators are ordered from the most radiant to the least radiant one. The array starts as a fully excited system (|ei⊗N) and converges to a fully ground state system (|gi⊗N) as photons are emitted. Circles represent the action of one operator Oνand the color indicates from most (white) to least radiant (black). Lines represent decay paths with its thickness representing how likely is the path. This figure has been taken from [19]. These operators can be used to understand how the photon emission process occurs in atomic arrays. As shown in Fig. 3.1, in the Dicke scenario (d= 0), only the most radiant operator plays a role in the decay. As the interatomic distance increases, less radiant operators start to matter in the decay process and superradiant channels become less dominant. Although superradiant operators are still present for large distances, they compete with each other and allow for less radiant states and even dark states to play a role. This grows up until superradiant channels are no longer important enough to materialize in a superradiant burst [19]. Looking for decay paths within jump operators gives a nice picture of how decay occurs in atomic arrays. However, as this is done by calculating trajectories, it is only possible for a small number of atoms because of the complexity of the system. That means we still need an easier way of looking at superradiance without taking into account the whole Hilbert space. 3.2 Second order correlation function Our method of looking for superradiance in an easier way consists on focusing on the first few emitted photons. One condition for superradiance is that, for the initial state, the intensity emitted by the system increases. That means that for very early times, the intensity must be increasing, i.e., its derivative 15 must be positive. In terms of photon emission, this means that the emission of the second photon is enhanced by the first photon. Thinking about the photon emission as a stochastic process, the emission of the first photon makes, on average, the emission of the second photon more likely to occur. This is captured by the second order correlation function, which gives information about statistics of the emitted field. Correlation functions are a common tool used in quantum optics, useful to characterize the coherence properties of the electromagnetic field. The normalized correlation functions are also called degree of coherence. The first order correlation function g(1)(τ) accounts for the amplitude-amplitude correlation of field and it is useful for looking at field interference. The second order correlation function g(2) captures the intensity-intensity correlation of the emitted field. This second order correlation function is given by [36] g(2)(r1, t1,r2, t2) = hE∗(r1, t1)E∗(r2, t2)E(r1, t1)E(r2, t2)i h|E(r1, t1)|2ih|E(r2, t2)|2i,(3.3) where the field is measured at two points r1and r2at times t1and t2. If the field is considered to be classical, and setting τ=t2−t1, the correlation function at the same point can also be expressed as [36] g(2)(t, τ) = hI(t)I(t+τ)i hI(t)2i.(3.4) When we are dealing with quantum fields, the above expressions can be written in terms of operators. As field operators are related with atomic coherence operators (see Eq. (2.17)), the second order correlation function at the initial time is [21,23] g(2)(0) = N P ν,µ=1 ΓνΓµO† νO† µOµOν N P ν=1 ΓνDO† νOνE2,(3.5) where the expectation values are taken at the initial state of the system, i.e., when |ϕ(t= 0)i=|ei⊗N, with the system fully excited. This function sets a way to characterize the presence or absence of a superradiant burst. When g(2)(0) >1, it means that the intensity of the emitted field increases at the early stages of the decay process, which we characterize as a superradiant burst. Otherwise, if g(2)(0) <1, intensity decreases immediately and therefore there is no superradiant burst. Also, g(2)(0) is directly related to other properties of the superradiant burst. As seen in Fig. 3.2, the time where the burst has the maximum emission rate is related to the second order correlation function. As soon as g(2)(0) >1, the time of peak deviates from zero (tmax >0) so there is a burst at a finite time. Also, the maximum emission rate of emission process increases with g(2)(0) only if g(2)(0) >1 (when there is 16 1.0 1.1 1.0 1.1 0.95 1.00 1.05 1.10 1.15 0.00 0.05 0.10 0.15 0.20 Figure 3.2: Time of peak emission rate (tmax) and maximum photon emission rate in terms of photon statistics at t= 0. Figure shows that tmax increases with g(2) when this is greater than one but it is equal to zero when g(2) <1. Also, the maximum photon emission rate increases with g(2) only if g(2) >1. All three arrays have N= 9 atoms. This figure has been taken from [21]. a superradiant burst). These comparisons can only be done for a low number of atoms as a full calculation of state dynamics is needed. The key to this work is that the second order correlation function g(2)(0) can be calculated analytically, such that an expression for a condition for superradiance can be derived. Starting from Eq. (3.5), and substituting the expression for the eigenvalue operators (3.2), we get the expression g(2)(0) = N P ν,µ=1 ΓνΓµ N P i,j,k,l=1 α∗ ν,iα∗ µ,jαµ,kαν,l σi egσj egσk geσl ge N P ν=1 Γν N P i,j=1 α∗ ν,iαν,j Dσi egσj geE!2,(3.6) where we require some decomposition properties of the constants in (3.2) that come from orthogonality which write N X i=1 α∗ ν,iαµ,i =δνµ and N X ν=1 Γν|αν,i|2= Γ0∀i, (3.7) where δνµ is the Kronecker delta function. As the initial state of the array is fully excited (i.e. |ei⊗N), we can simplify the expression (3.6) by pointing out that σi egσj ge=δij,(3.8) as an atom can only be excited again if the atom was already de-excited. In the case of the four operators expectation value, if we take into account all the 17 possible excitation/de-excitation pairs we get that σi egσj egσk geσl ge= (δikδjl +δilδjk)(1 −δij ),(3.9) where the factor 1 −δij accounts for the impossibility to excite two times a two-level atom. Then, we can develop expression (3.6) g(2)(0) = N P ν,µ=1 ΓνΓµ N P i,j,k,l=1 α∗ ν,iα∗ µ,jαµ,kαν,l σi egσj egσk geσl ge N P ν=1 Γν N P i,j=1 α∗ ν,iαν,j Dσi egσj geE!2 = N P ν,µ=1 ΓνΓµ N P i,j,k,l=1 α∗ ν,iα∗ µ,jαµ,kαν,l(δikδjl +δilδjk)(1 −δij) N P ν=1 Γν N P i,j=1 α∗ ν,iαν,jδij!2 =1 N P ν=1 Γν N P i=1 |αν,i|22 N X ν,µ=1 ΓνΓµ  N X i,j=1 |αν,i|2|αµ,j|2 + N X i,j=1 α∗ ν,iα∗ µ,jαµ,iαν,j −2 N X i=1 |αν,i|2|αµ,i|2! =1 N P i=1 N P ν=1 Γν|αν,i|22 N X ν,µ=1 ΓνΓµ" N X i=1 |αν,i|2! N X i=1 |αν,i|2! + N X i=1 α∗ ν,iαµ,i! N X j=1 α∗ µ,iαν,i!−2 N X i=1 |αν,i|2|αµ,i|2# = N P ν,µ=1 ΓνΓµ1 + δνµ −2 N P i=1 |αν,i|2|αµ,i|2 N2Γ2 0 = N2Γ2 0+ N P ν=1 Γ2 ν−2 N P i=1 N P ν=1 Γν|αν,i|2 N P µ=1 Γν|αµ,i|2! N2Γ2 0 = 1 + N X ν=1 Γν NΓ02 −2 N.(3.10) In this derivation we have used the fact that, as Γii = Γ0, Tr Γ=NΓ0and therefore, using the trace invariance, N X ν=1 Γν=NΓ0.(3.11) 18 Finally, using the fact that Var Γν Γ0= N X ν=1 Γ2 ν NΓ2 0−1,(3.12) we can simplify the expression for the second order correlation function to g(2)(0) = 1 + 1 NVar Γν Γ0−1,(3.13) where Var is the variance. With this, the condition for superradiance is found to be Var Γν Γ0>1.(3.14) We have an analytical general condition for superradiance which is valid for any atomic ordered configuration. Indeed, with this expression we see that the presence of a superradiant burst only depends on the decay rates, i.e., the eigenvalues of Γ. In fact, condition (3.11) states that the mean of the decay rates is always the free decay rate, and therefore, superradiance occurs only when decay rates are enough spread around Γ0. In terms of complexity, we have reduced an exponentially hard problem that scaled as 2Nto a problem of finding the eigenvalues of Γ, which is known to be o(N3) and Ω(N2). 3.3 Enhancement due to subsequent photon emission Knowing that g(2)(0) >1 is a sufficient condition for superradiance, one can ask if this condition is also necessary. To be so, we require that there is nothing that rephases the system. We demonstrate this claim in Fig. 3.3. The system can only be rephased by the action of the Hamiltonian or enhancement by future photons. In Fig. 3.3(a), evolution with and without Hamiltonian system is shown. This happens for all 1D, 2D and 3D arrays. The Hamiltonian does not affect the early evolution of the system. As a result, we see that coherent interactions due to the Hamiltonian do not affect the system until a time delay upon which the dephasing introduced suppresses photon emission. The other possibility is that even if the second photon is not enhanced, the third photon does get enhanced creating a rephasing of the system. To demonstrate that this is not the case, we look at the third order correlation function, which can also be found analytically [21]. In Fig. 3.3(b), we see that there is no region where the third photon is enhanced by the emission of the second photon while the second one is not. In all configurations considered here, this condition is never met. 19 (a) 1.2 1.1 1.0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.37 0.38 0.36 0.99 1.00 1.01 (b) 2.0 1.5 1.0 0.5 2.0 1.51.00.0 0.5 0.0 Figure 3.3: Impact of dissipative and coherent evolution. (a) The action of the Hamiltonian does not affect the early dynamics and therefore the condition of superradiance. Figure is done with arrays of 16 atoms with interatomic distance d= 0.1λ0. (b) The third photon is never enhanced when the second is neither as g(3) is never greater than one when g(2) is not. This is calculated for a 2D 36 atoms array. In all cases polarization is set to be perpendicular to the array. This figure has been taken from [21]. We have thus found a necessary and sufficient condition for superradiance. We can now proceed to study superradiance in arrays with different dimensions and different atomic configurations. 3.4 Infinite arrays The above captures the decay of finite atomic arrays. However, to look at the behavior in the infinite array limit, we need to adapt our approach to the infinite problem. To do so, we use the prescription N X ν=1 →Nd 2πnZ BZ1 dk,(3.15) 20 where nis the dimension in which we are working and BZ1 indicates that the integration is done over the first Brillouin zone with volume d 2πn. We assumed our array has Natoms (a limit is taken to get the infinite result). Note that this prescription is just the inverse of a discrete integral approximation with equally spaced points in momentum space. Then the second order correlation function from Eq. (3.5) can be calculated as g(2)(0) = N2d 2π2nR BZ1 dk1R BZ1 dk2Γk1Γk2Dˆ O† k1 ˆ O† k2 ˆ Ok2ˆ Ok1E Nd 2πnR BZ1 dkΓkDˆ O† kˆ OkE2 =R BZ1 dk1R BZ1 dk2Γk1Γk2P i,j,l,m≥1 α∗ k1,iα∗ k2,jαk2,lαk1,m ˆσi eg ˆσj eg ˆσl geˆσm ge R BZ1 dkΓkP i,j≥1 α∗ k,iαk,j Dˆσi eg ˆσj geE!2 =R BZ1 dk1R BZ1 dk2Γk1Γk21 + δ(k1−k2)− N P i=1 2|αk1,i|2|αk2,i|2 R BZ1 dkΓk2 = 1 + R BZ1 dk1R BZ1 dk2Γk1Γk2δ(k1−k2) R BZ1 dkΓk2 − 2 N P i=1 R BZ1 dk1Γk1|αk1,i|2R BZ1 dk2Γk2|αk2,i|2 R BZ1 dkΓk2,(3.16) where δis the Dirac delta function and we have used relations (3.8) and (3.9). To simplify this expression, we need to adapt some equations from the finite case N X ν=1 δνµ = 1 →Nd 2πnZ BZ1 dk2δ(k2−k1)=1,(3.17) N X ν=1 Γν|αν,i|2= Γ0→Nd 2πnZ BZ1 dkΓk|αk,i|2= Γ0,(3.18) N X ν=1 Γν Γ0 =N→Nd 2πnZ BZ1 dkΓk Γ0 =N(3.19) and N X i=1 |αk1,i|2= 1.(3.20) 21 can use expression (3.21) with decay rates (5.1) and (5.2). Now that we have the decay rates we can easily check Eq. (3.19) by simple 1D integration Z BZ1 dkΓ⊥ k Γ0 = π/d Z −π/d Γ⊥ k Γ0 dk =3π 4k0d π/d Z −π/d X gz |kz+gz|≤k0 1 + (kz+gz)2 k2 0!dkz =3π 4k0d k0 Z −k0 1 + ˜ k2 z k2 0 d˜ kz=3π 4k0d2k01 + 1 3=2π d,(5.3) where we have preformed the change of variables ˜ kz=kz+gz. As any point kz in the first Brillouin zone is uniquely represented, any point as kz+gzhas also a unique representation inside the region |k|< k0. Therefore, the summation can be seen as adding up integrals for all the possible Brillouin zones up to k0 which is the same as integrating f(k) = 1 + k2/k2 0through all the region. For parallel polarization the calculation is equivalent Z BZ1 dkΓk k Γ0 = π/d Z −π/d Γk k Γ0 dk =3π 2k0d π/d Z −π/d X gz |kz+gz|≤k0 1−(kz+gz)2 k2 0!dkz =3π 2k0d k0 Z −k0 1−˜ k2 z k2 0 d˜ kz=3π 2k0d2k01−1 3=2π d.(5.4) To get an analytical expression for Eq. (3.21) we need to integrate the squared decay rates throughout the first Brillouin zone. In the simplest case, when d/λ0<0.5, the region where decay rates are not zero is inside the first Brillouin zone. That eliminates the summation, as the only possible term is when gz= 0. In the case of perpendicular polarization we have that Z BZ1 dkΓ⊥ k Γ02 = π/d Z −π/d Γ⊥ k Γ02 dk =9π2 16k2 0d2 k0 Z −k01 + k2 k2 02 dk =9π2 16k2 0d2 k0 Z −k0 1+2k2 k2 0 +k4 k4 0 dk =9π2 16k2 0d22k01 + 2 3+1 5 =21 40k02π d2 .(5.5) The second order correlation function of the 1D infinite array reads g(2) ⊥(0) = 1 + 1 N21 40 2π k0d−2= 1 + 1 N21 40 d/λ0−2.(5.6) 28 For parallel polarization we can proceed in the same way and find Z BZ1 dk Γk k Γ0!2 = π/d Z −π/d Γk k Γ0!2 dk =9π2 4k2 0d2 k0 Z −k01−k2 k2 02 dk =9π2 4k2 0d2 k0 Z −k0 1−2k2 k2 0 +k4 k4 0 dk =9π2 4k2 0d22k01−2 3+1 5 =3 5k02π d2 ,(5.7) from which we can obtain the second order correlation function for parallel polarization g(2) k(0) = 1 + 1 N3 5 2π k0d−2= 1 + 1 N3 5d/λ0−2.(5.8) Both correlation functions are decreasing with distance at a rate 1/d and tend to one when Ntends to infinity. As we see in Fig. 5.4, the infinite g(2)(0) for both perpendicular and parallel polarization is greater than one until a critical distance, different for each polarization. For 0.5< d/λ0<1, the only possible values for gzare gz={−2π/d, 0,2π/d} and we can obtain an analytical expression. In the case of perpendicular polarization we have Z BZ1Γ⊥ k Γ02 dk =9π2 16k2 0d2 π/d Z −π/d X gz 1 + (kz+gz)2 k2 0!2 +X gz6=g0 z 1 + (kz+gz)2 k2 0! 1 + (kz+g0 z)2 k2 0!dkz =9π2 16k2 0d2  k0 Z −k01 + k2 k2 02 dk + 2 k0−2π/d Z −π/d 1 + k2 k2 01 + (k+ 2π/d)2 k2 0dk + 2 π/d Z 2π/d−k01 + k2 k2 01 + (k−2π/d)2 k2 0dk  , (5.9) where we have separated the sum between the repeating and no repeating terms and used the fact that decay rates are even under parity symmetry (k→ −k). 29 For simplicity, let’s now redefine some terms such that A=π/d and D= 1 2 1 d/λ0=π dk0=A k0. Then we have that k0−2π/d Z −π/d 1 + k2 k2 01 + (k+ 2π/d)2 k2 0dk = π/d Z 2π/d−k01 + k2 k2 01 + (k−2π/d)2 k2 0dk = k0−π/d Z 01 + (k−π/d)2 k2 01 + (k+π/d)2 k2 0dk = k0(1−D) Z 0 1+2k2+A2 k2 0 +(k2−A2)2 k4 0 dk = k0(1−D) Z 0 1+2D2+D4+ 2k2 k2 0 (1 −D2) + k4 k4 0 dk =k0(1 −D)(1 + D2)2+2 3(1 −D)3(1 + D) + 1 5(1 −D)4. (5.10) Now, using this and the result in (5.5) we get that Z BZ1Γ⊥ k Γ02 dk =9π2 16k2 0d22k0 28 15 + 4k0(1 −D)(1 + D2)2 +2 3(1 −D)3(1 + D) + 1 5(1 −D)4 =9AD 828 15 + 2(1 −D)(1 + D2)2 +2 3(1 −D)3(1 + D) + 1 5(1 −D)4 = 2AD −3 5D5−3D3+ 6D2−9 2D+63 20,(5.11) which leads to the second order correlation function for 0.5< d/λ0<1 g(2) ⊥(0) = 1 + 1 N−3 5D6−3D4+ 6D3−9 2D2+63 20D−2 30 = 1 + 1 320N  −3 d λ06−60 d λ04+240 d λ03−360 d λ02+504 d λ0−640  , (5.12) which satisfies the no-burst condition g(2)(0) <1 everywhere. In the case of parallel polarization we have that Z BZ1 Γk k Γ0!2 dk =9π2 4k2 0d2 π/d Z −π/d X gz 1−(kz+gz)2 k2 0!2 +X gz6=g0 z 1−(kz+gz)2 k2 0! 1−(kz+g0 z)2 k2 0!dkz =9π2 4k2 0d2  k0 Z −k01−k2 k2 02 dk + 2 k0−2π/d Z −π/d 1−k2 k2 01−(k+ 2π/d)2 k2 0dk + 2 π/d Z 2π/d−k01−k2 k2 01−(k−2π/d)2 k2 0dk  , (5.13) Similarly, in the case of parallel polarization, we have that k0−2π/d Z −π/d 1−k2 k2 01−(k+ 2π/d)2 k2 0dk = π/d Z 2π/d−k01−k2 k2 01−(k−2π/d)2 k2 0dk = k0−π/d Z 01−(k−π/d)2 k2 01−(k+π/d)2 k2 0dk = k0(1−D) Z 0 1−2k2+A2 k2 0 +(k2−A2)2 k4 0 dk = k0(1−D) Z 0 1−2D2+D4−2k2 k2 0 (1 + D2) + k4 k4 0 dk 31 =k0(1 −D)(1 −D2)2−2 3(1 −D)2(1 + D2) + 1 5(1 −D)4. (5.14) Now, using this and the result in (5.7) we get that Z BZ1 Γk k Γ0!2 dk =9π2 4k2 0d22k0 8 15 + 4k0(1 −D)(1 + D)2(1 −D)2 −2 3(1 −D)2(1 + D2) + 1 5(1 −D)4 =9AD 228 15 + 2(1 −D)3(1 + D)2 −2 3(1 + D2) + 1 5(1 −D)2 = 2AD −12 5D5+ 12D3−12D2−18 5,(5.15) which leads to the second order correlation function for 0.5< d/λ0<1 g(2) k(0) = 1 + 1 N−12 5D6+ 12D4−12D3−18 5D−2 = 1 + 1 80N  −3 d λ06+60 d λ04−120 d λ03+144 d λ0−160  ,(5.16) which also satisfies the no-burst condition g(2)(0) <1 everywhere. For higher distances, we can upper bound the integral to prove that g(2)(0) < 1 always. For example, for perpendicular polarization we have that X gz,g0 z 1 + (kz+gz)2 k2 0! 1 + (kz+g0 z)2 k2 0! =X gz 1 + (kz+gz)2 k2 0!2 +X gz6=g0 z 1 + (kz+gz)2 k2 0! 1 + (kz+g0 z)2 k2 0! <X gz 1 + (kz+gz)2 k2 0!2 + 2 X gz Ng 1 + (kz+gz)2 k2 0!2 ,(5.17) where Ngis a natural number such that gz=±Ng2π/d. Note that we have upper bounded the term with the lowest gzin terms of the Brillouin zone of the higher term. Then, Z BZ1Γ⊥ k Γ02 dk < 9π2 16k2 0d2 π/d Z −π/d X gz 1 + (kz+gz)2 k2 0!2 32 + 2 X gz Ng 1 + (kz+gz)2 k2 0!2 dkz <9π2 16k2 0d2 π/d Z −π/d X gz 1 + (kz+gz)2 k2 0!2 + 2 X gz|kz+gz|d 2π+1 2 1 + (kz+gz)2 k2 0!2 dkz =9π2 16k2 0d2 k0 Z −k01 + k2 k2 02 + 2 kd 2π+1 21 + k2 k2 02 dk =9π2 8k0d214 5+7 3d/λ0,(5.18) which satisfies the condition g(2)(0) <1 if d/λ0>0.586. Similarly, for parallel polarization we have that X gz,g0 z 1−(kz+gz)2 k2 0! 1−(kz+g0 z)2 k2 0! =X gz 1−(kz+gz)2 k2 0!2 +X gz6=g0 z 1−(kz+gz)2 k2 0! 1−(kz+g0 z)2 k2 0! <X gz 1−(kz+gz)2 k2 0!2 + 2 X gz 1−(kz+gz)2 k2 0!2 .(5.19) Then, Z BZ1 Γk k Γ0!2 dk < 9π2 4k2 0d2 π/d Z −π/d 3X gz 1 + (kz+gz)2 k2 0!2 =27π2 4k2 0d2 k0 Z −k01 + k2 k2 02 dk =9 5k02π d2 ,(5.20) as we have previously calculated in (5.7). Then, we have that g(2) <1 if d/λ0>9/10. With this, we have proven that for the 1D infinite array the only point where the function is above one is in the region d/λ0<0.5. That means that for both 33 polarizations, the critical burst distances for a 1D array are d⊥ critical/λ0=21 80 = 0.2625 (5.21) dk critical/λ0=3 10 = 0.3.(5.22) In the following section, we will see that this agrees with our numerical findings for large finite arrays. 5.2 Finite arrays For finite arrays, we need to numerically calculate the decay rates of the system, as this gets significantly large. Once we have them, we can somehow look at the system’s decay. All decay rates that satisfy Γν/Γ0>1 are said to be superradiant. Depending on the interatomic distance, the behavior of jump operators changes, as seen in Fig. 5.3. For small distances, there is a clear difference between superradiant operators and subradiant ones. Furthermore, at certain values of dthere are “revivals” caused by long range order (arising from the 1/r terms in Eq. (2.21)), which increase the contrast again. As expected, rates average Γ0and converge to this value when the distance tends to infinity, when atoms cease interacting and decay independently of one another at rate Γ0. 0 0.5 1 1.5 2 2.5 3 d/ 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 /0 Parallel polarization 0 0.5 1 1.5 2 2.5 3 d/ 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 /0 Perpendicular polarization (a) (b) Figure 5.3: Decay rates behavior with distance. Eigenvalues of the decay matrix Γ for their corresponding operators {Oν}for a 1D array of N= 10 atoms. Each line corresponds to the evolution of one decay rate as interatomic distance increases. The average of the decay rates is Γ0. For parallel polarization (a), decay rates present a really damped oscillatory behavior. In the case of perpendicular polarization (b), this periodic bumping is larger. At short distances, finite one dimensional arrays have a similar set of decay rates with atoms parallel and perpendicularly polarized. However, as the 34 distance increases, perpendicular polarization rates keep spread out, which increases the value of the second order correlation function (see Eq. (3.13)). This is due to the effect of periodic revivals present every d= (n/2)λ0. These revivals are significantly more pronounced for perpendicular polarization and is in this case where they significantly affect the value of g(2)(0). To compute g(2)(0) and efficiently count all possible decay rates Γij , we observe the following. Any interaction between atoms iand j, where atoms are array ordered from 1 to N, is equivalent to the interaction between atoms 1 and |j−i|+ 1, as the Green’s tensor in (2.21) is invariant to translations. Therefore we can count on all rates just by looking at the first atom. Noting these terms as am, with m=|j−i|, they can be calculated as am= 2 (N−m)Γ1(m+1) Γ02 ,(5.23) where the multiplying factor N−maccounts for all possible translations and the 2 accounts for the symmetry Γij = Γji. These terms are those present in Eq. (4.7) such that nβ= 2(N−m). With this, we account for all the decay rates of the system (N 2interactions) by just calculating a few and counting on repetitions (N−1 interactions). This is a key factor for speeding up calculations. As the second order correlation function is proportional to the spread of the decay rates, the pattern of g(2)(0) seen in Fig. 5.4 is pretty similar to the one seen in Fig. 5.3. We observe that revivals at distances that are multiples of the half wavelength are also present for perpendicular polarization. For d<λ0/2, the possible decaying modes (k < k0) are all inside the first Brillouin zone, in which they are uniquely represented. When the interatomic distance crosses the half wavelength limit, modes outside the first Brillouin zone start to decay. This means that the decay rates of some of the operators increase, as these additional modes can be folded into the first Brillouin zone. This increased contribution of some terms is responsible for the revivals in g(2)(0). Every multiple of the half wavelength, the mode region adds some extra contribution to the first Brillouin zone modes, increasing the second order correlation function. As the distance increases, additional contributions are less significant, so revivals diminish. This situation is analogous to the atom in a cavity problem, where the same revivals are also seen [38]. This does not happen in the case of parallel polarization because of the emitting pattern of a dipole. These revivals appear due to far-field interactions, i.e., the 1/r term in Green’s tensor Eq. (2.21) (when this term is suppressed, revivals disappear). This far-field emission from a dipole is zero in the direction of the dipole vector. This reduces the long-range interactions, which is reflected in no revivals, in contrast to perpendicular polarization. 35 0.2 0.4 0.6 0.8 1 1.2 d/ 0 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 g(2)(0) N=10 0.2 0.4 0.6 0.8 1 1.2 d/ 0 0.99 1 1.01 1.02 1.03 1.04 g(2)(0) N=100 0.2 0.4 0.6 0.8 1 1.2 d/ 0 0.999 1 1.001 1.002 1.003 1.004 g(2)(0) N=1000 (a) (b) (c) Figure 5.4: Evolution of the second order correlation function with respect to the normalized separation distance d/λ0. Both cases of perpendicular (red) and parallel (blue) polarization are plotted. Decaying process of 1D arrays with (a) N= 10, (b) N= 100 and (c) N= 1000 atoms. A superradiant burst is present in all cases up to a critical distance, from which g(2) is no longer greater than one. Whether the second order correlation function has periodic revivals or not, they do not affect the presence of a superradiant burst for one dimensional arrays. Although one could think that these revivals could increase the critical distance, g(2)(0) decays fast enough so that they do not cross the unit line in the 1D array case. In fact, Fig. 5.4 shows that the presence or not of a superradiant burst is easily defined for 1D arrays. The burst is present for low distances up to a threshold. The critical distance increases slowly with Nbut plateaus, as we expect from our calculation for infinite arrays. When the number of atoms increases, the second order correlation function gets compressed towards one. In fact, in the limit d→ ∞,g(2)(0) →1−1/N. This is because, as seen in Fig. 5.3, the variance of decay rates converges to 0. The critical distance over which the system can not exhibit superradiant behavior is a crucial factor for experimental realizations. Knowing which is the region for superradiance is a key clue for looking for superradiance in experiments. This distance is the greatest distance for which an atomic array with a fixed number of atoms exhibits a superradiant burst in its decay. This critical distance grows with Nas shown in Fig. 5.5. In case of 1D arrays, it grows up to a threshold. This is the maximum distance at which the system can exhibit a superradiant burst in emitted intensity. The distance threshold is higher for parallel polarization than for perpendicular polarization. This is because, as in the perpendicular, emission is more likely to happen in the direction of the array, interference effects are more important in this case that in the parallel case. This means that the strength of the 36 101102103104 Number of atoms N 0 0.05 0.1 0.15 0.2 0.25 0.2625 0.3 0.35 d/ 0 Figure 5.5: Behavior of the critical distance dcritical with the atom number Nfor 1D arrays in the cases of perpendicular (red) and parallel (blue) polarization. Both cases show a convergence to a specific distance d/λ0, different for every case, when the array approaches the infinite limit. The whole region below the points sets an array in which a superradiant burst is present on the decay process while above the points this burst is not possible. The axis on the number of atoms is in log scale. superradiant coherence is higher for parallel polarization than for perpendicular one. 37 Distances at which the revivals occur have changed from the 1D case. In that case the only possible distances for phase periodicity are multiples of the interatomic distance as all distances between atoms are multiples of it. However, in the 2D square array we have a lot of possibilities as diagonals start to play a role. If we order nearest neighbor distances we have that the closest one is the lattice distance d. Then we have the square diagonal that has a distance of d√2. And then, all the other possible rectangular diagonals such as d√5, d√10, d√13,etc. This means that photon enhancement effects occur at half these distances and their multiples, which implies revivals are present at distances 0.5, 0.707, 1, 1.118, 1.414, 1.5, etc, as we see in Fig. 6.4. 0.2 0.4 0.6 0.8 1 1.2 d/ 0 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 g(2)(0) N=9 0.2 0.4 0.6 0.8 1 1.2 d/ 0 0.99 1 1.01 1.02 1.03 g(2)(0) N=100 0.2 0.4 0.6 0.8 1 d/ 0 0.999 1 1.001 1.002 1.003 1.004 1.005 1.006 g(2)(0) N=961 (a) (b) (c) Figure 6.4: Evolution of the second order correlation function with respect to the normalized separation distance d/λ0. Both cases of perpendicular (red) and parallel (blue) polarization are plotted. Decaying process of 2D squared arrays with (a) N= 3 (3 ×3 array), (b) N= 100 (10 ×10 array) and (c) N= 961 atoms (31 ×31 array). In the first case, g(2) only crosses the unitary line once, while in the other two its crossed three times, increasing the critical distance. While these revivals do not have any effect on the burst distance in 1D arrays, they can have an impact here. For the parallel polarization, this periodic effect is suppressed in the polarization direction, eliminating one dimension in terms of long range interaction. This sets the system to one similar to the perpendicular 1D array, with revivals that do not contribute to increase the critical distance. However, for perpendicular polarization revivals are high enough to cross unity as the array gets larger. This shifts the critical distance to higher d as a new region in which a superradiant burst can be seen arises. This makes the maximum burst distance grow faster as Nincreases. Note that there are cases for which not all distances below the critical distance produce a superradiant burst. This can be seen in Fig. 6.4 plots (b) and (c), where below the critical distance there is a region where the second order correlation function is below one. 44 The revivals mean that the critical distance does not increase smoothly. As seen in Fig. 6.5, it keeps increasing with Nfor both polarizations. However, as expected, this growth is considerably large for perpendicular polarization than for parallel polarization because of the photon emission pattern. Also, the impact of revivals is clearly seen in the perpendicular case. Note that the enhancement distances listed above are in the middle of each discontinuity, as having a maximum just after them at g(2)(0) makes impossible for them to be a critical distance. Also, critical distances are considerable higher than in 1D arrays. For example, a superradiant burst could be seen in 2D squared arrays with interatomic distance d∼λ0if we have an array of N∼103×103atoms. 102104106 N 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 d/ 0 Figure 6.5: Behavior of the critical distance dcritical with the atom number Nfor 2D squared arrays in the cases of perpendicular and parallel polarization. The whole region above the points sets an array in which a superradiant burst does not occur while below the points this burst could be possible but it is not guaranteed. As shown in the above section, the critical distance does not seem to saturate with N. In contrast to the 1D case, the critical distance keeps increasing with the size of the array. In the 1D case we had a threshold distance above which superradiance was no longer possible, even in the infinite case. In this case, there is no threshold, which would mean that we can theoretically increase the critical distance arbitrarily if the array is large enough. 45 6.3 Non-square arrays In contrast with one dimensional arrays, where there is no other geometrical parameter rather than interatomic distance, in 2D we can change the angle of the unit cell to change the geometry of the array without breaking periodicity (see Fig. 6.1). Here, we consider how the system decays if the angle is changed and how this impacts the critical distance. We will only consider angles below 90◦because, although 90◦−βand 90◦+βarrays are not exactly the same, distances between atoms are conserved so the systems decay identically. Angles below 60◦will not be taken into account as interatomic distances start to be below the lattice constant which means that the lattice constant is no longer the shortest interatomic distance. We will only consider perpendicular polarization. As decay rates are similar to these of square arrays, there is little to directly extract from them. However, the second order correlation function does provide information about how the system changes with the angle. For distances lower than the transition wavelength, g(2)(0) has two revivals corresponding to spatial periodicities at distances d/λ = 0.5 and d/λ = 0.707. As seen in Fig. 6.6, when the angle diminishes the second revival gets divided in two, one moving to lower distances and the other one to higher distances. This second revival when d/λ0=√2/2, is the one corresponding to periodicity in the main diagonal direction. As the array is square, both diagonals are equivalent and overlap. However, as the angle diminishes, the two diagonal directions start to have different interatomic distances. One increases and the other decreases in distance dividing the original revival in two. Figure 6.6: Behavior of the second order correlation function with distance for different arrangenet angles α(in radiants). (a) The general behavior is almost identical for all α. The zoom in (b) shows that the position and number of revivals changes depending on the array. As we approach to α= 60◦, things change. While the higher of these two revivals disappears as the corresponding distance increases, the lower one overlaps the one at d/λ0= 0.5 creating a larger bump. This happens because when 46 α= 60◦, the array is highly symmetric and one of the diagonals has distance d, as the array is a tessellation of equilateral triangles that gives the array an hexagonal shape. This huge overlap of spatial periodicity increases the enhancement of photon emission with respect to the square case. The high symmetry accomplished in this geometric configuration permits the emission of a photon from a decaying atom to enhance other atoms in six periodic spatial directions in contrast with four in the case of the squared array. For higher distances this repeats at distances multiples of the first ones. This produces the second order correlation function for the hexagonal case to be almost always above the square case (except for some revivals of the square case). All these changes in g(2)(0) have a direct impact on the critical distance. The split of the revivals means a split on the discontinuities in the burst distance when the angle is less that 90◦. As we see in Fig. 6.7, the original discontinuity is split into two discontinuities that separate from each other as the angle decreases. The first one decreases until it meets with the one at d/λ0= 0.5, creating a larger combined discontinuity produced by the big revival we see in the second order correlation function for α= 60◦. As explained before, high symmetry plays an important role in this case, resulting in the highest critical distance for all atom numbers. Figure 6.7: Critical distance for different 2D arrays as a function of the number of atoms. The hexagonal array dominates over the others. The number of discontinuities change depending on the angle. To explore how the hexagonal and the square situations perform better, we fix the number of atoms and compute the critical distance, as shown in Fig. 6.8. As seen before, the best scenario is the hexagonal case with the square one not 47 so far away from it. However, what makes the square case practically better is that it is more robust; a little variation in the angle barely changes the critical distance or the response of the array, but in the hexagonal case a little variation on the angle (due to potential disorder in the experiment) completely breaks symmetry and causes the critical distance to quickly decrease. 1.1 1.2 1.3 1.4 1.5 1.6 (rad) 0.77 0.775 0.78 0.785 0.79 0.795 0.8 0.805 0.81 0.815 0.82 d/ 0 Angular burst limit with N=2500 60º 90º Figure 6.8: Critical burst distance with respect to the angle for an angular 2D array of N= 2500 atoms (50 array). The plot shows the zoom of the part beyond 60◦to 90◦, where we can see that the configuration that gives a greater distance is the hexagonal. Also, the quick drop seen in the middle of Fig. 6.8 corresponds to the absence of a discontinuity. This plot is a single realization of Fig. 6.7 with all the possible angles, which means that is a vertical cut on the graph for a given N. When we start from the square case and we diminish the angle we have three discontinuities until the point where the third discontinuity only occurs at higher N. This sudden drop in the critical distance is at the angle at which the third discontinuity ceases to occur for this array with Natoms. 48 Chapter 7 Three dimensional arrays Three dimensional arrays are the logical continuation after looking at 1D and 2D arrays. As 2D happens to be a fundamental advance in the presence of superradiance with respect to 1D, we might expect that 3D gives even better results. We will only consider finite arrays. Infinite arrays are physically meaningless as, if the array occupies all space. there is no “reservoir” onto which the photons can escape. We consider a periodic cubic 3D array with interatomic distance d(see Fig. 7.1). As in the cases shown before, the array is initially fully excited, with twolevel atoms whose transition has frequency ω0, with corresponding wavelength λ0and wavevector k0. We will consider only one polarization as the 3D array is isotropic, so any direction of polarization is equivalent. Figure 7.1: General picture of a cubic 3D array. The array is set to be in free space with interatomic distance din all directions. In this scenario, to efficiently count all possible decay rates Γij, we observe the following. Any interaction between atoms (mi, ni, li) and (mj, nj, lj), where 49 (mi, ni, li) indicates the coordinates of the atom in the array, is equivalent to the interaction between atoms (m1, n1, l1) and (m|j−i|+1, n|j−i|+1, l|j−i|+1), as the Green’s tensor in (2.21) is invariant to translations. Therefore we can count all rates just by looking at the corners. Noting these terms as am,n,l, with m=|mj−mi|,n=|nj−ni|and l=|lj−li|, they can be calculated as am,n,l = 2 3 √N−m3 √N−n3 √N−lΓij Γ02 ,(7.1) where the multiplying factor 3 √N−m3 √N−n3 √N−laccounts for all the possible translations and the 2 accounts for the symmetry Γij = Γji. This terms are those present in Eq. (4.7) such that nβ= 2 3 √N−m3 √N−n3 √N−l.(7.2) Note that only one diagonal direction is taken into account. In cubic arrays, we have 6 face diagonal directions, two for each x-y,y-zand z-xplane. So when we are counting on face diagonal directions we need to multiply them by 2. Also, a cube has 4 possible interior diagonal directions. Therefore, when counting these terms we have to multiply them by 4. This means that in 3D arrays we account for all the decay rates of the system (N 2interactions) by just calculating a few and counting on repetitions (Ninteractions). This is a key factor for speeding up calculations. Decay rates in 3D arrays are similar to 2D arrays. As there is one more dimension, emitted photons are more likely to interact with other atoms. In 0 0.5 1 1.5 2 2.5 3 d/ 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 /0 Decay rates 0.4 0.6 0.8 1 1.2 d/ 0 0.9995 1 1.0005 1.001 1.0015 1.002 1.0025 1.003 1.0035 1.004 g(2)(0) Second order correlation function (a) (b) Figure 7.2: Decay characterization of a 3D cubic array. (a) Shows the behavior of the decay rates with distance in a N= 27 atomical array (3 ×3×3 array). (b) Shows the second order correlation function with respect to distance for a 3D cubic array of N= 103atoms (10 ×10 ×10 array). 50 Fig. 7.2 we can see that decay rates are even more spread than the 2D case but still converge to one in the limit d→ ∞. Revivals are more frequent, both in the decay rates and in the second order correlation function. In 3D we have all the spatial periodicities we have in 2D plus the ones corresponding to cubic diagonals. Therefore, we have also revivals at distances √3/2λ0,√6/2λ0,etc, and all its multiples. As in other dimensions, the revivals get more important when we increase the number of atoms. As they are due to far field interactions, they dominate in large arrays. In general terms, the importance of these revivals is similar to 2D arrays: they increase the critical distance when Nis large enough. This is because as these far field interactions have an emission with a dipole spatial pattern, only two dimensions are effective as the other one gets suppressed by the polarization direction. That means that the photon enhancement is similar in the 2D perpendicular array to that in 3D arrays. The second order correlation function is above one for much longer distances than in 2D arrays, as shown in Fig. 7.3. It takes less atoms to obtain a critical distance over the wavelength. Equivalently, with the same number of atoms, creating a 3D array makes superradiance survive for longer interatomic distances than in 2D arrays. As in 2D arrays, the critical distance seems to keep increasing without a threshold as long as we increase the number of atoms. 101102103104105 N 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 d/ 0 Figure 7.3: Behavior of the critical distance dcritical with the atom number Nfor 3D squared arrays. The plot shows a divergence when the array approaches the infinite limit, with a critical distance over 2λ0for less than N= 106atoms. The whole region above the points sets an array in which a superradiant burst does not occur while below the points this burst could be possible but it is not guaranteed. 51 Chapter 8 Conclusions Motivated by the increasing interest in many-body phisics in light-matter interaction settings, this work is a theoretical investigation of superradiance in atomic arrays. Dealing with initially fully excited arrays of two-level atoms distributed in 1D, 2D and 3D systems, we give a full description of how geometry and dimensionality impact superradiance. A semi-analytical condition for a superradiant burst to occur makes it possible to study large systems, simply by the use of statistics of the first two emitted photons. Numerical calculations of the second order correlation function makes it possible to establish if a given finite system will produce a superradiant burst in its decay process or not. A theoretical deduction for infinite arrays helps to see what is the behavior of array systems with a high number of atoms. Superradiance in arrays occurs when atoms decay collectively with an enhanced emission that is externally seen as a burst in radiated intensity. Although the size of the burst is reduced when the interatomic distance is increased, we show that the burst survives nevertheless. For 1D arrays, a burst can be seen up to an interatomic distance of d= 0.3λ0. For 2D and 3D arrays, the critical distance scales with system size and does not saturate. For N∼106atoms, superradiance occurs for d>λ0. Recent progress in experimental treatment of atoms show that atomic arrays can be created in a lab with interatomic distance on the order of λ0[12, 39]. There are also theoretical proposals for bosonic strontium that enable distances as small as d=λ0/16.3 [17, 40]. This makes the experimental confirmation of our predictions feasible within a short time-scale. The most significant impact on superradiance in arrays comes from dimensionality. This makes the change between 1D and 2D really significant, even to a fundamental level as superradiance for 1D arrays has a maximum distance, while in 2D there is no such threshold. Another important factor is polarization, or more specifically, quantization axis. Due to the periodic structure of 52 arrays, long range interactions are crucial for superradiance. Due to the dipolar spatial profile, photon enhancement is highly suppressed in the direction of polarization. Furthermore, geometry also impacts the critical distance. Spatial periodicity patterns with the lowest period are the ones that characterize the superradiant burst. This means that array geometries that minimize this distance are more likely to experience superradiance. That is why, theoretically, the best configuration in 2D is the hexagonal, as is the periodic structure that minimizes distances between atoms. This work constitutes a significant advance in the understanding of superradiance and will enable future experimental realizations and applications of superradiance in atomic arrays. Our approach of looking at photon statistics and highly reducing the complexity of an exponential growing problem could be applied to other many-body problems in quantum optics. 53