Thermalization of an isotropic quantum harmonic oscillator

Trabajo Fin de Máster Máster en Física Thermalization of an Isotropic Quantum Harmonic Oscillator Autor: María del Pilar Morales Rodríguez Tutores: Luis Miguel Nieto Calzada Blas Manuel Rodríguez Lara Abstract This work describes the most important states used in the field of quantum information: coherent states. We start with the definition of a Hamiltonian for a two-dimesional harmonic oscillator from which we construct the eigenstates called Hermite-Gauss states and Laguerre-Gauss states, which allow to describe the modes of a beam system. We continue describing the main coherent and thermal states that can be defined from the same Hamiltonian, this time they are visualized as eigenstates of the SU(1,1) group generators and then for SU(2). Subsequently, a theoretical system involving the input of a thermal state and an empty state in a beam splitter is discussed and experimentally compared. Resumen El presente trabajo cubre los estados m´as destacados que se utilizan en el ´ambito de la informaci´on cu´antica:los estados coherentes. Se inicia definiendo un hamiltoniano para un oscilador arm´onico bidimensional del cual se construyen los eigenestados llamados estados de Hermite- Gauss yestados de Laguerre-Gauss, los cuales permiten describir los modos de un sistema de haces. Continuamos describiendo los principales estados coherentes y termicos que se pueden definir a partir del mismo hamiltoniano, esta vez se visualizan como eigenestados de los generadores del grupo SU(1,1) y luego para SU(2). Posteriormente se aborda un sistema te´orico que implica el ingreso de un estado t´ermico y un estado vac´ıo en un divisor de haces el cual se logra comparar experimentalmente. Contents 1 Introduction 1 2 Two Dimensional Isotropic Harmonic Oscillator 5 2.1 Eigenvalues ....................................... 6 2.2 Hermite-Gaussstates.................................. 7 2.3 Angularmomentum .................................. 9 2.3.1 Right and left circular quanta . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.2 Eigenvectors .................................. 9 2.3.3 Eigenvalues................................... 9 2.4 Laguerre-Gaussstates ................................. 11 3 States Associated with the Group SU(1,1) 13 3.1 Radialrepresentation.................................. 13 3.2 Gilmore-Perelomov coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Barut-Girardello coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Thermalstates ..................................... 17 4 States Associated with the Group SU(2) 19 4.1 Radialrepresentation.................................. 19 4.2 Gilmore-Perelomov coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Barut-Girardello coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4 Thermalstates ..................................... 21 5 An Experiment with Thermal States 23 5.1 BeamSplitter ..................................... 23 5.2 Thermalstates ..................................... 24 5.3 Beam Splitter in a thermal state . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6 Conclusions and perspectives 29 v List of Figures 2.1 Intensity distribution for Hermite-Gauss modes, |ψn1,n2(q1, q2)|2. The first number corresponds to n1and the second to n2, so the mode shown in n1= 0, n2= 0 is the fundamental mode. High-order modes, had a distribution most spread out radially than the fundamental mode. ................................. 8 2.2 Intensity distribution for Laguerre-Gauss modes, |ψp,ℓ(r, ϕ)|2. The first number corresponds to the radial number pand the second to the azimuthal number ℓ, so the mode shown in p= 0, ℓ= 0 is a Gaussian mode. ........................ 12 3.1 Probability distribution in the constant azimuthal number basis {|k;m⟩} for a Gilmore- Perelomov coherent states |k;ζ⟩for Bargmann parameter a) k=1 2and b) k= 4. In c) the probability density function, |⟨r, ϕ|k;ζ⟩|2, of the wave function in dimensionless configuration space for different coherent phase values θ= 0, π/2, π, for Bargmann parameter k=1 2,3 2,4. ........................................ 15 3.2 Phase distribution for a Gilmore-Perelomov coherent states, arg(⟨r, ϕ|k;ζ, θ⟩), in dimensionless configuration space for different coherent phase values θ= 0, π/2, π, for Bargmann parameter k=1 2,3 2,4. ............................ 16 3.3 Probability distribution in the constant azimuthal number basis, {|k;m⟩} for a Barut- Girardello coherent states, |k;z⟩for Bargmann parameter a) k= 1/2 and b) k= 4. In c) probability density function, |⟨r, ϕ|k;z⟩|2, of the wave function in dimensionless configuration space for different coherent phase values θ= 0, π/2, π, for Bargmann parameter k= 1/2,3/2,4. ..................................... 17 3.4 Phase distribution for a Barut-Girardello coherent states, arg(⟨r, ϕ|k;z, θ⟩), in dimensionless configuration space for different coherent phase values θ= 0, π/2, π, for Bargmann parameter k= 1/2,3/2,4. ................................ 18 4.1 Probability distribution in the constant excitation basis, {|j;m⟩}, for Gilmore-Perelomov coherent states, |j;ξ⟩, for the parameter (a) j= 1/2 and (b) j= 4. In c) probability density function, arg(⟨r, ϕ|j;ξ, θ⟩), in dimensionless configuration space for different coherent phase values θ= 0, π/2, π for j= 1/2,3/2,4. ...................... 21 4.2 Phase distribution for a Gilmore-Perelomov coherent states, arg(⟨r, ϕ|j;ξ, θ⟩), in dimensionless configuration space for different coherent phase values θ= 0, π/2, π, for j= 1/2,3/2,4. ..................................... 21 5.1 Experimental setup where a light is input from a thermal source, and an empty state is input from the other input, resulting in two output beams. ............... 25 5.2 Theoretical probability distribution (left) and experimental probability distribution (right) for a 50 : 50 symmetric beam splitter. .......................... 28 5.3 Theoretical probability distribution for a 35 : 65 symmetric beam splitter. ........ 28 vii Chapter 1 Introduction Quantum mechanics has taken a new approach in recent years, showing its potential to extend into domains traditionally governed by classical systems [1]. Quantum computation, quantum communication, and quantum information are some of the new areas of growing interest, where the former mainly seeks to overcome the limitations of classical computers by designing more efficient algorithms [2, 3], communication seeks to transmit qubits between remote locations, and quantum information perfects the speed of transmission and information processing [1,2,4]. These three areas are interrelated, and many of their advances, have been achieved through the application of quantum optics, given its focus on how to achieve control, manipulation and measurement of the properties of light [5,6]. In this work, we will focus on examining one of the essential aspects of quantum optics and quantum information, specifically the quantum states from the perspective of the harmonic oscillator. This decision is due to the importance of the quantum harmonic oscillator in numerous areas, including quantum information processing [6–9]. The quantization of the electromagnetic field can be analogously analyzed as a many-particle system description [10]. Systems where there are well-defined numbers of photons for each mode of the field are described by the called Fock states or number states [11]. Since the quantization of the electromagnetic field in terms of ladder operators can be described as the Hamiltonian harmonic oscillator, Fock states can be obtained by repeated application of the creation operator on the ground state, which allow determines the energy [10–12]. While the Fock state possesses well-defined energy, it lacks a well-defined electric field; consequently, the expectation value of the field operator for a Fock state is zero [12]. Mathematically these are essentially eigenstates of the number operator, indicated its eigenvalue in number of photons excited in the calculated mode [11]. The interest in these states resides in the fact that they represent the fundamental states of the quantum theory of light and constitute a complete set of one-mode states that are easy to manipulate. However, it’s important to note that their experimental implementation is not straightforward [12]. Moreover, one of the most extensively researched states is the coherent state [10]. Coherent states can be expressed as a superposition of Fock states [11], with the property that a coherent state can repeatedly absorb photons from an electromagnetic field without changing in any way [13,14]. The coherent states, which are the closest equivalents to classical fields in the classical limit, were initially discovered by Schr¨odinger and constructed by Glauber while studying electromagnetic correlation functions [6,15]. These states can be represented as eigenstates of the annihilation operators [11,13]. The construction of coherent states can be achieved through two equivalent definitions [16,17], one of which involves identifying them as eigenstates of the algebra’s annihilation operator, called Barut-Girardello coherent state. The coherent states, defined by Gilmore-Perelomov coherent states, involve using a displacement operator on the system’s empty state [14], in addition to coherent Perelomov states, squeezed states are obtained that correspond to states of minimum uncertainty [37,38]. Coherent states are significant as a laser’s output mimics them, and a single-mode laser that operates above its threshold 1 CHAPTER 2. TWO DIMENSIONAL ISOTROPIC HARMONIC OSCILLATOR where Hn(q) are the Hermite polynomials, Hn(q) = (−1)neq2dn d(q)ne−q2,(2.23) and, q−d dqn exp−q2 2= exp−q2 2Hn(q).(2.24) So the states ψn, defined in eq.(2.19), can be written in terms of the Hermite polynomials, ψn(q) = 1 4 √π 1 √2nn!exp−q2 2Hn(q),(2.25) the functions ψ2n(q) are even (i.e. ψ2n(−q) = ψ2n) and ψ2n+1(q)are odd (i.e., ψ2n+1(−q) = −ψ2n+1(q)) so, the Hermite polynomials H2nare even and H2n+1 are odd. The meaning of this is the wave functions of even one-dimensional potentials have definite parity [42]. So back to our two dimensional problem that the state eq.(2.15), can be written in terms of Hermite polynomials, ψn1;n2(q1, q2) = 1 √π2n1+n2n1!n2!exp−q2 1+q2 2 2Hn1(q1)Hn2(q2),(2.26) Figure 2.1 shows the intensity distribution, |ψn1,n2(q1, q2)|2, for some Hermite-Gauss modes in dimensionless canonical phase space, (q1, q2), for transversal excitation numbers n1;n2= 0,1,2,3. For highe-oders modes we can note that if the value of n1remains invariant, the increase of n2will be seen as an upward separation of the mode and otherwise if the n2axis remains invariant, the mode will extend in space to the right. The mean value of the transversal excitation numbers is associate with the number of modes in the corresponding direction [44,45]. Solutions of in terms of Hermite polynomials aren’t unique, it’s possible to construct alternate Figure 2.1: Intensity distribution for Hermite-Gauss modes, |ψn1,n2(q1, q2)|2. The first number corresponds to n1and the second to n2, so the mode shown in n1= 0, n2= 0 is the fundamental mode. High-order modes, had a distribution most spread out radially than the fundamental mode. solutions due to the isotropic dimensionless Hamiltonian of the harmonic oscillator, guarantee the invariance of any of the axes compared to infinitesimal rotations generated by the angular momentum operator, which will be conserved [13]. So in the next section we’ll focus in obtain circular quanta operators. 8 2.3. ANGULAR MOMENTUM 2.3 Angular momentum To describe the system more broadly it is sometimes necessary to expand the problem about its behavior on other physical elements that it affects, in this case the angular momentum. Let’s start by considering the ˆ L3component of angular momentum defined as ˆ L3= ˆq1ˆp2−ˆq2ˆp1=iℏˆa1ˆa† 2−ˆa† 1ˆa2.(2.27) Using the Hamiltonian seen in eq.(2.5), it is easy see hˆ H, ˆ L3i= 0, this proves that both operators have a basis of eigenvectors in common. So it is possible to write new operators as a combination of the ladder operators ˆa1,ˆa2(ˆa† 1,ˆa† 2), where these operators will act as azimutal operators. 2.3.1 Right and left circular quanta We define new ladder operators, defined as, ˆa±=1 √2(ˆa1∓iˆa2),ˆa† ±=1 √2ˆa† 1±iˆa† 2.(2.28) The operators ˆa±are commonly referred to as destruction operators right (−) and left (+) “circular quanta”,, where ˆa† ±the corresponding creation operators [43]. Knowing the action of the ˆaj,j= 1,2 operators and their respective adjoints operators for a state |ψn1;n2⟩. It is expected the action of the operators ˆa±on a state |ψn1;n2⟩, generates a combination of state |ψn1−1,n2⟩, with |ψn1;n2−1⟩. Thus ˆa±and ˆa† ±are analogous to the ladder operators ˆaj, ˆa† j,j= 1,2. Being similar in their commutation properties. hˆa+,ˆa† +i=hˆa−,ˆa† −i= 1.(2.29) 2.3.2 Eigenvectors To obtain the eigenvectors of the harmonic oscillator in this new base, consider the raising operators ˆa† +, ˆa† −on the state |0; 0⟩, which will generate a new state, denoted by |n+;n−⟩, such that |n+;n−⟩=1 p(n+)! (n−)! ˆa† +n+ˆa† −n− |n1= 0; n2= 0⟩.(2.30) Because of the way ladder operators are defined, the state |n+;n−⟩, will be a composition of the states generated by |n1= 0; n2= 0⟩. More generally, since we have two options, n+> n−and n−> n+. If consider (ˆa† +)n+(ˆa† +)−n+=I, eq.( 2.30), could rewrite, |n+;n−⟩=1 p(n+)! (n−)!     ˆa† +ˆa† −n−ˆa† +(n+−n−)|n1= 0; n2= 0⟩, n+> n− ˆa† +ˆa† −n+ˆa† −(n−−n+)|n1= 0; n2= 0⟩, n−> n+ (2.31) 2.3.3 Eigenvalues We are now interested in how n+and n−are related to the eigenvalues of the angular momentum operator and those of the harmonic oscillator Hamiltonian, so it is convenient to define new numbers operators, ˆ N+= ˆa† +ˆa+=1 2ˆa† 1ˆa1+ ˆa† 2ˆa2−iˆa† 1ˆa2+iˆa1ˆa† 2,(2.32a) ˆ N−= ˆa† −ˆa−=1 2ˆa† 1ˆa1+ ˆa† 2ˆa2+iˆa† 1ˆa2−iˆa1ˆa† 2(2.32b) 9 CHAPTER 2. TWO DIMENSIONAL ISOTROPIC HARMONIC OSCILLATOR The eq.(2.5) and eq.(2.27) can be written as, ˆ L3=ℏˆ N+−ˆ N−,(2.33) ˆ H=ℏωˆ N++ˆ N−+ 1.(2.34) Respectively. Operating ˆ L3and ˆ Hon |n+;n−⟩, ˆ L3|n+;n−⟩=ℏ(n+−n−)|n+;n−⟩,(2.35a) ˆ H|n+;n−⟩=ℏω(n++n−+ 1) |n+;n−⟩.(2.35b) Where the eigenvalue of ˆ His similar to that found in the previous section (eq.(2.9)). For the case of eigenvalues of ˆ L3, it is convenient to definite, ℓ=n+−n−,(2.36) to get ˆ L3|n+;n−⟩=ℏℓ|n+;n−⟩.(2.37) Here we note that ˆa† +increases the angular momentum by a factor ℏ, ˆ L3ˆa† +|n+;n−⟩=ℏ[(n++ 1) −n−)] |n++ 1; n−⟩,(2.38) Doing the same for ˆa† −we find that the angular momentum increases by a factor −ℏ With this we have, as expected, that the values of ℓ= 0,±1,±2,±3, . . . . However, it should not be forgotten that Hand Lhave the same base in common, therefore it is convenient to see how the values n±are, with respect to the system, we can do n=n++n−, resulting in ˆ H|n+;n−⟩=ℏω(n+ 1) |n+;n−⟩,(2.39) Since the values of n±must satisfy the form of the eigenvalues of ˆ H, we have Dˆ N++ˆ N−E=n++n−=n, (2.40) where ncan be any positive integer, but composed of any possible combination of n+and n−, that is, n+could be zero and n−could be n, or perhaps n+=n−2 while n−= 2. The only condition is that n±be positive integers or zero and their sum results in a number n. For the case of ˆ L3, for an energy level (n+ 1)ℏωand knowing that ℓ=n+−n−, the possible values of lare ℓ=n , n −2, n −4,...,−n+ 2,−n. (2.41) A way of to relate the possible values of land the values of n±can be given from its construction, If So Dˆa† +ˆa+E=n+=j;Dˆa† −ˆa−E=n−=n−j, (2.42) with j= 0,1, . . . , n. Being the angular component Dˆ L3E=ℏ(n+−n−) = ℏ(n−2j),(2.43) Likewise the states |n+=n;n−= 0⟩,|n+= 0; n−=n⟩, correspond to the maximum and minimum value for ˆ L3, which are visualized as the vectors of circular polarization (circular movements to the right or to the left) of the classical field associated with a given value of the total energy [43] Since circular motions are described, it is convenient to change our system to polar coordinates. 10 2.4. LAGUERRE-GAUSS STATES 2.4 Laguerre-Gauss states It is convenient to study the harmonic oscillator operators defined in eq.(2.28) in polar coordinates: q1=rcos(ϕ), r ≥0,(2.44) q2=rsin(ϕ),0≤ϕ < 2π, (2.45) it implies that, ψn1, n2(q1, q2)−→ ψn1;n2(r, ϕ) = e−r2 2Hn1(rcos(ϕ)) Hn2(rsin(ϕ)) √π√n1!n2!2n1+n2,(2.46) and if we choose ⟨q(r, ϕ)|n+;n−⟩=χn+,n−(r, ϕ),(2.47) we have a new set of states, which are related to angular momentum [41,46] χn+,n−(r, ϕ) = (−1)n−n−! √πpn−!(ℓ+n−)! rℓexp −1 2(r)2Lℓ n−r2e(iℓϕ)(2.48) Here Lℓ n−is the generalized Laguerre polynomial of order n−and degree ℓ, where ℓ= n+−n−and ℓ=n , n −2, n −4,...,−n+ 2,−n, with nthe excited level. These states have the form of the Laguerre–Gaussian modes [47], and allow to study the structure of the system which consists of pdark concentric rings for a given azimuthal number ℓ[46]. If we define the azimuthal mode index ℓand the radial mode index p[9], p=n−, ℓ=n+−n−,(2.49) we may rewrite eq.(2.31) as |p;ℓ⟩=1 p(p)! (p+|ℓ|)! ˆa† +ˆa† −p    ˆa† +|ℓ||0; 0⟩, ℓ > 0, ˆa† −|ℓ||0; 0⟩, ℓ < 0. (2.50) by projecting over the complete basis |r;ϕ⟩we get the corresponding wave function ψp,ℓ(r, ϕ) in the transverse (dimensionless) parameters. ψp,ℓ(r, ϕ) = ⟨r;ϕ|p;ℓ⟩, =1 √π(−1)psp! (p+|ℓ|)! r|ℓ|e−1 2r2L|ℓ| p(r2)eiℓϕ,(2.51) in terms of the associated Laguerre polynomials L|ℓ| p(ρ2) [46]. Figure 2.2 shows the intensity distribution, |ψp,ℓ(r, ϕ)|2, for some Laguerre-Gauss modes, for radial and azimuthal numbers p, ℓ = 0,1,2,3. For highe-oders modes we can note that for p= 0, (first column) with ℓ= 1,2,3 exists only a single-ring mode called doughnut modes, for p= 1 and ℓ= 1,2,3 we observe two rings so in general for ℓ= 0, the intensity pattern of LG beams had p+ 1 concentric rings, if ℓ= 0 there are prings around the center Gaussian mode, then ℓ determines its size [23,48]. Now operating ˆ L3and ˆ Hon |p;ℓ⟩, ˆ L3|p;ℓ⟩=ℏℓ|p;ℓ⟩,(2.52a) ˆ H|p;ℓ⟩=ℏω(2p+ℓ+ 1) |p;ℓ⟩,(2.52b) 11 CHAPTER 2. TWO DIMENSIONAL ISOTROPIC HARMONIC OSCILLATOR Figure 2.2: Intensity distribution for Laguerre-Gauss modes, |ψp,ℓ(r, ϕ)|2. The first number corresponds to the radial number pand the second to the azimuthal number ℓ, so the mode shown in p= 0, ℓ= 0 is a Gaussian mode. from ˆ Hwe obtain the relation n=n1+n2= 2p+|ℓ|,(2.53) the absolute value in ℓindicate that nis always a positive number. Some authors work with the so-called radial number operator ⟨ˆp⟩=p= 0,1,2, . . . [9, 46]. Both representations, with the same Hilbert space bases (H=H1× H2, spanned by the eigenstates |n1;n2⟩, ) are in essence equivalent and equally valid due to there is no structurally determined symmetry in free space. So in rectangular coordinates the mode fields are described by a set of Hermite-Gaussian functions resulting in Hermite-Gaussian modes being appropriate for treat problems with square symmetry, whereas in polar coordinates they are described by Laguerre-Gaussian functions, so the Laguerre-Gauss modes are appropriate to treat problems with axial symmetry [44, 45, 49]. In quantum physics and even more in quantum optics there are two families of states that are of great interest, the pure states, which are those that are completely coherent and the mixed states that are a superposition of the coherent states, both analyzed theoretical and experimental [50]. So far, we have approached the eigenstates of the harmonic oscillator from the perspective of the creation and destruction operators and their representation in the linear or polarized basis, however, using the harmonic oscillator we can study the coherent states, which are the most classical of the states of the oscillator, these means constitutes a state of minimum uncertainty. However, coherent states may be constructed for an angular momentum system with some differences of the oscillator coherent states [51]. And consider that the Laguerre- Gauss states cover the whole Hilbert space, where the transverse momentum and the orbital angular momentum are conserved. This Hilbert space can be partitioned and each partition can be represented in terms of the Lie group generators SU(1,1) and SU(2) [51, 52], different representations of the system can be studied in coherent and mixed states. 12 Chapter 3 States Associated with the Group SU(1,1) The circular quanta right and left creation and annihilation operators given in (2.28) allow the construction of a dimensionless representation of the algebra su(1,1) ˆ K+= ˆa† +ˆa† −,ˆ K−= ˆa+ˆa−,ˆ K0=1 2ˆa† +ˆa++ ˆa† −ˆa−+ 1,(3.1) where ˆ K0,ˆ K+yˆ K−are the generators of this algebra, being ˆ K±ladder operators, and ˆ K2=1 4ˆa† +ˆa+−ˆa† −ˆa−2−1(3.2) the Casimir operator. They satisfy the commutation relations, hˆ K+,ˆ K−i=−2ˆ K0,hˆ K0,ˆ K±i=±ˆ K±,hˆ K2,ˆ Kji= 0.(3.3) The corresponding group SU(1,1) is the simplest non-abelian (i.e. non-commutative) group, so the Lie algebra corresponding to this group considers different representations of irreducible, discrete, continuous and supplementary unitary series. However, the discrete representations of this algebra are used for the description of physical systems [37,38,53], where the standard orthonormal basis for this representation is ˆ K+|k;m⟩=p(2k+m)(m+ 1) |k;m+ 1⟩,(3.4) ˆ K−|k;m⟩=p(2k+m−1)m|k;m−1⟩,(3.5) ˆ K0|k;m⟩= (k+m)|k;m⟩,(3.6) ˆ K2|k;m⟩=k(k−1) |k;m⟩,(3.7) where k > 0 is the so-called Bargmann parameter. Knowing this, we can now study some different representations generated by this Lie algebra. 3.1 Radial representation Using the standard representation (3.4)–(3.7) |k;m⟩=k(k−1) |k;m⟩, to obtain the eigenvalue of the Casimir operator, and applying ˆ K2=1 4(ˆ N+−ˆ N−)2−1to the state |n+;n−⟩, we get ˆ K2|n+;n−⟩=1 4h(n+−n−)2−1i|n+;n−⟩=1 4ℓ2−1|n+;n−⟩,(3.8) 13 CHAPTER 3. STATES ASSOCIATED WITH THE GROUP SU(1,1) Comparing (3.7) and (3.8), we have k2−k−1 4ℓ2+ 1= 0,and it is easy to get k=1 2(|ℓ|+ 1).(3.9) Similarly, for the rest of the operators. It’s convenient to define new states in terms of pand k given in (2.49), such that |n+;n−⟩→|k;p⟩.(3.10) Then ˆ K2|k;p⟩=k(k−1) |k;p⟩,(3.11) ˆ K0|k;p⟩= (k+p)|k;p⟩,(3.12) ˆ K+|k;p⟩=p(2k+p)(p+ 1) |k;p+ 1⟩,(3.13) ˆ K−|k;p⟩=p(2k+p−1)p|k;p−1⟩.(3.14) The raising and lowering operators ˆ K±acting on a Laguerre-Gauss mode |p, ℓ⟩increase or decrease the radial number p, however don’t change the value of ℓ, this implies that, they conserve the Bargmann parameter for any value of p, with p≥0, [9,46]. The Bargmann parameter k= (|ℓ|+ 1) /2=1/2,1,3/2 depends only on the azimuthal number operator, implies there will be two subspaces labeled with the same Bargmann parameter, one for each sign of the mean value of the azimuthal number except for zero. This allows us to define a relation between the Bargmann subspace basis and the Laguerre-Gauss states: ⟨r;ϕ|k;p⟩≡⟨r, ϕ||ℓ|= 2k−1; p⟩=ψ2k−1, p(r, ϕ),(3.15) with ⟨r;ϕ|k;p⟩defined in eq(2.51). The ladder operators ˆ K±acting on |k;p⟩increase or decrease the radial number p. Thus, each infinite dimensional Bargmann subspace consists of states with equal azimuthal excitation number and progressive radial excitation number. If now the group generators are used to define a unitary operator, acting on the lowest state |k;m= 0⟩, we obtain the Gilmore-Peremolov coherent states. 3.2 Gilmore-Perelomov coherent states The Gilmore-Perelomov coherent states for the su(1,1) algebra are defined [37], |k;ζ⟩= eξˆ K+−ξ∗ˆ K−|k; 0⟩= (1 −|ζ|2)keξˆ K+|k; 0⟩= (1 −|ζ|2)k ∞ X p=0 sΓ(p+ 2k) p! Γ(2k)ζp|k;p⟩,(3.16) where ξ=1 2θe−iω,ζ= ( ξ |ξ|) tanh |ξ|=−tanh1 2θe−iω. With ω∈[0,2π), −∞ < θ < ∞, and the parameter ζis restricted by |ζ|<1, this condition shows that these states are defined in the interior of a unit disk [37,38]. Being the states |k;ζ⟩normalized but not orthogonal: ⟨k;ζ1|k;ζ2⟩= (1 −|ζ1|2)k(1 −|ζ2|2)k(1 −|ζ∗ 1ζ2|2)−2k.(3.17) In dimensionless canonical space, ⟨r;ϕ|k;ζ⟩= (1 −|ζ|2)k ∞ X p=0 sΓ(p+ 2k) p!Γ(2k)ζp⟨r;ϕ|k;p⟩,(3.18) =1−|ζ|2k pπ(2k−1)!r2k−1e−1 2r2eiℓϕ ∞ X p=0 (−ζ)pL2k−1 pr2,(3.19) 14 3.2. GILMORE-PERELOMOV COHERENT STATES where we have used the identity (1 −α)m+1 ∞ X n=0 αnLm n(x) = expα α−1x(3.20) and make ψk;ζ=⟨r, ϕ|k;ζ⟩, where ψk,ζ =   Ak,θ ei(2k−1)ϕ, ℓ > 0, Ak,θ, ℓ = 0, Ak,θ e−i(2k−1)ϕ, ℓ < 0, (3.21) being Ak,θ ="1 + tanhθ 2 1−tanhθ 2#k r2k−1s1 π(2k−1) exp(−1 2 1 + tanhθ 2 1−tanhθ 2!r2).(3.22) These states are also eigenstates of the OAM, and present the same characteristics as those introduced in [46]. However, we present their analysis in terms of the Bargmann parameter, given the relation |ℓ|= 2k−1, so they are shape invariant in the time evolution. The average number of sharp rings in the state ψk,ξ is p= 2k|ζ|2 |ζ|2−1.(3.23) Figure 3.1 shows the probability of distribution and probability for (3.21). We can highlight that the densities are equivalent to the intensity distribution for Laguerre-Gauss modes with p= 0, ℓ (see Figure 2.2). This is due to the fact that it is a coherent state that maintains the value of pbut varies ℓby varying the Bargamm’s parameter given its relationship (3.9). Figure 3.1: Probability distribution in the constant azimuthal number basis {|k;m⟩} for a Gilmore- Perelomov coherent states |k;ζ⟩for Bargmann parameter a) k=1 2and b) k= 4. In c) the probability density function, |⟨r, ϕ|k;ζ⟩|2, of the wave function in dimensionless configuration space for different coherent phase values θ= 0, π/2, π, for Bargmann parameter k=1 2,3 2,4. On the other hand, in Figure 3.2 we show the phase distribution of (3.21). Here we can emphasize that the number of vortices is directly proportional to the value of ℓ, in similar way there exists a rotation that depends of the θvalue, which will be cyclic in a period of π. Other states in the su(1,1) representation are the coherent states of Barut and Girardelo, which are constructed as eigenstates of ladder operators representing discrete series [54], as we will show in the following section. 15 CHAPTER 3. STATES ASSOCIATED WITH THE GROUP SU(1,1) Figure 3.2: Phase distribution for a Gilmore-Perelomov coherent states, arg(⟨r, ϕ|k;ζ, θ⟩), in dimensionless configuration space for different coherent phase values θ= 0, π/2, π, for Bargmann parameter k=1 2,3 2,4. 3.3 Barut-Girardello coherent states The Barut-Girardello coherent states for the su(1,1) algebra, [37,54], are such that ˆ K−|k;z⟩=z|k;z⟩,(3.24) with z∈Cand kthe Bargmann parametrer. The expansion of these states on an orthonormal basis is, |k;z⟩=zk−1/2 pI2k−1(2 |z|) ∞ X n=0 zn pn!Γ(2k+n)|k;z⟩,(3.25) with Iν(x), a modified Bessel function of the first kind of ν–order: Iν(z) = ∞ X n=0 1 2zν+2n n!Γ(ν+n+ 1).(3.26) The Barut-Girardello coherent states are normalized but not orthogonal [37]. For real negative coherent parameter, z < 0, the Barut-Girardello coherent state is given by ⟨r, ϕ|k;z⟩=(−1)−k+1/2e−z pπI2k−1(2|z|)e−1 2r2J2k−12√−z rei(2k−1)ϕ,(3.27) and therefore takes a Bessel function form in dimensionless canonical space. Figure 3.3 shows the probability distribution and probability density for eq.(3.27), similar to the Gilmore-Peremolov states presented previously. The density is equivalent to the intensity distribution for Laguerre-Gauss modes, see (2.2), with p= 0, ℓ, but in reverse. Figure 3.4 shows the phase distribution of (3.27), where anew the number of vortices is directly proportional to the value of ℓ. However, remarkable differences are shown at θ=π, the process being cyclic with a period of 2π. Coherent states are highly efficient as they represent the most classical states attainable. Nevertheless, in laboratory settings, mixed states are often created. These states cannot be characterized by state vectors, but instead by a density operator [12]. Given their nature, these are usually found as thermal states, and given their importance for this work, we will analyze their representation in the group basis SU(1,1). 16 3.4. THERMAL STATES Figure 3.3: Probability distribution in the constant azimuthal number basis, {|k;m⟩} for a Barut- Girardello coherent states, |k;z⟩for Bargmann parameter a) k= 1/2 and b) k= 4. In c) probability density function, |⟨r, ϕ|k;z⟩|2, of the wave function in dimensionless configuration space for different coherent phase values θ= 0, π/2, π, for Bargmann parameter k= 1/2,3/2,4. 3.4 Thermal states We now considered the thermal state ρth, with βthe Boltzmann’s constant, Tthe temperature of the system, and ˆ Hthe Hamiltonian [11,12], ρth =e−βˆ H Tr e−βˆ H, β =1 kBT,(3.28) For our case, we take the Hamiltonian related to the harmonic oscillator eq.(2.34) in terms of the su(1,1) algebra operators, ˆ H= 2ℏωˆ K0,(3.29) and has a diagonal form, ˆ H= 2ℏω"∞ X u=1 ∞ X v=0 u 2+v u 2;vEDu 2;v++ ∞ X u=2 ∞ X v=0 u 2+|v| u 2;vEDu 2;v−#,(3.30) in the basis |k;p⟩±, with k=u/2 and v=p, and where we use the positive subscript to refer to zero and positive azimuthal numbers, ℓ≥0, and the negative subscripts to negative azimuthal numbers, ℓ < 0. Thus, the thermal state in this basis is ρth =eℏωβ −12 eℏωβ (∞ X u=1 ∞ X v=0 e−ℏω(u+2v)β u 2;vEDu 2;v++ + ∞ X u=2 ∞ X v=0 e−ℏω(u+2v)β u 2;vEDu 2;v−) =1 n(n+ 1) (∞ X u=1 ∞ X v=0 n (n+ 1)u+2v u 2;vEDu 2;v++ + ∞ X u=2 ∞ X v=0 e−ℏω(u+2v)β u 2;vEDu 2;v−) in terms of the Boltzmann parameter βor the average excitation number [12] e−ℏωβ =n n+ 1.(3.31) 17 CHAPTER 5. AN EXPERIMENT WITH THERMAL STATES agree for coherent and thermal beams (“classical”-like light beams), but with a single or few photons, the classical approach to beam splitting produces erroneous results [12]. Here we will focus on the thermal states affecting a quantum beam splitter. But here we can ask ourselves what we call the thermal state, which is what we will describe in the next section. 5.2 Thermal states A thermal state is a description of a thermal light, this means, a light beam emitted by a thermal sourse (a source in thermal equilibrium) can be described by a density matrix [12,32], ˆρ=X i pi|ψi⟩⟨ψi|,(5.5) where ψiare the states vectors, and piis the probability of the system being in the ith state of the ensamble |ψi⟩, considering the temperature Tof a system described by its Hamiltonian ˆ H, the thermal state can be written ˆρth =e−βˆ H Tr e−βˆ H,(5.6) with β= 1/(kBT) the Boltzmann’s constant and Tr the trace: Tr e−βˆ H= ∞ X n=0 e−Enβ.(5.7) From (5.5) and (5.6), is easy to deduce that pi=⟨ψi|ˆρ|ψi⟩=e−βEi Pie−βEi,(5.8) more in particulary, here piin the probability that the mode is thermally excited in the ith level [12]. If we now consider a two-dimensional isotropic harmonic oscillator in thermal equilibrium, we have, e−βˆ H= ∞ X n1;n2=0 e−ℏω(n1+n2+1)β|n1;n2⟩⟨n1;n2|,(5.9) and Tr e−βˆ H=eℏωβ eℏωβ −1−2.(5.10) Hence, finally we have ˆρth =1−e−ℏωβ2∞ X n1;n2=0 e−ℏω(n1+n2)β|n1;n2⟩⟨n1;n2|.(5.11) If we consider the average number of photons, then n=1 e−ℏωβ−1(5.12) and the thermal state is ˆρth = ∞ X n1;n2=0 nn1+n2 (n+ 1)n1+n2+2 |n1;n2⟩⟨n1;n2|.(5.13) This is how thermal states are described by a density operator, and are considered “classical” beams, since they are indistinguishable mixed states, knowing their probability helps to characterize the system [11, 12]. In the following we will make use of a thermal state for a system described by the Hamiltonian of a harmonic oscillator, which will enter on one of the inputs of a BS, which will allow us to study the correlations for each output arm. 24 5.3. BEAM SPLITTER IN A THERMAL STATE 5.3 Beam Splitter in a thermal state Considerer a general beam splitter with two inputs, 1 and 2, whose unitary transformation is described by ˆ Ubs(ζ) = exphζˆa† 1a2−ζ∗ˆa† 2a1i.(5.14) Now if we introduce a thermal state ρin one of the inputs in the beam splitter and an empty state in the other input, as shown in Figure 5.1, a new thermal state ˆρth−bs will be obtained ˆρbs =ˆ Ubs(ζ)ˆρinput th ˆ U† bs(ζ),(5.15) the normalized thermal state ˆρinput th is, ˆρinput th =1−e−ℏωβ∞ X n=0 e−nℏωβ |0; n⟩N N ⟨0; n|,(5.16) where the subscripts on the bra and ket denote the base in which we are using, in this case N denoted Fock’s basis. Figure 5.1: Experimental setup where a light is input from a thermal source, and an empty state is input from the other input, resulting in two output beams. Considering the Schwinger adimensionless relations, with ladder operators in ‘Cartesian’ coordinates [65], ˆ J+≡ˆa† 1ˆa2,ˆ J−≡ˆa† 2ˆa1,(5.17) it is possible to write the beam splitter operator in terms of the rising and lowering angular momentum operators ˆ Ubs(ζ) = exphζˆ J+−ζ∗ˆ J−i,(5.18) so it is convenient to write ˆρinput th in the base of Schwinger J. We observe that the operators in this representation equation (5.17) are equivalent to those presented in the previous section equation (4.2). Therefore, so we can write j=n1+n2 2, m =n1−n2 2.(5.19) Considering the scenario where n1= 0 and n2=n, where nrepresents the total number of excitations of the system, we can express the states |0; n⟩Nits terms of jand m, as: |n1;n2⟩N|j;m⟩J |0; 0⟩N|0; 0⟩J |0; 1⟩N 1 2;−1 2J |0; 2⟩N|1; −1⟩J . . .. . . 25 CHAPTER 5. AN EXPERIMENT WITH THERMAL STATES The subscript Jin the bras and kets will represent the base of the angular momentum. These relations must satisfy, ˆ J±|j;±j⟩J= 0.(5.20) So rewritting equation (5.16) ˆρinput th =1−e−ℏωβX j=0,1/2,1,... e−2jℏωβ |j;−j⟩J J ⟨j;−j|,(5.21) which can be expressed in terms of nbut in Jbases, this is, ˆρinput th =1−e−ℏωβ∞ X n=0 e−nℏωβ  n 2;−n 2EJ J Dn 2;−n 2,(5.22) Now it is enough to act the unitary operator of the BS to the left ˆ Ubs(ζ)|j;−j⟩J, to later obtain its conjugate this is J⟨j;−j|ˆ U† bs(ζ). To achieve this we will follow a series of steps. First using the disentangling relations, we can rewritten eq.(5.18), like ˆ Ubs(ζ) = exphζˆ J+−ζ∗ˆ J−i= exphA+ˆ J+iexphln(A0)ˆ J0iexphA−ˆ J−i,(5.23) Where A+,A−,A0are coefficients that can be determined by means of the Masashi-Ban formulas described in [66]. Which indicates that the coefficients for the normal order decomposition are A+= e−iϕ tan(θ), A−=−eiϕ tan(θ), A0= cos−2(θ),(5.24) so that ˆ Ubs(θ, ϕ) = exphe−iϕ tan(θ)ˆ J+iexph−lncos2(θ)ˆ J0iexph−eiϕ tan(θ)ˆ J−i.(5.25) Now we can operate on a state |j, −j⟩J, ˆ Ubs(θ, ϕ)|j;−j⟩J= (cos2(θ))jexphe−iϕ tan(θ)ˆ J+i|j;−j⟩J. Let κ= cos(θ) and λ= e−iϕ tan(θ) κ2jexphλˆ J+i|j;−j⟩J=κ2j ∞ X l=0 (λ)l l!ˆ J+|j;−j⟩J(5.26) =κ2j 2j X k=0 (λ)k(2j)! (2j−k)!(k)!1/2 |j;−j+k⟩BS J.(5.27) Finally, with ˆ Ubs(θ, ϕ)≡ˆ Ubs ˆ Ubs |j;−j⟩J=κ2j 2j X k=0 (λ)k2j k1/2 |j;−j+k⟩BS J,(5.28) in terms of n, ˆ Ubs  n 2;−n 2EJ=κn n X k=0 (λ)kn k1/2 n 2;−n 2+kEBS J.(5.29) 26 5.3. BEAM SPLITTER IN A THERMAL STATE Now we have everything necessary to obtain the thermal state ˆρtbs, defined in (5.15): ˆρbs = (1 −γ) ∞ X n=0 γnˆ Ubs  n 2;−n 2EJ J Dn 2;−n 2 ˆ U† bs = (1 −γ) ∞ X n=0 γnκ2n n X k=0 (λ)kn k1/2n X l=0 (λ∗)ln l1/2 n 2;−n 2+kEBS J BS JDn 2;−n 2+l = (1 −γ) ∞ X n=0 γnκ2n n,l X k=0 (λ)k(λ∗)ln kn l1/2 n 2;−n 2+kEBS J BS JDn 2;−n 2+l,(5.30) where γ= e−ℏωβ. Therefore ˆρbs given by eq.(5.30) is the output thermal state of the beam splitter. In the Fock basis, ˆρbs = (1 −γ) ∞ X n=0 γnκ2n n X k,l=0 (λ)k(λ∗)ln kn l1/2 |k;n−k⟩BS N BS N⟨l;n−l|,(5.31) Substituting κand λ, and κ2n n X k,l=0 (λ)k(λ∗)l= [cos(θ)]2n[tan(θ)]k+le−ikϕeilϕ =µk+lκ2n−k−lνk+l.(5.32) where κ= cos(θ), µ= e−iϕ,ν= sin(θ), ˆρbs = (1 −γ) ∞ X n=0 γn n X k,l=0 n kn l1/2 µk−lκ2n−k−lνk+l|k;n−k⟩BS N BS N⟨l;n−l|,(5.33) In terms of the mean number of photons, nor γ= e−ℏωβ =n n+1,we have [12] ˆρbs = ∞ X n=0 nn (n+ 1)n+1 n X k,l=0 n kn l1/2 µk−lκ2n−k−lνk+l|k;n−k⟩BS N BS N⟨l;n−l|,(5.34) With this information we now ask ourselves what is the probability of obtaining a certain state at the output of the beam splitter? To answer this we will make use of an arbitrary state |a, b⟩BS N, so the probability of that state will be, Pa,b =BS N⟨a;b|ˆρbs |a;b⟩BS N(5.35) Then Pa,b = (1 −γ) ∞ X n=0 γn n X k,l=0 n kn l1/2 µk−lκ2n−k−lνk+lBS N⟨a, b||k;n−k⟩BS NBS N⟨l;n−l|a;b⟩BS N, = (1 −γ) ∞ X n=0 γn n X k,l=0 n kn l1/2 µk−lκ2n−k−lνk+lδa,kδb,n−kδa,lδb,n−l = (1 −γ)γa+ba+b aκ2bν2a(5.36) Replacing the terms Pa,b =na+b (n+ 1)a+b+1 a+b a[cos θ]2b[sin θ]2a.(5.37) Figure 5.2 shows the probability distribution for a 50 : 50 symmetric beam splitter making use of (5.37), as well as the experimental probability, in both cases with n≈6 and θ=π/4. 27 CHAPTER 5. AN EXPERIMENT WITH THERMAL STATES Figure 5.2: Theoretical probability distribution (left) and experimental probability distribution (right) for a 50 : 50 symmetric beam splitter. By fitting the experimental data with the theoretical data, we find that the experimental system behaves best as the one corresponding to a 35 : 65 beam splitter, as shown in Figure 5.3. Figure 5.3: Theoretical probability distribution for a 35 : 65 symmetric beam splitter. Given the information provided by other experimental systems of thermal sources [67], the discrepancies observed between the theoretical and experimental data can be attributed to fluctuations, since the grouping effect presented by thermal light increases the probability that the fluctuations exceed the expected average in the number of photons measured in the detectors. 28 Chapter 6 Conclusions and perspectives In this work, from an isotropic harmonic oscillator Hamiltonian in two dimensions, we have obtained Hermite-Gauss and Laguerre-Gauss states. The Hermite-Gauss states are described by transverse excitation numbers and describe the modes of beams that have a square structure, called Hermite-Gauss modes. The intensity distribution shows radial dispersion. Compared to with the fundamental mode, the mean value of the transverse excitation numbers is associated with the number of modes in the corresponding direction. The Laguerre-Gauss modes are described by the radial and azimuthal numbers and describe the modes of beams with radial structure. The intensity distribution is constructed using rings, where the number of rings is related to the radial number and the width of the rings is defined by the azimuthal number. With these results, we use the known Gilmore-Perelomov and Barut-Girardelo coherent states in terms of the radial and azimuthal number. First for the SU(1,1) group representation, we find that it is possible to define new coherent states that have a relation with a Bargmann parameter defined in terms of the azimuthal number. This establishes that there are two subspaces with the same Bargmann parameter. In addition, the probability density has a similar structure to the intensity distribution of the Laguerre-Gauss modes. In the phase structure, the azimuthal number is directly proportional to the number of vortices. For the coherent states in the SU(2) representation, only Gilmore-Perelomov states are found. The spin quantum number jdescribed in terms of the radial and azimuthal numbers can be regarded as equivalent to the Bargann parameter. The phase transitions are proportional to the value of the spin number. The probability density are similar to the intensity distribution of the Hermite-Gauss modes, this implies that the phase structure does not contain any vortices. In addition to coherent states, for those systems that need a study that takes into account their statistical behavior, there are mixed states. Considered here are the thermal states, which can be written in terms of the Boltzmann constant or the average of photons, which have been discussed from the representations of SU(1,1) and SU(2). For the case of SU(1,1) group generators, we can obtain the thermal states, which are written as the sum of the density operator for ℓ≥0 with the density operator corresponding to ℓ < 0. The SU(2) thermal states allow to study theoretically in an efficient way the behavior of introducing a thermal state and a vacuum state into the corresponding inputs of beam splitter, resulting in a new output thermal state. However, when studying the correlations, it is ideal to write the output thermal state in the Fock basis to study the number of excitations of the system. This also provides a simple way to compare the theoretical and experimental data. An analysis of the results shows a discrepancy between the theoretical and experimental data. This can be produced by different factors, such as the sensitivity of the sensors. The fluctuations and the nature of the thermal light, that in the experiment increase the reading for the number of photons, giving an opportunity to extend the theoretical model developed here. 29 CHAPTER 6. CONCLUSIONS AND PERSPECTIVES In conclusion, the coherent states presented here are interesting due to their behavior and the versatility of their parameters to explain the system. The work presented is not complete and we are currently still expanding and developing the topic presented here, together with theoretical and experimental researchers from several universities. It is hoped that this analysis will provide the framework for future studies and allow us to go beyond coherent states and create a more complete analysis of thermal states. 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