Entanglement between atomic thermal states and coherent or squeezed photons in a damping cavity

Accepted Manuscript Entanglement between atomic thermal states and coherent or squeezed photons in a damping cavity F. Yadollahi, R. Safaiee, M.M. Golshan

PII: DOI: Reference:

S0378-4371(17)30939-1 https://doi.org/10.1016/j.physa.2017.09.047 PHYSA 18652

To appear in:

Physica A

Received date : 11 April 2017 Revised date : 27 August 2017 Please cite this article as: F. Yadollahi, R. Safaiee, M.M. Golshan, Entanglement between atomic thermal states and coherent or squeezed photons in a damping cavity, Physica A (2017), https://doi.org/10.1016/j.physa.2017.09.047 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Entanglement between Atomic Thermal States and Coherent or Squeezed Photons in a Damping Cavity F. Yadollahia , R. Safaieeb , M. M. Golshana,∗ a

Physics Department, College of Sciences, Shiraz University, Shiraz 71454, Iran b Faculty of Advanced Technologies, Shiraz University, Shiraz, Iran

Abstract In the present study, the standard Jaynes-Cummings model, in a lossy cavity, is employed to characterize the entanglement between atoms and photons when the former is initially in a thermal state (mixed ensemble) while the latter is described by either coherent or squeezed distributions. The whole system is thus assumed to be in equilibrium with a heat reservoir at a finite temperature T, and the measure of negativity is used to determine the time evolution of atom-photon entanglement. To this end, the master equation for the density matrix, in the secular approximation, is solved and a partial transposition of the result is made. The degree of atom-photon entanglement is then numerically computed, through the negativity, as a function of time and temperature. To justify the behavior of atom-photon entanglement, moreover, we employ the so obtained total density matrix to compute and analyze the time evolution of the initial photonic coherent or squeezed probability distributions and the squeezing parameters. On more practical points, our results demonstrate that as the initial photon mean number increases, the atom-photon entanglement decays at a faster pace for the coherent distribution compared to the squeezed one. Moreover, it is shown that the degree of atom-photon entanglement is much higher and more stable for the squeezed distribution than that for the coherent one. Consequently, we conclude that the time intervals during which the atom-photon entanglement is distillable is longer for the squeezed distribution. It is also illustrated that as the temperature increases the rate of approaching separability is faster ∗

Corresponding author Phone: +98 713 613 7014 , Fax: +98 713 646 0839 Email address: [email protected] (M. M. Golshan )

Preprint submitted to Physica A

October 27, 2017

for the coherent initial distribution. The novel point of the present report is the calculation of dynamical density matrix (containing all physical information) for the combined system of atom-photon in a lossy cavity, as well as the corresponding negativity, at a finite temperature. Keywords: Atom-photon dynamical entanglement, Photonic coherent or squeezed distributions, Leaky cavity, Master equation, Dynamical negativity. 1. Introduction Entanglement is of central importance for quantum computation [1], quantum teleportation [2] and certain types of quantum cryptography [3], etc [4, 5, 6]. Inevitable in such applications is the fact that information stored in the entangled systems become processed and transferred via communication channels. The communication or processing channels, in general, have destructive effects on entanglement behavior, especially on its maximum value [7]. An effective procedure to overcome such destructive effects is the idea of quantum distillation [8]. Recent studies show that, for example, use of distillable entangled states increases the capacity of the channels for transportation of information, in quantum teleportation and quantum cryptography [9]. To this end, it is well established that a 2 ⊗ N entangled state with nonvanishing negativity is necessarily distillable [8]. It is thus of crucial importance to study the behavior of negativity for a system of practical applications, namely, the system of two-level atoms and photons, described by the Jaynes-Cummings model (JCM) [10, 11]. Although the entanglement of the elements in such a composite system has been extensively investigated [12, 13, 14, 15, 16, 17], the measure of negativity, thereby the distillability of entanglement, has rarely been used [18, 19, 20, 21]. Moreover, what is missing in these rare treatments, however, is the effect of noisy channels, or in general the environment, on the behavior of entanglement at any temperature [20, 21]. The novel point of the present article is the inclusion of the effect of environment, modeled as a heat reservoir at a finite temperature, on atom-photon entanglement determined from the negativity. Specifically, in the present article we have combined the notions of JCM, cavity damping and negativity to investigate atom-photon entanglement. To this end, we call attention to reference [22] where the entanglement between field (atom) and the combination of atom (field)-reservoir is reported. 2

Although there have been implementations of producing entangled states through trapped ions [23], atomic ensembles [24], photon pairs [25] and superconducting qubits [26], to name a few [27], the system of atom-photon is of much interest. The vast interest in the entanglement of the system of atoms and photons stems from the fact that the latter form rapid carriers of quantum information, while the former are reliable units for long time storage and processing of information. In addition, generation of atom-photon entanglement has been experimentally implemented [28]. Common in such implementation is the initial presence of some sort of photonic distributions, normally a coherent one, which has to be accounted for. In the following, therefore, we assume explicitly that the photons interacting with atoms are initially injected into a lossy cavity (as a heat reservoir) [29, 30] with either a coherent [22, 31] or squeezed distribution. At this point attention is called to the state-of-art setups for generation and control of the coherent or squeezed states of electromagnetic fields as given in [32, 33]. Motivated by this and the previous points, in the present report we focus on the effects of temperature [22, 31, 34], as well as initial photonic distributions [20, 21], on the atom-photon entanglement in the realm of damping cavity-quantumelectrodynamics. Since the atom-photon interaction, as given by JCM, takes place in a damping cavity the state of total system (ensemble) is of a mixed type and thus has to be described by a mixed density operator [18]. The mixed density operator, as is well known [29, 30], satisfies the master equation in which the cavity (or energy) damping as well as phase damping enter [35, 36, 37]. The cavity damping is responsible for the loss of electromagnetic energy, while phase damping is concerned with the loss of atomic energy [38]. In our treatment, however, we neglect the process of phase damping, since the system of atoms is assumed to be much dilute. To this end, we first present, based on JCM, the so described master equation and the manner of solving for the general density matrix elements, as functions of time. We then proceed by adopting our general solutions to the case of coherent or squeezed initial photonic states, while the atoms are initially in a state determined by the Boltzmann factor. The inclusion of atomic Boltzmann distribution, which naturally arises in a lossy cavity at finite temperatures, is a point missing in the previous treatments [20, 21, 22]. At this stage it is worth emphasizing that, in addition to the determination of atom-photon entanglement, one can extract any desired physical information regarding the behavior of atoms and/or photons interacting inside a lossy cavity for the two initial photonic 3

distributions. To this end, we also calculate the time evolution of the initial photonic distributions as well as the squeezing parameters. The results of these calculations, in due turn, will assist us in understanding the behavior of atom-photon entanglement. As is well established [39, 40], to quantify the entanglement between atoms and photons in a mixed ensemble, the measure of negativity, derived from the partially transposed density matrix, is most appropriate [41, 42, 43, 44]. The behavior of negativity then illustrates how information is spread amongst the subsystems of atoms and photons [45]. At this point it is noted that the task of calculating negativity, because of participation of so many states in the initial distributions, demands a numerical computation at the final stage. The analytical expressions for the elements of the ensemble density matrix along with those of its partially transposed one, as we present here, furnish a rather time saving algorithm for such a task. In the following we give a brief account of the results. Our calculation of negativity, along with illustrative figures, confirm the general expectation that the atom-photon degree of entanglement exhibits oscillations while decaying with time and asymptotically vanishes. This point holds true for both distributions. The decay rate, however, is by far larger for the coherent distribution. Moreover, we demonstrate that an increase in the initial photon mean number causes an increase in both the degree, as well as the decay rate, of atom-photon entanglement. This point is more so for the coherent distribution. Under identical conditions, however, atoms and photons become more entangled for the squeezed distribution than the coherent one. To understand the aforementioned behavior of atom-photon entanglement, we also present the evolution of the initial photonic distributions. To this end, use is made of the so calculated total density operator to determine the probability distributions, as functions of time, for both initial photonic states. Representative figures and a full discussion of the results shall appear in section 4. Moreover, the density operator is employed to calculate the time evolution of fluctuations in photonic quadratures, thereby the squeezing parameters for the case of initial distributions. Again, we present the results in section 4. As for the effect of temperature, our results confirms that an increase in temperature decreases the degree of atom-photon entanglement, at any time. Last, but not least, our calculations reveal that the entanglement between atoms and photons in the presence of damping is distillable for both distributions. The time intervals that entanglement is distillable, however, is longer for squeezed states. 4

The remainder of this article is organized as follows: In section 2, the physical model, along with the corresponding Hamiltonian, and the master equation for the density matrix, in the secular approximation are presented. An efficient algorithm to evaluate the negativity, as a function of time, is developed in section 3. The numerical results for the two initial photonic distributions, coherent and squeezed ones, are presented in section 4. The temporal behavior of the initial photonic probability distributions, as well as the squeezing parameters, are also given and discussed in this section. In this section we also present illustrative figures from which one can easily analyze the effects of external agents, the temperature and initial mean photon number on the entanglement. A discussion of the temporal behavior of atom-photon entanglement is also provided in this section. Finally, in section 5, the main features of the article are summarized and some concluding remarks are made. 2. The Physical Model and Master Equation The system under consideration here consists of a single-mode quantized field of frequency ω, interacting with a two-level atom, inside a dissipative cavity, in equilibrium with the environment at a temperature T. The field is generated via an experimental setup [32, 33] that delivers photons of particular distributions, i.e., a specific superposition of photonic Fock states, {|ni}, into the cavity, as schematically illustrated in figure 1. In the figure the atomic ground and excited states are denoted by |gi and |ei, respectively. As in the JCM, the Hamiltonian of such an atom-field system, disregarding the dissipation, is given by: H = H0 + H,

(1)

where H0 is the sum of two parts, the atomic Hamiltonian and that of the field, ~ω0 (|eihe| − |gihg|) + ~ωa† a, (2) H0 = 2 where a (a† ) denotes the photonic annihilation (creation) operator. The interaction Hamiltonian that couples the atom and field, in the dipole and rotating wave approximations, is expressed as, H = ~λ (a|eihg| + a† |gihe|), 5

(3)

Figure 1: Schematic representation of a two-level atom interacting with a single-mode photonic distribution, in a lossy cavity held at a temperature T.

where λ determines the atom-photon coupling strength. To account for the field dissipation, it is customary to assume that the walls of cavity is a reservoir of simple harmonic oscillators which, in turn, is minimally coupled to the field. Eliminating the reservoir contributions from the dynamical equation for the total density operator, by partial tracing over the reservoir states, one obtains the master equation for the atom-field system, in the interaction picture, as [38, 46], i ρ(t) ˙ = − [H, ρ(t)] + D(ρ(t)) ~ i γ ¯ th (aa† ρ(t) − 2a† ρ(t) a + ρ(t) a a† ) = − [H, ρ(t)] − n ~ 2 γ − (¯ nth + 1) (a† a ρ(t) − 2a ρ(t) a† + ρ(t) a† a), 2

(4)

where ρ, the density operator (matrix) of the atom-photon, as well as the interaction Hamiltonian of equation (3), are in the interaction picture. For simplicity, the field is assumed to be in resonance with atom (ω = ω0 ). The decay constant, γ, is related to the quality factor of the cavity Q, by the relation γ = ω/Q and n ¯ th = 1/(e~ω/kT − 1) is the number of thermal bosons of the reservoir at a temperature T , where ~ and k are the Planck and 6

Boltzmann constants, respectively. To proceed, we suppose that the atom is initially in equilibrium with the cavity at a temperature T, so that the atom is initially in a (mixed) state described by the Boltzmann factor: e−~ω/2kT e~ω/2kT |gihg| + |eihe| ~ω ~ω ) ) 2 cosh( 2kT 2 cosh( 2kT ≡ A|gihg| + B|eihe|.

ρA (0) =

(5)

Physically, this assumption refers to a large (statistically) collection of noninteracting identical atoms whose initial state is a superposition of the atomic two states. The probability of occurrence for these states is of course specified by the Boltzmann factor [47]. On the other hand, the photonic state is initially determined by the specific distribution externally delivered into the cavity. The general form of the initial photonic state, therefor, is assumed to be a mixture of Fock states, |ψF (0)i =

∞ X n=0

Cn |ni,

(6)

so that the field is initially described by, ρF (0) = |ψF (0)ihψF (0)| ∞ X ∗ Cn Cm |nihm|. =

(7)

m,n=0

The initial state of the whole system is then given by ρ(0) =

∞ X

m,n=0

∗ Cn Cm [A|gihg| + B|eihe|] ⊗ |nihm|,

(8)

which is pure and separable with respect to the atomic and photonic subsystems. The manner of solving the master equation, with the initial condition of equation (8), follows. As described in Ref. [48], the solution of master equation of the type expressed in equation (4), is most conveniently obtained by invoking the unitary transformation, W (t) = eiHt/~ ρ(t) e−iHt/~ and the secular approximation (to be specified momentarily). Parenthetically we may mention that in similar, but not identical, situations the method of “effective Hamiltonian” 7

which is based on the assumption of large detuning, has been employed [22]. However, large detuning implies a small transition probability leading, consequently, to the reduction of atom-photon entanglement [22]. Our method is indeed more physical and avoids this shortcome. The dynamical equation to be solved then reads, ˙ (t) = eiHt/~ D(ρ(t)) e−iHt/~ , W

(9)

where ρ(t) = e−iHt/~ W (t) eiHt/~ is to be substituted in the last two terms of equation (4). It is noted that the two representations coincide at the initial time. The form of the latter equation clearly suggests the use of eigenstates of the interaction Hamiltonian, the “dressed” states, H|g, 0i = 0,

± ± H |Ψ± n i = εn |Ψn i,

n = 0, 1, 2, 3, ... ,

(10)

√ 1 ε± (11) |Ψ± n = ±~λ n + 1, n i = √ (|e, ni ± |g, n + 1i), 2 to form the matrix representation of the transformed density operator. In the dressed state representation and assuming a strong atom-photon coupling such that λ  γ (the well-known secular approximation [49]), the set of coupled equations for the transformed density matrix elements, resulting from equation (9), separates into a set of independent equations for the off-diagonal elements of W . On the other hand, the diagonal elements satisfy a set of coupled, recursive first-order differential equations. The sets of equations for the off-diagonal elements of W may be analytically solved, straightforwardly. Although we have used these solutions in the calculation of our final results, for the sake of brevity, we do not present them here. The off-diagonal elements of the original density operator then read, hg, m + 1|ρ(t)|g, n + 1i = e− γ t (m+n+1)/2 e− γ n¯ th t (m+n+2) √ √ ∗ ×[ACm+1 Cn+1 cos(λt m + 1) cos(λt n + 1) √ √ +BCm Cn∗ sin(λt m + 1) sin(λt n + 1)], (12) hg, m + 1|ρ(t)|e, ni = i e− γ t (m+n+1)/2 e− γ n¯ th t (m+n+2) √ √ ∗ ×[ACm+1 Cn+1 cos(λt m + 1) sin(λt n + 1) √ √ −BCm Cn∗ sin(λt m + 1) cos(λt n + 1)], 8

(13)

he, m|ρ(t)|e, ni = e− γ t (m+n+1)/2 e− γ n¯ th t (m+n+2) √ √ ∗ ×[ACm+1 Cn+1 sin(λt m + 1) sin(λt n + 1) √ √ +BCm Cn∗ cos(λt m + 1) cos(λt n + 1)],

(14)

√ ∗ cos(λt n + 1), hg, 0|ρ(t)|g, n + 1i = e− γ t(n+1/2)/2 e− γ n¯ th t AC0 Cn+1

(15)

√ ∗ hg, 0|ρ(t)|e, ni = ie− γ t(n+1/2)/2 e− γ n¯ th t AC0 Cn+1 sin(λt n + 1)

(16)

An interesting point we deduce from the temporal behavior of the off-diagonal elements is the fact that they vanish in time scales of a few γ −1 , the value of which will be specified later on. The rate of diminishing is by far larger for higher photon numbers. A drastic consequence of this fact is that the initial separable pure ensemble turns into a separable mixed one in a time scale of a few γ −1 , provided that the diagonal elements survive during that time (as will be seen shortly, it is indeed the case here). We shall elaborate on this point in section 4, where the atom-photon entanglement is discussed. To calculate the diagonal elements we define, Fn (t) = hψn+ |ρ(t)|ψn+ i + − hψn |ρ(t)|ψn− i, and use the forgoing procedure to express the corresponding elements of W(t) (or equivalently, ρ(t)) in terms of Fn (t), which then leads to, √ 1 hg, n + 1|ρ(t)|g, n + 1i = [Fn (t) + e− γ t (n+1/2) e−2 γ n¯ th t (n+1) ] cos(2λt n + 1) 2 × (A|Cn+1 |2 − B|Cn |2 ), (17) √ 1 he, n|ρ(t)|e, ni = [Fn (t) − e− γ t (n+1/2) e−2 γ n¯ th t (n+1) ] cos(2λt n + 1) 2 × (A|Cn+1 |2 − B|Cn |2 ) (18) for the diagonal elements of the original density matrix. The functions g(t) = hg, 0|ρ(t)|g, 0i and Fn (t) (introduced previously) satisfy the following first order recursive and coupled differential equations, γ g(t) ˙ +γ n ¯ th g(t) − (¯ nth + 1)F0 (t) = 0, (19) 2 1 3 F˙ 0 (t) + 2γ(¯ nth + )F0 (t) − γ(¯ nth + 1)F1 (t) − γ n ¯ th g(t) = 0, 4 2 9

(20)

for n = 0 and F˙ n (t) + [2γ n ¯ th (n + 1) + γ(n + 1/2)]Fn (t) −γ(¯ nth + 1)(n + 3/2)Fn+1 (t) − γ n ¯ th (n + 1/2)Fn−1 (t) = 0,

(21)

for n ≥ 1. The initial values for g(t) and Fn (t) are specified, through their constitutive definitions, by the initial condition placed on ρ(t). To the best of our knowledge, there is no analytical solution to the last three equations, so that one has to resort to numerical calculations for determination of diagonal elements of the original density matrix. Having described the manner of calculating the time-dependent density matrix for the whole system, we are in a position to discuss the entanglement between atoms and photons inside a dissipative cavity. It shall be done in the next section. 3. Atom-Field Entanglement, Calculation of the Negativity In the previous section the density matrix, describing the general behavior of the states (mixed) for the composite system of atom-field, was obtained. Since the partial transposition of the so-obtained 2 ⊗ (photonic number) dimensional density matrix directly defines the criterion for the atom-field distillable entanglement, we devote the present section to a brief discussion of the “positive partial transposed” (PPT) criterion [50, 51], followed by the manner of numerically computing the negativity. It is by now well established that matrix of a bipartite quanPif the density P B tum system can be written as ρ = i pi ρA ⊗ρ , where p ≥ 0, p = 1 and i i i i i A B ρi and ρi represents the density matrices for the subsystems (with bases, |ai and |bi, respectively), then the system is separable (disentangled), otherwise it is entangled. This criterion is equivalently expressed in terms of the partially transposed density matrix, ρP T = P T (ha, b|ρ|´ a, ´bi) = h´ a, b|ρ|a, ´bi = ha, ´b|ρ|´ a, bi: for the composite system to be separable it is necessary that ρP T has no negative eigenvalues. Conversely, if ρP T possessed even a single negative eigenvalue, then the quantum system would be entangled (inseparable), in which case the entanglement is distillable for 2 × N systems [8]. Quantitatively, the PPT criterion may be expressed in terms of the negativity, defined as [41, 42, 43], X EN = max(0, −λi ), (22) i

10

where λi ’s are the eigenvalues of ρP T . It then follows that the state of the composite system is necessarily entangled if EN > 0 but it may or may not be entangled if EN = 0. If in the latter case there were an entangled state it would also necessarily be an undistillable one. It is then of much interest to determine when our system of atoms and photons, being 2 × N dimentional, is entangled and can be distilled. For the case in hand, we note from equations (13) and (16) that under a partial transposition (P T ) on the atomic states we have, hg, m + 1|ρP T (t)|e, ni = he, m + 1|ρ(t)|g, ni = −ie− γ t (m+n+1)/2 e− γ n¯ th t (m+n+2) √ √ ×[ACm+2 Cn∗ sin(λt m + 2) cos(λt n) √ √ ∗ −BCm+1 Cn−1 cos(λt m + 2) sin(λt n)], (23) hg, 0|ρP T (t)|e, ni = he, 0|ρ(t)|g, ni = −ie− γ tn/2 e− γ n¯ th t (n+1) √ ×[AC1 Cn∗ sin(λt) cos(λt n) √ ∗ −BC0 Cn−1 cos(λt) sin(λt n)],

(24)

for n ≥ 1 and hg, m + 1|ρP T (t)|e, 0i = he, m + 1|ρ(t)|g, 0i √ = −ie− γ t (m+3/2)/2 e− γ n¯ th t A C0∗ Cm+2 sin(λt m + 2), (25) hg, 0|ρP T (t)|e, 0i = he, 0|ρ(t)|g, 0i = −ie− γ t /4 e− γ n¯ th t A C0∗ C1 sin(λt), (26) for n = 0, while the other elements remain unchanged (see equations (12), (14), (15), (17) and (18)). It is observed that the temporal behavior of the elements of ρP T , consequently the negativity, is one of simple oscillations, enveloped by a sinusoidal wave, while decaying exponentially. This point stems from the fact that energy delivered by the electromagnetic source, while dissipating through the cavity walls, forces the atom to make quantum transitions. The observations just made, as we shall see in the next section, where the behavior of atom-photon entanglement for coherent and squeezed 11

photonic distributions is discussed, prove to be of crucial importance. At this point it is worth emphasizing that for the case under our study, the dimensionality of ρP T can approach infinity, determined by the Ci ’s (see equation (8)) that specify the photonic distribution. In order to find the eigenvalues of ρP T , therefore, one has to resort to numerical computation for g(t) and Fn (t), then diagonalization of ρP T . To this end, a limitation on the dimensionality of the Hilbert space, consequently, the density matrix, should be enforced. We specify the restriction employed in our numerical computation in the following. In general, the photonic distributions delivered into the system contains an infinitely large number of states and, for numerical computations, should be cut off at some point. The cut-off point, in turn, limits the dimensionality of ρP T , or equivalently the density matrix itself, and its eigenvalues. It is, therefore, a reasonable criterion to terminate the initial photonic distributions and the dimensionality of ρ at a point where the trace of the bounded density matrix is arbitrarily close to the trace of the initial one at any time. Assuming that initially ρ is normalized to unity, in the following section we take this criterion as, | 1 − T r[ρM (t)] | ≤ 0.01,

(27)

where we have denoted the limiting value for the photon number present in the initial distributions by M. 4. Entanglement between Coherent or Squeezed Photonic and Atomic States The manner of computing negativity, presented in the last two sections, is quite general and may be employed for any photonic initial state. In this section, we make use of the atomic initial state given in equation (5) and specify the Cm s of equation (7) through the choice of coherent or squeezed distributions. To prepare for the calculation of negativity, we begin with the definition of coherent and squeezed photonic distributions. The Fock space representation of the normalized coherent state is given by [38]: ||α|, θi = e−|α|

2 /2

∞ X |α|n einθ √ |ni, n! n=0

12

(28)

where |α|2 gives the mean photon number, n ¯ . The initial photonic density operator of equation (7) is then specified from the explicit form, equation (28), by the identification, Cn = e−|α|

2 /2

|α|n einθ √ , n!

(29)

for the coherent distribution. When this identification, with θ = 0, is employed in solving (numerically) the master equation, the resulting density matrix straight forwardly gives the time evolution of any initial quantity, photonic, atomic or the composition, for the coherent probability distribution. Related to the present investigation, and as an example, one can easily compute the time evolution of the initial photonic probability distribution, Pn (t, T ) = T r[|nihn| ρ(t, T )]. Likewise, one can perform a calculation on statistical properties of the field, namely, the uncertainties in the field quadratures, X1 = 21 (a† + a) and X2 = 2i (a† − a) [38]. Such information, as is well known [53], can be readily extracted from the so called squeezing parameters, Sj (t) = 4h(∆Xj )2 i − 1. Mention of two points here is useful. Firstly, the expectation values are calculated from the total (reduced) density matrix through hXj i = T r[Xj ρ(t, T )] and secondly, −1 < Sj (t) < 0 implies squeezing in the jth direction. The case of Sj (t) = 0 corresponds to a coherent distribution. The result of our calculation of the evolution of the initial probability distribution (part a) along with the squeezing parameters (parts b and c) is illustrated in figure 2. In this typical figure we have set the temperature at 3K, ω = 1T Hz and γ = 0.01λ. In part (a) the initial mean photon number is fixed at 8. The scaled time in the figure (and the following ones) is defined as, τ = λt. From part (a) it is evident that the probability of any photonic state, except the vacuum, is a descending function of time and vanishes asymptotically. The diminishingly small initial probability of no photon in the distribution, however, is an ascending function of time and approaches unity asymptotically. Moreover, from parts (b) and (c) we note that the initial coherency (minimum uncertainty in both quadratures), S1 (0) = S2 (0) = 0, is lost during the process, S1,2 (t) > 0, returning back to the coherency asymptotically. For the latter case, the field has fallen into the vacuum state in accordance with part (a). These observations, as we shall see, assist us in understanding the behavior of atom-photon entanglement. We emphasize that this observation holds for any fixed set of parameters whose corresponding figures, for avoiding prorogation, are omitted here. It 13

is, moreover, worth mentioning that the distribution here is normalized to unity so that the criterion of equation (27) still holds. On the other hand, the explicit expression for the normalized vacuum (even) squeezed state is [52], p ∞ X (2n)! inθ 1 n (−1) e tanhn r |2ni. (30) |r, θi = √ n 2 n! cosh r n=0 In the vacuum squeezed state the parameter, r is related to the mean photon number through, n ¯ = sinh2 r. In the case of squeezed states the expansion coefficients that determine the initial density matrix in equation (7) read, p (2n)! inθ 1 n (−1) Cn = √ e tanhn r. (31) n 2 n! cosh r It is stressed that in the vacuum squeezed state only even number of photons participate, so that the indices of the Cx ’s in equations (12) through (21) and equations (23) through (26) should be replaced by Cx/2 . In the present case we have also set θ = 0. In the same manner as for the coherent case, the master equation can be solved (numerically) for the density matrix which paves the way for calculating any desired physical quantity. In conformity with the previous case, we have also computed the evolution of the photonic probability distribution along with the squeezing parameters. The results are depicted in figure 3, parts (a), (b) and (c), in the same order as in figure 2. The values of other parameters are also fixed as those specified for figure 2. From part (a) of the figure we again note a rapid increase in the probability of having no photons, making it approach unity. That is, after sufficiently long time the photonic state becomes that of the vacuum, being a coherent one. A comparison of part (a) of figures 2 and 3, however, reveals that the rate of increase in the probability of reaching the vacuum state is larger for the coherent distribution. Moreover, because of the choice, θ = 0, we see from parts (b) and (c) of the figure that the first quadrature, X1 , is squeezed because the corresponding squeezing parameter is negative (see the beginning of part (b)). The squeezing of the state, however, lasts for a very short while compared with the time scales involved. It is also noted that the distribution is normalized to unity so that the criterion of equation (27) is intact. Having discussed the temporal behavior of the two distributions, we are now in the position of exploring the properties of atom-photon entanglement, 14

in a lossy cavity. To this end, we focus on the effects of mean photon number and temperature on the corresponding negativity. We thus present the temporal behavior of the negativity in figure 4 for the coherent and squeezed distributions, parts a and b, respectively, with n ¯ = 8, 16 and 24, at a constant temperature T = 3K. Figure 5, in the meantime, is devoted to the variation of negativity with the scaled time for a fixed mean photon number, n ¯ = 16 and T = 0K, 4K and 8K. For extrapolation purposes, three dimensional graphs, along with density plots, of the demeanor of the negativity are presented in figures 6 through 11. In producing these figures we have taken the relevant parameters as those used in generating figure 2. Attention is called to the fact that the negativity vanishes at the starting time, since the initial state of the whole system is a separable one (as given in equation (8)). A glance at these figures reveals the general and common features of the atom-photon entanglement, which we enumerate in the following, for the two distributions. It is again stressed that whenever the negativity is nonzero the atom-photon entanglement is a distillable one. i) For every n ¯ , the atom-photon degree of entanglement exhibits oscillations while decaying with time and asymptotically vanishes. Such general features of atom-photon entanglement are attributed to the physical fact that atoms and photons periodically exchange energy, leading to oscillations. Meanwhile, the field energy is gradually lost through the dissipative cavity, resulting a decaying oscillation in the temporal behavior of the entanglement. It is also evident that for sufficiently long time, determined by the cavity damping constant (τ  λ/γ), there practically remains no energy in the field, in which case there is no photons and the atom falls into a combination of its ground and excited states, so that the total system is in the state, √12 (|gi + |ei) ⊗ |0i, which is again a separable one. This is the physical reason for the asymptotical behaviour of the entanglement. ii) It is, moreover, observed that in the coherent case the negativity decays at a faster rate for larger initial mean photon number. This result can be justified by making note of the fact that, due to dissipation, the initial photonic probability distribution, given by T r[|nihn|ρ(0)], becomes narrower and shifts towards states of smaller photon numbers, more so for smaller initial mean photon number, as time passes. Moreover, the symmetry of probability distribution is also preserved for longer time, as the initial photon mean number increases. As a result, more photonic states, with equal probabilities, participate so that the entanglement decreases at a faster rate for larger initial photon mean number. These conclusions are consistent with 15

the discussions regarding figure 2, presented above. On the other hand, for the squeezed state, the probability distributions for the three initial mean photon number used here are almost identical and include more photonic states compared to the coherent state. However, for small number of photons such states participate with unequal probabilities due to the asymmetry of probability distributions. Moreover, under the interaction with atoms the odd photon numbers also appear in the evolution of the initial vacuum squeezed state. Again attention is called to the discussions surrounding figure 3. These points give rise to a larger degree of entanglement, as well as a uniform decay rate, for the squeezed distribution. The foregoing conclusions are apparent from figures 4 and 5. iii) The next point of our considerations is the effect of temperature on the atom-photon entanglement for the two distributions. In this regard the temporal behavior of atom-photon entanglement for various temperatures is depicted in figures 5 through 11 for the coherent and squeezed initial states. From these figures, and as expected, an increase in the temperature results a reduction in the maximal atom-photon entanglement at any time. This holds true at any instant of time. The atom-photon entanglement then disappears asymptotically. This demeanor is physically due to the fact that an increase in the temperature gives rise to the excitation of more atom-photon states with almost equal probabilities. In the limit of extreme temperatures, they become exactly equal and the system approaches a fully mixed state, with no entanglement. iv) Along the above general features of the atom-photon entanglement, we further observe that the degree of entanglement, while oscillating, grows at earlier times reaching its maximal value. This is due to the fact that as the atoms start interacting with the reservoir modes, the ensemble tends towards a mixed one. Moreover, it is stressed that the degree of atom-photon entanglement is much higher and more stable for the squeezed distribution than that for the coherent one, so that the system remains distillable for longer time for the former one. 5. Conclusions The present report is devoted to the study of the entanglement, thus the distillability, for a mixed ensemble of two-level atoms and photons in a lossy cavity. The mixture is specified by the atomic Boltzman distribution at an equilibrium temperature, T, while the photons are externally injected 16

into the cavity with either coherent or squeezed initial distributions. The equilibrium temperature also takes effect in the photonic decay to the cavity walls. Since the ensemble is a mixed one, we take advantage of the concept of negativity to determine the temporal behavior of the degree of atom-photon entanglement. To this end, we first demonstrate how the master equation, involving the Jaynes-Cummings model and cavity damping, is solved for the total density matrix. We also develop the manner of calculating the negative eigenvalues of the corresponding partially transposed density matrix. We then employ the results to report the effects of initial photon mean number and temperature on the dynamical behavior of atom-photon entanglement. We further make a comparison between the degree of entanglement arising from the two initial photonic distributions. To make the comparison more vivid, we also make use of the density matrix so calculated to determine the time evolution of the initial photonic distributions and squeezing parameters. Even though a detailed description and physical discussion of the results have been presented in the previous section, we outline the more important points of our findings in the following. • At every temperature, the atom-photon degree of entanglement starts from zero, oscillates at short times reaching a maximum, for the coherent distribution. The entanglement diminishes as time passes and asymptotically vanishes. As a result, information is periodically shared by the atoms and the photons and eventually becomes local (separable). • For the squeezed distribution, the oscillations in the degree of entanglement begin after reaching the maximum and decay at a slower rate. Localization of information also occurs in the system eventually. This point also holds at every temperature. • The maximal value of atom-photon entanglement is larger for the squeezed distribution compared to that of the coherent one. Information can thus be distributed between atoms and photons more evenly with the squeezed states rather than the coherent one. • As the initial photon number is increased, the decay rate of entanglement increases for the coherent distribution, while it has negligible effect on the decay rate of the squeezed one. The distribution of information between the two subsystems remain intact for a longer while in the case of squeezed states. 17

• Our analysis also quantitatively confirms the well known fact that, at a fixed time, an increase in the temperature results a reduction in the maximal atom-photon entanglement, for both coherent and squeezed distributions. The atom-photon entanglement disappears at higher temperatures. This point is in conformity with the fact that at high enough temperatures information becomes localized. In short, we believe the present report that investigates the properties of atom-photon entanglement in a lossy cavity, may be advantageous for feasible applications. Acknowledgement This work has been in part supported by the grants from the Research Council of Shiraz University, under the contracts 95-GRSC-82 and 95GRD1M240190. [1] J. Miller, A. Miyake, npj Quantum Information 2 ( 2016) 16036. [2] Q.-C. Sun, Y.-L. Mao, S.-J. Chen, W. Zhang, Y.-F. Jiang, Y.-B. Zhang, W.-J. Zhang, S. Miki, T. Yamashita, H. Terai, X. Jiang, T.-Y. Chen, L.X. You, X.-F. Chen, Z. Wang, J.-Y. Fan, Q. Zhang, J.-W. Pan, Nature Photonics 10 (2016) 671. [3] M. Jacak, D. Melniczuk, J. Jacak, A. Janutka, I. J´o´zwiak, J. Gruber, P. J´o´zwiak, Opt Quant Electron 48 (2016) 363. [4] S. Steinlechner, J. Bauchrowitz, M. Meinders, H. M¨ uller-Ebhardt, K. Danzmann, R. Schnabel, Nature Photonics 7 (2013) 626. [5] A. Streltsov, Quantum Correlations Beyond Entanglement and Their Role in Quantum Information Theory, Springer, 2015. [6] R. Fickler, R. Lapkiewicz, M. Huber, M.P.J. Lavery, M.J. Padgett, A. Zeilinger, Nature Communications 5 (2014) 4502. [7] C. Antonellic, M. Shtaif, M. Brodsky, Phys. Rev. Lett. 106 (2011) 080404. [8] D. Bruß, J. Math. Phys. 43 (2002) 4237. [9] X.-F. Cai, X.-T. Yu, L.-H. Shi, Z.-C. Zhang, Front. Phys. 9 (2014) 646. [10] A. B. Klimov, S. M. Chumakov, Quantum Optics, Wiley-Vch, 2009. [11] B. W. Shore, J. Mod. Opt. 40 (1993) 1195. 18

[12] S. J. D. Phoenix, P. L. Knight, Phys. Rev. A 44 (1991) 6023. [13] S. M. Barnett, S. J. D. Phoenix, Phys. Rev. A 40 (1989) 2404. [14] J. Gea-Banacloche, Phys. Rev. A 44 (1991) 5913. [15] B. W. Shore, P. L. Knight, J. Mod. Opt. 40 (1993) 1195. [16] Z. Kordi, S. Ghanbari, M. Mahmoudi, Quantum Inf Process 15 (2016) 199. [17] M. R. Abbasi, M.M. Golshan, Phys. A 392 (2013) 6161. [18] N. Foroozani, M.M. Golshan, J. Stat. Mech. (2010) P02007. [19] V. Ceban, M.A. Macovei, J. Opt. Soc. Am. B 33 (2016) 942. [20] A.-S.F. Obada, H.A. Hessian, A.-B.A. Mohamed, J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 135503. [21] A.-S.F. Obada, H.A. Hessian, A.-B.A. Mohamed, J. Mod. Opt. 56 (2009) 881. [22] A.-S.F. Obada, H.A. Hessian, A.-B.A. Mohamed, Phys. Lett. A 372 (2008) 3699. [23] J.G. Bohnet, B.C. Sawyer, J.W. Britton, M.L. Wall, A.M. Rey, M. FossFeig, J.J. Bollinger, science 352 (2016) 1297. [24] L. Slodiˇcka, G. H´etet, N. R¨ock, P. Schindler, M. Hennrich, R. Blatt, Phys. Rev. Lett. 110 (2013) 083603. [25] Sh. Alizadeh, R. Safaiee, M.M. Golshan, Phys. A 428 (2015) 133. [26] D. Rist`e, M. Dukalski, C.A. Watson, G.de. Lange, M.J. Tiggelman, Ya.M. Blanter, K.W. lehnert, R.N. Schouten, L. DiCarlo, Nature 502 (2013) 350. [27] R. Safaiee, F. Aghel, M.M. Golshan, J. Stat. Mech. (2016) 023101. [28] J. Volz, M. Weber, D. Schlenk, W. Rosenfeld, J. Vrana, K. Saucke, C. Kurtsiefer, H. Weinfurter, Phys. Rev. Lett. 96 (2006) 030404.

19

[29] B. Klimov Andrei, M. Chumakov Sergei, A Group-Theoretical Approach to Quantum Optics: Models of Atom-Field Interactions, Wiley-VCH, 2009. [30] R.R. Puri, Mathematical Methods of Quantum Optics, Springer, 2001. [31] A.J.van. Wonderen, K. Lendi, J. Phys. A: Math. Gen. 35 (2002) 9889. [32] N. Bruno, A. Martin, R.T. Thew, Opt. Commun. 327 (2014) 17. [33] S.S.Y. Chua, Quantum Enhancement of a 4 Km Laser Interferometer Gravitational-Wave Detector, Springer, 2015. [34] M. Soltani, H. Ezatabadipour, J. Jalali, P. Darabi, E. Azizi, G. Rashedi, Eur. Phys. J. D 67 (2013) 256. [35] A. Reiserer, G. Rempe, Rev. Mod. Phys. 87 (2015) 1379. [36] S.-L. Su, X.-Q. Shao, H.-F. Wang, S. Zhang, Scientific Reports 4 (2014) 7566. [37] A.-S.F. Obada, H.A. Hessian, A.-B.A. Mohamed, Quantum Inf Process 12 (2013) 1947. [38] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, 2001. [39] Bo. Chong, Characterization of Multipartite Entanglement, 2006. [40] C. Wancang, L. Dan, P. Feng, L. Guilu, Science in China Series G: Physics, Mechanics and Astronomy 49 (2006) 606. [41] G. Vidal, R.F. Werner, Phys. Rev. A 65 (2002) 032314. [42] K. Zyczkowski, P. Horodecki, A. Sanpera, M. Lewenstein, Phys. Rev. A 58 (1998) 883. [43] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81 (2009) 865. [44] F. Amiri, M.M. Golshan, J. Nanopart Res 13 (2011) 6069. [45] N. Foroozani, M.M. Golshan, R. Safaiee, Journal of Computational and Theoretical Nanoscience 6 (2009) 1519. 20

[46] H.J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations, Springer, 2002. [47] R. K. Pathria, Paul D. Beale, Statistical Mechanics, Elsevier, 2011. [48] R.R. Puri, G.S. Agarwal, Phys. Rev. A 35 (1987) 3433. [49] H.-P. Breuer, F. Petruccinne, Irreversible Quantum Dynamics, SpringerVerlag Berlin Heidelberg, 2003. [50] A. Peres, Phys. Rev. Lett. 77 (1996) 1413. [51] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Lett. A 223 (1996) 1. [52] A. NoormandiPour, M.K. Tavassoly, Commun. Theor. Phys. 61 (2014) 521. [53] M.K. Tavassoly, F. Yadollahi, Int. J. Mod. Phys. B 26 (2012) 1250027.

21

Figure 2: (a) The probability distribution versus n and the scaled time, τ , for the coherent state. The inset depicts the corresponding density plot. In parts (b) and (c) evolution of squeezing in first and second quadratures are illustrated for n ¯ = 8 (solid curve, black), n ¯ = 16 (small dashed curve, red) and n ¯ = 24 (large dashed curve, blue), respectively. The insets show the evolution for very short times.

Figure 3: (a) The probability distribution versus n and the scaled time, τ , for the squeezed state. The inset depicts the corresponding density plot. In parts (b) and (c) evolution of squeezing in first and second quadratures are illustrated for n ¯ = 8 (solid curve, black), n ¯ = 16 (small dashed curve, red) and n ¯ = 24 (large dashed curve, blue), respectively. The insets show the evolution for very short times.

22

Figure 4: Time evolution of negativity for n ¯ = 8 (solid curve, black), n ¯ = 16 (small dashed curve, red) and n ¯ = 24 (large dashed curve, blue), for (a) the coherent state and (b) the squeezed state. The insets show the evolution at short times.

Figure 5: Time evolution of the negativity for T = 0K (solid curve, black), T = 4K (small dashed curve, red) and T = 8K (large dashed curve, blue) for (a) the coherent state and (b) the squeezed state. The insets show the evolution at short times.

23

Figure 6: Negativity versus τ and T, part (a), and the corresponding density plot, part (b), for the coherent state. The initial photon mean number is set at n ¯ = 8.

Figure 7: Negativity versus τ and T, part (a), and the corresponding density plot, part (b), for the coherent state. The initial photon mean number is set at n ¯ = 16.

24

Figure 8: Negativity versus τ and T, part (a), and the corresponding density plot, part (b), for the coherent state. The initial photon mean number is set at n ¯ = 24.

Figure 9: Negativity versus τ and T, part (a), and the corresponding density plot, part (b), for the squeezed state. The initial photon mean number is set at n ¯ = 8.

25

Figure 10: Negativity versus τ and T, part (a), and the corresponding density plot, part (b), for the squeezed state. The initial photon mean number is set at n ¯ = 16.

Figure 11: Negativity versus τ and T, part (a), and the corresponding density plot, part (b), for the squeezed state. The initial photon mean number is set at n ¯ = 24.

26

Highlights for the reviewers     

Time evolution of atom-photon entanglement in a lossy cavity, at T  0 , is reported. Jaynes-Cummings model and cavity damping are implemented in the master equation. Master equation for the total density matrix is solved in the secular approximation. Negativity is then used to characterize atom-photon entanglement. Characteristics of the entanglement are presented for coherent or squeezed states.