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Time evolution of entanglement in a cavity at finite temperature
Physica A 462 (2016) 1261–1272
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Physica A journal homepage: www.elsevier.com/locate/physa
Time evolution of entanglement in a cavity at finite temperature E.G. Figueiredo a,b,∗ , C.A. Linhares c , A.P.C. Malbouisson d , J.M.C. Malbouisson a a
Instituto de Física, Universidade Federal da Bahia, 40210-340, Salvador, BA, Brazil Centro de Ciência e Tecnologia em Energia e Sustentabilidade, Universidade Federal do Recôncavo da Bahia, 44085-132, Feira de Santana, BA, Brazil c Instituto de Física, Universidade do Estado do Rio de Janeiro, 20559-900, Rio de Janeiro, RJ, Brazil b
d
Centro Brasileiro de Pesquisas Físicas, MCTI, 22290-180, Rio de Janeiro, RJ, Brazil
highlights • Exact entanglement indicators are given for two oscillators in a cavity at finite T. • Entanglement survives at finite T for bipartite systems in a finite (small) cavity. • Sudden death of entanglement at finite T occurs for bipartite systems in free space.
article
info
Article history: Received 18 February 2016 Received in revised form 6 June 2016 Available online 12 July 2016 Keywords: Entanglement Time evolution Finite-temperature
abstract We consider two identical atoms, taken in the harmonic approximation, inside a spherical cavity filled with a field described by an infinite set of oscillators. The atoms are linearly coupled to the field. Using the dressed-state approach and considering the field modes populated with thermal-distribution, we study the effect of temperature on the time evolution of entangled states of the pair of atoms. We particularly analyze the cases of finite and very-large cavities showing that survival and sudden-death of entanglement can happen at finite temperature. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Quantum entanglement corresponds to correlations between separated parts of compound systems, which may even not interact directly, that cannot be conceived in any classical-physics description. Despite the nonintuitive character of quantum mechanics, particularly of entanglement, experimental implementations have demonstrated several successful applications in the realm of the theory of quantum information and in quantum computer science. The teleportation of information between two parts of a system, cryptographic protocols that cannot be violated and the increase in the processing speed of computations, as compared to classical algorithms, are some examples of these advances [1–3]. All these achievements rely heavily on the fact that entanglement is the fundamental resource to implement the processes. Despite the great progress in the theory of quantum entanglement [4], many fundamental problems are still unsolved; identify entanglement in multipartite systems and in mixed states of high-dimension bipartite systems are some of these
∗ Corresponding author at: Centro de Ciência e Tecnologia em Energia e Sustentabilidade, Universidade Federal do Recôncavo da Bahia, 44085-132, Feira de Santana, BA, Brazil. E-mail addresses: [email protected] (E.G. Figueiredo), [email protected] (C.A. Linhares), [email protected] (A.P.C. Malbouisson), [email protected] (J.M.C. Malbouisson). http://dx.doi.org/10.1016/j.physa.2016.06.128 0378-4371/© 2016 Elsevier B.V. All rights reserved.
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problems. Usually multipartite systems do not exist isolated and the effect of the interaction with the environment should be taken into account. In this case, survival of entanglement is not an easy task due to decoherence effects [1,2]. Some methods are known to preserve entanglement, for example, the quantum error correction [5]. The cases in which entanglement survives are of fundamental importance for processes in quantum information science. In various situations, the interaction with the environment leads to the disappearance of the entanglement as time evolves, either asymptotically or in a finite time (sudden death) [6–8]. For instance, in Refs. [9,10] the authors show that the decay of the entanglement of Bell-like superposition can occur asymptotically or by sudden death depending on the parameters of the superposition. For the environment at zero temperature, revival and induced entanglement are originated by system–reservoir correlations [11–14]. Entangled states have also been considered to explain some macroscopic properties of some solids, such as the magnetic susceptibility and the heat capacity [15,16]. Entanglement dynamics of gaussian states interacting with a non-zero temperature environment was studied in Refs. [17,18] and entanglement of Brownian particles were considered in Refs. [19,20]. At finite temperature, one also finds entanglement survival, oscillations between sudden deaths and revival and even induced entanglement [21–29]; most of these works use gaussian or two-level systems with the formalism of master equation for the reduced density matrix. The time evolution of entanglement for a system in contact with the environment is then an important question for physical applications; a review of the open-system dynamics of entanglement appear in Ref. [30]. In this paper, we address some of these questions by considering two families of states: gaussian, squeezed vacuum states and, non-gaussian, Bell-like states of two particles (atoms) confined in a spherical cavity of radius R, focusing on the time evolution of the entanglement. The atoms are approximated by harmonic oscillators and are coupled with an environment modeled by an infinite set of harmonic oscillators at finite temperature. Distinctly of the previous treatments, we consider the states of the atoms as ‘‘dressed’’ by the thermal field with the method presented in Refs. [31,32], which allows for analytical results. We show that the effect of the temperature on the system is rather distinct for finite and infinite cavities. In very large cavities (free space), the raising of temperature leads to a sudden death of the entanglement, while for zero temperature the decay is asymptotic. For small cavities, we find an oscillatory behavior with sudden death alternating with revivals of the entanglement and a limit temperature for which the system remains entangled. 2. Atoms in a cavity in the dressed state formalism 2.1. The model Let us consider two atoms A and B, taken in the harmonic approximation, coupled to the environment through a linear position–position interaction with the modes of a field in a perfectly reflecting spherical cavity of radius R. We do not consider any direct interaction between the atoms so that the Hamiltonian is given by H =
N N 2 1 1 2 pA + ωA2 q2A + p2B + ωB2 q2B + pk + ωk2 q2k − (ηA qA + ηB qB )ωk qk , 2 2 k=1 k=1
(1)
where ηA and ηB are constants and the limit N → ∞ will be undertaken. Our model is a slightly modified version of the model used in Ref. [21]. The coupling between the two atoms is indirect, mediated by the field in the cavity. √ We assume that the atoms A and B couple in the same way with the environment and take ηA = ηB = g 1ω, where g is a constant with the dimension of frequency measuring the strength of the coupling. The quantity 1ω = π c /R is the interval between neighboring frequencies of the field, which are given by ωk = kπ c /R [31]. Let us now define new coordinates, q+ (center of mass) and q− (relative position), such that 1 q+ = √ (qA + qB ), 2
1 q− = √ (qA − qB ), 2
(2)
√
with momenta p± = (pA ± pB )/ 2. In termsof {q± , p± }, for identical atoms (ωA = ωB = ω0 ), the Hamiltonian is written as H = H− + H+ , where H− = 21 p2− + ω02 q2− and H+ =
√
N N 1 2 1 2 p+ + ω02 q2+ + pk + ωk2 q2k − ηωk q+ qk , 2 2 k=1 k=1
(3)
with η = 2g 1ω, 1ω = π c /R and ωk = k1ω. We see that the center-of-mass oscillator q+ couples to the field while the relative-position oscillator q− evolves freely in time. Eq. (3) shows that H+ is identical to the Hamiltonian of a single oscillator coupled to the field. Thus, for the system composed by the center-of-mass oscillator and the field (q+ ⊕ field), we can use the concepts of dressed coordinates and dressed states introduced in Ref. [31]. This formalism has been employed to investigate the time evolution of superposed single-atom states [33], thermal effects on single atoms [34–37] and the time evolution of bipartite states [38,39].
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2.2. Dressing the center-of-mass oscillator The formalism developed in previous papers requires that first the Hamiltonian (3) be transformed to principal axis by means of a point transformation, N
qµ =
tµr Qr ,
pµ =
N
r =0
tµr Pr ,
(4)
r =0
where µ = {+, k }, k = 1, 2, . . . , N, through an orthonormal matrix (tµr ). Here, we use the subscripts µ = + (or 0) and µ = k to designate respectively the center-of-mass oscillator q+ and the harmonic modes of the environment field; the subscript r refers to the normal modes. Writing the transformed Hamiltonian in terms of normal momenta and coordinates, it becomes H =
N 1 2 Pr + Ωr2 Qr2 . 2 r =0
(5)
In this expression, the Ωr ’s are the normal frequencies related to the stable oscillation modes of the coupled system (q+ ⊕ field). It can be shown [31] that
ηωk tkr = 2 tr , ωk − Ωr2 +
r t+ =
1+
N k=1
with the condition ω02 − Ωr2 =
η2 ωk2 (ωk2 − Ωr2 )2
η2 ωk2
N
− 12 ,
k=1 ω2 −Ω 2 ; this sum diverges for N r k
(6)
→ ∞, so that we need to implement a renormalization
procedure to redefine the frequency [40]. It leads to
ω¯ 2 − Ωr2 = η2 Ωr2
∞
1
k=1
ωk2 − Ωr2
,
(7)
where ω ¯ 2 = limN →∞ (ω02 − N η2 ) is the renormalized frequency. Observe that the divergence of the bare frequency of the center-of-mass oscillator q+ in the limit N → ∞ is due to its interaction with the field. This does not occur for the frequency of the relative-position oscillator q− , which remains finite. However, we choose the (infinite) bare frequency of the q+ oscillator in such a way that the renormalized frequency ω ¯ equals the (finite) relative-position oscillator frequency. This has to be done for consistency, in considering two identical oscillators. Using that ωk = π ck/R and performing the summation, Eq. (7) becomes (without frequency labels)
cot
RΩ
c
Ω c = + πg RΩ
1−
Rω ¯2
π gc
;
(8)
the roots of Eq. (8) give the spectrum of eigenfrequencies Ωr (r = 0, 1, 2, . . .). To ensure the existence of the first root Ω0 , we assume that δ ω ¯ 2 /g 2 > 1 where δ = 2g 2 /η2 = gR/π c. We then consider the energy eigenstates of the system (q+ ⊕ field),
φn0 n1 n2 ... (Q ) =
√
1
2ns ns !
s
√ Hns Ωs Qs Γ0 ,
(9)
where Hns denotes the ns th Hermite polynomial and Γ0 is the normalized ground-state eigenfunction. For the system
(q+ ⊕ field) we define dressed coordinates by [31] √ ω¯ µ q′µ = tµr Ωr Qr ,
(10)
r
where q′µ = {q′+ , q′k } and ω ¯ µ = {ω, ¯ ωk }. Since the relative-position coordinate does not couple with the field, we have q′− = q− . In terms of the dressed coordinates, we define for a fixed initial instant, t = 0, dressed states, |κ+ , κ1 , κ2 , . . .⟩, by means of the complete orthonormal set of functions
ψκ+ κ1 ... (q ) = ′
1
µ
2κµ κµ !
Hκµ
ω¯ µ q′µ
Γ0 .
(11)
Distinctly from the energy eigenstates, which are stationary states of the system, the dressed states (with exception of the ground state) evolve in time.
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From the dressed coordinates and the dressed momenta, we define the dressed annihilation operator
aˆ ′µ
=
ω¯ µ
qˆ ′µ +
2
i 2ω ¯µ
pˆ ′µ ,
(12)
and its adjoint, the dressed creation operator aˆ ′µĎ . The time evolution of aˆ ′µ is given by [36] aˆ ′µ (t ) = fµν (t ) =
∞ ν=0 ∞
fµν (t )ˆa′ν ,
(13)
tµr tνr e−iΩr t .
(14)
r =0
On the other hand, the relative-position annihilation operator evolves in time with its free Hamiltonian, so that aˆ ′− (t ) = √ √ ′Ď e−iω¯ t aˆ ′− ; thus, qˆ ′− (t ) = (ˆa′− e−iω¯ t + aˆ − eiω¯ t )/ 2ω ¯ , pˆ ′− (t ) = −i(ˆa′− e−iω¯ t − aˆ ′−Ď eiω¯ t )/ 2/ω¯ , while
1 ∗ qˆ ′+ (t ) = √ f+µ (t )ˆa′µ + f+µ (t )ˆa′µĎ , 2ω ¯ µ pˆ ′+ (t )
= −i
ω¯ 2 µ
(15)
∗ f+µ (t )ˆa′µ − f+µ (t )ˆa′µĎ .
(16)
3. Temperature effects on the time evolution of entanglement We now address the question of the temperature dependence of the time-evolution of the entanglement for a bipartite system in contact with a thermal reservoir. First, we briefly recall methods to identifying entanglement in a bipartite mixed state. 3.1. Bipartite entanglement The notions of separability and entanglement (inseparability) can be directly settled for bipartite pure states: separable states can be written as direct products of states of the parts, while entangled states cannot. For mixed bipartite states, the definition of separability is more subtle: separable states are those for which the density matrix can be written as a convex 1 2 sum of direct products of density matrices of the two subsystems [41], that is ρˆ = p σ ˆ ⊗ σ ˆ , with p ≥ 0 and i i i i i i pi = 1; otherwise, the mixed states are said to be entangled. However, to demonstrate that a given bipartite density matrix does (or not) possess a convex decomposition is not an easy task and simpler criteria to identify inseparability are needed. We know from quantum mechanics that the density matrix associated to a given state is a non-negative matrix (i.e. does not have any negative eigenvalue) and the transposition of a state is also a state of the system. However, the partial transposition of a state of a system with two or more parts is not always a quantum state. Based on this fact, Peres [42] demonstrated that if a mixed bipartite state is separable then the partially transposed density matrix is a non-negative matrix. Horodecki [43] proved that the Peres separability criterion is a necessary and sufficient condition for 2 × 2 and 2 × 3 systems, but in higher dimensions it is only a necessary condition. For continuous systems, using the Wigner representation [44], Simon [45] showed that the operation of transposition is equivalent to a mirror reflection of the momentum in phase space and, based on this fact, formulated the Peres–Horodecki separability criterion as an inequality relation which is stronger than the Heisenberg uncertainty relation. The relation introduced by Simon for a bipartite separable state is V+
i 2
Ξ ≥ 0,
Ξ=
J 0
0 −J
,J=
0 −1
1 , 0
(17)
1 ⟨{1ζˆi , 1ζˆj }⟩, i, j = 1, . . . , 4, where 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1ζi = ζi − ⟨ζi ⟩ and {1ζi , 1ζj } = 1ζi 1ζj + 1ζj 1ζi , with ζ = qˆ 1 pˆ 1 qˆ 2 pˆ 2 . Representing the variance matrix V by blocks of 2 × 2 matrices, A C V = . (18) T
where V is the variance matrix. The matrix-elements of V are given by Vij =
C
B
Simon criterion claims that a state is separable if
det A det B +
1 4
2 1 − | det C | − Tr AJCJBJC T J − (det A + det B) ≥ 0. 4
For gaussian states, the violation of the inequality (19) is a necessary and sufficient condition for entanglement.
(19)
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Another way of investigating entanglement of bipartite oscillator states is by using the hierarchy of conditions for inseparability proposed by Shchukin and Vogel [46]. One starts from the observation that a Hermitian operator is nonnegative if and only if ⟨ψ|Aˆ |ψ⟩ = Tr[Aˆ |ψ⟩ ⟨ψ|] ≥ 0 for all states |ψ⟩. The operator |ψ⟩ ⟨ψ| can be represented by ϑˆ Ď ϑˆ , with ϑˆ = |0⟩ ⟨ψ| where |0⟩ is a vacuum state. Considering that any pure bipartite oscillator state can be written as |ψ⟩ = θˆ Ď |00⟩, where θˆ = θˆ (ˆa, bˆ ), the operator ϑˆ is given by ϑˆ = |00⟩ ⟨00| θˆ . Since the operator |00⟩ ⟨00| can be expressed in the normally ordered form, the operator ϑˆ has a normally ordered form. Therefore, a Hermitian operator is nonnegative if ˆ Aˆ = Tr[Aˆ ϑˆ Ď ϑ] ˆ ≥ 0 is satisfied for any operator ϑˆ for which the normally ordered form exists. and only if the inequality ⟨ϑˆ Ď ϑ⟩ Thus the Peres–Horodecki criterion can be stated as: if a mixed bipartite state ρˆ is separable then its partially transposed
ˆ ρˆ PT = Tr[ρˆ PT ϑˆ Ď ϑ] ˆ ≥ 0. Assuming that ϑˆ = density matrix ρˆ PT satisfies the inequality ⟨ϑˆ Ď ϑ⟩ can write ˆ ρˆ PT = ⟨ϑˆ Ď ϑ⟩
∞
∗ Cpqrs Cnmkl Mpqrs,nmkl ≥ 0,
∞
n,...,l=0
n
k
Cnmkl aˆ Ď aˆ m bˆ Ď bˆ l , we
(20)
p,...,l=0
where Mpqrs,nmkl = ⟨ˆaĎq aˆ p aˆ Ďn aˆ m bˆ Ďs bˆ r bˆ Ďk bˆ l ⟩ρˆ PT = ⟨ˆaĎq aˆ p aˆ Ďn aˆ m bˆ Ďl bˆ k bˆ Ďr bˆ s ⟩ρˆ . This inequality is valid for all Cnmkl if and only if all the main minors of the form (20) are nonnegative. Performing a convenient numbering of multiindices in order to obtain the elements Mpqrs,nmkl as elements of arrays, Shchukin and Vogel [46] demonstrated that the partially transposed matrix of a bipartite quantum state is nonnegative if and only if all the determinants
1 Ď ⟨ˆa ⟩ DN = ⟨ˆa⟩ ⟨bˆ ⟩ . . .
⟨ˆaĎ ⟩ 2 ⟨ˆaĎ ⟩ ⟨ˆaaˆ Ď ⟩ ⟨ˆaĎ bˆ ⟩ .. .
⟨ˆa⟩ ⟨ˆaĎ aˆ ⟩ ⟨ˆa2 ⟩ ⟨ˆabˆ ⟩ .. .
⟨bˆ Ď ⟩ ⟨ˆaĎ bˆ Ď ⟩ ⟨ˆabˆ Ď ⟩ ⟨bˆ Ď bˆ ⟩ .. .
· · · · · · · · · · · · . . .
(21)
are nonnegative; in other words, the state is entangled if and only if ∃ N such that DN < 0. Moreover, since one can restrict the operator ϑˆ by setting some coefficients Cnmkl to zero, the Shchukin–Vogel separability conditions can be extended for the nonnegativity of subdeterminants of DN . For example, in Ref. [46] the subdeterminant
⟨bˆ Ď ⟩ ⟨bˆ Ď bˆ ⟩ ⟨ˆaĎ bˆ Ď bˆ ⟩
1 S = ⟨bˆ ⟩ ⟨ˆaĎ bˆ ⟩
⟨ˆabˆ Ď ⟩ ⟨ˆabˆ Ď bˆ ⟩ ⟨ˆaĎ aˆ bˆ Ď bˆ ⟩
(22)
was used to demonstrate the entanglement of a non-gaussian bipartite state composed of two coherent states, |ψ⟩ ∼ (|α, β⟩ − | − α, −β⟩). 3.2. Time evolution of entanglement We now address the question of the temperature dependence of the time-evolution of the entanglement for a bipartite oscillator system in contact with a thermal reservoir. We consider the initial state of the system in the form ′ ρˆ 0′ = ρˆ +− ⊗ ρˆ β′ ,
(23)
where ρˆ β′ is the density matrix describing the thermal equilibrium state of the dressed-field modes in the cavity (the reservoir),
ρˆ β′ =
′
e−β Hk
k
Tr[
′
e−β Hk ]
,
(24)
k
′ and ρˆ +−
= Ψ +− Ψ +− accounts for the center-of-mass and the relative-position oscillators initial state. We consider two
distinct families of states: bipartite, Bell-like states of the form
+− Ψ = ξ |1+ , 0− ⟩ + 1 − ξ eiφ |0+ , 1− ⟩ ,
(25)
with 0 < ξ < 1, and two-mode squeezed vacuum (gaussian) states,
+− Ψ = exp(γ aˆ ′+Ď aˆ ′−Ď − γ ∗ aˆ ′+ aˆ ′− ) |0+ , 0− ⟩ , for 0 < γ < ∞.
(26)
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Let us start analyzing the state (23) in the case where the atoms are in the Bell-like state (25). Since it is a non-gaussian state, the Simon criterion cannot be faithfully used. In fact, it can be shown that the left hand side of the inequality (19) vanishes at T = 0, for all 0 < ξ < 1. Let us then use the Shchukin–Vogel criterion and calculate the determinant S, Eq. (22), for the state (23), (25), taking aˆ = aˆ ′+ (t ) and bˆ = aˆ ′− (t ). Using Eqs. (13) and (14), we find the entries in the determinant S:
⟨ˆa′− ⟩ = ⟨ˆa′+Ď aˆ ′−Ď aˆ ′− ⟩ = 0; ⟨ˆa′−Ď aˆ ′− ⟩ = 1 − ξ ; ¯ t ⟨ˆa′+ aˆ ′−Ď ⟩ = ξ (1 − ξ )f++ (t )e−i(φ−ω) ; ∞ |f+k (t )|2 n′k (β), ⟨ˆa′+Ď aˆ ′+ aˆ ′−Ď aˆ ′− ⟩ = (1 − ξ )
(27)
k=1
where 1 ′Ď n′k (β) = ⟨ˆak aˆ ′k ⟩ = βω k e −1
(28)
is the Bose–Einstein occupation of the kth dressed-mode of the field at equilibrium at temperature T = β −1 . Then, the determinant (22) for the state (23), (25) can be written as S (t , ξ , β) = (1 − ξ )2 E (t , ξ , β),
(29)
where E (t , ξ , β) =
∞
|f+k (t )|2 n′k (β) − ξ |f++ (t )|2 .
(30)
k =1
Thus the negativity of S is determined by the negativity of E (t , ξ , β). We see that the dependence of E (t , ξ , β) on the temperature appears only in the first term through the Bose–Einstein factor. In the zero-temperature limit (β → ∞), n′k (β) → 0 for all k, and we obtain S |T →0 = −ξ (1 − ξ )2 |f++ (t )|2 ,
(31)
showing that the state (23) is entangled, at T = 0, whatever the value 0 < ξ < 1, in agreement with the result obtained in Ref. [39] using a distinct way of identifying the entanglement regime at vanishing temperature. In the case of the state (23), with the atoms in a two-mode squeezed state (26) and the field modes at thermal equilibrium, Eq. (24), we can use the Simon criterion to identify separability. Taking qˆ 1 = qˆ ′+ (pˆ 1 = pˆ ′+ ) and qˆ 2 = qˆ ′− (ˆp2 = pˆ ′− ), and using Eqs. (15) and (16), we get the blocks composing the variance matrix (18) as A=
y(t , γ , β)
ω¯ z (γ ) 1 B= 0 ω¯
1 0
0
ω¯
0
ω¯ 2
,
(32)
(33)
2
and C =
x1 ( t , γ )
ω¯
1 0
0
−ω¯
2
+ x2 (t , γ )
0 1
1 , 0
(34)
where z (γ ) = sinh2 (γ ) + y(t , γ , β) = x1 ( t , γ ) = x2 ( t , γ ) =
1 2
1 2
,
+ |f++ (t )|2 sinh2 (γ ) +
sinh(2γ ) 2γ sinh(2γ ) 2γ
(35)
|f+k (t )|2 n′k (β),
(36)
k
Re(γ e−iω¯ t f++ (t )),
(37)
Im(γ e−iω¯ t f++ (t )),
(38)
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and 0 < γ < ∞. Thus, the left hand side of the Simon inequality (19) becomes the function ES (t , γ , β) = y2 (t , γ , β)z 2 (γ ) +
1 16
2
1 − 4 x21 (t , γ ) + x22 (t , γ )
1 2 − 2y(t , γ , β)z (γ ) x21 (t , γ ) + x22 (t , γ ) − y (t , γ , β) + z 2 (γ ) .
4
(39)
The squeezed state (23), (26) is entangled if and only if ES (t , γ , β) < 0. In the zero-temperature limit n′k (β) → 0 and we obtain ES |T →0 = −|f++ (t )|2 sinh2 (γ ) + sinh4 (γ ) ;
(40)
so, for T = 0 the state (23), (26) is entangled ∀γ > 0. The dynamics of entanglement is dictated by the functions f++ (t ) and f+k (t ), which are obtained from Eq. (14). The orthonormality of the matrix (tµr ) leads directly to f++ (0) = 1 and f+k (0) = 0. These initial conditions imply that the functions S (t , ξ , β) and ES (t , γ , β), given by Eqs. (29) and (39), at t = 0, are independent of the temperature and negative for 0 < ξ < 1 and γ > 0, respectively. Therefore, at t = 0, both initial states (23) we consider, with (25) and are (26), ∞ l l + 2 + + l 2 entangled. Orthonormality of the matrix (tµr ) also leads to the relations (t+ ) = 1− ∞ l=1 (t+ ) and t+ tk = − l=1 t+ tk , which allows us to write f++ (t ) = e−iΩ0 t +
∞ (t+l )2 e−iΩl t − e−iΩ0 t ,
(41)
l =1
f +k ( t ) =
∞
l l t+ tk e−iΩl t − e−iΩ0 t .
(42)
l =1
Furthermore, the matrix elements given by Eq. (6) can be written in terms of renormalized (physical) quantities [31] leading to
η2 Ωl2
(t+l )2 =
η2
(Ωl2 − ω¯ 2 )2 + 2 (3Ωl2 − ω¯ 2 ) + π 2 g 2 Ωl2 η ωk l l t+ (t l )2 . tk = 2 ωk − Ωl2 +
,
(43) (44)
To study the effect of temperature on the time evolution of the entanglement of the state (23), we consider two distinct situations, namely, finite (small) and very-large (free space) cavities. 3.2.1. Small cavity We now consider the system (atoms plus field) inside a finite cavity with a small radius R and assume the weak-coupling regime, g ≪ ω ¯ , for the atom–field interaction. The cavity is assumed to be small but sufficiently large to accommodate the dressed-field modes with the thermal Bose–Einstein distribution [36]. In a finite cavity the field modes are discrete. Since the eigenfrequencies Ωk are close to the frequencies ωk of the field modes, we can write g
(k + ϵk ), k = 1, 2, . . . , δ with 0 < ϵk < 1. Inserting Eq. (45) into Eq. (8) and assuming that ϵk is very small, we obtain Ωk =
ϵk ≈
δg 2k . − δ 2 ω¯ 2
g 2 k2
(45)
(46)
The first normal frequency Ω0 is calculated by assuming that Ω0 R/c is small and approximating cot(x) ≈ 1/x − x/3 in Eq. (8); we get
π 2δ Ω0 ≈ ω ¯ 1− . 6
(47)
Having the normal frequencies, we can use Eqs. (43) and (44) to find the functions (41) and (42). Inserting these functions into Eqs. (30) and (36)–(39), we find the functions E (t , ξ , β) and ES (t , γ , β) describing the time evolution of the entanglement of the states (23), with (25) and (26) being the two-atom oscillator states; no further simplification can be done analytically. However, since ωk and the leading term of Ωl are proportional to k and l, respectively, the series appearing in the expressions for E and ES are convergent and we can evaluate these functions to any desired degree of accuracy by truncating the series at sufficiently large values of l and k; in our calculations we take the truncation of the series being N = 500. In Fig. 1, we present the time evolution of the function E (t , ξ , β), fixing ξ = 1/2, g = 0.1, ω ¯ = 0.67, δ = 0.03, and taking various values of the temperature T = β −1 , including T = 0 (β = ∞). We find that, at T = 0 and
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Fig. 1. Time evolution of E (t ) for the state (23), (25), with ξ = 0.5, in a finite (small) cavity with δ = 0.03, g = 0.1 and ω ¯ = 0.67 for some values of β . Upper panel: β = ∞; 0.3; 0.1465; 0.1, full-, dashed-, dotdashed- and dotted-lines, respectively. Lower panel: β = 0.00315 and 0.001, full- and dashed-lines respectively; the inset shows a zoom of the first instants of the evolution. All parameters are written in arbitrary units.
for low temperatures, the state (23), with the atoms in the Bell-like state (25), remains entangled, with the function E (t , 1/2, β) being oscillatory (although not periodic) but negative for all times, as shown in the two lower curve in the upper panel of Fig. 1. But, as the temperature is raised, we reach a value of β1−1 = T1 ≈ 6.83 (see the dotdashed-curve) such that, above it, the system alternate sudden deaths and rebirths of entanglement as indicated by the dotted-line in the upper panel of Fig. 1. As shown in the lower panel of Fig. 1, further raising the temperature, one finds a value of β2−1 = T2 ≈ 317.5 (see the full-curve in the lower panel) above which the entanglement disappear at a finite time and is not regained anymore. As the temperature varies from T1 and T2 , the time intervals for which the system stays separable become longer while the time intervals for which the system remains entangled get shorter and shorter until disappear at T2 ; naturally, the values of T1 and T2 depend on the values of the parameters g, ω ¯ and δ . The behavior of the function ES (t , γ , β), describing the entanglement of the gaussian state (23), (26), follows a similar pattern, as shown in Fig. 2. One can determine the temperatures T1 and T2 , but since we are using arbitrary units these numbers are not important here. 3.2.2. Large cavity For a very large cavity, actually in the limit of R → ∞ corresponding to the free space, Ω0 → 0, Ωl → ωl and Ωl+1 − Ωl ≈ 1ω = π c /R is infinitesimal. In this case, the frequencies Ωl and Ωk can be taken as continuous variables and the sums in Eqs. (30), (36), (41) and (42) become Riemann summations over equally spaced partitions of the interval (0, ∞). Moreover, since η2 = 2g 1ω, we can neglect the second parcel in the denominator of Eqs. (43) and (41) can be rewritten as f++ (t ) = C1 (t ) + iS1 (t ),
(48)
where C1 (t ) = 1 + 2g
∞
dα 0
S1 (t ) = −2g
0
∞
dα
α 2 [cos(α t ) − 1] , (α 2 − ω¯ 2 )2 + π 2 g 2 α 2
α 2 sin(α t ) . (α − ω¯ 2 )2 + π 2 g 2 α 2 2
(49)
(50)
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Fig. 2. Time evolution of ES (t ) for the state (23), (26), with γ = 0.5, in a finite (small) cavity with δ = 0.03, g = 0.1 and ω ¯ = 0.67 for some values of β . Upper panel: β = ∞, 0.1 and 0.06, full-, dashed- and dotted-lines respectively. Lower panel: β = 0.004624 and 0.0035, full- and dashed-lines respectively; the inset shows a zoom of the first instants of the evolution.
Notice that C1 (0) = 1 and S1 (0) = 0, ensuring that f++ (0) = 1 as required by the orthonormality of the matrix (tµr ). By using the residue theorem, we can calculate the real part of f++ (t ), obtaining
πg sin kt , C1 (t ) = e−π gt /2 cos kt − 2k
(51)
where k2 = ω ¯ 2 − π 2 g 2 /4 and the weak-coupling regime (k2 > 0) was considered. For S1 (t ), however, the integral cannot be analytically evaluated and therefore a numerical solution is employed. Analogously, in the continuum limit, using Eqs. (43) and (44) with the replacements ωk → ω and Ωl → α , Eq. (42) can be rewritten as
√
f+ω (t ) = ω 1ω [C2 (ω, t ) + iS2 (ω, t )] ,
(52)
where [47] 2/3
C2 (ω, t ) = (2g )
∞
dα 0
S2 (ω, t ) = −(2g )2/3
α 2 [cos(α t ) − 1] , (ω2 − α 2 ) (α 2 − ω¯ 2 )2 + π 2 g 2 α 2
∞
dα 0
(53)
α 2 sin(α t ) . (ω2 − α 2 ) (α 2 − ω¯ 2 )2 + π 2 g 2 α 2
(54)
The integrals appearing in the definitions of the functions C2 and S2 , distinctly from the cases of C1 and S1 , have to be considered as Cauchy principal values due to the singularity in the integrand. As before, the integral involving cos(α t ) can be performed analytically and one finds C2 (ω, t ) =
(ω2 − ω¯ 2 ) cos(kt ) π g (ω2 + ω¯ 2 ) sin(kt ) − (ω2 − ω¯ 2 )2 + π 2 g 2 ω2 2k (ω2 − ω ¯ 2 )2 + π 2 g 2 ω2 2 2 π g ω sin ωt (ω − ω¯ ) + 2 − 2 , (ω − ω¯ 2 )2 + π 2 g 2 ω2 (ω − ω¯ 2 )2 + π 2 g 2 ω2
2g e−π gt /2
(55)
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Fig. 3. Time evolution of E (t ) for the state (23), (25), with ξ = 0.5, in an infinitely large cavity with g = 0.1 and ω ¯ = 1.0, for some values of β . Upper panel: β = ∞; 1.2; 0.9, full-, dashed- and dotted-lines, respectively. Lower panel: β = 0.175; 0.08, full- and dashed-lines respectively.
whereas the S2 integral can only be evaluated numerically. Thus, using Eq. (52), for an infinitely large cavity, the sum appearing in Eqs. (30) and (36) can be evaluated with the prescription
f+k 2 n′k
|
|
∞
→ 0
k
ω2 C22 (ω, t ) + S22 (ω, t ) . dω eβω − 1
(56)
Notice that, consistently with the orthogonality of the matrix (tµr ), both C2 and S2 functions vanish at t = 0 and, therefore, f+ω (0) = 0. Inserting Eqs. (48) and (56) into Eq. (30) we obtain the function describing the influence of the temperature in the dynamics of entanglement of the state (23), (25), E (t , ξ , β) = −ξ
C12
(t ) +
(t ) +
S12
∞
0
ω2 C22 (ω, t ) + S22 (ω, t ) dω . eβω − 1
(57)
The temperature dependence of the time-evolution of the function E (t , ξ , β) is illustrated in Fig. 3, considering ξ = 1/2, g = 0.1, ω ¯ = 1.0 and some values of β . For T = 0 (β → ∞), we find that the entanglement decays asymptotically, with the state becoming separable only for very long times (see the full line in the upper panel). However, for finite temperatures, the sudden death of entanglement occurs. If the temperature is small, entanglement dies and revives a few times before completely disappear (see dashed- and dotted-lines in the upper panel). Further increasing the temperature, we reach a value β1−1 = T1 ≈ 5.71 (full line in the lower panel) above which the sudden death occurs at a short time and entanglement is lost without any rebirth; the value of T1 depends on the parameters g and ω ¯. For the gaussian state (23), (26), in the limit of infinitely large cavity, we find y(t , γ , β) =
1 2
+ sinh (γ )
x21 (t , γ ) + x22 (t , γ ) =
2
C12
(t ) +
S12
(t ) +
∞
0
sinh2 (2γ ) 2 C1 (t ) + S12 (t ) ; 4
ω2 C22 (ω, t ) + S22 (ω, t ) dω , eβω − 1
(58)
(59)
inserting these expressions into Eq. (39), we obtain the function ES (t , γ , β) describing the effect of temperature on the dynamics of entanglement of the state (23), (26). As illustrated in Fig. 4, the entanglement decays asymptotically for T = 0
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Fig. 4. Time evolution of ES (t ) for the state (23), (26), with γ = 0.8, in an infinitely large cavity with g = 0.1 and ω ¯ = 1.0, for some values of β : β = ∞; 1.5; 0.88 full-, dashed- and dotted-lines, respectively.
while, at finite temperatures, the sudden death occurs in a similar way to the case of the Bell-like state: up to a temperature T1 , one gets an alternation between death and revival of entanglement before the true death is established but, above it, just a single sudden death appears. 4. Conclusions In the present work, we have studied the entanglement of two-atom states coupled with field modes confined in a cavity at finite temperature. We have used the dressed-state formalism to consider the time evolution of the entanglement of Bell-like and gaussian states of the dressed atoms; finite-temperature effects were taken into account by assuming that the dressed-field modes obey the Bose–Einstein distribution at temperature T = β −1 . Using the Shchukin–Vogel and the Simon criteria to identifying entanglement, we have analyzed the time evolution of the entanglement of these two family of states for some values of the temperature, in two different situations: a finite (small) cavity and a infinitely large cavity (free space); we found distinct behaviors for these two cases. In a small cavity, as shown in Figs. 1 and 2, for T = 0 the entanglement is maintained as time evolves; this also happens for low temperatures of the dressed-field modes. As the temperature is raised, one reaches a value (T1 ) above which the system evolves in time alternating between sudden death and rebirth of the entanglement. This behavior remains up to a temperature (T2 ) above which a single sudden death occurs and the entanglement does not revive anymore. In the case of a very large cavity, actually in the limit R → ∞, we find that the decay of entanglement is asymptotic only for the environment at zero temperature. For low temperatures, as illustrated in Figs. 3 and 4, one has few sudden deaths and rebirths of the entanglement before the system becomes separable. Above an specific value of the temperature (T1 ), a sudden death of the entanglement occurs at a short time and it is not recovered at any instant later. The survival of entanglement of the state (23) in a small cavity as the time evolves, at low temperatures, may be associated to a feedback process between the atoms and the field in the cavity that takes place in this confining situation; in contrast, this does not happens in a very large cavity. These aspects should be considered in attempts to implement devices that use entanglement as an important resource and work at finite temperatures. Acknowledgments The authors thank CNPq and CAPES, Brazilian Agencies, for partial financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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Physica A journal homepage: www.elsevier.com/locate/physa
Time evolution of entanglement in a cavity at finite temperature E.G. Figueiredo a,b,∗ , C.A. Linhares c , A.P.C. Malbouisson d , J.M.C. Malbouisson a a
Instituto de Física, Universidade Federal da Bahia, 40210-340, Salvador, BA, Brazil Centro de Ciência e Tecnologia em Energia e Sustentabilidade, Universidade Federal do Recôncavo da Bahia, 44085-132, Feira de Santana, BA, Brazil c Instituto de Física, Universidade do Estado do Rio de Janeiro, 20559-900, Rio de Janeiro, RJ, Brazil b
d
Centro Brasileiro de Pesquisas Físicas, MCTI, 22290-180, Rio de Janeiro, RJ, Brazil
highlights • Exact entanglement indicators are given for two oscillators in a cavity at finite T. • Entanglement survives at finite T for bipartite systems in a finite (small) cavity. • Sudden death of entanglement at finite T occurs for bipartite systems in free space.
article
info
Article history: Received 18 February 2016 Received in revised form 6 June 2016 Available online 12 July 2016 Keywords: Entanglement Time evolution Finite-temperature
abstract We consider two identical atoms, taken in the harmonic approximation, inside a spherical cavity filled with a field described by an infinite set of oscillators. The atoms are linearly coupled to the field. Using the dressed-state approach and considering the field modes populated with thermal-distribution, we study the effect of temperature on the time evolution of entangled states of the pair of atoms. We particularly analyze the cases of finite and very-large cavities showing that survival and sudden-death of entanglement can happen at finite temperature. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Quantum entanglement corresponds to correlations between separated parts of compound systems, which may even not interact directly, that cannot be conceived in any classical-physics description. Despite the nonintuitive character of quantum mechanics, particularly of entanglement, experimental implementations have demonstrated several successful applications in the realm of the theory of quantum information and in quantum computer science. The teleportation of information between two parts of a system, cryptographic protocols that cannot be violated and the increase in the processing speed of computations, as compared to classical algorithms, are some examples of these advances [1–3]. All these achievements rely heavily on the fact that entanglement is the fundamental resource to implement the processes. Despite the great progress in the theory of quantum entanglement [4], many fundamental problems are still unsolved; identify entanglement in multipartite systems and in mixed states of high-dimension bipartite systems are some of these
∗ Corresponding author at: Centro de Ciência e Tecnologia em Energia e Sustentabilidade, Universidade Federal do Recôncavo da Bahia, 44085-132, Feira de Santana, BA, Brazil. E-mail addresses: [email protected] (E.G. Figueiredo), [email protected] (C.A. Linhares), [email protected] (A.P.C. Malbouisson), [email protected] (J.M.C. Malbouisson). http://dx.doi.org/10.1016/j.physa.2016.06.128 0378-4371/© 2016 Elsevier B.V. All rights reserved.
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problems. Usually multipartite systems do not exist isolated and the effect of the interaction with the environment should be taken into account. In this case, survival of entanglement is not an easy task due to decoherence effects [1,2]. Some methods are known to preserve entanglement, for example, the quantum error correction [5]. The cases in which entanglement survives are of fundamental importance for processes in quantum information science. In various situations, the interaction with the environment leads to the disappearance of the entanglement as time evolves, either asymptotically or in a finite time (sudden death) [6–8]. For instance, in Refs. [9,10] the authors show that the decay of the entanglement of Bell-like superposition can occur asymptotically or by sudden death depending on the parameters of the superposition. For the environment at zero temperature, revival and induced entanglement are originated by system–reservoir correlations [11–14]. Entangled states have also been considered to explain some macroscopic properties of some solids, such as the magnetic susceptibility and the heat capacity [15,16]. Entanglement dynamics of gaussian states interacting with a non-zero temperature environment was studied in Refs. [17,18] and entanglement of Brownian particles were considered in Refs. [19,20]. At finite temperature, one also finds entanglement survival, oscillations between sudden deaths and revival and even induced entanglement [21–29]; most of these works use gaussian or two-level systems with the formalism of master equation for the reduced density matrix. The time evolution of entanglement for a system in contact with the environment is then an important question for physical applications; a review of the open-system dynamics of entanglement appear in Ref. [30]. In this paper, we address some of these questions by considering two families of states: gaussian, squeezed vacuum states and, non-gaussian, Bell-like states of two particles (atoms) confined in a spherical cavity of radius R, focusing on the time evolution of the entanglement. The atoms are approximated by harmonic oscillators and are coupled with an environment modeled by an infinite set of harmonic oscillators at finite temperature. Distinctly of the previous treatments, we consider the states of the atoms as ‘‘dressed’’ by the thermal field with the method presented in Refs. [31,32], which allows for analytical results. We show that the effect of the temperature on the system is rather distinct for finite and infinite cavities. In very large cavities (free space), the raising of temperature leads to a sudden death of the entanglement, while for zero temperature the decay is asymptotic. For small cavities, we find an oscillatory behavior with sudden death alternating with revivals of the entanglement and a limit temperature for which the system remains entangled. 2. Atoms in a cavity in the dressed state formalism 2.1. The model Let us consider two atoms A and B, taken in the harmonic approximation, coupled to the environment through a linear position–position interaction with the modes of a field in a perfectly reflecting spherical cavity of radius R. We do not consider any direct interaction between the atoms so that the Hamiltonian is given by H =
N N 2 1 1 2 pA + ωA2 q2A + p2B + ωB2 q2B + pk + ωk2 q2k − (ηA qA + ηB qB )ωk qk , 2 2 k=1 k=1
(1)
where ηA and ηB are constants and the limit N → ∞ will be undertaken. Our model is a slightly modified version of the model used in Ref. [21]. The coupling between the two atoms is indirect, mediated by the field in the cavity. √ We assume that the atoms A and B couple in the same way with the environment and take ηA = ηB = g 1ω, where g is a constant with the dimension of frequency measuring the strength of the coupling. The quantity 1ω = π c /R is the interval between neighboring frequencies of the field, which are given by ωk = kπ c /R [31]. Let us now define new coordinates, q+ (center of mass) and q− (relative position), such that 1 q+ = √ (qA + qB ), 2
1 q− = √ (qA − qB ), 2
(2)
√
with momenta p± = (pA ± pB )/ 2. In termsof {q± , p± }, for identical atoms (ωA = ωB = ω0 ), the Hamiltonian is written as H = H− + H+ , where H− = 21 p2− + ω02 q2− and H+ =
√
N N 1 2 1 2 p+ + ω02 q2+ + pk + ωk2 q2k − ηωk q+ qk , 2 2 k=1 k=1
(3)
with η = 2g 1ω, 1ω = π c /R and ωk = k1ω. We see that the center-of-mass oscillator q+ couples to the field while the relative-position oscillator q− evolves freely in time. Eq. (3) shows that H+ is identical to the Hamiltonian of a single oscillator coupled to the field. Thus, for the system composed by the center-of-mass oscillator and the field (q+ ⊕ field), we can use the concepts of dressed coordinates and dressed states introduced in Ref. [31]. This formalism has been employed to investigate the time evolution of superposed single-atom states [33], thermal effects on single atoms [34–37] and the time evolution of bipartite states [38,39].
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2.2. Dressing the center-of-mass oscillator The formalism developed in previous papers requires that first the Hamiltonian (3) be transformed to principal axis by means of a point transformation, N
qµ =
tµr Qr ,
pµ =
N
r =0
tµr Pr ,
(4)
r =0
where µ = {+, k }, k = 1, 2, . . . , N, through an orthonormal matrix (tµr ). Here, we use the subscripts µ = + (or 0) and µ = k to designate respectively the center-of-mass oscillator q+ and the harmonic modes of the environment field; the subscript r refers to the normal modes. Writing the transformed Hamiltonian in terms of normal momenta and coordinates, it becomes H =
N 1 2 Pr + Ωr2 Qr2 . 2 r =0
(5)
In this expression, the Ωr ’s are the normal frequencies related to the stable oscillation modes of the coupled system (q+ ⊕ field). It can be shown [31] that
ηωk tkr = 2 tr , ωk − Ωr2 +
r t+ =
1+
N k=1
with the condition ω02 − Ωr2 =
η2 ωk2 (ωk2 − Ωr2 )2
η2 ωk2
N
− 12 ,
k=1 ω2 −Ω 2 ; this sum diverges for N r k
(6)
→ ∞, so that we need to implement a renormalization
procedure to redefine the frequency [40]. It leads to
ω¯ 2 − Ωr2 = η2 Ωr2
∞
1
k=1
ωk2 − Ωr2
,
(7)
where ω ¯ 2 = limN →∞ (ω02 − N η2 ) is the renormalized frequency. Observe that the divergence of the bare frequency of the center-of-mass oscillator q+ in the limit N → ∞ is due to its interaction with the field. This does not occur for the frequency of the relative-position oscillator q− , which remains finite. However, we choose the (infinite) bare frequency of the q+ oscillator in such a way that the renormalized frequency ω ¯ equals the (finite) relative-position oscillator frequency. This has to be done for consistency, in considering two identical oscillators. Using that ωk = π ck/R and performing the summation, Eq. (7) becomes (without frequency labels)
cot
RΩ
c
Ω c = + πg RΩ
1−
Rω ¯2
π gc
;
(8)
the roots of Eq. (8) give the spectrum of eigenfrequencies Ωr (r = 0, 1, 2, . . .). To ensure the existence of the first root Ω0 , we assume that δ ω ¯ 2 /g 2 > 1 where δ = 2g 2 /η2 = gR/π c. We then consider the energy eigenstates of the system (q+ ⊕ field),
φn0 n1 n2 ... (Q ) =
√
1
2ns ns !
s
√ Hns Ωs Qs Γ0 ,
(9)
where Hns denotes the ns th Hermite polynomial and Γ0 is the normalized ground-state eigenfunction. For the system
(q+ ⊕ field) we define dressed coordinates by [31] √ ω¯ µ q′µ = tµr Ωr Qr ,
(10)
r
where q′µ = {q′+ , q′k } and ω ¯ µ = {ω, ¯ ωk }. Since the relative-position coordinate does not couple with the field, we have q′− = q− . In terms of the dressed coordinates, we define for a fixed initial instant, t = 0, dressed states, |κ+ , κ1 , κ2 , . . .⟩, by means of the complete orthonormal set of functions
ψκ+ κ1 ... (q ) = ′
1
µ
2κµ κµ !
Hκµ
ω¯ µ q′µ
Γ0 .
(11)
Distinctly from the energy eigenstates, which are stationary states of the system, the dressed states (with exception of the ground state) evolve in time.
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From the dressed coordinates and the dressed momenta, we define the dressed annihilation operator
aˆ ′µ
=
ω¯ µ
qˆ ′µ +
2
i 2ω ¯µ
pˆ ′µ ,
(12)
and its adjoint, the dressed creation operator aˆ ′µĎ . The time evolution of aˆ ′µ is given by [36] aˆ ′µ (t ) = fµν (t ) =
∞ ν=0 ∞
fµν (t )ˆa′ν ,
(13)
tµr tνr e−iΩr t .
(14)
r =0
On the other hand, the relative-position annihilation operator evolves in time with its free Hamiltonian, so that aˆ ′− (t ) = √ √ ′Ď e−iω¯ t aˆ ′− ; thus, qˆ ′− (t ) = (ˆa′− e−iω¯ t + aˆ − eiω¯ t )/ 2ω ¯ , pˆ ′− (t ) = −i(ˆa′− e−iω¯ t − aˆ ′−Ď eiω¯ t )/ 2/ω¯ , while
1 ∗ qˆ ′+ (t ) = √ f+µ (t )ˆa′µ + f+µ (t )ˆa′µĎ , 2ω ¯ µ pˆ ′+ (t )
= −i
ω¯ 2 µ
(15)
∗ f+µ (t )ˆa′µ − f+µ (t )ˆa′µĎ .
(16)
3. Temperature effects on the time evolution of entanglement We now address the question of the temperature dependence of the time-evolution of the entanglement for a bipartite system in contact with a thermal reservoir. First, we briefly recall methods to identifying entanglement in a bipartite mixed state. 3.1. Bipartite entanglement The notions of separability and entanglement (inseparability) can be directly settled for bipartite pure states: separable states can be written as direct products of states of the parts, while entangled states cannot. For mixed bipartite states, the definition of separability is more subtle: separable states are those for which the density matrix can be written as a convex 1 2 sum of direct products of density matrices of the two subsystems [41], that is ρˆ = p σ ˆ ⊗ σ ˆ , with p ≥ 0 and i i i i i i pi = 1; otherwise, the mixed states are said to be entangled. However, to demonstrate that a given bipartite density matrix does (or not) possess a convex decomposition is not an easy task and simpler criteria to identify inseparability are needed. We know from quantum mechanics that the density matrix associated to a given state is a non-negative matrix (i.e. does not have any negative eigenvalue) and the transposition of a state is also a state of the system. However, the partial transposition of a state of a system with two or more parts is not always a quantum state. Based on this fact, Peres [42] demonstrated that if a mixed bipartite state is separable then the partially transposed density matrix is a non-negative matrix. Horodecki [43] proved that the Peres separability criterion is a necessary and sufficient condition for 2 × 2 and 2 × 3 systems, but in higher dimensions it is only a necessary condition. For continuous systems, using the Wigner representation [44], Simon [45] showed that the operation of transposition is equivalent to a mirror reflection of the momentum in phase space and, based on this fact, formulated the Peres–Horodecki separability criterion as an inequality relation which is stronger than the Heisenberg uncertainty relation. The relation introduced by Simon for a bipartite separable state is V+
i 2
Ξ ≥ 0,
Ξ=
J 0
0 −J
,J=
0 −1
1 , 0
(17)
1 ⟨{1ζˆi , 1ζˆj }⟩, i, j = 1, . . . , 4, where 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 1ζi = ζi − ⟨ζi ⟩ and {1ζi , 1ζj } = 1ζi 1ζj + 1ζj 1ζi , with ζ = qˆ 1 pˆ 1 qˆ 2 pˆ 2 . Representing the variance matrix V by blocks of 2 × 2 matrices, A C V = . (18) T
where V is the variance matrix. The matrix-elements of V are given by Vij =
C
B
Simon criterion claims that a state is separable if
det A det B +
1 4
2 1 − | det C | − Tr AJCJBJC T J − (det A + det B) ≥ 0. 4
For gaussian states, the violation of the inequality (19) is a necessary and sufficient condition for entanglement.
(19)
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Another way of investigating entanglement of bipartite oscillator states is by using the hierarchy of conditions for inseparability proposed by Shchukin and Vogel [46]. One starts from the observation that a Hermitian operator is nonnegative if and only if ⟨ψ|Aˆ |ψ⟩ = Tr[Aˆ |ψ⟩ ⟨ψ|] ≥ 0 for all states |ψ⟩. The operator |ψ⟩ ⟨ψ| can be represented by ϑˆ Ď ϑˆ , with ϑˆ = |0⟩ ⟨ψ| where |0⟩ is a vacuum state. Considering that any pure bipartite oscillator state can be written as |ψ⟩ = θˆ Ď |00⟩, where θˆ = θˆ (ˆa, bˆ ), the operator ϑˆ is given by ϑˆ = |00⟩ ⟨00| θˆ . Since the operator |00⟩ ⟨00| can be expressed in the normally ordered form, the operator ϑˆ has a normally ordered form. Therefore, a Hermitian operator is nonnegative if ˆ Aˆ = Tr[Aˆ ϑˆ Ď ϑ] ˆ ≥ 0 is satisfied for any operator ϑˆ for which the normally ordered form exists. and only if the inequality ⟨ϑˆ Ď ϑ⟩ Thus the Peres–Horodecki criterion can be stated as: if a mixed bipartite state ρˆ is separable then its partially transposed
ˆ ρˆ PT = Tr[ρˆ PT ϑˆ Ď ϑ] ˆ ≥ 0. Assuming that ϑˆ = density matrix ρˆ PT satisfies the inequality ⟨ϑˆ Ď ϑ⟩ can write ˆ ρˆ PT = ⟨ϑˆ Ď ϑ⟩
∞
∗ Cpqrs Cnmkl Mpqrs,nmkl ≥ 0,
∞
n,...,l=0
n
k
Cnmkl aˆ Ď aˆ m bˆ Ď bˆ l , we
(20)
p,...,l=0
where Mpqrs,nmkl = ⟨ˆaĎq aˆ p aˆ Ďn aˆ m bˆ Ďs bˆ r bˆ Ďk bˆ l ⟩ρˆ PT = ⟨ˆaĎq aˆ p aˆ Ďn aˆ m bˆ Ďl bˆ k bˆ Ďr bˆ s ⟩ρˆ . This inequality is valid for all Cnmkl if and only if all the main minors of the form (20) are nonnegative. Performing a convenient numbering of multiindices in order to obtain the elements Mpqrs,nmkl as elements of arrays, Shchukin and Vogel [46] demonstrated that the partially transposed matrix of a bipartite quantum state is nonnegative if and only if all the determinants
1 Ď ⟨ˆa ⟩ DN = ⟨ˆa⟩ ⟨bˆ ⟩ . . .
⟨ˆaĎ ⟩ 2 ⟨ˆaĎ ⟩ ⟨ˆaaˆ Ď ⟩ ⟨ˆaĎ bˆ ⟩ .. .
⟨ˆa⟩ ⟨ˆaĎ aˆ ⟩ ⟨ˆa2 ⟩ ⟨ˆabˆ ⟩ .. .
⟨bˆ Ď ⟩ ⟨ˆaĎ bˆ Ď ⟩ ⟨ˆabˆ Ď ⟩ ⟨bˆ Ď bˆ ⟩ .. .
· · · · · · · · · · · · . . .
(21)
are nonnegative; in other words, the state is entangled if and only if ∃ N such that DN < 0. Moreover, since one can restrict the operator ϑˆ by setting some coefficients Cnmkl to zero, the Shchukin–Vogel separability conditions can be extended for the nonnegativity of subdeterminants of DN . For example, in Ref. [46] the subdeterminant
⟨bˆ Ď ⟩ ⟨bˆ Ď bˆ ⟩ ⟨ˆaĎ bˆ Ď bˆ ⟩
1 S = ⟨bˆ ⟩ ⟨ˆaĎ bˆ ⟩
⟨ˆabˆ Ď ⟩ ⟨ˆabˆ Ď bˆ ⟩ ⟨ˆaĎ aˆ bˆ Ď bˆ ⟩
(22)
was used to demonstrate the entanglement of a non-gaussian bipartite state composed of two coherent states, |ψ⟩ ∼ (|α, β⟩ − | − α, −β⟩). 3.2. Time evolution of entanglement We now address the question of the temperature dependence of the time-evolution of the entanglement for a bipartite oscillator system in contact with a thermal reservoir. We consider the initial state of the system in the form ′ ρˆ 0′ = ρˆ +− ⊗ ρˆ β′ ,
(23)
where ρˆ β′ is the density matrix describing the thermal equilibrium state of the dressed-field modes in the cavity (the reservoir),
ρˆ β′ =
′
e−β Hk
k
Tr[
′
e−β Hk ]
,
(24)
k
′ and ρˆ +−
= Ψ +− Ψ +− accounts for the center-of-mass and the relative-position oscillators initial state. We consider two
distinct families of states: bipartite, Bell-like states of the form
+− Ψ = ξ |1+ , 0− ⟩ + 1 − ξ eiφ |0+ , 1− ⟩ ,
(25)
with 0 < ξ < 1, and two-mode squeezed vacuum (gaussian) states,
+− Ψ = exp(γ aˆ ′+Ď aˆ ′−Ď − γ ∗ aˆ ′+ aˆ ′− ) |0+ , 0− ⟩ , for 0 < γ < ∞.
(26)
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Let us start analyzing the state (23) in the case where the atoms are in the Bell-like state (25). Since it is a non-gaussian state, the Simon criterion cannot be faithfully used. In fact, it can be shown that the left hand side of the inequality (19) vanishes at T = 0, for all 0 < ξ < 1. Let us then use the Shchukin–Vogel criterion and calculate the determinant S, Eq. (22), for the state (23), (25), taking aˆ = aˆ ′+ (t ) and bˆ = aˆ ′− (t ). Using Eqs. (13) and (14), we find the entries in the determinant S:
⟨ˆa′− ⟩ = ⟨ˆa′+Ď aˆ ′−Ď aˆ ′− ⟩ = 0; ⟨ˆa′−Ď aˆ ′− ⟩ = 1 − ξ ; ¯ t ⟨ˆa′+ aˆ ′−Ď ⟩ = ξ (1 − ξ )f++ (t )e−i(φ−ω) ; ∞ |f+k (t )|2 n′k (β), ⟨ˆa′+Ď aˆ ′+ aˆ ′−Ď aˆ ′− ⟩ = (1 − ξ )
(27)
k=1
where 1 ′Ď n′k (β) = ⟨ˆak aˆ ′k ⟩ = βω k e −1
(28)
is the Bose–Einstein occupation of the kth dressed-mode of the field at equilibrium at temperature T = β −1 . Then, the determinant (22) for the state (23), (25) can be written as S (t , ξ , β) = (1 − ξ )2 E (t , ξ , β),
(29)
where E (t , ξ , β) =
∞
|f+k (t )|2 n′k (β) − ξ |f++ (t )|2 .
(30)
k =1
Thus the negativity of S is determined by the negativity of E (t , ξ , β). We see that the dependence of E (t , ξ , β) on the temperature appears only in the first term through the Bose–Einstein factor. In the zero-temperature limit (β → ∞), n′k (β) → 0 for all k, and we obtain S |T →0 = −ξ (1 − ξ )2 |f++ (t )|2 ,
(31)
showing that the state (23) is entangled, at T = 0, whatever the value 0 < ξ < 1, in agreement with the result obtained in Ref. [39] using a distinct way of identifying the entanglement regime at vanishing temperature. In the case of the state (23), with the atoms in a two-mode squeezed state (26) and the field modes at thermal equilibrium, Eq. (24), we can use the Simon criterion to identify separability. Taking qˆ 1 = qˆ ′+ (pˆ 1 = pˆ ′+ ) and qˆ 2 = qˆ ′− (ˆp2 = pˆ ′− ), and using Eqs. (15) and (16), we get the blocks composing the variance matrix (18) as A=
y(t , γ , β)
ω¯ z (γ ) 1 B= 0 ω¯
1 0
0
ω¯
0
ω¯ 2
,
(32)
(33)
2
and C =
x1 ( t , γ )
ω¯
1 0
0
−ω¯
2
+ x2 (t , γ )
0 1
1 , 0
(34)
where z (γ ) = sinh2 (γ ) + y(t , γ , β) = x1 ( t , γ ) = x2 ( t , γ ) =
1 2
1 2
,
+ |f++ (t )|2 sinh2 (γ ) +
sinh(2γ ) 2γ sinh(2γ ) 2γ
(35)
|f+k (t )|2 n′k (β),
(36)
k
Re(γ e−iω¯ t f++ (t )),
(37)
Im(γ e−iω¯ t f++ (t )),
(38)
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and 0 < γ < ∞. Thus, the left hand side of the Simon inequality (19) becomes the function ES (t , γ , β) = y2 (t , γ , β)z 2 (γ ) +
1 16
2
1 − 4 x21 (t , γ ) + x22 (t , γ )
1 2 − 2y(t , γ , β)z (γ ) x21 (t , γ ) + x22 (t , γ ) − y (t , γ , β) + z 2 (γ ) .
4
(39)
The squeezed state (23), (26) is entangled if and only if ES (t , γ , β) < 0. In the zero-temperature limit n′k (β) → 0 and we obtain ES |T →0 = −|f++ (t )|2 sinh2 (γ ) + sinh4 (γ ) ;
(40)
so, for T = 0 the state (23), (26) is entangled ∀γ > 0. The dynamics of entanglement is dictated by the functions f++ (t ) and f+k (t ), which are obtained from Eq. (14). The orthonormality of the matrix (tµr ) leads directly to f++ (0) = 1 and f+k (0) = 0. These initial conditions imply that the functions S (t , ξ , β) and ES (t , γ , β), given by Eqs. (29) and (39), at t = 0, are independent of the temperature and negative for 0 < ξ < 1 and γ > 0, respectively. Therefore, at t = 0, both initial states (23) we consider, with (25) and are (26), ∞ l l + 2 + + l 2 entangled. Orthonormality of the matrix (tµr ) also leads to the relations (t+ ) = 1− ∞ l=1 (t+ ) and t+ tk = − l=1 t+ tk , which allows us to write f++ (t ) = e−iΩ0 t +
∞ (t+l )2 e−iΩl t − e−iΩ0 t ,
(41)
l =1
f +k ( t ) =
∞
l l t+ tk e−iΩl t − e−iΩ0 t .
(42)
l =1
Furthermore, the matrix elements given by Eq. (6) can be written in terms of renormalized (physical) quantities [31] leading to
η2 Ωl2
(t+l )2 =
η2
(Ωl2 − ω¯ 2 )2 + 2 (3Ωl2 − ω¯ 2 ) + π 2 g 2 Ωl2 η ωk l l t+ (t l )2 . tk = 2 ωk − Ωl2 +
,
(43) (44)
To study the effect of temperature on the time evolution of the entanglement of the state (23), we consider two distinct situations, namely, finite (small) and very-large (free space) cavities. 3.2.1. Small cavity We now consider the system (atoms plus field) inside a finite cavity with a small radius R and assume the weak-coupling regime, g ≪ ω ¯ , for the atom–field interaction. The cavity is assumed to be small but sufficiently large to accommodate the dressed-field modes with the thermal Bose–Einstein distribution [36]. In a finite cavity the field modes are discrete. Since the eigenfrequencies Ωk are close to the frequencies ωk of the field modes, we can write g
(k + ϵk ), k = 1, 2, . . . , δ with 0 < ϵk < 1. Inserting Eq. (45) into Eq. (8) and assuming that ϵk is very small, we obtain Ωk =
ϵk ≈
δg 2k . − δ 2 ω¯ 2
g 2 k2
(45)
(46)
The first normal frequency Ω0 is calculated by assuming that Ω0 R/c is small and approximating cot(x) ≈ 1/x − x/3 in Eq. (8); we get
π 2δ Ω0 ≈ ω ¯ 1− . 6
(47)
Having the normal frequencies, we can use Eqs. (43) and (44) to find the functions (41) and (42). Inserting these functions into Eqs. (30) and (36)–(39), we find the functions E (t , ξ , β) and ES (t , γ , β) describing the time evolution of the entanglement of the states (23), with (25) and (26) being the two-atom oscillator states; no further simplification can be done analytically. However, since ωk and the leading term of Ωl are proportional to k and l, respectively, the series appearing in the expressions for E and ES are convergent and we can evaluate these functions to any desired degree of accuracy by truncating the series at sufficiently large values of l and k; in our calculations we take the truncation of the series being N = 500. In Fig. 1, we present the time evolution of the function E (t , ξ , β), fixing ξ = 1/2, g = 0.1, ω ¯ = 0.67, δ = 0.03, and taking various values of the temperature T = β −1 , including T = 0 (β = ∞). We find that, at T = 0 and
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Fig. 1. Time evolution of E (t ) for the state (23), (25), with ξ = 0.5, in a finite (small) cavity with δ = 0.03, g = 0.1 and ω ¯ = 0.67 for some values of β . Upper panel: β = ∞; 0.3; 0.1465; 0.1, full-, dashed-, dotdashed- and dotted-lines, respectively. Lower panel: β = 0.00315 and 0.001, full- and dashed-lines respectively; the inset shows a zoom of the first instants of the evolution. All parameters are written in arbitrary units.
for low temperatures, the state (23), with the atoms in the Bell-like state (25), remains entangled, with the function E (t , 1/2, β) being oscillatory (although not periodic) but negative for all times, as shown in the two lower curve in the upper panel of Fig. 1. But, as the temperature is raised, we reach a value of β1−1 = T1 ≈ 6.83 (see the dotdashed-curve) such that, above it, the system alternate sudden deaths and rebirths of entanglement as indicated by the dotted-line in the upper panel of Fig. 1. As shown in the lower panel of Fig. 1, further raising the temperature, one finds a value of β2−1 = T2 ≈ 317.5 (see the full-curve in the lower panel) above which the entanglement disappear at a finite time and is not regained anymore. As the temperature varies from T1 and T2 , the time intervals for which the system stays separable become longer while the time intervals for which the system remains entangled get shorter and shorter until disappear at T2 ; naturally, the values of T1 and T2 depend on the values of the parameters g, ω ¯ and δ . The behavior of the function ES (t , γ , β), describing the entanglement of the gaussian state (23), (26), follows a similar pattern, as shown in Fig. 2. One can determine the temperatures T1 and T2 , but since we are using arbitrary units these numbers are not important here. 3.2.2. Large cavity For a very large cavity, actually in the limit of R → ∞ corresponding to the free space, Ω0 → 0, Ωl → ωl and Ωl+1 − Ωl ≈ 1ω = π c /R is infinitesimal. In this case, the frequencies Ωl and Ωk can be taken as continuous variables and the sums in Eqs. (30), (36), (41) and (42) become Riemann summations over equally spaced partitions of the interval (0, ∞). Moreover, since η2 = 2g 1ω, we can neglect the second parcel in the denominator of Eqs. (43) and (41) can be rewritten as f++ (t ) = C1 (t ) + iS1 (t ),
(48)
where C1 (t ) = 1 + 2g
∞
dα 0
S1 (t ) = −2g
0
∞
dα
α 2 [cos(α t ) − 1] , (α 2 − ω¯ 2 )2 + π 2 g 2 α 2
α 2 sin(α t ) . (α − ω¯ 2 )2 + π 2 g 2 α 2 2
(49)
(50)
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Fig. 2. Time evolution of ES (t ) for the state (23), (26), with γ = 0.5, in a finite (small) cavity with δ = 0.03, g = 0.1 and ω ¯ = 0.67 for some values of β . Upper panel: β = ∞, 0.1 and 0.06, full-, dashed- and dotted-lines respectively. Lower panel: β = 0.004624 and 0.0035, full- and dashed-lines respectively; the inset shows a zoom of the first instants of the evolution.
Notice that C1 (0) = 1 and S1 (0) = 0, ensuring that f++ (0) = 1 as required by the orthonormality of the matrix (tµr ). By using the residue theorem, we can calculate the real part of f++ (t ), obtaining
πg sin kt , C1 (t ) = e−π gt /2 cos kt − 2k
(51)
where k2 = ω ¯ 2 − π 2 g 2 /4 and the weak-coupling regime (k2 > 0) was considered. For S1 (t ), however, the integral cannot be analytically evaluated and therefore a numerical solution is employed. Analogously, in the continuum limit, using Eqs. (43) and (44) with the replacements ωk → ω and Ωl → α , Eq. (42) can be rewritten as
√
f+ω (t ) = ω 1ω [C2 (ω, t ) + iS2 (ω, t )] ,
(52)
where [47] 2/3
C2 (ω, t ) = (2g )
∞
dα 0
S2 (ω, t ) = −(2g )2/3
α 2 [cos(α t ) − 1] , (ω2 − α 2 ) (α 2 − ω¯ 2 )2 + π 2 g 2 α 2
∞
dα 0
(53)
α 2 sin(α t ) . (ω2 − α 2 ) (α 2 − ω¯ 2 )2 + π 2 g 2 α 2
(54)
The integrals appearing in the definitions of the functions C2 and S2 , distinctly from the cases of C1 and S1 , have to be considered as Cauchy principal values due to the singularity in the integrand. As before, the integral involving cos(α t ) can be performed analytically and one finds C2 (ω, t ) =
(ω2 − ω¯ 2 ) cos(kt ) π g (ω2 + ω¯ 2 ) sin(kt ) − (ω2 − ω¯ 2 )2 + π 2 g 2 ω2 2k (ω2 − ω ¯ 2 )2 + π 2 g 2 ω2 2 2 π g ω sin ωt (ω − ω¯ ) + 2 − 2 , (ω − ω¯ 2 )2 + π 2 g 2 ω2 (ω − ω¯ 2 )2 + π 2 g 2 ω2
2g e−π gt /2
(55)
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Fig. 3. Time evolution of E (t ) for the state (23), (25), with ξ = 0.5, in an infinitely large cavity with g = 0.1 and ω ¯ = 1.0, for some values of β . Upper panel: β = ∞; 1.2; 0.9, full-, dashed- and dotted-lines, respectively. Lower panel: β = 0.175; 0.08, full- and dashed-lines respectively.
whereas the S2 integral can only be evaluated numerically. Thus, using Eq. (52), for an infinitely large cavity, the sum appearing in Eqs. (30) and (36) can be evaluated with the prescription
f+k 2 n′k
|
|
∞
→ 0
k
ω2 C22 (ω, t ) + S22 (ω, t ) . dω eβω − 1
(56)
Notice that, consistently with the orthogonality of the matrix (tµr ), both C2 and S2 functions vanish at t = 0 and, therefore, f+ω (0) = 0. Inserting Eqs. (48) and (56) into Eq. (30) we obtain the function describing the influence of the temperature in the dynamics of entanglement of the state (23), (25), E (t , ξ , β) = −ξ
C12
(t ) +
(t ) +
S12
∞
0
ω2 C22 (ω, t ) + S22 (ω, t ) dω . eβω − 1
(57)
The temperature dependence of the time-evolution of the function E (t , ξ , β) is illustrated in Fig. 3, considering ξ = 1/2, g = 0.1, ω ¯ = 1.0 and some values of β . For T = 0 (β → ∞), we find that the entanglement decays asymptotically, with the state becoming separable only for very long times (see the full line in the upper panel). However, for finite temperatures, the sudden death of entanglement occurs. If the temperature is small, entanglement dies and revives a few times before completely disappear (see dashed- and dotted-lines in the upper panel). Further increasing the temperature, we reach a value β1−1 = T1 ≈ 5.71 (full line in the lower panel) above which the sudden death occurs at a short time and entanglement is lost without any rebirth; the value of T1 depends on the parameters g and ω ¯. For the gaussian state (23), (26), in the limit of infinitely large cavity, we find y(t , γ , β) =
1 2
+ sinh (γ )
x21 (t , γ ) + x22 (t , γ ) =
2
C12
(t ) +
S12
(t ) +
∞
0
sinh2 (2γ ) 2 C1 (t ) + S12 (t ) ; 4
ω2 C22 (ω, t ) + S22 (ω, t ) dω , eβω − 1
(58)
(59)
inserting these expressions into Eq. (39), we obtain the function ES (t , γ , β) describing the effect of temperature on the dynamics of entanglement of the state (23), (26). As illustrated in Fig. 4, the entanglement decays asymptotically for T = 0
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Fig. 4. Time evolution of ES (t ) for the state (23), (26), with γ = 0.8, in an infinitely large cavity with g = 0.1 and ω ¯ = 1.0, for some values of β : β = ∞; 1.5; 0.88 full-, dashed- and dotted-lines, respectively.
while, at finite temperatures, the sudden death occurs in a similar way to the case of the Bell-like state: up to a temperature T1 , one gets an alternation between death and revival of entanglement before the true death is established but, above it, just a single sudden death appears. 4. Conclusions In the present work, we have studied the entanglement of two-atom states coupled with field modes confined in a cavity at finite temperature. We have used the dressed-state formalism to consider the time evolution of the entanglement of Bell-like and gaussian states of the dressed atoms; finite-temperature effects were taken into account by assuming that the dressed-field modes obey the Bose–Einstein distribution at temperature T = β −1 . Using the Shchukin–Vogel and the Simon criteria to identifying entanglement, we have analyzed the time evolution of the entanglement of these two family of states for some values of the temperature, in two different situations: a finite (small) cavity and a infinitely large cavity (free space); we found distinct behaviors for these two cases. In a small cavity, as shown in Figs. 1 and 2, for T = 0 the entanglement is maintained as time evolves; this also happens for low temperatures of the dressed-field modes. As the temperature is raised, one reaches a value (T1 ) above which the system evolves in time alternating between sudden death and rebirth of the entanglement. This behavior remains up to a temperature (T2 ) above which a single sudden death occurs and the entanglement does not revive anymore. In the case of a very large cavity, actually in the limit R → ∞, we find that the decay of entanglement is asymptotic only for the environment at zero temperature. For low temperatures, as illustrated in Figs. 3 and 4, one has few sudden deaths and rebirths of the entanglement before the system becomes separable. Above an specific value of the temperature (T1 ), a sudden death of the entanglement occurs at a short time and it is not recovered at any instant later. The survival of entanglement of the state (23) in a small cavity as the time evolves, at low temperatures, may be associated to a feedback process between the atoms and the field in the cavity that takes place in this confining situation; in contrast, this does not happens in a very large cavity. These aspects should be considered in attempts to implement devices that use entanglement as an important resource and work at finite temperatures. Acknowledgments The authors thank CNPq and CAPES, Brazilian Agencies, for partial financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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