Monitoring atom-atom entanglement and decoherence in a solvable

PHYSICAL REVIEW A 77, 033839 共2008兲

Monitoring atom-atom entanglement and decoherence in a solvable tripartite open system in cavity QED Matteo Bina, Federico Casagrande,* and Alfredo Lulli Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy

Enrique Solano Physics Department, ASC, and CeNS, Ludwig-Maximilians-Universität, Theresienstrasse 37, 80333 Munich, Germany and Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Apartado 1761, Lima, Peru 共Received 8 November 2007; published 20 March 2008兲 We present a fully analytical solution of the dynamics of two strongly driven atoms resonantly coupled to a dissipative cavity field mode. We show that an initial atom-atom entanglement cannot be increased. In fact, the atomic Hilbert space divides into two subspaces, one of which is decoherence free so that the initial atomic entanglement remains available for applications, even in presence of a low enough atomic decay rate. In the other subspace a measure of entanglement, decoherence, and also purity, are described by a similar functional behavior that can be monitored by joint atomic measurements. Furthermore, we show the possible generation of Schrödinger-cat-like states for the whole system in the transient regime, as well as of entanglement for the cavity field and the atom-atom subsystems conditioned by measurements on the complementary subsystem. DOI: 10.1103/PhysRevA.77.033839

PACS number共s兲: 42.50.Pq, 03.67.Mn

I. INTRODUCTION

The interaction of a two-level system with a quantized single mode of a harmonic oscillator, called the JaynesCummings 共JC兲 model 关1兴, is arguably the most fundamental quantum system describing the interaction of matter and light. The JC model has found its natural playground in the field of cavity quantum electrodynamics 共CQED兲, in the microwave 关2,3兴 and in the optical regime 关4兴, as well as in other physical systems such as trapped ion 关5兴 or circuit 关6兴 and solid-state 关7兴 QED. Extensions of the JC model to more atoms and more modes, externally driven or not, have been developed and, presently, we enjoy a vast number of theoretical and experimental developments. Unfortunately, a many of them are not easy to handle and most of the interesting physics has to be extracted from heavy numerical solutions and calculations, especially when realistic dissipative processes are taking into account. The advent of quantum information 关8兴 has been a fresh input in the field of CQED 关9兴, reshaping concepts and using it for fundamental tests and initial steps in the demanding field of quantum information processing. Here, coherence and the generation of entanglement play an important role and, in particular, the manner in which they are affected by the presence of a dissipative environment 关10兴. In spite of their relevance, most of the attractive quantum models do not enjoy analytical solutions and the few available ones are not as close to realistic conditions as desired. In this paper, we consider a system composed by two coherently driven two-level atoms trapped inside a cavity and coupled to one of its quantized modes. The experimental implementation seems to be feasible due to the recent advances in deterministic trapping of atoms in optical cavities 关11,12兴. We show that under full resonance conditions and

*[email protected] 1050-2947/2008/77共3兲/033839共12兲

negligible atomic decays the system dynamics can be solved analytically also in the presence of cavity field dissipation. With the solutions at hand we are able to monitor the purity of the cavity field and atom-atom subsystems, as well as the entanglement and decoherence dynamics of the latter. Atomic entanglement cannot be generated. However, the quantum correlations of suitable entangled states can remain completely protected in a decoherence-free subspace. In case of atomic states outside this subspace, the decays of quantum entanglement, coherence, and purity are remarkably described by the same function. In Sec. II we introduce the master equation of the considered open system. In Sec. III we present the analytical solutions of the system evolution. In Sec. IV we study the dynamics of entanglement in the atom-atom subsystem. In Sec. V we consider the conditional generation of subsystem states. In Sec. VI we present fully numerical results that confirm our analytical developments. In Sec. VII we conclude with a summary of our results and physical discussions. In the Appendix we discuss the solution of the master equation presented in Sec. II. II. OPEN SYSTEM MASTER EQUATION

We consider a pair of two-level atoms interacting inside a cavity with a field mode also coupled to the environment 共see Fig. 1兲. A coherent external field of frequency ␻D drives

FIG. 1. Two driven two-level atoms interacting with a dissipative cavity mode. 033839-1

©2008 The American Physical Society


BINA et al.

both atoms during the interaction with the cavity mode of frequency ␻ f 关13,14兴. The transition frequency ␻a between excited and ground states 兩e典 j and 兩g典 j 共j = 1 , 2兲, respectively, is the same for the two-level atoms. This system could be experimentally implemented with two-level Rydberg atoms in a microwave cavity 关9兴 or with three-level atoms reduced effectively to two levels interacting with an optical cavity. Relevant advances in cooling and trapping atoms in optical cavities have been recently achieved 关11,12兴. Similar dynamics could be also implemented in trapped ions interacting with a vibrational mode instead of the cavity mode 关9兴. The Hamiltonian which describes the whole system unitary dynamics is 2

2

ˆ 共t兲 = ប␻a 兺 ␴ˆ z + ប␻ aˆ†aˆ + ប⍀ 兺共e−i␻Dt␴ˆ † + ei␻Dt␴ˆ 兲 H f j j 2 j=1 j j=1 2

+ បg 兺共␴ˆ †j aˆ + ␴ˆ jaˆ†兲,

共1兲

j=1

where ⍀ is the Rabi frequency associated with the coherent driving field amplitude, g is the atom-cavity mode coupling constant 共taken equal for both atoms兲, aˆ 共aˆ†兲 the field annihilation 共creation兲 operator, ␴ˆ j = 兩g典 j具e兩共␴ˆ †j = 兩e典 j具g兩兲 the atomic lowering 共raising兲 operator, and ␴ˆ zj = 兩e典 j具e兩 − 兩g典 j具g兩 the inversion operator. In the perspective of possible experimental implementation of our scheme we must include the effects of cavity mode dissipation and the decay of the atomic upper level. Therefore, we must solve the following master equation 共ME兲 for the statistical density operator ␳ˆ ⬘ of the whole tripartite system i ˆ ˆ ␳ˆ ⬘ + L ˆ ␳ˆ ⬘ , , ␳ˆ ⬘兴 + L ␳˙ˆ ⬘ = − 关H f a ប

共2兲

2

ˆ = − ប␦aˆ†aˆ + ប⍀ 兺共␴ˆ † + ␴ˆ 兲, H j 0 j j=1

2

ˆ = បg 兺共␴ˆ †aˆ + ␴ˆ aˆ†兲, H 1 j j j=1

共5兲 where we introduced the atom-cavity field detuning parameter ␦ = ␻a − ␻ f , and from now on we consider a resonance condition between the atoms and the external field 共␻D = ␻a兲. We remark that Hamiltonian 共5兲 was derived by a standard technique, whereas in the case of less simple systems, e.g., three-level atoms, more refined treatments are necessary to derive a time-independent Hamiltonian such as adiabatic elimination or nonlinear rotations, as e.g., in Ref. 关15兴. The ME 共4兲 can be solved only by numerical techniques, as we discuss in Sec. VI, where we present some numerical results for the whole system dynamics. On the other hand, if we consider negligible atomic decay 共␥ = 0兲 it is possible to solve Eq. 共4兲 analytically. First of all, we consider the unitary ˆ t其 and we derive for the dentransformation Uˆ 共t兲 = exp兵 បi H 0 sity operator ␳ˆ 共t兲 = Uˆ 共t兲␳ˆ I共t兲Uˆ †共t兲 the following ME: i ˆ Uˆ †, ␳ˆ 兴 + L ˆ ␳ˆ , ␳˙ˆ = − 关Uˆ H 1 f ប

共6兲

where the transformed Hamiltonian can be written as 2

ˆ Uˆ † = បg 兺关兩 + 典具+ 兩 − 兩− 典具− 兩 + e2i⍀t兩 + 典 Uˆ H 1 j j j 2 j=1 ⫻具− 兩 − e−2i⍀t兩− 典 j具+ 兩兴ae−i␦t + H.c.

共7兲

Here, 兵兩 + 典 j , 兩−典 j其共j = 1 , 2兲 is a rotated basis connected to the 兩g典 ⫾兩e典 standard basis 兵兩e典 j , 兩g典 j其 via 兩 ⫾ 典 j = j冑2 j . In the strongdriving regime for the interaction between the atoms and the external coherent field ⍀ g, we can use the rotating-wave approximation 共RWA兲 obtaining the effective Hamiltonian 关13,14兴 2

ˆ 共t兲 = បg 兺共␴ˆ † + ␴ˆ 兲共aˆei␦t + aˆ†e−i␦t兲. H eff j 2 j=1 j

where ˆ ␳ˆ ⬘ = k 关2aˆ␳ˆ ⬘aˆ† − aˆ†aˆ␳ˆ ⬘ − ␳ˆ ⬘aˆ†aˆ兴, L f 2 2

ˆ ␳ˆ ⬘ = ␥ 兺关2␴ˆ ␳ˆ ⬘␴ˆ † − ␴ˆ †␴ˆ ␳ˆ ⬘ − ␳ˆ ⬘␴ˆ †␴ˆ 兴, L a j j j j j j 2 j=1

共3兲

where k and ␥ are the cavity and atomic decay rates, respectively, and the environment is modeled by a thermal bath at zero temperature. Changing to the interaction picture the dissipative terms remain unchanged and the ME 共2兲 can be rewritten as i ˆ ˆ ␳ˆ + L ˆ ␳ˆ , ˆ I兴 + L ␳˙ˆ I = − 关H a I I, ␳ f I ប

Equation 共8兲 outlines the presence of Jaynes-Cummings 共␴ˆ †j aˆ + ␴ˆ jaˆ†兲 as well as anti–Jaynes-Cummings 共␴ˆ †j aˆ† + ␴ˆ jaˆ兲 interaction terms of each coherently driven atom with the cavity mode. We remark that in the framework of trapped ions a similar Hamiltonian can be found but the cavity mode is replaced by the vibrational mode of the ions system 关16兴. In the following we will describe the solution of the effective ME i ˆ ˆ ␳ˆ . ˆ兴 + L ␳ˆ˙ = − 关H eff共t兲, ␳ f ប

共4兲

where Hamiltonian 共1兲 has been replaced by the timeˆ with ˆ +H ˆ =H independent Hamiltonian H I 0 1

共8兲

共9兲

III. EXACT SOLUTION AT RESONANCE FOR ATOMS PREPARED IN THE GROUND STATE

In this section we consider the solution of the ME 共9兲 in the case of exact resonance ␦ = 0, cavity field initially in the

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MONITORING ATOM-ATOM ENTANGLEMENT AND ...

’;⬎?vacuum state, and both atoms prepared in the ground state as an example of separable initial state. We leave to future treatments the cases of cavity field prepared in more general states. The exact solution of the ME for any atomic preparation is described in details in the Appendix. It is based on the following decomposition for the density operator ␳ˆ 共t兲 of the whole system 4

␳ˆ 共t兲 =

兺

4

具i兩␳ˆ 共t兲兩j典兩i典具j兩 =

i,j=1

兺 ␳ˆ ij共t兲兩i典具j兩,

共10兲

i,j=1

subspace for system dynamics. It is the component of the initial state 兩0典丢兩gg典 in that subspace, Eq. 共14兲, which does not evolve. We note that starting from atoms prepared in the other elements of the standard basis we obtain solutions quite analogous to Eq. 共11兲, as can be easily verified from the Appendix. In the transient regime 共kt 1兲, the decoherence function 2 can be approximated by f 1共t兲 ⯝ e−兩˜␣共t兲兩 /2 where ˜␣共t兲 = igt, and the whole system is described by a pure Schrödinger-cat-like state

4 where 兵兩i典其i=1 = 兵兩++典 , 兩+−典 , 兩−+典 , 兩−−典其 is the rotated basis of the atomic Hilbert space. Here we report only the final expressions for the field operators ␳ˆ ij in the present case:

1 ␳ˆ 共11,44兲共t兲 = 兩⫿ ␣共t兲典具⫿ ␣共t兲兩, 4

1 兩˜␺共t兲典 = 关兩− ˜␣共t兲典丢兩+ +典 + 兩0典丢共兩+ − 典 + 兩− +典兲 2 + 兩˜␣共t兲典丢兩− − 典兴. If we rewrite this result in the standard atomic basis 1 兩˜␺共t兲典 = 兵关兩− ˜␣共t兲典 − 2兩0典 + 兩˜␣共t兲典兴丢兩ee典 4

1 f 1共t兲 ␳ˆ 共12,13兲共t兲 = 兩− ␣共t兲典具0兩, 4 e−兩␣共t兲兩2/2

␳ˆ 14共t兲 =

+ 关兩− ˜␣共t兲典 − 兩˜␣共t兲典兴丢共兩eg典 + 兩ge典兲 + 关兩− ˜␣共t兲典 + 2兩0典 + 兩˜␣共t兲典兴丢兩gg典其

1 f 2共t兲兩− ␣共t兲典具␣共t兲兩, 4 e−2兩␣共t兲兩2

1 ␳ˆ 共22,23,33兲共t兲 = 兩0典具0兩, 4

␳ˆ 共24,34兲共t兲 =

1 f 1共t兲兩0典具␣共t兲兩, 4 e−兩␣共t兲兩2/2

共11兲

where we introduced the time-dependent coherent field amplitude

␣共t兲 = i and the function

再

共15兲

2g 共1 − e−k/2t兲 k

showing the onset of correlations between atomic states and cavity field catlike states. For the terms related to the field subsystem we recoverexpressions analogous to those derived in Ref. 关17兴 for a strongly driven micromaser system. Unlike the present system, in that case the atoms pump the cavity mode with a Poissonian statistics, interacting for a very short time such that the cavity dissipation is relevant only in the time intervals between atomic injections. Furthermore, the cavity field states are conditioned on atomic measurements. At steady state 共kt → ⬁兲 the density operator is mixed and given by

共12兲

冎

2g2 4g2 f 1共t兲 = exp − t + 2 共1 − e−k/2t兲 . k k

1 ␳ˆ SS = 关兩− ␣SS典具− ␣SS兩丢兩+ +典具+ +兩 + 兩␣SS典具␣SS兩 4 丢

兩− − 典具− − 兩 + 兩0典具0兩丢共兩+ − 典具+ − 兩 + 兩+ − 典

⫻具− +兩 + 兩− +典具+ − 兩 + 兩− +典具− +兩兲兴

共13兲

共17兲

with ␣SS = 2i k . Interestingly, the steady state has not a fully diagonal structure, i.e., it is not completely mixed, in agreement with the previous discussion on the time-dependent solution. The change from a pure state to a mixed one and the degree of mixedness can be we evaluated by the purity of the whole system ␮共t兲 = Tr关␳ˆ 2共t兲兴: g

We recall that ␳ˆ ji共t兲 = ␳ˆ †ij共t兲. We notice the presence of single atom-cavity field coherences whose evolution is ruled by the function f 1共t兲关15兴, as well as full atom-atom-field coherences ruled by f 2共t兲 = f 41共t兲, that we shall discuss later on. There are also two one-atom coherences, and two diagonal terms, which do not evolve in time, corresponding to a pure state 兩0典丢共兩+ − 典 + 兩− +典兲 = 兩0典丢共兩gg典 − 兩ee典兲,

共16兲

␮共t兲 =

共14兲

where we recognize 共up to normalization兲 a Bell atomic state 兩⌽−典. The explanation goes as follows. First of all, if we start with atoms prepared in states of the rotated basis used in the decomposition of Eq. 共10兲, we obtain much simpler results due to the structure of Hamiltonian 共8兲 on resonance and the obvious uninfluence of dissipation on the cavity vacuum. Actually, either the field states are coherent, 兩0典丢兩⫾⫾典哫兩⫿␣共t兲典丢兩⫾⫾典, or there is no evolution at all for the states 兩0典丢兩⫾⫿典, showing the presence of an invariant

冋

册

f 2共t兲 f 2共t兲 1 3+4 1 2 + 2 2 . 8 e−兩␣共t兲兩 e−4兩␣共t兲兩

共18兲

In Fig. 2 we show ␮共t兲 as a function of the dimensionless time kt for different values of the dimensionless coupling constant g / k. We see that the purity decays rather fast in the strong coupling regime g / k ⲏ 1. Subsystems dynamics. Now we consider the time evolution of the atomic and cavity field subsystems in the case of both atoms initially prepared in the ground state. The cavity field reduced density operator ␳ˆ f 共t兲 = Tra关␳ˆ 共t兲兴 can be derived by tracing over both atoms:

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BINA et al.

1 (1)

µ (2)

0.6 (3) (4)

(a)

0.2 0

5

kt 10

FIG. 2. Purity ␮共t兲 of the whole system density operator vs dimensionless time kt for dimensionless coupling constant g / k: 共1兲 0.05, 共2兲 0.2, 共3兲 0.5, 共4兲 2.

1 ␳ˆ f 共t兲 = 关兩− ␣共t兲典具− ␣共t兲兩 + 2兩0典具0兩 + 兩␣共t兲典具␣共t兲兩兴. 共19兲 4 Hence the cavity field mean photon number is 具Nˆ典 = 21 兩␣共t兲兩2 ˆ 典 = 2 g22 . These results hold for and its steady state value 具N SS k atoms prepared in any state of the standard basis. At any time Eq. 共19兲 describes a mixed state, whose purity ␮ f 共t兲 = Tr f 关␳ˆ 2f 共t兲兴 is 1 2 2 ␮ f 共t兲 = 关3 + 4e−兩␣共t兲兩 + e−4兩␣共t兲兩兴. 8

(b)

共20兲

In Fig. 3 we show the purity ␮ f 共t兲 as a function of dimensionless time kt for different values of the ratio g / k, showing a better survival of the purity than for the global state of Fig. 2 except in the strong coupling regime. The reduced atom-atom density operator ␳ˆ a共t兲 can be obtained by tracing over the field variables. In the rotated basis 兵兩++典 , 兩+−典 , 兩−+典 , 兩−−典其, we obtain

␳⫾ a 共t兲 =

冢

1

1 f 1共t兲 4 f 1共t兲

f 1共t兲 f 1共t兲 f 2共t兲 1

1

f 1共t兲

1

1

f 1共t兲

f 2共t兲 f 1共t兲 f 1共t兲

1

冣

.

共21兲

(c)

FIG. 4. Correlation functions 共a兲 Cgg, 共b兲 Ceg, 共c兲 Cee vs dimensionless time kt and coupling constant g / k, showing atomic bunching and antibunching.

␮a共t兲 = Tra关␳ˆ 2a共t兲兴 of the biatomic subsystem is

The presence of six time-independent matrix elements is in agreement with the remarks below Eq. 共14兲. The purity

µf

1 (2)

1 Pe,g共t兲 = 关1 ⫿ f 1共t兲兴. 2

(3)

(4)

0.2 0

5

共22兲

Its behavior is quite similar to the one of Fig. 2 for the whole system purity. From Eq. 共21兲 we can derive the singleatom density matrices and evaluate the probability to measure one atom in the excited or ground state

(1)

0.6

1 ␮a共t兲 = 关3 + 4f 21共t兲 + f 22共t兲兴. 8

kt 10

FIG. 3. Purity ␮ f 共t兲 of the cavity field subsystem vs dimensionless time kt for dimensionless coupling constant g / k: 共1兲 0.05, 共2兲 0.2, 共3兲 0.5, 共4兲 2.

共23兲

Quite similar expressions hold for atoms prepared in any state of the standard basis. We see that from measurements of the atomic inversion I共t兲 = pg共t兲 − pe共t兲 we can monitor the one-atom decoherence function f 1共t兲 as in Ref. 关15兴. By rewriting the atomic density matrix 共21兲 in the standard basis we evaluate the joint probabilities Plm共t兲 = 具lm兩␳ˆ 共e,g兲 a 共t兲兩lm典 with 兵l , m其 = 兵e , g其. The corresponding correlation functions at a given time t are

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Cee共t兲 =

3 − 4f 1共t兲 + f 2共t兲 , 2关1 − f 1共t兲兴2

Ceg共t兲 = Cge共t兲 =

Cgg共t兲 =

1 − f 2共t兲 2关1 − f 21共t兲兴

,

3 + 4f 1共t兲 + f 2共t兲 . 2关1 + f 1共t兲兴2

共24兲

In Figs. 4共a兲–4共c兲 we show the correlation functions Clm共t兲 versus dimensionless time kt and coupling constant gk . At steady state we see that Cgg共t兲 , Cee共t兲哫 3 / 2, that is, positive atom-atom correlation or bunching, whereas Ceg共t兲哫 1 / 2, indicating negative correlation or antibunching. These results generalize those in Ref. 关17兴, which can be derived from Eqs. 共24兲 in the limit of negligible dissipation kt 1. We notice that from the joint atomic probability Peg共t兲 = Pge共t兲 = 81 关1 − f 2共t兲兴 one can monitor the two-atom decoherence described by f 2共t兲. IV. DYNAMICS OF ENTANGLEMENT AND DECOHERENCE OF THE ATOMIC SUBSYSTEM

In Sec. III we discussed the solution of the ME 共9兲 for both atoms prepared in the ground state as an example of

␳mB a =

冢

关1 + f 2兴兩c1兩2 + 关1 − f 2兴兩c3兩2 2

separable state. In order to describe also initially entangled atoms now we consider the general solution derived in the Appendix for atoms prepared in a superposition of Bell states 4 兩␺a共0兲典 = 兺i=1 ci兩␯i典, where the coefficients ci are normalized 4 as 兺i=1兩ci兩2 = 1, and 兵兩␯i典其i=1,..,4 = 兵兩⌽+典 , 兩⌽−典 , 兩⌿+典 , 兩⌿−典其 is the Bell basis where 兩⌽⫾典 = 兩ee典⫾兩gg典 and 兩⌿⫾典 = 兩eg典⫾兩ge典 . Tracing 冑2 冑2 over the atomic variables the solution for the whole density operator 共Eqs. 共10兲 and 共A6兲兲 we derive the cavity field density operator generalizing Eq. 共19兲:

1 ␳ˆ f 共t兲 = 关兩c1 + c3兩2兩− ␣共t兲典具− ␣共t兲兩 + 2共兩c2兩2 + 兩c4兩2兲兩0典具0兩 2 + 兩c1 − c3兩2兩␣共t兲典具␣共t兲兩兴.

ˆ 典共t兲 = 共兩c 兩2 For the mean photon number we obtain 具N 1 2 2 + 兩c3兩兲兩␣共t兲兩 that is independent of the coefficients of states 兩⌽−典 and 兩⌿−典. On the other hand, tracing over the field variables we obtain the atomic density matrix. We report the solution in the so-called “magic basis” 关18兴 that can be obtained from the Bell basis simply multiplying 兩␯2典 and 兩␯3典 by the imaginary unit

− if 1c1cⴱ2 − i

关1 + f 2兴c1cⴱ3 + 关1 − f 2兴cⴱ1c3 2

f 1c1cⴱ4

if 1cⴱ1c2

兩c2兩2

f 1c2cⴱ3

ic2cⴱ4

关1 + f 2兴cⴱ1c3 + 关1 − f 2兴c1cⴱ3 i 2

f 1cⴱ2c3

关1 − f 2兴兩c1兩2 + 关1 + f 2兴兩c3兩2 2

if 1c3cⴱ4

f 1cⴱ1c4

− icⴱ2c4

− if 1cⴱ3c4

兩c4兩2

where we omitted the time dependence for brevity. To evaluate the entanglement properties of the atomic subsystem we consider the entanglement of formation ⑀F共t兲关19兴 defined as

⑀F共t兲 = −

共25兲

共26兲

1 Pee,gg共t兲 = 兵兩c1兩2关1 + f 2共t兲兴 + 2兩c2兩2 4 + 兩c3兩2关1 − f 2共t兲兴 ⫾ 4f 1共t兲Re共c1cⴱ2兲其,

1 − 冑1 − C2共t兲 1 − 冑1 − C2共t兲 log2 2 2

1 + 冑1 − C2共t兲 1 + 冑1 − C2共t兲 log2 , − 2 2

冣

,

1 Peg,ge共t兲 = − 兵兩c1兩2关1 − f 2共t兲兴 + 2兩c4兩2 4 共27兲

where C共t兲 is the concurrence that can be evaluated as C共t兲 = max兵0 , ⌳4共t兲 − ⌳3共t兲 − ⌳2共t兲 − ⌳1共t兲其 (⌳i are the square roots of the eigenvalues of the non hermitian matrix mB ⴱ ␳mB a 共t兲关␳a 共t兲兴 taken in decreasing order). From Eq. 共26兲 we can derive the probabilities for joint atomic measurements in the standard basis

+ 兩c3兩2关1 + f 2共t兲兴 ⫾ 4f 1共t兲Re共c3cⴱ4兲其.

共28兲

Also, we can derive the density matrix corresponding to a single atom and obtain the atomic probabilities generalizing Eq. 共23兲: 1 Pe,g共t兲 = 关1 ⫾ 2f 1共t兲Re共cⴱ1c2 + cⴱ3c4兲兴. 2

共29兲

First we consider the case of a superposition of Bell states 兩⌽⫾典共i.e., c1 = a , c2 = bei␪ , c3 = c4 = 0 with a , b real numbers兲. The initial atomic state 兩gg典共兩ee典兲 can be obtained if a = b

033839-5


BINA et al.

Let us summarize and discuss the main results in the case of atoms prepared in entangled states. If we consider a superposition of states 兩⌽−典 and 兩⌿−典共i.e., c2 = a , c4 = bei␪ , c1 = c3 = 0兲 we find that the atomic density matrix of Eq. 共26兲 does not evolve. Actually, the atomic subspace spanned by 兩⌽−典 and 兩⌿−典 coincides with the time-invariant subspace spanned by 兩+−典 and 兩−+典, so that it remains protected from dissipation during the system evolution. It provides an example of decoherence free subspace 共DFS兲关20兴. Atomic entanglement injected in the system can thus remain available for long storage times for applications in quantum information processing 关21兴. If we consider a superposition of states 兩⌽+典 and 兩⌿+典共i.e., c1 = a , c3 = bei␪ , c2 = c4 = 0兲 the concurrence is given by

1 ε

F

0.5

(1)

(2) (4)

(3)

0

(a) 0

5

kt

10

1 εF

C共t兲 = 冑a4 + b4 − 2a2b2 cos共2␪兲f 2共t兲 (1)

and we find the remarkable result that the initial entanglement is progressively reduced by the decoherence function f 2共t兲. Note that C共t兲 = f 2共t兲 for states 兩⌽+典, 兩⌿+典, and any superposition as a兩⌽+典 ⫾ ib兩⌿+典. To understand this point let us consider the specific example of the initial state 兩0典丢兩⌽+典. The evolved density operator of the whole system is

0.5 (2) (3) (4)

0 (b) 0

5

共30兲

kt 10

␳ˆ 共t兲 =

FIG. 5. Entanglement of formation ⑀F共t兲 as a function of kt and for different values of g / k: 0.1 共1兲, 0.5 共2兲, 1 共3兲, 5 共4兲. 共a兲 Atoms 兩⌽+典+ei␲/4兩⌽−典 , 共b兲 atoms preprepared in a partially entangled state 冑2 + pared in the Bell states 兩⌽ 典.

= 冑12 and ␪ = ␲ exp共−2g2t2兲. We find that the concurrence C共t兲 vanishes for any time and every value of g / k, so that it is not possible to entangle the atoms. In the case of atoms prepared in a partially entangled state we find that the entanglement of formation can only decrease during the system evolution as shown, for example, in Fig. 5共a兲 in the case a = b = 冑12 and ␪ = ␲ / 4. We also see that the progressive loss of entanglement is faster for large values of the parameter g / k. For atoms prepared in the maximally entangled state 兩⌽+典共i.e., ␪ = 0 , a = 1 , b = 0兲 we derive that the concurrence simply reduces to f 2共t兲 that also describes the whole system decoherence. This important point will be discussed later. In Fig. 5共b兲 we show the entanglement of formation as a function of dimensionless time kt. For atoms prepared in the maximally entangled state 兩⌽−典共i.e., ␪ = 0 , a = 0 , b = 1兲 the concurrence is always maximum 关C共t兲 = 1兴. In fact, we can see from Eq. 共26兲 that the atomic density matrix is always the one of the initial state, as expected because 兩⌽−典 is a linear combination of the invariant states 兩+−典 and 兩−+典. Starting from a superposition of Bell states 兩⌿⫾典共i.e., c3 = a , c4 = bei␪ , c1 = c2 = 0兲 we obtain analogous results. In particular entanglement cannot be generated for atoms prepared in states 兩eg典 and 兩ge典, for the state 兩⌿+典 the concurrence is given by f 2共t兲, and the entanglement of state 兩⌿−典 is preserved during system evolution. We find analogous results also for the concurrence of atoms prepared in a superposition of 兩⌽−典 and 兩⌿+典共i.e., c2 = a , c3 = bei␪ , c1 = c4 = 0兲 or in a superposition of 兩⌽+典 and 兩⌿−典共i.e., c1 = a , c4 = bei␪ , c2 = c3 = 0兲.

再

1 兩− ␣共t兲典具− ␣共t兲兩丢兩+ +典具+ +兩 + 兩␣共t兲典具␣共t兲兩丢兩− − 典 2 ⫻具− − 兩 +

f 2共t兲 e

−2兩␣共t兲兩2

关兩− ␣共t兲典具␣共t兲兩丢兩+ +典具− − 兩 + 兩␣共t兲典

冎

⫻具− ␣共t兲兩丢兩− − 典具+ +兩兴 .

共31兲

In the limit, kt 1, of short time and/or negligible dissipa2 tion, where f 2共t兲 ⯝ e−2兩˜␣共t兲兩 with ˜␣共t兲 = igt, the system evolves into a pure cat-like state where the atoms are correlated with coherent states 兩0典丢

兩+ +典 + 兩− − 典

冑2

哫

兩− ˜␣典丢兩+ +典 + 兩˜␣典丢兩− − 典

冑2

. 共32兲

For longer times and larger dissipation, the system coherence decays as described by the function f 2共t兲. Let us now consider the atomic dynamics disregarding the field subsystem. The reduced atomic density operator is 1 ␳ˆ a共t兲 = 关兩+ +典具+ +兩 + 兩− − 典具− − 兩 2 + f 2共t兲共兩+ +典具− − 兩 + 兩− − 典具+ +兩兲兴.

共33兲

Hence, as the quantum coherence reduces, simultaneously the atoms lose their inseparability, and the state becomes maximally mixed 共in the relevant subspace兲. In fact, the 1+f 2共t兲 atomic purity is given by ␮a共t兲 = 22 . Hence the time evolution of decoherence, concurrence, and purity is described by the function f 2共t兲, which can be monitored via a measurement of joint atomic probabilities 关see Eq. 共28兲兴

033839-6



V. CONDITIONAL GENERATION OF STATES

0.5 P

ee (1)

0.375 (2) (3) (4)

0.25 0

5

kt

10

FIG. 6. Atoms prepared in the Bell state 兩⌽+典: joint atomic probability Pee共t兲 vs kt for different values of g / k: 0.1 共1兲, 0.5 共2兲, 1 共3兲, 5 共4兲.

1 Pee共t兲 = Pgg共t兲 = 关1 + f 2共t兲兴, 4

1 Peg共t兲 = Pge共t兲 = 关1 − f 2共t兲兴. 4

In this section we seek information about the states of one of the two subsystems conditioned by a projective measurement on the other one. If the system is implemented in the optical domain and the cavity field is accessible to measurements, a null measurement by a on/off detector implies the generation of a maximally entangled atomic Bell state 关14兴. In the case of atoms prepared in state 兩gg典共or 兩ee典兲, the atomic conditioned state will be the Bell state 兩⌽−典关see Eq. 共14兲兴. Analogously, for initial atomic state 兩eg典 or 兩ge典 the atomic conditioned state will be 兩⌿−典. Now we consider the evolution of the field subsystem conditioned by a projective atomic measurement on the bare basis 兵兩ee典 , 兩eg典 , 兩ge典 , 兩gg典其. Starting, e.g., from the initial state 兩0典丢兩gg典, the cavity field will be in the conditioned states at a given time t 共omitted for brevity in this section兲

␳ˆ f,共ee,gg兲 =

2

␥D ⯝ k兩␣SS兩 = 2g,

+ 兩0典具␣兩 + 兩− ␣典具0兩 + 兩␣典具0兩兲兴,

␳ˆ f,共eg,ge兲 =

1 关兩− ␣典具− ␣兩 + 兩␣典具␣兩 2共1 − f 2兲 2

− f 2e2兩␣兩共兩␣典具− ␣兩 + 兩− ␣典具␣兩兲兴.

共36兲

Note that in the limit kt 1 the conditioned field state is a Schrödinger-cat-like state 兩− ˜␣典 ⫿ 2兩0典 + 兩˜␣典

兩␺典 f,共ee,gg兲 =

冑2共e−2兩˜␣兩

兩␺典 f,共eg,ge兲 =

冑2共1 − e−2兩˜␣兩兲 ,

共35兲

that is again independent of 共and faster than兲 k. A physical interpretation of this result is that the more coupled the two atoms are to the dissipative cavity mode, the more effective becomes the decay of both the environment-induced decoherence and the initial entanglement. For g / k 1, that is in a weak coupling regime, the exponential decay of coherence and concurrence starts later and its rate ␥D ⯝ 8g2 / k, is slower than the dissipative rate k. We have shown that under full resonance conditions it is not possible to generate or increase the initial atomic entanglement. This can be explained looking at the initial requirements that allow us to write Eq. 共A1兲. In particular, the strong driving condition ⍀ g, the resonance condition ␦ = 0 and the choice of negligible atomic decays ␥ = 0 are necessary to obtain an independent set of equations for the operators ␳ˆ ij and to exactly solve the system dynamics. In Sec. VI we will show that removing the strong driving condition it is possible to slightly entangle the atoms. On the other hand, it can be shown 关22兴 that with off-resonant atomscavity field interaction it is possible to generate maximally entangled atomic states also under strong driving conditions. Also we recall that in the resonant case and without driving field, two atoms prepared in a separable state can partially entangle by coupling to a thermal cavity field 关23兴.

2

+ f 2e2兩␣兩共兩␣典具− ␣兩 + 兩− ␣典具␣兩兲 ⫿ 2f 1e兩␣兩 /2共兩0典具− ␣兩

共34兲 The probability Pee共t兲 is shown in Fig. 6 as a function of kt for different values of the ratio g / k. For kt 1, when the whole system is in the cat-like state 共32兲, f 2共t兲 quadratically decreases as exp共−2g2t2兲, independent of the dissipative rate k. The subsequent behavior is approximately an exponential decay whose start and rate depend on the atom-cavity field coupling. For g / k ⲏ 0.5, that is also below the strong coupling regime, we can introduce a decoherence and disentanglement rate

1 关兩− ␣典具− ␣兩 + 兩␣典具␣兩 + 4兩0典具0兩 2共3 + f 2 ⫿ 4f 1兲

2

⫿ 4e−兩˜␣兩

2/2

,

+ 3兲

兩− ˜␣典 − 兩˜␣典

2

共37兲

where ˜␣共t兲 = igt. The Wigner functions representing the states 共36兲 in phase space at a given time, are 2

W f,共ee,gg兲共␤兲 =

2e−2兩␤兩 2 关2 + e−2兩␣兩 cosh共4兩␣兩Im␤兲 ␲共3 + f 2 ⫿ 4f 1兲 2

+ f 2e2兩␣兩 cos共4兩␣兩Re␤兲 ⫿ 4f 1 cosh共2兩␣兩Im␤兲cos共2兩␣兩Re␤兲兴, 2

W f,共eg,ge兲共␤兲 =

2e−2兩␤兩 2 关e−2兩␣兩 cosh共4兩␣兩Im␤兲 ␲共1 − f 2兲 2

− f 2e2兩␣兩 cos共4兩␣兩Re␤兲兴.

共38兲

In Fig. 7 we illustrate two of the Wigner functions 共38兲 in the transient kt 1 and in the strong coupling regime g k. At steady state the Wigner functions are positive, corresponding to the states

033839-7

1 SS ␳ˆ F,共ee,gg兲 = 关兩− ␣SS典具− ␣SS兩 + 兩␣SS典具␣SS兩 + 4兩0典具0兩兴, 6


BINA et al.

(a)

(b)

FIG. 7. Wigner function of the cavity field W关␳ˆ f,共ee兲兴共a兲 and g g W关␳ˆ f,共eg兲兴共b兲, for kt = 0.05, and for k = 80 共a兲 and k = 40 共b兲.

1 SS ␳ˆ F,共eg,ge兲 = 关兩− ␣SS典具− ␣SS兩 + 兩␣SS典具␣SS兩兴, 2

共39兲

2g

where ␣SS = i k . VI. NUMERICAL RESULTS

To confirm our theoretical analysis as well as to investigate system dynamics without the strong driving condition and including the effect of atomic decay, where analytical results are not available, we numerically solve, by Monte Carlo wave function 共MCWF兲 method 关24兴, the ME 共4兲 in the resonant case ␦ = 0, that we rewrite in the Lindblad form

First we consider negligible atomic decay ˜␥ = 0 to confirm the analytical solutions and to evaluate the effect of the driv˜ . For numerical convenience we consider the ing parameter ⍀ ˜ case k = 1 so that the steady state mean photon number assumes small enough values. We consider the atoms prepared in the maximally entangled states 兩⌽⫾典. We recall that for the state 兩⌽+典 the theoretical mean photon number is 具Nˆ共t˜兲典 = 兩␣共t˜兲兩2, the atomic populations Pe,g共t˜兲 = 0.5, the atomic pu1+f 2共t˜兲 rity ␮a共t˜兲 = 22 , and the entanglement of formation ⑀F共t兲 is given by Eq. 共27兲, where the concurrence C共t兲 coincides with f 2共t兲. In Fig. 8 we show, e.g., the mean photon number and ˜ the atomic probability pg共t兲. In the strong driving limit, ⍀ = 20, we find an excellent agreement with the predicted theoretical behavior. We note that we simulated the system dynamics without the RWA approximation so that pg共t˜兲 exhibits oscillations due to the driving field. We remark that the entanglement of formation in the case of state 兩⌽+典 evolves ˜ and becomes neglialmost independently of parameter ⍀ gible after times gt ⬇ 2. In the case of state 兩⌽−典 ⑀F共t兲 decays ˜ , but it remains close to in a similar way for small values of ⍀ 1 for large enough values of the driving parameter as predicted in our analysis. Finally, we consider the effect of the atomic decay. For example, we consider the atoms prepared in the Bell state 兩⌽−典 in the strong driving limit and for the cavity field decay rate ˜k = 1. In Fig. 9 we show the mean photon number and the entanglement of formation. We see that for ˜␥ up to 10−3 the effect of atomic decays is negligible and the results of our treatment still apply. For larger decay rates the atomic dynamics becomes no more restricted within the decoherence free subspace. VII. CONCLUSIONS

3

i ˆ ˆ †兲 + 兺 C ˆ ␳ˆ C ˆ† ˆ I − ␳ˆ IH ␳˙ˆ I = − 共H e␳ i I i, e ប i=1

共40兲

ˆ is given where the non-Hermitian effective Hamiltonian H e by ˆ = H e

ˆ H I

3

iប ˆ †C ˆ , − 兺C i g 2 i=1 i

共41兲

ˆ is that of Eq. 共5兲 for ␦ = 0, and the colthe Hamiltonian H I ˆ = 冑˜␥␴ˆ , C ˆ = 冑˜kaˆ. We have introlapse operators are C 1,2 1,2 3 duced the scaled time ˜t = gt so that the relevant dimensionless system parameters are ˜ = ⍀, ⍀ g

˜k = k , g

˜␥ =

␥ . g

共42兲

The system dynamics can be simulated by a suitable number Ntr of trajectories, i.e., stochastic evolutions of the whole system wave function 兩␺ j共t˜兲典共j = 1 , 2 , . . . , Ntr兲. Therefore, the statistical operator of the whole system can be approximated by averaging over the Ntr trajectories, i.e., ␳ˆ I共t˜兲 ⬵ N1tr 兺Ni=jtr兩␺ j共t˜兲典具␺ j共t˜兲兩.

We have described the dynamics of a system where a pair of two-level atoms, strongly driven by an external coherent field, are resonantly coupled to a cavity field mode that is in contact with an environment. There are different available or forthcoming routes to the implementation of our model. In the microwave regime of cavity QED pairs of atoms excited to Rydberg levels cross a high-Q superconductive cavity with negligible spontaneous emission during the interaction 关9兴. Difficulties may arise due to the 共ideal兲 requirements on atomic simultaneous injection, equal velocity, and equal coupling rate to the cavity mode. In the optical regime the application of cooling and trapping techniques in cavity QED 关11兴 allows the deterministic loading of single atoms in a high-finesse cavity, with accurate position control and trapping times of many seconds 关12兴. In this regime laserassisted three-level atoms can behave as effective two-level atoms with negligible spontaneous emission 关15兴. On the other hand, trapped atomic ions can remain in an optical cavity for an indefinite time in a fixed position, where they can couple to a single mode without coupling rate fluctuations 关25兴. These systems are quite promising for our purposes and could become almost ideal in case of achievement of the strong coupling regime.

033839-8



4

1.5

(4)

〈N〉

〈N〉

(3)

(3)

2

0.75 (2)

(2)

(1)

(1)

0

(a) 0

10

gt 20

0

(a) 0

10

gt

20

1

p

(1)

g

1

(1)

ε

(2)

F

0.65

(2) (3)

0.5

(4)

0.3 0 (b)

5

gt

(3)

10

0

(b) 0

2.5 〈N〉

1.25 (2)

(1) (4)

10

1

gt 20

(1)

p

g (2)

0.65

(3) (4)

0.3 (d) 0

5

gt 20

FIG. 9. Effect of atomic decay ˜␥ in the strong driving condition ˜ ⍀ = 20, atoms prepared in the Bell state 兩⌽−典 and ˜k = 1: ˜␥ = 0.001 共1兲, 0.01 共2兲, 0.1 共3兲. 共a兲 Mean photon number 具Nˆ共t˜兲典, 共b兲 entanglement of formation ⑀F共t˜兲. The number of trajectories is Ntr = 500.

(3)

0 (c) 0

10

gt 10

˜ for negligible atomic FIG. 8. Effect of driving parameter ⍀ ˜ decay ˜␥ = 0, k = 1, atoms prepared in the Bell state 兩⌽+典关共a兲,共b兲兴 and ˜ : 0.5 共1兲, 1 共2兲, 2 共3兲, 20 共4兲. The 兩⌽−典关共c兲,共d兲兴, for values of ⍀ theoretical functions are the dashed lines. We show in 共a兲,共c兲 the ˆ 共t˜兲典, and in 共b兲,共d兲 the atomic probability mean photon number 具N pg共t˜兲. The number of trajectories is Ntr = 500.

Under full resonance conditions and starting from the vacuum state of the cavity field, and for negligible atomic decays and thermal fluctuations, we solved exactly the system dynamics for any initial preparation of the atom pair, thus deriving a number of analytical results on the whole system as well as the different subsystems. These results are confirmed and extended by numerical simulations, e.g., including also atomic decays and investigating regimes under weaker driving conditions. Here we discuss some results mainly concerning the biatomic subsystem. First of all we find that if atoms are prepared in a separable state, no atom-atom entanglement can be generated, in agreement with previous results that showed the classical nature of atom-atom correlations under purely unitary dynamics 关26兴. If atoms are initially quantum correlated, their dynamics can be quite different as can be appreciated in the Bell basis. If the initial state is any linear combination of the states 兩⌽−典 and 兩⌿−典, it does not evolve in time. The atomic subspace is free from cavity dissipation and decoherence, hence the initial entanglement can be preserved for storage times useful for quantum computation/communication purposes 关21兴. If the atoms are prepared in any superposition of the other Bell states 兩⌽+典 , 兩⌿+典 we find, remarkably, that the same function describes the decay of atom-atom concurrence, that

033839-9


BINA et al.

is an entanglement measure, as well as of environmentinduced decoherence and purity. Also we show that joint atomic measurements allow monitoring all these fundamental quantities. By generalizing the one-atom analysis of Ref. 关15兴 we can describe the nontrivial decay process. First of all, in a short time/low dissipation limit 共kt 1兲 the whole system is in a pure, entangled, catlike state, as e.g., in Eq. 共31兲 starting from 兩0典丢兩⌽+典. In this initial stage the decay is quadratic in time and independent of the cavity dissipative rate k. The subsequent behavior can be well approximated by an exponential decay. In a weak coupling regime 共g / k 1兲 the decoherence and disentanglement rate is ␥D ⯝ 8g2 / k. In a strong coupling regime 共g / k ⲏ 1兲 the rate is ␥D ⯝ 2g, i.e., twice the JC coupling frequency. In this regime the whole process is independent of the dissipative rate k. It is in fact the strong coupling of each strongly driven atom to the cavity field, which is in turn entangled with the environment, that rules the decay of both entanglement and quantum coherence of the biatomic subsystem.

enough for our purposes. Equations 共A1兲 become in phase space

␹˙ 11 = ig共␤ + ␤ⴱ兲␹11 −

␹˙ 12,13 = − ig −

−

ACKNOWLEDGMENTS

E.S. is thankful for financial support from EuroSQIP and DFG SFB 631 projects, as well as from German Excellence Initiative via NIM. APPENDIX: SOLUTION OF THE MASTER EQUATION

ˆ ␳ˆ ␳ˆ˙ 12,13 = − ig共aˆ† + aˆ兲␳ˆ 12,13 + L f 12,13 , ˆ ␳ˆ , ␳˙ˆ 14 = − ig兵aˆ† + aˆ, ␳ˆ 14其 + L f 14

冊

冉冉

冋

册

⳵ ⳵ ␹14 − ⳵ ␤ ⳵ ␤ⴱ

冊冊

⳵ ⳵ k ␤ − ␤ⴱ ⴱ + 兩␤兩2 ␹22,23,33 , 2 ⳵␤ ⳵␤

冋

冉

册

1 ⳵ ⳵ − + 共␤ + ␤ⴱ兲 ␹24,34 ⳵ ␤ ⳵ ␤ⴱ 2

冊

⳵ ⳵ k ␤ − ␤ⴱ ⴱ + 兩␤兩2 ␹24,34 , 2 ⳵␤ ⳵␤

冉

冊

⳵ ⳵ k ␤ − ␤ⴱ ⴱ + 兩␤兩2 ␹44 . 2 ⳵␤ ⳵␤

␹˙ 44 = − ig共␤ + ␤ⴱ兲␹11 −

共A2兲 For the initial atomic preparation we consider the general 4 ci兩␯i典, where the coefficients ci are pure state 兩␺a共0兲典 = 兺i=1 4 兩ci兩2 = 1 and 兵兩␯i典其i=1,..,4 normalized as 兺i=1 + − + − = 兵兩⌽ 典 , 兩⌽ 典 , 兩⌿ 典 , 兩⌿ 典其 is the Bell basis. By this choice we can describe atoms in a maximally or partially entangled state, as well as separable states with both atoms in a superposition state 具⌿兩a共0兲 = 共a1兩e典1 + b1兩g典1兲丢共a2兩e典2 + b2兩g典2兲, where

ˆ ␳ˆ ␳˙ˆ 22,23,33 = L f 22,23,33 ,

c1 =

a 1a 2 + b 1b 2

冑2

,

c2 =

a 1a 2 − b 1b 2

冑2

, 共A3兲

ˆ ␳ˆ ␳˙ˆ 24,34 = − ig␳ˆ 24,34共aˆ† + aˆ兲 + L f 24,34 , c3 = ˆ ␳ˆ , ␳˙ˆ 44 = ig关aˆ† + aˆ, ␳ˆ 44兴 + L f 44

册

1 ⳵ ⳵ − − 共␤ + ␤ⴱ兲 ␹12,13 ⳵ ␤ ⳵ ␤ⴱ 2

⳵ ⳵ k ␤ − ␤ⴱ ⴱ + 兩␤兩2 ␹14 , 2 ⳵␤ ⳵␤

␹˙ 24,34 = − ig −

ˆ ␳ˆ , ␳˙ˆ 11 = − ig关aˆ† + aˆ, ␳ˆ 11兴 + L f 11

冉

冊

⳵ ⳵ k ␤ − ␤ⴱ ⴱ + 兩␤兩2 ␹12,13 , 2 ⳵␤ ⳵␤

␹˙ 14 = − 2ig

␹˙ 22,23,33 = −

We illustrate how to solve the master Eq. 共9兲 in the case of exact resonance, ␦ = 0. For the density operator ␳ˆ 共t兲 of the whole system, we introduce the decomposition of Eq. 共10兲. Therefore, the ME is equivalent to the following set of uncoupled equations for the field operators ␳ˆ ij共t兲 = 具i兩␳ˆ 共t兲兩j典:

冋

冉

⳵ ⳵ k ␤ − ␤ⴱ ⴱ + 兩␤兩2 ␹11 , 2 ⳵␤ ⳵␤

a 1b 2 + b 1a 2

共A1兲

where the brackets and braces denote the standard commutator and anticommutator symbols and ␳ˆ˙ j,i共t兲 = 关␳ˆ˙ i,j共t兲兴†. In the phase space associated to the cavity field we introduce ˆ 共␤兲兴. We note that the the functions ␹ij共␤ , t兲 = Tr f 关␳ˆ ij共t兲D functions ␹ij共␤ , t兲 cannot be interpreted as characteristic functions 关27兴 for the cavity field, because the operators ␳ˆ ij do not exhibit all required properties of a density operator. As a consequence the functions ␹ij共␤ , t兲 do not fulfill all conditions for quantum characteristic functions. Nevertheless, they are continuous and square integrable, which is

冑2

,

c4 =

a 1b 2 − b 1a 2

冑2

.

For the cavity field we consider the vacuum state, so that for the whole system the initial state is ␳ˆ 共0兲 = 兩0典具0兩丢兩␺a共0兲典具␺a共0兲兩, corresponding to the functions

冉兺 4

␹ij共␤,0兲 =

k,l=1

冊

ckcⴱl 具i兩␯k兩典具␯l兩j兩典 exp兵兩␤兩2/2其.

共A4兲

Introducing the real and imaginary part of the variable ␤ we can derive from Eq. 共共A2兲 a system of uncoupled equations such that each of them can be solved by applying the method of characteristics 关28兴. The solutions are

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再

冎

兩␤兩2 1 ⫿ ␣ⴱ共t兲␤ ⫾ ␣共t兲␤ⴱ , ␹11,44共␤,t兲 = 兩c1 ⫾ c3兩2exp − 2 2

f 2共t兲 = f 41共t兲. From the above expressions we can recognize that the corresponding field operators ␳ˆ ij共t兲 are 1 ␳ˆ 11,44共t兲 = 兩c1 ⫾ c3兩2兩⫿ ␣共t兲典具⫿ ␣共t兲兩, 2

1 ␹12,13共␤,t兲 = − 共c1 + c3兲共c2 ⫿ c4兲ⴱ f 1共t兲 2

再

⫻exp −

冎

1 ␹14共␤,t兲 = 共c1 + c3兲共c1 − c3兲ⴱ f 2共t兲 2

再

⫻exp −

1 f 1共t兲 ␳ˆ 12,13共t兲 = 共c1 + c3兲共c2 ⫿ c4兲ⴱ 2 兩− ␣ 共t兲典 2 e−兩␣共t兲兩 /2

兩␤兩2 + ␣共t兲␤ⴱ , 2

⫻具0兩, 1 f 2共t兲 ␳ˆ 14共t兲 = 共c1 + c3兲共c1 − c3兲ⴱ 2 兩− ␣ 共t兲典 −2兩 2 e ␣共t兲兩

冎

兩␤兩 + ␣共t兲ⴱ␤ + ␣共t兲␤ⴱ , 2 2

⫻具␣共t兲兩,

再冎再冎

兩␤兩2 1 ␹22,33共␤,t兲 = 兩c2 ⫿ c4兩2 exp − , 2 2

1 ␳ˆ 22,33共t兲 = 兩c2 ⫿ c4兩2兩0典具0兩, 2

兩␤兩2 1 ␹23共␤,t兲 = 共c2 − c4兲共c2 + c4兲ⴱ exp − , 2 2

1 ␳ˆ 23共t兲 = 共c2 − c4兲共c2 + c4兲ⴱ兩0典具0兩, 2 1 f 1共t兲 ␳ˆ 24,34共t兲 = − 共c2 ⫿ c4兲共c1 − c3兲ⴱ 兩0典 −兩␣共t兲兩2/2 2 e

1 ␹24,34共␤,t兲 = − 共c2 ⫿ c4兲共c1 − c3兲ⴱ f 1共t兲 2

再

⫻exp −

冎

兩␤兩2 + ␣ⴱ共t兲␤ , 2

⫻具␣共t兲兩.

共A5兲

共A6兲

where ␣共t兲 and f 1共t兲 are defined in Eqs. 共12兲 and 共13兲 and

Hence, by remembering that ␳ˆ ji = ␳ˆ †ij, we can reconstruct the whole system density operator ␳ˆ 共t兲 of Eq. 共10兲.

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