Decoherence of two qubits in a non-Markovian reservoir without
The decoherence of two initially entangled qubits in a non-Markovian reservoir has been investigated exactly without Born Markovian approximation and rotating-wave approximation(RWA). The non-perturbative quantum master equation is derived and its exact solution is obtained. The decoherence behaviors of two qubits, initially entangled in Bell states, has been investigated in three different cases of parameters. The results show that the counter-rotating wave terms have great influence on the decoherence behavior, and there are differences between the exact solution of the Hamiltonian with RWA and that of the exact Hamiltonian without RWA.
In recent years, the phenomenon , termed as “entanglement sudden death”(ESD), has been found theoreticallyyu1 (); yu2 () and shown experimentallymp (); mf (). It is shown that spontaneous disentanglement may take only a finite-time to be completed, while local decoherence (the normal single-atom transverse and longitudinal decay) takes an infinite timeyu2 (). It’s quite different from the case of continuous variable two-atom model discussed by Dodd and Halliwelldod (). Thereafter, many works have been devoted to the related topicsikr (); hu (); asm (); sch (); daj (), and some authors extended the results in Markovian regime to non-Markovian casecao (); bel ().
The RWA, which neglecting counter rotating terms corresponding to the emission and absorption of virtual photon without energy conservation, is widely used in quantum optics. Generally, the RWA is justified for small detunings and small ratio of the atom-field coupling divided by the atomic transition frequencykli (); Lou (); pur (). In atom-field cavity systems, this ratio is typically of the order . Recently, cavity systems with very strong couplings have been discussedmei (). The ratio may also become order of magnitudes larger in solid state systems, and the full Hamiltonian, including the virtual processes (counter-rotating terms), must be considerediri (). The neglect of counter-rotating wave terms in Hamiltonian strongly simplifies the mathematical treatment of the problem and usually give exact solution of the approximate Hamiltonian, while the perturbative approach to the systems beyond the RWA usually is complicated and gives approximate solutionpur (). Otherwise, with the same parameters, the approximation in Hamiltonian might lead to different result from the approximation of exact solution with exact Hamiltonian. Fortunately, non-perturbative master equation could be obtained by path integral and it is beneficial to study, exactly, the systems beyond the RWAIsh ().
In this paper, we will focus on the influence of counter-rotating wave terms on the decoherence behavoir of two qubits in a non-markovian reservoir from the exact solution of the system beyond the RWA and Born-Markovian approximation. In section II , the reduced non-perturbative non-Markovian quantum master equation of an atom in a non-Markovian reservoir is derived and its the exact solution is obtained by algebraic approach of Lie superoperator. In section III, the decoherence of two initially entangled atoms coupled with two cavities separately has been discussed. The conclusion will be given in section IV.
Ii Model and exact solution
ii.1 Hamiltonian and non-perturbative master equation
Now we restrict our attention to two noninteracting two-level atoms A and B coupled individually to two environment reservoirsbel (). To this aim, we first consider the Hamiltonian of the subsystem of a single qubit coupled to its reservoir as
where is the atomic transition frequency between the ground state and excited state . , and are pseudo-spin operators of atom. The index labels the field modes of the reservoir with frequency , and are the modes’ creation and annihilation operators, and is the frequency-dependent coupling constant between the transition and the field mode .
The reduced non-perturbative non-Markovian quantum master equation of atom could be obtained by path integralIsh ()
where , and are Liouvillian operators defined as
and stands for partial trace of the reservoir.
where respents the width of the spectral distribution of the reservoir modes and is related to the correlation time of the noise induced by the reservoir, . The parameter is related to the subsystem-reservoir coupling strength. There are two correlation functions in this modelccq ()
comes from the rotating-wave interaction and from the counter-rotating wave interaction.
So, we could obtain the non-perturbative master equation of the subsystem from Eq.(5)
where , , , , and are superoperators defined as
, and are real part, image part and conjugate of , respectively. .
From Eq.(9), we find that there is a frequency shift caused by the interaction between the atom and the non-Markovian reservoirbreu (). Eq.(9) reveals that the contribution of counter-rotating terms is in order of besides the rapidly oscillating terms, which is quite different from the result of Born and Markovian approximation of atom in vacuum agr ().
ii.2 Exact solution of master equation
where is time time order operator, , , , , , , , and
where , , and , , satisfy the following equation
for and for .
Using the results in Appendix, the exact solution of the master equation Eq.(9) is obtained
In order to investigate the entanglement dynamics of the bipartite system, we use Wootters concurrencewoott (). For simplicity, we assume the two subsystems having the same parameters. The concurrence of the whole system could be obtainedikr ()
corresponding to the initial states of and , respectively. Where is real and , and . The reduced joint density matrix of the two atoms, in the standard product basis , could be written asbel ()
here the diagonal elements are
and the nondiagonal elements are
Iii Numerical results and discussion
First, we focus on the decoherence of two qubits with an initial state . For the RWA model in Ref.bel (), Fig.1 shows that the concurrence periodically vanishes with a damping of its revival amplitude. For the non-RWA model in this paper, the decoherence of the system were categorized into three cases
(A) . Fig.2 reveals that the concurrence decreases exponentially to zero with very small amplitude oscillation and the decoherence time is about , which corresponds to the correlation time of coming from the rotating-wave terms. As for the correlation function , it has no memory effect because it’s value averages to zero on time if . Compared with Fig.1 for RWA model, there is no revival of entanglement for non-RWA model, which exhibits that the counter-rotating wave terms could not be neglected in Hamiltonian because the contribution of counter-rotating terms in master equation is in the order of which influence the decoherence behavior in long time scale. This characteristic will hold on for more weaker coupling constant.
(B) . From Fig.3, we could find that the concurrence first decreases to a finite value and maintain it for a period of time, then periodically vanishes with a damping of its revival amplitude, like that for RWA model. This is the result of coaction of rotating wave process and counter-rotating process of atom with a non-Markovian reservoir because the contribution of counter-rotating wave process to the system is bigger than the former case and its memory effect is nozero.
(C) . Fig.4 exhibits that the concurrence decreases to zero, then periodically vanishes with a damping of its revival amplitude, which resulted from the strong interaction between atom and the non-Markovian reservoir throuth virtual photon process.
From the discussion above, we find that the decoherence behavior of is symmetric with because of the symmetry of initial state . And the revival of entanglement could be found only with big ratio of coupling constant to the atom transition frequency. For no-RWA model, the decoherence behaviors are richer than that for RWA model.
Then, we focus on the decoherence of two qubits with initial state of . For the RWA model in Ref.bel (), Fig.5 shows that the entanglement represented by has a similar behavior to for in Fig.1. In contrast, for , there is ESD because vanishes permanently after a finite time, similar to the Markovian caseyu2 (). Second, revival of entanglement appears after periods of times when disentanglement is complete. For the non-RWA model, the decoherence of the system were also categorized into three cases
(A) . Fig.6 reveals that the concurrence decreases exponentially to zero for , while vanishes permanently after a finite time for .
(B) . Fig.7 shows that the entanglement represented by has a similar behavior to for in Fig.3. In contrast, for , the concurrence first decreases to zero and maintain it for a long period of time, then revives with very small amplitude before vanishes permanently.
(C) . The evolution dynamics of concurrence is almost the same as that in Fig.4. Unlike the two cases above, the evolution behavior of concurrence becomes symmetric, like that of , because of the strong interaction of atom with reservoir through the emission and absorption of virtual photon.
The above characteristics of also verify that the revival of entanglement could be found only with big ratio of coupling constant to the atom transition frequency.
The reduced non-perturbative non-Markovian quantum master equation of atom in non-Markovian reservoir has been derived and its exact solution is obtained by algebraic approach of Lie superoperator. The decoherence of two initially entangled atoms, coupled with two vacuum cavities separately, has been discussed.
The results show that the decoherence behavior of two qubits in a non-markovian reservoir is dependent on the ratio of the coupling strength to atomic frquency. First, with the increasing of coupling strength, the decoherence behavior becomes more and more complicated, and there is revival of entanglement after a period of time of disentanglement, due to the strong interaction and the memory effect of the non-markovian reservoir. Second, the strong coupling and the counter-rotating wave interaction could make the decoherence of even parity Bell entanglement state become symmetrical. Third, with no-RWA model, one could find much more decoherence behaviors than that of RWA model under different conditions of system parameters. The case for finite temperature could also be obtained by the method in this paper and will be published elsewhere.
Acknowledgements.This work was supported by the National Natural Science Foundation of China Grants No.60578055, the State Key Program for Basic Research of China under Grant No. 2007CB925204 and No.2007CB307001.
Defining two superoperators as
and using the following relations
one could get
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