Decoherence suppression of a dissipative qubit by non-Markovian effect
We evaluate exactly the non-Markovian effect on the decoherence dynamics of a qubit interacting with a dissipative vacuum reservoir and find that the coherence of the qubit can be partially trapped in the steady state when the memory effect of the reservoir is considered. Our analysis shows that it is the formation of a bound state between the qubit and its reservoir that results in this residual coherence in the steady state under the non-Markovian dynamics. A physical condition for the decoherence suppression is given explicitly. Our results suggest a potential way to decoherence control by modifying the system-reservoir interaction and the spectrum of the reservoir to the non-Markovian regime in the scenario of reservoir engineering.
pacs:03.65.Yz, 42.50.Dv, 42.50.Lc
Any realistic quantum system inevitably interacts with its surrounding environment, which leads to the loss of coherence, or decoherence, of the quantum system . The decoherence of quantum bit (qubit) is deemed as a main obstacle to the realization of quantum computation and quantum information processing . Understanding and suppressing the decoherence are therefore a major issue in quantum information science. For a Markovian environment, it is well known that the coherence of a qubit experiences an exponential decrease . To beat this unwanted degradation, many controlling strategies, passive or active, have been proposed [3, 4, 5, 6, 7].
In recent years much attention has been paid to the non-Markovian effect on the decoherence dynamics of open quantum system [8, 9, 10, 11, 12]. The significance of the non-Markovian dynamics in the study of open quantum system is twofold. i) It is of fundamental interest to extend the well-developed methods and concepts of Markonian dynamics to non-Markovian case [1, 13] for the open quantum system in its own right. ii) There are many new physical situations in which the Markovian assumption usually used is not fulfilled and thus the non-Markovian dynamics has to be introduced. In particular, many experimental results have evidenced the existence of the non-Markovian effect [14, 15, 16], which indicates that one can now approach the non-Markovian regime via tuning the relevant parameters of the system and the reservoir. The non-Markovian effect means that the environment, when its state is changed due to the interaction with the quantum system, in turn, exerts its dynamical influence back on the system. Consequently one can expect decoherence dynamics of the quantum system could exhibit a dramatic deviation from the exponential decaying behavior. In 2005, DiVincenzo and Loss studied the decoherence dynamics of the spin-boson model for the Ohmic heat bath in the weak-coupling limit. They used the Born approximation and found that the coherence dynamics has a power-law behavior at long-time scale , which greatly prolongs the coherence time of the quantum system. Such power-law behavior suggests that the non-Markovian effect may play a constructive role in suppressing decoherence of the system. Nevertheless, in many cases the finite extension of the coherence time of the system is not sufficient for the quantum information processing, a question arises whether the coherence of the system can be preserved in the long-time limit, even partially. Theoretically, the answer is positive if the environment has a nontrivial structure. It has been shown that some residual coherence can be preserved in the long-time steady state when the environment is a periodic band gap material [18, 19, 20, 21] or leaky cavity . It is stressed that the residual coherence is due to the confined structured environment. A natural question is: Whether the coherence of the system can be dynamically preserved or not by the non-Markovian effect if the environment has no any special structure, e.g., a vacuum reservoir?
In this paper, we study the exact decoherence dynamics of a qubit interacting with a vacuum reservoir and examine the possibility of decoherence suppression using the non-Markovian effect. The main aim of this work is to analyze if and how the coherence present in the initial state can be trapped with a noticeable fraction in the steady state even when the environment is consisted of a vacuum reservoir with trivial structure. We show that the non-Markovian effect manifests its action on the qubit not only in the transient dynamical process, but also in the asymptotical behavior. Our analysis shows that the physical mechanism behind this dynamical suppression to decoherence is the formation of a bound state between the qubit and the reservoir. The no-decaying character of the bound state leads to the inhibition of the decoherence and the residual coherence trapped in the steady state. A similar vacuum induced coherence trapping in the continuous variable system has been reported in [23, 24]. Such coherence trapping phenomenon provides an alternative way to suppress decoherence. This could be realized by controlling and modifying the system-reservoir interaction and the properties of the reservoir  by the recently developed reservoir engineering technique [25, 26, 27].
Our paper is organized as follows. In Sec. 2, we introduce the model of a qubit interacting with a vacuum reservoir and derive the exact master equation. In Sec. 3, two quantities, i.e. purity and decoherence factor, to characterize the decoherence dynamics are introduced. In Sec. 4 we give the numerical study for the time evolution of decay rate, purity and decoherence factor in terms of coupling constant and cutoff frequency and the physical mechanism of the dynamical decoherence suppression. Finally, discussions and summary are given in Sec. 5
2 The model and exact dynamics of the qubit
We consider a qubit interacting with a reservoir which is consisted of a quantized radiation field. The Hamiltonian of the total system is given by
where and are the inversion operators and transition frequency of the qubit, and are the creation and annihilation operators of the -th mode with frequency of the radiation field. The coupling strength between the qubit and the radiation field has the form 
where and are the unit polarization vector and the normalization volume of the radiation field, is the dipole moment of the qubit, and is the free space permittivity. Throughout this paper we assume . This model has been well studied under the Born-Markovian approximation in quantum optics . However, what is the physical condition under which such approximation is applicable and how the non-Markovian effect affects the decoherence dynamics in the different parameter regimes have not been quantitatively investigated.
If there is no correlation between the qubit and the radiation field initially, then the initial state of the whole system can be factorized into a product of the states of qubit and the field. If the initial state is , with and , respectively, denoting the exited state of the qubit and the vacuum state of the radiation field, then governed by the Hamiltonian (1), the state will evolve to the following form
where is the field state containing one photon only in the -th mode. From the Schrödinger equation, we can get the time evolution of the probability amplitudes in Eq. (3). On substituting the formal solution of into the equation of motion satisfied by , we obtain
where the kernel function is . The integro-differential equation (4) renders the dynamics of the qubit non-Markovian, with the memory effect of the reservoir registered in the time-nonlocal kernel function . In the continuous limit of the environment frequency, one can verify from the coupling strength (2) that the kernel function has the form
where is called the spectral density with the coupling constant . Here the in is introduced to make dimensionless. To eliminate the infinity in frequency integration, we have introduced the cutoff frequency . Physically, the introducing of the cutoff frequency means that not all of the infinite modes of the reservoir contribute to the interaction with the qubit, and one always expects the spectral density going to zero for the modes with frequencies higher than certain characteristic frequency. It is just this characteristic frequency which determines the specific behavior and the properties of the reservoir. One can see that in our model, the spectral density has a super-Ohmic form .
From the time evolution of Eq. (3) and the fact that the ground state of the qubit is immune to the radiation field, one can get the time evolution of any given initial state of the system readily. For a general state of the qubit described by
the time evolution of the total system can be calculated explicitly. In fact, what is needed is the reduced density matrix of the qubit, which is obtained by tracing over the reservoir variables
Differentiating Eq. (6) with respect to time, we arrive at the equation of motion of the reduced density matrix
where and . plays the role of time-dependent shifted frequency and that of time-dependent decay rate . It is worth mentioning that in the derivation of the master equation (7), we have not resorted to the Born-Markovian approximation, that is, Eq. (7) is the exact master equation of the qubit system. Compared with the master equation derived in Ref.  under the condition that the initial state of the qubit is pure, our derivation shows that Eq. (7) can describe the dynamics not only for the initial pure state, but also for any mixed state of the qubit.
It is interesting to note that one can reproduce the conventional Markovian one from our exact non-Markovian master equation under certain approximations. By redefining the probability amplitude as , one can recast Eq. (4 ) into
Then, we take the Markovian approximation , namely, approximately taking the dynamical variable to the one that depends only on the present time so that any memory effect regarding the earlier time is ignored. The Markovian approximation is mainly based on the physical assumption that the correlation time of the reservoir is very small compared with the typical time scale of system evolution. Also under this assumption we can extend the upper limit of the integration in Eqs. (8) to infinity and use the equality
where and the delta-function denote the Cauchy principal value and the singularity, respectively. The integro-differential equation in (8) is thus reduced to a linear ordinary differential equation. The solutions of as well as can then be easily obtained as , where . Thus one can verify that,
which are just the coefficients in the Markovian master equation of the two-level atom system .
3 Purity and decoherence factor
To quantify the decoherence dynamics of the qubit, we introduce the following two quantities. The first one is the purity, which is defined as 
Clearly for pure state and for mixed state. The second quantity describing the decoherence is the decoherence factor of the qubit, which is determined by the off-diagonal elements of the reduced density matrix
The decoherence factor maintains unity when the reservoir is absent and vanishes for the case of completely decoherence.
For definiteness, we consider the following initial pure state of the qubit
in which and satisfy the normalization condition. Using Eq. (6), the exact time evolution of the qubit is easily obtained
With Eq. (14), the purity and decoherence factor can be expressed explicitly
It is easy to verify, under the Born-Markovian approximation, the purity and decoherence factor have the following forms
where the time-independent decay rate is given in Eq. (10). Obviously, the system asymptotically loses its quantum coherence () and approaches a pure steady state () irrespective of the form of the initial state under the Markovian approximation. One can also find from Eqs. (15-18) that the probability amplitude of excited state plays key role in the decoherence dynamics.
4 Numerical results and analysis
In this section, by numerically solving Eq. (4), we study the influence of memory effect of reservoir on the exact dynamics of the qubit. Noticing the fact that the memory effect registered in the kernel function is essentially determined by the spectrum density , one can expect that plays an major role in the exact dynamics of the qubit. In the following, we show how the decoherence of the qubit can be fully suppressed under the non-Markovian dynamics in terms of the relevant parameters of .
4.1 The influence of coupling constant
In the following, we numerically analyze the exact decoherence dynamics of the qubit with respect to decay rate , purity and decoherence factor in terms of the coupling constant .
In Fig. 1 we plot the time evolution of decay rate , purity , decoherence factor and their Markovian correspondences in the weak coupling and low cutoff frequency case. We can see that shows distinct difference from its Markovian counterpart over a very short time interval. With time, tends to a definite positive value. The small “jolt” of in the short time interval just evidences the backaction of the memory effect of the reservoir exerted on the qubit [29, 30]. It manifests that the reservoir does not exert decoherence on the qubit abruptly, just as the result based on Markovian approximation, but dynamically influences the qubit and gradually establishes a stable decay rate to the qubit. Furthermore, it is also shown that the decay rate is positive in the full range of evolution, which results in any initial qubit state evolving to the ground state irreversibly. Consequently the decoherence factor monotonously decreases to zero with time and the purity approaches unity in the long-time limit, which is consistent with the result under Markovian approximation. The result indicates that although the reservoir has backaction effect on the qubit, it is quite small. And the dissipation effect of the reservoir dominates the dynamics of the qubit. Thus no qualitative difference can be expected between the exact result and the Markovian one with the backaction effect ignored. Therefore the widely used Markovian approximation is applicable in this case. Nevertheless, at the short and immediate time scales the overall behavior is still quite different from that of the Markovian dynamics. The decoherence factor shown in the righ-hand panel of Fig. 1 shows non-exponential decay, which is in agreement with the result obtained previously in the spin-boson model in the weak-coupling limit . However, the situation is dramatically changed if the coupling is strengthened as discussed below.
With the same cutoff frequency as in Fig. 1 but a larger coupling constant, we plot in Fig. 2 the decay rate, purity and decoherence factor in the strong coupling case. In this case the non-negligible backaction of the reservoir has a great impact on the dynamics of the qubit. Firstly, we can see that the decay rate not only exhibits oscillations, but also takes negative values in the short time scale. Physically, the negative decay rate is a sign of strong backaction induced by the non-Markovian memory effect of the reservoir. And the oscillations of the decay rate between negative and positive values reflect the exchange of excitation back and forth between qubit and the reservoir . Consequently both the decoherence factor and the purity exhibit oscillations in a short-time scale, which shows dramatic deviation to the Markovian result. Therefore, entirely different to the weak coupling case in Fig. 1, the reservoir in the strong coupling case here has strong backaction effect on the qubit. Secondly, we also notice that the decay rate approaches zero in the long-time limit. The vanishing decay rate means, after several rounds of oscillation, the qubit ceases decaying asymptotically. The non-Markovian purity maintains a steady value asymptotically, which is less then unity. This indicates that the steady state of the qubit is not the ground state anymore, but a mixed state. The decoherence factor also tends to a non-zero value, which implies that the coherence of the qubit is preserved with a noticeable fraction in the long-time steady state. These phenomena, which are qualitatively different to the Markovian situation, manifest that the memory effect has a considerable contribution not only to the short-time, but also to the long-time behavior of the decoherence dynamics. The presence of the residual coherence in the steady state also suggests a potential active control way to protect quantum coherence of the qubit from decoherence via the non-Markovian effect.
4.2 The influence of cutoff frequency
The cutoff frequency , on the one hand, is introduced to eliminate the infinity in the frequency integration. On the other hand it also determines the frequency range in which the power form is valid . In the following, we elucidate the influence of cutoff frequency on the exact decoherence dynamics.
Fixing as the value in Fig. 1 and increasing the cutoff frequency, we plot in Fig. 3 the dynamics of the qubit in a high cutoff frequency case. It shows that a similar decoherence behavior as the strong coupling case in Fig. 2 can be obtained. After several rounds of oscillation, the decay rate tends to zero in the long-time limit. The negative decay rate makes the lost coherence partially recovered. The vanishing decay rate in the long-time limit results in the decoherence frozen before the qubit gets to its ground state. Thus there is some residual coherence trapped in the steady state. Similar to the strong coupling case, it is essentially the interplay between the backaction and the dissipation on the dynamics of qubit which results in the inhibition of decoherence. We argue that in this high cutoff frequency regime, the widely used Markovian approximation is not applicable because of the strong backaction effect of the reservoir.
4.3 The physical mechanism of the decoherence inhibition
From the analysis above we can see clearly that the decoherence can be inhibited in the non-Markovian dynamics. A natural question is: What is physical mechanism to cause such dynamical decoherence inhibition? To answer this question, let us find the eigen solution of Eq. (1) in the sector of one-excitation in which we are interested. The eigenequation reads , where . After some algebraic calculation, we can obtain a transcendental equation of
From the fact that decreases monotonically with the increase of when we can say that if the condition , i.e.
is satisfied, always has one and only one intersection in the regime with the function on the right-hand side of Eq. (19). Then the system will have an eigenstate with real (negative) eigenvalue, which is a bound state , in the Hilbert space of the qubit plus its reservoir. While in the regime of , one can see that is divergent, which means that no real root can make Eq. (19) well-defined. Consequently Eq. (19) does not have positive real root to support the existence of a further bound state. It is noted that Eq. (19) may possess complex root. Physically this means that the corresponding eigenstate experiences decay contributed from the imaginary part of the eigenvalue during the time evolution, which causes the excited-state population approaching zero asymptotically and the decoherence of the reduced qubit system.
The formation of bound state is just the physical mechanism responsible for the inhibition of decoherence. This is because a bound state is actually a stationary state with a vanishing decay rate during the time evolution. Thus the population probability of the atomic excited state in bound state is constant in time, which is named as “population trapping” [18, 20]. This claim is fully verified by our numerical results. The parameters in Fig. 1 do not satisfy the condition (20) to support the existence of a bound state, then the dynamics experiences a severe decoherence. While with the increase of either (in Fig. 2) or (in Fig. 3), the bound state is formed. Then the system and its environment is so correlated that it causes the decay rate of the system in the non-Markovian dynamics exhibiting: 1) transient negative value due to the backaction of the environment; 2) vanishing asymptotic value. Such interesting phenomenon, i.e. the vanishing asymptotical decay rate in the large cutoff frequency regime for super-Ohmic spectrum density, was also revealed in Ref. . This effect of course is missing in the conventional Born-Markovian decoherence theory, where the reservoir is memoryless.
In order to understand the exact decoherence dynamics more completely, we plot in Fig. 4 the crossover from coherence destroying to coherence trapping via increasing either the coupling constant or the cutoff frequency. Coherence trapping can be achieved as long as the bound state is formed. Therefore, one can preserve coherence via tuning the relevant parameters of system and the reservoir, e.g. the qubit-reservoir coupling constant and the property of the reservoir so that the condition (20) is satisfied.
5 Summary and discussions
In summary, we have investigated the exact decoherence dynamics of a qubit in a dissipative vacuum reservoir. We have found that even in a vacuum environment without any nontrivial structure, we can still get the decoherence suppression of the qubit owing to the dynamical mechanism of the non-Markovian effect. From our analytic and numerical results, we find that the non-Markovian reservoir has dual effects on the qubit: dissipation and backaction. The dissipation effect exhausts the coherence of the qubit, whereas the backaction one revives it. In the strong coupling and/or high cutoff frequency regimes, a bound state between the qubit and its reservoir is formed. It induces a strong backaction effect in the dynamics because the reservoir is strongly correlated with the qubit in the bound state. Furthermore, because of the non-decay character of the bound state the decay rate in this situation approach zero asymptotically. The vanishing of the decay rate causes the decoherence to cease before the qubit decays to its ground state. Thus the qubit in the non-Markovian dynamics would evolve to a non-ground steady state and there is some residual coherence preserved in the long-time limit. Our results make it clear how the non-Markovian effect shows its effects on the decoherence dynamics in different parameter regimes.
The presence of such coherence trapping phenomenon actually gives us an active way to suppress decoherence via non-Markovian effect. This could be achieved by modifying the properties of the reservoir to approach the non-Markovian regime via the potential usage of the reservoir engineering technique [25, 26, 27, 34]. Many experimental platforms, e.g. mesoscopic ion trap [25, 26], cold atom BEC , and the photonic crystal material  have exhibited the controllability of decoherence behavior of relevant quantum system via well designing the size (i.e. modifying the spectrum) of the reservoir and/or the coupling strength between the system and the reservoir. It is also worth mentioning that a proposal aimed at simulating the spin-Boson model, which is relevant to the one considered in this paper, has been reported in the trapped ion system . On the other side many practical systems can now be engineered to show the novel non-Markovian effect [14, 15, 16, 36]. All these achievements show that the recent advances have paved the way to experimentally simulate the paradigmatic models of open quantum system, which is one part of the new-emergent field, quantum simulators . Our work sheds new light on the way to indirectly control and manipulate the dynamics of quantum system in this experimental platforms.
This work is supported by NSF of China under Grant No. 10604025, Gansu Provincial NSF of China under Grant No. 0803RJZA095, Program for NCET, and NUS Research Grant No. R-144-000-189-305.
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