Wang Xiao-Lan, Ren Yu-Kun, Zeng Hao-Sheng. Dynamical control of population and entanglement for open Λ-type atoms by engineering the environment. Chinese Physics B, 2019, 28(3): 030301
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Dynamical control of population and entanglement for open Λ-type atoms by engineering the environment
Wang Xiao-Lan, Ren Yu-Kun, Zeng Hao-Sheng †
Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, and Department of Physics, Hunan Normal University, Changsha 410081, China
Project supported by the National Natural Science Foundation of China (Grant No. 11275064), the Specialized Research Fund for the Doctoral Program of Higher Education, China (Grant No. 20124306110003), and the Construct Program of the National Key Discipline, China.
Abstract
The exactly analytical solution for the dynamics of the dissipative Λ-type atom in the zero-temperature Lorentzian environment is presented. On this basis, we study the evolution of the population and entanglement. We find that the stable populations on the two lower levels of the Λ-type atom can be effectively adjusted by the combination of the relative decay rate and the environmental spectral frequency. However, for the initial Werner-like state, the stable entanglement between the two Λ-type atoms has very little tunability. Furthermore, the stable entanglement for the bilateral environment case is larger than that of the unilateral environmental case. A nonintuitive relation between the stable entanglement and stable population is found.
The dynamics of open quantum systems is very sophisticated. Early study of these dynamics involved the application of the Born–Markov approximation, that is, neglecting all the memory effects, which led to the well-known Lindblad master equation.[1,2] The corresponding dynamics is attributed to the quantum Markovian processes. However, quantum non-Markovian processes are more realistic in principle. Many relevant physical systems, such as the quantum optical system,[3] the nanoscale solid-state quantum system,[4,5] quantum chemistry,[6] and the excitation transfer in the biological system,[7] should be treated by quantum non-Markovian processes. Recently, quantum non-Markovian processes have been studied extensively, from the measure of non-Markovianity of the processes[8–18] to the properties[19–29] and applications[30–37] of the dynamical processes.
The research of quantum non-Markovian processes is very much in favor of exact dynamics study because any approximation can possibly wipe out the memory effects. Due to this reason, the problem of a two-level atom dissipating in the zero-temperature environment becomes the paradigm for the investigation of non-Markovian processes, while the dissipative multilevel systems are relatively seldom involved. For the dynamics of V-type atoms dissipating in zero-temperature environments, the method of finding an exact solution has been discussed.[38,39] For the dissipative dynamics of Λ-type atoms, however, we have not yet found the relevant discussions, even though it has been touched on already.[40] One motivation of this paper is to present the exactly analytical solution for a three-level Λ-type atom dissipating in a zero-temperature Lorentzian environment.
The control of population and entanglement in multilevel quantum systems is very important for quantum information processing. In the ideal situation, the population transfer could be realized by optical pumping techniques.[41,42] For open quantum multilevel systems, it could be realized through the stimulated Raman adiabatic method.[43,44] There were many studies on the control of entanglement in dissipative environments, but most of them focused on qubit systems.[45–49] Although a few studies have involved the evolution of entanglement of dissipative V-type atoms,[50,51] similar research for the dissipative Λ-type atoms, without Born–Markov approximation, have not been reported yet. In this paper, based on the exact solution of the open Λ-type atom, we investigate the control of population and entanglement through the engineering of the decay rate and environmental spectral frequency.
The paper is organized as follows. In Section 2, we introduce the microscopic model for a Λ-type three-level atom interacting with a zero-temperature Lorentzian environment, and present the exact analytical solution. In Section 3, we study the problem of controlling population and entanglement. Finally, we give conclusions in Section 4.
2. Dynamical model and its solution
Our model is constituted by a Λ-type three-level atom embedded in an optical vacuum cavity with decay rate λ (Fig. 1(a)). The dissipative optical vacuum cavity can be described equivalently by a zero-temperature Lorentzian environment that is a bosonic reservoir with infinite quantum harmonic oscillators. The configuration of energy levels of the Λ-type atom is depicted in Fig. 1(b), where |1⟩, |2⟩, and |3⟩ denote the ground, meta-stable, and excited states, respectively. The transition frequencies from level |3⟩ to levels |1⟩ and |2⟩ are denoted by ω1 and ω2, respectively, while the transition between |2⟩ and |1⟩ is forbidden. The Hamiltonian for the whole system may be written as
where bk, , and ωk are the annihilation, creation operators, and frequency for the k-th harmonic oscillator of the reservoir, with g1k and g2k being the coupling strengths between the reservoir and the two transition channels. In the above Hamiltonian, we have set the energy of level |3⟩ to be zero and ħ = 1.
Fig. 1. (a) A Λ-type atom embedded in a vacuum cavity with decay rate λ; the latter may be described by a zero-temperature Lorentzian environment. (b) Energy levels of the Λ-type atom with ground, meta-stable, and excited states (|1⟩, |2⟩, and |3⟩), respectively. The transitions |1〉 ↔ |3〉 and |2〉 ↔ |3〉 are allowed, but the transition between |1⟩ and |2⟩ is forbidden.
Suppose that the atom is initially in a general pure state, so that the state of the compound system is
where |0⟩R denotes the vacuum state of the reservoir. The evolved state at any time t may be written as
where |1k⟩R indicates that there is one photon in the k-th mode of the reservoir, and all other modes are empty. Tracing over the environmental degrees of freedom, one obtains the reduced state of the atom in its natural bases as
Here the evolution of the coefficients is determined by the following set of equations:
The evolution of c1(t) and c2(t) can be easily obtained,
To obtain the other coefficients, we formally integrate Eqs. (5d) and (5e) in the condition of ck(0) = dk(0) = 0 and obtain
Plugging them into Eq. (5c), in the continuum limitation ∑kgikgjk → ∫ dω Jij(ω) (i,j = 1,2), we obtain
where the correlation function is defined by fj(t − τ) = ∫d ωJjj(ω) ei (ωj − ω)(t − τ).
We use the Lorentzian distribution
to describe the spectral density of the dissipative optical vacuum cavity. Here, ω0 is the central frequency and the decay rate λ defines the spectral width. The parameter γjj ≡ γj (j = 1,2) describes the spontaneous decay of level |3⟩ to level |j〉, and γij with i ≠ j describes the correlation between the two transitions. When the dipole moments of the two transitions are parallel, the relation is satisfied. In this paper, we consider only this case.
Under the condition of the Lorentzian spectrum, the correlation function becomes with Mj = λ + i (ω0 − ωj), and the solution of Eq. (8) may be written as
with ξ(t) = (D1eb1t + D2eb2t + D3 eb3t). Here, bi are the roots of equation
which are assumed non-degenerate. Since the degenerative probability is very small and can always be avoided by adjusting the structure parameters, we no longer consider the case here. The coefficients Di are given by
Having c3(t) in hand, we can then obtain the evolution of ck(t) and dk(t). Equations (7a) and (7b), in transition to the continuum limitation, lead to
where
δj = ω0 − ωj, and we use
in the second equality of Eq. (14).
The double integrals in Eqs. (12)–(14) can be solved further. Using
to divide the integral with respect to τ′ into two parts, after tedious but straightforward calculation, we finally obtain
where
with
The other four coefficients can be obtained through the following replacement:
In addition, the coefficient α2(t) in Eq. (15b) can be obtained by making the replacement
Finally, the reduced density matrix equation (4) can be written as
where , , with
The populations of the three levels of the Λ-type atom are, respectively,
From Eq. (19), we can easily obtain the quantum map ε for the dynamics of the open Λ-type atom
which will be used in the study of entanglement evolution.
3. Control of population and entanglement
To study the evolution of population, we assume that the atom initially populates in the excited level |3⟩, i.e., initially the atomic state is given by Eq. (2) with c1(0) = c2(0) = 0 and c3(0) = 1. In Fig. 2(a), we show the evolution of the three populations over dimensionless time for γ1 = γ2 = γ. As expected, the excited population P3 decays monotonically and the populations P1,2 increase continuously. To avoid overlap of curves, we set the spectral frequency to be ω0 = 84γ, which deviates slightly from the central frequency 85γ between ω1 and ω2, leading to a quicker increase in P2 (red line) than P1 (blue line). This result implies that the stable populations of levels |1⟩ and |2⟩ should be related to both the spectral frequency ω0 and the decay rates γ1,2.
Fig. 2. Population versus (a) dimensionless time γ t, (b) relative decay rate γ2/γ1, and (c) spectral frequency ω0/γ for the initial state given by Eq. (2) with c1(0) = c2(0) = 0 and c3(0) = 1. The blue, red, and black lines correspond to populations P1, P2, and P3, respectively. Other parameters are set as γ1 = γ, λ = 2γ, ω1 = 90 γ, and ω2 = 80γ. (a) γ2 = γ, ω0 = 84γ; (b) γ t = 50, ω0 = 85γ; (c) γ t = 50, γ2 = γ (dash lines) or γ2 = 9γ (solid lines).
In Fig. 2(b), we show the evolution of the stable populations of the three levels versus the relative decay rate γ2/γ1 for γ t = 50, where the spectral frequency ω0 is set to be the center between ω1 and ω2. It is shown that in the stable state, the population P3 of the excited level is zero, and P1 decreases and P2 increases with the change of γ2/γ1. The intersection of the two curves corresponds to P1 = P2 = 0.5 and γ2/γ1 = 1. Figure 2(b) tells us that if the relative decay rate γ2/γ1 can be adjusted, we can then control the stable populations of levels |1⟩ and |2⟩.
In Fig. 2(c), we show the evolution of the stable populations versus the spectral frequency ω0/γ. It is shown that, in the case of γ2 = γ1, when ω0 resonates approximately with ω2, the stable population P2 approaches one and P1 approaches zero, and the result exchanges when ω0 approaches ω1 (see the dash lines). However in the case that the difference between γ1 and γ2 is large (γ2 = 9 γ1 for the solid lines in the figure), the situation is different: when ω0 approaches ω2, the stable value of P2 is more close to one and P1 is more close to zero; but when ω0 approaches ω1, although the stable value of P1 is larger than P2, they are far away from one or zero. This is the result of the cooperative effect between resonance and decay: both a large decay rate and resonance can improve the stable population. When ω0 resonates with ω2, the cooperative effect is constructive, i.e., both resonance and decay improve the stable value of P2; when ω0 approaches ω1, the cooperative effect is destructive, i.e., the resonant effect improves but the smaller γ1 reduces the stable value of P1. These results remind us that we should use the combination of spectral frequency and decay rates to control stable populations.
To study the evolution of entanglement, we employ the notion of entanglement negativity. For a bipartite system state ρAB, the entanglement negativity is defined as[52,53]
where and are, respectively, the negative and all eigenvalues of the partial transpose of ρAB with respect to subsystem A.
Having Eq. (22) in hand, we can calculate in principle the evolution of any quantum entangled state. Here we take the Werner-like state[54]
as the exemplary example. Here, I denotes the three-dimensional identity matrix, and is a maximally entangled state of two qutrits A and B. The Werner-like state is separable for 0 ≤ ε ≤ 1/4, and entangled for 1/4 < ε ≤ 1.
We discuss the problem in two cases: the unilateral environment and the bilateral environment. The former means that only atom A is influenced by the noisy environment while atom B remains noise-free; the later means that both atoms are influenced by noises. In Fig. 3, we show the time evolution of the entanglement negativity of the Werner-like state for these two cases. It is shown that the entanglement negativity has similar decay behaviors for the unilateral and bilateral environments: after the decaying at the initial stage, the entanglement negativity reaches a stable value that depends on the parameter ε. The difference between the unilateral and bilateral environments is mainly reflected in three aspects. 1) In the initial stage, the entanglement for the bilateral environment case reduces faster than that for the unilateral environment case. 2) The stable entanglement for the bilateral environment case is larger than that for the unilateral environment case. 3) The entanglement for the bilateral environment case has a momentary restoration after decaying in the initial stage. The first difference is intuitive, because the bilateral environment has a larger noisy effect than the unilateral environment. The second difference may be explained as follows. The stable entanglement for the unilateral environment case is formed between an effective two-level system and a three-level system, but for the bilateral environment case it is formed between two effective two-level systems. When the populations of the excited levels of both entangled Λ-type atoms decay into their lower levels, there is more entanglement. Finally, the momentary restoration of entanglement for the bilateral environment case implies that there are some intermediate states in the decay process, which have less entanglement than the stable state.
Fig. 3. Evolution of entanglement versus dimensionless time γ t for the Werner-like state with ε = 1 (dot-dash red lines), ε = 0.7 (solid black lines), and ε = 0.5 (dash blue lines): (a) bilateral environment case and (b) unilateral environment case. The parameters are chosen as γ1 = γ2 = γ, λ = 2γ, ω1 = 92γ, ω2 = 90γ, and ω0 = 91γ.
In Fig. 4, we show the evolution of the stable entanglement versus the relative decay rate γ2/γ1 for the unilateral and bilateral environments. It is shown that the stable entanglement for the unilateral environment case almost remains unchanged, and only has some tiny changes in the beginning stage of γ2/γ1 for the bilateral environment case. This is sharply contrasted with the decaying of populations where the stable populations P1,2 change remarkably with γ2/γ1. In addition, it is seen more clearly that the stable entanglement for the bilateral environment case is lager than that for the unilateral environment case, which agrees with the result presented by Fig. 3.
Fig. 4. Asymptotic entanglement versus relative decay rate γ2/γ1 for the Werner-like state with ε = 1 (dot-dash red lines), ε = 0.7 (solid black lines), and ε = 0.5 (dash blue lines): (a) bilateral environment case and (b) unilateral environment case. Here, γ1 = γ, λ = 2γ, ω1 = 92γ, ω2 = 90γ, ω0 = 91γ, and γ t = 50.
Figure 5 shows the evolution of stable entanglement versus the environmental spectral frequency for the Werner-like state with ε = 1, i.e., the maximally entangled state |ΨAB〉. It again shows that the stable entanglement for the bilateral environment case is lager than that for the unilateral environment case. For the unilateral environment, the stable entanglement is almost unchanged when ω0/γ changes (lower dash lines). For the bilateral environment, it also has limited adjustability via the change in ω0 (upper solid lines). The minimum stable entanglement for the case of γ1 = γ2 takes place at ω0/γ = 85, i.e., at the center between ω1 and ω2 (red lines). For γ2 = 9γ1, the minimum points move to the right (blue lines). The results of Figs. 4 and 5 imply that the stable entanglement for the open Λ-type atoms has only little adjustability by engineering the environmental spectral frequency and the relative decay rate.
Fig. 5. Asymptotic entanglement versus dimensionless frequency ω0/γ for the Werner-like state with ε = 1, where ω1 = 90γ, ω2 = 80γ, λ = 2γ, and γt = 50. Red lines correspond to γ1 = γ2 = γ and blue lines correspond to γ1 = γ, γ2 = 9γ. The upper solid lines correspond to the bilateral environment and the lower dash lines correspond to the unilateral environment.
There is a nonintuitive result that needs to be mentioned. For the initial Werner-like state, and in the conditions of γ1 = γ2 and ω0 = (ω1 + ω2)/2, the stable populations of each atom should satisfy P1 = P2 (refer to Fig. 2(b)), where the corresponding stable entanglement is just the minimum (red lines in Fig. 5). On the contrary, when ω0 resonates with ω1 or ω2, the stable populations P1,2 have great disparity (dash lines in Fig. 2(c)), where the stable entanglement is the maximum (red lines in Fig. 5). In other words, the smaller the population difference, the smaller the entanglement. This result can also be seen from Fig. 4(b), where the stable entanglement has its minimum for γ2/γ1 = 1 (corresponding to P1 = P2), and increases when γ2/γ1 deviates from this point. This is sharply contrastive with the fact that a maximal entangled pure state, i.e., Bell state, has equal weight between the two levels.
4. Conclusion
We have presented the exact solution of the dynamics for an open Λ-type atom dissipating in the zero-temperature Lorentzian environment. Special attention has been paid to the study of the stable population and stable entanglement. We have found that the stable populations on the two lower levels of the Λ-type atom can be effectively adjusted by the combination of relative decay rate and environmental spectral frequency. Both a large decay rate and resonance effect between the atomic transition and environmental frequency can improve the stable population.
In the discussion of entanglement, we have distinguished two cases of unilateral and bilateral environments. We have found that the entanglement for the bilateral environment case decays faster but its stable value is larger than that of the unilateral environment case. Unlike the evolution of populations, the stable entanglement for both unilateral and bilateral environments only has very restrictive adjustability by engineering the decay rates or environmental spectral frequency. A nonintuitive relation between the stable entanglement and stable population has been found: the smaller the population difference between the two lower levels of the Λ-type atom, the smaller the entanglement.
