We take a Lorentzian spectral density of reservoir^[38]

where δ is the detuning between ω ₀ and the center frequency ω of the reservoir. The parameter λ defines the spectral width of the coupling, which is connected to the reservoir correlation time τ _R by τ _R = λ ^{− 1} and the parameter γ ₀ is related to the relaxation time scale τ _S by . In the subsequent analysis of dynamical evolution of the system, typically a weak and a strong coupling regime can be distinguished. For a weak regime, λ > 2 γ ₀, that is, τ _S > 2 τ _R. In this regime the relaxation time is greater than the reservoir correlation time and the behavior of dynamical evolution of the system is essentially a Markovian exponential decay controlled by γ ₀. In the strong coupling regime, that is, for λ < 2 γ ₀, or τ _S < 2 τ _R, the reservoir correlation time is greater than or of the same order as the relaxation time and non-Markovian effects become relevant.^{[39– 41]}

Inserting Eq.  (11) to Eqs.  (9) and (10), we obtain the correlation functions

and

Figures  1(a) and 1(b) present the population dynamics of the excited atoms in Lorentz reservoirs at zero temperature. Figure  1(a) gives the influence of the spectral width λ on the P₁₁ with the detuning (δ = 4γ ₀). It can be seen that the behaviors of the P₁₁ have clear differences in both the Markovian and the non-Markovian regimes. When λ = 5γ ₀ (i.e., in the Markovian regime), as time goes on, the P₁₁ reduces exponentially and quickly to zero, as shown with the dotted line in Fig.  1(a). When λ = 0.5γ ₀ (i.e., with the weak non-Markovian effect), due to the feedback and memory effect of environment, the P₁₁ will oscillate damply and its decay obviously becomes slow, as shown with the dashed line in Fig.  1(a). When λ = 0.1γ ₀ (i.e., with the strong non-Markovian effect), the P₁₁ can be effectively trapped for a long time, as shown with the solid line in Fig.  1(a). Therefore the P₁₁ relies on the non-Markovian effect under δ = 4γ ₀. Increasing the non-Markovian effect, the P₁₁ can be effectively protected. The time evolution of the P₁₁ for different δ in the non-Markovian regime (λ = 0.05γ ₀) is shown in Fig.  1(b). It can be found that the P₁₁ with or without detuning exhibits different behaviors in the non-Markovian regime. When δ = 0 (i.e., without detuning), the P₁₁ also reduces exponentially and quickly to zero, shown as the dotted line in Fig.  1(b). However, when δ = γ ₀ (i.e., with small detuning), due to the feedback and memory effect of the environment, the P₁₁ will oscillate damply, as shown with the dashed line in Fig.  1(b). When δ = 4γ ₀ (i.e., with large detuning), the P₁₁ can also be effectively preserved for a long time, as shown with the solid line in Fig.  1(b). Therefore, the P₁₁ is obviously dependent on detuning, and the bigger the value of δ is, the more the P₁₁ can be effectively protected.

Therefore, from the above analyses, we find that, only if the detuning and the non-Markovian effect are present simultaneously at zero temperature, can the atomic population be effectively trapped in the excited state for a very long time by increasing the detuning and the non-Markovian effect. The physical interpretation is that, when λ > 2 γ ₀ or δ = 0, the quantum information only flows out from the atom and is not returned from its reservoir, so the population dynamics of the excited atoms will reduce exponentially, as shown with the dotted lines in Figs.  1(a) and 1(b). However, when λ < 2 γ ₀ and δ ≠ 0, due to the memory and feedback effect of the reservoir, the quantum information flowing to the reservoir will be partly returned to the atom, so the population dynamics of the excited atoms will oscillate damply. And with δ increasing and λ decreasing, the information returned will be more and the atomic decay rate will become smaller, so that the atomic population can be effectively trapped in the excited state for a very long time, as shown with the dashed lines and the solid lines in Figs.  1(a) and 1(b).

Fig.  1.

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	Fig.  1. Population of the excited atoms as a function of γ 0t in Lorentz reservoirs at zero temperature. The parameter ω 0 = 10γ 0.

For a given temperature such that k_BT/ħ ω ₀ = 10, the P₁₁ dynamics is plotted in Figs.  2(a)– 2(d), where figures  2(c) and 2(d) show the very long time behaviors of the P₁₁. Comparing Fig.  1(a) and Fig.  2(a), it can be seen that the influence of temperature on the P₁₁ is clearly visible when δ = 4γ ₀. From the dotted lines in Fig.  1(a) and Fig.  2(a), we find that, in the Markovian regime, the difference is in the decay rate of the P₁₁ and in the amount of trapping. The decay of the latter is quicker than that of the former. The amount of trapping in the excited state, in the first case, reaches zero while in the second case it is close to 18%. Comparing the dashed lines and the solid lines in Fig.  1(a) and Fig.  2(a) respectively, we know that, in the non-Markovian regime, the decay rates and the oscillating amplitudes in the second case are larger than those in the first case. From the long time behavior, under zero temperature and the non-Markovian regime, the amount of trapping in the excited state eventually reaches zero (It is not shown in Fig.  1(a)), while it is close to 21% (see the dashed and solid lines in Fig.  2(c)) under k_BT/ħ ω ₀ = 10 and the non-Markovian regime. Comparing Fig.  1(b) and Fig.  2(b), it can be seen that the influence of temperature on the P₁₁ is also very evident when λ = 0.05γ ₀. From the dotted lines in Fig.  1(b) and Fig.  2(b), we find that, without detuning, the difference is also in the decay rate of the P₁₁ and in the amount of trapping. The decay in the second case is quicker than that in the first one. In the first case, the amount of trapping in the excited state reaches zero while it is close to 23% in the second case. Comparing the dashed lines and the solid lines in Fig.  1(b) and Fig.  2(b) respectively, we know that the decay rates and the oscillating amplitudes of the P₁₁ with the detuning are greater than those without the detuning. From the long time behavior, under zero temperature and the detuning, the amount of the P₁₁ eventually reaches zero (it is not shown in Fig.  1(b)), while it is close to 23% (see the dashed and solid lines in Fig.  2(d)) under k_BT/ħ ω ₀ = 10 and the detuning.

Fig.  2.

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	Fig.  2. Population of the excited atoms as a function of γ 0t in Lorentz reservoirs at finite temperature kBT/ħ ω 0 = 10. Panels  (c) and (d) show very long time behaviors. The parameter ω 0 = 10γ 0.

Thus, nonzero temperature can accelerate the decay of P₁₁, enlarge the oscillating amplitude of the P₁₁ and increase the amount of trapping in the excited state after a long time, whether in the Markovian regime or in the non-Markovian regime, and whether with detuning or without detuning. The physical explanation is that, at finite temperature, due to the thermal effect of the environment, the influence of environment on the atom becomes large and the atomic dissipation becomes strong so that the oscillating amplitude of the P₁₁ becomes big and the population of the excited atoms decay more quickly. On the other hand, as the mean photon number of the reservoir is not equal to zero, the population of the excited atoms will reduce to a nonzero value after a long time, which is induced by the thermal equilibrium of the total system.

We consider an Ohmic spectral density with a Lorentz– Drude cutoff function^[34]

where ω is the frequency of the reservoir, and ω _c is the cut-off frequency, which depends on the coupling strength. In this section, we will compare the non-Markovian P₁₁ dynamics with the Markovian one in an Ohmic reservoir in the following three conditions: ω _c ≪ ω ₀, ω _c ≈ ω ₀, and ω _c ≫ ω ₀, where ω ₀ is the characteristic frequency of the atom. ω _c ≪ ω ₀ implies that the spectrum of the reservoir does not completely overlap with the frequency of the atom, that is, the reservoir is effectively adiabatic, so that the dynamical evolution of the system is essentially non-Markovian. While ω _c ≫ ω ₀ indicates the converse case, which corresponds to the fact that the environmental fluctuations evolve on the shortest possible timescale so that the quantum information is quickly dissipated, the dynamical evolution of the system is Markovian.^{[42, 43]}

Utilizing Eq.  (14), the correlation functions in Eqs.  (9) and  (10) have the following forms

and

Figure  3(a) displays the P₁₁ time evolution of the excited atoms in two independent Ohmic reservoirs at zero temperature. From Fig.  3, it can be observed that the P₁₁ is related to the ratio ω _c/ω ₀. When ω _c/ω ₀ = 10 (in the Markovian regime), the P₁₁ reduces quickly to zero, as shown with the dotted line in Fig.  3(a). When ω _c/ω ₀ = 0.1 (in the non-Markovian regime), the P₁₁ decreases very slowly, as shown with the solid line in Fig.  3(a). Namely, the smaller the value of the ratio ω _c/ω ₀ is, the stronger the non-Markovian effect is, the more the P₁₁ can be effectively trapped.

Fig.  3.

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	Fig.  3. In Ohmic reservoirs with a Lorentz– Drude cutoff function, the population of the atomic excited state versus γ 0t for different temperatures. (a) T = 0; (b) kBT/ħ ω 0 = 10.

Figure  3(b) depicts the P₁₁ time evolution of the excited atoms in two independent Ohmic reservoirs at k_BT/ħ ω ₀ = 10. Comparing Fig.  3(a) and Fig.  3(b), it can be seen that the P₁₁ is obviously dependent on the temperature. From the dotted lines in Fig.  3(a) and Fig.  3(b), we find that, the decay in the second case is quicker than that in the first one, the amount of the P₁₁ in the first case can reach zero while in the second case it is close to 22%. Comparing the three curves in Fig.  3(b), it is clear that, when the ratio ω _c/ω ₀ decreases, the amount of population trapping in the excited state increases a little and the decay of the P₁₁ can become slow.

Hence, when the ratio ω _c/ω ₀ decreases, the atomic decay rate can become small. When the temperature rises, the atomic decay rate can become large and the amount of population trapping in the excited state will increase. The physical mechanism is that, the environmental fluctuations evolve on the shortest possible timescale if ω _c ≫ ω ₀ so that the quantum information is quickly dissipated, while the reservoir is effectively adiabatic if ω _c ≪ ω ₀ so that the atomic decay becomes slow. So the decay rate of the P₁₁ will become small with the ratio ω _c/ω ₀ reducing. At finite temperature, the thermal effect of the environment will accelerate the atomic dissipation so that the decay rate of the population of the excited atoms increases, but the population of the excited atoms will tend to a nonzero value after a time, which is dependent on the thermal equilibrium of the total system.