Non-Markovianity measure using two-time correlation functions

Md. Manirul Ali, Ping-Yuan Lo, Matisse Wei-Yuan Tu, and Wei-Min Zhang

Phys. Rev. A 92, 062306 – Published 3 December 2015

Abstract

We investigate non-Markovianity measure using two-time correlation functions for open quantum systems. We define non-Markovianity measure as the difference between the exact two-time correlation function and the one obtained from quantum regression theorem in the Born-Markov approximation. Such non-Markovianity can easily be measured in experiments. We found that the non-Markovianity dynamics in different time scale crucially depends on the system-environment coupling strength and other physical parameters such as the initial temperature of the environment and the initial state of the system. In particular, we obtain the short-time and long-time behaviors of non-Markovianity for different spectral densities. We find that the thermal fluctuation always reduce the non-Markovian memory effect. Also, the non-Markovianity measure shows nontrivial initial state dependence in different time scales.

Received 26 May 2015

DOI:https://doi.org/10.1103/PhysRevA.92.062306

Authors & Affiliations

Md. Manirul Ali^*, Ping-Yuan Lo, Matisse Wei-Yuan Tu, and Wei-Min Zhang^†

Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan

^*mani@mail.ncku.edu.tw
^†wzhang@mail.ncku.edu.tw

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Vol. 92, Iss. 6 — December 2015

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Images

Figure 1
The non-Markovianity measure $N (t, τ たう)$ is plotted as a function of $τ たう$ with $t = 0$ at a fixed temperature ( $T = 1$ K) for different system-environment coupling strengths: $η いーた = 0.1 {η いーた}_{c}$ (black dotted line), $η いーた = 0.5 {η いーた}_{c}$ (blue dashed line), $η いーた = 1.0 {η いーた}_{c}$ (dashed double-dotted red line), $η いーた = 1.2 {η いーた}_{c}$ (dashed dotted violet line), and $η いーた = 1.5 {η いーた}_{c}$ (solid green line). Four different spectral densities are considered (a) sub-Ohmic $(s = 1 / 2)$ (b) Ohmic $(s = 1)$ , (c) super-Ohmic $(s = 2)$ , and (d) super-Ohmic $(s = 3)$ for the bosonic reservoir. The other parameters are taken as ${ω おめが}_{0} = 10$ GHz, ${ω おめが}_{c} = 5 {ω おめが}_{0}$ , and $n (t_{0}) = 1$ .
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Figure 2
Temperature dependence of non-Markovianity measure is shown as a function of $τ たう$ at three different temperatures: $T = 1$ K (solid blue line), $T = 5$ K (red dashed line), and $T = 50$ K (dotted green line) with two type of coupling strengths: (a), (c), (e), and (g) weak coupling $(η いーた = 0.5 {η いーた}_{c})$ and (b), (d), (f), and (h) strong coupling $(η いーた = 1.5 {η いーた}_{c})$ . Four different spectral densities for the bosonic environment are considered: (a) and (b) sub-Ohmic $(s = 1 / 2)$ , (c) and (d) Ohmic $(s = 1)$ , (e) and (f) super-Ohmic $(s = 2)$ , and (g) and (h) super-Ohmic $(s = 3)$ . The other parameters are taken as ${ω おめが}_{0} = 10$ GHz, ${ω おめが}_{c} = 5 {ω おめが}_{0}$ , and $n (t_{0}) = 1$ , with a fixed value of $t = 0$ .
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Figure 3
Initial state dependence of non-Markovianity measure is shown as a function of $τ たう$ for three different $n (t_{0})$ : $n (t_{0}) = 1$ (solid blue line), $n (t_{0}) = 10$ (red dashed line), and $n (t_{0}) = 50$ (green dotted line) with two type of coupling strengths: (a), (c), (e), and (g) weak coupling $(η いーた = 0.5 {η いーた}_{c})$ and (b), (d), (f), and (h) strong coupling $(η いーた = 1.5 {η いーた}_{c})$ . Four different spectral densities for the bosonic environment is investigated: (a) and (b) sub-Ohmic $(s = 1 / 2)$ , (c) and (d) Ohmic $(s = 1)$ , (e) and (f) super-Ohmic $(s = 2)$ , and (g) and (h) super-Ohmic $(s = 3)$ . The other parameters are taken as ${ω おめが}_{0} = 10$ GHz, ${ω おめが}_{c} = 5 {ω おめが}_{0}$ , and $T = 1$ K, with a fixed value of $t = 0$ .
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Figure 4
We show $η いーた$ dependence of the trace distance $D ({ρ ろー}_{1} (t), {ρ ろー}_{2} (t))$ and dimensionless decay rate $Γ がんま (t) / (2 {ω おめが}_{0})$ in $(a_{1})$ and $(a_{2})$ , respectively, with a fixed temperature of the reservoir $k_{B} T = 0.1 {ω おめが}_{0}$ . Different curves in $(a_{1})$ and $(a_{2})$ represent different coupling strengths, namely $η いーた = 0.02$ (blue dashed line), $η いーた = 0.05$ (red dashed dotted line), and $η いーた = 0.10$ (solid green line). The temperature dependence of the same quantities $D ({ρ ろー}_{1} (t), {ρ ろー}_{2} (t))$ and dimensionless $Γ がんま (t) / (2 {ω おめが}_{0})$ are shown in $(b_{1})$ and $(b_{2})$ , respectively, with a fixed coupling strength $η いーた = 0.1$ . Different curves in $(b_{1})$ and $(b_{2})$ represent different temperatures, namely $k_{B} T = 0.1 {ω おめが}_{0}$ (dashed violet line), $k_{B} T = 0.2 {ω おめが}_{0}$ (dashed dotted black line), and $k_{B} T = 0.5 {ω おめが}_{0}$ (solid cyan line). The cutoff frequency ${ω おめが}_{c} = 5 {ω おめが}_{0}$ .
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