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⇦Chinese Physics Letters, 2018, Vol. 35, No. 6, Article code 060301⇨ Preservation of Quantum Coherence for Gaussian-State Dynamics in a Non-Markovian Process * Jun Wen(文军)1,2, Guan-Qiang Li(李冠强)3,4** Affiliations 1State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071 2University of the Chinese Academy of Sciences, Beijing 100049 3School of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an 710021 4Department of Applied Physics and Key Laboratory for Quantum Information and Quantum Optoelectronic Devices of Shaanxi Province, Xi'an Jiaotong University, Xi'an 710049 Received 8 December 2017, online 19 May 2018 ^*Supported by the National Natural Science Foundation of China under Grant Nos 11405100, 11404377 and 11674360, the Natural Science Basic Research Plan of Shaanxi Province of China under Grant No 2015JM1032, and the Doctoral Research Fund of Shaanxi University of Science and Technology of China under Grant No 2018BJ-02.
^**Corresponding author. Email: liguanqiang@sust.edu.cn Citation Text: Wen J and Li G Q 2018 Chin. Phys. Lett. 35 060301 Abstract Coherence is a key resource in quantum information science. Exactly understanding and controlling the variation of coherence are vital for implementation in realistic quantum systems. Using

$P$ -representation of density matrix, we obtain the analytical solution of the master equation for the classical states in the non-Markovian process and investigate the coherent dynamics of Gaussian states. It is found that quantum coherence can be preserved in such a process if the coupling strength between system and environment exceeds a threshold value. We also discuss the characteristic function of the Gaussian states in the non-Markovian process, which provides an inevitable bridge for the control and operation of quantum coherence. DOI:10.1088/0256-307X/35/6/060301 PACS:03.65.Yz, 03.65.Xp, 42.25.Kb, 42.50.Ar © 2018 Chinese Physics Society Article Text Quantum coherence is an important resource in quantum information science, and it can be applied to quantum computation, quantum metrology and quantum cryptography.[1] Intensive investigations of rigorously quantifying quantum coherence have been carried out only recently,[2] and there have various results from different perspectives of studies. To date, many new and feasible measurement schemes of quantum coherence have been proposed,[3] and coherently transforming and freezing quantum coherence have been explored under incoherent channels.[4,5] Quantifying quantum coherence is associated with quantum correlation, which can be thought of as another quantum resource.[6-8] For concrete estimates, most treatments for measuring quantum coherence are restricted to finite-dimensional systems rather than infinite-dimensional ones. A relevant result published recently for an infinite-dimensional system was obtained under a certain condition.[9] Particularly, for Gaussian states which are usually employed experimentally, a quantification of quantum coherence was investigated very recently in an infinite-dimensional system.[10] However, a more realistic quantum system in experiments must be open. The dynamics of quantum coherence for a general open quantum system has not yet been understood completely. In this Letter, we exemplify the coherence of Gaussian states and focus on the dynamics under a non-Markovian environment. Compared with a Markovian environment, a non-Markovian environment can echo information back to the system, which reflects the real physical process in the strong noisy environment.[11] Due to this reason, the process is also called the non-Markovian process. As far as we know, the non-Markovian process has been studied by the non-equilibrium Green function, quantifying non-Markovianity.[12-15] Especially, the non-Markovianity of Gaussian channels has been studied to quantify non-Markovianity.[16] The key point of our work is for an analytical study of the dynamics of quantum coherence for Gaussian states in a non-Markovian process and we find that the coherence can be frozen if the coupling strength is greater than a threshold value. We discuss the characteristic function of the Gaussian states in the non-Markovian process, which provides an inevitable bridge for the control and operation of quantum coherence. To begin, we consider an open system consisting of a simple harmonic oscillator and linear coupling to the environment, which is modeled as a multi-mode boson. The Hamiltonian in units of

$\hbar =1$ is given by[12,13,17-20]

$H=\omega _{0}a^{+}a+\sum_{k}\omega_{k}b_{k}^{+}b_{k} +\sum_{k}g_{k}(ab_{k}^{+}+a^{+}b_{k}),~~ \tag {1}$ where

$a^{+}$ (

$a$ ) and

$b_{k}^{+}$ (

$b_{k}$ ) are, respectively, the creation (annihilation) operators of the system and the

$k$ th boson with frequency

$\omega _{k}$ ,

$\omega _{0}$ is the frequency of the system, and

$g_{k}$ represents the coupling strength between the system and the

$k$ th boson. Tracing over the environment's degrees of freedom by

$\varrho (t)={\rm Tr}_{\rm R}[\varrho _{\rm tot}(t)]$ , we can obtain the master equation of this system in a non-Markovian process[12,15,21,22]

$\begin{align} \frac{d\varrho (t)}{dt} =\,&-i\omega (t)[a^{+}a,\varrho (t)]+\gamma (t)[2a\varrho (t)a^{+}\\ &-a^{+}a\varrho (t)-\varrho (t)a^{+}a]+\widetilde{\gamma}(t)[a\varrho (t)a^{+} \\ &+a^{+}\varrho (t)a-a^{+}a\varrho(t)-\varrho (t)aa^{+}],~~ \tag {2} \end{align}$ where

$\omega (t)$ is the renormalized energy regarding to the system, and

$\gamma (t)$ and

$\widetilde{\gamma}(t)$ are dissipation and fluctuation coefficients, respectively. They are time-dependent with specific forms as follows:[23,24]

$\begin{align} \omega (t) =\,&-\Im \Big[\frac{du(t)}{dt}\frac{1}{u(t)}\Big], \\ \gamma (t) =\,&-\Re \Big[\frac{du(t)}{dt}\frac{1}{u(t)}\Big], \\ \widetilde{\gamma}(t) =\,&\frac{d\upsilon (t)}{dt}-2\Re\Big[\upsilon (t)\frac{du(t)}{dt}\frac{1}{u(t)}\Big],~~ \tag {3} \end{align}$ where

$u(t)$ and

$\upsilon (t)$ are related to the non-equilibrium Green function by the equations

$\begin{align} &\frac{du(t)}{dt}+i\omega _{0}u(t)+\int_{0}^{t}g(t-\tau)u(\tau)d\tau=0, \\ &\upsilon (t)=\int_{0}^{t}d\tau _{1}\int_{0}^{t}d\tau _{2}u(t-\tau _{1})\widetilde{g}(\tau _{1}-\tau _{2})u^{+}(t-\tau _{2}),~~ \tag {4} \end{align}$ with

$u(t)$ and

$\upsilon (t)$ being obtained numerically once the initial solution

$u(0)$ is given. The self-energy corrections

$g(t)$ and

$\widetilde{g}(t)$ involving the backaction from the environment are given by

$\begin{align} g(t) =\,&\int \frac{d\omega}{2\pi}J(\omega)e^{-i\omega t}, \\ \widetilde{g}(t)=\,&\int \frac{d\omega}{2\pi}J(\omega)e^{-i\omega t}f(\omega),~~ \tag {5} \end{align}$ where

$J(\omega)=2\pi \sum_{k}|g_{k}|^{2}\delta (\omega -\omega _{k})$ is the spectral density, and

$f(\omega)=1/[\exp(\hslash \omega /kT)-1]$ is the distribution function of bosons.[15] Therefore, seeking the solution to Eq. (2) is possible provided that

$u(t)$ and

$J(\omega)$ are given. For our purpose of analytical treatment, however, we have to introduce

$P$ -representation of density matrix

$\varrho (t)$ , i.e.,

$P(\alpha,\alpha^{\ast},t)={\rm Tr}[\varrho (t)\delta (\alpha^{\ast}-a^{+})\delta (\alpha -a)],~~ \tag {6}$ which is a classical number rather than an operator.[25,26] Putting the inverse transform

$\varrho (t)=\int P(\alpha,\alpha^{\ast},t)|\alpha\rangle \langle \alpha |d\alpha ^{2}$ into Eq. (2), we can transform the master equation into the Fokker–Planck equation as follows:

$\begin{align} \frac{\partial P(\alpha,\alpha^{\ast},t)}{\partial t} =\,&[\gamma (t)+i\omega (t)]\frac{\partial}{\partial \alpha}[\alpha P(\alpha,\alpha ^{\ast},t)] \\ &+[\gamma (t)-i\omega (t)]\frac{\partial}{\partial \alpha^{\ast}}[\alpha ^{\ast}P(\alpha,\alpha^{\ast},t)] \\ &+\widetilde{\gamma}(t)\frac{\partial ^{2}P(\alpha,\alpha^{\ast},t)}{\partial \alpha \partial \alpha^{\ast}}.~~ \tag {7} \end{align}$ Due to the time-dependence of the coefficients, the Fokker–Planck equation is for the non-Markovian process. Following the standard steps,[25,26] once the conditions

$\begin{align} &\frac{\partial}{\partial \alpha}\alpha^{\ast}=\frac{\partial}{\partial\alpha^{\ast}}\alpha =0, \\ &a|\alpha \rangle \langle \alpha |=(|\alpha \rangle \langle \alpha |a^{+})^{+}=\alpha |\alpha \rangle \langle \alpha|, \\ &a^{+}|\alpha \rangle \langle \alpha |=(|\alpha \rangle \langle \alpha |a)^{+}=\Big(\frac{\partial}{\partial \alpha}+\alpha^{\ast}\Big)|\alpha\rangle \langle \alpha |, \\ &\int P(\alpha,\alpha^{\ast},t)\Big(\alpha \frac{\partial}{\partial \alpha}|\alpha \rangle \langle \alpha |\Big)d\alpha^{2}\\ =\,& -\int |\alpha \rangle \langle \alpha |\Big[\frac{\partial}{\partial \alpha}\alpha P(\alpha,\alpha^{\ast},t)\Big]d\alpha ^{2},\\ &\int P(\alpha,\alpha^{\ast},t)(\frac{\partial ^{2}}{\partial \alpha\partial \alpha^{\ast}}|\alpha \rangle \langle \alpha |)d\alpha ^{2}\\ =\,&\int |\alpha \rangle \langle \alpha |\Big[\frac{\partial ^{2}}{\partial \alpha \partial \alpha^{\ast}}P(\alpha,\alpha^{\ast},t)\Big]d\alpha ^{2} \end{align}$ are satisfied, we can obtain Eq. (7) relying on Eq. (2) and the solution to Eq. (7) is

$P_{\alpha_{0}}(\alpha,\alpha^{\ast},t)=\frac{1}{\pi A(t)}\exp \Big\{-\frac{|\alpha -\alpha_{0}e^{-B(t)-if(t)}|^{2}}{A(t)}\Big\},$ where

$A(t)=\int_{0}^{t}dt'\widetilde{\gamma}(t')\exp [2\int_{t}^{t'}\gamma (t'')dt'']$ , and

$B(t)=\int_{0}^{t}dt'\gamma (t')$ and

$f(t)= \int_{0}^{t}dt'\omega (t')$ . We have assumed that the initial state is a coherent state

$|\alpha_{0}\rangle\langle \alpha_{0}|$ , which is denoted by the subscript

$\alpha_{0}$ . Assisted by Eq. (3), we further obtain

$\begin{align} P_{\alpha_{0}}(\alpha,\alpha^{\ast},t) =\,&\frac{1}{\pi A(t)}\exp \Big\{-\frac{|\alpha -\alpha_{0}u(t)|^{2}}{A(t)}\Big\}, \\ A(t) =\,&\int_{0}^{t}dt'\Big|\frac{u(t)}{u(t')}\Big|^{2}\widetilde{\gamma}(t').~~ \tag {8} \end{align}$ Note that

$A(t)\geq 0$ is required or

$P_{\alpha_{0}}(\alpha,\alpha^{\ast},t)$ will diverge, and the condition can be proved by employing the expression

$\langle a^{+}a\rangle _{t}=\langle a^{+}a\rangle _{0}|u(t)|^{2}+A(t)$ if considering

$\frac{d\langle O\rangle}{dt}={\rm Tr}[O\dot{\varrho}]$ , where

$O$ can be an arbitrary time-independent operator under the Schrödinger picture.[27] In addition, one can verify that

$P(\alpha,\alpha^{\ast},t)$ is the Green function of the Fokker–Planck equation under the initial condition

$P(\alpha,\alpha^{\ast},0)=\delta ^{2}(\alpha -\alpha_{0})$ . That is, the Green function

$P(\alpha,\alpha^{\ast},t|\alpha_{0},\alpha_{0}^{\ast},0)=\frac{1}{\pi A(t)}\exp \Big\{-\frac{|\alpha -\alpha_{0}u(t)|^{2}}{A(t)}\Big\}.~~ \tag {9}$ Obviously, the result is the same as the one given by Eq. (8). In fact, for

$P(\alpha,\alpha^{\ast},t)$ with respect to an arbitrary classical state, we always have[26]

$P(\alpha,\alpha^{\ast},t)=\int d\alpha_{0}^{2}P(\alpha,\alpha ^{\ast },t|\alpha_{0},\alpha_{0}^{\ast},0)P(\alpha_{0},\alpha_{0}^{\ast},0).~~ \tag {10}$ This expression can be seen as the general solution of Eq. (7), from which the density matrix

$\varrho (t)$ for arbitrary classical state can be computed. This implies that Eq. (2) can be solved analytically using

$P$ -representation based on the Green function. Next, we concentrate on qualifying quantum coherence based on the above solution of the master equation. As an example, we consider Gaussian states which are usually employed in experiments. The characteristic function of the Gaussian state is

$\chi (\varrho, \lambda)={\rm Tr}[\varrho D(\lambda)]$ with the displacement operator

$D(\lambda)=e^{\lambda a^{+}-\lambda^{\ast}a}$ .[10,28] The characteristic function is represented by

$\begin{align} \chi (\varrho,\lambda) =\,&\exp \Big\{-\frac{1}{2} \begin{pmatrix} x_{\lambda} & y_{\lambda} \end{pmatrix} {\it \Omega} V{\it \Omega} ^{\rm T} \begin{pmatrix} x_{\lambda}\\ y_{\lambda}\end{pmatrix} \\ &-i\Big[{\it \Omega} \begin{pmatrix} d_{1}\\ d_{2}\end{pmatrix}\Big]^{\rm T} \begin{pmatrix} x_{\lambda}\\ y_{\lambda}\end{pmatrix} \Big\},~~ \tag {11} \end{align}$ where

$x_{\lambda}$ and

$y_{\lambda}$ are the real and imaginary parts of

$\lambda$ ,

$\lambda^{\ast}$ is the complex conjugate of

$\lambda$ ,

${\it \Omega}=\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}$ , and superscript T denotes transpose. The displacement vector

${\boldsymbol d}=(d_{1},d_{2})$ , where

$d_{1}$ and

$d_{2}$ are all real numbers. The covariance matrix of Gaussian state

$V= \begin{pmatrix} V_{11} & V_{12} \\ V_{21} & V_{22}\end{pmatrix}$ is real symmetric positive-definite and satisfies

$V+i{\it \Omega} \geq 0$ with

$\det (V)\geq 1$ .[30,31] If and only if

$\varrho$ is pure,

$\det(V)=1$ . We can see that

$\varrho$ can be described by

$V$ and

${\boldsymbol d}$ completely. For any one-mode Gaussian state

$\varrho (V, {\boldsymbol d})$ , we define a coherence measure by using the fixed orthonormal basis

$\{|n\rangle\}_{n=0}^{+\infty}$ as follows:

$C(\varrho)=\underline{\rm inf}\{S(\varrho\|\delta)\}_{\delta},~~ \tag {12}$ where

$\delta$ represents an incoherent Gaussian state,

$S(\varrho\|\delta)={\rm Tr}(\varrho \log _{2}\varrho)-{\rm Tr}(\varrho \log _{2}\delta)$ is the relative entropy, and

$\underline{\rm inf}$ runs over all incoherent Gaussian state.[10] The von Neumann entropy is

$S(\varrho)=-{\rm Tr}(\varrho \log _{2}\varrho)$ , yielding

$S(\varrho)=\frac{\nu +1}{2}\log _{2}\frac{\nu +1}{2}-\frac{\nu -1}{2}\log _{2}\frac{\nu -1}{2},~~ \tag {13}$ with

$\nu =\sqrt{\det (V)}$ .[29] Straightforward calculation shows

$\begin{alignat}{1} C[\varrho (V,d)] =\,&\frac{\nu -1}{2}\log _{2}\frac{\nu -1}{2}-\frac{\nu +1}{2}\log _{2}\frac{\nu +1}{2} \\ &+(\bar{n}+1)\log _{2}(\bar{n}+1)-\bar{n}\log _{2}\bar{n},~~ \tag {14} \end{alignat}$ where

$\bar{n}=\frac{1}{4}(V_{11}+V_{22}+d_{1}^{2}+d_{2}^{2}-2)$ . The above equation implies that the quantum coherence can be qualified by given

$V$ and

${\boldsymbol d}$ . Specifically, employing Eq. (8) we can obtain the characteristic function for the initial coherent state

$|\alpha_{0}\rangle \langle \alpha_{0}|$ as

$\begin{align} \chi _{\alpha_{0}}(\varrho,\lambda)=\,&\exp \Big\{-\Big[\frac{1}{2}+A(t)\Big]|\lambda |^{2}+\lambda \alpha_{0}^{\ast}u^{\ast}(t)\\ &-\lambda^{\ast}\alpha_{0}u(t)\Big\}.~~ \tag {15} \end{align}$ Correspondingly, we have

$\begin{align} V_{11} =\,&V_{22}=1+2A(t),\\ V_{12} =\,&V_{21}=0, \\ d_{1} =\,&2\Re [\alpha_{0}u(t)],d_{2}=2\Im [\alpha_{0}u(t)].~~ \tag {16} \end{align}$ Based on this, the coherence measure given by Eq. (14) can be calculated by

$\begin{align} C_{\alpha_{0}}(t) =\,&[1+A(t)+|\alpha_{0}u(t)|^{2}]\log _{2}[1+A(t) \\ &+|\alpha_{0}u(t)|^{2}]+A(t)\log _{2}[A(t)]-[A(t)\\ &+|\alpha_{0}u(t)|^{2}]\log _{2}[A(t)+|\alpha_{0}u(t)|^{2}] \\ &-[A(t)+1]\log _{2}[A(t)+1].~~ \tag {17} \end{align}$ However, some Gaussian states are non-classical ones which cannot be described by

$P$ -representation. To solve this problem, we must solve

$\chi (\varrho,\lambda)$ under the Heisenberg picture, which can describe all the quantum states, and

$\varrho$ is independent of time. However, the evolution operator must be time-dependent. In this case, obtaining the annihilation operator of the system

$a(t)$ becomes the key issue and we can assume

$a(t)=u(t)a_{0}+f_{\rm R}(t),~~ \tag {18}$ where

$f_{\rm R}(t)=-i\sum_{k}g_{k}b_{k}(0)\int_{0}^{t}d\tau e^{-i\omega _{k}\tau}u(t-\tau)$ ,

$a_{0}=a(0)$ .[13,19] We can see that

$f_{\rm R}(t)$ only relies on the initial state of the environment and is not related with our system. Thus if the initial state of the total system is uncorrelated,

$\chi(\varrho,\lambda)$ has the following form

$\chi (\varrho,\lambda)={\rm Tr}\{\varrho \exp [^{\lambda u^{\ast }(t)a^{+}-\lambda^{\ast}u(t)a}]\}\times E,~~ \tag {19}$ where

$E={\rm Tr}\{\varrho _{\rm R}\exp [\lambda f_{\rm R}^{+}(t)-\lambda^{\ast}f_{\rm R}(t)]\}$ . For the initial coherent state of our system

$|\alpha_{0}\rangle \langle \alpha_{0}|$ , after combining Eqs. (15) and (19), we can obtain

$E=\exp \Big\{-\Big[\frac{1}{2}-\frac{1}{2}|u(t)|^{2}+A(t)\Big]|\lambda |^{2}\Big\},~~ \tag {20}$ where

$E$ does not depend on the initial state of the system. It can be seen as a constant for our analysis since the environment cannot be changed. We must point out two very important issues: (1) Eq. (19) only offers

$\chi (\varrho,\lambda)$ , which does not illustrate whether

$\varrho$ is an Gaussian state or not. We need judgment according to Eq. (11); (2) Eqs. (2) and (18) must have the same bath for the calculation of

$E$ . That is, the coherence measure for the non-classical Gaussian states can be determined by Eq. (14) once obtaining the characteristic function of the corresponding states from Eq. (19). For convenience, we analyze this issue based on the spectral density

$J(\omega)=2\pi \eta \omega (\frac{\omega}{\omega_{\rm c}})^{s-1}\exp (-\frac{\omega}{\omega _{\rm c}})$ , where

$\eta$ is the coupling strength between the system and environment, and

$\omega _{\rm c}$ is the frequency cutoff of the environmental spectra.[15] When

$s=1$ ,

$< 1$ and

$>1$ , the corresponding environments are Ohmic, sub-Ohmic, and super-Ohmic, respectively. Figure 1 presents the measure of the quantum coherence with the coherent state

$|\alpha_{0}\rangle \langle \alpha_{0}|$ as the initial state, where different coupling strengths have been considered. As shown in Fig. 1(a), if the coupling strength is smaller than the threshold value and the measure of coherence approaches zero after long time evolution, which implies that the information of system will be lost to the environment and there is no back action to the system since both

$\gamma (t)$ and

$\widetilde{\gamma}(t)$ are greater than zero. Increasing the coupling strength (but the coupling strength is still smaller than the threshold value) will not be conducive to coherence, which can be seen in Fig. 1(b). In Fig. 1(c), when the coupling strength equals the threshold, the absolute value of the coherent measure over time

$|\frac{dC(t)}{dt}|$ decreases as time goes on. In Fig. 1(d), the measure of coherence approaches to a nonzero constant after the finite time. The measure of coherence is not monotonous function due to the interplay between the system and the environment in Figs. 1(c) and 1(d). This shows that the information exchange between the system and the environment occurs and it reaches a balance when the coupling strength exceeds the threshold value. It also means that the coherence can be preserved by adjusting the coupling strength. It is noted that the similar tendency occurs for the function

$u(t)$ in Refs. [15,19], but here the coherence measure is concentrated on.

Fig. 1. The value of

$C(t)$ versus time

$t$ and the initial state

$|\alpha_{0}\rangle \langle \alpha_{0}|$ . This figure expresses the influence resulting from the coupling strength

$\eta$ : (a)

$\eta=0.1$ , (b)

$\eta=0.5$ , (c)

$\eta=1$ , and (d)

$\eta=3$ . The other parameters are set as:

$|\alpha_{0}|^{2}=4$ ,

$s=1$ ,

$\omega_{0}=\omega _{\rm c}=\frac{kT}{\hbar}=1$ , where

$k$ is the Boltzmann constant, and

$T$ is the thermodynamic temperature.

For other non-classical Gaussian states, similar results can also be found. In Fig. 2, we give the corresponding results with the squeezed coherent state as the initial condition. When the coupling strength is greater than the threshold value, the measure of coherence is a nonzero constant after a finite time. Figure 2 shows the contribution of different parameters to the system which offer a basis for the coherence control. In Figs. 2(a) and 2(b), the nonzero value of the coherence measure increases with

$r$ and

$|\alpha |^{2}$ . At this time,

$|u(t)|$ and

$\upsilon (t)$ are both constant. In Ref. [19], the authors pointed that

$|u(t)|$ will be a constant which is greater than zero after a long time if the coupling strength is greater than the threshold value or the final value of

$|u(t)|=0$ . From Eq. (18), we can see that the nonzero constant for the measure of coherence can be ensured if we extend the time to infinity. For the results given in Figs. 1 and 2, we can also obtain that the time from which the coherence measure keeps constant is unnecessarily very long.

Fig. 2. (Color online) The value of

$C(t)$ versus time

$t$ . The initial state is a squeezed coherent state

$|\alpha',\xi\rangle \langle \alpha',\xi |$ (where

$|\alpha',\xi \rangle =\exp (\frac{1}{2}\xi^{\ast}a^{2}-\frac{1}{2}\xi a^{+2})|\alpha' \rangle$ ),

$\alpha'$ are set as real numbers, and the argument of

$\xi$ is

$\frac{\pi}{2}$ . This figure expresses the influence resulting from

$|\alpha' |^{2}$ and

$r$ (

$r=|\xi |$ ): (a)

$|\alpha'|^{2}=2$ , and (b)

$r=2$ . The other parameters are set as

$s=1$ ,

$\eta =2$ ,

$\omega =\omega_{\rm c}=\frac{kT}{\hbar}=1$ , where

$k$ is the Boltzmann constant, and

$T$ is the thermodynamic temperature.

In conclusion, we have provided the solution of the master equation under the non-Markovian environment for the single-mode system analytically. Based on the solution, we investigate the properties of the coherence measure for the Gaussian states. To let the measure be suitable for all types of the Gaussian states expediently, we offer the general form of the characteristic function by the Heisenberg picture. We have shown that the existence of frozen coherence and studied the effects of parameters on the coherence. It is found that the coherence measure can be controlled by the parameters of the states. Thus our investigation provides the basis for the control and operation of the coherence in the non-Markovian process. The system studied in this work is essentially the infinite-dimensional system with a continuous variable. Since the creation and maintenance of quantum coherence in such a system are vital problems in quantum communication and computation, we think that the results presented in this work are meaningful. We thank Mang Feng for his helpful discussion. References Observable Measure of Quantum Coherence in Finite Dimensional SystemsQuantifying CoherenceFidelity and trace-norm distances for quantifying coherenceConditions for coherence transformations under incoherent operationsFrozen Quantum CoherenceQuantum coherence in multipartite systemsMeasuring Quantum Coherence with EntanglementUnified view of quantum correlations and quantum coherenceQuantifying coherence in infinite-dimensional systemsQuantifying coherence of Gaussian statesMeasure for the Degree of Non-Markovian Behavior of Quantum Processes in Open SystemsNon-Markovianity measure using two-time correlation functionsNon-Markovian dynamics of an open quantum system with initial system-reservoir correlations: A nanocavity coupled to a coupled-resonator optical waveguideQuantum Metrology in Open Systems: Dissipative Cramér-Rao BoundGeneral Non-Markovian Dynamics of Open Quantum SystemsNon-Markovianity of Gaussian ChannelsLocalized Magnetic States in MetalsEffects of Configuration Interaction on Intensities and Phase ShiftsThreshold for nonthermal stabilization of open quantum systemsQuantum tunnelling in a dissipative systemNon-Markovian decoherence theory for a double-dot charge qubitNon-equilibrium quantum theory for nanodevices based on the Feynman–Vernon influence functionalBrownian Motion of a Quantum OscillatorSubnatural linewidth averaging for coupled atomic and cavity-mode oscillatorsGaussian quantum informationCapacity of quantum Gaussian channelsQuantum optics in the phase spaceQuantum information with continuous variables

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