Effects of non-Markovian squeezed bath on the dynamics of open systems

Open quantum system consists of system and environment, its dynamics is a basic research topic in the field of quantum information and quantum computing Ralph ; Reichental ; Wisniacki ; Eleuch ; Venuti . Normally dissipation or decoherence of the system will occur due to the system-bath coupling, which is one of the main problems to be solved in the manipulation of quantum devices. Based on the memory effects of the environment, the dynamics of an open quantum system is defined as Markovian or non-Markovian Vega . For the memoryless Markovian process, the dynamical is usually described by the Lindblad master equation with semigroup property. In the non-Markovian processes where the memory effects can not be neglected, the information backflow from the environment to the system leads to the deviation of the system dynamics from that of a dynamical semigroup, and in this case Lindblad master equation fails. A lot of methods have been developed to tackle the non-Markovian dynamics, such as non-Markovian quantum jump Piilo , Feynman-Vernon influence functional path-integral Hu1992 ; zhang , the time-evolving matrix product operator Strathearn and QSD equation approach Diosi97 .

For a non-Markovian environment, normally the environment is assumed to be a bosonic or fermionic coherent state zhang ; Luoma ; Liu . It is the smallest uncertain state in the orthogonal component, and the quantum fluctuations of the coherent state in the phase space are equal in the direction of the orthogonal components $x$ and $p$, i.e., $\Delta x^{2}=\Delta p^{2}=\hbar/2$. With the development of the reservoir engineering technology Diehl ; Li ; Cavaliere , the issue of protecting quantum information through environmental actions has attracted lots of attention. Theoretical studies have shown that a bath consisting of a squeezed vacuum can partially suppress the two level atom phase decay Gardiner . Experimentally people have used resonant fluorescence on superconducting circuits to characterize the squeezing phenomenon in a cryogenic environment Toyli . The squeezed bosonic bath can be generated by pumping a lumped-element Josephson parametric amplifier in superconducting circuit Murch . For a quantum squeezed state, the quantum fluctuations of an orthogonal component can be reduced below the symmetry limit $\hbar/2$ and the quantum fluctuations of its conjugate variables rise to satisfy the Heisenberg uncertainty principle $\Delta x^{2}\Delta p^{2}\geq\hbar/4$. This provides us more adjustable parameters in controlling the system by the reservoir. Link et al. demonstrated that the system-bath coupling strength can be controlled by the squeezed direction, the model is the two level atom in a non-Markovian bath Link .

The QSD equation approach proposed by Diósi et al. Diosi97 ; Diosi98 has shown its unique computational characteristics in the study of non-Markovian dynamics of open quantum systems Ren20 ; wang2020 ; wangenergy ; Flannigan ; Arapat ; Chen ; MChen . In this paper, by using this approach we derive a group of closed equations for solving the dynamics of the system in non-Markovian squeezed baths. The baths are assumed to be bosonic with weak system-bath couplings. We use the bath correlation function to describe the effects of bath squeezing on quantum fluctuations, and give the containing historical memory effect $O$ operators. Finally, as an demonstration, we analyze two examples: the adiabatic evolution and quantum state transmission. The effects of the non-Markovianity $\gamma$, the squeezing direction $\theta$ and squeezing strength $r$ are analyzed in these two models, respectively.

This paper is organized as follows. In Sec. II, we brief introduce the two mode squeezed bath and derive the non-Markovian master equation using the QSD approach. In Sec. III, we numerically calculate the adiabatic evolution of a three level atom, and the state transmission in the Heisenberg spin chain. Finally, conclusion is presented in Sec. IV.

In the open quantum system model, the total Hamiltonian can be written as

where $H_{s}$ and $H_{b}$ are the system and bath Hamiltonian, respectively. Suppose in multi-qubit system each qubit is coupled to its own environment. For the bosonic squeezed bath $H_{b}=\sum_{j=1}^{N}H_{b}^{j}$, $H_{b}^{j}=\sum_{k}\omega_{k}^{j}b_{k}^{j\dagger}b_{k}^{j}$ (setting $\hbar=1$). $b_{k}^{j\dagger}$, $b_{k}^{j}$ are the $k$th mode bosonic creation and annihilation operators of the $j$th bath with frequency $\omega_{k}^{j}$. The system-bath interaction $H_{int}$ in the rotation wave approximation can be written as

where $L_{j}$ is the Lindblad operator characterizing the mutual couplings between the system and the $j$th reservoir. $f_{k}^{j}$ stands for the coupling strength between the system and the $k$th mode of the $j$th bath. In the interaction picture with respect to the bath $H_{b}$, the total Hamiltonian becomes

Now the state vector of the total system satisfies the Schr$\ddot{o}$dinger equation $\frac{\partial}{\partial t}\left|\psi(t)\right\rangle=-iH^{I}_{tot}\left|\psi(t)\right\rangle$.

In this paper, we only consider the symmetric two mode squeezed state bath as in Ref. Link . In this case we take $L_{j}=L_{j}^{\dagger}$, and use the squeezed vacuum state $\left|\phi\right\rangle=S(\xi)\left|0\right\rangle$ to describe the initial state of the environment. Here $S(\xi)$ is the unitary two mode squeezing operator, for the $k$th mode of the $j$th bath it can be written as

where $r_{j}\in[0,1]$ is the squeezing strength. When $r_{j}=0$ the $j$th bath corresponds to the non-squeezed vacuum state $\left|0\right\rangle$. $\xi_{j}=e^{i\theta_{j}}$, and $\theta_{j}$ denotes the squeezing direction. When $\theta_{j}=0,(\pi)$ the $j$th bath squeezes the $p,(x)$-quadrature. $k_{0}^{j}=\omega_{0}^{j}/c$, and $\omega_{0}$ is the squeezed center frequency. $b_{k}^{j\dagger}$, $b_{k}^{j}$ and $S^{j}_{k}(\xi_{j})$ satisfy

where $u_{j}=cosh(r_{j})$, $v_{j}=w_{j}\xi_{j}$, $w_{j}=sinh(r_{j})$ and $u_{j}^{2}-\left|v_{j}\right|^{2}=1$.

For the non-Markovian bosonic baths, the time-local QSD equation can be written as Diosi97 ; Diosi98 ; wang2018

where $z^{j\ast}_{t}$ is the stochastic environmental noise, and $\overline{O}^{j}(t,z^{j\ast}_{t})=\intop_{0}^{t}ds\alpha^{j}(t,s)O^{j}(t,s,z^{j\ast}_{t})$. The $O^{j}(t,s,z^{j\ast}_{t})$ operator is defined as $O^{j}(t,s,z^{j\ast}_{t})\left|\psi(t,z^{j\ast}_{t})\right\rangle=\delta\left|\psi(t,z^{j\ast}_{t})\right\rangle/\delta z^{j*}_{s}$.

In the Markovian limit, the $O$ operator is the Lindblad operator $L$. Then the initial condition of the $O$ operator is $O^{j}(t,t,z^{j\ast}_{t})=L_{j}$. $\alpha^{j}(t,s)$ is the bath correlation function. For the two mode symmetric squeezed bath it can be determined by

where $B_{j}(t)=\sum_{k}f^{j}_{k}b^{j}_{k}e^{-iw^{j}_{k}t}$. Assume the bath spectral density has the Lorentzian form $J_{j}(\omega_{j})=\frac{\Gamma_{j}}{2}\frac{\gamma_{j}^{2}}{\gamma_{j}^{2}+\left(\omega^{j}_{0}-\omega^{j}\right)^{2}}$ when $r_{j}=0$, which corresponds to the Ornstein-Uhlenbeck type bath correlation function $\alpha^{j}_{0}(t,s)=\frac{\Gamma_{j}\gamma_{j}}{2}e^{-i\omega^{j}_{0}(t-s)-\gamma_{j}|t-s|}$. $\Gamma_{j}$ and $\gamma_{j}$ denote the system-bath coupling strength and bandwidth of the environmental spectral density, respectively. The bandwidth $\gamma_{j}$ determines the magnitude of the environmental backaction. $\gamma_{j}\rightarrow\infty$ ($\gamma_{j}\rightarrow\ 0$) corresponds to a white (colored) noise situation, and the environment reaches the Markovian (non-Markovian) limit. As in Ref. Link , Eq. (8) can be decomposed as $\alpha^{j}(t,s)=\alpha^{j}_{1}(t,s)+\alpha^{j}_{2}(t,s)$, its specific form is

We can see that these correlation functions are not only functions of $t-s$, but also functions of $t$ and $s$, indicating that the bath is initially non-stationary. Since the two-mode squeezed state is an entangled state, the bath correlation functions $\alpha^{j}_{1}(t,s)$ and $\alpha^{j}_{2}(t,s)$ contain both the effects of each mode on the system. Now the non-Markovian QSD equation (Eq. (7)) can be written as

In the non-Markovian QSD method finite , the density operator of the system is obtained by taking the statistical average of all possible random realizations of the environmental noise $z^{j\ast}_{t}$, i.e., $\rho_{s}=\mathcal{M}\left[P_{t}\right]$. $P_{t}=\left|\psi(t,z^{j\ast}_{t})\right\rangle\left\langle\psi(t,z^{j\ast}_{t})\right|$, $\mathcal{M}\left[...\right]=\int e^{-z^{j*}_{t}z^{j}_{t}}\left[...\right]d^{2}z^{j}_{t}$. This process encodes large environmental effects with a huge dimensional Hilbert space as random variables. Actually, taking statistical averages of random variables is equivalent to taking trajectories in a huge Hilbert space. Therefore the time-local QSD equation has its physical meaning of statistical average. The non-Markovian master equation of the system can be written as

In Eq. (12), the first term is the unitary part of the reduced dynamics, it is independent of noise variable $z^{j\ast}_{t}$ and only ruled by the system Hamiltonian $H_{s}$. The Lindblad operator and its associated operators $\overline{O}^{j}_{1}(t,z^{j\ast}_{t})$, $\overline{O}^{j}_{2}(t,z^{j\ast}_{t})$ containing the memory kernel describe relaxation and decoherence in open systems with different decay modes. This equation of motion for the density matrix $\rho_{s}$ contains all the dynamics of the system $H_{s}$ in snon-Markovian baths. However, it is a hard task to determine the $O$ operator with the noise term $z^{j\ast}_{t}$, so we invoke a perturbation technique to obtain the approximate $O$ operators Yu99 ,

When the system is weakly coupled to the bath, we can take an approximation $O^{j}(t,s,z^{j\ast}_{t})$=$O^{j}(t,s)$ shiwf ; zhao2 , i.e., $\mathcal{M}[P_{t}\overline{O}_{\eta}^{j\dagger}(t,z^{j\ast}_{t})]=\rho_{s}\overline{O}_{\eta}^{j\dagger}(t)$($\eta=1,2$). The non-Markovian master equation in Eq. (12) takes a simple form nie ; luo ; finite

The time dependent operators $\overline{O}^{j}_{1}(t)$, $\overline{O}^{j}_{2}(t)$ can be numerically calculated by the following closed equations

Now by using Eqs. (15) and (16), the non-Markovian master equation in Eq. (14) can be numerically solved. In the Markovian limit, $\overline{O}^{j}_{1}(t)=\frac{1}{2}(u_{j}^{2}-v_{j}u_{j})L_{j}$, $\overline{O}^{j}_{2}(t)=\frac{1}{2}(\left|v_{j}\right|^{2}-v_{j}^{*}u_{j})L_{j}$. The Eq. (14) reduces to the Lindblad form,

In this paper, by using the QSD approach under weak coupling approximation, we obtain a non-Markovian master equation of the system in squeezed baths. Then for two examples: adiabatic evolution in a three-level system and quantum state transmission in a spin chain, we calculate the dynamics of the system. The effects of the non-Markovianity, squeezing direction and squeezing strength on the adiabatic and state transmission fidelity are analyzed, respectively. Our results show that, the two fidelities always decrease due to the presence of the environment, but they both can be enhanced for more non-Markovian baths or bigger $p$-quadrature squeezing. The $x$-quadrature squeezing leads to the enhancement of the system-bath coupling, thus decreasing the two fidelities. In addition, we find that at $0<\theta<\pi/2$, the fidelity peak appears when the squeezed strength $r$ reach a critical value $r=1-2\theta/\pi$, and its effects reverse after exceeding this critical value. We conjecture that this critical value is universal for different system models, but the only drawback is that it is obtained by numerical simulation and needs to be theoretically clarified in the following research. By using reservoir engineering, these results can be potentially used to combat the environment noises in performing quantum information processing tasks.