On the non-Markovian quantum control dynamics

Haijin Ding, Nina H. Amini, John E. Gough, Guofeng Zhang Haijin Ding was with the Laboratoire des Signaux et Systèmes (L2S), CNRS-CentraleSupélec-Université Paris-Sud, Université Paris-Saclay, 3, Rue Joliot Curie, 91190, Gif-sur-Yvette, France. He is now with the Department of Applied Mathematics, the Hong Kong Polytechnic University, Hung Hom, Kowloon, SAR, China (e-mail: [email protected]). Nina H. Amini is with the Laboratoire des Signaux et Systèmes (L2S), CNRS-CentraleSupélec-Université Paris-Sud, Université Paris-Saclay, 3, Rue Joliot Curie, 91190, Gif-sur-Yvette, France (e-mail: [email protected]). John E. Gough is with the Institute of Mathematics and Physics, Aberystwyth University, SY23 3BZ, Wales, United Kingdom (e-mail: [email protected]). Guofeng Zhang is with the Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, SAR, China, and The Hong Kong Polytechnic University Shenzhen Research Institute, Shenzhen, 518057, China (e-mail: [email protected]).Corresponding author: Haijin Ding.

Abstract

In this paper, we study both open-loop control and closed-loop measurement feedback control of non-Markovian quantum dynamics resulting from the interaction between a quantum system and its environment. We use the widely studied cavity quantum electrodynamics (cavity-QED) system as an example, where an atom interacts with the environment composed of a collection of oscillators. In this scenario, the stochastic interactions between the atom and the environment can introduce non-Markovian characteristics into the evolution of quantum states, differing from the conventional Markovian dynamics observed in open quantum systems. As a result, the atom’s decay rate to the environment varies with time and can be described by nonlinear equations. The solutions to these nonlinear equations can be analyzed in terms of the stability of a nonlinear control system. Consequently, the evolution of quantum state amplitudes follows linear time-varying equations as a result of the non-Markovian quantum transient process. Additionally, by using measurement feedback through homodyne detection of the cavity output, we can modulate the steady atomic and photonic states in the non-Markovian process. When multiple coupled cavity-QED systems are involved, measurement-based feedback control can influence the dynamics of high-dimensional quantum states, as well as the resulting stable and unstable subspaces.

Index Terms:

quantum non-Markovian dynamics, open quantum system, quantum open-loop control, quantum measurement feedback control.

I Introduction

Quantum control has garnered much attention due to its potential applications in quantum optics [1, 2], quantum information processing [3, 4, 5, 6], quantum engineering [7] and others [8, 9, 10, 11, 12, 13]. Control methods for quantum systems can be categorized into the open-loop control and closed-loop control, similar to that in classical systems [1]. In open-loop control, designed or iteratively optimized control pulses are utilized to produce required gate operations in quantum computations [14, 15, 16, 17]. When a control field is applied upon an atom, the atom can be excited, leading to the generation of single or multiple photons for quantum networking [18]. On the other hand, closed-loop quantum control can be realized through coherent feedback or measurement feedback methods. For instance, when an atom or cavity quantum electrodynamics (cavity-QED) system is coupled with a waveguide, a coherent feedback channel can be established using photons transmitted in the waveguide [19]. Subsequently, the atomic dynamics and photonic states can be modulated by tuning the parameters of the coherent feedback loop [20, 21, 19, 22, 23]. This form of coherent feedback dynamics can be represented as a linear control system with delays determined by the length of the feedback loop [21, 19]. Then the quantum state dynamics can be interpreted in terms of the stability of a linear control system with delays [22, 23, 24].

In addition to quantum coherent feedback, quantum measurement feedback is another commonly employed method for feedback control in quantum systems [1]. This approach involves designing feedback control based on the measurement outcomes of the quantum system [25, 24]. The measurement of quantum states can be categorized into quantum non-demolition (QND) measurement and non-QND measurement, depending on whether the measurement operator commutes with the quantum system’s Hamiltonian [1, 24]. These measurement methods have an impact on the steady quantum states [26, 27, 28, 29]. In the realm of control theory, quantum measurements can be represented using stochastic equations due to measurement and detection noise [30, 31, 32], distinguishing if from feedback control without measurement [33, 30]. By employing these measurement techniques and feedback control, quantum measurement feedback can influence the evolution of quantum states and facilitate the generation of desired quantum states. Consequently, quantum measurement feedback control holds significant promise for various applications in open quantum systems where the quantum states are affected by the environment [1]. One notable application is the use of measurement-based quantum feedback control in the quantum error correction (QEC) to rectify error bits in quantum computations [34, 35, 36]. Additionally, it can help preserve the coherence of a quantum state when the quantum system interacts with its environment[37].

In the realm of open quantum system control, it is typically assumed that the environmental evolution timescale is considerably shorter than the atomic system, and further can be assumed as static [38]. This assumption allows for the modeling of the interaction between the quantum system and the environment using a master equation with a static decaying rate, which is a widely employed Markovian approximation method for studying quantum dynamics in open systems. However, in numerous scenarios, this Markovian approximation proves inadequate for comprehensively analyzing the dynamics of open quantum systems [38, 39]. For instance, in the experimental implementation utilizing nuclear magnetic resonance (NMR) [40], where the NMR qubit interacts with a non-Markovian environment characterized by a randomized configuration of modulated radio-frequency fields. In such setups, the interaction between the NMR and the environment can lead to information backflow from the environment to the qubit, thus the decoherence of the qubit can be nonmonotonic, which is different from the monotonic decoherence of a qubit in the Markovian environment [41]. Moreover, experimental evidence of information backflow induced by non-Markovianity has also been observed in optical systems [42]. Given that traditional quantum control strategies relying on Markovian approximation are not directly applicable in these instances, it is required to explore the open-loop and closed-loop quantum control dynamics for the non-Markovian open quantum systems. Additionally, it is worth noting that the traditional Morkovian scenario can be encompassed within the broader framework of non-Markovian settings as a simplified special case.

The interaction between the quantum system and its environment, characterized as non-Markovian, can be effectively represented by the stochastic Schrödinger equation [43]. This approach incorporates the influence of the environment on quantum states through a complex-valued stochastic process [44, 43]. Alternatively, the non-Markovian dynamics can be described using a master equation featuring time-varying Lindblad components [45, 46], which also represents the memory effect inherent in non-Markovian dynamics, distinguish it from the Markovian master equation [47]. Consequently, various non-Markovian quantum control techniques have been developed, expanding upon the foundation laid by Markovian quantum control methods [48, 49, 50].

In this paper, we utilize the commonly employed cavity-QED system depicted in Fig. 1 as an example to investigate the dynamics of non-Marovkovian quantum control in a novel manner. We approach the evolution of parameters in quantum non-Markovian control as a nonlinear control problem, resulting in the introduction of time-varying parameters in the linear evolution of quantum states due to non-Markovianity. Initially, we analyze the non-Markovian interactions between the atom and the environment from a control perspective, and for the first time analyze the transition from non-Markovian transient dynamics to steady Markovian dynamics using stability theory in nonlinear control. Subsequently, we describe the evolution of atomic and photonic states using linear time varying (LTV) control equations, where external drives or the measurement feedback can influence the equation dynamics. Lastly, we extend our analysis to encompass non-Markovian dynamics in systems where atoms in multiple coupled cavities interact with a non-Markovian environment. Then the quantum states can be categorized into the stable and unstable subspaces based on the application of measurement feedback controls.

The rest of the paper is organized as follows. Section II concentrates on the nonlinear parameter dynamics of the non-Markovian interaction between the cavity-QED system and the environment. In Section III, the open-loop quantum control with the above non-Markovian interactions is analyzed from the perspective of LTV control theory. In Section IV, the effects of quantum measurement feedback control on the quantum states are considered. In Section V, we generalize to the circumstance of multiple coupled cavities with non-Markovian interactions with the environment, and analyze its open loop and closed loop control dynamics. Section VI concludes this paper.

II Non-Markovian quantum control for open systems

Refer to caption — Figure 1: Quantum control based on the Markovian interactions between a multi-level atom and a cavity, as well as the non-Markovian interactions between the atom and the environment modeled as a collection of oscillators.

As illustrated in Fig. 1, a resonant cavity is constructed between two mirrors, one multi-level atom with the energy levels represented with the state vectors $|0\rangle,|1\rangle,\cdots,|N-1\rangle$ is coupled with the cavity. The multi-level atom is simultaneously coupled with the environment modeled as a group of oscillators represented with the yellow circles. The cavity can be driven by the field represented with the red arrow, and can be detected or measured via its output channel, which is represented with the blue arrow. Using the measurement information, the feedback operator $G$ can be designed and applied upon the quantum system, thus the closed-loop control with measurement feedback can be realized.

For a simplified case without feedback, the above cavity-QED system can be modeled with the Hamiltonian

H=\omega_{c}a^{{\dagger}}a+\sum_{\omega}b_{\omega}^{{\dagger}}b_{\omega}+\sum_{n=0}^{N-1}\omega_{n}|n\rangle\langle n|+H_{t},

(1)

where $\omega_{c}$ is the resonant frequency of the cavity with the annihilation (creation) operator $a$ ( $a^{{\dagger}}$ ), the second component represents a group of oscillators in the environment modeled by the annihilation (creation) operator $b_{\omega}$ ( $b_{\omega}^{{\dagger}}$ ) with different frequencies $\omega$ , the third component is the atom Hamiltonian with $\omega_{n}$ represents the energy of the $n$ -th level represented with the multiplication of the vector $|n\rangle$ and $\langle n|$ , $H_{t}$ is composed with the interaction Hamiltonian between the atom and cavity, as well as the interaction Hamiltonian between the atom and environment. According to Ref. [51], the environment can be represented with the oscillator $\left|z_{\omega}\right\rangle=\exp\left(z_{\omega}b_{\omega}^{{\dagger}}\right)\left|0\right\rangle$ .

The analysis on the dynamics among the atom, cavity and environment in this paper, as well as the meaning of the $H_{t}$ in Eq. (1), is based on the following assumptions.

Assumption 1.

The interaction between the cavity and atom is Markovian, and the interaction between the atom and environment is initially non-Markovian.

Assumption 2.

The transition frequency between two neighborhood energy levels are identical, i.e., $\omega_{n}-\omega_{n-1}\equiv\omega_{a}$ for arbitrary $n$ .

For the $N$ -level atom coupled with the cavity, as in Fig. 1, the Hamiltonian $H_{t}$ in Eq. (1) can be equivalently represented in the interaction picture as [19, 52]

\displaystyle\bar{H}_{t}=

\displaystyle\sum_{n=1}^{N-1}g_{n}\left(e^{-i\delta t}\sigma^{-}_{n}a^{{\dagger}}+e^{i\delta t}\sigma^{+}_{n}a\right)+\sum_{n=1}^{N-1}\sum_{\omega}\chi_{\omega}^{(n)}\left(\sigma^{-}_{n}b_{\omega}^{{\dagger}}+\sigma^{+}_{n}b_{\omega}\right),

(2)

where $g_{n}$ represents the coupling strengths between the cavity and different atom energy levels, $\delta=\omega_{a}-\omega_{c}$ represents the detuning between the cavity resonant frequency and the transition frequency of two neighborhood atom energy levels, the lowering operator $\sigma^{-}_{n}=|n-1\rangle\langle n|$ , and the rising operator $\sigma^{+}_{n}=|n\rangle\langle n-1|$ . The first part at RHS represents the interaction between the atom and cavity, $\sigma^{-}_{n}a^{{\dagger}}$ represents that when the atom decays from the $n$ -th energy level to the $(n-1)$ -th energy level, it can emit one photon into the cavity. The following Hermitian conjugate $\sigma^{+}_{n}a$ represents that the atom can absorb one photon from the cavity and be excited to a higher energy level. The second part at RHS similarly represents the interaction between the multi-level atom and environment, and the coupling strength between the $n$ -th energy level and the environmental oscillator with the frequency $\omega$ is $\chi_{\omega}^{(n)}$ . $\sigma^{-}_{n}b_{\omega}^{{\dagger}}$ represents that an emitted filed generated by the atom’s decay from the $n$ -th to $(n-1)$ -th energy level can be absorbed by the environment, similar for the following Hermitian conjugate.

The overall coupling between the multi-level atom and the environment can be represented with the operator [53]

\displaystyle L=\sum_{n=1}^{N-1}\kappa_{n}|0\rangle\langle n|,

(3)

where $\kappa_{n}$ represents the decaying of the atomic state $|n\rangle$ to the environment. More details are introduced in Appendix A.

The dynamics of the above quantum system can be modeled with the Schrödinger equation $\frac{\mathrm{d}}{\mathrm{d}t}|\psi(t)\rangle=-iH|\psi(t)\rangle$ , and the components for the interaction between the quantum system and the environment can be written as a stochastic format, as introduced in Refs. [44, 43, 54, 51]. Combined with Eq. (3), the Schrödinger equation can be equivalently written as [44, 43, 54, 51]

\frac{\mathrm{d}}{\mathrm{d}t}|\psi(t)\rangle=-i\bar{H}|\psi(t)\rangle+L|\psi(t)\rangle z_{t}-L^{{\dagger}}\int_{0}^{t}\alpha(t,s)\frac{\delta|\psi(t)\rangle}{\delta z_{s}}\mathrm{d}s,

(4)

where $\bar{H}$ represents the interaction component between the atom and cavity in Eq. (2), $z_{t}$ is a complex stochastic process with $E(z_{t}^{*}z_{s})=\alpha(t,s)$ , $E(z_{t}z_{s})=0$ , and

\alpha(t,s)=\frac{\gamma}{2}e^{-\gamma|t-s|-i\Omega(t-s)},

(5)

where $\gamma^{-1}$ represents the environmental memory time scale and $\Omega$ represents the environmental central frequency for the modeled oscillators [44].

Then the density matrix representation of quantum states is governed by the following master equation according to the derivation in Appendix B,

$\displaystyle\dot{\rho}(t)=$	$\displaystyle-i[\bar{H},\rho(t)]$	(6)
	$\displaystyle+\left(\int_{0}^{t}\alpha^{}(t,s)f^{}(t,s)\mathrm{d}s+\int_{0}^{t}\alpha(t,s)f(t,s)\mathrm{d}s\right)L\rho(t)L^{{\dagger}}$
	$\displaystyle-\int_{0}^{t}\alpha(t,s)f(t,s)\mathrm{d}sL^{{\dagger}}L\rho(t)-\int_{0}^{t}\alpha^{}(t,s)f^{}(t,s)\mathrm{d}s\rho(t)L^{{\dagger}}L,$

where we denote

F(t)=\int_{0}^{t}\alpha(t,s)f(t,s)\mathrm{d}s,

(7)

and the integration in Eq. (6) represents the memory effect of the non-Markovian process based on the interaction between the atom and the environment. We can distinguish the Markovian and non-Markvoian quantum dynamics in the open systems according to the following definition.

Definition 1.

The quantum dynamics described by the master equation (6) is non-Markovian when $F(t)$ is a time-dependent integration, and Markovian when $F(t)$ is a constant.

Remark 1.

When $\gamma\rightarrow\infty$ , the interaction between the system and the environment is Markovian and $\alpha(t,s)=\delta(t-s)$ [44]. Then

F(t)=\int_{0}^{t}\delta(t-s)f(t,s)\mathrm{d}s=\frac{1}{2}f(t,t)\triangleq\frac{1}{2}\chi,

(8)

is a constant and the dynamics reduces to the case in open quantum systems with Markovian interactions between the atom and environment.

II-A Single multi-level atom coupled with the cavity

According to the non-Markovian master equation (6), the dynamics of the mean-value of an arbitrary operator O is governed by $\dot{\langle\textbf{O}\rangle}={\rm Tr}\left[\dot{\rho}\textbf{O}\right]$ [30]. To clarify the atomic dynamics, we study the dynamics of the operator $\langle\sigma^{+}_{n}\sigma^{-}_{n}\rangle$ representing the population or probability that the atom is excited at the $n$ -th energy level. Because of the coupling between the atom and the cavity, the dynamics of $\langle\sigma^{+}_{n}\sigma^{-}_{n}\rangle$ is affected by the exchange of energy between the atom and the cavity, which can be evaluated by the dynamics of the operator $\langle\sigma^{+}_{n}a\rangle$ , representing that the atom can absorb one photon from the cavity to be excited to a higher energy level, or the reverse process evaluated by $\langle\sigma^{-}_{n}a^{{\dagger}}\rangle$ . Besides, the number of photons in the cavity can be evaluated with the operator $\langle a^{{\dagger}}a\rangle$ [55]. Then we can derive the equation of the mean values of the operators with $n=1,2,\dots,N-1$ as [56, 57, 58, 59]


		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\langle\sigma^{+}_{n}\sigma^{-}_{n}\rangle=-ig_{n}e^{i\delta t}\langle\sigma^{+}_{n}a\rangle+ig_{n}e^{-i\delta t}\langle\sigma^{-}_{n}a^{{\dagger}}\rangle$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+ig_{n+1}e^{i\delta t}\langle\sigma^{+}_{n+1}a\rangle-ig_{n+1}e^{-i\delta t}\langle\sigma^{-}_{n+1}a^{{\dagger}}\rangle$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left(F(t)+F^{*}(t)\right)\|\kappa_{n}\|^{2}\langle\sigma^{+}_{n}\sigma^{-}_{n}\rangle,$			(9a)
		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\langle\sigma^{+}_{n}a\rangle=-i\delta\langle\sigma^{+}_{n}a\rangle-ig_{n}e^{-i\delta t}\langle\sigma^{+}_{n}\sigma^{-}_{n}\rangle$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+ig_{n}e^{-i\delta t}\langle a^{{\dagger}}a\rangle-F^{*}(t)\|\kappa_{n}\|^{2}\langle\sigma^{+}_{n}a\rangle,$			(9b)
		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\langle\sigma^{-}_{n}a^{{\dagger}}\rangle=i\delta\langle\sigma^{-}_{n}a^{{\dagger}}\rangle+ig_{n}e^{i\delta t}\langle\sigma^{+}_{n}\sigma^{-}_{n}\rangle$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-ig_{n}e^{i\delta t}\langle a^{{\dagger}}a\rangle-F(t)\|\kappa_{n}\|^{2}\langle\sigma^{-}_{n}a^{{\dagger}}\rangle,$			(9c)
		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\langle a^{{\dagger}}a\rangle=i\sum_{n}g_{n}\left(e^{i\delta t}\langle\sigma^{+}_{n}a\rangle-e^{-i\delta t}\langle\sigma^{-}_{n}a^{{\dagger}}\rangle\right),$			(9d)
		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}F(t)=\sum_{n}\|\kappa_{n}\|^{2}F^{2}(t)-(\gamma+i\Omega-i\omega_{a})F(t)+\frac{\gamma\chi}{2},$			(9e)

where $\chi$ is a constant as in Appendix A. The first two lines of RHS of Eq. (9a) represent that the atomic state excited at the $n$ -th energy level can be acquired from a lower energy level by absorbing one photon from the cavity, or from a higher energy level by emitting one photon into the cavity, and the third line of Eq. (9a) represents the non-Markovian interaction between the atom and environment, which can make the atom decay to its ground state. Eq. (9b) and Eq. (9c) represents the interface between the atom and the cavity via the emitting and absorbing processes of a photon. This process is influenced by the detuning between the atom and the cavity (i.e., the first component of the RHS of Eq. (9b) and Eq. (9c)), the coupling strengths $g_{n}$ , and the non-Markovian decaying to the environment represented by the last component of the RHS of Eq. (9b) and Eq. (9c), respectively. Eq. (9d) represents how the number of photons in the cavity is influenced by the coupling between the atom and cavity. The nonlinear Eq. (9e) represents the non-Markovian decaying amplitude of the atom to the environment, which is derived in detail in Appendix A.

Additionally, the atom populations are normalized as $\langle\sigma_{00}\rangle+\sum_{n=1}^{N-1}\langle\sigma^{+}_{n}\sigma^{-}_{n}\rangle=1$ , where $\langle\sigma_{00}\rangle$ represents the population that the atom is at the ground state and its dynamics is governed as

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\langle\sigma_{00}\rangle=$	$\displaystyle\left(F(t)+F^{*}(t)\right)\sum_{n=1}^{N-1}\|\kappa_{n}\|^{2}\langle\sigma^{+}_{n}\sigma^{-}_{n}\rangle$		(10)
		$\displaystyle+ig_{1}e^{i\delta t}\langle\sigma^{+}_{1}a\rangle-ig_{1}e^{-i\delta t}\langle\sigma^{-}_{1}a^{{\dagger}}\rangle.$		(10)

We define the state vector as

\displaystyle X(t)=\left[X_{1}(t),X_{2}(t),\cdots,X_{N-1}(t),\langle a^{{\dagger}}a\rangle\right]^{T},

(11)

with $X_{n}(t)=\left[\langle\sigma^{+}_{n}\sigma^{-}_{n}\rangle,\langle\sigma^{+}_{n}a\rangle,\langle\sigma^{-}_{n}a^{{\dagger}}\rangle\right]^{T}$ . Then Eq. (9) can be rewritten as

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}X$	$\displaystyle=\begin{bmatrix}A_{1}(t)&B_{1}(t)&\cdots&0&D_{1}(t)\\ 0&A_{2}(t)&\cdots&0&D_{2}(t)\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&A_{N-1}(t)&D_{N-1}(t)\\ C_{1}(t)&C_{2}(t)&\cdots&C_{N-1}(t)&0\end{bmatrix}X$		(12)
		$\displaystyle\triangleq A(t)X,$		(12)

where

\displaystyle A_{n}(t)=\begin{bmatrix}-\left(F(t)+F^{*}(t)\right)|\kappa_{n}|^{2}&-ig_{n}e^{i\delta t}&ig_{n}e^{-i\delta t}\\ -ig_{n}e^{-i\delta t}&-i\delta-F^{*}(t)|\kappa_{n}|^{2}&0\\ ig_{n}e^{i\delta t}&0&i\delta-F(t)|\kappa_{n}|^{2}\end{bmatrix},

(13)

\displaystyle B_{n}(t)=\begin{bmatrix}0&ig_{n+1}e^{i\delta t}&-ig_{n+1}e^{-i\delta t}\\ 0&0&0\\ 0&0&0\end{bmatrix},

(14)

\displaystyle C_{n}(t)

\displaystyle=\begin{bmatrix}0&ig_{n}e^{i\delta t}&-ig_{n}e^{-i\delta t}\end{bmatrix},

(15)

\displaystyle D_{n}(t)

\displaystyle=\begin{bmatrix}0\\ ig_{n}e^{-i\delta t}\\ -ig_{n}e^{i\delta t}\end{bmatrix},

(16)

with $n=1,2,\cdots,N-1$ .

The analysis in Section II and Section III is based on the following assumption for initial settings.

Assumption 3.

Assume initially the atom is excited at the highest energy level and $\langle\sigma^{+}_{n}a\rangle=\langle\sigma^{-}_{n}a^{{\dagger}}\rangle=\langle a^{{\dagger}}a\rangle=0$ .

Remark 2.

Eq. (12) is a time-varying linear equation and reduces to the time-invariant case only when $\delta=0$ and $F(t)$ becomes a constant, thereby converging to the Markovian case.

In the following subsection, we clarify the dynamics of $F(t)$ from the perspective of nonlinear dynamics, and then analyze how the time-varying $F(t)$ can influence the quantum control performance.

II-B Non-Markovian parameter dynamics

In Eq. (9e), we denote


		$\displaystyle P=\sum_{n}\|\kappa_{n}\|^{2},$			(17a)
		$\displaystyle Q=-\left(\gamma+i\Omega-i\omega_{a}\right),$			(17b)
		$\displaystyle R=\frac{\gamma\chi}{2},$			(17c)

then Eq. (9e) can be simply rewritten as

\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}F(t)=PF^{2}(t)+QF(t)+R,

(18)

and $F(t)$ can be solved according to following different parameter settings.

II-B1 $\Omega=\omega_{a}$ and $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}<0$

When $\Omega=\omega_{a}$ and $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}<0$ , namely $4PR<Q^{2}$ and $2\chi\sum_{n}|\kappa_{n}|^{2}<\gamma$ , which means that the coupling between the atom and the environment can be relatively small and bounded by $\gamma$ , then

\displaystyle F(t)=\frac{2\sqrt{\left(\frac{Q}{2P}\right)^{2}-\frac{R}{P}}}{1-e^{C+\sqrt{Q^{2}-4PR}t}}-\left(\frac{Q}{2P}+\sqrt{\left(\frac{Q}{2P}\right)^{2}-\frac{R}{P}}\right),

(19)

where $C$ is a constant and can be determined as

\displaystyle e^{C}=1-2\sqrt{\left(\frac{Q}{2P}\right)^{2}-\frac{R}{P}}\left[\frac{Q}{2P}+\sqrt{\left(\frac{Q}{2P}\right)^{2}-\frac{R}{P}}\right]^{-1},

(20)

based on the initial condition that $F(0)=0$ .

II-B2 $\Omega=\omega_{a}$ and $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}>0$

When $\Omega=\omega_{a}$ and $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}>0$ , namely $2\chi\sum_{n}|\kappa_{n}|^{2}>\gamma$ , which means that the coupling between the atom and environment is relatively stronger. Then by Eq. (18),

\displaystyle F(t)=\sqrt{\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}}\tan\left[\sqrt{\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}}\left(Pt+C\right)\right]-\frac{Q}{2P},

(21)

where $C$ can be similarly determined by $F(0)=0$ and the solution is not unique.

II-B3 $\Omega=\omega_{a}$ and $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}=0$

When $\Omega=\omega_{a}$ and $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}=0$ , namely $2\chi\sum_{n}|\kappa_{n}|^{2}=\gamma$ . Then by Eq. (18), $C=-2P/Q$ and

\displaystyle F(t)=-\frac{Q}{2P}-\frac{1}{Pt+C}.

(22)

We have the following proposition for the relationship between the parameter settings above and the non-Markovian dynamics.

Proposition 1.

When $\Omega=\omega_{a}$ and $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}\leq 0$ , the master equation for open quantum system finally converges to the Markovian format when $t\rightarrow\infty$ .

Proof.

According to the calculations above, when $\Omega=\omega_{a}$ and $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}\leq 0$ ,

\lim_{t\rightarrow\infty}F(t)=-\left(\frac{Q}{2P}+\sqrt{\left(\frac{Q}{2P}\right)^{2}-\frac{R}{P}}\right)\triangleq F_{\infty}.

(23)

When $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}>0$ , $\lim_{t\rightarrow\infty}F(t)$ doesn’t exist because $\tan\left[\sqrt{\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}}\left(Pt+C\right)\right]$ is infinite when

\sqrt{\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}}\left(Pt+C\right)=n\pi+\pi/2

with $n=0,1,2,\cdots$ . ∎

For the general case that $F(t)$ is complex when $\Omega\neq\omega_{a}$ , we denote $F(t)=F_{R}(t)+iF_{I}(t)$ with $F_{R}(t)$ and $F_{I}(t)$ representing the real and imaginary part of $F(t)$ respectively, then we can derive the following real-valued nonlinear equation


		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}F_{R}(t)=\sum_{n}\|\kappa_{n}\|^{2}\left(F_{R}^{2}(t)-F_{I}^{2}(t)\right)-\gamma F_{R}(t)$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\left(\Omega-\omega_{a}\right)F_{I}(t)+\frac{\gamma\chi}{2},$			(24a)
		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}F_{I}(t)=2\sum_{n}\|\kappa_{n}\|^{2}F_{R}(t)F_{I}(t)-\left(\Omega-\omega_{a}\right)F_{R}(t)-\gamma F_{I}(t).$			(24b)

Denote $X_{F}(t)=\left[F_{R}(t),F_{I}(t)\right]^{T}$ and $X_{F}(0)=\left[0,0\right]^{T}$ . Then Eq. (24) can be rewritten as

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}X_{F}(t)$	$\displaystyle=\begin{bmatrix}\sum_{n}\|\kappa_{n}\|^{2}F_{R}(t)-\gamma&-\sum_{n}\|\kappa_{n}\|^{2}F_{I}(t)\\ \sum_{n}\|\kappa_{n}\|^{2}F_{I}(t)&\sum_{n}\|\kappa_{n}\|^{2}F_{R}(t)-\gamma\end{bmatrix}\begin{bmatrix}F_{R}(t)\\ F_{I}(t)\end{bmatrix}$	(25)
	$\displaystyle~{}~{}~{}~{}+\begin{bmatrix}0&\left(\Omega-\omega_{a}\right)\\ -\left(\Omega-\omega_{a}\right)&0\end{bmatrix}\begin{bmatrix}F_{R}(t)\\ F_{I}(t)\end{bmatrix}+\begin{bmatrix}\gamma\chi/2\\ 0\end{bmatrix}$
	$\displaystyle\triangleq\textbf{f}\left(X_{F}(t)\right)+\textbf{f}_{\Omega}X_{F}(t),$

where $\textbf{f}\left(X_{F}(t)\right)$ represents the sum of the first and third terms after the first equal sign, $\textbf{f}_{\Omega}=\begin{bmatrix}0&\left(\Omega-\omega_{a}\right)\\ -\left(\Omega-\omega_{a}\right)&0\end{bmatrix}$ .

Based on the contraction analysis for arbitrary initial condition of $X_{F}(t)$ [60], consider the differential relation in Eq. (25), then

\displaystyle\delta\dot{X}_{F}(t)=\left[\frac{\partial\textbf{f}\left(X_{F}(t)\right)}{\partial X_{F}}+\textbf{f}_{\Omega}\right]\delta X_{F}(t),

(26)

and

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left(\delta X_{F}(t)^{T}\delta X_{F}(t)\right)$	$\displaystyle=2\delta X_{F}(t)^{T}\delta\dot{X}_{F}(t)$	(27)
	$\displaystyle=2\delta X_{F}(t)^{T}\left[\frac{\partial\textbf{f}}{\partial X_{F}}\left(X_{F}(t)\right)+\textbf{f}_{\Omega}\right]\delta X_{F}(t)$
	$\displaystyle=2\delta X_{F}(t)^{T}\frac{\partial\textbf{f}}{\partial X_{F}}\delta X_{F}(t)+2\textbf{f}_{\Omega}\delta X_{F}(t)^{T}\delta X_{F}(t).$

The stability of Eq. (25) is mainly determined by the Jacobian $\textbf{J}=\frac{\partial\textbf{f}}{\partial X_{F}}+\textbf{f}_{\Omega}$ , according to Eq. (25),

\displaystyle\textbf{J}\left(X_{F}(t)\right)=\begin{bmatrix}PF_{R}(t)-\gamma&-PF_{I}(t)+\left(\Omega-\omega_{a}\right)\\ PF_{I}(t)-\left(\Omega-\omega_{a}\right)&PF_{R}(t)-\gamma\end{bmatrix},

(28)

which reduces to $\frac{\partial\textbf{f}}{\partial X_{F}}$ when $\Omega=\omega_{a}$ .

For the nonlinear system in Eq. (24), we regard $u=\Omega-\omega_{a}$ as the control,

\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}X_{F}(t)=\textbf{f}\left(X_{F}(t)\right)+\begin{bmatrix}0&u\\ -u&0\end{bmatrix}X_{F}(t),

(29)

and define the output of the system as

\displaystyle y_{F}(t)

\displaystyle=\begin{bmatrix}1&1\\ 1&-1\end{bmatrix}\begin{bmatrix}F_{R}(t)\\ F_{I}(t)\end{bmatrix}\triangleq CX_{F}(t).

(30)

Because the norm of matrix $C$ is finite, the norm of the output vector $y_{F}(t)$ is determined by the state vector $X_{F}(t)$ . We further analyze the control input and output properties according to the following definition.

Definition 2.

(BIBO) [61] The system in Eq. (29) is said to be bounded-input bounded-output (BIBO) if for every $a_{u}\geq 0$ and $\alpha_{X}\geq 0$ there is a finite number $\beta_{X}=\beta_{X}\left(a_{u},\alpha_{X}\right)$ such that for every initial condition $X_{F}\left(t_{0}\right)$ with $X_{F}\left(t_{0}\right)\leq\alpha_{X}$ and every control sequence $u$ with $\|u\|\leq a_{u}$ ,

X_{F}\left(u,t^{\prime};X_{F}\left(t_{0}\right),t_{0}\right)\leq\beta_{X},

for $t^{\prime}>t_{0}$ .

Additionally, a discrete generalization of Definition 2 is given in Ref. [62].

We rewrite Eq. (25) as

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}X_{F}(t)$	$\displaystyle=\mathbf{A}\left(X_{F}(t)\right)X_{F}(t)+\mathbf{B}X_{F}(t)u+\begin{bmatrix}\gamma\chi/2\\ 0\end{bmatrix}$		(31)
		$\displaystyle=\left(\mathbf{A}\left(X_{F}(t)\right)+\mathbf{B}u\right)X_{F}(t)+\begin{bmatrix}\gamma\chi/2\\ 0\end{bmatrix},$		(31)

where $\textbf{f}\left(X_{F}(t)\right)=\mathbf{A}\left(X_{F}(t)\right)X_{F}(t)+\begin{bmatrix}\gamma\chi/2\\ 0\end{bmatrix}$ , $\mathbf{B}=\begin{bmatrix}0&1\\ -1&0\end{bmatrix}$ . In the second line of Eq. (31), the parameter $u$ can be regarded as a constant perturbation or random unknown uncertainty when the environment cannot be precisely modeled.

According to Refs. [63, 60], the Lyapunov function can be defined as $V\left(X_{F}\right)=\textbf{f}^{T}\left(X_{F}\right)\mathbf{M}\left(X_{F}\right)\textbf{f}\left(X_{F}\right)$ , where $\mathbf{M}\left(X_{F}\right)$ is a contraction matrix to be determined. When $u=0$ ,

\displaystyle\dot{V}(X_{F})=\textbf{f}^{T}\left(X_{F}(t)\right)\left[\frac{\partial\textbf{f}^{T}}{\partial X_{F}}\mathbf{M}\left(X_{F}\right)+\mathbf{M}\left(X_{F}\right)\frac{\partial\textbf{f}}{\partial X_{F}}+\dot{\mathbf{M}}\right]\textbf{f}\left(X_{F}(t)\right).

(32)

For the simplest case with $\Omega=\omega_{a}$ and $F_{I}(t)\equiv 0$ , Eq. (25) reduces to $\frac{\mathrm{d}}{\mathrm{d}t}F_{R}(t)=PF_{R}^{2}(t)-\gamma F_{R}(t)+\frac{\gamma\chi}{2}.$ More explanations are given combined with the following example and propositions.

II-B4 Example

Take the parameter settings in Section II-B3) as an example. When $\Omega=\omega_{a}$ and $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}=0$ , $F_{I}(t)\equiv 0$ , $Q=-\gamma$ , $R=\frac{\gamma^{2}}{4P}$ , $F_{R}(t)=\frac{\gamma}{2P}-\frac{\gamma}{\gamma Pt+2P}$ , then $PF_{R}(t)-\gamma<0$ for arbitrary $t$ and $\textbf{J}\left(X_{F}(t)\right)$ is definitely negative. If we take $\mathbf{M}=\mathbf{I}$ for simplification in Eq. (32), then $\dot{V}(X_{F})<0$ and Eq. (24) is globally stable at this parameter setting.

Definition 3 (see [64]).

A set $S$ is said to be invariant if each solution starting in $S$ remains in $S$ for all $t$ .

Generalized from the stability property in Ref. [64], we can derive the following proposition.

Proposition 2.

For the set $\Omega_{\alpha}={X_{F}\in\textbf{R}^{2}:V\leq\alpha_{V}}$ is bounded for $\alpha_{V}>0$ , when $2\sum_{n}|\kappa_{n}|^{2}\alpha_{V}+\gamma\chi-2\gamma<0$ and $\mathbf{M}=\mathbf{I}$ , then a local invariant set contained in the region $\dot{V}=0$ and the scale of the invariant set is determined by the parameters as $\alpha_{V}\leq\frac{\gamma(2-\chi)}{2\sum_{n}|\kappa_{n}|^{2}}$ with $0<\chi<2$ .

Proof.

Assume initially $X_{F}\in\Omega_{\alpha}$ and $\Gamma_{F}$ represents an invariant set of $X_{F}$ satisfying that $\dot{V}(X_{F})=0$ . When $2\sum_{n}|\kappa_{n}|^{2}\alpha_{V}+\gamma\chi-2\gamma<0$ , then $\dot{V}<0$ can be satisfied according to Eqs. (32), (24a) with $\Omega=\omega_{a}$ , and $V(x)\geq 0$ by its definition. Then there exists a positive value $v_{l}$ that $\lim_{t\rightarrow\infty}V(X_{F})=v_{l}\geq 0$ and $v_{l}<\alpha_{v}$ . Thus $V(X_{F})$ is always within the invariant set. ∎

Remark 3.

Proposition 2 means that larger $\gamma$ or smaller $\sum_{n}|\kappa_{n}|^{2}$ can induce a larger invariant set with stronger Markovian property. This is why the Markovian approximation can be applied in the circumstance that the coupling between the quantum system and the environment is weak.

Proposition 3 ([61]).

Suppose for each $u\geq 0$ , there exists a positive Lyapunov function $V_{u}$ for the system with the control $u$ and

\dot{V}_{u}(t,X_{f})\leq 0.

When the control satisfies $\|u\|\leq a_{u}$ , then the system is BIBO stable.

Proposition 4.

The system in Eq. (24) is BIBO stable when the initial condition $\left\|F(0)\right\|$ is bounded and the system starts in the invariant set $\Omega_{\alpha}={X_{F}\in\textbf{R}^{2}:V\leq\alpha_{V}},$ satisfying $2\sum_{n}|\kappa_{n}|^{2}\alpha_{V}+\gamma\chi-2\gamma<0$ .

Proof.

When the condition is satisfied, then $\dot{V}<0$ according to Eq. (32), then the system in Eq. (24) is BIBO based on Proposition 3. ∎

Remark 4.

When the system is BIBO, the Jacobian in Eq. (28) is also bounded, then the Lyapunov function can also be bounded according to Ref. [65].

Lemma 1.

When $\dot{V}<0$ , then $\lim_{t\rightarrow\infty}F_{R}(t)$ and $\lim_{t\rightarrow\infty}F_{I}(t)$ exist when $\left|\Omega-\omega_{a}\right|\leq\epsilon$ .

Proof.

When $\dot{V}<0$ and $V(t)>0$ , then $\lim_{t\rightarrow\infty}V(t)$ exists and $\lim_{t\rightarrow\infty}\dot{V}(t)=0$ . Then $\lim_{t\rightarrow\infty}F_{R}(t)$ and $\lim_{t\rightarrow\infty}F_{I}(t)$ can be solved by the first line of Eq. (32). ∎

Then we can derive the following proposition based on the results above and the following assumption [66].

Assumption 4.

Assume that the Lyapunov function in Eq. (32) satisfies $\gamma_{1}\left|X_{F}(t)\right|^{2}\leq V\left(X_{F}(t)\right)\leq\gamma_{2}\left|X_{F}(t)\right|^{2}$ .

Proposition 5.

If the system with $\Omega=\omega_{a}$ is BIBO stable and $\left|\Omega-\omega_{a}\right|\leq\epsilon$ , then the system will finally converge to the Markovian case as $t\rightarrow\infty$ .

Proof.

The Lyapunov function in Eq. (32) is independent from $\left(\Omega-\omega_{a}\right)$ . When the system is BIBO stable and $\left|\Omega-\omega_{a}\right|\leq\epsilon$ , then $F_{R}(t)$ and $F_{I}(t)$ are solvable with steady values according to Lemma 1 and Eq. (28). ∎

The real and imaginary parts of $F(t)$ are compared in Fig. 2(a), where we take $\chi=0.5$ and $\gamma=1$ . For the solid lines in (a), $P=1$ , $\Omega=\omega_{a}=50$ , while for the dashed lines, $P=1$ , $\Omega=50$ and $\omega_{a}=45$ , which means that there is a detuning between the atom and the environment. Besides, the purple dot line is for the case that $P=0.2$ and $\Omega=\omega_{a}=50$ , which illustrates that smaller $P$ can induce faster convergence to Markovian interactions with the environment. The bounded evolution of the real and imaginary parts in (a) are further compared in (b) with the red line for $\gamma=1$ , the green line for $\gamma=0.5$ , the black line for $\gamma=3$ , and the other parameters are the same as the dashed lines in (a). The comparisons in (b) show that larger $\gamma$ is better for the realization of Markovian interactions with the environment, which agrees with Remark 3.

II-C An example to clarify how $F(t)$ influences the atomic dynamics

In this subsection, we take the two-level atom as an example. One initially excited two-level atom is coupled with a cavity, and the atom is also coupled with the environment. The dynamics of the system can be represented by Eq. (9) with $N=2$ , and the dynamics is compared as follows.

As shown in Fig. 3, initially the atom is excited with $\left\langle\sigma^{+}_{1}\sigma^{-}_{1}\right\rangle=1$ and the other amplitudes of quantum states in Eq. (9) are zero. We take the atom and cavity parameters as $\omega_{a}=50$ , $g_{1}=2$ and $\delta=0$ . When the atom is coupled with a non-Markovian environment with the parameters $\Omega=45$ , $\kappa_{1}=4$ , $\chi=1$ and $\gamma=1$ , then the population of the atom’s two different states and the number of photons in the cavity are shown as the solid lines in Fig. 3. Besides, a simplified case that the atom interacts with the Markovian environment is simulated as the dashed lines with $F(t)\equiv 0.0792$ , which is the final steady amplitude of $F(t)$ in the non-Moarkovian case. The larger oscillations of the solid lines are induced by the detuning between the two-level atom and the environment valued by $\Omega-\omega_{a}$ as in Eq. (9e), then as a result, $F(t)$ can be complex values but always time-varying in the non-Markovian parameter settings.

II-D Nonlinear dynamics with uncertain control input

Practically, we usually cannot know the exact value of the environment parameter $\Omega$ , thus the component $\Omega-\omega_{a}$ in Eq. (24) can be regarded as an uncertain control input. Then Eq. (25) can be re-written as

\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}X_{F}(t)

\displaystyle=\textbf{f}\left(X_{F}(t)\right)+\textbf{u},

(33)

where we denote $\textbf{u}=\Delta AX_{F}(t)$ with $\Delta A=\begin{bmatrix}0&\Omega-\omega_{a}\\ \omega_{a}-\Omega&0\end{bmatrix}$ , and there always exists $\epsilon>0$ satisfying $\|\Delta A\|\leq\epsilon$ . Then Eq. (33) can be regarded as a perturbed nonlinear system.

Assumption 5.

The nonlinear equation (24) with $\Omega=\omega_{a}$ is stable according to the parameter settings in Proposition 1.

According to Eqs. (18)-(22), when $\Omega=\omega_{a}$ , $F_{I}(t)\equiv 0$ , and we denote $\lim_{t\rightarrow\infty}F_{R}(t)=\bar{F}_{R}$ . Define a new state vector $\tilde{X}_{F}(t)=\left[F_{R}(t)-\bar{F}_{R},F_{I}(t)\right]^{T}\triangleq\left[\tilde{F}_{R}(t),F_{I}(t)\right]^{T}$ and $\lim_{t\rightarrow\infty}\tilde{X}_{F}(t)=\left[0,0\right]^{T}\triangleq\tilde{X}_{F}^{*}$ when Assumption 5 is satisfied and $\Omega=\omega_{a}$ . Then


		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\tilde{F}_{R}(t)=\sum_{n}\|\kappa_{n}\|^{2}\left[\left(\tilde{F}_{R}(t)+\bar{F}_{R}\right)^{2}-F_{I}^{2}(t)\right]$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\gamma\left(\tilde{F}_{R}(t)+\bar{F}_{R}\right)+\left(\Omega-\omega_{a}\right)F_{I}(t)+\frac{\gamma\chi}{2},$			(34a)
		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}F_{I}(t)=2\sum_{n}\|\kappa_{n}\|^{2}\left(\tilde{F}_{R}(t)+\bar{F}_{R}\right)F_{I}(t)$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left(\Omega-\omega_{a}\right)\left(\tilde{F}_{R}(t)+\bar{F}_{R}\right)-\gamma F_{I}(t),$			(34b)

and can be simplified as

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\tilde{X}_{F}(t)=$	$\displaystyle\tilde{\textbf{f}}\left(\tilde{X}_{F}(t)\right)+\begin{bmatrix}\frac{\gamma\chi}{2}+\sum_{n}\|\kappa_{n}\|^{2}\bar{F}_{R}^{2}-\gamma\bar{F}_{R}\\ \left(\omega_{a}-\Omega\right)\bar{F}_{R}\end{bmatrix}$		(35)
		$\displaystyle+\begin{bmatrix}0&\Omega-\omega_{a}\\ \omega_{a}-\Omega&0\end{bmatrix}\tilde{X}_{F}(t),$		(35)

where $\tilde{\textbf{f}}\left(\tilde{X}_{F}(t)\right)$ is for the nonlinear component at the RHS of Eq. (34). Obviously, $\tilde{\textbf{f}}\left(\tilde{X}_{F}(t)\right)=0$ when $\tilde{X}_{F}=0$ . When $\Omega=\omega_{a}$ and $X_{F}(t)$ converges by Proposition 1, the RHS of Eq. (35) converges to zero.

Proposition 6 (see [67]).

For the nonlinear system

\dot{\textbf{X}}(t)=\textbf{f}(\textbf{X}(t),t)+\textbf{h}(\textbf{X}(t),t),

where f is continuously differentiable with $\textbf{f}(0,t)=0$ and $\textbf{h}(\textbf{X}(t),t)$ is a persistent perturbation: if the system is uniformly and asymptotically stable about its equilibrium $\textbf{X}^{*}=0$ when $\textbf{h}(\textbf{X}(t),t)=0$ , and there are two positive constants $\tilde{\delta}_{1}$ and $\tilde{\delta}_{2}$ such that $\|\textbf{h}(\textbf{X}(t),t)\|<\tilde{\delta}_{1}$ for $t\in[0,\infty]$ and $\|\textbf{h}(\textbf{X}(0),0)\|<\tilde{\delta}_{2}$ , then the persistently perturbed system remains to be stable in the sense of Lyapunov.

Proposition 7.

When $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}\leq 0$ , there exists $\epsilon>0$ such that the quantum system approaches a Markovian behavior when $\left|\omega_{a}-\Omega\right|\leq\epsilon$ .

Proof.

When $\omega_{a}=\Omega$ , by Proposition 1, the quantum dynamics approaches Markovian behavior as $t\rightarrow\infty,$ and the nonlinear dynamics $\frac{\mathrm{d}}{\mathrm{d}t}\tilde{X}_{F}(t)=\tilde{\textbf{f}}\left(\tilde{X}_{F}(t)\right)$ is uniformly and asymptotically stable around its equilibrium $\tilde{X}_{F}=0$ . When $\left|\omega_{a}-\Omega\right|\leq\epsilon$ and $\gamma\chi$ is finite, Proposition 6 ensures that $\tilde{F}_{R}(t)$ and $F_{I}(t)$ in Eq. (34) remain stable. Then the quantum dynamics will approach Markovian behavior as $t\rightarrow\infty$ . ∎

III Time-varying linear quantum control dynamics

In the following, we only consider the circumstance that $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}\leq 0$ , thus the interaction between the atom and environment finally converges to a Markovian form. Return to Eq. (12), we divide the following analysis into two parts according to whether there are detunings between the atom and cavity, as clarified in Remark 2.

III-A $\delta=0$ , $g_{n}\neq 0$

At this parameter setting, $C_{n}$ and $D_{n}$ in Eqs. (15) and (16) are all time-invariant for arbitrary $n$ , while $A_{n}$ is time-varying with $F(t)$ converges to its steady values according to Eqs. (19) and (22). $A(t)$ can be separated into the time-invariant component and time-varying components as

$\displaystyle A_{n}(t)=$	$\displaystyle\bar{A}_{n}+\hat{A}_{n}(t)$	(36)
$\displaystyle=$	$\displaystyle\begin{bmatrix}-L_{n}^{(1)}&-ig_{n}&ig_{n}\\ -ig_{n}&-L_{n}^{(2)}&0\\ ig_{n}&0&-L_{n}^{(3)}\end{bmatrix}$
	$\displaystyle+\begin{bmatrix}L_{n}^{(1)}-\left(F(t)+F^{}(t)\right)\|\kappa_{n}\|^{2}&0&0\\ 0&L_{n}^{(2)}-F^{}(t)\|\kappa_{n}\|^{2}&0\\ 0&0&L_{n}^{(3)}-F(t)\|\kappa_{n}\|^{2}\end{bmatrix},$

where $L_{n}^{(1)}=\lim_{t\rightarrow\infty}\left(F(t)+F^{*}(t)\right)|\kappa_{n}|^{2}$ , $L_{n}^{(2)}=\lim_{t\rightarrow\infty}F^{*}(t)|\kappa_{n}|^{2}$ and $L_{n}^{(3)}=\lim_{t\rightarrow\infty}F(t)|\kappa_{n}|^{2}$ , then $\lim_{t\rightarrow\infty}\hat{A}_{n}(t)=\textbf{0}_{3\times 3}$ .

For the special case where $g_{n}=0$ , meaning the multi-level atom only interacts with the non-Markovian environment with $C_{n}=D_{n}^{T}=\textbf{0}_{1\times 3}$ (where $T$ denotes the matrix transpose), we then construct the vector representing the populations of a multi-level atom as

\tilde{X}=\left[\langle\sigma^{+}_{1}\sigma^{-}_{1}\rangle,\cdots,\langle\sigma^{+}_{n}\sigma^{-}_{n}\rangle,\cdots,\langle\sigma^{+}_{N-1}\sigma^{-}_{N-1}\rangle\right]^{T},

which is governed by the following real-value equation if initially $\tilde{X}(0)$ is real,

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\tilde{X}$	$\displaystyle=\begin{bmatrix}-L_{1}^{(1)}+P_{1}&0&\cdots&0\\ 0&-L_{2}^{(1)}+P_{2}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&-L_{N-1}^{(1)}(t)+P_{N-1}\\ \end{bmatrix}\tilde{X}(t)$		(37)
		$\displaystyle\triangleq\left(\textbf{L}+\textbf{P}(t)\right)\tilde{X}(t),$		(37)

where $\textbf{L}=\mathrm{diag}\left(-L_{1}^{(1)},-L_{2}^{(1)},\cdots,-L_{N-1}^{(1)}\right)$ , $\textbf{P}(t)=\mathrm{diag}\left(P_{1},P_{2},\cdots,P_{N-1}\right)$ with $P_{n}=L_{n}^{(1)}-\left(F(t)+F^{*}(t)\right)|\kappa_{n}|^{2}$ . The dynamics of Eq. (37) has been analyzed in Ref. [68]. We denote $\tilde{X}(t)=\Phi\left(t,t_{0}\right)\tilde{X}(t_{0})$ for $t\geq t_{0}$ , and

\Phi\left(t,t_{0}\right)=e^{\textbf{L}\left(t-t_{0}\right)}+\int_{t_{0}}^{t}e^{\textbf{L}\left(t-\tau\right)}\textbf{P}(\tau)\Phi\left(\tau,t_{0}\right)\mathrm{d}\tau,

then we can derive the following results.

Lemma 2.

When the open system converges to a Markvoian regime, the time-varying components in Eq. (37) converges to zero, and the transition matrix is bounded.

Proof.

The convergence of quantum dynamics to a Markovian regime means the convergence of $F(t)$ according to Definition 1. Consequently, the time-varying components in Eq. (37) converge to zero. In this case, $\lim_{t\rightarrow\infty}\textbf{P}(t)=\textbf{0}_{(N-1)\times(N-1)}$ . Thus $\Phi\left(t,t_{0}\right)$ is bounded because $\lim_{t\rightarrow\infty}\frac{\partial\Phi\left(t,t_{0}\right)}{\partial t}=0$ . ∎

Obviously, when $\textbf{P}(t)\equiv\textbf{0}_{(N-1)\times(N-1)}$ , Eq. (37) reduces to a linear time-invariant system and is exponentially stable because L is a diagonal matrix whose eigenvalues are all negative. Then $\tilde{X}$ can be solved as

\displaystyle\tilde{X}(t)=e^{\textbf{L}t}\tilde{X}(0)+\int_{0}^{t}e^{\textbf{L}(t-\tau)}\textbf{P}(\tau)\tilde{X}(\tau)\mathrm{d}\tau,

(38)

where the non-Markovian interaction between the atom and the environment can influence the integration kernel in Eq. (38), and further influence the convergence rate and steady value of $\tilde{X}$ . The dynamics of the equation with the format of Eq. (38) has been introduced in Ref. [69]. Generalized from Theorem 2 in Ref. [69] (Chapter 2, Page 36), we have the following proposition.

Proposition 8.

Provided that $\|\textbf{P}(t)\|\leq c_{1}$ for $t\geq t_{0}$ , where $c_{1}$ is determined by $\max(L_{n}^{(1)})$ , it follows that $\lim_{t\rightarrow\infty}\tilde{X}(t)=0$ in Eq. (38).

Proof.

The proof is similar to that in Ref. [69] and is thus omitted due to page limitations. ∎

Proposition 8 means that when the interaction between the atom and environment converges to a Markovian regime, $\tilde{X}(t)$ will converge to zero. Besides, Proposition 8 is further generalized to linear time-varying system in Ref. [69], and this generalization will be used in the following analysis.

III-B $\delta\neq 0$ , $g_{n}\neq 0$

In the following, we consider the most general case by separating $X(t)$ by its real and imaginary parts. Considering that in Eq. (9), Eq. (9a) and Eq. (9d) are always real, Eq. (9b) and Eq. (9c) are conjugate complex values, then we rewrite the state vector in a simplified format as

\mathbb{X}(t)=\left[\mathbb{X}_{1}(t),\mathbb{X}_{2}(t),\cdots,\mathbb{X}_{N-1}(t),a^{R}\right]^{T},

with the dimension $3N-2$ , where $\mathbb{X}_{n}(t)=\left[X_{n}^{R}[1],X_{n}^{R}[2],X_{n}^{I}[2]\right]$ , $X_{n}^{R}$ and $X_{n}^{I}$ , $a^{R}$ and $a^{I}$ represent the real and imaginary parts of $X_{n}(t)$ and $\langle a^{{\dagger}}a\rangle$ in Eq. (11) respectively, $X_{n}^{R}[1]$ represents the first element of the vector $X_{n}^{R}$ , and similar for other elements in the vector $\mathbb{X}(t)$ . $X_{n}^{I}[1]=a^{I}=0$ , $X_{n}^{R}[3]=X_{n}^{R}[2]$ and $X_{n}^{I}[3]=-X_{n}^{I}[2]$ . Then the evolution of $\mathbb{X}(t)$ reads

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathbb{X}$	$\displaystyle=\begin{bmatrix}\mathbb{A}_{1}(t)&0&\cdots&0&\mathbb{D}_{1}(t)\\ 0&\mathbb{A}_{2}(t)&\cdots&0&\mathbb{D}_{2}(t)\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&\mathbb{A}_{N-1}(t)&\mathbb{D}_{N-1}(t)\\ \mathbb{C}_{1}(t)&\mathbb{C}_{2}(t)&\cdots&\mathbb{C}_{N-1}(t)&0\end{bmatrix}\mathbb{X}$		(39)
		$\displaystyle\triangleq\mathbb{A}(t)\mathbb{X},$		(39)

which is a real-value equation. We take the $n$ -th energy level as an example, $\frac{\mathrm{d}}{\mathrm{d}t}\mathbb{X}_{n}(t)=\mathbb{A}_{n}(t)\mathbb{X}_{n}(t)+\mathbb{D}_{n}(t)a^{R}$ , and

\displaystyle\mathbb{A}_{n}(t)=\begin{bmatrix}-2R_{f}(t)&2g_{n}\sin\delta t&2g_{n}\cos\delta t\\ -g_{n}\sin\delta t&-R_{f}(t)&I_{f}(t)-\delta\\ -g_{n}\cos\delta t&\delta-I_{f}(t)&-R_{f}(t)\\ \end{bmatrix},

(40)

\displaystyle\mathbb{C}_{n}(t)

\displaystyle=\begin{bmatrix}0&-2g_{n}\sin\delta t&-2g_{n}\cos\delta t\end{bmatrix},

(41)

\displaystyle\mathbb{D}_{n}(t)

\displaystyle=\begin{bmatrix}0\\ -g_{n}\sin\delta t\\ -g_{n}\cos\delta t\end{bmatrix},

(42)

where we take $F(t)|\kappa_{n}|^{2}\triangleq R_{f}(t)+iI_{f}(t)$ , $F^{*}(t)|\kappa_{n}|^{2}\triangleq R_{f}(t)-iI_{f}(t)$ for simplification.

In summary, when there are detunings between the atom and cavity, the dynamics of atomic states and the photon number in the cavity will always be linear and time-varying.

Upon this, when the interaction between the atom and the environment becomes asymptotically Markovian, we denote $\lim_{t\rightarrow\infty}R_{f}(t)=\bar{R}_{f}$ and $\lim_{t\rightarrow\infty}I_{f}(t)=\bar{I}_{f}$ . Then Eq. (40) becomes

$\displaystyle\mathbb{A}_{n}(t)=$	$\displaystyle\begin{bmatrix}-2\bar{R}_{f}&2g_{n}\sin\delta t&2g_{n}\cos\delta t\\ -g_{n}\sin\delta t&-\bar{R}_{f}&\bar{I}_{f}-\delta\\ -g_{n}\cos\delta t&\delta-\bar{I}_{f}&-\bar{R}_{f}\\ \end{bmatrix}$	(43)
	$\displaystyle+\begin{bmatrix}2\bar{R}_{f}(t)-2R_{f}(t)&0&0\\ 0&\bar{R}_{f}-R_{f}(t)&I_{f}(t)-\bar{I}_{f}\\ 0&\bar{I}_{f}-I_{f}(t)&\bar{R}_{f}-R_{f}(t)\\ \end{bmatrix}$
$\displaystyle=$	$\displaystyle\bar{\mathbb{A}}_{n}(t)+\tilde{\mathbb{A}}_{n}(t),$

where $\bar{\mathbb{A}}_{n}$ represents the first matrix in Eq. (43) and $\tilde{\mathbb{A}}_{n}$ represents the second matrix. Obviously, $\bar{\mathbb{A}}_{n}(t)$ is periodic with $\bar{\mathbb{A}}_{n}(t)=\bar{\mathbb{A}}_{n}(t+T_{\delta}),$ where $T_{\delta}=\frac{2\pi}{\delta}$ , and the norm of $\tilde{\mathbb{A}}_{n}(t)$ finally converges to zero.

To clarify the stability of the above linear time-varying system, we first introduce the following definition and propositions.

Definition 4 ([70]).

A linear time varying system is uniformly exponential stable (UES) if and only if there exist positive constants $K$ and $\tilde{\alpha}$ such that

\|\Phi\left(t,\tau\right)\|\leq Ke^{-\tilde{\alpha}\left(t-\tau\right)},for~{}t_{0}\leq\tau<t<\infty.

Definition 5 ([70]).

For a real-valued continuous matrix $A(t)\in\textbf{R}^{n\times n}$ , the logarithmic norm can be defined when $t\geq 0$ as

\mu[A(t)]=\lim_{h\rightarrow 0^{+}}\frac{\|I_{n}+hA(t)\|-1}{h},

and

\displaystyle\Pi^{+}(t)\triangleq\int_{t_{0}}^{t}\mu[A(\tau)]\mathrm{d}\tau.

(44)

As a combination of the conclusion in Ref. [70] on the relationship between $\Pi^{+}(t)$ and the stability of a periodic linear time-varying system, and the generalization of Proposition 8 to the linear time-varying system in Ref. [69], we can derive the following stability proposition for the linear time-varying system in Eq. (39) with the time-varying matrix in Eq. (43).

Proposition 9.

For the linear time-varying system with the matrix represented as a combination of time-varying periodic and time-varying converging matrices, as given in Eq. (43), the solutions approach zero when
(a) $\bar{\Pi}^{+}(t_{0}+T)=\int_{t_{0}}^{t_{0}+T}\mu\left[\mathbb{A}(\tau)\right]\mathrm{d}\tau<0$ ,
(b) $\int_{0}^{\infty}\|\tilde{\mathbb{A}}_{n}(t)\|\mathrm{d}t<\infty$ for arbitrary $n$ .

Proof.

The proof is based on two separated parts as discussed in Refs. [70, 69]. According to Ref. [70], when (a) is satisfied, the linear time-varying system $\dot{\hat{\mathbb{X}}}=\bar{\mathbb{A}}_{n}(t)\hat{\mathbb{X}}$ is uniformly exponential stable, and $\lim_{t\rightarrow\infty}\hat{\mathbb{X}}=0$ . Based on this, when (b) is satisfied, the state vector in Eq. (39) approaches zero, as shown in Ref. [69] (Chapter 2, Theorem 4). ∎

III-C With external drives

When there is a drive field with the amplitude $\mathcal{E}$ applied upon the cavity, the Hamiltonian $H_{t}$ in Eq. (2) should be replaced by

\displaystyle H_{t}^{\prime}

\displaystyle=H_{t}-i\mathcal{E}\left(a-a^{{\dagger}}\right).

(45)

Then the evolution of the mean values of operators reads

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\langle\sigma^{+}_{n}a\rangle=$	$\displaystyle-i\delta\langle\sigma^{+}_{n}a\rangle-ig_{n}e^{-i\delta t}\langle\sigma^{+}_{n}\sigma^{-}_{n}\rangle+ig_{n}e^{-i\delta t}\langle a^{{\dagger}}a\rangle$		(46)
		$\displaystyle-F^{*}(t)\|\kappa_{n}\|^{2}\langle\sigma^{+}_{n}a\rangle+\mathcal{E}\left\langle\sigma^{+}_{n}\right\rangle,$		(46)

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\langle a^{{\dagger}}a\rangle=$	$\displaystyle i\sum_{n}g_{n}\left(e^{i\delta t}\langle\sigma^{+}_{n}a\rangle-e^{-i\delta t}\langle\sigma^{-}_{n}a^{{\dagger}}\rangle\right)$		(47)
		$\displaystyle+\mathcal{E}\langle a+a^{{\dagger}}\rangle,$		(47)

\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left\langle\sigma^{-}_{n}\right\rangle=-ig_{n}e^{i\delta t}\langle a\rangle-F(t)|\kappa_{n}|^{2}\left\langle\sigma^{-}_{n}\right\rangle.

(48)

Obviously, the drive field directly influences the evolution of the operators containing the cavity operator. We further denote $R_{a}=\left\langle a+a^{{\dagger}}\right\rangle$ , and $I_{a}=i\left\langle a-a^{{\dagger}}\right\rangle$ with real values, then

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}R_{a}$	$\displaystyle=-i\sum_{n}g_{n}e^{-i\delta t}\left\langle\sigma^{-}_{n}\right\rangle+i\sum_{n}g_{n}e^{i\delta t}\left\langle\sigma^{+}_{n}\right\rangle+2\mathcal{E}$		(49)
		$\displaystyle=2\sum_{n=1}^{N-1}g_{n}\left(s_{n}^{I}\cos\delta t-s_{n}^{R}\sin\delta t\right)+2\mathcal{E},$		(49)

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}I_{a}$	$\displaystyle=\sum_{n}g_{n}e^{-i\delta t}\left\langle\sigma^{-}_{n}\right\rangle+\sum_{n}g_{n}e^{i\delta t}\left\langle\sigma^{+}_{n}\right\rangle$		(50)
		$\displaystyle=2\sum_{n=1}^{N-1}g_{n}\left(s_{n}^{R}\cos\delta t+s_{n}^{I}\sin\delta t\right).$		(50)

where $s_{n}^{R}$ and $s_{n}^{I}$ represent the real and imaginary parts of $\left\langle\sigma^{-}_{n}\right\rangle$ , respectively.

Generalized from the state vector $\mathbb{X}(t)$ in Eq. (39), we define another real-valued vector $\tilde{\mathbb{X}}$ as

\tilde{\mathbb{X}}(t)=\left[\tilde{\mathbb{X}}_{1}(t),\tilde{\mathbb{X}}_{2}(t),\cdots,\tilde{\mathbb{X}}_{N-1}(t),a^{R},R_{a},I_{a}\right]^{T},

where $\tilde{\mathbb{X}}_{n}(t)$ is generalized from $\mathbb{X}_{n}(t)$ in Eq. (39) as

\tilde{\mathbb{X}}_{n}(t)=\left[X_{n}^{R}(1),X_{n}^{R}(2),X_{n}^{I}(2),s_{n}^{R},s_{n}^{I}\right],

and $a^{R}$ has been defined in Eq. (39). Then the evolution of $\tilde{\mathbb{X}}(t)$ reads

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathbb{X}}$	$\displaystyle=\begin{bmatrix}\mathbb{B}_{1}(t)&0&\cdots&0&\tilde{\mathbb{D}}_{1}(t)\\ 0&\mathbb{B}_{2}(t)&\cdots&0&\tilde{\mathbb{D}}_{2}(t)\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&\mathbb{B}_{N-1}(t)&\tilde{\mathbb{D}}_{N-1}(t)\\ \tilde{\mathbb{C}}_{1}(t)&\tilde{\mathbb{C}}_{2}(t)&\cdots&\tilde{\mathbb{C}}_{N-1}(t)&\tilde{\mathbb{P}}\end{bmatrix}\tilde{\mathbb{X}}+2\mathcal{E}\begin{bmatrix}0\\ 0\\ \vdots\\ 1\\ 0\end{bmatrix}$		(51)
		$\displaystyle\triangleq\mathbb{B}(t)\tilde{\mathbb{X}}+2\mathcal{E}\begin{bmatrix}0\\ 0\\ \vdots\\ 1\\ 0\end{bmatrix},$		(51)

where

\displaystyle\mathbb{B}_{n}(t)=\begin{bmatrix}-2R_{f}(t)&2g_{n}\sin\delta t&2g_{n}\cos\delta t&0&0\\ -g_{n}\sin\delta t&-R_{f}(t)&I_{f}(t)-\delta&\mathcal{E}&0\\ -g_{n}\cos\delta t&\delta-I_{f}(t)&-R_{f}(t)&0&-\mathcal{E}\\ 0&0&0&-R_{f}(t)|\kappa_{n}|^{2}&I_{f}(t)|\kappa_{n}|^{2}\\ 0&0&0&-I_{f}(t)|\kappa_{n}|^{2}&-R_{f}(t)|\kappa_{n}|^{2}\end{bmatrix},

(52)

\displaystyle\tilde{\mathbb{C}}_{n}(t)

\displaystyle=\begin{bmatrix}0&-2g_{n}\sin\delta t&-2g_{n}\cos\delta t&0&0\\ 0&0&0&-2g_{n}\sin\delta t&2g_{n}\cos\delta t\\ 0&0&0&2g_{n}\cos\delta t&2g_{n}\sin\delta t\end{bmatrix},

(53)

\displaystyle\tilde{\mathbb{D}}_{n}(t)

\displaystyle=\begin{bmatrix}0&0&0\\ -g_{n}\sin\delta t&0&0\\ -g_{n}\cos\delta t&0&0\\ 0&\frac{g_{n}\sin\delta t}{2}&-\frac{g_{n}\cos\delta t}{2}\\ 0&-\frac{g_{n}\cos\delta t}{2}&\frac{g_{n}\sin\delta t}{2}\end{bmatrix},

(54)

and

\displaystyle\tilde{\mathbb{P}}

\displaystyle=\begin{bmatrix}0&\mathcal{E}&0\\ 0&0&0\\ 0&0&0\end{bmatrix}.

(55)

Above all, when there are no external drives, namely $\mathcal{E}=0$ , the initially excited atom can decay to the ground state and the cavity will finally be empty, which can also be interpreted with the quantum system’s exponential stability similarly as in Ref. [23] by regarding the components outside of the cavity as an environment. However, when $\mathcal{E}>0$ , the exponential stability explaining the atom and cavity’s decaying can be destructed, especially when $\mathcal{E}$ is large compared with the non-Markovian exchanging rate between the atom and the environment.

We take the two-level atom as an example, to clarify how the dynamics can be influenced by the non-Markovian environment, coupling strength between the atom and cavity, and the external drive applied on the cavity, which are compared in Fig. 4. In all the four simulations, $\omega_{a}=50$ , $\delta=10$ , and $\Omega=45$ , $\kappa_{1}=4$ , $\chi=1$ , $\gamma=1$ for the non-Markvoian parameter settings, which is the same as that in solid lines of Fig. 3. In Fig. 4(a-b), $\mathcal{E}=0.02$ , and in (c-d), $\mathcal{E}=0.2$ . When the drive applied on the cavity is weak, as in (a-b), the atom can decay to the ground state and the cavity is finally almost empty, because of the non-Markovian Lindblad components. However, when $\mathcal{E}$ is larger and the coupling between the atom and cavity is not so strong, then there can be multiple photons in the cavity, as in (c). When both $\mathcal{E}$ and $g_{1}$ are larger, the exchanging between the atom and cavity is faster, and the atom and cavity states are both oscillating apart from decaying to the environment, thus the population of the excited atom and photon numbers in the cavity are not exponentially stable, as shown in (d).

IV Quantum measurement feedback control in the non-Markovian cavity-QED system

When the quantum system is measured via the cavity output as in Fig. 1, the feedback control can be designed according to the measurement result [71]

\displaystyle I_{c}(t)

\displaystyle=\sqrt{2}\langle x\rangle+\frac{1}{\sqrt{\eta}}\xi(t),

(56)

where $x=\left(a+a^{{\dagger}}\right)/\sqrt{2}$ , $\xi(t)$ is the measurement-induced time-independent random term satisfies that $\langle\xi(t)\rangle=0$ and $\langle\xi(t)\xi(t^{\prime})\rangle=\delta(t-t^{\prime})$ , and $\eta$ is the measurement detection efficiency. Then the feedback Hamiltonian reads

\displaystyle H_{\rm fb}(t)

\displaystyle=GI_{c}(t)=G\left(\sqrt{2}\langle x\rangle+\frac{1}{\sqrt{\eta}}\xi(t)\right),

(57)

where $G=G^{{\dagger}}$ is the feedback operator.

Assumption 6.

Assume that the detection efficiency of the measurement apparatus is ideal as $\eta=1$ .

The Markovian stochastic master equation with measurement feedback can be rewritten as [34, 72]

	$\displaystyle\mathrm{d}\rho$	$\displaystyle=-i[H,\rho(t)]\mathrm{d}t+\kappa\mathcal{L}_{a}\left[\rho(t)\right]\mathrm{d}t-ig_{f}\left[G,a\rho(t)+\rho(t)a^{{\dagger}}\right]\mathrm{d}t$		(58)
		$\displaystyle~{}~{}~{}~{}-\frac{1}{2}g_{f}^{2}\left[G,\left[G,\rho(t)\right]\right]\mathrm{d}t+\mathcal{H}[a]\rho(t)dW-ig_{f}\left[G,\rho(t)\right]\mathrm{d}W,$		(58)

where $\mathcal{L}_{a}$ represents the decaying of the cavity with the rate $\kappa$ , $\mathcal{L}_{O}\left[\rho\right]=O\rho O^{{\dagger}}-\frac{1}{2}\rho O^{{\dagger}}O-\frac{1}{2}O^{{\dagger}}O\rho$ for an arbitrary operator $O$ , $\mathcal{H}[O]\rho=O\rho(t)+\rho(t)O^{{\dagger}}-\rm{Tr}\left[\right(O+O^{{\dagger}}\left)\rho(t)\right]\rho(t)$ , $g_{f}$ is the feedback strength, $\mathrm{d}W(t)=\xi(t)\mathrm{d}t$ is a Wiener process representing the Homodyne detection noise in the quantum measurement, $E\left[\mathrm{d}W(t)^{2}\right]=\mathrm{d}t$ , and $\xi(t)$ can be approximated with the white noise [73].

Generalized from Eq. (58), the stochastic quantum measurement feedback control equation when there are non-Markovian interactions between the atom and environment reads

$\displaystyle\mathrm{d}\rho=$	$\displaystyle\left\{-i[\bar{H}+H_{D},\rho(t)]+\left(F^{*}(t)+F(t)\right)L\rho(t)L^{{\dagger}}-F(t)L^{{\dagger}}L\rho(t)\right.$	(59)
	$\displaystyle\left.-F^{*}(t)\rho(t)L^{{\dagger}}L+\kappa\mathcal{L}_{a}\left[\rho(t)\right]\right\}\mathrm{d}t-ig_{f}\left[G,a\rho(t)+\rho(t)a^{{\dagger}}\right]\mathrm{d}t$
	$\displaystyle-\frac{1}{2}g_{f}^{2}\left[G,\left[G,\rho(t)\right]\right]\mathrm{d}t+\mathcal{H}[a]\rho(t)dW-ig_{f}\left[G,\rho(t)\right]\mathrm{d}W,$

where $H_{D}$ represents external drives applied upon the cavity, $F(t)$ is for the non-Markovian interaction between the atom and the environment, and the other components have the same meaning as in Eq. (58).

In the following, we study two circumstances. One is that the feedback operator $G$ is an atomic operator, the other is that $G$ is a cavity operator.

IV-A Feedback control with atomic operators

The feedback operator for the $N$ -level system can be represented as

\displaystyle G=\sum_{i\neq j}G_{ij}=\sum_{i\neq j}\left(|i\rangle\langle j|+|j\rangle\langle i|\right),

(60)

where $1\leq i,j\leq N-1$ . In the following, we take the two-level atom case as an example.

IV-A1 Example

When $N=2$ , we choose $G=\left(|0\rangle\langle 1|+|1\rangle\langle 0|\right)$ , and assume initially the atom is excited. The atomic state can be evaluated with the vector of the operators as $S=\left[\langle a\rangle,\langle\sigma_{-}\rangle,\langle\sigma_{z}\rangle\right]^{T}\triangleq\left[\bar{\alpha},\bar{s},\bar{w}\right]^{T}$ , where $\sigma_{-}=|0\rangle\langle 1|$ and $\sigma_{z}=|1\rangle\langle 1|-|0\rangle\langle 0|$ . When the external drive is $H_{D}=-i\mathcal{E}\left(a-a^{{\dagger}}\right)$ , the dynamics of the cavity or atomic operator reads


		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\bar{\alpha}=-ig_{1}e^{-i\delta t}\bar{s}-\frac{\kappa}{2}\bar{\alpha}+\mathcal{E},$			(61a)
		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\bar{s}=ig_{1}e^{i\delta t}\bar{\alpha}\bar{w}-F(t)\|\kappa_{1}\|^{2}\bar{s}+ig_{f}\left(\bar{\alpha}+\bar{\alpha}^{*}\right)\bar{w}$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-g_{f}^{2}\left(\bar{s}-\bar{s}^{*}\right)+ig_{f}\bar{w}\xi(t),$			(61b)
		$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\bar{w}=-2ig_{1}e^{i\delta t}\bar{s}^{}\bar{\alpha}+2ig_{1}e^{-i\delta t}\bar{s}\bar{\alpha}^{}$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left(F(t)+F^{}(t)\right)\|\kappa_{1}\|^{2}\left(\bar{w}+1\right)-2ig_{f}\left(\bar{s}^{}-\bar{s}\right)\left(\bar{\alpha}+\bar{\alpha}^{*}\right)$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}-2g_{f}^{2}\bar{w}-2ig_{f}\left(\bar{s}^{*}-\bar{s}\right)\xi(t),$			(61c)

with the initially normalization condition that $\bar{w}^{2}+4\left|\bar{s}\right|^{2}=1$ [74], and the stochastic component with higher order amplitude is omitted in Eq. (61a). We consider the simplified steady states satisfying that $\dot{\bar{\alpha}}=\dot{\bar{s}}=\dot{\bar{w}}=0$ after averaging the measurement noise $\xi(t)=0$ , and take $\delta=0$ for simplification.

The performance for different non-Markovian parameter settings is compared in Fig. 5, where we take $\kappa_{1}=1$ , $\chi=0.5$ , $\omega_{a}=50$ , $\Omega=45$ , $\mathcal{E}=0.1$ , $g_{1}=0.2$ , $\delta=0$ , $\kappa=0.2$ and $\mathrm{d}t=0.01$ . It can be seen that the measurement feedback control can enhance the probability that the atom is excited, both when the interface between atom and environment is Markovian and non-Markovian.

IV-B Feedback control with cavity operators

Additionally, the feedback operator can be designed according to the cavity operator as [75]

\displaystyle G

\displaystyle=\beta_{x}x+\beta_{p}p=\frac{\beta_{x}-i\beta_{p}}{\sqrt{2}}a+\frac{\beta_{x}+i\beta_{p}}{\sqrt{2}}a^{{\dagger}},

(62)

where the position operator $x$ is given by Eq. (56) and the momentum operator $p=i\left(a^{{\dagger}}-a\right)/\sqrt{2}$ with $xp-px=i$ . Besides, Eq. (62) is in the rotating frame with respect to the feedback driving frequency $\omega_{d}$ from the original format $\tilde{G}=\frac{\beta_{x}-i\beta_{p}}{\sqrt{2}}ae^{i\omega_{d}t}+\frac{\beta_{x}+i\beta_{p}}{\sqrt{2}}a^{{\dagger}}e^{-i\omega_{d}t}$ [76]. Denote $\Delta=\omega_{c}-\omega_{d}$ , then the dynamics of the mean-value of the operators $x$ and $p$ can be derived based on Eq. (59) without $H_{D}$ as


		$\displaystyle\langle\dot{x}\rangle=\Delta\langle p\rangle-\frac{\kappa}{2}\langle x\rangle-i\sum_{n}\frac{g_{n}e^{-i\delta t}}{\sqrt{2}}\left\langle\sigma^{-}_{n}\right\rangle+i\sum_{n}\frac{g_{n}e^{i\delta t}}{\sqrt{2}}\left\langle\sigma^{+}_{n}\right\rangle$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\sqrt{2}g_{f}\beta_{p}\langle x\rangle+\sqrt{2}\left(V_{x}-\frac{1}{2}\right)\xi(t)+g_{f}\beta_{p}\xi(t),$			(63a)
		$\displaystyle\langle\dot{p}\rangle=-\Delta\langle x\rangle-\frac{\kappa}{2}\langle p\rangle-\sum_{n}\frac{g_{n}e^{-i\delta t}}{\sqrt{2}}\left\langle\sigma^{-}_{n}\right\rangle-\sum_{n}\frac{g_{n}e^{i\delta t}}{\sqrt{2}}\left\langle\sigma^{+}_{n}\right\rangle$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}-\sqrt{2}g_{f}\beta_{x}\langle x\rangle+\sqrt{2}V_{xp}\xi(t)-g_{f}\beta_{x}\xi(t),$			(63b)
		$\displaystyle\langle\dot{\sigma}^{-}_{n}\rangle=-ig_{n}e^{i\delta t}\langle a\rangle-F(t)\|\kappa_{n}\|^{2}\langle\sigma^{-}_{n}\rangle,$			(63c)
		$\displaystyle\langle\dot{a}\rangle=-i\Delta\langle a\rangle-i\sum_{n}g_{n}e^{-i\delta t}\langle\sigma^{-}_{n}\rangle-\frac{\kappa}{2}\langle a\rangle-ig_{f}\left(\beta_{x}+i\beta_{p}\right)\langle x\rangle$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\left(\left\langle\left(a+a^{{\dagger}}\right)a\right\rangle-\left\langle a+a^{{\dagger}}\right\rangle\left\langle a\right\rangle\right)\xi(t)-\frac{ig_{f}}{\sqrt{2}}\left(\beta_{x}+i\beta_{p}\right)\xi(t),$			(63d)

where $V_{x}$ is the variance of the position operator as $V_{x}=\left\langle x^{2}\right\rangle-\langle x\rangle^{2}$ , $V_{xp}$ is the covariance between position and momentum as $V_{xp}=\left(\langle xp\rangle+\langle px\rangle\right)/2-\langle x\rangle\langle p\rangle$ [75], and $F(t)$ is governed by Eq. (9e).

Then based on the noise components in Eqs. (63a) and (63b), we can derive the following proposition on the relationship between feedback control and the effct of noise in quantum dynamics.

Proposition 10.

The feedback with the cavity operator can cancel the noise when $g_{f}\beta_{p}=-\sqrt{2}\left(V_{x}-1/2\right)$ and $g_{f}\beta_{x}=\sqrt{2}V_{xp}$ .

Proof.

The proposition can be easily derived by the noise components in Eqs. (63a) and (63b), thus further details are omitted. ∎

Besides, $V_{x}$ and $V_{xp}$ are governed by the following nonlinear equation [77, 75, 78, 79]


		$\displaystyle\dot{V}_{x}=2\Delta V_{xp}-\kappa V_{x}^{\prime}+2\sqrt{2}g_{f}\beta_{p}V_{x}^{\prime}+g_{f}^{2}\beta_{p}^{2}-\left(\sqrt{2}V_{x}^{\prime}+g_{f}\beta_{p}\right)^{2},$			(64a)
		$\displaystyle\dot{V}_{xp}=\Delta V_{p}-\Delta V_{x}-\kappa V_{xp}+\sqrt{2}g_{f}\beta_{p}V_{xp}-\sqrt{2}g_{f}\beta_{x}V_{x}$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\sqrt{2}}{2}g_{f}\beta_{x}-g_{f}^{2}\beta_{x}\beta_{p}-\left(\sqrt{2}V_{xp}-g_{f}\beta_{x}\right)\left(\sqrt{2}V_{x}^{\prime}+g_{f}\beta_{p}\right),$			(64b)
		$\displaystyle\dot{V}_{p}=-2\Delta V_{xp}-\kappa V_{P}+\frac{\kappa}{2}-2\sqrt{2}g_{f}\beta_{x}V_{xp}+g_{f}^{2}\beta_{x}^{2}$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}-\left(\sqrt{2}V_{xp}-g_{f}\beta_{x}\right)^{2},\$			(64c)

where we denote $V_{x}^{\prime}=V_{x}-1/2$ , the last component of Eqs. (64a), (64b), and (64c) comes from the stochastic components in $\left(\mathrm{d}\langle x\rangle\right)^{2}$ , $\mathrm{d}\langle x\rangle\mathrm{d}\langle p\rangle,$ and $\left(\mathrm{d}\langle p\rangle\right)^{2}$ , respectively.

Remark 5.

Proposition 10 as well as Eq. (64) reveals that the final steady states and the covariance can be modulated by choosing the measurement operator in Eq. (62) with tunable $\beta_{x}$ , $\beta_{p}$ , and the feedback control field.

For the open loop control without feedback, Eq. (64) can finally converge to the steady values $V_{x}(\infty)=V_{p}(\infty)=\frac{1}{2}$ and $V_{xp}(\infty)=0$ , similar as in Ref. [79].

For arbitrary $\beta_{x}$ , $\beta_{p}$ and $g_{f}$ , Eq. (64) can be simplified as


		$\displaystyle\dot{V}_{x}=2\Delta V_{xp}-\kappa V_{x}^{\prime}-2V_{x}^{\prime 2},$			(65a)
		$\displaystyle\dot{V}_{xp}=\Delta V_{p}-\Delta V_{x}-\kappa V_{xp}-2V_{x}^{\prime}V_{xp},$			(65b)
		$\displaystyle\dot{V}_{p}=-2\Delta V_{xp}-\kappa V_{P}^{\prime}-2V_{xp}^{2},$			(65c)

where $V_{p}^{\prime}=V_{p}-1/2$ similar as the definition of $V_{x}^{\prime}$ . It can be seen that the steady values for valience can be influenced by detuning $\Delta$ of the feedback field. Obviously, when $\Delta=0$ , $V_{xp}(\infty)=0$ , $V_{x}(\infty)=V_{p}(\infty)=1/2$ , which reduces to the circumstance in Ref. [79].

V Quantum control for the multiple Jaynes-Cummings model

When there are overall $M$ nearest neighbor coupled cavities with one two-level atom in each cavity, the free Hamiltonian for the atoms and cavities reads

\displaystyle H_{M}

\displaystyle=\sum_{m=1}^{M}\omega_{c}^{(m)}a^{{\dagger}}_{m}a_{m}+\sum_{m=1}^{M}\omega_{a}^{(m)}\sigma_{+}^{(m)}\sigma_{-}^{(m)},

(66)

where $a_{m}$ ( $a^{{\dagger}}_{m}$ ) is the annihilation (creation) operator for the $m$ -th cavity with the resonant frequency $\omega_{c}^{(m)}$ , $\sigma_{-}^{(m)}$ ( $\sigma_{+}^{(m)}$ ) represents the lowering (raising) operator for the two-level atom with frequency $\omega_{a}^{(m)}$ in the $m$ -th cavity, and the detunings $\delta_{m}=\omega_{a}^{(m)}-\omega_{c}^{(m)}$ .

Assumption 7.

Assume the resonant frequencies and decay rates of the cavities are identical as $\omega_{c}^{(1)}=\cdots=\omega_{c}^{(M)}=\omega_{c}$ , $\kappa_{1}=\cdots=\kappa_{M}=\kappa$ , while the resonant frequency of atoms $\omega_{a}^{(m)}$ in different cavities can be different. Initially the cavity is not empty such that $\left\langle a_{m}\right\rangle\neq 0$ and $\left\langle\sigma_{-}^{(m)}\right\rangle\neq 0$ .

The interaction Hamiltonian among the atoms and cavities reads

	$\displaystyle H_{M}$	$\displaystyle=\sum_{m=1}^{M}g_{m}\left(e^{-i\delta_{m}t}\sigma_{-}^{(m)}a^{{\dagger}}_{m}+e^{i\delta_{m}t}\sigma_{+}^{(m)}a_{m}\right)$		(67)
		$\displaystyle~{}~{}~{}~{}+\sum_{m}J_{c}\left(a^{{\dagger}}_{m}a_{m+1}+a_{m}a^{{\dagger}}_{m+1}\right),$		(67)

where $g_{m}$ is the coupling strength between the atom and cavity in the $m$ -th cavity, $J_{c}$ is the coupling between two neighborhood cavities, the atom in each cavity is coupled with the environment via non-Markovian dynamics as

\displaystyle H_{A-E}

\displaystyle=\sum_{m=1}^{M}\sum_{\omega}\chi_{\omega}^{(m)}\left(\sigma_{+}^{(m)}b_{\omega}+\sigma_{-}^{(m)}b_{\omega}^{{\dagger}}\right),

(68)

and the atomic system is driven by the non-Markovian master equation as

	$\displaystyle\mathrm{d}\rho=$	$\displaystyle\left\{-i\left[H_{M},\rho(t)\right]+\sum_{m}\left[\left(F_{m}^{*}(t)+F_{m}(t)\right)\sigma_{-}^{(m)}\rho(t)\sigma_{+}^{(m)}\right.\right.$		(69)
		$\displaystyle\left.\left.-F_{m}(t)\sigma_{+}^{(m)}\sigma_{-}^{(m)}\rho(t)-F_{m}^{*}(t)\rho(t)\sigma_{+}^{(m)}\sigma_{-}^{(m)}+\kappa_{m}\mathcal{L}_{a_{m}}\left[\rho(t)\right]\right]\right\}\mathrm{d}t,$		(69)

where $\sigma_{-}^{(m)}=I_{1}\otimes\cdots\otimes I_{m-1}\otimes\sigma_{-}\otimes I_{m+1}\cdots\otimes I_{M}$ and

\frac{\mathrm{d}}{\mathrm{d}t}F_{m}(t)=\left|\kappa^{(m)}\right|^{2}F_{m}^{2}(t)-\left(\gamma+i\Omega-i\omega_{a}^{(m)}\right)F_{m}(t)+\frac{\gamma\chi}{2},

where $\kappa^{(m)}$ represents the coupling between the atom in the $m$ -th cavity and environment.

V-A Dynamics without drives

When there are no drives applied either to the cavities or atoms, generalized from Eq. (9), the open-loop dynamics in the $m$ -th cavity reads [76, 80]

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\langle a_{m}^{{\dagger}}a_{m}\rangle=$	$\displaystyle ig_{m}\left(e^{i\delta t}\langle\sigma_{+}^{(m)}a_{m}\rangle-e^{-i\delta t}\langle\sigma_{-}^{(m)}a^{{\dagger}}_{m}\rangle\right)-\kappa\langle a_{m}^{{\dagger}}a_{m}\rangle$		(70)
		$\displaystyle-iJ_{c}\langle a_{m}^{{\dagger}}a_{m+1}\rangle+iJ_{c}\langle a_{m}a_{m+1}^{{\dagger}}\rangle+iJ_{c}\langle a_{m-1}^{{\dagger}}a_{m}\rangle-iJ_{c}\langle a_{m-1}a_{m}^{{\dagger}}\rangle,$		(70)

\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\langle a_{m}a_{m+1}^{{\dagger}}\rangle=iJ_{c}\left(\langle a_{m}^{{\dagger}}a_{m}\rangle-\langle a_{m+1}^{{\dagger}}a_{m+1}\rangle\right).

(71)

Besides, the dynamics of the operators in the $m$ -th cavity $\langle\sigma_{+}^{(m)}a_{m}\rangle$ and $\langle\sigma_{-}^{(m)}a_{m}^{{\dagger}}\rangle$ are the same as that of $\langle\sigma^{+}_{n}a\rangle$ and $\langle\sigma^{-}_{n}a^{{\dagger}}\rangle$ in Eq. (9) respectively by taking $n=N-1=1$ and omitting the higher order components.

Denote $\mathbf{x}_{m}=\left[\langle\sigma_{-}^{(m)}\rangle,\langle a_{m}\rangle\right]^{T}$ , and $\tilde{\mathbf{X}}=\left[\mathbf{x}_{1}^{T},\mathbf{x}_{2}^{T},\cdots,\mathbf{x}_{M}^{T}\right]^{T}$ , then

	$\displaystyle\dot{\mathbf{x}}_{m}=$	$\displaystyle\begin{bmatrix}-F_{m}(t)\|\kappa^{(m)}\|^{2}&-ig_{m}e^{i\delta_{m}t}\\ -ig_{m}e^{-i\delta_{m}t}&-\kappa\end{bmatrix}\mathbf{x}_{m}$		(72)
		$\displaystyle+\begin{bmatrix}0\\ -iJ_{c}\end{bmatrix}\langle a_{m+1}\rangle+\begin{bmatrix}0\\ -iJ_{c}\end{bmatrix}\langle a_{m-1}\rangle.$		(72)

V-A1 Multi-cavity case

Based on Eq. (72), we define the real-valued vector $\mathbf{\mathcal{X}}_{M}$ representing the real and imaginary components of $\tilde{\mathbf{X}}$ as

\mathbf{\mathcal{X}}_{M}=\left[\mathbf{s}^{R}_{1},\mathbf{s}^{I}_{1},\cdots,\mathbf{s}^{R}_{M},\mathbf{s}^{I}_{M},\mathbf{a}^{R}_{1},\mathbf{a}^{I}_{1},\cdots,\mathbf{a}^{R}_{M},\mathbf{a}^{I}_{M}\right]^{T},

where $\mathbf{s}^{R}_{m}$ and $\mathbf{s}^{I}_{m}$ are the real and imaginary parts of $\langle\sigma_{-}^{(m)}\rangle$ , $\mathbf{a}^{R}_{m}$ and $\mathbf{a}^{I}_{m}$ are the real and imaginary parts of $\langle a_{m}\rangle$ , respectively, and $M\geq 2$ . Then

\displaystyle\dot{\mathbf{\mathcal{X}}}_{M}

\displaystyle=\begin{bmatrix}\mathbf{F}(t)&\mathbf{G}(t)\\ \mathbf{R}(t)&\mathbf{S}\end{bmatrix}\mathbf{\mathcal{X}}_{M}\triangleq\mathbf{\mathcal{A}}_{M}\mathbf{\mathcal{X}}_{M},

(73)

where

\displaystyle\mathbf{F}(t)=\begin{bmatrix}-\mathbf{\mathcal{A}}^{F}_{1}(t)|\kappa^{(1)}|^{2}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&-\mathbf{\mathcal{A}}^{F}_{M}(t)|\kappa^{(M)}|^{2}\end{bmatrix},

(74)

\displaystyle\mathbf{G}(t)=\begin{bmatrix}\mathbf{\mathcal{A}}^{G}_{1}(t)&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&\mathbf{\mathcal{A}}^{G}_{M}(t)\end{bmatrix},

(75)

\displaystyle\mathbf{R}(t)=\begin{bmatrix}\mathbf{\mathcal{A}}^{g}_{1}(t)&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&\mathbf{\mathcal{A}}^{g}_{M}(t)\end{bmatrix},

(76)

	$\displaystyle\mathbf{S}$	$\displaystyle=\begin{bmatrix}\mathbf{\mathcal{A}}_{\kappa}&\mathbf{\mathcal{A}}_{J}&0&\cdots&0\\ \mathbf{\mathcal{A}}_{J}&\mathbf{\mathcal{A}}_{\kappa}&\mathbf{\mathcal{A}}_{J}&\cdots&0\\ 0&\mathbf{\mathcal{A}}_{J}&\mathbf{\mathcal{A}}_{\kappa}&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&\mathbf{\mathcal{A}}_{\kappa}\end{bmatrix}$		(77)
		$\displaystyle=I_{M}\otimes\mathbf{\mathcal{A}}_{\kappa}+\begin{bmatrix}0&1&0&\cdots&0\\ 1&0&1&\cdots&0\\ 0&1&0&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&0\end{bmatrix}\otimes\mathbf{\mathcal{A}}_{J},$		(77)

where $\mathbf{\mathcal{A}}^{F}_{j}(t)=\begin{bmatrix}F^{R}_{j}(t)&-F^{I}_{j}(t)\\ F^{I}_{j}(t)&F^{R}_{j}(t)\end{bmatrix}$ , $\mathbf{\mathcal{A}}^{G}_{j}(t)=\begin{bmatrix}g_{j}\sin\left(\delta_{j}t\right)&g_{j}\cos\left(\delta_{j}t\right)\\ -g_{j}\cos\left(\delta_{j}t\right)&g_{j}\sin\left(\delta_{j}t\right)\end{bmatrix}$ , $\mathbf{\mathcal{A}}^{g}_{j}(t)=\begin{bmatrix}-g_{j}\sin\left(\delta_{j}t\right)&g_{j}\cos\left(\delta_{j}t\right)\\ -g_{j}\cos\left(\delta_{j}t\right)&-g_{j}\sin\left(\delta_{j}t\right)\end{bmatrix}$ , $\mathbf{\mathcal{A}}_{\kappa}=\begin{bmatrix}-\kappa&0\\ 0&-\kappa\end{bmatrix}$ and $\mathbf{\mathcal{A}}_{J}=\begin{bmatrix}0&J_{c}\\ -J_{c}&0\end{bmatrix}$ .

Remark 6.

When the interaction between the atom and the environment is Markovian or converges from non-Markovian to Markovian interactions, $F_{j}(t)$ are constants and $\mathbf{\mathcal{A}}_{M}$ in Eq. (73) is periodic as $\mathbf{\mathcal{A}}_{M}\left(t+T_{M}\right)=\mathbf{\mathcal{A}}_{M}\left(t\right)$ and $T_{M}=\frac{2\pi}{\delta_{1}}$ when $\delta_{1}=\cdots=\delta_{M}\neq 0$ . Further when $\delta_{1}=\cdots=\delta_{M}=0$ , $\mathbf{\mathcal{A}}_{M}\left(t\right)$ converges to be time invariant.

Remark 7.

The stability of the high-dimensional equation (73) can be similarly clarified by Proposition 9.

V-A2 An example with $M=2$

When there are two coupled cavities, the dynamics based on Markovian interaction between the quantum system and the environment has been analyzed in Ref. [80]. For the non-Markovian circumstance with $M=2$ , $\mathbf{X}=\left[\langle\sigma_{-}^{(1)}\rangle,\langle\sigma_{-}^{(2)}\rangle,\langle a_{1}\rangle,\langle a_{2}\rangle\right]^{T}$ , then

\displaystyle\dot{\mathbf{X}}=

\displaystyle\begin{bmatrix}-F_{1}(t)|\kappa^{(1)}|^{2}&0&-ig_{1}e^{i\delta_{1}t}&0\\ 0&-F_{2}(t)|\kappa^{(2)}|^{2}&0&-ig_{2}e^{i\delta_{2}t}\\ -ig_{1}e^{-i\delta_{1}t}&0&-\kappa&-iJ_{c}\\ 0&-ig_{2}e^{-i\delta_{2}t}&-iJ_{c}&-\kappa\end{bmatrix}\mathbf{X},

(78)

which can be interpreted as a time-varying linear system. Similar as in Section II, we separate the vector $\mathbf{X}$ by its real and imaginary parts as $\mathbf{\mathcal{X}}=\left[\mathbf{s}^{R}_{1},\mathbf{s}^{I}_{1},\mathbf{s}^{R}_{2},\mathbf{s}^{I}_{2},\mathbf{a}^{R}_{1},\mathbf{a}^{I}_{1},\mathbf{a}^{R}_{2},\mathbf{a}^{I}_{2}\right]^{T}$ , and denote $F_{j}(t)=F^{R}_{j}(t)+iF^{I}_{j}(t)$ , which is governed by Eq. (24) after replacing $F_{R}(t)$ and $F_{I}(t)$ with $F^{R}_{j}(t)$ and $F^{I}_{j}$ respectively, then Eq. (78) can be equivalently represented with the following real-value equation as

	$\displaystyle\dot{\mathbf{\mathcal{X}}}$	$\displaystyle=\begin{bmatrix}-\mathbf{\mathcal{A}}^{F}_{1}(t)\|\kappa^{(1)}\|^{2}&0&\mathbf{\mathcal{A}}^{G}_{1}(t)&0\\ 0&-\mathbf{\mathcal{A}}^{F}_{2}(t)\|\kappa^{(2)}\|^{2}&0&\mathbf{\mathcal{A}}^{G}_{2}(t)\\ \mathbf{\mathcal{A}}^{g}_{1}(t)&0&\mathbf{\mathcal{A}}_{\kappa}&\mathbf{\mathcal{A}}_{J}\\ 0&\mathbf{\mathcal{A}}^{g}_{2}(t)&\mathbf{\mathcal{A}}_{J}&\mathbf{\mathcal{A}}_{\kappa}\end{bmatrix}\mathbf{\mathcal{X}}$		(79)
		$\displaystyle\triangleq\mathbf{\mathcal{A}}\mathbf{\mathcal{X}},$		(79)

by taking $M=2$ in Eq. (73).

Firstly, denote $\mathbf{\mathcal{X}}_{\mathbf{s}}=\left[\mathbf{a}^{R}_{1},\mathbf{a}^{I}_{1},\mathbf{a}^{R}_{2},\mathbf{a}^{I}_{2}\right]^{T}$ , as a subspace of $\mathbf{\mathcal{X}}$ . Obviously, when $g_{m}=0$ for $m=1,2$ ,

\displaystyle\dot{\mathbf{\mathcal{X}}_{\mathbf{s}}}

\displaystyle=\begin{bmatrix}\mathbf{\mathcal{A}}_{\kappa}&\mathbf{\mathcal{A}}_{J}\\ \mathbf{\mathcal{A}}_{J}&\mathbf{\mathcal{A}}_{\kappa}\end{bmatrix}\mathbf{\mathcal{X}}_{\mathbf{s}}\triangleq\mathbf{\mathcal{A}}_{\mathbf{s}}\mathbf{\mathcal{X}}_{\mathbf{s}},

(80)

which is linear and time-invariant. $\mathbf{\mathcal{X}}_{\mathbf{s}}$ is exponentially stable because in Eq. (78) the real part of roots determined by $(s+\kappa)^{2}+J_{c}^{2}=0$ are negative, where $s$ is for Laplace transformation. Further by Proposition 8, $\lim_{t\rightarrow\infty}\mathbf{\mathcal{X}}_{\mathbf{s}}(t)=0$ . This can be further generalized to the circumstance that $M>2$ .

V-B Dynamics with drives

The drive field can be applied via the feedforward or feedback channel. Take the measurement feedback as an example, as a generalization of Section IV-B, the measurement information can be collected from an arbitrary cavity of the coupled cavity array, and the measurement information from the $m$ -th cavity reads

\displaystyle I_{c}^{(m)}(t)

\displaystyle=\sqrt{2}\left\langle x_{m}\right\rangle+\frac{1}{\sqrt{\eta}}\xi_{m}(t),

(81)

where $x_{m}=\left(a_{m}+a^{{\dagger}}_{m}\right)/\sqrt{2}$ . Then we can derive the feedback dynamics according to the feedback operator similar as in Section IV.

For multiple coupled cavities, we take the feedback operator $G=\sum_{m}\left[\beta_{x}^{(m)}x+\beta_{p}^{(m)}p\right]$ with the feedback strength $g_{f}$ , the dynamics of the operators can be generalized by Eq. (63) after averaging the homodyne detection noise as


		$\displaystyle\left\langle\dot{\sigma}_{-}^{(m)}\right\rangle=-ig_{m}e^{i\delta t}\left\langle a_{m}\right\rangle-F_{m}(t)\|\kappa_{1}\|^{2}\left\langle\sigma_{-}^{(m)}\right\rangle,$			(82a)
		$\displaystyle\left\langle\dot{a}_{m}\right\rangle=-i\Delta\left\langle a_{m}\right\rangle-ig_{m}e^{-i\delta t}\left\langle\sigma_{-}^{(m)}\right\rangle-\kappa\left\langle a_{m}\right\rangle$
		$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}-ig_{f}\left(\beta_{x}^{(m)}+i\beta_{p}^{(m)}\right)\left\langle x_{m}\right\rangle-iJ_{c}\left(\left\langle a_{m-1}\right\rangle+\left\langle a_{m+1}\right\rangle\right),$			(82b)

where $\left\langle x_{m}\right\rangle=\left(\left\langle a_{m}\right\rangle+\left\langle a_{m}\right\rangle^{*}\right)/\sqrt{2}$ . Obviously, the feedback drive dynamics will not directly influence the dynamics of $\left\langle\sigma_{-}^{(m)}\right\rangle$ according to Eq. (82a), but can influence the atomic dynamics via the atom-cavity couplings.

Remark 8.

When $g_{m}=0$ , $\left\langle\sigma_{-}^{(m)}\right\rangle$ can stably converge to zero, while $\left\langle a_{m}\right\rangle$ can be unstable if the feedback parameter $g_{f}$ is large enough compared with the values of $F_{j}(t)$ and $\kappa$ .

Denote $X_{u}=\left[\mathbf{a}^{R}_{1},\mathbf{a}^{I}_{1},\cdots,\mathbf{a}^{R}_{M},\mathbf{a}^{I}_{M}\right]$ , and $X_{s}=\left[\mathbf{s}^{R}_{1},\mathbf{s}^{I}_{1},\cdots,\mathbf{s}^{R}_{M},\mathbf{s}^{I}_{M}\right]$ , then Eq. (82) can be equivalently written as

\displaystyle\begin{bmatrix}\dot{X}_{u}\\ \dot{X}_{s}\end{bmatrix}

\displaystyle=\left(\begin{bmatrix}\mathbf{\mathcal{A}}^{u}&0\\ 0&\mathbf{\mathcal{A}}^{s}\end{bmatrix}+\begin{bmatrix}P_{1}(t)&P_{2}(t)\\ P_{3}(t)&P_{4}(t)\end{bmatrix}\right)\begin{bmatrix}X_{u}\\ X_{s}\end{bmatrix},

(83)

where

	$\displaystyle\mathbf{\mathcal{A}}^{u}=$	$\displaystyle\begin{bmatrix}\sqrt{2}g_{f}\beta_{p}^{(1)}-\kappa&\Delta&\cdots&0&0\\ \sqrt{2}g_{f}\beta_{x}^{(1)}-\Delta&-\kappa&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&\sqrt{2}g_{f}\beta_{p}^{(M)}-\kappa&\Delta\\ 0&0&\cdots&\sqrt{2}g_{f}\beta_{x}^{(M)}-\Delta&-\kappa\end{bmatrix}$		(84)
		$\displaystyle+\begin{bmatrix}0_{2\times 2}&\mathbf{\mathcal{A}}_{J}&0_{2\times 2}&\cdots&0_{2\times 2}&0_{2\times 2}\\ \mathbf{\mathcal{A}}_{J}&0_{2\times 2}&\mathbf{\mathcal{A}}_{J}&\cdots&0_{2\times 2}&0_{2\times 2}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0_{2\times 2}&0_{2\times 2}&0_{2\times 2}&\cdots&0_{2\times 2}&\mathbf{\mathcal{A}}_{J}\\ 0_{2\times 2}&0_{2\times 2}&0_{2\times 2}&\cdots&\mathbf{\mathcal{A}}_{J}&0_{2\times 2}\end{bmatrix},$		(84)

\displaystyle\mathbf{\mathcal{A}}^{s}

\displaystyle=-\begin{bmatrix}\bar{R}_{f}^{(1)}&-\bar{I}_{f}^{(1)}&\cdots&0&0\\ \bar{I}_{f}^{(1)}&\bar{R}_{f}^{(1)}&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&\bar{R}_{f}^{(M)}&-\bar{I}_{f}^{(M)}\\ 0&0&\cdots&\bar{I}_{f}^{(M)}&\bar{R}_{f}^{(M)}\end{bmatrix},

(85)

$\mathbf{\mathcal{A}}_{J}$ is firstly given by Eq. (77), $\bar{R}_{f}^{(m)}=\lim_{t\rightarrow\infty}F^{R}_{m}(t)$ , $\bar{I}_{f}^{(m)}=\lim_{t\rightarrow\infty}F^{I}_{m}(t)$ , $m=1,2,\cdots,M$ , $P_{1}(t)=0$ , $P_{2}(t)=\mathbf{R}(t)$ , $P_{3}(t)=\mathbf{G}(t)$ , $P_{4}(t)=\mathbf{F}(t)-\mathbf{\mathcal{A}}^{s}$ , $\mathbf{R}(t)$ , $\mathbf{G}(t)$ and $\mathbf{F}(t)$ are given by Eqs. (76), (75), and (74), respectively.

For a special case that $g_{m}=0$ , then $P_{2}(t)=P_{3}(t)=0$ , the dynamics of $X_{u}$ and $X_{s}$ are decoupled in Eq. (83). Obviously, $X_{s}$ will converge to zero if only the interaction between the atom and the environment becomes Markovian. However, $X_{u}$ can be unstable because of the feedback.

When $g_{m}\neq 0$ , $X_{u}$ and $X_{s}$ are always coupled, the stability of Eq. (83) can be determined by the following proposition.

Proposition 11.

When the interaction between the atom and the environment becomes asymptotically Markovian, $X_{u}$ approaches zero when $t\rightarrow\infty$ provided that

\int_{t_{0}}^{t_{0}+T}\mu\begin{Bmatrix}\begin{bmatrix}\mathbf{\mathcal{A}}^{u}&P_{2}(t)\\ P_{3}(t)&\mathbf{\mathcal{A}}^{s}\end{bmatrix}\end{Bmatrix}\mathrm{d}\tau<0.

(86)

Proof.

When the interaction between the atom and the environment becomes asymptotically Markovian, $\lim_{t\rightarrow\infty}P_{4}(t)=0$ in Eq. (83), and the matrix $\begin{bmatrix}\mathbf{\mathcal{A}}^{u}&P_{2}(t)\\ P_{3}(t)&\mathbf{\mathcal{A}}^{s}\end{bmatrix}$ is periodic. Then the proposition holds according to the proof of Proposition 9. ∎

Remark 9.

As in Ref. [70], a choice of the norm in Proposition 11 can be

		$\displaystyle\frac{1}{2}\lambda_{\max}\left(\begin{bmatrix}\mathbf{\mathcal{A}}^{u}+{\mathbf{\mathcal{A}}^{u}}^{T}&P_{2}(t)+P_{3}^{T}(t)\\ P_{3}(t)+P_{2}^{T}(t)&\mathbf{\mathcal{A}}^{s}+{\mathbf{\mathcal{A}}^{s}}^{T}\end{bmatrix}\right)$		(87)
	$\displaystyle=$	$\displaystyle\frac{1}{2}\lambda_{\max}\left(\begin{bmatrix}\bar{\mathbf{a}}_{1}&\cdots&0&0&\cdots&0\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ 0&\cdots&\bar{\mathbf{a}}_{M}&0&\cdots&0\\ 0&\cdots&0&\bar{\mathbf{b}}_{1}&\cdots&0\\ \vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\ 0&\cdots&0&0&\cdots&\bar{\mathbf{b}}_{M}\end{bmatrix}\right),$		(87)

where $\bar{\mathbf{a}}_{m}=\begin{bmatrix}2\sqrt{2}g_{f}\beta_{p}^{(m)}-2\kappa&\sqrt{2}g_{f}\beta_{x}^{(m)}\\ \sqrt{2}g_{f}\beta_{x}^{(m)}&-2\kappa\end{bmatrix}$ , $\bar{\mathbf{b}}_{m}=\begin{bmatrix}-2\bar{R}_{f}^{(m)}&0\\ 0&-2\bar{R}_{f}^{(m)}\end{bmatrix}$ . When $g_{f}$ and $g_{x}$ are small, the inequality in (86) can be easier to be satisfied, then both $X_{u}$ and $X_{s}$ can converge to zero. An extreme case is that $\beta_{x}^{(m)}=0$ and $\kappa>\sqrt{2}g_{f}\beta_{p}^{(m)}$ , then the matrix in Eq. (LABEL:con:MPMtr) is a diagonal matrix with all the elements on the diagonal being negative.

As compared in Fig. 6, there are two coupled cavities, $J_{c}=0.1$ , $\delta=2$ , $\kappa=1$ , $g_{1}=0.2$ , $g_{2}=0.4$ , $\Delta=5$ , the two atoms in the two cavities are coupled identically with the non-Markovian environment with $\omega_{a}=50$ , $\Omega=45$ , $\chi=0.5$ , $\kappa_{1}=1$ and $\gamma=2$ . In the measurement feedback design, $\beta_{x}^{(1)}=\beta_{x}^{(2)}=0$ , $\beta_{p}^{(1)}=\beta_{p}^{(2)}=0.2$ . The solid lines represent the simulation results with $g_{f}=1$ and $\sqrt{2}g_{f}\beta_{p}^{(1)}-\kappa=-0.7172<0$ . The dotted lines represent the simulations with $g_{f}=7$ and $\sqrt{2}g_{f}\beta_{p}^{(1)}-\kappa=0.9799>0$ . The simulations agree with Remark 9, showing that the stable and unstable subspaces can be modulated by tuning the feedback parameters.

VI Conclusion

In this paper, we study the quantum control dynamics within a cavity-QED system, focusing on the non-Markovian interactions between the atom and the environment. The evolution of parameters representing the atom’s non-Markovian decay into the environment is described by nonlinear equations. The transition to Markovian interactions between the quantum system and the environment is explained by the stability of these nonlinear equations. Consequently, the dynamics of the multi-level system with non-Markovian interactions with the environment are represented by a set of linear time-varying equations. Manipulation of atomic states and photons in the cavity is then achieved using open-loop and closed-loop control methods that utilize quantum measurement feedback. This approach can be extended to scenarios involving multiple coupled cavities described by high-dimensional linear time-varying equations, where feedback control can further influence the dynamics between stable and unstable subspaces.

Appendix A Derivation of the non-Markovian parameter dynamics

Take the atom and cavity as a whole represented with the state $|\psi(t)\rangle$ , then the combined system interacts with the bath via the multi-level atom. The interaction between the atom and environment can be represented with the Hamiltonian

H_{I}=\sum_{n=1}^{N-1}\sum_{\omega}\chi_{\omega}^{(n)}\left(|n+1\rangle\langle n|b_{\omega}+|n\rangle\langle n+1|b_{\omega}^{{\dagger}}\right),

with the detailed meaning introduced after Eq. (2) in the main text. Because the coupling strengths between the atom and environment, and the number of environmental oscillator modes cannot be precisely clarified, the evolution of quantum states is influenced by its stochastic interaction with the environment as [44, 43, 54, 51]

\frac{\delta|\psi(t)\rangle}{\delta z_{s}}=f(t,s)L|\psi(t)\rangle,

(88)

where the operator $L$ is defined in the main text as Eq. (3), $z_{s}$ is a complex value Wiener process at the time $s$ , $f(t,s)$ is a two-time function, to be determined, related to the two time points $t$ and $s$ [44].

Combined with Eq. (3), the differential of Eq. (88) upon the time $t$ reads

\frac{\mathrm{d}}{\mathrm{d}t}\frac{\delta|\psi(t)\rangle}{\delta z_{s}}=\frac{\partial f(t,s)}{\partial t}L|\psi(t)\rangle+f(t,s)L|\dot{\psi}(t)\rangle,

(89)

where

$\displaystyle\frac{\partial f(t,s)}{\partial t}L\|\psi(t)\rangle$	$\displaystyle=\frac{\delta}{\delta z_{s}}\|\dot{\psi}(t)\rangle-f(t,s)L\|\dot{\psi}(t)\rangle$	(90)
	$\displaystyle=-iH\frac{\delta\|\psi(t)\rangle}{\delta z_{s}}+L\frac{\delta\|\psi(t)\rangle}{\delta z_{s}}z_{t}-\frac{\delta}{\delta z_{s}}L^{{\dagger}}\int_{0}^{t}\alpha(t,s)\frac{\delta\|\psi(t)\rangle}{\delta z_{s}}\mathrm{d}s$
	$\displaystyle~{}~{}~{}~{}-f(t,s)L\left(-iH\|\psi(t)\rangle+L\|\psi(t)\rangle z_{t}-L^{{\dagger}}\int_{0}^{t}\alpha(t,s)\frac{\delta\|\psi(t)\rangle}{\delta z_{s}}\mathrm{d}s\right)$
	$\displaystyle=i[L,H]f(t,s)\|\psi(t)\rangle+F(t)\left(LL^{{\dagger}}-L^{{\dagger}}L\right)f(t,s)L\|\psi(t)\rangle,$

with the time-dependent function $F(t)$ defined by Eq. (7) and $\alpha(t,s)$ given by Eq. (5) in the main text.

Then by Eq. (90),

\displaystyle\frac{\partial f(t,s)}{\partial t}L|\psi(t)\rangle=i\omega_{a}f(t,s)L|\psi(t)\rangle+F(t)\sum_{n}|\kappa_{n}|^{2}f(t,s)L|\psi(t)\rangle,

(91)

combined with Eq. (7), we have

$\displaystyle\dot{F}(t)$	$\displaystyle=\alpha(t,t)f(t,t)+\int_{0}^{t}\frac{\partial\alpha(t,s)}{\partial t}f(t,s)\mathrm{d}s+\int_{0}^{t}\alpha(t,s)\frac{\partial f(t,s)}{\partial t}\mathrm{d}s$	(92)
	$\displaystyle=\frac{\gamma\chi}{2}-(\gamma+i\Omega)F(t)$
	$\displaystyle~{}~{}~{}~{}+\int_{0}^{t}\alpha(t,s)\left[i\omega_{a}f(t,s)+F(t)\sum_{n}\|\kappa_{n}\|^{2}f(t,s)\right]\mathrm{d}s$
	$\displaystyle=\sum_{n}\|\kappa_{n}\|^{2}F^{2}(t)-(\gamma+i\Omega-i\omega_{a})F(t)+\frac{\gamma\chi}{2},$

where $f(t,t)=\chi$ is a constant, $F(0)=0$ , and the analytic solution of generalized $F(t)$ is given in Ref. [44].

Appendix B Derivation of the non-Markovian master equation

The non-Markovian master equation has been introduced in Refs. [44, 46]. In this Appendix, we briefly introduce the derivation of the non-Markovian control equation (6) from the main text, which is the control equation we focus on in this paper.

The evolution of $f(t,s)$ has been clarified in Appendix A. Combining Eq. (4) with Eq. (88) after the normalization, we have the following equation for the state vector,

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\|\psi(t)\rangle=$	$\displaystyle-iH\|\psi(t)\rangle+\left(L-\langle L\rangle\right)\|\psi(t)\rangle z_{t}$		(93)
		$\displaystyle-\left(L^{{\dagger}}-\langle L^{{\dagger}}\rangle\right)\int_{0}^{t}\alpha(t,s)f(t,s)L\|\psi(t)\rangle\mathrm{d}s.$		(93)

Considering that $\rho(t)=E_{z}(|\psi(t)\rangle\langle\psi(t)|)$ , and denote $P_{t}\left(z_{t}^{*}\right)=|\psi(t)\rangle\langle\psi(t)|$ , then $E_{z}\left(P_{t}z_{t}\right)=\int_{0}^{t}E_{z}\left[z_{t}z_{s}^{*}\right]f^{*}(t,s)\rho(t)L^{{\dagger}}\mathrm{d}s$ and $E_{z}\left(P_{t}z_{t}^{*}\right)=\int_{0}^{t}E_{z}\left[z_{t}^{*}z_{s}\right]f(t,s)L\rho(t)\mathrm{d}s$ [51]. Then the equation for the density matrix reads

$\displaystyle\dot{\rho}(t)=$	$\displaystyle-i[H,\rho(t)]$	(94)
	$\displaystyle+E_{z}\left\{L\|\psi(t)\rangle\langle\psi(t)\|z_{t}-L^{{\dagger}}\int_{0}^{t}\alpha(t,s)f(t,s)L\|\psi(t)\rangle\langle\psi(t)\|\mathrm{d}s\right\}$
	$\displaystyle+E_{z}\left\{\|\psi(t)\rangle\left[\langle\psi(t)\|L^{{\dagger}}z_{t}^{}-\langle\psi(t)\|L^{{\dagger}}L\int_{0}^{t}\alpha^{}(t,s)f^{*}(t,s)L\mathrm{d}s\right]\right\}$
$\displaystyle=$	$\displaystyle-i[H,\rho(t)]$
	$\displaystyle+\left\{\int_{0}^{t}\mathrm{d}sE_{z}\left[z_{t}z_{s}^{}\right]f^{}(t,s)+\int_{0}^{t}\mathrm{d}sE_{z}\left[z_{t}^{*}z_{s}\right]f(t,s)\right\}L\rho(t)L^{{\dagger}}$
	$\displaystyle-\int_{0}^{t}\alpha(t,s)f(t,s)\mathrm{d}sL^{{\dagger}}L\rho(t)-\rho(t)L^{{\dagger}}L\int_{0}^{t}\alpha^{}(t,s)f^{}(t,s)\mathrm{d}s,$

and can be further written as Eq. (6) in the main text, which agrees with the format in Ref. [46].

Acknowledgements

This work is supported by the ANR project “Estimation et controle des systèmes quantiques ouverts” Q-COAST Project ANR- 19-CE48-0003 and the ANR project IGNITION ANR-21-CE47-0015.

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$\displaystyle\frac{\partial f(t,s)}{\partial t}L\|\psi(t)\rangle$	$\displaystyle=\frac{\delta}{\delta z_{s}}\|\dot{\psi}(t)\rangle-f(t,s)L\|\dot{\psi}(t)\rangle$	(90)
	$\displaystyle=-iH\frac{\delta\|\psi(t)\rangle}{\delta z_{s}}+L\frac{\delta\|\psi(t)\rangle}{\delta z_{s}}z_{t}-\frac{\delta}{\delta z_{s}}L^{{\dagger}}\int_{0}^{t}\alpha(t,s)\frac{\delta\|\psi(t)\rangle}{\delta z_{s}}\mathrm{d}s$
	$\displaystyle~{}~{}~{}~{}-f(t,s)L\left(-iH\|\psi(t)\rangle+L\|\psi(t)\rangle z_{t}-L^{{\dagger}}\int_{0}^{t}\alpha(t,s)\frac{\delta\|\psi(t)\rangle}{\delta z_{s}}\mathrm{d}s\right)$
	$\displaystyle=i[L,H]f(t,s)\|\psi(t)\rangle+F(t)\left(LL^{{\dagger}}-L^{{\dagger}}L\right)f(t,s)L\|\psi(t)\rangle,$

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\|\psi(t)\rangle=$	$\displaystyle-iH\|\psi(t)\rangle+\left(L-\langle L\rangle\right)\|\psi(t)\rangle z_{t}$		(93)
		$\displaystyle-\left(L^{{\dagger}}-\langle L^{{\dagger}}\rangle\right)\int_{0}^{t}\alpha(t,s)f(t,s)L\|\psi(t)\rangle\mathrm{d}s.$		(93)

On the non-Markovian quantum control dynamics

Abstract

Index Terms:

I Introduction

II Non-Markovian quantum control for open systems

Assumption 1.

Assumption 2.

Definition 1.

Remark 1.

II-A Single multi-level atom coupled with the cavity

Assumption 3.

Remark 2.

II-B Non-Markovian parameter dynamics

II-B1 Ω=ωa\Omega=\omega_{a} and RP−(Q2​P)2<0\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}<0

II-B2 Ω=ωa\Omega=\omega_{a} and RP−(Q2​P)2>0\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}>0

II-B3 Ω=ωa\Omega=\omega_{a} and RP−(Q2​P)2=0\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}=0

Proposition 1.

Proof.

Definition 2.

II-B4 Example

Definition 3 (see [64]).

Proposition 2.

Proof.

Remark 3.

Proposition 3 ([61]).

Proposition 4.

Proof.

Remark 4.

Lemma 1.

Proof.

Assumption 4.

Proposition 5.

Proof.

II-C An example to clarify how F​(t)F(t) influences the atomic dynamics

II-D Nonlinear dynamics with uncertain control input

Assumption 5.

Proposition 6 (see [67]).

Proposition 7.

Proof.

III Time-varying linear quantum control dynamics

III-A δ=0\delta=0, gn≠0g_{n}\neq 0

Lemma 2.

Proof.

Proposition 8.

Proof.

III-B δ≠0\delta\neq 0, gn≠0g_{n}\neq 0

Definition 4 ([70]).

Definition 5 ([70]).

Proposition 9.

Proof.

III-C With external drives

IV Quantum measurement feedback control in the non-Markovian cavity-QED system

Assumption 6.

IV-A Feedback control with atomic operators

IV-A1 Example

IV-B Feedback control with cavity operators

Proposition 10.

Proof.

Remark 5.

V Quantum control for the multiple Jaynes-Cummings model

Assumption 7.

V-A Dynamics without drives

V-A1 Multi-cavity case

Remark 6.

Remark 7.

V-A2 An example with M=2M=2

V-B Dynamics with drives

Remark 8.

Proposition 11.

Proof.

Remark 9.

VI Conclusion

Appendix A Derivation of the non-Markovian parameter dynamics

Appendix B Derivation of the non-Markovian master equation

Acknowledgements

References

II-B1 $\Omega=\omega_{a}$ and $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}<0$

II-B2 $\Omega=\omega_{a}$ and $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}>0$

II-B3 $\Omega=\omega_{a}$ and $\frac{R}{P}-\left(\frac{Q}{2P}\right)^{2}=0$

II-C An example to clarify how $F(t)$ influences the atomic dynamics

III-A $\delta=0$ , $g_{n}\neq 0$

III-B $\delta\neq 0$ , $g_{n}\neq 0$

V-A2 An example with $M=2$