Quantum non-stationary phenomena of spin systems in collision models

The quantum system that is subjected to coupling to an uncontrollable environment is referred to as an open quantum system. The inevitable interaction with the environment always leads to the dissipative effect of the open quantum system and the non-unitary feature of its time-evolution Breuer2002 ; Rivas2012 ; Breuer2016 . In the presence or absence of memory effects, the dynamics of open quantum systems can be classified into the memoryless Markovian process and the non-Markovian master process which allows the backflow of information during the time-evolution. In dissipative open quantum systems, in comparison with their closed counterparts, plenty of intriguing phenomena emerge especially in the quantum many-body systems, such as the steady-state phase transition Banchi2014 ; Maile2018 ; Hwang2018 ; Prasad2022 , information spreading Zhangyongliang2019 ; Styliaris2021 ; Zanardi2021 , quantum many-body delocalization Potter2015 ; Hyatt2017 ; Rubio2019 ; Xian2021 , etc. Generally, the state of an open quantum system asymptotically evolves to one or more time-independent steady states in the long-time limit. However, there are exceptional cases in which the system evolves to non-stationary states. The time-dependent long-time behavior of the system intimately related to the quantum time crystals Wilczek2012 ; Kozin2019 ; Georg2021 , quantum chaos Zhuang2013 ; Bruzda2010 , and quantum synchronization Laskar2020PRL ; Jaseem2020PRR ; Solanki2022PRAL ; Hajdusek2022PRL ; Krithika2022PRA .

As the extension of classical synchronization in the quantum regime, quantum synchronization has attracted much attention in the last few years. The quantum synchronization between self-sustained oscillations can be established through either the external driving or the internal coupling ZhirovPRL2008 ; ZhirovPRB2009 ; GalvePRL2010 ; WalterPRL2014; AmeriPRA2015 . The former is known as the forced synchronization or entrainment Karpat2019PRA while the latter is called spontaneous synchronization which stems from the quantum correlations in the systems. Meanwhile, the quantum correlations in the systems is not limited to direct interactions between subsystems, but also may emerge indirectly through the interactions of subsystems together with their surrounding environment. This quantum correlations can eventually lead to non-stationary phenomena depicted by the quantum synchronization measure. In recent years, quantum synchronization in open quantum systems has already been explored at length in a variety of quantum systems, such as optomechanical arrays Cabot2017NJP ; Ludwig2013PRL ; RodriguesNC2021 , van der Pol (VdP) oscillators Walter2014PRL ; Tilley2018NJP , atomic ensembles Xu2014PRL , and superconducting circuit systems Vinokur2008Nature ; Quijan2013PRL ; Fistul2017SR .

In general, theoretic descriptions for the dynamics of open quantum systems as well as the non-stationary behavior in the long-time limit are carried out by calculating the expectation values of the local observables and correlations over time through the dynamical equations of the system such as the quantum master equation and quantum Langevin equation. In recent years, investigations on the open quantum system within the framework of collision model (CM) are reported Rau1963 ; Ziman2002 ; Scarani2002 ; Ziman2005 ; Ciccarello2013PRAR . The CM approach can provide intuitive pictures of the interactions between the system and its environment as well as the strategies of the information flows in the time-evolution of the state of the system. Recently, it is shown that the CM can efficiently reproduce the dissipative collective phenomena of multipartite open quantum systems cattaneo2021prl ; cattaneo2022arXiv . In the CM framework, the coupling of the system and environment is simulated by repeated collisions between system and a set of environmental particles. Since the flexibility and scalability in designing the collision details, the CM becomes a powerful tool for investigating non-Markovian dynamics Ciccarello2013 ; McCloskey2014 ; Jinjiasen2015 ; Lorenzo2017 ; Jinjiasen2018 ; Camasca2021 ; Filippov2022 ; Francesco2022 ; cattaneo2022osid , quantum information scrambling Liyan2020 ; Liyan2022 , quantum steering Beyer2018 , quantum friction Grimmer2019 , multipartite entanglement generation celse2019 , and quantum synchronization Karpat2019PRA ; Karpat2021 in complicated open quantum systems. Recently, the experimental realization of an all-optical collision model has already been demonstrated Cuevas2019 .

In this work, we utilize the CM to investigate the long-time behavior of a composite spin system consisting of three spin-1/2 subsystems subjected to thermal environment. By varying the strength of the interactions within the environment blocks, we can obtain the Markovian dynamics and non-Markovian dynamics, respectively. We find that, without additional external driving, the system is able to reach a stable non-stationary steady state only through its internal interactions and dissipative processes. At this point synchronization phenomena are also constructed between subsystems which reminisce the results reported by Karpat, et al. in Ref. Karpat2020PRA . Furthermore, after taking the continuous time limit for the CM, we establish the connection between the CM and the Markovian master equation in Lindblad form. We interpret the appearance of the underlying long-time oscillations of the subsystems by analyzing the Liouvillian spectrum of the associated quantum master equation. In addition, we investigate the thermodynamical properties and correlations between the system and the environment when the dynamics of the subsystems are synchronized.

The paper is organized as follows. We first explain the idea of the our specific model in the CM framework and its utility in simulating the dynamics of the composite systems consisting of three spins in Sec.II. In Sec.III we then focus on the Markovian dynamics to show the temporal expectation values of the local observables. In particular for the case that the subsystems enter into the long-time oscillations we discuss the quantum synchronization among the subsystems. We explore the connection between the CM and Lindblad master equation in describing the underlying system in the continuous time limit. We present our understanding on the appearance of oscillations in the dynamics from the Liouvillian spectrum. The thermodynamical properties and the effects of imperfect collective dissipation on the dynamics. In Sec. IV, we discuss the time-evolution of the systems in the non-Markovian case. Finally, we summarize in Sec. V.

In the generic CM framework, the entire installation consists of two parts: the system $\mathcal{S}$ and its environment $E$. Usually, the environment is represented by a set of identical particles denoted by $\{E_{1},E_{2},\cdots,E_{j},\cdots\}$ which are initialized in the same state. The interaction between the system and the environment is simulated by the successive collisions between system particles and environmental particles in a stroboscopic manner. In our CM, both the system and environment are consisted of a set of spin-$1/2$ particles. In particular, the system of interest is tripartite with interacting subsystems $\mathcal{S}_{1}$, $\mathcal{S}_{2}$ and $\mathcal{S}_{3}$. The inner interactions between the subsystems are given by (set $\hbar=1$ hereinafter),

where $\sigma_{m}^{\alpha}$ ($\alpha=x,y$ and $z$) are the Pauli matrices of the $m$-th subsystem and $J_{\alpha}$ is the coupling strength between system spins. Notice that the periodic boundary condition is imposed in Eq. (1).

We consider the case that the environmental spins are uncorrelated and are prepared in the identical thermal state $\eta^{j}_{\text{th}}=e^{-\beta H_{j}}/\text{tr}(e^{-\beta H_{j}})$, $\forall j$. The free Hamiltonian of each environmental particle is $H_{j}=\omega\sigma_{j}^{z}/2$ where $\omega$ describes the self-energy and $\beta=1/k_{B}T$ is the inverse of temperature. Although the environment spins are initialized in the uncorrelated state, interactions between environment particles are allowed to take place. As will be seen in the next sections, it is the intra-environment interactions that generate the memory effect of the stroboscopic evolution of the system.

Now let us illustrate the setup of our CM which is schematically shown in Fig. 1(a). The CM works through the following steps:

After the $E_{n}$-$E_{n+1}$ collision, the spin $E_{n+1}$ carries part of the information that flows into the environment in step 2. The system together with the $(n+1)$-th environment particle enters into the next loop and the spin $E_{n}$ is discarded as shown in Fig.1(a). Therefore, the state of the system of interest after the $n$-th loop ($n\geq 2$) is transformed into

where $\rho^{n}_{\mathcal{S}E_{n}}$ is the joint state of the system and the $n$-th environment unit after collision. The system is initialized in the state $\rho^{\text{ini}}_{\mathcal{S}}=|\psi\rangle_{\mathcal{S}}\langle\psi|$ with the separable state $|\psi\rangle_{\mathcal{S}}=|\psi\rangle_{\mathcal{S}_{1}}\otimes|\psi\rangle_{\mathcal{S}_{2}}\otimes|\psi\rangle_{\mathcal{S}_{3}}$. More precisely, since the Hamiltonian has the symmetry along the spin $z$-axis we choose the initial state of each subsystem as the $120^{\circ}$-state on the equatorial plane of Bloch sphere, i.e., $\psi_{\mathcal{S}_{m}}=\left(|\uparrow\rangle+e^{im\frac{2}{3}\pi}|\downarrow\rangle\right)/\sqrt{2}$, with $m=1,2$, and $3$. The chosen $120^{\circ}$-state will facilitate the discussion on the evolution of the phase differences among the state of subsystems. For simplicity, we set the interaction time for the each collision as $\tau=\tau_{\mathcal{S}}=\tau_{I}=\tau_{E}$ in the rest of the analysis.

In this section, we focus on the case that the intra-environment collision is absent in the CM, i.e. $U_{E}$ reduces to an identity matrix implying that the dynamics of the system of interest is Markovian. We choose the temporal expectation value of local observable $\langle\sigma^{x}_{m}\rangle=\text{tr}(\sigma^{x}_{m}\rho_{\mathcal{S}})$ to monitor the dynamics of the system. We first focus on the effect of the environmental temperature $\beta$ on the time-evolution of the state of the system. Note that $\beta$ is proportional to $1/T$.

In Figs.2(a) and (b), we show the time-dependence of the local observables $\langle\sigma^{x}_{m}\rangle$ for different $\beta$. One can see that the magnetizations of all the spins trivially approach to zero after a sufficient large number of collisions for $J_{x}=3$ and $\beta=2$. As the environmental temperature goes down, the asymptotically values of the magnetiaztions of all the spins become unstable and eventually the time-evolution of the magnetizations enter into the oscillating trajectories. The time-dependent oscillations of $\langle\sigma^{x}_{m}(n)\rangle$ for $\beta=10$ can be observed in Fig.2(b). Moreover, as shown in Fig.2(c), the Fast Fourier Transform analysis on the numerics of $\langle\sigma^{x}_{m}(n)\rangle$ reveals that the oscillations share the same dominant frequency. A zoom-in at the vicinity of the peak in the frequency domain shows that the dominant frequency is $f^{d}\approx 0.32$.

The phase-locking feature of the oscillations can be verified by checking the cross-correlation between the time-evolutions of $\langle\sigma^{x}_{m}(n)\rangle$. Generally, the cross-correlation quantifies the degree of the association between these two time-dependent functions $f(n)$ and $g(n)$ and is defined as their convolution-like function. Here, we analogously express the cross-correlation for the discrete evolutions of $\langle\sigma^{x}_{m}(n)\rangle$ as

where $\Delta n$ is the amount of translation and $N$ is the total number of collisions. The cross-correlation function is actually an inner product of two vectors representing the projection of one onto another in the linear space. Therefore the cross-correlation can faithfully capture the similarity of two vectors under different amounts of translation within a period. The maximum of $X_{jk}(\Delta n)$ in Eq. (6) reveals the optimal translation that shifts the trajectory of $\langle\sigma^{x}_{j}\rangle$ closest to $\langle\sigma^{x}_{k}\rangle$ and thus in turn reflects the phase difference $\Delta\phi_{\mathcal{S}_{j}\mathcal{S}_{k}}$. In Fig. 2(d), we show the cross-correlation $X_{jk}(\Delta n)$ produced by the data in Fig. 2(b), one can see that the maxima of the cross-correlation $X_{12}(\Delta n_{1})$ and $X_{13}(\Delta n_{2})$ appear at $\Delta n_{1}=-101$ and $\Delta n_{2}=-205$, respectively. Recall the previously calculated dominant frequency $f^{d}\approx 0.32$, the phase differences can be obtained as $\Delta\phi_{\mathcal{S}_{1}\mathcal{S}_{2}}=2\pi f^{d}/\Delta n_{1}\tau\approx-2\pi/3$, $\Delta\phi_{\mathcal{S}_{1}\mathcal{S}_{3}}=2\pi f^{d}/\Delta n_{2}\tau\approx-4\pi/3$. The initial phase differences between the sublattices are conserved after the dissipative evolution. Note that we have rescaled the number of collisions $n$ into time $n\tau$ with time interval $\tau=0.01$ .

The phase-locking oscillation in the long-time dynamics of the system indicates the emergence of the spontaneous synchronization among the subsystems. In the following we will employ the nonlocal synchronization measure proposed by Es’haqi-Sani et al. in Ref. NajmehPRR2020 to quantify the synchronization features in our CM. The mentioned measure is a temporal complex-valued correlator and defined as follows,

where $\langle O\rangle_{n}=\text{Tr}[O\rho^{n}_{\mathcal{S}}]$ is the expectation value of the observable $O$ after the $n$-th collision. The modulus of $Q_{jk}(n)$ characterizes the degree of non-local correlation, for instance, two subsystems are completely correlated when $|Q_{jk}(t)|\to 1$. Combining the dynamics of local observable of the system and the behavior of this non-local correlation, we are able to investigate the synchronization properties. When the dynamics of the subsystems are oscillating and the non-local correlation function tends to a stable value, the subsystem can be considered to be synchronized. In Fig. 2(e), we show the modulus for the steady-state $|Q_{12}|_{\text{ss}}$ in the $\beta$-$J_{x}$ plane. One can see that the lower environmental temperature ($\beta\rightarrow\infty$) could facilitate the synchronization of among the subsystems. Moreover, for $\beta\gtrsim 4$ the long-time oscillation of the subsystems becomes almost fully synchronized at an optimal interaction strength $J_{x}=J_{y}$. Namely, the anisotropy of the system Hamiltonian tends to drive the dynamics of the subsystems far way from the synchronization.

On the other hand, the coherence property is also related to synchronization Galve2017 which has already been considered as a synchronization measure Jaseem2020PRE ; Karpat2020PRA . In Fig. 2(f) we present the $l_{1}$ norm of coherence of the state of the system in the $\beta-J_{x}$ plane. The $l_{1}$ norm of coherence is defined as follows,

where $\{|p\rangle\}$ are the computational basis of $\rho_{\mathcal{S}}$ Baumgratz2014PRL . Indeed, we find that there are many similar behaviors in Figs. 2(e) and (f). For lower environmental temperature or an anisotropic Hamiltonian, the coherence of the system are poorly present in the system which implies that the synchronization is more likely to be established when the state of system $\rho_{\mathcal{S}}$ contains more coherence.

We turn to the non-Markovian case in this section by switching on interaction between neighboring environmental spins. As we mentioned in Sec.II, in the CM framework, the non-Markovian dynamic can be implemented by introducing the inner collision between the environment spins $U_{E}$. The idea is the following: At the collision step $n$, the system $\mathcal{S}$ collides with the environment spin $E_{n}$, and the information from the system partially flows into the environment. Then the environment inner collision takes place, the environment spin $E_{n}$ collides with the fresh environment spin $E_{n+1}$. In this way, the spin $E_{n+1}$ also partially carries the system information. In the next collision step $n+1$, when a collision happens between the system $\mathcal{S}$ and the environment $E_{n+1}$, the information that flowed into environment has the possibility to flow back to the system.

Although the $E_{n}$-$E_{n+1}$ collision enables the information backflow, a measure still need to quantify the degree of the non-Markovianity of the dynamics of the system. To this aim, we employ the well-known BLP measure Breuer2009PRL and generalize it to the discrete time evolution.

The idea of the BLP measure is to quantify the non-Markovianity through the trace distance change of the system state as follows,

with $\Delta\mathcal{D}(n)=\mathcal{D}[\rho_{1}(n+1),\rho_{2}(n+1)]-\mathcal{D}[\rho_{1}(n),\rho_{2}(n)]$ and $\mathcal{D}(\rho_{1},\rho_{2})=\text{tr}(\sqrt{(\rho_{1}-\rho_{2})^{\dagger}(\rho_{1}-\rho_{2}})/2$ is the trace distance between the two density matrices $\rho_{1}$ and $\rho_{2}$. The trace distance quantifies the degree of distinguishability between the two states with $0\leq\mathcal{D}(\rho_{1},\rho_{2})\leq 1$ and $\mathcal{D}(\rho_{1},\rho_{2})=1$ corresponds to the case where the two states are orthogonal. In the Markovian process, because the information of the system is one-way flows into the environment, all distinct initial states will eventually approach to a unique steady state manifested by $\Delta\mathcal{D}(n)<0$ for all $n$. However, if it exists $\Delta\mathcal{D}(n)>0$ for some $n$, namely the distinguishability of the two initial states increases, it is signal that the information have flowed back to the system at some point thus the dynamics is non-Markovian. The non-Markovianity is obtained by the sum of all the increasing of the trace distance during the time-evolution.

In principle, the maximization in Eq. (24) should run over all the possible initial states. Here we performed $K=200$ simulations with $\rho_{1}$ being a random state and $\rho_{2}=\rho_{1}^{\perp}$ being the orthogonal state of $\rho_{1}$. The non-Markovianity $\mathcal{N}$ is thus given by the optimal pair of $\rho_{1}$ and $\rho_{2}$ within the all performed simulations. The non-Markovianity $\mathcal{N}$ as a function of the strength of $E_{n}$-$E_{n+1}$ collision $\theta$ is shown in Fig.6. Note that according to the equation $\theta=2g_{E}\tau$ in Sec.II, the coupling strength between environmental spins is proportional to $\theta$. One can find that the non-Markovianity of the dynamics is not induced as soon as the $E_{n}$-$E_{n+1}$ collision is switched on. When the $E_{n}$-$E_{n+1}$ collision is weak, e.g. $\theta/\pi<0.2$, after $n=500$ collisions, the non-Markovianity $\mathcal{N}$ is always zero. As we continue to increase the strength of $E_{n}$-$E_{n+1}$ collision, the dynamics becomes non-Markovian and $\mathcal{N}$ increases monotonic with $\theta$ increasing.

For the non-Markovian region ($\theta/\pi=0.2$), we discuss through the following three cases:

Meanwhile, the unitary evolution operator within the environment blocks $U_{E_{n}E_{n+1}}$ is the SWAP gate when $\theta=\pi/2$. The state of the system is initialized to the $120^{\circ}$-state. As for initial states of the system in the calculation of trace distance $\mathcal{D}(\rho_{1},\rho_{2})$, the state $\rho_{1}$ is the $120^{\circ}$-state and the other one is the orthogonal state of $\rho_{2}=\rho_{1}^{\perp}$.

Since the correlation between environment blocks has been established in the non-Markovian case, we have to redefine the entropy production before proceeding to a specific discussion. What is noteworthy in the non-Markovian case is that in each collision cycle, there involves two environment blocks. To be more precisely, the detail expression of interval entropy flux in each collision cycle has to contain the total energy of those two environment parts PezzuttoNJP2016 , that is

For the sake of understanding, let us recall the entail collision evolution introduced before

Here once again, to avoid misunderstanding, we emphasize $H$ is the free Hamiltonian of the thermal environment particle.

In Fig. 7, we show the time-evolution of $\langle\sigma^{x}_{m}\rangle$ and the modulus of the synchronization measure $Q_{12}$. For the weak non-Markovian case, one can find that the dynamics of the system do not suffer from non-Markovianity and oscillations can be rapidly built up in the evolution. As shown in the left-middle panel of Fig. 7, although the dynamics is non-Markovian, the information backflow ($\Delta\mathcal{D}>0$) only occurs in the early stage of the evolution and then the trace distance decreases monotonically. As for the entropy production, in Case I, the entropy production is always a positive value, which is far from zero value and rapidly reaches a constant. The phenomena are similar with the Markovian case, a subspace of the system will rapidly decouple will rapidly decouple from the environment during evolution and the weak non-Markovianity does not have a significant effect on the system dynamics.

In Cases II, by contrast, both the expectation values of $\langle\sigma_{m}^{x}\rangle$ and $|Q_{12}|$ show irregular oscillations at the beginning of the evolution, see the second column of Fig. 7. In spite of the early-stage haphazard behavior, the system evolution gradually becomes regular and stabilizes around $n=2500$ (not shown in Fig.7), and the system reaches stable oscillations eventually. The modulus of the quantum synchronization measure $|Q_{12}|$ also reaches a constant value in the long-time evolution which hints at the existence of good synchronization. The time dependence of trace distance for Case II shows a rapid oscillation until $n>1000$ which is a typical evidence for the strong non-Markovian dynamics. In particular, at some moments (marked by red dashed lines), the trace distance instantaneously recovers the initial value.

The strong non-Markovianity significantly impacts the entropy production in the time-evolution. One can see that the entropy production does not monotonically increase and may decrease drastically close to to zero at some $n$ in the discrete time-evolution. This indicates the entropy production at the $n$-th step $\Sigma^{n}$ becomes negative. This can be understood by, for example, focusing on the steps $n=9$ and $17$ ( the red dashed lines in Fig. 7) at which the trace distance (almost) recovers to unity, it is the strong information backflow that diminishes the vague of different states and decrease the entropy of the system and thus leads to the negative $\Sigma^{n}$ at $n=9$ and $17$. But when we look at $\Sigma(n)$ through the entire time-evolution, the boundary of violation of the second law of thermodynamics is untouchable PezzuttoNJP2016 .

In Case III, the collision between neighboring environmental spins is set to be the SWAP operation. As shown in the right column of Fig.7, one can find that the dynamics of the system always exhibit irregular oscillations even in the long-time limit $(n\sim 5000)$. In addition, the measure $|Q_{12}|$ fails to reach an asymptotic steady-state value implying the absence of synchronization in such ultrastrong non-Markovian dynamics. The time-dependence of both the trace distance as well as the entropy production oscillate irregularly. Again the non-monotonic behavior of the entropy production is observed in the ultrastrong non-Markovian case. The negative entropy production $\Sigma^{n}$ during the time-evolution is due to the creation of the correlations in the system and environmental qubits Manzhongxiao2019 ; PezzuttoNJP2016 .

We would finish by emphasizing again that, in CM scenario, since the thermal environment is represented by a series of qubits, the thermodynamic quantities should be the cumulative of the same quantities at each step to ensure all the changes in the involved environmental qubits are taken into account. In this sense, the second law of thermodynamics still holds in our model from a macroscopic point of view.

In summary, we have investigated the dissipative dynamics of a tripartite spin system in the framework of CM. With the introduction of successive collisions between the system and environment spins, collective dissipation has been simulated. We have found that when the environment temperature is low, the dynamics of the system exhibit a well-defined oscillation and the mutual synchronization can be established among the subsystems. We proceeded the discussion by classifying the present CM into the Markovian and the non-Markovian cases, according to allow or not the collision between neighboring environmental spins.

In the Markovian case, we have discussed the effects of the coupling strength and environment temperature on the dynamics of the systems as well as the property of synchronization among subsystems. We find that the dynamics of system may show periodical oscillations when the environmental temperature is low and the anisotropy of the interactions between the subsystems tends to destroy the stable oscillation. In addition, we have extracted the frequency of the oscillation by means of FFT.

To understand the establishment of the stable oscillations in the Markovian dynamics, we have connected the descriptions of CM and master equation by virtue of the taking the continuous time limit on the interval collisions. We then analyzed the Liouvillian spectrum of the master equation and found that there indeed exist purely imaginary eigenvalues when the environment temperature is low. In particular the inverse of the modulus of the pure imaginary eigenvalues is consistent with oscillation frequency accessed by the FFT. We further investigated the structure of the density matrix of the system and found that the system’s oscillatory behavior is actually a process of leapfrogging between degenerate Hamiltonian eigenstates and other energy eigenstates within a subspace consisted of eigenoperators. Finally, we investigated the temporal behaviors of entropy production and quantum correlations which shows the oscillation of the system is induced not only by the coupling to the environment but also a result of self-adjusting of the correlations among the subsystems. We also discussed the effects of the imperfection of the collective interactions on the long-time oscillations.

For the non-Markovian case, we utilized the BLP measure of non-Markovianity in the stroboscopic time-evolution in the CM. We found in the strong (and ultrastrong) non-Markovian dynamics, the entropy production is always positive and may exhibit nomonotonic behavior in the time-evolution implying that the entropy production in some collisions steps become negative. This can be understood by the fact that the backflow of information in strong non-Markovianity dynamics compensates the energy in erasing the information. Finally, the dynamic of the system exhibits chaos-like behavior even after sufficient long time in the ultrastrong non-Markovian case. In the future work, it will be interesting to check the chaotic behavior of the system and investigate the influence of the system environment correlations on the system dynamics behavior.

This work is supported by National Natural Science Foundation of China under Grant No. 11975064.

In this appendix, we take the collision sequence given in the previous Sec.III.5 as an example to show the specific form of the master equation for sequential collision case. Here we recall the unitary-evolution operator $U_{I,123}^{\text{seq}}=U_{\mathcal{S}_{3}E_{n}}U_{\mathcal{S}_{2}E_{n}}U_{\mathcal{S}_{1}E_{n}}$, with $U_{\mathcal{S}_{m}E_{n}}=\exp{[-ig\tau(\sigma_{\mathcal{S}_{m}}^{+}\sigma_{E_{n}}^{-}+\text{h.c.})]}$, ($m=1,2,3$) and the environment particles are still located in the thermal state without off-diagonal matrix elements.

The dynamical map for the system also can be given by,

Based on the requirements of the continuous time limit, which hints $\tau\to 0$, we still expand the unitary operators as follows,

and

Then we can directly obtain the mapping of the collision,

Recall the time scale $\tau\to 0$ and $g^{2}\tau=\text{constant}$, we have