Dynamics of a quantum interacting system: Extended global approach beyond the Born-Markov and secular approximations

In various fields from quantum physics to biology, the open quantum dynamics of a system consisting of interacting subsystems emphasizes its fundamental functionality. The local approach, deriving a dissipator in a master equation by ignoring the intersubsystem interaction, has been widely used to describe the reduced dynamics due to its robustness to keep the positivity of a density operator. However, one critique is that a stationary state obtained by the approach in the limit of weak system-environment coupling is written in the form of the Gibbs state for the partial Hamiltonian by excluding the intersubsystem interaction from the total one of the relevant system. As an alternative, the global approach, deriving a dissipator with including the intersubsystem interaction, under the Born-Markov and secular approximations has attracted much attention, and there is debate concerning its violation of positivity in the short-time region and/or limited parameter region for the Bohr frequencies of the subsystems. In this paper, we present a formalism that leads to the time-convolutionless (time-local) master equation obtained by extending the global approach beyond the Born-Markov and secular approximations. We apply it to the excitation energy transfer between interacting sites in which only the terminal site weakly interacts with a bosonic environment of finite temperature in a manner beyond the rotating-wave approximation. We find that the formulation (1) gives the short-time behavior while preserving positivity, (2) shows the oscillatory features that the secular approximation would obscure, and (3) leads to a stationary state very near to the Gibbs state for the total Hamiltonian of the relevant system.

Figure 1

Time evolution of each element of $\rho \left[\tilde{t}(={\omega }_{2}t\right)]$ obtained by the GA beyond the SA and the BMA for the initial condition, ${\rho }_{11}\left(0\right)=1$, with parameter settings ${\tilde{\omega }}_1(\equiv {\omega }_1/{\omega }_2)=1/2, {\tilde{\omega }}_3(\equiv {\omega }_3/{\omega }_2)=0, {\tilde{V}}_{12}(\equiv V_{12}/{\omega }_2)=3/10, {\tilde{\mathrm{\Omega }}}_c(\equiv {\mathrm{\Omega }}_c/{\omega }_2)=1, \tilde{\beta }[\equiv \hslash {\omega }_2/\left(k_{B}T\right)]=2$, and $s=1/100$. We find that the asymptotes obtained from the GA give different values, and the stationary value of coherence $\mathrm{Re}\left[{\rho }_{G,12}\right]$ has a finite value, the features of which differ from those obtained from the LA given in Appendix pp4.

Figure 2

(a) Dependence of the stationary values of the density matrix obtained by the GA beyond the SA on the inverse temperature $\tilde{\beta }$ and interaction strength between sites ${\tilde{V}}_{12}$. (b) Dependence of the trace distance between the stationary state ${\rho }_G$ and the expected Gibbs state $D({\rho }_G,{\rho }_{\mathrm{Gibbs}})$. We find that this difference remains in the order of the interaction strength $O\left(s\right)$ in the range from $\tilde{\beta }$ to ${\tilde{V}}_{12}$ even beyond the SA. Other settings are the same as given in Fig. 1.

Figure 3

Comparison of the non-Markovian dynamics of the sink site, ${\rho }_{33}\left(\tilde{t}\right)$, obtained by the GA with and without the SA shown as dashed and solid lines, and labeled “NM SA” and “NM BS,” respectively. We show the dynamics for three sets of parameters $\{{\tilde{\omega }}_1,\tilde{\beta }\}$ as $\{\frac{1}{2},2\}, \{\frac{1}{2},\frac{1}{2}\}$, and $\{\frac{19}{20},\frac{1}{2}\}$ and marked by stars, asterisks, and circles, respectively. Other settings are fixed as in Fig. 1. We find that the SA worsens in the short-time region and/or high temperature other than for the large difference between eigenvalues of the relevant system.

Figure 4

Comparison of the dynamics of the sink site, ${\rho }_{33}\left(\tilde{t}\right)$, obtained by the GA with and without the BMA and/or the SA for $\{{\tilde{\omega }}_1,\tilde{\beta }\}=\{\frac{1}{2},2\}$; the other parameters are fixed as in Fig. 1. The dotted and dot-dashed lines, labeled “M SA” and “M BS,” respectively, show the dynamics under the BMA with and without the SA. The arrow below the horizontal axis indicates the point where the sign of the values from “M BS” changes from negative to positive. For comparison, we show the non-Markovian dynamics with and without the SA (dashed and solid lines labeled “NM SA” and “NM BS,” respectively.

Figure 5

Dependence of ${\rho }_{33}(\tilde{t}=0.1)$ obtained by the GA beyond the SA on the inverse temperature $\tilde{\beta }$ and the interaction strength between sites ${\tilde{V}}_{12}$; all other parameters are fixed as in Fig. 1. (a) Under BMA beyond the SA (“M BS”) the values are negative, and (b) under non-Markovian dynamics beyond the SA (“NM BS”) the values are strictly positive. We find that the positivity in the short-time region is recovered by the treatment “NM BS.”

Figure 6

Time evolution of each element of $\rho \left[\tilde{t}(={\omega }_{2}t\right)]$ obtained by the LA beyond the BMA and the SA with the same settings as given in Fig. 1. We find that the stationary values of the populations at the sites 1 and 2 coincide, ${\rho }_{L,11}={\rho }_{L,22}$, and the stationary value of coherence $\mathrm{Re}\left[{\rho }_{L,12}\right]$ vanishes.

Figure 7

(a) Dependence of the stationary values obtained by the LA on the inverse temperature $\tilde{\beta }$ and interaction strength between sites ${\tilde{V}}_{12}$. (b) Dependence of the trace distance, $D({\rho }_L,{\rho }_{\mathrm{Gibbs}})$, on $\tilde{\beta }$ and ${\tilde{V}}_{12}$. The difference between the obtained stationary state, ${\rho }_L$, and the Gibbs state for the total relevant system, ${\rho }_{\mathrm{Gibbs}}$, is much larger compared with $D({\rho }_G,{\rho }_{\mathrm{Gibbs}})$ [Fig. 2].

Figure 8

Dependence of the smallest eigenvalue of the stationary state obtained by the GA beyond the SA on the inverse temperature $\tilde{\beta }$ and the interaction strength between sites ${\tilde{V}}_{12}$ with setting ${\tilde{\omega }}_1=5/4$. The white line marks the boundary between positive and negative values; below the line, the eigenvalues are positive and in crossing the boundary change to very small negative values. Other parameters are set the same as given in Fig. 1.