Time-correlated blip dynamics of open quantum systems

The non-Markovian dynamics of open quantum systems is still a challenging task, particularly in the nonperturbative regime at low temperatures. While the stochastic Liouville–von Neumann equation (SLN) provides a formally exact tool to tackle this problem for both discrete and continuous degrees of freedom, its performance deteriorates for long times due to an inherently nonunitary propagator. Here we present a scheme that combines the SLN with projector operator techniques based on finite dephasing times, gaining substantial improvements in terms of memory storage and statistics. The approach allows for systematic convergence and is applicable in regions of parameter space where perturbative methods fail, up to the long-time domain. Findings are applied to the coherent and incoherent quantum dynamics of two- and three-level systems. In the long-time domain sequential and superexchange transfer rates are extracted and compared to perturbative predictions.

Figure 1

Illustration of a distinct quantum system embedded in a thermal reservoir. The coupling to the environment provokes phenomena such as fluctuating forces that act on the system and the loss of energy due to dissipation.

Figure 2

Scheme of “memory recycling” in the time-correlated blip dynamics.

Figure 3

Comparison of the TCBD and the full SLED [Eq. (11)] for expectation values ${\langle {\sigma }_x\rangle }_t$ and ${\langle {\sigma }_z\rangle }_t$ of a spin-boson model with $\varepsilon =0,\beta =0.7,K=0.24,{\omega }_c=10,{\tau }_m=2$, and $n_{\mathrm{samp}}=2500$; frequencies are in units of $\mathrm{\Delta }$.

Figure 4

Comparison of the numerical variance for TCBD and SLED for ${\langle {\sigma }_z\rangle }_t$; parameters are the same as in Fig. 3.

Figure 5

NIBA and TCBD relaxation dynamics towards equilibrium of the population difference ${\langle {\sigma }_z\rangle }_t$ for $\varepsilon =0.5,\beta =5,K=0.1,{\omega }_c=10$, and $n_{\mathrm{samp}}=100$; frequencies are in units of $\mathrm{\Delta }$.

Figure 6

Symmetric three-state donor-bridge-acceptor system with donor $\left|1\right.〉$ and acceptor $\left|3\right.〉$ states and a bridge state $\left|2\right.〉$ elevated by an energy $\hslash \varepsilon$. In the case of incoherent population transfer, two transfer channels with transition rate ${\mathrm{\Gamma }}_{DB}$ for sequential hopping and ${\mathrm{\Gamma }}_{SQM}$ for superexchange govern the dynamics.

Figure 7

Population dynamics $p_j\left(t\right)=\text{Tr}\left\{\left|j\right.〉\left.〈j\right|\rho \left(t\right)\right\},j=\{1,2,3\}$, for $\varepsilon =1,K=0.24,\beta =5,n_{\mathrm{samp}}=2\times {10}^4$, and ${\omega }_c=10$; frequencies are in units of $\mathrm{\Delta }$.

Figure 8

Same as in Fig. 7 but for $\varepsilon =3$.

Figure 9

Same as in Fig. 7 but for $\varepsilon =17$.

Figure 10

Perturbative rates ${\mathrm{\Gamma }}_{DB}^{GR}\left(\varepsilon \right)$ and ${\mathrm{\Gamma }}_{SQM}^{GR}\left(\varepsilon \right)$. The changeover from the regime where sequential hopping dominates to the regime of superexchange occurs around $\varepsilon \approx 2.5$. The other parameters are $K=0.24$ and $\beta =0.7$; frequencies are in units of $\mathrm{\Delta }$.

Figure 11

Rate for sequential hopping ${\mathrm{\Gamma }}_{DB}\left(\varepsilon \right)$ and superexchange rate ${\mathrm{\Gamma }}_{SQM}\left(\varepsilon \right)$ as extracted from TCBD simulations for the same parameters as in Fig. 10. Error bars from the curve fitting procedure are indicated by vertical lines.

Figure 12

Sequential hopping rate ${\mathrm{\Gamma }}_{DB}\left(\varepsilon \right)$ as extracted from TCBD simulations for the same parameters as in Fig. 10. The linear fit $e^{m·\varepsilon }$ with slope $m\approx -0.6$ demonstrates good agreement with a thermally activated process with $-\beta \equiv m_{\beta }=-0.7$.

Figure 13

Superexchange rate ${\mathrm{\Gamma }}_{SQM}\left(\varepsilon \right)$ as extracted from TCBD simulation data for the same parameters as in Fig. 10. An algebraic dependence $\frac{1}{{\varepsilon }^{\left|m\right|}}$ with $\left|m\right|\approx 1.96$ is in agreement with a superexchange mechanism for which the expected asymptotic prediction is $\left|m\right|=2$.

Figure 14

Population dynamics for the DBA complex via the ME and the TCBD method (inset) for $K=0.08,\beta =7,{\omega }_c=50$, and $\varepsilon =1$; frequencies are in units of $\mathrm{\Delta }$.

Figure 15

Same as in Fig. 14 but for $K=0.24,\beta =0.7,{\omega }_c=50$, and $\varepsilon =2$.

Figure 16

Sequential transfer rates ${\mathrm{\Gamma }}_{DB}^{\left(ME\right)}\left(\varepsilon \right)$ as derived via the rate model (32) for various damping strengths $K=0.08,0.16,0.24$ and $\beta =0.7$. A linear fit $e^{m^{\left(ME\right)}\varepsilon }$ with $m^{\left(ME\right)}\approx -0.24$ reveals strong deviations from the classical expectation for thermal activation $-\beta \equiv m_{\beta }=-0.7$ and the TCBD result $m\approx -0.6$ in Fig. 12; frequencies are in units of $\mathrm{\Delta }$.

Figure 17

Same as in Fig. 16 but for the superexchange rates ${\mathrm{\Gamma }}_{SQM}^{\left(ME\right)}\left(\varepsilon \right)$. These results do not show the expected algebraic dependence on $\varepsilon$ (see the text for details).

Figure 18

Decoherence rates according to the ME approach (44) in the energy basis compared to the numerically extracted superexchange rate ${\mathrm{\Gamma }}_{SQM}^{\left(ME\right)}$ in the site representation (48). While the rate ${\mathrm{\Gamma }}_R$ for energy relaxation decreases with increasing bridge height, the rate for the decay of coherences in the energy basis (47) tends to dominate and determines ${\mathrm{\Gamma }}_{SQM}^{\left(ME\right)}$ (see the text for details); frequencies are in units of $\mathrm{\Delta }$.