Unraveling a Brownian particle’s memory with effective mode chains

Memory effects in quantum dynamical processes involving structured environments are presently difficult, if not impossible, to investigate using standard approaches. Progress can be made by transforming the environmental variables to a suitable chain representation which effectively performs a Markovian embedding of the dynamics. Here, we show that this effective-mode chain representation provides a unique way of unraveling the memory kernel $\kappa \left(t\right)$ as a function of time. Truncated or Markov-closed chains with $n$ effective modes exactly reproduce $\kappa \left(t\right)$ to the $4n$th order in time, up to an irrelevant constant of order $\kappa \left(0\right)/n$. These favorable convergence properties pave the way for efficient quantum simulations of fast (non-Markovian) processes by reduced dynamical models.

Figure 1

Schematic illustration of the linear-chain representation of the bath. $X_n$ are the effective modes, ${\Omega }_n$ their frequencies, and $D_n$ the couplings $X_{n-1}-X_n$ ($X_0=s$). The spectral densities $J_n\left(\omega \right)=\text{Im}{W}_n^{+}\left(\omega \right)$ describe the interaction of the $n$th effective modes with their residual baths.

Figure 2

A typical, structured spectral density [panel (a)] and its effective mode parameters ${\Omega }_n^2$ (squares) and $D_n$ (circles) [panel (b)].

Figure 3

The memory kernel offset $\kappa \left(t\right)-\kappa \left(0\right)$ for the spectral density shown in Fig. 2a (thick black line) and for chains with $n=1$–10 effective modes. In panel (a) the chains have been truncated and results are shown up to $t\sim 2\pi /{\Omega }_1$, where ${\Omega }_1$ is the lowest eigenfrequency of the chain, with $n$ increasing from left to right. In panel (b) a Rubin closure has been applied to the chains, and results have been color and symbol coded as in the left panel (for clarity for $n=1$–5 only). The insets show the dependence of $\kappa \left(0\right)$ on the chain length, compared to the exact value (thick horizontal line).

Figure 4

Linear-chain approximants to the real [$S_{\beta }\left(t\right)$] and imaginary [$A\left(t\right)$] parts of the force correlator given by the spectral density reported in Fig. 2a. The results for Rubin-closed chains (for clarity, for $n=1$–5 only; symbols as in Fig. 3) are compared to the exact results, plotted as thick black lines, for $T={10}^{-9}$ K.