How Quantum Evolution with Memory is Generated in a Time-Local Way

Two widely used but distinct approaches to the dynamics of open quantum systems are the Nakajima-Zwanzig and time-convolutionless quantum master equation, respectively. Although both describe identical quantum evolutions with strong memory effects, the first uses a time-nonlocal memory kernel $\mathcal{K}$, whereas the second achieves the same using a time-local generator $\mathcal{G}$. Here we show that the two are connected by a simple yet general fixed-point relation: $\mathcal{G}=\widehat{\mathcal{K}}[\mathcal{G}]$. This allows one to extract nontrivial relations between the two completely different ways of computing the time evolution and combine their strengths. We first discuss the stationary generator, which enables a Markov approximation that is both nonperturbative and completely positive for a large class of evolutions. We show that this generator is not equal to the low-frequency limit of the memory kernel, but additionally “samples” it at nonzero characteristic frequencies. This clarifies the subtle roles of frequency dependence and semigroup factorization in existing Markov approximation strategies. Second, we prove that the fixed-point equation sums up the time-domain gradient or Moyal expansion for the time-nonlocal quantum master equation, providing nonperturbative insight into the generation of memory effects. Finally, we show that the fixed-point relation enables a direct iterative numerical computation of both the stationary and the transient generator from a given memory kernel. For the transient generator this produces nonsemigroup approximations which are constrained to be both initially and asymptotically accurate at each iteration step.

Popular Summary

Predicting the behavior of quantum devices requires a thorough understanding of how they are disturbed by their immediate environment. Unfortunately, in many cases of scientific and technological interest, the environment’s reaction to the device is not instantaneous but occurs with some time delay or memory, which quickly complicates the analysis. Surprisingly, it is possible, without introducing any error, to describe such devices as if there were no time delay at all. Here, we provide precise insight into how this very useful workaround can be performed.

The “memory” introduced by the environment can be explicitly characterized by a mathematical function—a “filter” function—that determines what noise frequencies are passed on to the system, leading to damping. In contrast, when modeling the system as if there were no memory, we use a different function that directly determines the characteristic frequency at which the system evolves.

Our main result is simply to express this latter function in terms of the filter function: The characteristic frequency must match the damping, which, in turn, depends on this characteristic frequency. Although this is very similar to a resonance condition of ordinary mechanical and electrical systems, it applies to much more complicated quantum devices.

The practical necessity to choose a description either with explicit memory or without explicit memory has resulted in the development of two disconnected lines of research on open quantum systems. Our result reunites these efforts, allowing for a new exchange of independently gained insights and technical expertise. We explicitly demonstrate how our elegant equation can be used to solve quantum evolution with strong memory effects in new ways.

Figure 1

Graphical representation of the derivation of the functional fixed-point equation (10). (a) Equivalent expressions for $d\mathrm{\Pi }(t,t_0)/dt$ as given by the two QMEs. (b) Insertion of canceling backward and forward propagation to initial time $t_0$. (c) Evolution ${\mathcal{T}}_{\leftarrow }\exp \{{\int }_{t_0}^{t}d\tau [-i\mathcal{G}(\tau ,t_0)]\}={\lim }_{N\to \infty }[\mathcal{I}-i\mathcal{G}(t_1)\mathrm{\Delta }{t}_1]\dots [\mathcal{I}-i\mathcal{G}(t_N)\mathrm{\Delta }{t}_N]$ expressed as product of infinitesimal steps for the sake of illustration. (d) Backward propagation to memory time $s$ expressed in terms of $\mathcal{G}$ using the divisor. The self-consistency expressed by the functional fixed-point equation (10) arises from backward propagation that is needed to enforce the time-local structure of QME (2) onto the QME (1). For time-translational systems in the stationary limit the generator becomes $\mathcal{G}(\tau ,t_0)\to \mathcal{G}(\infty )$ and literally takes on the role of the complex frequency at which $\widehat{\mathcal{K}}(\omega )$ is sampled in Eq. (15).

Figure 2

Jaynes-Cummings model, overdamped regime $\mathrm{\Gamma }/\gamma =0.495$ (${\gamma }^{\prime }/\gamma =0.1$). Decay of the probability $⟨1|\rho (t)|1⟩$ for the excited state when it is initially occupied, $\rho (0)=|1⟩⟨1|$. (a) Solutions obtained from the Markovian approximation $\dot{\rho }(t)\approx -i\widehat{\mathcal{K}}(0)\rho (t)$ (red line) together with the first iteration ${\mathcal{G}}^{(1)}$ (cyan line) of the transient fixed-point equation starting from ${\mathcal{G}}^{(0)}=\widehat{\mathcal{K}}(0)$. The exact solution is shown in black. (b) Solutions obtained from the Markovian approximation $\dot{\rho }(t)\approx -i\mathcal{G}(\infty )\rho (t)$ (blue line) together with two iterations of the transient fixed-point equation (30), ${\mathcal{G}}^{(1)}$ (cyan line) and ${\mathcal{G}}^{(2)}$ (green line), when started from ${\mathcal{G}}^{(0)}=\mathcal{G}(\infty )$ and exact solution (black line).

Figure 3

Jaynes-Cummings model, underdamped regime $\mathrm{\Gamma }/\gamma =13$ ($\mathrm{\Omega }/\gamma =5$) and $ϵ=20$. (a) Decay of the excited state occupation $⟨1|\rho (t)|1⟩$ and (b) decay of the real part of the coherence $⟨0|\rho (t)|1⟩$. The initial state is $\rho (0)=0.1[|00)+|01)+|10)]+0.9|11)$. Shown are the solution for the Markovian approximations $\dot{\rho }(t)\approx -i\mathcal{G}(\infty )\rho (t)$ (blue line) with generator obtained by iteration of the stationary fixed point equation (15), and $\dot{\rho }(t)\approx -i\widehat{\mathcal{K}}(0)\rho (t)$ (gray line). The exact solution is shown in black.

Figure 4

Jaynes-Cummings model, underdamped regime. Decay of the probability $⟨1|\rho (t)|1⟩$ obtained from generators of the transient iteration for $\mathrm{\Gamma }/\gamma =13$ and $ϵ=20$ using the starting point $\widehat{\mathcal{K}}(0)$.

Figure 5

Decay of the occupation $⟨1|\rho (t)|1⟩$ in the resonant level model for $ϵ-\mu =2\pi \mathrm{\Gamma }$, $T=0.1\times \mathrm{\Gamma }/(2\pi )$ obtained from ${\mathcal{G}}^{(0)}(t)=\widehat{\mathcal{K}}(0)$ and ${\mathcal{G}}^{(1)}(t)=\mathcal{G}(t)$. The former corresponds to a Markov semigroup approximation using $\widehat{\mathcal{K}}(0)$, which is never able to describe the initial growth of the occupation away from the stationary value. When used as initial guess in the transient iteration the exact solution is recovered by the fixed-point equation (10) after a single step.