Weak-Memory Dynamics in Discrete Time

Abstract

Discrete dynamics arise naturally in systems with broken temporal translation symmetry and are typically described by first-order recurrence relations representing classical or quantum Markov chains. When memory effects induced by hidden degrees of freedom are relevant, however, higher-order discrete evolution equations are generally required. Focusing on linear dynamics, we identify a well-delineated weak-memory regime where such equations can, on an intermediate time scale, be systematically reduced to a unique first-order counterpart acting on the same state space. We formulate our results as a mathematical theorem and work out two examples showing how they can be applied to stochastic Floquet dynamics under coarse-grained and quantum collisional models.

Figures (3)

• Figure 1: Top: Mesoscopic charge pump. The device consists of two quantum dots, each coupled to a thermochemical reservoir, and admits three microstates corresponding to the left dot, right dot, or neither being occupied. Charge pumping proceeds in a three-stroke cycle: a particle is absorbed by the left dot (a), tunnels between the dots (b), and is ejected from the right dot (c), with respective probabilities $L_+$, $L_0$ and $L_-$. An observer that measures the total charge of the system distinguishes two mesostates, whose dynamics are non-Markovian. Bottom: Collisional model with memory. A quantum system $S$ interacts sequentially with a stream of identical ancillas $A_i$. Ancilla $A_1$ first collides with the system (d), then with $A_2$ (e), before leaving the scattering region (f). System-ancilla and ancilla-ancilla interactions are described by bipartite maps $\mathcal{U}_i$ and $\mathcal{Q}_{ij}$, respectively.
• Figure 2: Mesoscopic charge pump. Left: For $L_+ = L_- = L$, the shaded area indicates the region in the parameter space of the model, where the weak-memory conditions \ref{['eq:WMC1']} and \ref{['eq:WMC2']} are satisfied with $v=1/\lVert \mathsf{V}^{-1} \rVert$ and $M=\lVert \mathsf{K}_1 \rVert$. Right: Error of the long-time approximation $Y_n=\mathsf{G}^n\mathsf{D} X_0$ with respect to the exact solution $X_n$ of Eq. \ref{['eq:starting_point']} (solid), for $X_0 = [1,0]^\mathsf{T}$ and selected parameter sets, indicated with dots in the left panel. For comparison, we also show the error bound \ref{['eq:BndErrLTA']} (dashed).
• Figure 3: Collisional model. Left: Weak-memory regime in the parameter space defined by the swap probabilities $u$ and $k$, with $v = 1/\lVert \mathcal{K}_0^{-1} \rVert$ and $M=k-k^2$. The ancillas are initially in fully mixed states. Right: Error of the long-time approximation \ref{['eq:LTACM']} for different approximate generators $\mathcal{G}_\ell = T(\mathcal{G}_{\ell-1})$ (solid), together with the bound \ref{['eq:BndLTACM']} (dashed). For all plots, we used the model parameters indicated by the dot in the left panel, and set $\rho_0 = \lvert 0_S \rangle\langle 0_S \rvert$.