Fingerprints of classical memory in quantum hysteresis

TL;DR

This paper introduces a framework for classical and quantum memory effects in Hamiltonian dynamics, emphasizing the role of past control influences and illustrating how hysteresis manifests in driven qubits.

Contribution

It provides a novel, simple framework for modeling classical and quantum memory in Hamiltonian systems, distinguishing control memory from non-Markovian dynamics, with explicit examples and properties.

Findings

Hysteresis appears in the response of driven qubits.

The framework distinguishes control memory from non-Markovian effects.

An equivalent time-local description exists for exponential kernels.

Abstract

We present a simple framework for classical and quantum ``memory'' in which the Hamiltonian at time $t$ depends on past values of a control Hamiltonian through a causal kernel. This structure naturally describes finite-bandwidth or filtered control channels and provides a clean way to distinguish between memory in the control and genuine non-Markovian dynamics of the state. We focus on models where $H(t)=H_0+\int_{-\infty}^{t}K(t-s)\,H_1(s)\,ds$, and illustrate the framework on single-qubit examples such as $H(t)=\sigma_z+\Phi(t)\sigma_x$ with $\Phi(t)=\int_{-\infty}^{t}K(t-s)\,u(s)\,ds$. We derive basic properties of such dynamics, discuss conditions for unitarity, give an equivalent time-local description for exponential kernels, and show explicitly how hysteresis arises in the response of a driven qubit.