Fingerprints of classical memory in quantum hysteresis

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 $H(t)=\ddot{I}{ƒ}_z+\hat{I}¦(t)\ddot{I}{ƒ}_x$ with $\hat{I}¦(t)={\int }_{-\infty }^{t}K(t-s)u(s)ds$. 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.

22 pages double column; 26 pages appendix