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.
Categories: quant-ph cond-mat.other