Fingerprints of classical memory in quantum hysteresis

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 $H(t)=σ_z+Φ(t)σ_x$ with $Φ(t)=\int_{-\infty}^{t}K(t-s),u(s),ds$. 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.

💡 Research Summary

The authors introduce a unified framework for incorporating classical memory effects into quantum control Hamiltonians by convolving the commanded control signal u(t) with a causal kernel K(t‑s). The resulting realized field Φ(t)=∫_{‑∞}^{t}K(t‑s)u(s)ds appears in the Hamiltonian as H(t)=H₀+Φ(t)M, where M is a fixed Hermitian operator (e.g., a Pauli matrix). By requiring K to be real‑valued and integrable on ℝ⁺, the instantaneous Hamiltonian remains Hermitian for any u(t), guaranteeing unitary evolution of the closed quantum system.

A key technical contribution is the time‑local embedding of kernels that are finite sums of decaying exponentials, K(τ)=∑{k=1}^{K_max}c_k e^{‑ν_k τ}Θ(τ). Introducing auxiliary variables Φ_k(t)=∫{‑∞}^{t}c_k e^{‑ν_k (t‑s)}u(s)ds, each satisfies a first‑order differential equation \dot{Φ}_k=‑ν_k Φ_k + c_k u. The full dynamics become a coupled set of ordinary differential equations: the Schrödinger equation for the state and the linear ODEs for the filter variables. This representation mirrors the behavior of passive RC networks and makes the otherwise non‑local convolution computationally tractable.

To diagnose memory effects, the paper defines loop‑area measures in the (u, Φ), (u, O) and (Φ, O) planes, where O(t) is any observable. In the adiabatic regime, O(t)≈f(Φ(t)) so that the area A_{ΦO}=∮O dΦ vanishes, yet the area A_{uO}=∮O du can remain non‑zero because the control channel’s memory imprints a deterministic hysteresis on the observable. Conversely, a non‑zero A_{ΦO} at fixed Φ indicates genuine quantum non‑Markovian behavior arising from system‑environment coupling. Simultaneous measurement of A_{uO} and A_{ΦO} therefore provides a clear operational separation between classical control memory and true quantum memory.

The authors also discuss the instantaneous‑limit K→g δ(t), obtained by sending all decay rates ν_k→∞ while keeping the weighted gains c_k/ν_k finite. In this limit the filter collapses to a memoryless gain, and the Hamiltonian reduces to the familiar H(t)=H₀+g u(t) M.

Concrete single‑qubit examples illustrate the theory. For H(t)=σ_z+Φ(t)σ_x with a sinusoidal command u(t)=u₀ sin(ωt), numerical simulations show that low‑frequency drives are followed adiabatically, whereas higher frequencies produce phase lag, amplitude attenuation, and sizable hysteresis loops. By varying the kernel parameters ν_k and c_k, the authors demonstrate how to tune the strength of the hysteresis, effectively controlling the amount of classical memory injected into the quantum dynamics.

Overall, the work provides a practical, analytically transparent method to model finite‑bandwidth control lines, to quantify the resulting hysteresis, and to distinguish it from genuine open‑system non‑Markovian effects. This framework can guide the design of control hardware with appropriate bandwidth, improve calibration protocols, and enable more reliable identification of true quantum memory in experimental platforms.