Fingerprints of classical memory in quantum hysteresis

TL;DR

The paper develops a memory-kernel framework for quantum control where the device Hamiltonian contains a history-dependent term $H(t)=H_A+\int_{-\infty}^{t} K(t-s)\,H_c(s)\,ds$, yielding a realized field $\Phi(t)$ that encodes classical control memory. It introduces loop-based hysteresis measures $\mathcal{A}_{u\Phi}$, $\mathcal{A}_{\ΦO}$, and $\mathcal{A}_{uO}$ to distinguish memory in the control channel from genuine quantum memory, and demonstrates how adiabatic dynamics map $O(t)$ to a single-valued function $f(\Phi)$ so that $\mathcal{A}_{\Phi O}\approx 0$. The authors justify the exponential-mode embedding by connecting the kernel to passive RC ladder physics and Kubo linear response, showing that RC-type memory yields a finite set of auxiliary modes with $\dot{\Phi}_k=-\nu_k\,\Phi_k+c_k u$, enabling a time-local description while preserving unitary evolution of the system. Numerical and analytic results in single-qubit models illustrate how classical memory and nonadiabatic dynamics shape the observed loops, and provide practical diagnostics for separating control-memory effects from true quantum information backflow. The work offers a principled framework for interpreting hysteresis in driven quantum devices and for designing control protocols that explicitly account for hardware-induced memory.

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 $H(t)=σ_z+Φ(t)σ_x$ 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.

Figures (5)

• Figure 1: Representative steady-cycle parametric loops for a filtered transverse drive. A periodic command $u(t)$ (triangle protocol in the example shown) is applied for many cycles; after discarding transients, the final cycle is plotted parametrically as three loops: (left) the control-channel loop$(u,\Phi)$ quantifying classical filtering and phase lag; (middle) the commanded observable loop$(u,O)$ with $O(t)=\langle\sigma_z\rangle_t$; (right) the realized-drive loop$(\Phi,O)$ which isolates state history dependence at fixed realized field. The $(u,\Phi)$ loop is nontrivial whenever the control channel has memory ($\mathcal{A}_{u\Phi}\neq 0$). In contrast, a nontrivial $(\Phi,O)$ loop indicates nonadiabatic quantum response under the realized drive; in the adiabatic-following regime it collapses toward a single-valued curve and $\mathcal{A}_{\Phi O}\approx 0$.
• Figure 2: Testing the adiabatic "classical-memory imprint" prediction. The numerically extracted commanded-area $\tilde{\mathcal{A}}_{uO}$ (markers) is compared to the adiabatic small-amplitude prediction obtained from the control-channel area $\tilde{\mathcal{A}}_{u\Phi}$ via $\tilde{\mathcal{A}}_{uO}\approx f'(0)\,\tilde{\mathcal{A}}_{u\Phi}$ (dashed curve), where $f(\Phi)$ is the adiabatic response function of the qubit observable. Points satisfying the adiabaticity diagnostic $\epsilon_{\rm ad}^{\max}<\epsilon_0$ are highlighted. Agreement holds in the regime where both (i) the realized field varies slowly compared to the instantaneous gap and (ii) the amplitude is sufficiently small for linearization of $f(\Phi)$ about $\Phi=0$.
• Figure 3: Schematic connection between a phenomenological classical control filter and a microscopic quantum control-channel model. Top: the room-temperature controller outputs a command $u(t)$, which is distorted by a passive control line (e.g. an RC ladder) into a realized in-situ field $\Phi(t)=(K*u)(t)$ that drives the device $A$ via $\hat{H}_A+\Phi(t)\hat{M}$. Bottom: the same architecture is modeled microscopically by a channel $B$ with Hamiltonian $\hat{H}_B$, driven at the input port by the classical source $u(t)$ through an operator $\hat{F}$ and coupled at the device port through $g\,\hat{M}\otimes\hat{L}$. In the weak-coupling regime (Born/Oppenheimer) limit and linear response (Kubo), the delivered field is the channel expectation $\Phi(t)=\langle \hat{L}\rangle_t=\int_{-\infty}^t \chi_{LF}(t-s)\,u(s)\,ds$, yielding the effective filtered Hamiltonian $\hat{H}_A+\int^t K(t-s)u(s)\,ds\,\hat{M} \equiv \hat{H}_A+\Phi(t)\hat{M}$ with $K$ identified as the retarded susceptibility of the control.
• Figure 4: In practice, the filter at the target device is the result of various stages of the stack.
• Figure 5: Discrete RC ladder (lossy line) model of the control channel. The commanded input voltage $u(t)$ drives a chain of series resistors with shunt capacitors to ground. The delivered device-node voltage is $\Phi(t)=V_N(t)$.

Theorems & Definitions (14)

• Lemma 1: Parameterization formula
• proof
• Proposition 1: Single-valued $O=f(\Phi)$ $\Rightarrow$ zero loop integral
• proof
• Remark 1
• Theorem 1: Variation bound
• proof
• Corollary 1: Sup-norm and Cauchy--Schwarz bounds
• proof
• Lemma 2: Two-branch formula for a single turning point