Control of memory effects in a spin-boson system by periodic driving

We study the emergence of quantum memory effects in a spin-boson system at finite temperature driven by an external time-periodic force. Quantifying memory effects by the trace-distance based measure for non-Markovianity and performing numerical simulations employing the hierarchical equations of motion approach, we find a pronounced peak structure when plotting the non-Markovianity measure as a function of the driving amplitude. This distinctive feature is interpreted using Floquet theory and the Floquet-Lindblad master equation, associating the peaks with the degeneracies of the quasienergy spectrum which lead to a strong enhancement of the relaxation times of the system. These results suggest strategies for the efficient control of non-Markovianity in open quantum systems by periodic driving.

💡 Research Summary

In this work the authors investigate how a periodic external drive can be used to control memory effects—i.e., non‑Markovian behavior—in a spin‑boson system at finite temperature. The system consists of a two‑level spin coupled linearly to a bosonic bath with a Lorentz‑Drude spectral density. A monochromatic drive of the form (\Omega\cos(\omega_0 t)) is applied to the spin, making the system Hamiltonian time‑periodic with period (T=2\pi/\omega_0).

Non‑Markovianity is quantified using the trace‑distance based measure introduced by Breuer, Laine and Piilo (BLP). The authors compute the BLP measure (\mathcal{N}) by numerically maximizing the integrated positive time‑derivative of the trace distance over a large set of orthogonal initial state pairs. To obtain the exact reduced dynamics they employ the hierarchical equations of motion (HEOM) method, which is numerically exact for the chosen Lorentz‑Drude bath and allows extraction of the full time‑dependent density matrix of the spin.

Scanning the driving amplitude (\Omega) while keeping temperature, coupling strength (\alpha), and cutoff frequency (\omega_c) fixed, they observe a striking series of peaks in (\mathcal{N}). Simultaneously, the overall relaxation time (\tau) of the spin—defined as the longest exponential decay time among all matrix elements—exhibits peaks at exactly the same values of (\Omega). This correlation suggests that the enhancement of memory effects is directly linked to a slowdown of the dissipative dynamics.

To explain the phenomenon, the authors turn to Floquet theory. Because the Hamiltonian is periodic, the evolution operator can be expressed in terms of Floquet states (|u_k(t)\rangle) and quasienergies (\epsilon_k). By numerically diagonalizing the one‑period propagator they obtain the quasienergy spectrum as a function of (\Omega). The peaks in (\mathcal{N}) coincide with points where the two quasienergies become degenerate (crossings). At these degeneracies the structure of the Floquet‑Lindblad master equation changes dramatically.

In the non‑degenerate regime the Floquet‑Lindblad dissipator contains three jump operators: (\Sigma_+=|u_1\rangle\langle u_2|), (\Sigma_-=\Sigma_+^\dagger), and (\Sigma_z=|u_1\rangle\langle u_1|-|u_2\rangle\langle u_2|). Their associated rates (\gamma_{\downarrow},\gamma_{\uparrow},\gamma_z) are of comparable magnitude, leading to relaxation times for populations and coherences that are all of the same order.

When a quasienergy crossing occurs ((\epsilon_1=\epsilon_2)), the transition frequencies between the two Floquet states collapse to zero. Consequently the two transition operators (\Sigma_+) and (\Sigma_-) combine coherently into a single operator (\Sigma_x=|u_1\rangle\langle u_2|+|u_2\rangle\langle u_1|). The Lindblad dissipator now contains only two channels: (\Sigma_x) (with a very small rate (\gamma_x) originating from the low‑frequency limit of the spectral density) and (\Sigma_z) (with the same rate as before). This reduction of dissipative channels produces a markedly longer decay time for the real part of the off‑diagonal element, (\tau_{\text{off-diag}}^{\Re}=25/(2\gamma_z)), while other components still decay on the faster scales set by (\gamma_x) or (\gamma_z). Effectively a nearly decoherence‑free subspace aligned with the (\Sigma_x) direction emerges.

Because the information backflow that defines non‑Markovianity accumulates over the longest surviving coherence, the extended relaxation time at quasienergy degeneracies yields the observed peaks in the BLP measure. The optimal pair of initial states that maximizes (\mathcal{N}) are eigenstates of (\Sigma_x), precisely the states whose coherence decays with the longest time constant.

Importantly, the role of the specific spectral density is limited to setting the relative magnitudes of the rates; the mechanism—quasienergy degeneracy leading to a reduced set of jump operators and a long‑lived coherence—is generic and should persist for other bath models. Thus, by tuning the drive amplitude (or frequency) one can deliberately bring the system into or out of quasienergy degeneracy, thereby switching the degree of non‑Markovianity on demand.

The paper concludes that periodic driving offers a powerful, experimentally accessible knob for engineering memory effects in open quantum systems. This capability could be exploited in quantum information processing (e.g., protecting coherence or generating controlled information backflow), quantum metrology (enhancing sensitivity via non‑Markovian reservoirs), and quantum thermodynamics (designing engines that benefit from memory). The combination of numerically exact HEOM simulations with analytical Floquet‑Lindblad insight provides a robust framework for future studies of driven open quantum systems.