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

Technical Summary: Control of Memory Effects in a Spin-Boson System by Periodic Driving

Problem Statement
Realistic quantum systems are rarely isolated; they interact with environments, leading to dissipation, decoherence, and potentially non-Markovian dynamics where information flows back from the environment to the system. While non-Markovianity in undriven spin-boson models is well-studied, its behavior in periodically driven variants remains largely unexplored. This work investigates the emergence and control of quantum memory effects in a finite-temperature, periodically driven spin-boson system. Specifically, the authors aim to understand how external periodic driving influences the non-Markovianity of the open system and whether specific driving parameters can enhance or suppress these memory effects.

Methodology
The study employs a driven spin-boson model where a two-level system (Hamiltonian $H_S(t)$) is coupled to a bath of harmonic oscillators via a linear interaction term. The system is subjected to a time-periodic driving force of the form $-\Omega \cos(\omega_0 t)\sigma_x$, with $\Omega$ representing the driving amplitude. The environment is characterized by a Lorentz-Drude spectral density $J(\omega)$ at finite temperature.

To analyze the dynamics, the authors utilize two complementary approaches:

1. Numerical Simulation: They employ the Hierarchical Equations of Motion (HEOM) approach, which provides a numerically exact treatment of the reduced system dynamics. This allows for the direct computation of the non-Markovianity measure $\mathcal{N}$, defined via the trace distance between two quantum states. The measure quantifies information backflow by integrating the positive time derivative of the trace distance over all possible orthogonal pairs of initial states.
2. Theoretical Analysis: The authors apply Floquet theory to derive a Floquet-Lindblad master equation. This framework describes the system in the weak-coupling limit using quasienergies and Floquet states. They analyze the structure of the Lindblad jump operators and decay rates in both non-degenerate and degenerate (quasienergy crossing) regimes to explain the numerical findings.

Key Results
The numerical simulations reveal a distinctive peak-like structure in the non-Markovianity measure $\mathcal{N}$ as a function of the driving amplitude $\Omega$. These peaks correspond one-to-one with peaks in the relaxation times of the system.

The theoretical analysis attributes these peaks to degeneracies in the quasienergy spectrum of the driven system. The key findings regarding the mechanism are:

• Non-Degenerate Regime: Away from quasienergy crossings, the Floquet-Lindblad master equation yields three active dissipative channels (jump operators corresponding to $\Sigma_+$, $\Sigma_-$, and $\Sigma_z$). This leads to standard relaxation times.
• Degenerate Regime (Quasienergy Crossings): At specific driving amplitudes where quasienergies cross ($\epsilon_1 = \epsilon_2$), the structure of the jump operators changes fundamentally. The transition operators for upward and downward transitions coalesce into a single operator $\Sigma_x$, while the elastic scattering operator $\Sigma_z$ remains. Consequently, only two dissipative channels ($\Sigma_x$ and $\Sigma_z$) remain active.
• Decoherence-Free Subspace: This reduction in active channels creates a nearly decoherence-free subspace along the $x$-component in the Floquet basis. Specifically, the relaxation time associated with the real part of the coherences becomes significantly longer than all other time scales ($\tau_{\text{off-diag}}^D(\Re) \gg \tau_{\text{off-diag}}^D(\Im), \tau_{\text{diag}}^D, \tau_{\text{diag}}^{ND}$).
• Information Backflow: The extended relaxation times allow for persistent, albeit small, information backflow processes to accumulate over longer durations. Since the non-Markovianity measure is determined by the maximum relaxation time (specifically for initial states that are eigenstates of $\Sigma_x$), this leads to the observed peaks in $\mathcal{N}$.

Significance and Claims
The paper claims to provide a theoretical explanation for the control of non-Markovianity in open quantum systems via periodic driving. The authors assert that the observed peaks are not merely artifacts but result from a robust physical mechanism: the emergence of a protected channel due to quasienergy degeneracies.

Key aspects of the paper's contribution include:

• Mechanism Identification: Unlike previous studies that relied on high-frequency approximations or specific noise sources, this work identifies a mechanism where the interplay between quasienergy degeneracies and exact Fourier coefficients of the coupling operator leads to a decoherence-free channel.
• Universality: The mechanism is described as robust and not dependent on specific parameter choices or low-temperature regimes, distinguishing it from prior suggestions in the literature.
• Control Strategy: The results suggest a strategy for controlling memory effects: by tuning the driving amplitude to match quasienergy crossings, one can significantly extend coherence times and enhance non-Markovianity. This offers a potential pathway for preserving coherence in open quantum systems, such as solid-state qubits, without requiring extreme environmental isolation.

The authors conclude that their findings offer complementary insights into coherence protection, linking extended coherence times directly to enhanced non-Markovianity through a universally valid mechanism driven by periodic forcing.