Engineer coherent oscillatory modes in Markovian open quantum systems

TL;DR

This work tackles the challenge of achieving persistent oscillations in open quantum systems described by the time-independent Lindblad master equation $\frac{d}{dt}\hat{\rho}=\mathscr{L}[\hat{\rho}]$. It introduces a generalized fragmentation framework in which the Hamiltonian $\hat{H}$ and jump operators $\hat{L}_i$ are simultaneously block-diagonal, placing Liouvillian eigenvalues on the imaginary axis and enabling long-lived coherent dynamics even when the dissipators are nonzero. The approach unifies and extends the decoherence-free subspace picture, introducing weak and strong $\boldsymbol{\Delta}_H$ conditions that guarantee persistent oscillatory modes; it demonstrates this with four models, including dephasing harmonic oscillators and spin chains with local and collective dissipation, and shows tunability of the oscillation frequencies via system parameters. The results provide a constructive route to robust nonstationary quantum dynamics, with potential implications for autonomous quantum clocks and scalable coherent control in open quantum devices, and open avenues for exploring many-body limit cycles in larger systems.

Abstract

We develop a novel framework to engineer persistent oscillatory modes in Markovian open quantum systems governed by a time-independent Lindblad master equation. We show that oscillatory modes can be created when the Hamiltonian and jump operator can be expressed in the same block-diagonal form. A key feature of the framework is that the dissipator of the Lindblad master equation are generally non-zero. We identify the weak and strong conditions, where the onset of the oscillatory modes is dependent and independent of the parameters of the system, respectively. Our method extends beyond the typical decoherence-free subspace approach, in which the dissipator is zero. We demonstrate the applicability of this framework using various models, showing how carefully tailored system-environment interactions can produce sustained coherent oscillations.

Figures (4)

• Figure 1: (a) Liouvillian eigenspectrum. We only show the spectrum near the steady state. The red, blue, and gray points indicate oscillatory, steady, and damped modes, respectively. Notably, the spectrum features an eigenvalue pair at $\pm \mathbbm{i} 4$. (b) Time evolution of fidelities. The dashed curve shows the fidelity between $\rho_\text{osc}(t)$ and $\rho_\text{osc}(0)$. Solid curve shows the fidelity between $\rho_\text{damp}(t)$ and $\rho_\text{damp}(0)$. The parameters $\{\nu, \gamma\} = \{2, 1\}$ are used throughout for this Dephasing Quantum Harmonic Oscillator Example, yielding $T = 0.5\pi$.
• Figure 2: Liouvillian eigenspectrum. The red, blue, and gray points indicate the oscillatory, steady, and damped modes, respectively. Notably, the spectrum includes eigenvalue pairs located at $\pm \mathbbm{i} 4.4$, $\pm \mathbbm{i} 3.2$, $\pm \mathbbm{i} 0.8$, and $\pm \mathbbm{i} 0.4$. The parameters $\{ \chi, J_{12}, J_{23}, \gamma_1, \gamma_2 \} = \{ 0.3,0.9,1.0,1.0,1.0 \}$ are used in the calculation.
• Figure 3: (a) Liouvillian eigenspectrum. The red, blue, and gray points indicate the persistent oscillatory, steady, and damped modes, respectively. Notably, the spectrum features an eigenvalue pair at $\pm \mathbbm{i} 20$. (b) Time evolution of the fidelities. Dashed curve shows the fidelity between $\rho_\text{osc}(t)$ and $\rho_\text{osc}(0)$. Solid curve shows the fidelity between $\rho_\text{damp}(t)$ and $\rho_\text{damp}(0)$. The parameters $\{ J, h_x, h_y, h_z, \gamma \} = \{ 5,5,5,5,1 \}$ are used in the calculation. This yields a period $T = 0.1\pi$ in the oscillation.
• Figure 4: Liouvillian eigenspectra. The red, blue, and gray points indicate persistent oscillatory, steady, and damped modes, respectively. (a) Parameters $\{E,\gamma_1,\gamma_2\} = \{1,1,8\}$. In this case, the spectrum features an eigenvalue pair at $\pm \mathbbm{i} 4$. (b) Parameters $\{E,\gamma_1,\gamma_2\} = \{1,1,1\}$. In this case, the spectrum features no purely imaginary eigenvalues.