Dynamics and Control of Two Coupled Quantum Oscillators: An Analytical Approach

TL;DR

This work addresses decoherence in a minimal open-quantum-system model: two directly coupled bosonic oscillators sharing a common Lorentzian bath. It develops an exact, approximation-free framework yielding a closed-form, probability-conserving propagator and exact average excitation numbers, enabling precise assessment of detuning-based dynamical decoupling. By implementing leakage-elimination–inspired detuning with both regular and irregular schedules, the authors quantify how off-resonant detuning and duty cycle suppress non-Markovian revivals, revealing design rules: large detuning $\omega_D$, high duty cycle $\eta$, and short control periods relative to bath memory improve protection, especially in non-Markovian baths. The results provide an exact benchmark for controlled non-Markovian dynamics and actionable guidance for engineering decoherence suppression in structured reservoirs.

Abstract

We analyze two coupled quantum oscillators in a common Lorentzian environment and control them by detuning (temporarily shifting) their frequencies. The reduced dynamics are solved exactly, without Born or Markov approximations, by propagating each detuning segment in closed form. We study two control schedules: regular detuning, with perfectly periodic on and off pulses of fixed period, width, and amplitude; and irregular detuning, with the same on/off structure but cycle-to-cycle jitter in period, width, and amplitude. Our main observable is the average excitation number (AEN) of each mode. Detuning moves the system away from the bath's spectral peak, suppressing decoherence and damping non-Markovian revivals; in effectively Markovian baths the benefit is small. We quantify performance with a simple time-domain suppression factor. Larger detuning amplitudes and higher duty cycles yield stronger protection. Irregular control is slightly weaker at low duty cycle but becomes comparable to regular control as the duty cycle approaches one. These results give practical design rules linking detuning, duty cycle, and bath width, and provide an exact benchmark for controlled non-Markovian dynamics.

Figures (11)

• Figure 1: Schematic of the model: two bosonic oscillators with direct coupling, both interacting with a common thermal bath. The shared reservoir mediates additional indirect interactions and induces non-Markovian memory effects. In our analysis, dynamical decoupling (DD) is implemented as fast detuning of the mode frequencies to move the system off resonance with the bath’s spectral peak; inter-mode coherence is used only as a secondary diagnostic observable mandel1995opticalwalls2008quantum.
• Figure 2: Regular vs. irregular detuning–based dynamical–decoupling (DD) pulse trains. (a) Regular (periodic) rectangular DD with fixed detuning amplitude $\omega_D$, pulse width $\delta$, and period $\tau$ (duty cycle $\eta=\delta/\tau$). (b) Irregular (randomized) DD in which the width $\delta_k$, period $\tau_k$, and detuning amplitude $\omega_{D,k}$ fluctuate from cycle to cycle (with $0<\delta_k<\tau_k$). Vertical dashed lines mark pulse-start times; double-headed arrows show the parameters on the first interval. Irregular DD models realistic timing/amplitude noise and is used to assess the robustness of the scheme.
• Figure 3: Time evolution of the mode occupation numbers in the Markovian regime. The upper panel shows $n_1(t)$ versus $\omega_1 t$ and the lower panel shows $n_2(t)$ versus $\omega_2 t$ for different bath temperatures $T_B$ (in units of $\omega_1$ and $\omega_2$, respectively). Parameters: $\Gamma = 1.0$, $\Omega = 1.0$, $\omega_1 = \omega_2 = 1.0$, and $\gamma = 15.0$. The large spectral width with $\gamma > \Gamma$ ensures a short reservoir correlation time, so the bath acts as a memoryless sink, leading to smooth relaxation without significant revivals or backflow.
• Figure 4: Time evolution of the mode occupation numbers in the non-Markovian regime. The upper panel shows $n_1(t)$ versus $\omega_1 t$ and the lower panel shows $n_2(t)$ versus $\omega_2 t$ for different bath temperatures $T_B$. Parameters: $\Gamma = 15.0$, $\Omega = 1.0$, $\omega_1 = \omega_2 = 1.0$, and $\gamma = 1.0$. Here, $\Gamma > \gamma$, resulting in a long reservoir correlation time and strong non-Markovian memory effects, which enable partial revivals and backflow of energy.
• Figure 5: Time evolution of the mode occupation numbers at fixed bath temperature $T_B=1.0$ for varying bath spectral widths $\gamma$. Top: $n_1(t)$ vs. $\omega t$; bottom: $n_2(t)$ vs. $\omega t$. Curves correspond to $\gamma=0.1$ (blue), $0.5$ (orange), $1.0$ (green), and $5.0$ (red). Parameters: $\Gamma=5.0$, $\Omega=1.0$, $\omega_1=\omega_2=1.0$. The bath correlation time scales as $\tau_c\!\sim\!1/\gamma$: small $\gamma$ (0.1, 0.5) produces pronounced non-Markovian oscillations and short-time overshoots, whereas as $\gamma$ increases to $1.0$ and $5.0$ the oscillation amplitude decreases and the approach becomes monotonic, indicating that a broadband reservoir quenches backflow. Both modes converge to the same thermal occupation $n_B$ set by $T_B$.