[Paper Review] Quantum-coherent phase oscillations in synchronization

Talitha Weiß, Stefan Walter+1 more|Max Planck Digital Library|Aug 11, 2016

Mechanical and Optical Resonators4 citations

TL;DR

This paper demonstrates quantum-coherent phase oscillations in synchronized quantum limit-cycle oscillators, specifically the quantum Van der Pol oscillator under external driving. By deriving an effective quantum model, the authors identify regimes of underdamped, coherent phase dynamics where dephasing is slower than oscillation frequency, enabling long-lived quantum superpositions and phase self-oscillations—offering a pathway to observe non-classical synchronization effects like quantum tunnelling and entanglement in phase space.

ABSTRACT

Recently, several studies have investigated synchronization in quantum-mechanical limit-cycle oscillators. However, the quantum nature of these systems remained partially hidden, since the dynamics of the oscillator's phase was overdamped and therefore incoherent. We show that there exist regimes of underdamped and even quantum-coherent phase motion, opening up new possibilities to study quantum synchronization dynamics. To this end, we investigate the Van der Pol oscillator (a paradigm for a self-oscillating system) synchronized to an external drive. We derive an effective quantum model which fully describes the regime of underdamped phase motion and additionally allows us to identify the quality of quantum coherence. Finally, we identify quantum limit cycles of the phase itself.

Motivation & Objective

• To identify conditions under which quantum synchronization exhibits coherent, underdamped phase dynamics rather than overdamped, classical-like behavior.
• To develop an effective quantum model that captures coherent phase motion in driven quantum Van der Pol oscillators.
• To quantify the quality of quantum coherence in phase dynamics through a coherence quality factor.
• To demonstrate the existence of quantum limit cycles in the phase degree of freedom.
• To enable experimental observation of non-classical synchronization effects such as superposition states and phase self-oscillations.

Proposed method

• Derives an effective quantum model for the driven quantum Van der Pol oscillator using a linearized approximation around the classical steady state.
• Introduces a squeezing Hamiltonian to describe the dynamics of phase fluctuations, capturing the interplay between gain, loss, and nonlinear damping.
• Solves the quantum Langevin equations in Fourier space to analytically compute the fluctuation spectrum and identify effective oscillation frequencies.
• Uses the Wigner quasiprobability distribution to visualize quantum coherence, including interference fringes and dephasing dynamics.
• Defines a coherence quality factor based on the ratio of oscillation frequency to dephasing rate, identifying conditions for long-lived quantum coherence.
• Compares results from the full master equation with the effective model, validating the approximation across varying detuning and driving strength.

Experimental results

Research questions

• RQ1 Under what conditions does quantum synchronization exhibit coherent, underdamped phase dynamics instead of overdamped, classical-like behavior?
• RQ2 Can quantum limit cycles emerge in the phase degree of freedom of a synchronized oscillator?
• RQ3 What is the effective quality factor of quantum coherence in the phase dynamics of a driven quantum oscillator?
• RQ4 How does the interplay between nonlinear damping and external driving lead to quantum-coherent self-oscillations in phase?
• RQ5 Can the effective quantum model accurately reproduce the spectral features of the full quantum system?

Key findings

• Quantum-coherent phase oscillations are possible when the dynamically generated dephasing rate is smaller than the oscillation frequency, enabling long-lived quantum superpositions in phase space.
• The effective quantum model successfully captures underdamped phase dynamics and reproduces the spectral peaks near ±Ω_eff = ±√(Δ² - γ₂²|β_ss|⁴), validating its accuracy.
• Initial superposition states in phase space exhibit slowly decaying interference fringes due to slow dephasing, with the Wigner function showing rotation around the classical steady state before relaxation.
• The coherence quality factor is determined by the ratio of oscillation frequency to dephasing rate, with higher values indicating longer quantum coherence times.
• Phase self-oscillations are identified in the quantum regime, analogous to classical phase trapping, but with quantum coherence preserved over extended timescales.
• The effective model fails to capture a third spectral peak at large detuning, indicating limitations in regimes of strong nonlinearity or high driving.