Critical Response of a Quantum van der Pol Oscillator

Abstract

Classical nonequilibrium systems close to a dynamical critical point are known to exhibit a strongly nonlinear response, resulting in very high sensitivity to weak external perturbations. This attribute is key to the operation of important biological sensors. Here, we explore such systems in the quantum regime by modeling a driven van der Pol oscillator with a master equation approach. We find the classical response survives well into the quantum regime of low system energy. At very weak drives, genuine quantum features arise, including a diverging linear susceptibility and negative susceptibility. Further, deep in the limit cycle phase, the linear response is only limited by the weak two-particle decay, yielding a greatly enhanced sensitivity over a passive system. These generic results can be probed in current experimental setups and could have important applications in quantum sensing.A key insight from the theory of phase transitions is that systems close to a critical point are highly sensitive to perturbations [1]. This is evident, for example, in a diverging compressibility near a liquid-gas phase transition. Such divergences in susceptibility also occur in classical dynamical systems close to a bifurcation [2]. A particularly important prototype is an oscillator with nonlinear damping, which can transition from a dormant to a limitcycle phase across a Hopf bifurcation [3]. Generically, the response of such a van der Pol (vdP) oscillator [4] to a resonant drive Ω grows as Ω 1/3 at the critical point [5]. This nonlinearity is what enables the ear and other biological sensors to detect very weak stimuli and maximally process environmental inputs [5][6][7][8][9].In this paper, we ask if this increased sensitivity persists into the quantum regime and whether it could be useful for quantum sensing. At the semiclassical level, the vdP oscillator shows up in the physics of lasers [10,11]. Recent theoretical studies have examined a quantum vdP oscillator in the context of synchronization [12][13][14][15][16][17]. However, its critical properties and response characteristics are as yet unexplored. Besides technological applications, understanding the dynamics of such an oscillator is also of fundamental interest, as it represents an intrinsically nonequilibrium open system [18] which can be probed in state-of-the-art experimental setups [19][20][21][22].We model a driven quantum vdP oscillator by a master equation, finding a feature-rich response. In particular, the classical response survives down to one quantum of excitation. At very weak drives, we find both divergent and negative susceptibilities which arise from competing energy scales. Further, in the limit-cycle phase, the susceptibility is only limited by two-particle decay, providing a strong enhancement over a passive system. After briefly reviewing the iconic classical vdP oscillator, we present results for the quantum response, concluding with a discussion of possible experimental realizations. The classical vdP model describes a harmonic oscillator wit...

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