Exact steady state of quantum van der Pol oscillator: critical phenomena and enhanced metrology

TL;DR

This paper derives the exact steady state of the quantum van der Pol oscillator, revealing critical phenomena, phase transitions, and conditions for Heisenberg-limited quantum metrology.

Contribution

It provides the first exact analytical steady state solution for the quantum van der Pol oscillator, linking dissipative phase transitions to enhanced quantum metrology.

Findings

Dissipative phase transition at the threshold with abrupt property changes

Divergent quantum Fisher information at critical point and in the time crystal phase

Photon number as the optimal observable for high-precision estimation

Abstract

Quantum criticality of open many-body systems has attracted lots of interest for emergent phenomena and universality. Here we present the exact steady state of the quantum van der Pol oscillator using the complex $P$-representation. We show the threshold corresponds to a dissipative phase transition with abrupt changes of steady-state properties and enhanced metrology. The critical behaviors and finite-size effects are investigated through the analytical steady state. Moreover, we obtain divergent quantum Fisher information (QFI) in the thermodynamic limit both at the critical point and in the time crystal phase, but only the QFI at the critical point approaches the Heisenberg limit. We further prove that the steady-state photon number is the optimized estimated observable with the largest signal-to-noise ratio. We show that the Heisenberg-limited metrology originates from the larger enhancement of the susceptibility than the standard deviation of the photon number. Our work reveals the underlying relation between the time crystal, dissipative phase transition, and enhanced metrology.