# General linearized theory of quantum fluctuations around arbitrary limit cycles

## TL;DR

This paper develops a linearization method for analyzing quantum fluctuations around time-dependent limit cycles in nonlinear quantum systems, extending standard techniques to non-stationary states like spontaneous oscillations.

## Contribution

It introduces a novel linearization scheme tailored for systems with time-dependent classical solutions, such as limit cycles, and demonstrates its effectiveness through the driven Van der Pol oscillator.

## Key findings

- The new method accurately captures quantum fluctuations around limit cycles.
- It maintains computational simplicity and scalability for large systems.
- Comparison with numerical simulations validates the approach.

## Abstract

The theory of Gaussian quantum fluctuations around classical steady states in nonlinear quantum-optical systems (also known as standard linearization) is a cornerstone for the analysis of such systems. Its simplicity, together with its accuracy far from critical points or situations where the nonlinearity reaches the strong coupling regime, has turned it into a widespread technique, which is the first method of choice in most works on the subject. However, such a technique finds strong practical and conceptual complications when one tries to apply it to situations in which the classical long-time solution is time dependent, a most prominent example being spontaneous limit-cycle formation. Here we introduce a linearization scheme adapted to such situations, using the driven Van der Pol oscillator as a testbed for the method, which allows us to compare it with full numerical simulations. On a conceptual level, the scheme relies on the connection between the emergence of limit cycles and the spontaneous breaking of the symmetry under temporal translations. On the practical side, the method keeps the simplicity and linear scaling with the size of the problem (number of modes) characteristic of standard linearization, making it applicable to large (many-body) systems.