# Frequency Fluctuations in Tunable and Nonlinear Microwave Cavities

## TL;DR

This paper develops a model for frequency fluctuations in tunable and nonlinear microwave cavities, revealing their impact on scattering measurements and providing methods to extract true damping rates and fluctuation scales.

## Contribution

It introduces a comprehensive model for frequency fluctuations in microwave cavities and demonstrates how to distinguish these from other effects in experimental data.

## Key findings

- Frequency fluctuations can mimic damping rate variations.
- A fundamental quantum limit sets a lower bound on fluctuation scale.
- The model enables extraction of true damping rates from scattering data.

## Abstract

We present a model for how frequency fluctuations comparable to the total cavity linewidth may arise in tunable and nonlinear microwave cavities, and how these fluctuations affect the measurement of scattering matrix elements. Applying this model to the specific case of a two-sided cavity, we obtain closed-form expressions for the average scattering matrix elements in several important cases. A key signature of our model is the subtle deformation of the trajectories swept out by scattering matrix elements in the complex plane. Despite this signature, the fluctuating and non-fluctuating models are qualitatively similar enough to be mistaken for one another. In the case of tunable cavities we show that if one fails to account for these fluctuations then one will find damping rates that appear to depend on the tuning parameter, which is a common observation in such systems. In the case of a Kerr cavity, we show that there exists a fundamental lower bound to the scale of these frequency fluctuations in the steady state, imposed by quantum mechanical uncertainty, which can appreciably affect the apparent damping rates of the cavity as the strength of the nonlinearity approaches the single-photon level. By using the model we present as a fitting function for experimental data, however, one can extract both the true damping rates of the cavity and the effective scale of these frequency fluctuations over the scattering measurement's bandwidth. Lastly, we compare this new method for observing frequency fluctuations to other methods, one of which we extend beyond the regime of small fluctuations.