# Schr\"odinger cats and steady states in subharmonic generation with Kerr nonlinearities

## TL;DR

This paper investigates the properties of steady states in nonlinear quantum circuits, revealing that true Schr"odinger cats cannot persist under realistic loss conditions, and characterizes the complex mixed states formed.

## Contribution

It provides a detailed analysis of the steady states in Kerr nonlinear systems, highlighting the limitations of Schr"odinger cat states and describing the nature of the resulting mixed states.

## Key findings

- True Schr"odinger cats cannot survive with photon loss.
- A 'cat-like' steady state exists only at very high nonlinearity.
- Steady states are complex mixed states with reduced purity.

## Abstract

We discuss general properties of the equilibrium state of parametric down-conversion in superconducting quantum circuits with detunings and Kerr anharmonicities, in the strongly nonlinear regime. By comparing moments of the steady state and those of a Schr\"odinger cat, we show that true Schr\"odinger cats cannot survive in the steady state if there is any single-photon loss. A delta-function 'cat-like' steady-state distribution can be formed, but this only exists in the limit of an extremely large nonlinearity. The steady state is a mixed state, which is more complex than a mixture or linear combination of delta-functions, and whose purity is reduced by driving. We expect this general behaviour to occur in other driven, dissipative quantum subharmonic non-equilibrium open systems.