# Discrete time symmetry breaking in quantum circuits: exact solutions and tunneling

## TL;DR

This paper provides exact solutions and analysis of discrete time symmetry breaking in quantum circuits, revealing how nonlinearities and anharmonicities influence stability, tunneling, and multistability in superconducting quantum systems.

## Contribution

It offers the first exact analytic solutions for steady states and tunneling times in discrete time quantum symmetry breaking, incorporating effects of anharmonicities in superconducting circuits.

## Key findings

- Anharmonic terms increase tunneling rates.
- Exact solutions for steady states and tunneling times are derived.
- Superconducting circuits exhibit regimes of tristability and bistability.

## Abstract

We discuss general properties of discrete time quantum symmetry breaking in degenerate parametric oscillators. Recent experiments in superconducting quantum circuit with Josephson junction nonlinearities give rise to new properties of strong parametric coupling and nonlinearities. Exact analytic solutions are obtained for the steady-state of this single-mode case of subharmonic generation. We also obtain analytic solutions for the tunneling time over which the time symmetry-breaking is lost above threshold. We find that additional anharmonic terms found in the superconducting case increase the tunneling rate, and can also lead to new regimes of tristability as well as bistability. Our analytic results are confirmed by number state calculations.