# Dissipative quantum state preparation and metastability in two-photon micromasers

## TL;DR

This paper explores how to prepare and utilize coherent quantum states, including Schrödinger cat states, in a two-photon micromaser for enhanced quantum metrology, accounting for realistic imperfections and metastability.

## Contribution

It introduces a method to generate pure and metastable quantum states in a two-photon micromaser using additional atomic levels to cancel Stark shifts, with applications in phase estimation.

## Key findings

- Degenerate stationary states include Schrödinger cat states with tunable photon number.
- Quantum Fisher information exceeds the standard limit in both pure and metastable states.
- Metastability persists despite cavity losses and atomic decay, maintaining quantum advantage.

## Abstract

We study the preparation of coherent quantum states in a two-photon micromaser for applications in quantum metrology. While this setting can be in principle realized in a host of physical systems, we consider atoms interacting with the field of a cavity. We focus on the case of the interaction described by the Jaynes-Cummings Hamiltonian, which cannot be achieved by the conventional approach with three-level atoms coupled to the cavity field at two-photon resonance. We find that additional levels are required in order to cancel Stark shifts emerging in the leading order. Once this is accomplished, the dynamics of the cavity features a degenerate stationary state manifold of pure states. We derive the analytic form of these states and show that they include Schr\"odinger cat states with a tunable mean photon number. We also confirm these states can be useful in phase estimation protocols with their quantum Fisher information exceeding the standard limit. To account for realistic imperfections, we consider single-photon losses from the cavity, finite lifetime of atom levels, and higher order corrections in the far-detuned limit, which result in metastability of formerly stationary cavity states and long-time dynamics with a unique mixed stationary state. Despite being mixed, this stationary state can still feature quantum Fisher information above the standard limit. Our work delivers a comprehensive overview of the two-photon micromaser dynamics with particular focus on application in phase estimation and, while we consider the setup with atoms coupled to a cavity, the results can be directly translated to optomechanical systems.