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Quantum trajectory of the one atom maser

Tristan Benoist, Laurent Bruneau, Clément Pellegrini

TL;DR

This work analyzes quantum trajectories for the infinite-dimensional one-atom maser, proving that in the non-resonant, positive-temperature setting there exists a unique invariant Markov measure for the trajectory and that the distribution converges in Wasserstein distance. A purification theorem shows that, almost surely, the quantum state collapses to a pure Fock state along trajectories, with the limiting Fock index distributing according to the Gibbs-type invariant. The analysis connects the quantum trajectory dynamics to a classical birth–death process on Fock states and employs a key martingale to establish purification and invariant-state change of measure. In resonant scenarios, multiple invariant measures can occur, and purification may fail in degenerate cases, while a tuned cavity ($\eta=0$) restores purification and uniqueness of the asymptotic state. Overall, the paper extends finite-dimensional purification results to an infinite-dimensional maser model, clarifying how non-resonance, purification, and invariant measures interact in this physically relevant quantum system.

Abstract

The evolution of a quantum system undergoing repeated indirect measurements naturally leads to a Markov chain on the set of states which is called a quantum trajectory. In this paper we consider a specific model of such a quantum trajectory associated to the one-atom maser model. It describes the evolution of one mode of the quantized electromagnetic field in a cavity interacting with two-level atoms. When the system is non-resonant we prove that this Markov chain admits a unique invariant probability measure. We moreover prove convergence in the Wasserstein metric towards this invariant measure. These results rely on a purification theorem: almost surely the state of the system approaches the set of pure states. Compared to similar results in the literature, the system considered here is infinite dimensional. While existence of an invariant measure is a consequence of the compactness of the set of states in finite dimension, in infinite dimension existence of an invariant measure is not free. Furthermore usual purification criterions in finite dimension have no straightforward equivalent in infinite dimension.

Quantum trajectory of the one atom maser

TL;DR

This work analyzes quantum trajectories for the infinite-dimensional one-atom maser, proving that in the non-resonant, positive-temperature setting there exists a unique invariant Markov measure for the trajectory and that the distribution converges in Wasserstein distance. A purification theorem shows that, almost surely, the quantum state collapses to a pure Fock state along trajectories, with the limiting Fock index distributing according to the Gibbs-type invariant. The analysis connects the quantum trajectory dynamics to a classical birth–death process on Fock states and employs a key martingale to establish purification and invariant-state change of measure. In resonant scenarios, multiple invariant measures can occur, and purification may fail in degenerate cases, while a tuned cavity () restores purification and uniqueness of the asymptotic state. Overall, the paper extends finite-dimensional purification results to an infinite-dimensional maser model, clarifying how non-resonance, purification, and invariant measures interact in this physically relevant quantum system.

Abstract

The evolution of a quantum system undergoing repeated indirect measurements naturally leads to a Markov chain on the set of states which is called a quantum trajectory. In this paper we consider a specific model of such a quantum trajectory associated to the one-atom maser model. It describes the evolution of one mode of the quantized electromagnetic field in a cavity interacting with two-level atoms. When the system is non-resonant we prove that this Markov chain admits a unique invariant probability measure. We moreover prove convergence in the Wasserstein metric towards this invariant measure. These results rely on a purification theorem: almost surely the state of the system approaches the set of pure states. Compared to similar results in the literature, the system considered here is infinite dimensional. While existence of an invariant measure is a consequence of the compactness of the set of states in finite dimension, in infinite dimension existence of an invariant measure is not free. Furthermore usual purification criterions in finite dimension have no straightforward equivalent in infinite dimension.
Paper Structure (17 sections, 26 theorems, 166 equations)

This paper contains 17 sections, 26 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Description of the model and main results
  3. The one-atom maser model
  4. The quantum trajectory
  5. Dynamical system properties
  6. Purification
  7. Purification of the trajectory
  8. The quantum trajectory on Fock states
  9. A key martingale
  10. Purification
  11. Proof of Theorem \ref{['thm:invariantmeasure']}
  12. The resonant cases
  13. Rabi sectors
  14. Invariant measures
  15. Purification
  16. ...and 2 more sections

Key Result

Theorem 2.2

Suppose the non-resonant condition holds and $\beta>0$. Then $\rho_{\rm inv}:= \frac{{\rm e}^{-\beta\varepsilon N}}{{\rm Tr}\left({\rm e}^{-\beta\varepsilon N} \right)}$ is the unique invariant state of ${\mathcal{L}}$ and for any initial state $\rho$ one has

Theorems & Definitions (72)

  • Definition 2.1
  • Theorem 2.2: Bru14, Theorem 3.4
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Theorem 2.9
  • Proposition 2.10
  • ...and 62 more