Spectral theory of Liouvillians for dissipative phase transitions (1804.11293v2)

Abstract: A state of an open quantum system is described by a density matrix, whose dynamics is governed by a Liouvillian superoperator. Within a general framework, we explore fundamental properties of both first-order dissipative phase transitions and second-order dissipative phase transitions associated with a symmetry breaking. In the critical region, we determine the general form of the steady-state density matrix and of the Liouvillian eigenmatrix whose eigenvalue defines the Liouvillian spectral gap. We illustrate our exact results by studying some paradigmatic quantum optical models exhibiting critical behavior.

• The paper shows that the Liouvillian gap closure marks non-analytical changes and hysteresis in first-order dissipative phase transitions.
• It employs analytical methods and numerical simulations of driven-dissipative Kerr resonators to link microscopic quantum fluctuations with macroscopic symmetry breaking.
• The research provides a framework for engineering novel quantum phases with potential applications in quantum simulation and information processing.

Spectral Theory of Liouvillians for Dissipative Phase Transitions

This paper presents an in-depth exploration of the spectral properties of Liouvillian superoperators and their vital role in understanding dissipative phase transitions in open quantum systems. The research focuses on both first-order and second-order transitions, addressing the intricacies of critical behavior in quantum optical systems.

The authors begin by grounding their investigation in the behavior of open quantum systems described by Lindblad master equations. The dynamics of such systems are dictated by the Liouvillian superoperator, $L$, which governs the evolution of the density matrix $\hat{\rho}(t)$. A steady state corresponds to the zero eigenvalue of $L$, while the Liouvillian gap, defined by the real part of the first non-zero eigenvalue, dictates the relaxation dynamics.

First-Order Dissipative Phase Transitions

The paper illustrates that a first-order phase transition involves a non-analytical change in the steady state. In the thermodynamic limit, characterized by a large number of excitations, the Liouvillian gap closes at the critical point, resulting in two distinct phases. The steady state near criticality is expressed as an equal mixture of eigenstates associated with $L$'s degenerate eigenvalues. This provides insights into phenomena like hysteresis, which occurs due to the metastable states involved in the transition.

Second-Order Phase Transitions and Symmetry Breaking

For second-order transitions, associated with symmetry breaking, the paper underscores the importance of spectral properties. The symmetry operations of the system map its density matrices into eigenmatrices of the Liouvillian, indicating a structured degeneracy. A $\mathcal{Z}_n$ symmetry leads to a series of eigenmatrices that allow for symmetry-broken solutions even when the system stabilizes in a symmetric state on average. This crucial insight connects microscopic quantum fluctuations to macroscopic quantum phases.

Numerical and Analytical Illustrations

Applying these theoretical concepts, the paper analyses paradigmatic models such as driven-dissipative Kerr resonators. These systems, exhibiting both linear and quadratic drive terms, demonstrate the theoretical predictions, especially in the context of phase bistability and symmetry-breaking transitions. The numerical results show excellent agreement with the theoretical framework developed and highlight the transition from exponential to non-exponential decay modes near criticality.

Implications and Future Directions

This research has significant implications for understanding non-equilibrium quantum dynamics. The spectral theory elucidated here not only advances the comprehension of dissipative phase transitions but also aids in the engineering of novel phases of matter through external and system parameters. By explaining the role of the Liouvillian spectrum, the findings lay the groundwork for applying these concepts to more complex systems and potentially leveraging them for technological applications in quantum information processing and quantum simulation.

Future investigations may extend this framework to consider more intricate systems where interactions and topology play pivotal roles. Exploring the spectral properties in lattices of superconducting circuits, Rydberg atom arrays, and other cutting-edge platforms could open new avenues in the paper of quantum criticality. Additionally, the integration of numerical techniques and variational approaches with the spectral theory of Liouvillians could enhance the computational tractability of addressing quantum many-body phenomena in real-world systems.