Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1804.11293 · Physical Review A

Spectral theory of Liouvillians for dissipative phase transitions

Fabrizio Minganti, Alberto Biella, Nicola Bartolo, Cristiano Ciuti

Published Oct 17, 2018• 412 citations
arXivHTMLPDF

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.

DOI
10.1103/physreva.98.042118

Related papers

Dissipative quantum dynamics, phase transitions, and non-Hermitian random matrices

Mahaveer Prasad, Hari Kumar Yadalam, Camille Aron +1 more

We explore the connections between dissipative quantum phase transitions and non-Hermitian random matrix theory. For this, we work in the framework of the dissipative Dicke model which is archetypal of symmetry-breaking phase transitions in open quantum systems. We establish that the Liouvillian describing the quantum dynamics exhibits distinct spectral features of integrable and chaotic character on the two sides of the critical point. We follow the distribution of the spacings of the complex Liouvillian eigenvalues across the critical point. In the normal and superradiant phases, the distributions are two-dimensional Poisson and that of the Ginibre unitary random matrix ensemble, respectively. Our results are corroborated by computing a recently introduced complex-plane generalization of the consecutive level-spacing ratio distribution. Our approach can be readily adapted for classifying the nature of quantum dynamics across dissipative critical points in other open quantum systems.

arXiv:2112.05765•Physical Review A•May 11, 2022•35 citations

Critical dynamical properties of a first-order dissipative phase transition

Wim Casteels, Rosario Fazio, Cristiano Ciuti

We theoretically investigate the critical properties of a single driven-dissipative nonlinear photon mode. In a well-defined thermodynamical limit of large excitation numbers, the exact quantum solution describes a first-order phase transition in the regime where semiclassical theory predicts optical bistability. We study the behavior of the complex spectral gap associated with the Liouvillian superoperator of the corresponding master equation. We show that in this limit the Liouvillian gap vanishes exponentially and that the bimodality of the photon Wigner function disappears. The connection between the considered thermodynamical limit of large photon numbers for the single-mode cavity and the thermodynamical limit of many cavities for a driven-dissipative Bose-Hubbard system is discussed.

arXiv:1608.00717•Physical Review A•Jan 20, 2017•179 citations

Superoperator structures and no-go theorems for dissipative quantum phase transitions

Thomas Barthel, Yikang Zhang

In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville superoperator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The superoperator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems and are also demonstrated for various spin models.

arXiv:2012.05505•Physical Review A•May 26, 2022•10 citations

Oscillating-mode gap: An indicator of phase transitions in open quantum many-body systems

Taiki Haga

It presents a significant challenge to elucidate the relationship between the phases of open quantum many-body systems and the spectral structure of their governing Liouvillian, which determines how the density matrix evolves. Previous studies have focused on the Liouvillian gap, defined as the decay rate of the most slowly decaying mode, as a key indicator of dissipative phase transition, noting its closure in symmetry-broken phases and opening in disordered phases. In this work, we propose an additional spectral gap, termed the oscillating-mode gap, defined as the decay rate of the most slowly decaying oscillating mode. Through the analysis of a prototype dissipative boson system, we demonstrate the necessity of both the Liouvillian gap and the oscillating-mode gap for the comprehensive characterization of the system's phases and the transitions between them.

arXiv:2405.07132•Physical review. B./Physical review. B•Sep 9, 2024•6 citations

Exceptional spectral phase in a dissipative collective spin model

Álvaro Rubio-García, Ángel L. Corps, Armando Relaño +4 more

We study a model of a quantum collective spin weakly coupled to a spin-polarized Markovian environment and find that the spectrum is divided into two regions that we name normal and exceptional Liouvillian spectral phases. In the thermodynamic limit, the exceptional spectral phase displays the unique property of being made up exclusively of second-order exceptional points. As a consequence, the evolution of any initial density matrix populating this region is slowed down and cannot be described by a linear combination of exponential decays. This phase is separated from the normal one by a critical line in which the density of Liouvillian eigenvalues diverges, a phenomenon analogous to that of excited-state quantum phase transitions observed in some closed quantum systems. In the limit of no bath polarization, this criticality is transferred onto the steady state, implying a dissipative quantum phase transition and the formation of a boundary time crystal.

arXiv:2202.09337•Physical Review A•Jul 7, 2022•11 citations

Liouvillian spectral collapse in Scully-Lamb lasing: Non-equilibrium second-order phase transition with or without $U(1)$ symmetry breaking

Fabrizio Minganti, Ievgen I. Arkhipov, Adam Miranowicz +1 more

The paradigm of second-order phase transition and symmetry breaking for thermal and quantum systems is a pillar of modern physics that has been fruitfully applied to out-of-equilibrium open quantum systems. Despite the non-equilibrium nature of lasing, the laser threshold was compared with thermal phase transitions systems more than forty years ago. For instance, the Landau theory -- based on thermodynamical consideration -- has been used to characterize the Scully-Lamb laser model (SLLM), describing a single-mode laser without saturable absorbers. Accordingly, the laser threshold is a second-order phase transition, associated with a spontaneous symmetry breaking (SSB) of the $U(1)$ symmetry of the model. To capture a genuine non-equilibrium, quantum, and dissipative phase transition of the SLLM, here we employ a Liouvillian superoperator formalism derived from a Lindblad master equation. Our results confirm that the $U(1)$ SSB can occur at the lasing threshold but, in contrast to the Landau theory and semiclassical approximation, they signal that the SLLM fundamental transition is a different phenomenon, which we call Liouvillian spectral collapse; that is, the emergence of diabolic points of infinite degeneracy. By considering a generalized SLLM with additional dephasing, we witness a second-order phase transition, with a Liouvillian spectral collapse, but in the absence of symmetry breaking. This demonstrates the difference between the Liouvillian spectral collapse and a standard second-order phase transition with symmetry breaking. We characterize the new phenomena associated with the spectral collapse, such as the onset of a rather unusual multistability enabled (and not suppressed) by the very presence of fluctuations. This multistability, characterizing the dissipative and quantum origin of lasing, can be experimentally revealed by the presence of a dynamical hysteresis.

•Mar 9, 2021•2 citations
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.
We explore the connections between dissipative quantum phase transitions and non-Hermitian random matrix theory. For this, we work in the framework of the dissipative Dicke model which is archetypal of symmetry-breaking phase transitions in open quantum systems. We establish that the Liouvillian describing the quantum dynamics exhibits distinct spectral features of integrable and chaotic character on the two sides of the critical point. We follow the distribution of the spacings of the complex Liouvillian eigenvalues across the critical point. In the normal and superradiant phases, the distributions are two-dimensional Poisson and that of the Ginibre unitary random matrix ensemble, respectively. Our results are corroborated by computing a recently introduced complex-plane generalization of the consecutive level-spacing ratio distribution. Our approach can be readily adapted for classifying the nature of quantum dynamics across dissipative critical points in other open quantum systems.
We theoretically investigate the critical properties of a single driven-dissipative nonlinear photon mode. In a well-defined thermodynamical limit of large excitation numbers, the exact quantum solution describes a first-order phase transition in the regime where semiclassical theory predicts optical bistability. We study the behavior of the complex spectral gap associated with the Liouvillian superoperator of the corresponding master equation. We show that in this limit the Liouvillian gap vanishes exponentially and that the bimodality of the photon Wigner function disappears. The connection between the considered thermodynamical limit of large photon numbers for the single-mode cavity and the thermodynamical limit of many cavities for a driven-dissipative Bose-Hubbard system is discussed.
In the thermodynamic limit, the steady states of open quantum many-body systems can undergo nonequilibrium phase transitions due to a competition between coherent and driven-dissipative dynamics. Here, we consider Markovian systems and elucidate structures of the Liouville superoperator that generates the time evolution. In many cases of interest, an operator-basis transformation can bring the Liouvillian into a block-triangular form, making it possible to assess its spectrum. The spectral gap sets the asymptotic decay rate. The superoperator structure can be used to bound gaps from below, showing that, in a large class of systems, dissipative phase transitions are actually impossible and that the convergence to steady states follows an exponential temporal decay. Furthermore, when the blocks on the diagonal are Hermitian, the Liouvillian spectra obey Weyl ordering relations. The results apply, for example, to Davies generators and quadratic systems and are also demonstrated for various spin models.
It presents a significant challenge to elucidate the relationship between the phases of open quantum many-body systems and the spectral structure of their governing Liouvillian, which determines how the density matrix evolves. Previous studies have focused on the Liouvillian gap, defined as the decay rate of the most slowly decaying mode, as a key indicator of dissipative phase transition, noting its closure in symmetry-broken phases and opening in disordered phases. In this work, we propose an additional spectral gap, termed the oscillating-mode gap, defined as the decay rate of the most slowly decaying oscillating mode. Through the analysis of a prototype dissipative boson system, we demonstrate the necessity of both the Liouvillian gap and the oscillating-mode gap for the comprehensive characterization of the system's phases and the transitions between them.
We study a model of a quantum collective spin weakly coupled to a spin-polarized Markovian environment and find that the spectrum is divided into two regions that we name normal and exceptional Liouvillian spectral phases. In the thermodynamic limit, the exceptional spectral phase displays the unique property of being made up exclusively of second-order exceptional points. As a consequence, the evolution of any initial density matrix populating this region is slowed down and cannot be described by a linear combination of exponential decays. This phase is separated from the normal one by a critical line in which the density of Liouvillian eigenvalues diverges, a phenomenon analogous to that of excited-state quantum phase transitions observed in some closed quantum systems. In the limit of no bath polarization, this criticality is transferred onto the steady state, implying a dissipative quantum phase transition and the formation of a boundary time crystal.
Liouvillian spectral collapse in Scully-Lamb lasing: Non-equilibrium second-order phase transition with or without U(1)U(1)U(1) symmetry breaking
The paradigm of second-order phase transition and symmetry breaking for thermal and quantum systems is a pillar of modern physics that has been fruitfully applied to out-of-equilibrium open quantum systems. Despite the non-equilibrium nature of lasing, the laser threshold was compared with thermal phase transitions systems more than forty years ago. For instance, the Landau theory -- based on thermodynamical consideration -- has been used to characterize the Scully-Lamb laser model (SLLM), describing a single-mode laser without saturable absorbers. Accordingly, the laser threshold is a second-order phase transition, associated with a spontaneous symmetry breaking (SSB) of the U(1)U(1)U(1) symmetry of the model. To capture a genuine non-equilibrium, quantum, and dissipative phase transition of the SLLM, here we employ a Liouvillian superoperator formalism derived from a Lindblad master equation. Our results confirm that the U(1)U(1)U(1) SSB can occur at the lasing threshold but, in contrast to the Landau theory and semiclassical approximation, they signal that the SLLM fundamental transition is a different phenomenon, which we call Liouvillian spectral collapse; that is, the emergence of diabolic points of infinite degeneracy. By considering a generalized SLLM with additional dephasing, we witness a second-order phase transition, with a Liouvillian spectral collapse, but in the absence of symmetry breaking. This demonstrates the difference between the Liouvillian spectral collapse and a standard second-order phase transition with symmetry breaking. We characterize the new phenomena associated with the spectral collapse, such as the onset of a rather unusual multistability enabled (and not suppressed) by the very presence of fluctuations. This multistability, characterizing the dissipative and quantum origin of lasing, can be experimentally revealed by the presence of a dynamical hysteresis.