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:2304.02367 · Journal of Physics A Mathematical and Theoretical

Third quantization for bosons: symplectic diagonalization, non-Hermitian Hamiltonian, and symmetries

Steven Kim, Fabian Hassler

Published Aug 17, 2023• 7 citations
arXivHTMLPDF

Abstract

Abstract Open quantum systems that interact with a Markovian environment can be described by a Lindblad master equation. The generator of time-translation is given by a Liouvillian superoperator acting on the density matrix of the system. As the Fock space for a single bosonic mode is already infinite-dimensional, the diagonalization of the Liouvillian has to be done on the creation- and annihilation-superoperators, a process called ‘third quantization’. We propose a method to solve the Liouvillian for quadratic systems using a single symplectic transformation. We show that the non-Hermitian effective Hamiltonian of the system, next to incorporating the dynamics of the system, is a tool to analyze its symmetries. As an example, we use the effective Hamiltonian to formulate a -‘symmetry’ of an open system. We describe how the inclusion of source terms allows us to obtain the cumulant generating function for observables such as the photon current.

DOI
10.1088/1751-8121/acf177

Related papers

Third quantization of open quantum systems: Dissipative symmetries and connections to phase-space and Keldysh field-theory formulations

Alexander McDonald, Aashish A. Clerk

The connections between standard theoretical tools used to study open quantum systems can sometimes seem opaque. Whether it is a Lindblad master equation, the equation of motion for the Wigner function, or a dissipative Keldysh action, features evident in one formalism are often masked in another. Here, we reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. This symmetry can then be used to easily diagonalize these models, and it provides an intuitive way to demonstrate the separation of dissipation and fluctuations in linear systems. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ``wave functions'' of the density matrix in the eigenbasis of the third-quantized superoperators we introduce. The field-theory representation of the time-evolution operator in this basis is then the Keldysh path integral. To highlight the utility of our approach, we apply our version of third quantization to a dissipative nonlinear oscillator, and we use it to obtain new exact results.

arXiv:2302.14047•Physical Review Research•Aug 16, 2023•33 citations

Solving quasi-free and quadratic Lindblad master equations for open fermionic and bosonic systems

Thomas Barthel, Yikang Zhang

Abstract The dynamics of Markovian open quantum systems are described by Lindblad master equations. For fermionic and bosonic systems that are quasi-free, i.e. with Hamiltonians that are quadratic in the ladder operators and Lindblad operators that are linear in the ladder operators, we derive the equation of motion for the covariance matrix. This determines the evolution of Gaussian initial states and the steady states, which are also Gaussian. Using ladder super-operators (a.k.a. third quantization), we show how the Liouvillian can be transformed to a many-body Jordan normal form which also reveals the full many-body spectrum. Extending previous work by Prosen and Seligman, we treat fermionic and bosonic systems on equal footing with Majorana operators, shorten and complete some derivations, also address the odd-parity sector for fermions, give a criterion for the existence of bosonic steady states, cover non-diagonalizable Liouvillians also for bosons, and include quadratic systems. In extension of the quasi-free open systems, quadratic open systems comprise additional Hermitian Lindblad operators that are quadratic in the ladder operators. While Gaussian states may then evolve into non-Gaussian states, the Liouvillian can still be transformed to a useful block-triangular form, and the equations of motion for k -point Green’s functions form a closed hierarchy. Based on this formalism, results on criticality and dissipative phase transitions in such models are discussed in a companion paper.

arXiv:2112.08344•Journal of Statistical Mechanics Theory and Experiment•Nov 1, 2022•81 citations

Lindblad dynamics of open multi-mode bosonic systems: Algebra of bilinear superoperators, exceptional points and speed of evolution

Andrei Gaidash, Alexei D. Kiselev, Козубов Антон Владимирович +1 more

We develop the algebraic method based on the Lie algebra of quadratic combinations of left and right superoperators associated with matrices to study the Lindblad dynamics of multimode bosonic systems coupled a thermal bath and described by the Liouvillian superoperator that takes into account both dynamical (coherent) and environment mediated (incoherent) interactions between the modes. Our algebraic technique is applied to transform the Liouvillian into the diagonalized form by eliminating jump superoperators and solve the spectral problem. The temperature independent effective non-Hermitian Hamiltonian, $\hat{H}_{eff}$, is found to govern both the diagonalized Liouvillian and the spectral properties. It is shown that the Liouvillian exceptional points are represented by the points in the parameter space where the matrix, $H$, associated with $\hat{H}_{eff}$ is non-diagonalizable. We use our method to derive the low-temperature approximation for the superpropagator and to study the special case of a two mode system representing the photonic polarization modes. For this system, we describe the geometry of exceptional points in the space of frequency and relaxation vectors parameterizing the intermode couplings and, for a single-photon state, evaluate the time dependence of the speed of evolution as a function of the angles characterizing the couplings and the initial state.

arXiv:2412.13890•Dec 18, 2024•0 citations

Symmetry of Open Quantum Systems: Classification of Dissipative Quantum Chaos

Kohei Kawabata, Anish Kulkarni, Jiachen Li +2 more

We develop a theory of symmetry in open quantum systems.Using the operator-state mapping, we characterize symmetry of Liouvillian superoperators for the open quantum dynamics by symmetry of operators in the double Hilbert space and apply the 38-fold internal-symmetry classification of non-Hermitian operators.We find rich symmetry classification due to the interplay between symmetry in the corresponding closed quantum systems and symmetry inherent in the construction of the Liouvillian superoperators.As an illustrative example of open quantum bosonic systems, we study symmetry classes of dissipative quantum spin models.For open quantum fermionic systems, we develop the Z 4 classification of fermion parity symmetry and antiunitary symmetry in the double Hilbert space, which contrasts with the Z 8 classification in closed quantum systems.We also develop the symmetry classification of open quantum fermionic many-body systems-a dissipative generalization of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation.We establish the periodic tables of the SYK Lindbladians and elucidate the difference from the SYK Hamiltonians.Furthermore, from extensive numerical calculations, we study its complex-spectral statistics and demonstrate dissipative quantum chaos enriched by symmetry.

arXiv:2212.00605•PRX Quantum•Aug 29, 2023•73 citations

The Irreversible Quantum Dynamics of the Three-Level su(1, 1) Bosonic Model

N. M. Bogoliubov, Andrei V. Rybin

We study the quantum dynamics of the opened three-level su(1, 1) bosonic model. The effective non-Hermitian Hamiltonians describing the system of the Lindblad equation in the short time limit are constructed. The obtained non-Hermitian Hamiltonians are exactly solvable by the Algebraic Bethe Ansatz. This approach allows representing biorthogonal and nonorthogonal bases of the system. We analyze the biorthogonal expectation values of a number of particles in the zero mode and represent it in the determinantal form. The time-dependent density matrix satisfying the Lindblad master equation is found in terms of the nonorthogonal basis.

Symmetry•Dec 1, 2022•0 citations

Conformal symmetry in quasifree Markovian open quantum systems

Anatoliy I. Lotkov, Denis V. Kurlov, Aleksey K. Fedorov +2 more

Conformal symmetry governs the behavior of closed systems near second-order phase transitions and is expected to emerge in open systems going through dissipative phase transitions. We propose a framework allowing for a manifest description of conformal symmetry in open Markovian systems. The key difference from the closed case is that both conformal algebra and the algebra of local fields are realized on the space of superoperators. We illustrate the framework by a series of examples featuring systems with quadratic Hamiltonians and linear jump operators, where the Liouvillian dynamics can be efficiently analyzed using the formalism of third quantization. We expect that our framework can be extended to interacting systems using an appropriate generalization of the conformal bootstrap.

arXiv:2305.01629•Physical review. B./Physical review. B•Aug 25, 2023•3 citations
Abstract Open quantum systems that interact with a Markovian environment can be described by a Lindblad master equation. The generator of time-translation is given by a Liouvillian superoperator acting on the density matrix of the system. As the Fock space for a single bosonic mode is already infinite-dimensional, the diagonalization of the Liouvillian has to be done on the creation- and annihilation-superoperators, a process called ‘third quantization’. We propose a method to solve the Liouvillian for quadratic systems using a single symplectic transformation. We show that the non-Hermitian effective Hamiltonian of the system, next to incorporating the dynamics of the system, is a tool to analyze its symmetries. As an example, we use the effective Hamiltonian to formulate a -‘symmetry’ of an open system. We describe how the inclusion of source terms allows us to obtain the cumulant generating function for observables such as the photon current.
The connections between standard theoretical tools used to study open quantum systems can sometimes seem opaque. Whether it is a Lindblad master equation, the equation of motion for the Wigner function, or a dissipative Keldysh action, features evident in one formalism are often masked in another. Here, we reformulate the technique of third quantization in a way that explicitly connects all three methods. We first show that our formulation reveals a fundamental dissipative symmetry present in all quadratic bosonic or fermionic Lindbladians. This symmetry can then be used to easily diagonalize these models, and it provides an intuitive way to demonstrate the separation of dissipation and fluctuations in linear systems. For bosons, we then show that the Wigner function and the characteristic function can be thought of as ``wave functions'' of the density matrix in the eigenbasis of the third-quantized superoperators we introduce. The field-theory representation of the time-evolution operator in this basis is then the Keldysh path integral. To highlight the utility of our approach, we apply our version of third quantization to a dissipative nonlinear oscillator, and we use it to obtain new exact results.
Abstract The dynamics of Markovian open quantum systems are described by Lindblad master equations. For fermionic and bosonic systems that are quasi-free, i.e. with Hamiltonians that are quadratic in the ladder operators and Lindblad operators that are linear in the ladder operators, we derive the equation of motion for the covariance matrix. This determines the evolution of Gaussian initial states and the steady states, which are also Gaussian. Using ladder super-operators (a.k.a. third quantization), we show how the Liouvillian can be transformed to a many-body Jordan normal form which also reveals the full many-body spectrum. Extending previous work by Prosen and Seligman, we treat fermionic and bosonic systems on equal footing with Majorana operators, shorten and complete some derivations, also address the odd-parity sector for fermions, give a criterion for the existence of bosonic steady states, cover non-diagonalizable Liouvillians also for bosons, and include quadratic systems. In extension of the quasi-free open systems, quadratic open systems comprise additional Hermitian Lindblad operators that are quadratic in the ladder operators. While Gaussian states may then evolve into non-Gaussian states, the Liouvillian can still be transformed to a useful block-triangular form, and the equations of motion for k -point Green’s functions form a closed hierarchy. Based on this formalism, results on criticality and dissipative phase transitions in such models are discussed in a companion paper.
We develop the algebraic method based on the Lie algebra of quadratic combinations of left and right superoperators associated with matrices to study the Lindblad dynamics of multimode bosonic systems coupled a thermal bath and described by the Liouvillian superoperator that takes into account both dynamical (coherent) and environment mediated (incoherent) interactions between the modes. Our algebraic technique is applied to transform the Liouvillian into the diagonalized form by eliminating jump superoperators and solve the spectral problem. The temperature independent effective non-Hermitian Hamiltonian, H^eff\hat{H}_{eff}H^eff​, is found to govern both the diagonalized Liouvillian and the spectral properties. It is shown that the Liouvillian exceptional points are represented by the points in the parameter space where the matrix, HHH, associated with H^eff\hat{H}_{eff}H^eff​ is non-diagonalizable. We use our method to derive the low-temperature approximation for the superpropagator and to study the special case of a two mode system representing the photonic polarization modes. For this system, we describe the geometry of exceptional points in the space of frequency and relaxation vectors parameterizing the intermode couplings and, for a single-photon state, evaluate the time dependence of the speed of evolution as a function of the angles characterizing the couplings and the initial state.
We develop a theory of symmetry in open quantum systems.Using the operator-state mapping, we characterize symmetry of Liouvillian superoperators for the open quantum dynamics by symmetry of operators in the double Hilbert space and apply the 38-fold internal-symmetry classification of non-Hermitian operators.We find rich symmetry classification due to the interplay between symmetry in the corresponding closed quantum systems and symmetry inherent in the construction of the Liouvillian superoperators.As an illustrative example of open quantum bosonic systems, we study symmetry classes of dissipative quantum spin models.For open quantum fermionic systems, we develop the Z 4 classification of fermion parity symmetry and antiunitary symmetry in the double Hilbert space, which contrasts with the Z 8 classification in closed quantum systems.We also develop the symmetry classification of open quantum fermionic many-body systems-a dissipative generalization of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation.We establish the periodic tables of the SYK Lindbladians and elucidate the difference from the SYK Hamiltonians.Furthermore, from extensive numerical calculations, we study its complex-spectral statistics and demonstrate dissipative quantum chaos enriched by symmetry.
We study the quantum dynamics of the opened three-level su(1, 1) bosonic model. The effective non-Hermitian Hamiltonians describing the system of the Lindblad equation in the short time limit are constructed. The obtained non-Hermitian Hamiltonians are exactly solvable by the Algebraic Bethe Ansatz. This approach allows representing biorthogonal and nonorthogonal bases of the system. We analyze the biorthogonal expectation values of a number of particles in the zero mode and represent it in the determinantal form. The time-dependent density matrix satisfying the Lindblad master equation is found in terms of the nonorthogonal basis.
Conformal symmetry governs the behavior of closed systems near second-order phase transitions and is expected to emerge in open systems going through dissipative phase transitions. We propose a framework allowing for a manifest description of conformal symmetry in open Markovian systems. The key difference from the closed case is that both conformal algebra and the algebra of local fields are realized on the space of superoperators. We illustrate the framework by a series of examples featuring systems with quadratic Hamiltonians and linear jump operators, where the Liouvillian dynamics can be efficiently analyzed using the formalism of third quantization. We expect that our framework can be extended to interacting systems using an appropriate generalization of the conformal bootstrap.