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

Journal of Mathematical Physics

SU(1,1) Lie algebraic approach to linear dissipative processes in quantum optics

Masashi Ban

Published Sep 1, 1992• 68 citations
Publisher

Abstract

An SU(1,1) Lie algebraic formulation is presented for investigating the linear dissipative processes in quantum optical systems. The Liouville space formulation, thermo field dynamics, and the disentanglement theorem of SU(1,1) Lie algebra play essential roles in this formulation. In the Liouville space, the time-evolution equation for the state vector of a system is solved algebraically by using the decomposition formulas of SU(1,1) Lie algebra and the thermal state condition of thermo field dynamics. The presented formulation is used for investigating a dissipative nonlinear oscillator, the quantum mechanical model of phase modulation, and the photon echo in the localized electron–phonon system. This algebraic formulation gives a systematic treatment for investigating the phenomena in quantum optical systems.

DOI
10.1063/1.529540

Related papers

Lie-algebra methods in quantum optics: The Liouville-space formulation

Masashi Ban

Lie-algebra methods for investigating quantum optical systems are presented within the framework of the Liouville-space formulation. The generalized decomposition formulas for exponential functions of the generators of su(2) and su(1,1) Lie algebras are derived and their expectation values are calculated for typical states in quantum optics. The general procedure for using Lie algebras in the Liouville space to treat quantum optical processes is given in terms of generalized decomposition formulas and their use is demonstrated by calculating the absorption line shape and photon echo signal in a localized electron-phonon system. It is also shown that the photon-counting probability can be calculated by using the su(1,1) Lie algebra and that the electron-counting probability can be calculated by using the su(2) Lie algebra. The su(1,1) Lie algebra is also used to investigate a quantum-nondemolition measurement of photon number in the four-wave-mixing model.

Physical Review A•Jun 1, 1993•64 citations

Plasmonics

Maxim Sukharev

When we investigate properties of a quantum optical system, we notice that many systems including a interferometer and a parametric amplifier can be described in terms of the generators of the SU(1, 1) and SU( 2 ) Lie algebras [ 1 – 12 ]. Hence, the Lie algebra method [ 13 – 15 ] is very useful for investigating such quantum optical systems [16,17]. It is also well known that the Liouville-space method [ 18 – 21 ] is suitable for investigating nonequilibrium dynamics of an open quantum system which is placed under the influence of a large environmental system. Indeed, the Markovian quantum master equation of the Lindblad form [22,23] can be represented in terms of the generators of the Lie algebra. Therefore, the combination of the Lie algebra method and the Liouville-space method provides a powerful tool for investigating irreversible time-evolution of a quantum optical system. Furthermore, since the nonequilibrium thermo field dynamics [ 24 – 28 ], which is abbreviated as NETFD, can disentangle operator algebra in the Liouville space, it gives a useful calculational technique in the Liouville-space method. Therefore, in this chapter, we review a synthetic method of Lie algebra and NETFD and its applications to quantum optical systems [ 31 – 38 ].

CRC Press eBooks•Dec 14, 2012•0 citations

Utility of su(1, 1)-Algebra in a Schematic Nuclear su(2)-Model

H. Akaike, Y. Tsue, Constança Providência +3 more

The su(2)-algebraic model interacting with an environment is investigated from the viewpoint of treating a dissipative system. By using a time-dependent variational approach with a coherent state and with the help of the canonicity condition, the time-evolution of this quantum many-body system is described in terms of the canonical equations of motion in the classical mechanics. Then, it is shown that the su(1, 1)-algebra plays an essential role in treating this model. An exact solution with appropriate initial conditions is obtained in terms of Jacobi's elliptic function. The implications for the dissipative process are discussed.

arXiv:nucl-th/0102014•Progress of Theoretical Physics•May 1, 2001•1 citation

Thermal Effects and Dissipation in su(1, 1)-Algebraic Model by Means of Time-Dependent Variational Approach

Y. Tsue, A. Kuriyama, Masatoshi Yamamura

The appearance of thermal effects is investigated in a simple system composed of quantum mechanical damped harmonic oscillator governed by su(1, 1)-algebra within a framework of the time-dependent variational approach in terms of mixed-mode coherent states. It is shown that certain fluctuations originated from the use of the mixed-mode coherent state can be interpreted as thermal fluctuations which appear in the form of the bose distribution function. Thus, our time-dependent variational approach with mixed-mode coherent states leads to a possible simultaneous description of classical motions and thermal fluctuations. It is also shown that the energy concentrated in a pure state at initial time dissipates to the other states after time-evolution. It is then remarked that the thermal effects play a significant role in the above process.

Progress of Theoretical Physics•Mar 1, 1994•5 citations

Thermal Effects and Dissipation in su(1, 1)-Algebraic Model by Means of Time-Dependent Variational Approach

Y. Tsue, A. Kuriyama, M. Yamamura

The appearance of thermal effects is investigated in a simple system composed of quantum mechanical damped harmonic oscillator governed by su(1, 1)-algebra within a framework of the time-dependent variational approach in terms of mixed-mode coherent states. It is shown that certain fluctuations originated from the use of the mixed-mode coherent state can be interpreted as thermal fluctuations which appear in the form of the bose distribution function. Thus, our time-dependent variational approach with mixed-mode coherent states leads to a possible simultaneous description of classical motions and thermal fluctuations. It is also shown that the energy concentrated in a pure state at initial time dissipates to the other states after time-evolution. It is then remarked that the thermal effects play a significant role in the above process.

Progress of Theoretical Physics•Mar 1, 1994•9 citations

Solving the quantum master equation of coupled harmonic oscillators with Lie-algebra methods

Lucas Teuber, Stefan Scheel

Based on a Liouville-space formulation of open systems, we present two methods to solve the quantum dynamics of coupled harmonic oscillators experiencing Markovian loss. The starting point is the quantum master equation in Liouville space which is generated by a Liouvillian that induces a Lie algebra. We show how this Lie algebra allows to define ladder operators that construct Fock-like eigenstates of the Liouvillian. These eigenstates are used to decompose the time-evolved density matrix and, together with the accompanying eigenvalues, provide insight into the transport properties of the lossy system. Additionally, a Wei-Norman expansion of the generated time evolution can be found by a structure analysis of the algebra. This structure analysis yields a construction principle to implement effective non-Hermitian Hamiltonians in lossy systems.

arXiv:1912.07320•Physical Review A•Apr 27, 2020•24 citations
An SU(1,1) Lie algebraic formulation is presented for investigating the linear dissipative processes in quantum optical systems. The Liouville space formulation, thermo field dynamics, and the disentanglement theorem of SU(1,1) Lie algebra play essential roles in this formulation. In the Liouville space, the time-evolution equation for the state vector of a system is solved algebraically by using the decomposition formulas of SU(1,1) Lie algebra and the thermal state condition of thermo field dynamics. The presented formulation is used for investigating a dissipative nonlinear oscillator, the quantum mechanical model of phase modulation, and the photon echo in the localized electron–phonon system. This algebraic formulation gives a systematic treatment for investigating the phenomena in quantum optical systems.
Lie-algebra methods for investigating quantum optical systems are presented within the framework of the Liouville-space formulation. The generalized decomposition formulas for exponential functions of the generators of su(2) and su(1,1) Lie algebras are derived and their expectation values are calculated for typical states in quantum optics. The general procedure for using Lie algebras in the Liouville space to treat quantum optical processes is given in terms of generalized decomposition formulas and their use is demonstrated by calculating the absorption line shape and photon echo signal in a localized electron-phonon system. It is also shown that the photon-counting probability can be calculated by using the su(1,1) Lie algebra and that the electron-counting probability can be calculated by using the su(2) Lie algebra. The su(1,1) Lie algebra is also used to investigate a quantum-nondemolition measurement of photon number in the four-wave-mixing model.
When we investigate properties of a quantum optical system, we notice that many systems including a interferometer and a parametric amplifier can be described in terms of the generators of the SU(1, 1) and SU( 2 ) Lie algebras [ 1 – 12 ]. Hence, the Lie algebra method [ 13 – 15 ] is very useful for investigating such quantum optical systems [16,17]. It is also well known that the Liouville-space method [ 18 – 21 ] is suitable for investigating nonequilibrium dynamics of an open quantum system which is placed under the influence of a large environmental system. Indeed, the Markovian quantum master equation of the Lindblad form [22,23] can be represented in terms of the generators of the Lie algebra. Therefore, the combination of the Lie algebra method and the Liouville-space method provides a powerful tool for investigating irreversible time-evolution of a quantum optical system. Furthermore, since the nonequilibrium thermo field dynamics [ 24 – 28 ], which is abbreviated as NETFD, can disentangle operator algebra in the Liouville space, it gives a useful calculational technique in the Liouville-space method. Therefore, in this chapter, we review a synthetic method of Lie algebra and NETFD and its applications to quantum optical systems [ 31 – 38 ].
The su(2)-algebraic model interacting with an environment is investigated from the viewpoint of treating a dissipative system. By using a time-dependent variational approach with a coherent state and with the help of the canonicity condition, the time-evolution of this quantum many-body system is described in terms of the canonical equations of motion in the classical mechanics. Then, it is shown that the su(1, 1)-algebra plays an essential role in treating this model. An exact solution with appropriate initial conditions is obtained in terms of Jacobi's elliptic function. The implications for the dissipative process are discussed.
The appearance of thermal effects is investigated in a simple system composed of quantum mechanical damped harmonic oscillator governed by su(1, 1)-algebra within a framework of the time-dependent variational approach in terms of mixed-mode coherent states. It is shown that certain fluctuations originated from the use of the mixed-mode coherent state can be interpreted as thermal fluctuations which appear in the form of the bose distribution function. Thus, our time-dependent variational approach with mixed-mode coherent states leads to a possible simultaneous description of classical motions and thermal fluctuations. It is also shown that the energy concentrated in a pure state at initial time dissipates to the other states after time-evolution. It is then remarked that the thermal effects play a significant role in the above process.
The appearance of thermal effects is investigated in a simple system composed of quantum mechanical damped harmonic oscillator governed by su(1, 1)-algebra within a framework of the time-dependent variational approach in terms of mixed-mode coherent states. It is shown that certain fluctuations originated from the use of the mixed-mode coherent state can be interpreted as thermal fluctuations which appear in the form of the bose distribution function. Thus, our time-dependent variational approach with mixed-mode coherent states leads to a possible simultaneous description of classical motions and thermal fluctuations. It is also shown that the energy concentrated in a pure state at initial time dissipates to the other states after time-evolution. It is then remarked that the thermal effects play a significant role in the above process.
Based on a Liouville-space formulation of open systems, we present two methods to solve the quantum dynamics of coupled harmonic oscillators experiencing Markovian loss. The starting point is the quantum master equation in Liouville space which is generated by a Liouvillian that induces a Lie algebra. We show how this Lie algebra allows to define ladder operators that construct Fock-like eigenstates of the Liouvillian. These eigenstates are used to decompose the time-evolved density matrix and, together with the accompanying eigenvalues, provide insight into the transport properties of the lossy system. Additionally, a Wei-Norman expansion of the generated time evolution can be found by a structure analysis of the algebra. This structure analysis yields a construction principle to implement effective non-Hermitian Hamiltonians in lossy systems.