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Physical Review A

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

Masashi Ban

Published Jun 1, 1993• 64 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.47.5093

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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.
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.
Generalized normal- and antinormal-order decomposition formulas are derived for exponential functions of the generators of su(1, 1) and su(2) Lie algebras. The expectation values of analytic functions of these generators are calculated in terms of the generalized decomposition formulas, allowing su(1, 1) and su(2) fluctuations and their squeezing properties to be discussed. The second-order correlation functions are also considered. It is shown that the Mach–Zehnder interferometer, which is described by su(2) Lie algebra, is equivalent to a three-dimensional rotation and is characterized by the quantum counterparts of the classical Euler angles.
The theory of generalized coherent states associated with the SU(2) and SU(1,1) Lie algebras is discussed and applied in an investigation of possible reductions of fluctuations. In the framework of a system of N two-level atoms the squeezing of angular-momentum [SU(2)] fluctuations is exhibited for optical coherent transients involving the photon echo. The SU(1,1) fluctuations are discussed and established for general two-photon processes involving dynamical variables different from the creation and annihilation operators.
Decomposition formulas of general exponential operators in a Banach algebra and in a Lie algebra are presented that yield a basis of Monte Carlo simulation of quantum systems. They are applied to study the relaxation and fluctuation from the initially unstable point and to confirm algebraically the scaling theory of transient phenomena. A global approximation method of transient phenomena is also formulated on the basis of decomposition formulas. It is applied to the laser model as a simple example.
This work is aiming to show the advantage of using the Lie algebraic decomposition technique to solvefor Schrödinger’s wave equation for a quantum model, compared with the direct method of solution. The advantageis a two-fold: one is to derive general form of solution, and, two is relatively manageable to deal with the case oftime-dependent system Hamiltonian. Specifically, we consider the model of 2-level optical atom and solve for thecorresponding Schrödinger’s wave equation using the Lie algebraic decomposition technique. The obtained formof solution for the wave function is used to examine computationally the atomic localization in the coordinate space.For comparison, the direct method of solution of the wave function is analysed in order to show its complicationwhen dealing with time-dependent Hamiltonian.The possibility of using the Lie algebraic method for a qubit model(a driven quantum dot model) is briery discussed, if Schrödinger’s wave function is to be examined for the qubitlocalization.
Lie algebraic techniques are powerful and widely used tools for studying dynamics and metrology in quantum optics. When the Hamiltonian generates a Lie algebra with finite dimension, the unitary evolution can be expressed as a finite product of exponentials using the Wei-Norman expansion. The system is then exactly described by a finite set of scalar differential equations, even if the Hilbert space is infinite. However, the differential equations provided by the Wei-Norman expansion are nonlinear and often have singularities that prevent both analytic and numerical evaluation. We derive a new Lie algebra expansion for the quantum Fisher information, which results in linear differential equations. Together with existing Lie algebra techniques this allows many metrology problems to be analysed entirely in the Heisenberg picture. This substantially reduces the calculations involved in many metrology problems, and provides analytical solutions for problems that cannot even be solved numerically using the Wei-Norman expansion. It also allows us to study general features of metrology problems, valid for all quantum states. We provide detailed examples of these methods applied to problems in quantum optics and nonlinear optomechanics. Published by the American Physical Society 2024