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:quant-ph/9711018

Spectra and generalized eigenfunctions of the one- and two-mode squeezing operators in quantum optics

Bengt Nagel

Published Nov 14, 1997• 0 citations
arXivHTMLPDF

Abstract

The spectra and generalized eigenfunctions of the hyperbolic and parabolic generators of the standard representation of SU(1,1) in the one-mode boson Hilbert space are derived. The eigenfunctions are given in three different forms, corresponding to the coordinate, photon number, and Fock-Bargmann representations of the state vectors. The possible spectra of general second degree Hamiltonians are determined. Some corresponding results in the two-mode case are also given. - In the Appendix we prove completeness and orthonormality relations for the polynomials giving the number representation expansion coefficients of the generalized eigenfunctions of the hyperbolic generator (= squeezing generator). These polynomials are special cases of Pollaczek polynomials.

DOI
10.48550/arxiv.quant-ph/9711018

Related papers

Solving the two-mode squeezed harmonic oscillator and thekth-order harmonic generation in Bargmann–Hilbert spaces

Yao-Zhong Zhang

We analyze the two-mode squeezed harmonic oscillator and the k th-order harmonic generation within the framework of Bargmann–Hilbert spaces of entire functions. For displaced single-mode squeezed and two-mode squeezed harmonic oscillators, we derive exact closed-form expressions for their energies and wave functions. For the k th-order harmonic generation with k ⩾ 3, our result indicates that it does not have eigenfunctions and is thus ill-defined in the Bargmann–Hilbert space.

arXiv:1304.3979•Journal of Physics A Mathematical and Theoretical•Oct 24, 2013•17 citations

MINIMUM UNCERTAINTY SU(1,1) COHERENT STATES FOR NUMBER DIFFERENCE — TWO-MODE PHASE SQUEEZING

Hong-Yi Fan, Zhihu Sun

For the two-mode photon number-difference operator D = a†a -b†b and the Noh, Fougers and Mandel (NFM) operational phase operator [Formula: see text], we introduce the concept of number-difference-phase squeezing. We then find a new minimum-uncertainty state for number-difference-phase squeezing, which turns out to be a special combination of generalized SU(1,1) coherent state. As a by-product, a new orthonormal and complete representation in two-mode Fock space made up of R†n|m, m> is found.

Modern Physics Letters B•Feb 28, 2000•5 citations

Comment on ‘Solving the two-mode squeezed harmonic oscillator and thekth-order harmonic generation in Bargmann–Hilbert spaces’

Chi‐Fai Lo

Recently Zhang (2013 J. Phys. A: Math. Theor. 46 455302) proposed an analytical approach to solve the time-independent Schrödinger equation for the single-mode and two-mode squeezed harmonic oscillators in the Bargmann space of entire functions. In this comment we show that the eigenfunctions of these two systems exist in closed form and are expressed in terms of the Hermite polynomials. Moreover, since both oscillators exhibit the SU(1,1) dynamical symmetry, the eigenvalue problem can be tackled in a unified manner. In the Hilbert space of analytic functions of a complex variable in the unit disc, the energy eigenvalue equations involve first-order ordinary differential equations only, so we can easily solve these equations to obtain simple closed-form solutions.

Journal of Physics A Mathematical and Theoretical•Jan 30, 2014•4 citations

Uncorrelated two-mode SU (1,1) coherent states

Hou Li-Zhen, Gao-Jian Zeng

We have constructed a new kind of two-mode bosonic realization of SU (1,1) Lie algebra, on the basis of which the SU (1,1) generalized coherent states in the two-mode Fock space are derived. These two-mode SU (1,1) coherent states, which are called uncorrelated two-mode SU (1,1) coherent states, include three special cases. For these states, we study the mean photon number distribution and their non-classical properties, which are photon anti-bunching, violations of Cauchy-Schwarz inequality and two-mode squeezing.

Chinese Physics•Aug 28, 2001•3 citations

Multi-mode, multi-photon coherent states

Jacob Katriel, A. I. Solomon

A two-mode photon operator satisfying the canonical boson commutation relations is used to define two-mode coherent states for the Weyl, SU(2) and SU(1,l) groups. In all these cases squeezing is determined by an appropriate correlation, which we evaluate. The generalisation to multi-mode, many-photon states is indicated. In this letter we introduce a formalism for describing coherent states, tailored to the multi-mode case. We shall present the two-mode case for definiteness, although the generalisation to several modes will be obvious. Similarly, we show at the end of this note how to accommodate multi-photon states with little extra effort. States associ- ated with the Weyl group (two-mode coherent states), and the groups SU.(2) and SU(1,l) will be given; and we show that (for a specific representation) the last case reduces to the two-mode sqaeezed states of Caves and Schumaker (1986). In the conventional quantum optics approach, a mode of the electromagnetic field is represented by a creation (annihilation) operator a(at), with commutation relation (a, a') = 1. Our starting point in considering the two-mode case is to introduce a two-mode operator A analogous to the single-mode operator a. We do this by defining (Katriel and Hummer 1981) A = (max(Al, A2) + 1)'2ala2 , (1) Here al, a2 are the (annihilation) operators referring to each mode; they obey the basic commutation relation (ai, ait) = 6, but commute with each other. The number operators AI, A2 are defined in the usual way, Ai = aitai and functions of Ai will be evaluated in simultaneous eigenstates of rii, where of course they equal the functions of the corresponding eigenvalues. Now, we may show easily that (A, A') = 1. We note that AA = AtA = min(A1, h2). (The generalisation of (1) to three (or more) modes, which we shall not pursue, is immediate on replacing one of the two single-mode operators by a two-mode one.) Note that all the states defined here give variance-free single-mode operators; since At always creates equal numbers of 1 and 2 modes

Quantum Optics Journal of the European Optical Society Part B•Dec 1, 1989•3 citations

Generalized two-mode harmonic oscillator: dynamical group and squeezed states

José M. Cerveró, Juan D Lejarreta

In this paper we generalize our previous results of the generalized one-mode harmonic oscillator to the generalized two-mode case. Systematic use is made of the SO(3,2) dynamical group and we are able to write a general form for the exact time evolution operator in terms of squeezing operators of one and two modes. A complete classification of the exact solutions is made and we derive them explicitly whenever possible. The relevant results on algebraic decomposition, coherent state generators and classification of the solutions are shown in tables. A plethora of soluble Hamiltonians already treated in the literature, which appear to be particular cases of the general formalism presented herein, are analysed as well as new cases, which to the authors' knowledge, have not yet been considered.

Journal of Physics A Mathematical and General•Dec 7, 1996•26 citations
The spectra and generalized eigenfunctions of the hyperbolic and parabolic generators of the standard representation of SU(1,1) in the one-mode boson Hilbert space are derived. The eigenfunctions are given in three different forms, corresponding to the coordinate, photon number, and Fock-Bargmann representations of the state vectors. The possible spectra of general second degree Hamiltonians are determined. Some corresponding results in the two-mode case are also given. - In the Appendix we prove completeness and orthonormality relations for the polynomials giving the number representation expansion coefficients of the generalized eigenfunctions of the hyperbolic generator (= squeezing generator). These polynomials are special cases of Pollaczek polynomials.
For the two-mode photon number-difference operator D = a†a -b†b and the Noh, Fougers and Mandel (NFM) operational phase operator [Formula: see text], we introduce the concept of number-difference-phase squeezing. We then find a new minimum-uncertainty state for number-difference-phase squeezing, which turns out to be a special combination of generalized SU(1,1) coherent state. As a by-product, a new orthonormal and complete representation in two-mode Fock space made up of R†n|m, m> is found.
Recently Zhang (2013 J. Phys. A: Math. Theor. 46 455302) proposed an analytical approach to solve the time-independent Schrödinger equation for the single-mode and two-mode squeezed harmonic oscillators in the Bargmann space of entire functions. In this comment we show that the eigenfunctions of these two systems exist in closed form and are expressed in terms of the Hermite polynomials. Moreover, since both oscillators exhibit the SU(1,1) dynamical symmetry, the eigenvalue problem can be tackled in a unified manner. In the Hilbert space of analytic functions of a complex variable in the unit disc, the energy eigenvalue equations involve first-order ordinary differential equations only, so we can easily solve these equations to obtain simple closed-form solutions.
We have constructed a new kind of two-mode bosonic realization of SU (1,1) Lie algebra, on the basis of which the SU (1,1) generalized coherent states in the two-mode Fock space are derived. These two-mode SU (1,1) coherent states, which are called uncorrelated two-mode SU (1,1) coherent states, include three special cases. For these states, we study the mean photon number distribution and their non-classical properties, which are photon anti-bunching, violations of Cauchy-Schwarz inequality and two-mode squeezing.
A two-mode photon operator satisfying the canonical boson commutation relations is used to define two-mode coherent states for the Weyl, SU(2) and SU(1,l) groups. In all these cases squeezing is determined by an appropriate correlation, which we evaluate. The generalisation to multi-mode, many-photon states is indicated. In this letter we introduce a formalism for describing coherent states, tailored to the multi-mode case. We shall present the two-mode case for definiteness, although the generalisation to several modes will be obvious. Similarly, we show at the end of this note how to accommodate multi-photon states with little extra effort. States associ- ated with the Weyl group (two-mode coherent states), and the groups SU.(2) and SU(1,l) will be given; and we show that (for a specific representation) the last case reduces to the two-mode sqaeezed states of Caves and Schumaker (1986). In the conventional quantum optics approach, a mode of the electromagnetic field is represented by a creation (annihilation) operator a(at), with commutation relation (a, a') = 1. Our starting point in considering the two-mode case is to introduce a two-mode operator A analogous to the single-mode operator a. We do this by defining (Katriel and Hummer 1981) A = (max(Al, A2) + 1)'2ala2 , (1) Here al, a2 are the (annihilation) operators referring to each mode; they obey the basic commutation relation (ai, ait) = 6, but commute with each other. The number operators AI, A2 are defined in the usual way, Ai = aitai and functions of Ai will be evaluated in simultaneous eigenstates of rii, where of course they equal the functions of the corresponding eigenvalues. Now, we may show easily that (A, A') = 1. We note that AA = AtA = min(A1, h2). (The generalisation of (1) to three (or more) modes, which we shall not pursue, is immediate on replacing one of the two single-mode operators by a two-mode one.) Note that all the states defined here give variance-free single-mode operators; since At always creates equal numbers of 1 and 2 modes
In this paper we generalize our previous results of the generalized one-mode harmonic oscillator to the generalized two-mode case. Systematic use is made of the SO(3,2) dynamical group and we are able to write a general form for the exact time evolution operator in terms of squeezing operators of one and two modes. A complete classification of the exact solutions is made and we derive them explicitly whenever possible. The relevant results on algebraic decomposition, coherent state generators and classification of the solutions are shown in tables. A plethora of soluble Hamiltonians already treated in the literature, which appear to be particular cases of the general formalism presented herein, are analysed as well as new cases, which to the authors' knowledge, have not yet been considered.