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

Modern Physics Letters B

SU(1,1) SQUEEZED STATES AS DAMPED OSCILLATORS

E. Celeghini, Mario Rasetti, M. Tarlini, Giuseppe Vitiello

Published Nov 10, 1989• 37 citations
Publisher

Abstract

The conventional squeezed states of quantum optics, which can be thought of as generalized coherent states for the algebra SU(1,1), are dynamically generated by single-mode hamiltonians characterized by two-photon process interactions. By the explicit construction of a (highly non-linear) faithful realization of the group [Formula: see text] of automorphisms of SU(1,1), such hamiltonians are shown to be equivalent — up just to elements of [Formula: see text] — to that describing quantum mechanically a damped oscillator.

DOI
10.1142/s0217984989001850

Related papers

Which Q-analogue of the squeezed oscillator?

A. I. Solomon

The noise (variance squared) of a component of the electromagnetic field - considered as a quantum oscillator - in the vacuum is equal to one half, in appropriate units (taking Planck's constant and the mass and frequency of the oscillator all equal to 1). A practical definition of a squeezed state is one for which the noise is less than the vacuum value - and the amount of squeezing is determined by the appropriate ratio. Thus the usual coherent (Glauber) states are not squeezed, as they produce the same variance as the vacuum. However, it is not difficult to define states analogous to coherent states which do have this noise-reducing effect. In fact, they are coherent states in the more general group sense but with respect to groups other than the Heisenberg-Weyl Group which defines the Glauber states. The original, conventional squeezed state in quantum optics is that associated with the group SU(1,1). Just as the annihilation operator a of a single photon mode (and its hermitian conjugate a, the creation operator) generates the Heisenberg Weyl algebra, so the pair-photon operator a(sup 2) and its conjugate generates the algebra of the group SU(1,1). Another viewpoint, more productive from the calculational stance, is to note that the automorphism group of the Heisenberg-Weyl algebra is SU(1,1). Needless to say, each of these viewpoints generalizes differently to the quantum group context. Both are discussed. The following topics are addressed: conventional coherent and squeezed states; eigenstate definitions; exponential definitions; algebra (group) definitions; automorphism group definition; example: signal-to-noise ratio; q-coherent and q-squeezed states; M and P q-bosons; eigenstate definitions; exponential definitions; algebra (q-group) definitions; and automorphism q-group definition.

•Jan 1, 1993•0 citations

Coherent states, squeezed fluctuations, and the SU(2) am SU(1,1) groups in quantum-optics applications

K. Wódkiewicz, Joseph H. Eberly

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.

Journal of the Optical Society of America B•Mar 1, 1985•498 citations

On the Squeezed States fornObservables

D. A. Trifonov

Three basic properties (eigenstate, orbit and intelligence) of the canonical squeezed states (SS) are extended to the case of n arbitrary observables. The SS for n observables X i can be constructed as eigenstates of their linear complex combinations or as states which minimize the Robertson uncertainty relation. When X i close a Lie algebra L the generalized SS could also be introduced as orbit of Aut ( L C ). It is shown that for the nilpotent algebra h N the three generalizations are equivalent. For the simple su (1, 1) the family of eigenstates of uK - + vK + ( K ± being lowering and raising operators) is a family of ideal K 1 – K 2 SS, but it cannot be represented as an Aut ( su C (1, 1) orbit although the SU (1, 1) group related coherent states (CS) with symmetry are contained in it. Eigenstates | z, u, v, w ; k ⟩ of general combination of uK - + vK + + wK 3 the three generators K j of SU (1, 1) in the representations with Bargman index k = 1/2, 1, ... , and k = 1/4, 3/4 are constructed and discussed in greater detail. These are ideal SS for K 1,2,3 . In the case of the one mode realization of su (1, 1) the nonclassical properties (sub-Poissonian statistics, quadrature squeezing) of the generalized even CS | z, u, v ; + ⟩ are demonstrated. The states | z, u, v, w ; k = 1/4, 3/4 ⟩ can exhibit strong both linear and quadratic squeezing.

arXiv:quant-ph/9705001•Physica Scripta•Sep 1, 1998•13 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

Photon-mediated spin-mixing dynamics

Emily J. Davis, Avikar Periwal, Eric S. Cooper +4 more

We analyze the dynamics of spin-mixing interactions generated by coupling spin-1 atoms to the mode of a high-finesse optical cavity. We show that the dynamics can be understood in terms of generators of the noncompact Lie group SU(1, 1) and introduce a set of SU(1, 1) coherent states which are preserved under Hamiltonian evolution. In terms of these coherent states the resulting dynamics may be interpreted as classical motion on the unit disk. We explicitly compute the trajectories of this classical motion and show that the motion is equivalent to spin-nematic squeezing in the atomic ensemble. Non-uniform coupling between the atomic ensemble and the cavity mode leads to departures from this simple behavior; we introduce a toy model that captures this non-uniformity and solve it exactly.

Optical and Quantum Sensing and Precision Metrology II•Mar 1, 2019•0 citations

Damped harmonic oscillator and squeezed state

王晓光

Study of the two-photon coherent state or squeezed state has made great progress both in theory and in experiment for the generalization of usual coherent state of the electromagnetic field. The squeezed state can be produced by some nonlinear process and the Hamiltonian which governs the dynamics of squeezed state usually has a quatratic form. The time-dependent Bateman Hamiltonian of damped harmonic oscillator just has this

Chinese Science Bulletin•Feb 28, 1995•0 citations
The conventional squeezed states of quantum optics, which can be thought of as generalized coherent states for the algebra SU(1,1), are dynamically generated by single-mode hamiltonians characterized by two-photon process interactions. By the explicit construction of a (highly non-linear) faithful realization of the group [Formula: see text] of automorphisms of SU(1,1), such hamiltonians are shown to be equivalent — up just to elements of [Formula: see text] — to that describing quantum mechanically a damped oscillator.
The noise (variance squared) of a component of the electromagnetic field - considered as a quantum oscillator - in the vacuum is equal to one half, in appropriate units (taking Planck's constant and the mass and frequency of the oscillator all equal to 1). A practical definition of a squeezed state is one for which the noise is less than the vacuum value - and the amount of squeezing is determined by the appropriate ratio. Thus the usual coherent (Glauber) states are not squeezed, as they produce the same variance as the vacuum. However, it is not difficult to define states analogous to coherent states which do have this noise-reducing effect. In fact, they are coherent states in the more general group sense but with respect to groups other than the Heisenberg-Weyl Group which defines the Glauber states. The original, conventional squeezed state in quantum optics is that associated with the group SU(1,1). Just as the annihilation operator a of a single photon mode (and its hermitian conjugate a, the creation operator) generates the Heisenberg Weyl algebra, so the pair-photon operator a(sup 2) and its conjugate generates the algebra of the group SU(1,1). Another viewpoint, more productive from the calculational stance, is to note that the automorphism group of the Heisenberg-Weyl algebra is SU(1,1). Needless to say, each of these viewpoints generalizes differently to the quantum group context. Both are discussed. The following topics are addressed: conventional coherent and squeezed states; eigenstate definitions; exponential definitions; algebra (group) definitions; automorphism group definition; example: signal-to-noise ratio; q-coherent and q-squeezed states; M and P q-bosons; eigenstate definitions; exponential definitions; algebra (q-group) definitions; and automorphism q-group definition.
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.
Three basic properties (eigenstate, orbit and intelligence) of the canonical squeezed states (SS) are extended to the case of n arbitrary observables. The SS for n observables X i can be constructed as eigenstates of their linear complex combinations or as states which minimize the Robertson uncertainty relation. When X i close a Lie algebra L the generalized SS could also be introduced as orbit of Aut ( L C ). It is shown that for the nilpotent algebra h N the three generalizations are equivalent. For the simple su (1, 1) the family of eigenstates of uK - + vK + ( K ± being lowering and raising operators) is a family of ideal K 1 – K 2 SS, but it cannot be represented as an Aut ( su C (1, 1) orbit although the SU (1, 1) group related coherent states (CS) with symmetry are contained in it. Eigenstates | z, u, v, w ; k ⟩ of general combination of uK - + vK + + wK 3 the three generators K j of SU (1, 1) in the representations with Bargman index k = 1/2, 1, ... , and k = 1/4, 3/4 are constructed and discussed in greater detail. These are ideal SS for K 1,2,3 . In the case of the one mode realization of su (1, 1) the nonclassical properties (sub-Poissonian statistics, quadrature squeezing) of the generalized even CS | z, u, v ; + ⟩ are demonstrated. The states | z, u, v, w ; k = 1/4, 3/4 ⟩ can exhibit strong both linear and quadratic squeezing.
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.
We analyze the dynamics of spin-mixing interactions generated by coupling spin-1 atoms to the mode of a high-finesse optical cavity. We show that the dynamics can be understood in terms of generators of the noncompact Lie group SU(1, 1) and introduce a set of SU(1, 1) coherent states which are preserved under Hamiltonian evolution. In terms of these coherent states the resulting dynamics may be interpreted as classical motion on the unit disk. We explicitly compute the trajectories of this classical motion and show that the motion is equivalent to spin-nematic squeezing in the atomic ensemble. Non-uniform coupling between the atomic ensemble and the cavity mode leads to departures from this simple behavior; we introduce a toy model that captures this non-uniformity and solve it exactly.