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

1996 ◽  
Vol 29 (23) ◽  
pp. 7545-7560 ◽  
Author(s):  
José M Cerveró ◽  
Juan D Lejarreta
Keyword(s):  
Harmonic Oscillator ◽  
Squeezed States ◽  
Dynamical Group

Quantum algebras and parity-dependent spectra

2007 ◽  
Vol 463 (2086) ◽  
pp. 2415-2427 ◽  
Author(s):  
C.V Sukumar ◽  
Andrew Hodges
Keyword(s):  
Quantum Optics ◽  
Harmonic Oscillator ◽  
Combinatorial Theory ◽  
Squeezed States ◽  
Quantum Algebras ◽  
Matrix Elements ◽  
Euler Numbers ◽  
Simple Harmonic Oscillator ◽  
Non Gaussian ◽  
Special Case

We study the structure of a quantum algebra in which a parity-violating term modifies the standard commutation relation between the creation and annihilation operators of the simple harmonic oscillator. We discuss several useful applications of the modified algebra. We show that the Bernoulli and Euler numbers arise naturally in a special case. We also show a connection with Gaussian and non-Gaussian squeezed states of the simple harmonic oscillator. Such states have been considered in quantum optics. The combinatorial theory of Bernoulli and Euler numbers is developed and used to calculate matrix elements for squeezed states.

Coherent and squeezed states for the 3D harmonic oscillator

2017 ◽  
Vol 31 (03) ◽  
pp. 1750019
Author(s):  
Amel Mazouz ◽  
Mustapha Bentaiba ◽  
Ali Mahieddine
Keyword(s):  
Harmonic Oscillator ◽  
Uncertainty Relation ◽  
Three Dimensional ◽  
Heisenberg Algebra ◽  
Squeezed States ◽  
Ladder Operators ◽  
Compression Effect ◽  
Precise Knowledge ◽  
Generalized Coherent States ◽  
Heisenberg Uncertainty

A three-dimensional harmonic oscillator is studied in the context of generalized coherent states. We construct its squeezed states as eigenstates of linear contribution of ladder operators which are associated to the generalized Heisenberg algebra. We study the probability density to show the compression effect on the squeezed states. Our analysis reveals that squeezed states give us some freedom on the precise knowledge of position of the particle while maintaining the Heisenberg uncertainty relation minimum, squeezed states remains squeezed states over time.

q-ANALOGUE OF BOSON COMMUTATOR AND THE QUANTUM GROUPS SUq(2) AND SUq(1, 1)

1990 ◽  
Vol 05 (04) ◽  
pp. 237-242 ◽  
Author(s):  
HARUO UI ◽  
N. AIZAWA
Keyword(s):  
Harmonic Oscillator ◽  
Quantum Group ◽  
Symmetry Group ◽  
Quantum Groups ◽  
Heisenberg Algebra ◽  
Positive Definiteness ◽  
Two Dimensional ◽  
Number Operator ◽  
Boson Operator ◽  
Dynamical Group

We propose a defining set of commutation relations to a q-analogue of boson operator; [Formula: see text], [Formula: see text] and [N, aq]=−aq, which contracts to the Heisenberg algebra of boson operators in the limit of q=1. Here, N is the number operator, [N]q being its q-analogue operator. By making use of this set, we construct a new realization of the “noncompact” quantum group SUq(1, 1) in addition to that of the SUq(2) recently proposed by Biedenharn. The explicit form of the number operator is given in terms of aq and [Formula: see text] and its positive definiteness is proved. A uniqueness of our commutators is also discussed. It is shown that the quantum group SUq(2) appears as a true symmetry group of a q-analogue of the two-dimensional harmonic oscillator and the SUq(1, 1) as its dynamical group.

TIME-DEPENDENT WIGNER DISTRIBUTION FUNCTION EMPLOYED IN SQUEEZED SCHRÖDINGER CAT STATES: |Ψ(t)〉 = N-1/2(|β〉+eiϕ|-β〉)

2009 ◽  
Vol 23 (25) ◽  
pp. 5049-5066
Author(s):  
JEONG RYEOL CHOI ◽  
KYU HWANG YEON
Keyword(s):  
Distribution Function ◽  
Harmonic Oscillator ◽  
Wigner Distribution ◽  
Time Dependent ◽  
Squeezed States ◽  
Wigner Distribution Function ◽  
Practical Applications ◽  
Schrödinger Cat ◽  
Cat States ◽  
Theory Of Invariants

The Wigner distribution function (WDF) for the time-dependent quadratic Hamiltonian system is investigated in the squeezed Schrödinger cat states with the use of Lewis–Riesenfeld theory of invariants. The nonclassical aspects of the system produced by superposition of two distinct squeezed states are analyzed with emphasis on their application into special systems beyond simple harmonic oscillator. An application of our development to the measurement of quantum state by reconstructing the WDF via Autler–Townes spectroscopy is addressed. In addition, we considered particular models such as Cadirola–Kanai oscillator, frequency stable damped harmonic oscillator, and harmonic oscillator with time-variable frequency as practical applications with the object of promoting the understanding of nonclassical effects associated with the WDF.

A dynamical group for the harmonic oscillator

1973 ◽  
Vol 334 (1599) ◽  
pp. 541-551 ◽  
Keyword(s):  
Harmonic Oscillator ◽  
Symplectic Group ◽  
Three Dimensional ◽  
Matrix Elements ◽  
Dynamical Group ◽  
Group A ◽  
Complete Set ◽  
Direct Product Group ◽  
Real Symplectic Group ◽  
Creation Operators

A dynamical group is constructed for the isotropic three-dimensional harmonic oscillator by forming the semi-direct product group W (3)⊗ Sp (6, R ), where W (3) is the Weyl group and Sp (6, R ) the real symplectic group. A single representation of W (3)⊗ Sp (6, R ) is con­structed, using the usual harmonic oscillator annihilation and creation operators for W (3) and their anticommutators for Sp (6, R ), which can be spanned by the complete set of harmonic oscillator states. The group W (3)⊗ Sp (6, R ) is found to simplify the calculation of matrix elements of operators acting on harmonic oscillator states and to have a rich and useful subgroup structure.

Dynamical group of the anisotropic harmonic oscillator

1966 ◽  
Vol 43 (4) ◽  
pp. 1203-1207 ◽  
Author(s):  
F. Duimio ◽  
G. Zambotti
Keyword(s):  
Harmonic Oscillator ◽  
Dynamical Group

EXACT SOLUTIONS, LADDER OPERATORS AND BARUT–GIRARDELLO COHERENT STATES FOR A HARMONIC OSCILLATOR PLUS AN INVERSE SQUARE POTENTIAL

2005 ◽  
Vol 19 (28) ◽  
pp. 4219-4227 ◽  
Author(s):  
SHI-HAI DONG ◽  
M. LOZADA-CASSOU
Keyword(s):  
Harmonic Oscillator ◽  
Exact Solutions ◽  
Coherent States ◽  
Factorization Method ◽  
Ladder Operators ◽  
Dynamical Group ◽  
One Dimensional ◽  
Hidden Symmetry ◽  
The One ◽  
Analytical Expressions

We present exact solutions of the one-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square potential. The ladder operators are constructed by the factorization method. We find that these operators satisfy the commutation relations of the generators of the dynamical group SU(1, 1). Based on those ladder operators, we obtain the analytical expressions of matrix elements for some related functions ρ and [Formula: see text] with ρ=x2. Finally, we make some comments on the Barut–Girardello coherent states and the hidden symmetry between E(x) and E(ix) by substituting x→ix.

Sign in / Sign up

Export Citation Format

Share Document