The quantum harmonic oscillator on a circle and a deformed Heisenberg algebra

We show that the Heisenberg-type algebra describing the first levels of the quantum harmonic oscillator on a circle of large length L is a deformed Heisenberg algebra. The successive energy levels of this quantum harmonic oscillator on a circle of large length L are interpreted, similarly to the standard quantum one-dimensional harmonic oscillator on an infinite line, as being obtained by the creation of a quantum particle of frequency w at very high energies. Received: 29 March 2001 / Revised version: 17 July 2001 / Published online: 31 August 2001     

Generalized two-mode harmonic oscillator model: squeezed number state solutions and nonadiabatic Berry’s phase

A generalized two-mode harmonic oscillator model is investigated within the framework of its general dynamical algebra so(3,2). Two types of eigenstates, formulated as extended su(1,1), su(2) squeezed number states are found respectively. The nonadiabatic Berrys phase for this system with the cranked time-dependent Hamiltonian is also given.Received: 16 January 2004, Published online: 10 August 2004PACS: 42.50.Dv Nonclassical states of the electromagnetic field, including entangled photon states; quantum state engineering and measurements - 03.65.Fd Algebraic methods - 03.65.Vf Phases: geometric; dynamic or topological     

Realizations ofU q(su(1, 1)) by deformed quantum harmonic oscillator algebras and associated deformed Heisenberg algebras

Forsu(1, 1)-symmetric Hamiltonians of quantum mechanical systems (e.g. single-mode quantum harmonic oscillator, radial Schrödinger equation for Coulomb problem or isotropic quantum harmonic oscillator, etc.), the Heisenberg algebra of phase-space variables in two dimensions satisfy the bilinear commutation relation [ip,x]=1 (in normal units). Also there are different realizations ofsu(1, 1) by the generators of quantum harmonic oscillator algebra. We seek here the forms of deformed Heisenberg algebras (bilinear in deformedx and ip) associated with deformedsu(1, 1)-symmetric Hamiltonians. These forms are not unique in contrast to the undeformed case; and these forms are obtained here by considering different realizations of the deformedsu(1, 1) algebra by deformed oscillator algebras (satisfying different bilinear relations in deformed creation and annihilation operators), and then imposing different conditions (e.g. the deformed Heisenberg algebra of the form of the undeformed one, the form of realizations of the deformedsu(1, 1) algebra by deformed phase-space variables being the same as that ofsu(1, 1) algebra by undeformed phase-space variables, etc.), assuming linear relations between deformed phase-space variables and deformed creation-annihilation operators (as it is done in the undeformed case), we get different Heisenberg algebras. These facts are revealed in the case of a two-body Calogero model in its centre of mass frame (and for no other integrable systems in one-dimension having potential of the formV(x _i ? x_j).     

Coherent states and squeezed states of realq-deformed quantum oscillators

A detailed physical characterisation of the coherent states and squeezed states of a realq-deformed oscillator is attempted. The squeezing andq-squeezing behaviours are illustrated by three different model Hamiltonians, namely i) Batemann Hamiltonian ii) harmonic oscillator with time dependent mass and frequency and iii) a system with constant mass and time-dependent frequency.     

Quantum mechanics of a general driven time-dependent oscillator

Summary In this paper we investigate the time evolution of a general driven time-dependent oscillator using the evolution operator method developed by Chenget al. We obtain an exact form of the time evolution operator which, in turn, enables us to find the exact wave functions and coherent states at any timet. Our analyses indicate that the time-dependent coherent state is equivalent to the well-known squeezed state, while the time-dependent number state is equivalent to the displaced and squeezed number state. Besides, we also calculate the time-dependent transition probabilities among the coherent states and number states of a simple harmonic oscillator associated with the initial HamiltonianH(0).     

Exact solvability of open stochastic systems

It is shown that the boundary vectors in the matrix-product states approach to open stochastic diffusion processes are deformed coherent states of a deformed harmonicoscillator algebra. A unified deformed coherent states solution to the partially and totally asymmetric diffusion boundary problem is proposed and studied.     

SU(1, 1) and SU(2) Perelomov number coherent states: algebraic approach for general systems

We study some properties of the SU(1, 1) Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator K₀ of the su(1, 1) Lie algebra. Analogous results for the SU(2) Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.     

Propagator for the Chebyshev q object

We propose an alternative role of the harmonic oscillator algebra. Observing that the q-deformed harmonic oscillator algebra defines the Chebyshev q object, we show that the q-free particle and the pulsed oscillator are special cases of the Chebyshev q object, characterized by a common deformation parameter q and reduced to a usual free particle as q tends to unity. For the deformed free particle, q is a real number, whereas for the pulsed oscillator it belongs to S ¹. Then, we derive the propagator for the Chebyshev q object, from which we obtain the propagators for the deformed free particle and the pulsed oscillator.     

Interference in Phase Space of Squeezed States for the Time-Dependent Hamiltonian System

We showed that the idea of Schleich and Wheeler (1987, Nature 326, 574) for the semiclassical approach of the interference in phase space of harmonic oscillator squeezed states can be extended to that of general time-dependent Hamiltonian system. The quantum phase properties of squeezed states for the general time-dependent Hamiltonian system are investigated by using the quantum distribution function. The weighted overlaps A _n and phases θ _n for the system are evaluated in the semiclassical limit.     

Multi-parameter deformed fermionic oscillators

In this paper, we study a quantum group covariant deformed fermion algebra. This system can be formulated in n dimensions and posesses two deformation parameters. The undeformed fermion algebra is obtained when both deformation parameters are unity. When both parameters are zero the deformed fermionic oscillator algebra reduces to the orthofermion algebra. If the quantum group symmetry is not preserved, then the number of parameters in n dimensions can be increased to 2n-2. Received: 6 December 2001 / Revised version: 18 June 2002 / Published online: 20 September 2002     

Dynamical symmetries inq-deformed quantum mechanics

The dynamical algebra of theq-deformed harmonic oscillator is constructed. As a result, we find the free deformed Hamiltonian as well as the Hamiltonian of the deformed oscillator as a complicated, momentum dependent interaction Hamiltonian in terms of the usual canonical variables. Furthermore we construct a welldefined algebraSU _q(1,1) with consistent conjugation properties and comultiplication. We obtain non lowest weight representations of this algebra.     

Even and odd q-coherent states in a finite-dimensional basis and their squeezing properties

The even and odd coherent states of a deformed harmonic oscillator in a finites-dimensional Hilbert space are studied. It is shown that both fors even ands odd, the even q-coherent states exhibit quadrature and amplitude-squared squeezing, while the odd q-coherent states show an antibunching effect and amplitude-squared squeezing.     

The driven-oscillator evolution in the tomographic-probability representation

We consider the problem of the driven harmonic oscillator in the probability representation of quantum mechanics, where the oscillator states are described by fair nonnegative probability distributions of position measured in rotated and squeezed reference frames in the system??s phase space. For some specific oscillator states like coherent states and nth excited states, the tomographic-probability distributions (called the state tomograms) are found in an explicit form. The evolution equation for the tomograms is discussed for the classical and quantum driven oscillators, and the tomographic propagator for this equation is studied.     

量子光学中的非经典态

本文扼要地介绍了光子数态、热光场态、相干态、压缩态、相位态和中间态等。重点是介绍它们的物理性质。例如,指出相干态在谐振子座标表象中的表示就是带电谐振子在均匀电场中的基态波函数;它的时间演化波包的概率密度分布,形状不随时间变但中心位置随时间作周期振荡。文中对相干态和压缩态等提供了也许是一点新的看法：将相干态、压缩真空态、压缩相干态和相干压缩态等看作是一准玻色子的基态或相干态。而实现的手段可以是原来的幺正算符也可以是投影算符。这样的好处是：（１）对相干态和压缩态间的联系有更深的认识;（２）便于计算和进一步展开等等。文中还对各个态的压缩性、统计性等作了介绍,有的还用图表等演示了它们的非类经典特性。最后,文中还介绍了准概率分布函数、相空间技术以及它们的应用并给出了示例     

Deformed Heisenberg algebras, a Fock-space representation and the Calogero model

We describe generally deformed Heisenberg algebras in one dimension. The condition for a generalized Leibniz rule is obtained and solved. We analyze conditions under which deformed quantum-mechanical problems have a Fock-space representation. One solution of these conditions leads to a q-deformed oscillator already studied by Lorek et al., and reduces to the harmonic oscillator only in the infinite-momentum frame. The other solution leads to the Calogero model in ordinary quantum mechanics, but reduces to the harmonic oscillator in the absence of deformation. Received: 27 April 2000 / Published online: 8 September 2000     

SU(1,1) Lie Algebra Applied to the General Time-dependent Quadratic Hamiltonian System

Exact quantum states of the time-dependent quadratic Hamiltonian system are investigated using SU(1,1) Lie algebra. We realized SU(1,1) Lie algebra by defining appropriate SU(1,1) generators and derived exact wave functions using this algebra for the system. Raising and lowering operators of SU(1,1) Lie algebra expressed by multiplying a time-constant magnitude and a time-dependent phase factor. Two kinds of the SU(1,1) coherent states, i.e., even and odd coherent states and Perelomov coherent states are studied. We applied our result to the Caldirola–Kanai oscillator. The probability density of these coherent states for the Caldirola–Kanai oscillator converged to the center as time goes by, due to the damping constant γ. All the coherent state probability densities for the driven system are somewhat deformed. PACS Numbers: 02.20.Sv, 03.65.-w, 03.65.Fd     

New q‐Hermite polynomials: characterization,operators algebra and associated coherent states

This paper addresses a construction of new q‐Hermite polynomials with a full characterization of their main properties and corresponding raising and lowering operator algebra. The three‐term recursive relation as well as the second‐order differential equation obeyed by these new polynomials are explicitly derived. Relevant operator actions, including the eigenvalue problem of the deformed oscillator and the self‐adjointness of the related position and momentum operators, are investigated and analyzed. The associated coherent states are constructed and discussed with an explicit resolution of the induced moment problem. The phase collapse in a q‐deformed boson system is studied.     

关于双参数量子代数SUqs(2)和双参数形变谐振子的若干结果

构造了双参数形变量子代数SU_qs(2)的Holstein-Primakoff实现和Nodvik实现,并给出了量子代数SU_qs(2)和双参数形变谐振子的形变映射.     

Overlaps of deformed and non-deformed harmonic oscillator basis states

A systematic approach for expanding non-deformed harmonic oscillator basis states in terms of deformed ones, and vice versa, is presented. The objective is to provide analytical results for calculating these overlaps (transformation brackets) between deformed and non-deformed basis states in spherical, cylindrical, and Cartesian coordinates. These overlaps can be used for reducing the complexity of different research problems that employ three-dimensional harmonic oscillator basis states, for example as used in coherent state theory and the nuclear shell-model, especially within the context of ab initio symmetry-adapted no-core shell model.     

A Two-Parameter Deformed N = 2 SUSY Algebra

A two-parameter deformed N = 2 SUSY algebra is constructed by using the q-deformed bosonic and fermionic Newton oscillator algebras. The Fock space representation of the (q ₁,q ₂)-deformed N = 2 SUSY algebra is analyzed. The comparison between the algebra constructed and earlier versions of deformed N = 2 SUSY algebras is also made.