Deformed coherent and squeezed states of multiparticle processes

Deformed squeezed states are introduced as the q-analogues of the conventional undeformed harmonic oscillator algebra squeezed states. It is shown that the boundary vectors in the matrix-product states approach to multiparticle diffusion processes are deformed coherent or squeezed states of a deformed harmonic oscillator algebra. A deformed squeezed and coherent-states solution to the n-species stochastic diffusion boundary problem is proposed and studied.Received: 31 January 2003, Published online: 10 October 2003     

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     

Deformed quantum harmonic oscillator with diffusion and dissipation

A master equation for the deformed quantum harmonic oscillator interacting with a dissipative environment, in particular with a thermal bath, is derived in the microscopic model by using perturbation theory. The coefficients of the master equation and of equations of motion for observables depend on the deformation function. The steady-state solution of the equation for the density matrix in the number representation is obtained and the equilibrium energy of the deformed harmonic oscillator is calculated in the approximation of small deformation.     

Quantum Dynamics of a Harmonic Oscillator in a Deformed Bath

The dissipative quantum dynamics of a harmonic oscillator in the presence of a deformed bath is investigated. The deformed bath is modelled by a collection of deformed quantum harmonic oscillators as a generalization of Hopfield model. The transition probabilities between energy levels of the oscillator are obtained perturbatively and discussed.     

Fun and frustration with Calogero model

We review some algebraical (oscillator) aspects of N-body single-species and multispecies Calogero models in one dimension. We treat them as a particular cases of deformed harmonic oscillators and discuss the corresponding Fock spaces. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

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).     

Heat Transport in Ordered Harmonic Lattices

We consider heat conduction across an ordered oscillator chain with harmonic interparticle interactions and also onsite harmonic potentials. The onsite spring constant is the same for all sites excepting the boundary sites. The chain is connected to Ohmic heat reservoirs at different temperatures. We use an approach following from a direct solution of the Langevin equations of motion. This works both in the classical and quantum regimes. In the classical case we obtain an exact formula for the heat current in the limit of system size N→∞. In special cases this reduces to earlier results obtained by Rieder, Lebowitz and Lieb and by Nakazawa. We also obtain results for the quantum mechanical case where we study the temperature dependence of the heat current. We briefly discuss results in higher dimensions.     

变形谐振子势的能谱方程

讨论了3种变形谐振子势:左右两边不同参数的谐振子势、左边方形势加右边谐振子势和谐振子势中间加δ势中的能量本征态函数.这些函数都可以由厄米函数表示.由波函数及其一次导数在原点的衔接条件,得到了能谱方程.     

Extending Newton's law from nonlocal-in-time kinetic energy

We study a new equation of motion derived from a context of classical Newtonian mechanics by replacing the kinetic energy with a form of nonlocal-in-time kinetic energy. It leads to a hypothetical extension of Newton's second law of motion. In a first stage the obtainable solution form is studied by considering an unknown value for the nonlocality time extent. This is done in relation to higher-order Euler-Lagrange equations and a Hamiltonian framework. In a second stage the free particle case and harmonic oscillator case are studied and compared with quantum mechanical results. For a free particle it is shown that the solution form is a superposition of the classical straight line motion and a Fourier series. We discuss the link with quanta interpretations made in Pais-Uhlenbeck oscillators. The discrete nature emerges from the continuous time setting through application of the least action principle. The harmonic oscillator case leads to energy levels that approximately correspond to the quantum harmonic oscillator levels. The solution to the extended Newton equation also admits a quantization of the nonlocality time extent, which is determined by the classical oscillator frequency. The extended equation suggests a new possible way for understanding the relationship between classical and quantum mechanics.     

Supersymmetric Pair of q-Deformed Nonlocal Operators

A simple version of the q-deformed calculus is used to generate a pair ofq-nonlocal, second-order difference operators by means of deformed counterpartsof Darboux intertwining operators for the Schrödinger—Hermite oscillators atzero factorization energy. These deformed nonlocal operators may be consideredas supersymmetric partners and their structure contains contributions originatingin both the Hermite operator and the quantum harmonic oscillator operator. Thereare also extra ±x contributions. The undeformed limit, in which allq-nonlocalities wash out, corresponds to the usual supersymmetric pair of quantum mechanicalharmonic oscillator Hamiltonians. The more general case of negative factorizationenergy is briefly discussed as well.     

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.     

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.     

Quantum Dynamics of a Dissipative Deformed Harmonic Oscillator

The quantum dynamics of a dissipative deformed harmonic oscillator is investigated in the framework of the minimal coupling method. The reduced density matrix of the deformed oscillator is obtained and the decay transitions are calculated.     

Classical Interpretation of a Deformed Quantum Oscillator

Following the same procedure that allowed Shcrödinger to construct the (canonical) coherent states in the first place, we investigate on a possible classical interpretation of the deformed harmonic oscillator. We find that, these oscillator, also called q-oscillators, can be interpreted as quantum versions of classical forced oscillators with a modified q-dependant frequency.     

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.     

“Optimal” bounds on the ground-state energy ofN-body systems of bosons and fermions interacting by attractive forces

We present inequalities on the ground state energy ofN-body systems which reduce, for bosons and fermions, to the exact solution in the limit where forces approach harmonic oscillator forces.     

外加磁场下抛物型量子线中的带电激子

在一维等效模型下采用有效差分法对抛物型量子阱线中带电激子的束缚能进行了计算,分析了约束势以及磁场对带电激子束缚能的影响,并对带正电激子(X⁺)和带负电激子(X^-)的情况进行了比较.结果表明:电子和空穴的振子强度对带电激子的稳定性有重要影响,X⁺的束缚能不总是比X^-的大,随着空穴振子强度的增加束缚能的函数曲线将会出现交叉,这同实验得到的结果符合;磁场的存在会增加粒子间的束缚,并且磁场对束缚能的影响同振子强度大小有关. 关键词： 带电激子 量子线 束缚能 磁场     

Quantum Dynamics of a Harmonic Oscillator in a Defomed Bath in the Presence of Lamb Shift

In this paper, we investigate the dissipative quantum dynamics of a harmonic oscillator in the presence a deformed bath by considering the Lamb shift term. The deformed bath is modelled by a collection of deformed quantum harmonic oscillators as a generalization of Hopfield model. The Langevin equation for both the photon number and the fluctuation spectrum under the Weisskopf–Winger approximation are obtained and discussed.