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     

Solutions of Two-Mode Bosonic and Transformed Hamiltonians

We have constructed the quasi-exactly-solvable two-mode bosonic realization of SU(2). Two-mode boson Hamiltonian is defined through a differential equation which is solved by quantum Hamilton-Jacobi formalism. The squeezed states of two-mode boson systems are characterized through canonical transformation. The illustrated concept of squeezed boson systems has been applied two-mode bosonic Hamiltonian which is a squeezed one and is determined through a differential equation. This differential equation is solved and energy eigenvalues are found approximately.     

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.     

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     

Josephson array of mesoscopic objects. Modulation of system properties through the chemical potential

The phase diagram of a two-dimensional Josephson array of mesoscopic objects (superconducting granules, superfluid helium in a porous medium, traps with Bose-condensed atoms, etc.) is examined. Quantum fluctuations in both the modulus and phase of the superconducting order parameter are taken into account within a lattice boson Hubbard model. Modulating the average occupation number n ₀ of the sites in the system (the “number of Cooper pairs” per granule, the number of atoms in a trap, etc.) leads to changes in the state of the array, and the character of these changes depends significantly on the region of the phase diagram being examined. In the region where there are large quantum fluctuations in the phase of the superconducting order parameter, variation of the chemical potential causes oscillations with alternating superconducting (superfluid) and normal states of the array. On the other hand, in the region where the bosons interact weakly, the properties of the system depend monotonically on n ₀. Lowering the temperature and increasing the particle interaction force lead to a reduction in the width of the region of variation in n ₀ within which the system properties depend weakly on the average occupation number. The phase diagram of the array is obtained by mapping this quantum system onto a classical two-dimensional XY model with a renormalized Josephson coupling constant and is consistent with our quantum path-integral Monte Carlo calculations. Zh. éksp. Teor. Fiz. 114, 591–604 (August 1998)     

Order parameter quantum fluctuations in a two-dimensional system of mesoscopic Josephson junctions

The boson lattice Hubbard model is used to study the role of quantum fluctuations of the phase and local density of the superfluid component in establishing a global superconducting state for a system of mesoscopic Josephson junctions or grains. The quantum Monte Carlo method is used to calculate the density of the superfluid component and fluctuations in the number of particles at sites of the two-dimensional lattice for various average site occupation numbers n ₀ (i.e., number of Cooper pairs per grain). For a system of strongly interacting bosons, the phase boundary of the ordered superconducting state lies above the corresponding boundary for its quasiclassical limit—the quantum XY-model—and approaches the latter as n ₀ increases. When the boson interaction is weak in the boson Hubbard model (i.e., the quantum fluctuations of the phase are small), the relative fluctuations of the order parameter modulus are significant when n ₀<10, while quantum fluctuations in the phase are significant when n ₀<8; this determines the region of mesoscopic behavior of the system. Comparison of the results of numerical modeling with theoretical calculations show that mean-field theory yields a qualitatively correct estimate of the difference between the phase diagrams of the quantum XY-model and the Hubbard model. For a quantitative estimate of this difference the free energy and thermodynamic averages of the Hubbard model are expanded in powers of 1/n ₀ using the method of functional integration. Zh. éksp. Teor. Fiz. 113, 261–277 (January 1998)     

Time Evolution and Squeezing of Degenerate and Non-degenerate Coupled Parametric Down-Conversion with Driving Term

By properly selecting the time-dependent unitary transformation for the linear combination of the number operators, we construct a time-dependent invariant and derive the corresponding auxiliary equations for the degenerate and non-degenerate coupled parametric down-conversion system with driving term. By means of this invariant and the Lewis-Riesenfeld quantum invariant theory, we obtain closed formulae of the quantum state and the evolution operator of the system. We show that the time evolution of the quantum system directly leads to production of various generalized one- and two-mode combination squeezed states, and the squeezed effect is independent of the driving term of the Hamiltonian. In somespecial cases, the current solution can reduce to the results of the previous works.     

Higher-order <Emphasis Type="Bold">?</Emphasis> corrections in the semiclassical quantization of chaotic billiards

In the periodic orbit quantization of physical systems, usually only the leading-order ? contribution to the density of states is considered. Therefore, by construction, the eigenvalues following from semiclassical trace formulae generally agree with the exact quantum ones only to lowest order of ?. In different theoretical work the trace formulae have been extended to higher orders of ?. The problem remains, however, how to actually calculate eigenvalues from the extended trace formulae since, even with ? corrections included, the periodic orbit sums still do not converge in the physical domain. For lowest-order semiclassical trace formulae the convergence problem can be elegantly, and universally, circumvented by application of the technique of harmonic inversion. In this paper we show how, for general scaling chaotic systems, also higher-order ? corrections to the Gutzwiller formula can be included in the harmonic inversion scheme, and demonstrate that corrected semiclassical eigenvalues can be calculated despite the convergence problem. The method is applied to the open three-disk scattering system, as a prototype of a chaotic system. Received 10 September 2001 and Received in final form 3 January 2002     

Semiclassical and quasiclassical calculation of reaction probabilities for collinear X + F2→XF + F (X = Mu,H, D,T)

Semiclassical calculations of reaction probabilities have been carried out for the collinear H + F₂ (n = 0, 1) reaction using the best extended LEPS surface No. II of Jonathan et al. Both real and complex valued classical trajectories have been included in the calculations for an energy range where the quasiclassical total reaction probability is unity. Comparison with quantum results shows the semiclassical reaction probabilities are accurate to about ± 0·05 provided only two real or complex stationary phase points make an important contribution to the S matrix element, so that the uniform Airy or integer Bessel approximations are valid. Real semiclassical calculations are also reported for the collinear Mu, D, T + F₂ (n = 0) reactions. For the D and T reactions, the semiclassical reaction probabilities are estimated to be accurate to ± 0·05, except close to the reaction threshold, but for the Mu reaction the estimated errors are much larger. In addition, quasiclassical calculations for the reaction probabilities have been carried out using half integer boxing and smooth sampling methods to quantize the product distributions. For the H + F₂ reaction, there are usually systematic deviations from the quantum reaction probabilities and the same is expected to be true for the Mu, D and T reactions.     

Quasiclassical Hamiltonians for large-spin systems

We extend and apply a previously developed method for a semiclassical treatment of a system with large spin S. A multisite Heisenberg Hamiltonian is transformed into an effective classical Hamilton function which can be treated by standard methods for classical systems. Quantum effects enter in form of multispin interactions in the Hamilton function. The latter is written in the form of an expansion in powers of J/(TS), where J is the coupling constant. Main ingredients of our method are spin coherent states and cumulants. Rules and diagrams are derived for computing cumulants of groups of operators entering the Hamiltonian. The theory is illustrated by calculating the quantum corrections to the free energy of a Heisenberg chain which were previously computed by a Wigner-Kirkwood expansion. Received 5 May 1999 and received in final form 24 September 1999     

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.     

Low Temperature Phase Diagrams¶of Fermionic Lattice Systems

We consider fermionic lattice systems with Hamiltonian H=H ^{(0)}+λH _Q , where H ^{(0)} is diagonal in the occupation number basis, while H _Q is a suitable “quantum perturbation”. We assume that H ^{(0)} is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitations, while H _Q is a finite range or exponentially decaying Hamiltonian that can be written as a sum of even monomials in the fermionic creation and annihilation operators. Mapping the d dimensional quantum system onto a classical contour system on a d+1 dimensional lattice, we use standard Pirogov–Sinai theory to show that the low temperature phase diagram of the quantum system is a small perturbation of the zero temperature phase diagram of the classical system, provided λ is sufficiently small. Particular attention is paid to the sign problems arising from the fermionic nature of the quantum particles. As a simple application of our methods, we consider the Hubbard model with an additional nearest neighbor repulsion. For this model, we rigorously establish the existence of a paramagnetic phase with commensurate staggered charge order for the narrow band case at sufficiently low temperatures. Received: 23 December 1996/ Accepted: 7 April 1999     

一维分子晶体系统的极化子-孤子压缩态、系统基态性质和量子涨落

基于一维分子晶体系统的 Holstein 模型,采用压缩-相干态展开方法,计及电子-声子间量子关联和重整化平移修正,分析和研究电子-双声子相互作用对极化子-孤子系统基态性质和量子涨落的影响.推导了一维极化子-孤子系统的封闭形式非线性方程.应用非线性项展开方法,给出非线性方程的解析解和相关基态特性结果.研究表明,仅当电子-双声子耦合强度 g₁<0时非线性方程才有孤波解,此时声子量子涨落效应随着压缩的增加,极化子-孤子系统基态能量变得更负,孤子局域减少,孤子态更加稳定;另一方面,电子密度涨落〈Δ²n〉和声子坐标-动量的不确定量〈Δ²p〉〈Δ²q〉比无声子压缩效应的大,极化子结合能变得更负.特别是,当g₁<0时,双声子效应的量子涨落〈Δ²n〉与〈Δ²p〉〈Δ²q〉的值比单声子情况有明显增加. 关键词： 压缩-相干态展开 极化子-孤子态与量子涨落 电子-双声子相互作用 非线性薛定谔方程     

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).     

Semiclassical bosonic <Emphasis Type="Italic">D</Emphasis>-brane boundary states in curved spacetime

In this paper we discuss the existence of quantum D-brane states in the strong gravitational field and in the presence of a constant Kalb-Ramond field. A semiclassical string quantization method in which the spacetime metric g _AB and the constant antisymmetric Kalb-Ramond field b _AB are treated exactly is employed. In this framework, the semiclassical D-branes are defined at the first order perturbation around the trajectory of the center-of-mass of a string. The set of equations the semiclassical D-branes must satisfy in a general strong gravitational field are given. These equations are solved in the AdS background where it is shown that a D-brane coherent state exists if the operators that project the string fields onto the corresponding Neumann and Dirichlet directions satisfy a set of algebraic constraints. A second set of equations that should be satisfied by the projectors in order that the semiclassical state be compatible with the global structure of the D-brane are derived in the particle limit of a string in the torsionless AdS background.     

Canonical Structure and Symmetries of the Schlesinger Equations

The Schlesinger equations S _(n,m) describe monodromy preserving deformations of order m Fuchsian systems with n + 1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations S _(n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates.     

On the dynamics of a quantum system which is classically chaotic

We investigate the dynamics of one anisotropic spin in an external time-dependent magnetic field. The classical dynamics of the system is nonintegrable (and very similar to the standard map). We present results on this model for a quantum spin (i.e. for finite values of the spin lengthS). In particular we discuss the semiclassical regime,S1, using the concept of Wigner functions to define a suitable probability distribution. In regular regions of phase space the time evolution of the probability distribution shows an algebraic decay of correlations as in quantum mechanics. In chaotic regions of phase space it is characterised by a positive Lyapunov exponent which depends onS. In these regions semiclassical trajectories coincide with classical ones fort <₀ where ₀InS.     

Quantum rotors with regular frustration and the quantum Lifshitz point

We have discussed the zero-temperature quantum phase transition in n-component quantum rotor Hamiltonian in the presence of regular frustration in the interaction. The phase diagram consists of ferromagnetic, helical and quantum paramagnetic phase, where the ferro-para and the helical-para phase boundary meets at a multicritical point called a (d,m) quantum Lifshitz point where (d,m) indicates that the m of the d spatial dimensions incorporate frustration. We have studied the Hamiltonian in the vicinity of the quantum Lifshitz point in the spherical limit and also studied the renormalisation group flow behaviour using standard momentum space renormalisation technique (for finite n). In the spherical limit ()one finds that the helical phase does not exist in the presence of any nonvanishing quantum fluctuation for m =d though the quantum Lifshitz point exists for all d > 1+m/2, and the upper critical dimensionality is given by d _u = 3 +m/2. The scaling behaviour in the neighbourhood of a quantum Lifshitz point in d dimensions is consistent with the behaviour near the classical Lifshitz point in (d+z) dimensions. The dynamical exponent of the quantum Hamiltonian z is unity in the case of anisotropic Lifshitz point (d>m) whereas z=2 in the case of isotropic Lifshitz point (d=m). We have evaluated all the exponents using the renormalisation flow equations along-with the scaling relations near the quantum Lifshitz point. We have also obtained the exponents in the spherical limit (). It has also been shown that the exponents in the spherical model are all related to those of the corresponding Gaussian model by Fisher renormalisation. Received: 23 December 1997 / Received in final form: 6 January 1998 / Accepted: 7 January 1998