Even and Odd Coherent States for Time-Dependent Harmonic Oscillator

The dynamical invariant for a general time-dependent harmonic oscillator is constructed by making use of two linearly independent solutions to the classical equation of motion. In terms of this dynamical invariant we define the time-dependent creation and annihilation operators and relevantly introduce even and odd coherent states for time dependent harmonic oscillator. The mathematical and quantum statistical properties of these states are discussed in detail. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.     

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.     

Even and Odd Coherent States for Time-Dependent Harmonic Oscillator

The dynamical invariant for a general time-dependent harmonic oscillator is constructed by making use of two linearly independent solutions to the classical equation of motion. In terms of this dynamical invariant we define the time-dependent creation and annihilation operators and relevantly introduce even and odd coherent states for time-dependent harmonic oscillator. The mathematical and quantum statistical properties of these states are discussed in detail. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.     

Coherent states of general time-dependent harmonic oscillator

By introducing an invariant operator, we obtain exact wave functions for a general time-dependent quadratic harmonic oscillator. The coherent states, both inx- andp-spaces, are calculated. We confirm that the uncertainty product in coherent state is always larger thankh/2 and is equal to the minimum of the uncertainty product of the number states. The displaced wave packet for Caldirola-Kanai oscillator in coherent state oscillates back and forth with time about the center as for a classical oscillator. The amplitude of oscillation with no driving force decreases due to the dissipation in the system. However, the oscillation with resonant frequency oscillates with a large amplitude, even after a sufficient time elapse.     

Coherent States and Modified de Broglie-Bohm Complex Quantum Trajectories

This paper examines the nature of classical correspondence in the case of coherent states at the level of quantum trajectories. We first show that for a harmonic oscillator, the coherent state complex quantum trajectories and the complex classical trajectories are identical to each other. This congruence in the complex plane, not restricted to high quantum numbers alone, illustrates that the harmonic oscillator in a coherent state executes classical motion. The quantum trajectories we consider are those conceived in a modified de Broglie-Bohm scheme. Though quantum trajectory representations are widely discussed in recent years, identical classical and quantum trajectories for coherent states are obtained only in the present approach. We may note that this result for standard harmonic oscillator coherent states is not totally unexpected because of their holomorphic nature. The study is extended to coherent states of a particle in an infinite potential well and that in a symmetric Poschl-Teller potential by solving for the trajectories numerically. For the Gazeau-Klauder coherent state of the infinite potential well, almost identical classical and quantum trajectories are obtained whereas for the Poschl-Teller potential, though classical trajectories are not regained, a periodic motion results as t→∞. Similar features were found for the SUSY quantum mechanics-based coherent states of the Poschl-Teller potential too, but this time the pattern of complex trajectories is quite different from that of the previous case. Thus we find that the method is a potential tool in analyzing the properties of generalized coherent states.     

Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach

This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.     

q变形非简谐振子奇偶广义相干态的高阶压缩效应及反聚束效应

研究了q变形非简谐振子奇偶广义相干态的高阶压缩效应和反聚束效应,并就q变量[χ]的两种不同表示情况进行了讨论.数值计算结果表明,q变形非简谐振子奇偶广义相干态均可呈现奇次方阶压缩效应和反聚束效应,这与谐振子情况的光学统计特性是不同的.     

Generalized path-integral solution and coherent states of the harmonic oscillator in D-dimensions

We construct a general form of propagator in arbitrary dimensions and give an exact wavefunction of a time- dependent forced harmonic oscillator in D（D ≥ 1） dimensions. The coherent states, defined as the eigenstates of annihilation operator, of the D-dimensional harmonic oscillator are derived. These coherent states correspond to the minimum uncertainty states and the relation between them is investigated.     

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.     

Radial coherent states—From the harmonic oscillator to the hydrogen atom

We construct spectrum generating algebras of SO(2, 1) ～ SU(1, 1) in arbitrary dimension for the isotropic harmonic oscillator and the Sturm-Coulomb problem in radial coordinates. Using these algebras, we construct the associated radial Barut-Girardello coherent states for the isotropic harmonic oscillator (in arbitrary dimension). We map these states into the Sturm-Coulomb radial coherent states and show that they evolve in a fictitious time parameter without dispersing.     

Photon-bunching, photon-antibunching and the nonclassical photon statistics of coherent light coupled to a driven harmonic oscillator of time dependent mass and frequency

The analytical solution of the quantum driven harmonic oscillator of time dependent mass and frequency (DHTDMF) is exploited to obtain the photon-bunching, photon-antibunching and the nonclassical photon statistics of the input coherent light coupled to the oscillator. We observe that the model Hamiltonian of a DHTDMF coincides with those of the model Hamiltonian of a two-photon coherent state due to H.P. Yuen [Phys. Rev. A 13 (1976) 226]. To substantiate our analytical results, we give a special case corresponding to a damped harmonic oscillator subject to an external periodic force. The quantum statistical properties of the radiation field are discussed for under-, over-, and for critically-damped situations of the oscillator coupled to an initial coherent field. The off-resonance and the near- or on-resonance of the oscillator with respect to the frequency of the external force are discussed. In the determination of quantum statistical properties of the radiation field, we observe that the effects of the driving force are important only when the frequency of the oscillator is in near- or in on-resonance condition with that of the frequency of the external periodic force.     

Quantum harmonic oscillator with time-dependent mass and frequency

The quantum harmonic oscillator with time-dependent mass and frequency is analyzed by using the canonical transformation method. The varying mass and frequency of the system are reduced to constant mass and frequency, and the corresponding eigenvalues and eigenvectors are derived. The exact time-dependent coherent state of the harmonic oscillator is constructed and shown to be equivalent to the squeezed state. Damped harmonic oscillators with different frictions and forced time-dependent harmonic oscillators are also discussed.     

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     

Coherence and stochastic resonance in a birhythmic van der Pol system

We consider the response to uncorrelated noise and harmonic excitation of a birhythmic van der Pol-type oscillator. This system, as opposed to the standard van der Pol oscillator, is characterized by two stable orbits. The noisy oscillator can be analytically mapped, with the technique of stochastic averaging, onto an ordinary bistable system with a bistable (quasi)potential. The birhythmic oscillator can also be numerically characterized through the diagnostics of coherent resonance and the signal-to-noise-ratio. The analysis shows the presence of noise-induced coherent states, influenced by the different time scales of the oscillator.     

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).     

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.     

Smorodinsky-Winternitz potentials: Coherent state approach

In this study, we construct the coherent states for a particle in the Smorodinsky-Winternitz potentials, which are the generalizations of the two-dimensional harmonic oscillator problem and the Kepler-Coulomb problem. In the first case we find the nonspreading wave packets by transforming the system into four oscillators in Cartesian, and also polar, coordinates. In the second case, the coherent states are constructed in Cartesian coordinates by transforming the system into three nonisotropic harmonic oscillators. All of these states evolve in physical-time. In the third case, the system is transformed into four oscillators and the parametric-time coherent states are constructed in two coordinate frames. In the fourth case, the system is transformed into two oscillators with the reflection symmetry and the parametrictime coherent states are constructed in two coordinate frames.     

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 driven oscillator in the photon-number probability representation of quantum mechanics

We study the evolution of the driven harmonic oscillator in the probability representation of quantum mechanics. We use the photon-number tomographic-probability-distribution function to describe the quantum states of the system. We give a general review of the photon-number tomographic framework, including a discussion on the connection with other representations of quantum mechanics. We find tomograms of coherent states as well as excited states of the harmonic oscillator in an explicit form. We discuss the time evolution of the photon-number tomograms and transforms of the propagators for different representations of quantum mechanics. We obtain the propagator for the photon-number tomographic-distribution function for the case of the driven oscillator in an explicit form.     

Interpretation of the Quantum Measurement: Example of a Linearly Coupled Harmonic Oscillator

We argue that it may be possible to consistently explain the quantum measurement by assuming that the wave function is in one-to-one correspondence with objective physical reality and has no probabilistic interpretation. In the context of such approach we consider the model of a harmonic oscillator linearly coupled to a heat bath and treat the oscillator as the system being measured. Three classes of initial pure states for the bath are considered. Exact expressions for the average values and variances of the oscillator coordinate and momentum as functions of time are considered for each class of pure states. It is shown that these quantities exhibit different asymptotic behavior for different classes of initial states of the bath. In particular, if each mode of the bath is initially in a coherent state, then for an arbitrary initial state of the oscillator the variances of the oscillator coordinate and momentum asymptotically approach the same values as for a coherent state of the free oscillator, while the averages of coordinate and momentum show a Brownian-like behavior. We argue that such behavior shows several features of the quantum measurement and supports our interpretation of the wave function.