Gauge covariant construction of the coherent states for a time-dependent harmonic oscillator by algebraic dynamical method

Time-dependent coherent states for a time-dependent harmonic oscillator are constructed in the framework of algebraic dynamics. These coherent states are gauge-covariant, and its time evolution is governed only by the solutions of a linear differential equation which describes the motion of the corresponding classical timedependent harmonic oscillator. Its non-classical and quantum statistical properties can thus be controlled by a proper choice of the frequency of the harmonic oscillator. Our coherent states reduce to Glauber coherent states in the case as the frequency is independent of time.     

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.     

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.     

Quanta and coherent states

The quantum harmonic oscillator can be considered as a composite system of indistinguishable Bose-Einstein symmetric two-level-systems (quanta). In analogy to the classical Poisson limit theorem, we show that a coherent state is the limit of a sequence of homogeneous product states (coherent spin states) and discuss statistical properties of the quanta in classical and nonclassical states.     

囚禁单离子的量子阻尼运动

用包含偶极和四极虚势能项的非厄米哈密顿算符来描述Paul阱中囚禁阻尼单离子在静电场下的量子运动.通过导出和分析系统的精确解,得到在PT对称和不对称情形下的不同实能谱与稳定量子态,以及PT不对称情形的虚能谱和衰减量子态,同时给出相应于不同态的参数区域和存活概率.结果发现该非厄米系统外场参数能惟一确定量子稳定态并导致波函数形态变化,据此提出非相干操控相应量子跃迁的方法.让量子态衰减导致的离子位置期待值的衰减与经典阻尼谐振子的衰减一致,得到虚势能参数与经典阻尼参数的对应关系.所得结果将进一步丰富具有广泛应用背景的囚禁离子动力学.     

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.     

Chaotic Motion of a Nucleon in Mean Field Potential With Octupole Deformation (I) The Formalism of the Propagation of Coherent States in Potential

The temporal variation characteristics of nonstationary wave functions are investigated, which enables us to carry out the study of quantum chaotic dynamics with the same starting point as in corresponding classical case, especially to realize the sensitivity of the quantum state with respect to the initial condition. Here the coherent states under the dynamical symmetry of asymmetrical two dimensional harmonic oscillator, in which the minimum uncertainty Principle is satisfied, are usedas an initial state. The formalism of the temporal variation of the expectation values and the uncertainty measurements of canonical variables of the quantum state under the broken symmetry by the additional octupole deformed potential is fulfilled.     

Infinite quantum well: A coherent state approach

A new family of 2-component vector-valued coherent states for the quantum particle motion in an infinite square well potential is presented. They allow a consistent quantization of the classical phase space and observables for a particle in this potential. We then study the resulting position and (well-defined) momentum operators. We also consider their mean values in coherent states and their quantum dispersions.     

氢原子相干态

由Kustannheimo-Stiefel变换,可将量子力学中的氢原子问题化为带有约束条件的四维各向同性谐振子。在此基础上定义相干态,并证明力学量坐标和动量对相干态的平均,给出经典开普勒运动轨道。同时也讨论该相干态中的测不准关系式。 关键词：     

Quantum Information in the Frame of Coherent States Representation

In this paper we have showed that the qubit can be expressed through the coherent states. Consequently, a message, i.e. a sequence of qubits, is expressed as a tensor product of coherent states. In the quantum information theory and practice, only the code and key message are expressed as a sequence of qubits, i.e. through a quantum channel, the properly information will be transmitted by using a classical channel. Even if the most used coherent states in the quantum information theory are the coherent states of the harmonic oscillator (particularly, expressing by them the Schrödinger “cat states” and the Bell states), several authors have been demonstrated that other kind of coherent states may be used in quantum information theory. For the ensembles of qubits, we must use the density operator, in order to describe the informational content of the ensemble. The diagonal representation of the density operator, in the coherent state representation, is also useful to examine the entanglement of the states.     

Symmetry Breaking Induced Geometric Surfaces with Topological Curves in Quantum and Classical Dynamics of the SU(2) Coupled Oscillators

In the three‐dimensional (3D) transversely symmetric oscillator, there are plentiful degeneracies and gaps in the quantum energy spectrum as a function of the ratio of the transverse to longitudinal frequency. It is theoretically verified that while the SU(2) interaction destroys the original degeneracies, numerous new degeneracies and gaps emerge around the original degeneracies to form a similar fine energy spectrum. The classical trajectories at the emergent degeneracies are analyzed to be localized on the 3D parametric surfaces which are constituted by the topologically invariant curves in the transverse tomography. The quantum coherent states are exploited to develop the wave functions that correspond to the 3D geometric surfaces in classical dynamics. Furthermore, the wave structures of the stationary coherent states at small quantum numbers are explored and found to display peculiar patterns with symmetries related to classical trajectories.     

The simplex structure of the classical states of the quantum harmonic oscillator

It is shown that the convex set of classical states of the quantum harmonic oscillator is a simplex generated as the closed convex hull of the coherent states in the weak topology of the Banach space of trace class operators.     

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.     

Construction of the Barut–Girardello quasi coherent states for the Morse potential

The Morse oscillator (MO) potential occupies a privileged place among the anharmonic oscillator potentials due to its applications in quantum mechanics to diatomic or polyatomic molecules, spectroscopy and so on. For this potential some kinds of coherent states (especially of the Klauder–Perelomov and Gazeau–Klauder kinds) have been constructed previously. In this paper we construct the coherent states of the Barut–Girardello kind (BG-CSs) for the MO potential, which have received less attention in the scientific literature. We obtain these CSs and demonstrate that they fulfil all conditions required by the coherent state. The Mandel parameter for the pure BG-CSs and Husimi’s and P$P$-quasi distribution functions (for the mixed-thermal states) are also presented. Finally, we show that all obtained results for the BG-CSs of MO tend, in the harmonic limit, to the corresponding results for the coherent states of the one dimensional harmonic oscillator (CSs for the HO-1D).     

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.     

单粒子在非轴对称八极形变势场中的混沌运动

针对单粒子处在长椭球谐振子势加上Y₃₂+Y_3－2形变势场中的情况, 从经典和量子两个角度分析了粒子在非轴对称八极形变势场中的混沌运动. 通过经典的轨道稳定性分析, 指出了系统等势能面性质对粒子运动特点的重要性. 又通过对体系存在的能级免交叉现象, 三维不对称谐振子相干态在势场中的演化特征的研究, 具体地说明了八级形变势的非轴对称性使单粒子更容易出现混沌运动.     

Quantum-classical correspondence for motion on a plane with deficit angle

A particle constrained to move on a cone and bound to its tip by harmonic oscillator and Coulomb-Kepler potentials is considered both in the classical as well as in the quantum formulations. The SU(2) coherent states are formally derived for the former model and used for showing some relations between closed classical orbits and quantum probability densities. Similar relations are shown for the Coulomb-Kepler problem. In both cases a perfect localization of the densities on the classical solutions is obtained even for low values of quantum numbers.     

Universal and deterministic manipulation of the quantum state of harmonic oscillators: a route to unitary gates for Fock state qubits

We present a simple quantum circuit that allows for the universal and deterministic manipulation of the quantum state of confined harmonic oscillators. The scheme is based on the selective interactions of the referred oscillator with an auxiliary three-level system and a classical external driving source, and enables any unitary operations on Fock states, two by two. One circuit is equivalent to a single qubit unitary logical gate on Fock states qubits. Sequences of similar protocols allow for complete, deterministic, and state-independent manipulation of the harmonic oscillator quantum state.     

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.