The mapping of the Wigner distribution function (WDF) for a given bound-state onto a semiclassical distribution function (SDF) satisfying the Liouville equation introduced previously by us is applied to the ground state of the Morse oscillator. Here we give results showing that the SDF gets closer to the corresponding WDF as the number of levels of the Morse oscillator increases. We find that for a Morse oscillator with one level only, the agreement between the WDF and the mapped SDF is very poor but for a Morse oscillator of ten levels it becomes satisfactory.
Using the Wigner distribution function, we analyze the behavior on phase space of generalized coherent states associated with the Morse potential (Morse-like coherent states). Within the f-deformed oscillator formalism, such states are constructed by means of the two following definitions: {\it i)} as deformed displacement operator coherent states (DOCSs) and {\it ii)} as deformed photon-added coherent states (DPACSs).
Analytical eigensolutions for Morse oscillators are used to investigate quantum resonance and revivals and show how Morse anharmonicity affects revival times. A minimum semi-classical Morse revival time Tmin-rev found by Heller is related to a complete quantum revival time Trev using a quantum deviation parameter that in turn relates Trev to the maximum quantum beat period Tmax-beat. Also, number theory of Farey and Thales-circle geometry of Ford is shown to elegantly analyze and display fractional revivals. Such quantum dynamical analysis may have applications for spectroscopy or quantum info…
Morse oscillator is one of the known solvable potentials which attracts many applications in quantum mechanics especially in quantum chemistry. One of the interesting results of this study is the generation of ladder operators for Morse potential. The operators are a representation of the shifting energy levels of the states exhibited by the wavefunction. From this result, we manipulate and deform the operators in such a way that it gives a noncommutative property to promote noncommutative quantum mechanics (NCQM). The resultant NC feature can be shown in the spatial coordinates and finally th…
The recently proposed PT-symmetric quantum mechanics works with complex potentials which possess, roughly speaking, a symmetric real part and an anti-symmetric imaginary part. We propose and describe a new exactly solvable model of this type. It is defined as a specific analytic continuation of the shape-invariant potential of Morse. In contrast to the latter well-known example, all the new spectrum proves real, discrete and bounded below. All its three separate subsequences are quadratic in n.
We make an extensive rigorous study on convergent behaviour of Morse-Feshbach nonlinear perturbation series (MFNPS) to find out energy levels of the anharmonic oscillator (AHO) in both weak and strong coupling limit. We develop a new method of multi step optimal splitting in order to get convergency in MFNPS for ground state of AHO and found that two step optimal splitting is sufficient to provide convergency in MFNPS. Unlike the ground state the optimal splitting parameters for excited states is modified according to their dependency on state in order to achieve convergency in MFNPS.
The physical origin is investigated of Robin boundary conditions for wave functions at an infinite reflecting wall. We consider both Schrödinger and phase-space quantum mechanics (a.k.a. deformation quantization), for this simple example of a contact interaction. A non-relativistic particle moving freely on the half-line is treated as moving on the full line in the presence of an infinite potential wall, realized as a limit of a Morse potential. We show that the wave functions for the Morse states can become those for a free particle on the half-line with Robin boundary conditions. However, Di…
Three-Dimensional Eigenvalues of Harmonic Oscillator- and Coulomb-type Potentials from One-Dimensional Generalized Morse Potential: Perturbative Analyse based on Generalized Laguerre Polynomials
We present the analysis of mathematical structure of SU(2) group, specifically the commutation relation between raising and lowering operators of the Morse oscillator. The relationship between the commutator of operators and other parameters of Morse oscillator is investigated. We show that the mathematical structure of operators which depends on the parameters of Morse oscillator may change our conventional expectation. The parameter space of Morse oscillator is visualized to scrutinize the mathematical relations that are related to the Morse oscillator. This parameter space is the space of p…
New time dependent Wigner functions for the quantum harmonic oscillator have been obtained in this work. The Moyal equation for the harmonic oscillator has been presented as the wave equation of a 2D membrane in the phase plane. The values of the Wigner function are equal to the deviation values of the points on the surface of the membrane from the equilibrium state. The positive and negative values of the Wigner function correspond to the direction of the deviation from the equilibrium state. As an example, a time dependent Wigner function corresponding to the standing wave of quasi-probabili…
For a quantum oscillator with the polynomial potential an explicit expression that describes the energy distribution as a coordinate (and momentum) function is obtained. The presence of the energy function poles is shown for the quantum system in the domains where the Wigner function has negative values.
The integral Wigner - Liouwille equation describing time evolution of the semi-relativistic quantum 1D harmonic oscillator have been exactly solved by combination of the Monte-Carlo procedure and molecular dynamics methods. The strong influence of the relativistic effects on the time evolution of the momentum, velocity and coordinate Wigner distribution functions and the average values of quantum operators have been studied. Unexpected 'protuberances' in time evolution of the distribution functions were observed. Relativistic proper time dilation for oscillator have been calculated.
The output field from a continuously driven linear parametric oscillator may exhibit considerably more squeezing than the intracavity field. Inspired by this fact, we explore the nonclassical features of the steady-state output field of a driven nonlinear Kerr parametric oscillator using a temporal wave packet mode description. Utilizing a new numerical method, we have access to the density matrix of arbitrary wave packet modes. Remarkably, we find that even though the steady-state cavity field is always characterized by a positive Wigner function, the output may exhibit Wigner negativity, dep…
This work presents the bound-state spectra of Morse oscillator, which remains one of the oldest important model potentials for molecules. Accurate ro-vibrational energies are obtained by means of a generalized pseudospectral method that offers an optimal, non-uniform discretization of the radial grid. Both s-wave ($\ell=0$) and rotational ($\ell \neq 0$) states for low and high quantum numbers are calculated for four representative diatomic molecules, namely H$_2$, LiH, HCl and CO. First nine states belonging to a maximum of $n, \ell =2$ are computed with good accuracy, along with nine other h…
We derive the steady state solution of the Fokker-Planck equation that describes the dynamics of the nondegenerate optical parametric oscillator in the truncated Wigner representation of the density operator. The adiabatic limit of strong pump damping is assumed. This phase space image provides a clear view of the intracavity two-mode entangled state valid in all operating regimes of the OPO. A nongaussian distribution is obtained for the above threshold solution.
A mixed supersymmetric-algebraic approach to construction of the minimum uncertainty coherent states of anharmonic oscillators is presented. It permits generating not only the well-known coherent states of the harmonic and Morse oscillators but also the so far unknown coherent states of the Wei Hua, Kratzer-Fues and generalized Morse and Kratzer-Fues oscillators. The method can be applied also to generate superpotentials indispensable for deriving the Schrödinger equation in the supersymmetric form amenable to direct solution in the SUSYQM scheme.
Using linear invariant operators in a constructive way we find the most general thermal density operator and Wigner function for time-dependent generalized oscillators. The general Wigner function has five free parameters and describes the thermal Wigner function about a classical trajectory in phase space. The contour of the Wigner function depicts an elliptical orbit with a constant area moving about the classical trajectory, whose eccentricity determines the squeezing of the initial vacuum.
The creation of quantum coherences requires a system to be anharmonic. The simplest such continuous 1D quantum system is the Kerr oscillator. It has a number of interesting symmetries we derive. Its quantum dynamics is best studied in phase space, using Wigner's distribution W and the associated Wigner phase space current J. Expressions for the continuity equation governing its time evolution are derived in terms of J and it is shown that J for Kerr oscillators follows circles in phase space. Using J we also show that the evolution's classical shear in phase space is quantum suppressed by an e…
The system of two $Q$-deformed oscillators coupled so that the total Hamiltonian has the su$_Q$(2) symmetry is proved to be equivalent, to lowest order approximation, to a system of two identical Morse oscillators coupled by the cross-anharmonicity usually used empirically in describing vibrational spectra of triatomic molecules. The symmetry also imposes a connection between the self-anharmonicity of the Morse oscillators and the cross-anharmonicity strength, which can be removed by replacing the $Q$-oscillators by deformed anharmonic oscillators. The generalization to $n$ oscillators is stra…
We present the quantum theory of coherent Ising machines based on networks of degenerate optical parametric oscillators (DOPOs). In a simple model consisting of two coupled DOPOs, both positive-$P$ representation and truncated Wigner representation predict quantum correlation and inseparability between the two DOPOs in spite of the open-dissipative nature of the system. Here, we apply the truncated Wigner representation method to coherent Ising machines with thermal, vacuum, and squeezed reservoir fields. We find that the probability of finding the ground state of a one-dimensional Ising model…
Exact Cahill-Glauber Hierarchy for the Double-Morse Ground State
Khalifa University of Science and Technology（哈利法科学技术大学）；KU Research Center for Intelligent Computing, Networks and Sensing (ICONS)（KU智能计算、网络与传感研究中心）；Institute of Physics of the Czech Academy of Sciences（捷克科学院物理研究所）
We consider the time-independent Wigner functions of phase-space quantum mechanics (a.k.a. deformation quantization) for a Morse potential. First, we find them by solving the $\ast$-eigenvalue equations, using a method that can be applied to potentials that are polynomial in an exponential. A Mellin transform converts the $\ast$-eigenvalue equations to difference equations, and factorized solutions are found directly for all values of the parameters. The symbols of both diagonal and off-diagonal density operator elements in the energy basis are found this way. The Wigner transforms of the dens…
Bound-state solutions of the singular harmonic oscillator and singular Coulomb potentials in arbitrary dimensions are generated in a simple way from the solutions of the one-dimensional generalized Morse potential. The nonsingular harmonic oscillator and nonsingular Coulomb potentials in arbitrary dimensions with their additional accidental degeneracies are obtained as particular cases. Added bonuses from these mappings are the straightforward determination of the critical attractive singular potential, the proper boundary condition on the radial eigenfunction at the origin and the inexistence…
We show how the recently discovered solvable rational extensions of Harmonic Oscillator and Morse potentials can be constructed in a direct and systematic way, without the need of supersymmetry, shape invariance, Darboux-Crum and Darboux-Bäcklund transformations
Motivated by quantum experiments with nanomechanical systems, the evolution of a Kerr oscillator with focus on creation of states with a negative Wigner function is investigated. Using the phase space formalism, results are presented that demonstrate an asymptotic behavior in the large squeezing regime for the negativity of a squeezed vacuum state under unitary evolution. The analysis and model are extended to squeezed vacuum states of open systems, adding the decoherence effects of damping and dephasing. To increase experimental relevance, the regime of strong damping is considered. These eff…
Starting with the quasi-Bell states of the qubit-oscillator system, we obtain time evolution of the density matrix under the adiabatic approximation. The composite density matrix leads to, via partial tracing of the qubit degree of freedom, the reduced density matrix of the oscillator that is utilized to obtain the quasi-probability distributions such as Glauber-Sudarshan P function, Wigner W function and Husimi Q function. The negativity of the Wigner function acts as a measure of the nonclassicality of the state. The negativity becomes particularly relevant in understanding a comparison betw…
We propose a Wigner quasiprobability distribution function for Hamiltonian systems in spaces of constant curvature --in this paper on hyperboloids--, which returns the correct marginals and has the covariance of the Shapiro functions under SO(D,1) transformations. To the free systems obeying the Laplace-Beltrami equation on the hyperboloid, we add a conic-oscillator potential in the hyperbolic coordinate. As an example, we analyze the 1-dimensional case on a hyperbola branch, where this conic-oscillator is the Poschl-Teller potential. We present the analytical solutions and plot the computed r…
We analyze the tunneling of a particle through a repulsive potential resulting from an inverted harmonic oscillator in the quantum mechanical phase space described by the Wigner function. In particular, we solve the partial differential equations in phase space determining the Wigner function of an energy eigenstate of the inverted oscillator. The reflection or transmission coefficients $R$ or $T$ are then given by the total weight of all classical phase space trajectories corresponding to energies below, or above the top of the barrier given by the Wigner function.
Using the f-deformed oscillator formalism, we introduce two types of squeezed coherent states for a Morse potential system (Morse-like squeezed coherent states) through the following definitions: i) as approximate eigenstates of a linear combination of f-deformed ladder operators and ii) as deformed photon-subtracted coherent states. For the states thus constructed we analyze their statistical properties, their uncertainty relations, and their temporal evolution on phase space.
The Wigner function is a quantum analogue of the classical joined distribution of position and momentum. As such is should be a good tool to study quantum-classical correspondence. In this paper, the classical limit of the Wigner function is shown using the quantum harmonic oscillator as an example. The Wigner function is found exactly for all states. The semi-classical wavefunctions for highly excited states are used as the approach to the classical limit. Therefore, one can found the classical limit of the Wigner function for highly excited states and shown that it gives the classical microc…
In this paper we review the basic results concerning the Wigner transform and then we completely solve the quantum forced harmonic/inverted oscillator in such a framework; eventually, the tunnel effect for the forced inverted oscillator is discussed.
We define a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum. The distribution function is thus defined on discrete phase-space, i.e. on a finite discrete square grid. These discrete Wigner functions possess a number of properties similar to the Wigner function for a continuous quantum system such as the quantum harmonic oscillator. As an example, we consider the so-called su(2) oscillator model in dimension 2j+1, which is known to tend to the canonical oscillator when j tends to in…
The bound-state solutions and the su(1,1) description of the $d$-dimensional radial harmonic oscillator, the Morse and the $D$-dimensional radial Coulomb Schrödinger equations are reviewed in a unified way using the point canonical transformation method. It is established that the spectrum generating su(1,1) algebra for the first problem is converted into a potential algebra for the remaining two. This analysis is then extended to Schrödinger equations containing some position-dependent mass. The deformed su(1,1) construction recently achieved for a $d$-dimensional radial harmonic oscillator i…
We show a simple mechanism to measure the Wigner function of a harmonic oscillator. For this system we also show that autocorrelation and Wigner functions are equivalent.
For a quantum harmonic oscillator an explicit expression that describes the energy distribution as a coordinate function is obtained. The presence of the energy function poles is shown for the quantum system in domains where the Wigner function has negative values.
The phase-space path-integral approach to the damped harmonic oscillator is analyzed beyond the Markovian approximation. It is found that pairs of nonclassical trajectories contribute to the path-integral representation of the Wigner propagating function. Due to the linearity of the problem, the sum coordinate of a pair still satisfies the classical equation of motion. Furthermore, it is shown that the broadening of the Wigner propagating function of the damped oscillator arises due to the time-nonlocal interaction mediated by the heat bath.
We propose a scheme to prepare nanomechanical oscillators in nonclassical steady states, characterized by a pronounced negative Wigner function. In our optomechanical approach, the mechanical oscillator couples to multiple laser driven resonances of an optical cavity. By lowering the resonance frequency of the oscillator via an inhomogeneous electrostatic field, we significantly enhance its intrinsic geometric nonlinearity per phonon. This causes the motional sidebands to split into separate spectral lines for each phonon number and transitions between individual phonon Fock states can be sele…
Exact nonlinear stationary solutions of the one-dimensional Wigner and Wigner-Poisson equations in the terms of the Wigner functions that depend not only on the energy but also on position are presented. In this way, the Bernstein-Greene-Kruskal modes of the classical plasma are adapted for the quantum formalism in the phase space. The solutions are constructed for the case of a quartic oscillator potential, as well as for the self-consistent Wigner-Poisson case. Conditions for well-behaved physically meaningful equilibrium Wigner functions are discussed.
The analysis of wave-packet dynamics may be greatly simplified when viewed in phase-space. While harmonic oscillators are often used as a convenient platform to study wave-packets, arbitrary state preparation in these systems is more challenging. Here, we demonstrate a direct measurement of the Wigner distribution of complex photon states in an anharmonic oscillator - a superconducting phase circuit, biased in the small anharmonicity regime. We test our method on both non-classical states composed of two energy eigenstates and on the dynamics of a phase-locked wavepacket. This method requires …
Dunkl derivative enriches solutions by discussing parity due to its reflection operator. Very recently, one of the authors of this manuscript presented one of the most general forms of Dunkl derivative that depends on three Wigner parameters to have a better tuning. In this manuscript, we employ the latter generalized Dunkl derivative in a relativistic equation to examine two-dimensional harmonic and anharmonic oscillators solutions. We obtain the solutions by Nikiforov-Uvarov and QES methods, respectively. We show that degenerate states can occur according to the Wigner parameter values.
Time evolution of the expectation values of various dynamical operators of the harmonic oscillator with dissipation is analitically obtained within the framework of the Lindblad theory for open quantum systems. We deduce the density matrix of the damped harmonic oscillator from the solution of the Fokker-Planck equation for the coherent state representation, obtained from the master equation for the density operator. The Fokker-Planck equation for the Wigner distribution function, subject to either the Gaussian type or the $δ$-function type of initial conditions, is also solved by using the Wa…
Based on the technique of integration within an ordered product (IWOP) of operators we introduce the Fresnel operator for converting Caldirola-Kanai Hamiltonian into time-independent harmonic oscillator Hamiltonian. The Fresnel operator with the parameters A,B,C,D corresponds to classical optical Fresnel transformation, these parameters are the solution to a set of partial differential equations set up in the above mentioned converting process. In this way the exact wavefunction solution of the Schrödinger equation governed by the Caldirola-Kanai Hamiltonian is obtained, which represents a squ…
We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. {\bf 62}, 092101 (2021)] in the classical and quantum formalisms, by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space $(x, p)$ to a deformed one $(x_γ, Π_γ)$. Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical …
Sufficient conditions for complete controllability of $N$-level quantum systems subject to a single control pulse that addresses multiple allowed transitions concurrently are established. The results are applied in particular to Morse and harmonic-oscillator systems, as well as some systems with degenerate energy levels. Morse and harmonic oscillators serve as models for molecular bonds, and the standard control approach of using a sequence of frequency-selective pulses to address a single transition at a time is either not applicable or only of limited utility for such systems.
We solve the parametric generalized effective Schrödinger equation with a specific choice of posi-tion-dependent mass function and Morse oscillator potential by means of the Nikiforov-Uvarov (NU) method combined with the Pekeris approximation scheme. All bound-state energies are found explicitly and all corresponding radial wave functions are built analytically. We choose the Weyl or Li and Kuhn ordering for the ambiguity parameters in our numerical work to calculate the energy spectrum for a few and diatomic molecules with arbitrary vibration and rotation quantum numbers and different positio…
In this paper we reconsider the notion of an optimal effective Hamiltonian for the semiclassical propagation of the Wigner distribution in phase space. An explicit expression for the optimal effective Hamiltonian is obtained in the short time limit by minimizing the Hilbert-Schmidt distance between the semiclassical approximation and the real state of the system. The method is illustrated for the quartic oscillator.
We analyze the quantization of the Pais-Uhlenbeck fourth order oscillator within the framework of deformation quantization. Our approach exploit the Noether symmetries of the system by proposing integrals of motion as the variables to obtain a solution to the -genvalue equation, namely the Wigner function. We also obtain, by means of a quantum canonical transformation the wave function associated to the Schrödinger equation of the system. We show that unitary evolution of the system is guaranteed by means of the quantum canonical transformation and via the properties of the constructed Wigner …
We obtain analytic solutions to various models of dissipation of the quantum harmonic oscillator, employing a simple method in the Wigner function Fourier transform description of the system; and study as an exemplification, the driven open quantum harmonic oscillator. The environmental models we use are based on optical master equations for the zero and finite temperature bath and whose open dynamics are described by a Lindblad master equation, and also we use the Caldeira-Leggett model for the high temperature limit, in the the under damped an the over damped case. Under the Wigner Fourier t…
We use a power-series expansion to calculate the eigenvalues of anharmonic oscillators bounded by two infinite walls. We show that for large finite values of the separation of the walls, the calculated eigenvalues are of the same high accuracy as the values recently obtained for the unbounded case by the inner-product quantization method. We also apply our method to the Morse potential. The eigenvalues obtained in this case are in excellent agreement with the exact values for the unbounded Morse potential.
Recently, a new definition for a Wigner distribution function for a one-dimensional finite quantum system, in which the position and momentum operators have a finite (multiplicity-free) spectrum, was developed. This distribution function is defined on discrete phase-space (a finite square grid), and can thus be referred to as the Wigner matrix. In the current paper, we compute this Wigner matrix (or rather, the pre-Wigner matrix, which is related to the Wigner matrix by a simple matrix multiplication) for the case of the $\mathfrak{su}(2)$ finite oscillator. The first expression for the matrix…
The symlectic quantum tomography for the general linear quantization is introduced. Using the approach based upon the Wigner function techniques the evolution equation of quantum tomograms is derived for a parametric driven oscillator.
In the Wigner framework, one abandons the assumption that the usual canonical commutation relations are necessarily valid. Instead, the compatibility of Hamilton's equations and the Heisenberg equations are the starting point, and no further assumptions are made about how the position and momentum operators commute. Wigner quantization leads to new classes of solutions, and representations of Lie superalgebras are needed to describe them. For the n-dimensional Wigner harmonic oscillator, solutions are known in terms of the Lie superalgebras osp(1|2n) and gl(1|n). For n=3N, the question arises …
