Quantum mechanics of a general driven time-dependent oscillator

The relationship between quantum degeneracies and the emergence of classical periodic orbits in the Fock-Darwin (FD) system is systematically explored by using the ladder operators in the Cartesian coordinates. The quantum-classical connection is analytically developed with the time-dependent coherent state that is theoretically verified to correspond to the Gaussian wave packet state with unitary transformation. The time-dependent coherent state is further used to derive the stationary coherent state that can be expressed as a superposition of degenerate eigenstates. More importantly, the stationary coherent state can alternatively be expressed as an integral of the Gaussian wave packet over a periodic orbit. With the integral representation, the quantum vortex structures of the stationary coherent states can be precisely characterized up to extremely high order. The vortex arrays are generally observed when the phase structures are near the crossing points and cusps of the classical trajectories. Finally, the quantum Fourier transform is employed to verify that the circularly symmetric eigenstate physically corresponds to an ensemble of classical hypotrochoid orbits. This work can provide a pedagogical insight into the formation of Laguerre-Gaussian eigenmodes and hypotrochoid geometric modes in degenerate spherical cavities.

Exact wave functions and coherent states for a forced damped harmonic oscillator

We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. 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 (xγ, Πγ). Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.

Exact coherent states in the Calogero–Sutherland models (of time-dependent parameters) which describe identical harmonic oscillators interacting through inverse-square potentials are constructed, in terms of the classical solutions of a harmonic oscillator. For quasi-periodic coherent states of the time-periodic systems, geometric phases are evaluated. For the AN-1 Calogero–Sutherland model, the phase is calculated for a general coherent state. The phases for other models are also considered.

Herein, a novel theoretical approach on the microwave‐induced resistance oscillations based on the coherent states of the quantum harmonic oscillator is reported. Within the theoretical model of microwave‐induced electron orbits, the coherent state expression for the quantum harmonic oscillator under a time‐dependent force (radiation) is calculated. Thus, it is found that the principle of minimum uncertainty of coherent states, involving time and energy, is the main responsible of photo‐oscillations and zero‐resistance states. Thus, essential experimental results of this remarkable effect are shed light (and explained) on. The origin of photo‐oscillations, their position with respect to the magnetic field, and their periodicity can be distinguished.

Coherent states for the harmonic oscillator with time-dependent mass and frequency

Displaced and squeezed number states

We show, by means of similarity transformations, that the time-dependent fermionic systems are associated to the time-dependent pseudofermionic systems. A general construction of time dependent fermionic coherent states (FCSs) describing the two-level dissipative system driven by a periodic electromagnetic field is developed, and a strict parallelism between FCS and the time dependent pseudofermionic coherent states (PFCSs) is established and examined. We discuss properties of the constructed FCS and PFCS.

In this paper, we introduce the generalized coherent states and Photon-added coherent states in the sence of Barut-Girardello for the nonlinear harmonic oscillator with position dependent effective mass.The study of the resolution of the unity as the fundamental property of coherent states, is done in Meijer Gfunction.Also, the non-classical features of these two states are calculated and analyized by using the probability distribution function and Mandel parameter.

We study the metric aspect of the Moyal plane from Connes’ noncommutative geometry point of view. First, we compute Connes’ spectral distance associated with the natural isometric action of $${\mathbb{R}^2}$$ on the algebra of the Moyal plane $${\mathcal{A}}$$ . We show that the distance between any state of $${\mathcal{A}}$$ and any of its translated states is precisely the amplitude of the translation. As a consequence, we obtain the spectral distance between coherent states of the quantum harmonic oscillator as the Euclidean distance on the plane. We investigate the classical limit, showing that the set of coherent states equipped with Connes’ spectral distance tends towards the Euclidean plane as the parameter of deformation goes to zero. The extension of these results to the action of the symplectic group is also discussed, with particular emphasis on the orbits of coherent states under rotations. Second, we compute the spectral distance in the double Moyal plane, intended as the product of (the minimal unitization of) $${\mathcal{A}}$$ by $${\mathbb{C}^2}$$ . We show that on the set of states obtained by translation of an arbitrary state of $${\mathcal{A}}$$ , this distance is given by the Pythagoras theorem. On the way, we prove some Pythagoras inequalities for the product of arbitrary unital and non-degenerate spectral triples. Applied to the Doplicher- Fredenhagen-Roberts model of quantum spacetime [DFR], these two theorems show that Connes’ spectral distance and the DFR quantum length coincide on the set of states of optimal localization.

The coherent states of a Hamiltonian linear in SU(1,1) operators are constructed by defining them, in analogy with the harmonic-oscillator coherent states, as the minimum-uncertainty states with equal variance in two observables. The proposed approach is thus based on a physical characteristic of the harmonic-oscillator coherent states which is in contrast with the existing ones which rely on the generalization of the mathematical methods used for constructing the harmonic-oscillator coherent states. The set of states obtained by following the proposed method contains not only the known SU(1,1) coherent states but also a different class of states. \textcopyright{} 1996 The American Physical Society.

Search for generalized, quantizable dynamical models

We apply the techniques of canonical transforms to equations of the type \[\left[ {A( t )\mathbb{P}^2 + B( t )\{ \mathbb{P}\mathbb{Q} + \mathbb{Q}\mathbb{P} \} + C( t )\mathbb{Q}^2 + D( t )\mathbb{Q} + E( t )\mathbb{P} + F( t )\mathbb{1}} \right]\psi ( q,t ) = - i\partial _t \psi ( q,t ),\]where $\mathbb{Q}$ and $\mathbb{P}$ are the quantum position and momentum operators. The time-dependent parameters of the ${\text{W}} \wedge {\text{SL}}( {{\text{2,}}R} )$ evolution operator are found through linear differential equations. In terms of these we give explicitly the Green’s function, all separating coordinates and similarity solutions of the equation. We analyze the behavior of Gaussian and coherent-state initial conditions in closed form and present a new interpretation of all the Lewis-Riesenfeld constants of motion.

We formulate radial coherent states for the isotropic harmonic oscillator from the unitary irreducible representations associated with the radial spectrum-generating group Sp(2) which is locally isomorphic to SU(1,1) and SO(2,1). We further consider the time evolution of a coherent-state wave packet and show that, in contrast to the ordinary oscillator coherent states, this wave packet does change its shape (periodically) as it evolves. This change of shape may be interpreted in terms of a generalized squeezing.

We consider various harmonic-oscillator states in a finite-dimensional Hilbert space. In greater detail we analyze finite-dimensional coherent states defined either by truncation of a number-state expansion of the standard coherent states or by the action of the generalized finite-dimensional displacement operator on vacuum. Within these two approaches other finite-dimensional states are analyzed, including displaced number states, and even and odd coherent states. We propose some new definitions of the states in a finitedimensional Hilbert space and construct explicitly all these states in Fock representation. The number-phase Wigner function is computed for several states. Analytical results and numerically computed graphs are presented.