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arXiv:2611.00280

Phase-space dynamics and multicomponent wave-packet structures in the Dunkl anharmonic oscillator

D. Ojeda-Guillén, R. D. Mota

math-phquant-phSubmitted Oct 30, 2026• Announced Nov 3, 2026
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Abstract

We study the phase-space dynamics and multicomponent wave-packet structures of the Dunkl anharmonic oscillator (Kerr medium). Starting from the time-evolved state, we derive analytical expressions for the Dunkl-Husimi $Q$-function and a normalized angular Husimi profile. The Dunkl parameter modifies the coherent-state excitation weights and the parity-dependent phase accumulation, producing angular anisotropy and asymmetric phase-space separation. For the half-integer deformation $μ=0.5$, the dynamics generate asymmetric multicomponent Husimi patterns with unequal peak heights and an early reconstruction at $t=π/λ$. For the deformation $μ=1.0$, the Husimi distribution exhibits broadening while retaining its multimodal structure and unitary evolution. At the fundamental nonlinear revival time, the wave packet is reconstructed for the selected parameter ratio. The standard Kerr dynamics are recovered in the limit $μ\rightarrow 0$.

Comments
17 pages, 4 figures

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We study the phase-space dynamics and multicomponent wave-packet structures of the Dunkl anharmonic oscillator (Kerr medium). Starting from the time-evolved state, we derive analytical expressions for the Dunkl-Husimi QQQ-function and a normalized angular Husimi profile. The Dunkl parameter modifies the coherent-state excitation weights and the parity-dependent phase accumulation, producing angular anisotropy and asymmetric phase-space separation. For the half-integer deformation μ=0.5μ=0.5μ=0.5, the dynamics generate asymmetric multicomponent Husimi patterns with unequal peak heights and an early reconstruction at t=π/λt=π/λt=π/λ. For the deformation μ=1.0μ=1.0μ=1.0, the Husimi distribution exhibits broadening while retaining its multimodal structure and unitary evolution. At the fundamental nonlinear revival time, the wave packet is reconstructed for the selected parameter ratio. The standard Kerr dynamics are recovered in the limit μ→0μ\rightarrow 0μ→0.
We study the Dunkl anharmonic oscillator (Kerr medium) Hamiltonian from an algebraic approach of the SU(1,1)SU(1,1)SU(1,1) group. In order to obtain the exact energy spectrum of this problem, we write its Hamiltonian in terms of the Dunkl creation and annihilation operators, which close the su(1,1)su(1,1)su(1,1) Lie algebra. This allows us to exactly solve this Hamiltonian and obtain its parity-dependent energy spectrum. Then, we investigate the quantum dynamics of the system, particularly the collapse and revival phenomena, by using an initial state given by a superposition of even and odd Dunkl coherent states. We compute the field quadrature and the survival probability, showing that the Dunkl parameter μμμ modulates the fractional revivals and produces perfect state reconstructions at half-periods for specific deformation values. We analyze the quadrature variance to show that the Dunkl deformation generates interference-induced squeezed states around t≈πt \approx πt≈π. The standard Kerr medium dynamics are exactly recovered in the limit μ→0μ\rightarrow 0μ→0.
Wave packet revivals and fractional revivals are hallmark quantum interference phenomena that arise in systems with nonlinear energy spectra, and their signatures in expectation values of observables have been studied extensively in earlier work. In this article, we build on these studies and extend the analysis in two important directions. First, we investigate fractional revival dynamics in angular momentum observables, deriving explicit expressions for the time evolution of their moments and demonstrating that higher-order angular momentum moments provide clear and selective signatures of fractional revivals. Second, we examine classical analogs of quantum revival phenomena and elucidate structural similarities between quantum fractional revivals and recurrence behavior in representative classical systems. Using the Kerr-type nonlinear Hamiltonian as a paradigmatic model, we analyze the autocorrelation function, moment dynamics, and phase-space structures, supported by visualizations such as quantum carpets. Our results broaden the range of experimentally accessible diagnostics of fractional revivals and provide a unified perspective on revival phenomena across quantum and classical dynamical systems.
The dynamics of an initially excital Gaussian wave packet (harmonic oscillator coherent state, HOCS) in a general one-dimensional anharmonic system is studied analytically with the aid of a second quantization formalism for times shorter than a classical oscillation period. The initial state population probability PQ(t), its classical counterpart PC(t), and the overlap probability PQC(t) between a time evolved wave packet and a classically moving Gaussian with stationary shape are calculated in order to analyze the spreading of the wave packet. It was found that, in general, PQ(t) and PC(t) agree very well if the potential energy function has no strongly changing low order derivatives. This agreement is held even if the wave packet spreads drastically within the time scale considered as can be demonstrated by analyzing PQC(t). The analytical expressions for PQ(t), PC(t), and PQC(t) are applied to a dissociative third-order oscillator and a strongly anharmonic fourth-order oscillator. While for a harmonic system, the initial decay time of PQ(t) and PC(t) (which are identical in this case) are only dependent on the energy expectation value 〈H〉 with respect to the initial HOCS, differences up to an order of magnitude occur for HOCS with the same energy in the model systems.
In this paper, we first analyze a parametric oscillator with both mass and frequency time-dependent. We show that the evolution operator can be obtained from the evolution operator of another parametric oscillator with a constant mass and time-dependent frequency followed by a time transformation [Formula: see text]. Then we proceed by investigating the quantum dynamics of a parametric oscillator with unit mass and time-dependent frequency in a Kerr medium under the influence of a time-dependent force along the motion of the oscillator. The quantum dynamics of the time-dependent oscillator is analyzed from both analytical and numerical points of view in two main regimes: (i) small Kerr parameter [Formula: see text], and (ii) small confinement parameter k. In the following, to investigate the characteristics and statistical properties of the generated states, we calculate the autocorrelation function, the Mandel Q parameter, and the Husimi Q-function.
We study theoretically optical tomograms of the time-evolved states generated by the evolution of different kinds of initial wave packets in a Kerr medium. An exact analytical expression for the optical tomogram of the quantum state at any instant during the evolution of a generic initial wave packet is derived in terms of Hermite polynomials. The time evolution of the optical tomogram is discussed for three kinds of initial states: a coherent state, an mmm-photon-added coherent state, and even and odd coherent states. We show the manifestation of revival and fractional revivals in the optical tomograms of the time-evolved states. We find that the optical tomogram of the time-evolved state at the instants of fractional revivals shows structures with sinusoidal strands. The number of sinusoidal strands in the optical tomogram of the time-evolved state at lll-subpacket fractional revivals is lll times the number of sinusoidal strands present in the optical tomogram of the initial state. We also investigate the effect of decoherence on the optical tomograms of the states at the instants of fractional revivals for the initial states considered above. We consider amplitude decay and phase damping models of decoherence and show the direct manifestations of decoherence in the optical tomogram.
Unique set of coherent states for the anharmonic oscillator is obtained by requiring i. under the quantum mechanical time evolution a coherent state evolves into another, governed by trajectory in the classical phase space (of a related hamiltonian); ii. the resolution of identity involves exactly the classical phase space measure.The rules are invariant under unitary transformations of the quantum theory and canonical transformations of the classical theory. The states are almost, but not quite, minimal uncertainty wave packets. The construction can be generalized to quantum versions of integrable classical theories.