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arXiv:quant-ph/9501010 · Modern Physics Letters B

DYNAMICS OF GENERALIZED COHERENT STATES

S. De Martino, Silvio De Siena, Fabrizio Illuminati

Published Jun 10, 1995• 5 citations
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Abstract

We show that generalized coherent states follow Schrödinger dynamics in time-dependent potentials. The normalized wave-packets follow a classical evolution without spreading; in turn, the Schrödinger potential depends on the state through the classical trajectory. This feedback mechanism with continuous dynamical re-adjustment allows the packets to remain coherent indefinitely.

DOI
10.1142/s0217984995000772

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We show that generalized coherent states follow Schrödinger dynamics in time-dependent potentials. The normalized wave-packets follow a classical evolution without spreading; in turn, the Schrödinger potential depends on the state through the classical trajectory. This feedback mechanism with continuous dynamical re-adjustment allows the packets to remain coherent indefinitely.
We introduce a general formalism, based on the stochastic formulation of quantum mechanics, to obtain localized quasi-classical wave packets as dynamically controlled systems, for arbitrary anharmonic potentials. The control is in general linear, and it amounts to introduce additional quadratic and linear time-dependent terms to the given potential. In this way one can construct for general systems either coherent packets moving with constant dispersion, or dynamically squeezed packets whose spreading remains bounded for all times. In the standard operatorial framework our scheme corresponds to a suitable generalization of the displacement and scaling operators that generate the coherent and squeezed states of the harmonic oscillator.
Coherent states play an important role in quantum mechanics because of their unique properties under time evolution. Here we explore this concept for one-dimensional repulsive nonlinear Schr\"odinger equations, which describe weakly interacting Bose-Einstein condensates or light propagation in a nonlinear medium. It is shown that the dynamics of phase-space translations of the ground state of a harmonic potential is quite simple: The center follows a classical trajectory whereas its shape does not vary in time. The parabolic potential is the only one that satisfies this property. We study the time evolution of these nonlinear coherent states under perturbations of their shape or of the confining potential. A rich variety of effects emerges. In particular, in the presence of anharmonicities, we observe that the packet splits into two distinct components. A fraction of the wavepacket is transferred toward incoherent high-energy modes, while the amplitude of oscillation of the remaining coherent component is damped toward the bottom of the well.