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Physical Review A

Squeezed states and uncertainty relations in rotating frames and Penning traps

Shahen Hacyan

Published Jun 1, 1996• 15 citations
Publisher

Abstract

The quantum harmonic oscillator in a nonuniformly rotating frame is studied. It is shown that squeezed states and particles are generated by inertial effects. The analytic forms of the squeezing and Bogoliubov coefficients are given explicitly in terms of the solutions of the classical equations of motion. The squeezed states have the minumum dispersion allowed by the Robertson uncertainty relations for an arbitrary number of observables (the explicit form of these relations is obtained). The problem of an ``abrupt twist'' is worked out in detail as an example. The formalism can also be applied to the problem of a charged particle in a Penning trap, where it is shown that squeezed states are produced, with a squeezing frequency of the order of the magnetron frequency. In both examples, analytic expressions for the variances of the position and momentum are obtained. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.4481

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The quantum harmonic oscillator in a nonuniformly rotating frame is studied. It is shown that squeezed states and particles are generated by inertial effects. The analytic forms of the squeezing and Bogoliubov coefficients are given explicitly in terms of the solutions of the classical equations of motion. The squeezed states have the minumum dispersion allowed by the Robertson uncertainty relations for an arbitrary number of observables (the explicit form of these relations is obtained). The problem of an ``abrupt twist'' is worked out in detail as an example. The formalism can also be applied to the problem of a charged particle in a Penning trap, where it is shown that squeezed states are produced, with a squeezing frequency of the order of the magnetron frequency. In both examples, analytic expressions for the variances of the position and momentum are obtained. \textcopyright{} 1996 The American Physical Society.
In this work, the dynamics of the deformed one-dimensional harmonic oscillator with minimal length uncertainty is examined and the analytical solutions for time evolution of position and momentum operators are presented in which the rough approximation that neglects the higher order terms in BakerHausdor lemma is avoided. Based on these analytical solutions the uncertainties for position and momentum operators are calculated in a coherent state, and an unexpected squeezing effect in both coordinate and momentum directions is found in comparison with ordinary harmonic oscillator. Obviously such a squeezing effect is induced by the minimal length uncertainty (gravitational effects). Our results are applied to the electrons trapped in strong magnetic fields to examine the degree of the existing squeezing effect in a real system, which shows the squeezing degree induced by minimal length uncertainty is very small.
A single closed-form expression is obtained for the probability amplitude ψ(x,t) for a harmonic oscillator whose initial state ψ(x,0) has a minimum value for the uncertainty product of position x and momentum p, Δx Δp=ℏ/2. This form describes both coherent and squeezed states. Oscillations in time for uncertainties in position and momentum, characteristic of squeezed states, are shown to follow directly from the derived form of ψ(x,t) and from Heisenberg equations of motion for operators x̂ and p̂. Expressions for average photon number and its uncertainty are obtained from calculations of average oscillator energy and fluctuations about this energy.
We describe a six-parameter family of the minimum-uncertainty squeezed states for the harmonic oscillator in nonrelativistic quantum mechanics. They are derived by the action of corresponding maximal kinematical invariance group on the standard ground state solution. We show that the product of the variances attains the required minimum value 1/4 only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied. The overlap coefficients between the squeezed, or generalized harmonic, and the Fock states are explicitly evaluated in terms of hypergeometric functions. The corresponding photons statistics are discussed and some applications to quantum optics, cavity quantum electrodynamics, and superfocusing in channeling scattering are mentioned. Explicit solutions of the Heisenberg equations for radiation field operators with squeezing are found.
Complete description of the classical and quantum dynamics of a particle in an anisotropic, rotating, harmonic trap is given. The problem is studied in three dimensions and no restrictions on the geometry are imposed. In the generic case, for an arbitrary orientation of the rotation axis, there are two regions of instability with different characteristics. The analysis of the quantum-mechanical problem is made simple due to a direct connection between the classical mode vectors and the quantum-mechanical wave functions. This connection is obtained via the matrix Riccati equation that governs the time evolution of squeezed states of the harmonic oscillator. It is also shown that the inclusion of gravity leads to a resonant behavior -- the particles are expelled from the trap.
We describe a multi-parameter family of the minimum-uncertainty squeezed states for the harmonic oscillator in nonrelativistic quantum mechanics. They are derived by the action of the corresponding maximal kinematical invariance group on the standard ground state solution. We show that the product of the variances attains the required minimum value 1/4 only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied. The overlap coefficients between the squeezed, or generalized harmonic, and the Fock states are explicitly evaluated in terms of hypergeometric functions and the corresponding photon statistics are discussed. Some applications to quantum optics, cavity quantum electrodynamics and superfocusing in channelling scattering are mentioned. Explicit solutions of the Heisenberg equations for radiation field operators with squeezing are found.
In this paper, results from the previous paper (I) are applied to calculations of squeezed states for such well-known systems as the harmonic oscillator, free particle, linear potential, oscillator with a uniform driving force, and repulsive oscillator. For each example, expressions for the expectation values of position and momentum are derived in terms of the initial position and momentum, as well as in the (α,z)- and in the (z,α)-representations described in I. The dependence of the squeezed-state uncertainty products on the time and on the squeezing parameters is determined for each system.