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Physical Review A

Higher-nsqueezed states and factorization of the squeezing operator

Jeffrey J. Hamilton

Published Jul 1, 1990• 8 citations

Abstract

It is shown that n>0 simple-harmonic-oscillator squeezed states have the same time-evolution behavior as the fundamental (n=0) squeezed state. That is, the probability density \ensuremath{\Vert}\ensuremath{\psi}(x,t)${\mathrm{\ensuremath{\Vert}}}^{2}$ retains its original shape as time progresses, except that the scales of the ordinate and abscissa oscillate in a reciprocal manner with frequency 2\ensuremath{\omega}. This allows the time-evolved squeezing operator S${\mathrm{^}}_{\mathit{H}}$(\ensuremath{\lambda},t) to be written as a product of simple factors. Moreover, if a thermal distribution with density matrix \ensuremath{\rho}^ is squeezed, then the mixed squeezed state with minimum free energy has a density matrix given by a unitary transformation \ensuremath{\rho}${\mathrm{^}}_{\mathit{s}}$=S^(\ensuremath{\lambda})\ensuremath{\rho}^S^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$(\ensuremath{\lambda}), where \ensuremath{\lambda} is a suitable squeezing parameter.

DOI: 10.1103/physreva.42.1

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It is shown that n>0 simple-harmonic-oscillator squeezed states have the same time-evolution behavior as the fundamental (n=0) squeezed state. That is, the probability density \ensuremath{\Vert}\ensuremath{\psi}(x,t)

{\mathrm{\ensuremath{\Vert}}}^{2}

retains its original shape as time progresses, except that the scales of the ordinate and abscissa oscillate in a reciprocal manner with frequency 2\ensuremath{\omega}. This allows the time-evolved squeezing operator S

{\mathrm{^}}_{\mathit{H}}

(\ensuremath{\lambda},t) to be written as a product of simple factors. Moreover, if a thermal distribution with density matrix \ensuremath{\rho}^ is squeezed, then the mixed squeezed state with minimum free energy has a density matrix given by a unitary transformation \ensuremath{\rho}

{\mathrm{^}}_{\mathit{s}}

=S^(\ensuremath{\lambda})\ensuremath{\rho}^S^

^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}

(\ensuremath{\lambda}), where \ensuremath{\lambda} is a suitable squeezing parameter.

We derive the density matrix ρ(x1, x2; x1', x2', β) for two coupled harmonic oscillators by taking a unitary transformation approach. The corresponding unitary operator U is characteristic of including frequency-jump squeezing transformation. By virtue of the technique of integration within ordered product and the coordinate representation of U we manifestly show that the ground state of the coupled oscillators is a squeezed state. Based on the derived ρ(x1, x2; x1', x2', β) we calculate the average coupling energy.

Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.

Physical consequences of the theorem of equivalence among harmonic oscillators of different frequencies, including a free particle as a special case, are explored. A new pictorial way of understanding time-dependent coherent states and squeezed states are presented and contrasted with the standard view. Three-dimensional spherical analogues of squeezed coherent states are constructed. It is also shown as a corollary of the theorem that a free wave packet which coincides initially with any one of the eigenfunctions of a harmonic oscillator maintains its shape, that is, the probability density of position does not change apart from a time-dependent scaling of spatial coordinates. Most of the results are derived without use of any sophisticated techniques but with the knowledge of free wave packets alone.