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Journal of Physics A Mathematical and General

A non-Gaussian single-mode squeezed state of the simple harmonic oscillator

C V Sukumar

Published Nov 21, 1988• 8 citations
Publisher

Abstract

A unitary operator U R , which generates a single-mode squeezed state of the simple harmonic oscillator from the vacuum, is discussed. The states generated by U R are of the form psi 0 ( alpha ) approximately Sigma n alpha n mod n). psi 0 ( alpha ) are not minimum-uncertainty states but have a small value for the uncertainty product even for large values of the mean number of quanta n; for example, var(q) var(p) approximately 0.74 for n=10 6 .

DOI
10.1088/0305-4470/21/22/005

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Revista Mexicana de Física E•Jan 1, 2013•4 citations
A unitary operator U R , which generates a single-mode squeezed state of the simple harmonic oscillator from the vacuum, is discussed. The states generated by U R are of the form psi 0 ( alpha ) approximately Sigma n alpha n mod n). psi 0 ( alpha ) are not minimum-uncertainty states but have a small value for the uncertainty product even for large values of the mean number of quanta n; for example, var(q) var(p) approximately 0.74 for n=10 6 .
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.
The phase operators P ± of the electromagnetic field, whose classical analogues are exp (∓ iθ), θ being the classical phase, and the number operator N associated with a harmonic oscillator mode are used to construct a class of operators T m = N(P −)m and T + m = (P +)m N, m = 1, 2,…. It is shown that T m, T + m and N satisfy a simple algebra. T m and T + m are used to construct model Hamiltonians H m describing the coupling of a two-level atom to a field that leads to decay and exact revivals in the atomic dynamics. In the second part of the paper, a family of unitary operators U m is constructed in terms of T m and T + m. For m = 1 and m = 2 the states generated by U m from the vacuum are non-Gaussian squeezed states. The variance properties of the squeezed states so constructed are analysed and compared with those of the Gaussian squeezed states. It is shown that the non-Gaussian squeezed states constructed in this paper have small values for the variance product that rise only logarithmically with for large values of , the mean number of quanta.
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.
"Harmonic oscillator squeezed states are states of minimum uncertainty, but unlike coherent states, in which the uncertainty in position and momentum are equal, squeezed states have the uncertainty reduced, either in position or in momentum, while still minimizing the uncertainty principle. It seems that this property of squeezed states would allow to obtain the position eigenstates as a limiting case, by doing null the uncertainty in position and infinite in momentum. However, there are two equivalent ways to define squeezed states, that lead to different expressions for the limiting states. In this work, we analyze both definitions and show the advantages and disadvantages of using them in order to find position eigenstates. With this in mind, but leaving aside the definitions of squeezed states, we find an operator that applied to the vacuum gives position eigenstates. We also analyze some properties of the squeezed states, based on the new expressions obtained for the eigenstates of the position."