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Journal of Physics A Mathematical and General

Squeezing and minimum uncertainty states in the supersymmetric harmonic oscillator

Miguel Orszag, S. Salamó

Published Nov 21, 1988• 31 citations
Publisher

Abstract

A displacement operator for a subset of the supercoherent states is found. It is also shown that a diagonal version of the squeezing operator reduces the fluctuations of one of the quadratures, but the reduction is not enough to go below the standard quantum noise level. New states are found, which are minimum uncertainty for the new operators X and P, and lead to coherence and squeezing, in the sense of equal fluctuations in the two quadrature and reduction in one of them, respectively. Moreover, these states are associated with the model of Jaynes and Cummings (1963) in the strong-coupling limit.

DOI
10.1088/0305-4470/21/22/004

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Wave functions have been determined which realize constant squeezing (steady-squeezing states) for a quadrature component of the oscillator with a quadratic self-coupling at an á priori given initial complex amplitude of the oscillator. The set of these wave functions forms a kernel of a special generating operator and is divided into two mutually orthogonal subsets: the subset of superpositions of the states with an even number of quanta of fluctuations around the classical value of the oscillator energy and the subset of analogous states but with odd number of such quanta. The value of the steady squeezing is not a monotonic function of the self-coupling constant and possesses an unexpected minimum. Steady-squeezing states do not minimize the uncertainty product and are not the Gaussian states at the same time.

International Journal of Modern Physics B•Jul 10, 1993•0 citations
A displacement operator for a subset of the supercoherent states is found. It is also shown that a diagonal version of the squeezing operator reduces the fluctuations of one of the quadratures, but the reduction is not enough to go below the standard quantum noise level. New states are found, which are minimum uncertainty for the new operators X and P, and lead to coherence and squeezing, in the sense of equal fluctuations in the two quadrature and reduction in one of them, respectively. Moreover, these states are associated with the model of Jaynes and Cummings (1963) in the strong-coupling limit.
In this paper, we have constructed even and odd superpositions of supercoherent states, similar to the standard even and odd coherent states of the harmonic oscillator. Then, their nonclassical properties, that is, squeezing and entanglement have been studied. We have observed that even supercoherent states show squeezing behavior for some values of parameters involved, while odd supercoherent states do not show squeezing at all. Also sub-Poissonian statistics have been observed for some ranges of the parameters in both states. We have also shown that these states may be considered as logical qubits which reduce to the Bell states at a limit, with concurrence equal to 1.
The authors present the supercoherent states. They are introduced as the eigenstates of the supersymmetric annihilation operator of the quantum mechanical supersymmetric harmonic oscillator. They have a compact expression in terms of the standard (bosonic) coherent states. For each value of the complex parameter z there are two linearly independent orthogonal supercoherent states. One of them has pure fermionic character and is the more classical, while the other is fully supersymmetric (i.e. the mean value of the Klein operator when the system is set in this state vanishes). There are no purely bosonic supercoherent states. The supercoherent fermionic states saturate both the Heisenberg uncertainty relation and the new entropic uncertainty recently introduced by Deutsch (1983), while the supersymmetric ones almost (but not exactly) make it. They give a classical (or almost classical) mean value for the energy of the system and do not spread along their time evolution.
The general parasupersymmetric annihilation operator of arbitrary order does not reduce to the Kornbluth–Zypman general supersymmetric annihilation operator for the first order. In this paper, we introduce an annihilation operator for a parasupersymmetric harmonic oscillator that in the first order matches with the Kornblouth–Zypman results. Then, using the latter operator, we obtain the parasupercoherent states and calculate their entanglement, uncertainties, and statistics. We observe that these states are entangled for any arbitrary order of parasupersymmetry and their entanglement goes to zero for the large values of the coherency parameter. In addition, we find that the maximum of the entanglement of parasupercoherent states is a decreasing function of the parasupersymmetry order. Moreover, these states are minimum uncertainty states for large and also small values of the coherency parameter. Furthermore, these states show squeezing in one of the quadrature operators for a wide range of the coherency parameter, while no squeezing in the other quadrature operator is observed at all. In addition, using the Mandel parameter, we find that the statistics of these new states are subPoissonian for small values of the coherency 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."
Wave functions have been determined which realize constant squeezing (steady-squeezing states) for a quadrature component of the oscillator with a quadratic self-coupling at an á priori given initial complex amplitude of the oscillator. The set of these wave functions forms a kernel of a special generating operator and is divided into two mutually orthogonal subsets: the subset of superpositions of the states with an even number of quanta of fluctuations around the classical value of the oscillator energy and the subset of analogous states but with odd number of such quanta. The value of the steady squeezing is not a monotonic function of the self-coupling constant and possesses an unexpected minimum. Steady-squeezing states do not minimize the uncertainty product and are not the Gaussian states at the same time.