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Journal of the Optical Society of America B

Multiphoton squeezed states

Jacob Katriel, A. I. Solomon, Giacomo Mauro D’Ariano, Mario Rasetti

Published Oct 1, 1987• 58 citations
Publisher

Abstract

The multiphoton squeezed states defined in this paper are generalizations of the conventional coherent (Glauber) and squeezed (Yuen) states previously discussed by many authors. We define multiphoton generalizations of the latter by a unified class of states that includes the Holstein–Primakoff realizations of SU(2) and SU(1, 1) as well as the standard harmonic oscillator coherent states (Weyl–Heisenberg group) and squeezed states in a general framework that allows also non-Hermitian realizations. We determine the squeezing properties of these states in a unified formalism and study numerically their dependence on the parameter classifying the states.

DOI
10.1364/josab.4.001728

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The multiphoton squeezed states defined in this paper are generalizations of the conventional coherent (Glauber) and squeezed (Yuen) states previously discussed by many authors. We define multiphoton generalizations of the latter by a unified class of states that includes the Holstein–Primakoff realizations of SU(2) and SU(1, 1) as well as the standard harmonic oscillator coherent states (Weyl–Heisenberg group) and squeezed states in a general framework that allows also non-Hermitian realizations. We determine the squeezing properties of these states in a unified formalism and study numerically their dependence on the parameter classifying the states.
We define a new set of squeezed states using group-theoretical methods. The definition is based on the Holstein-Primakoff realization of both SU(2) and SU(1,1). Generalizations of these realizations are presented, connected with the Brandt-Greenberg generalized Bose operators. The new states exhibit interesting squeezing properties, depending in a characteristic way on the dimension of the irreducible unitary representation adopted. We also discuss the asymptotic behavior and present a set of relevant numerical results. Unexpected and interesting scaling behavior appears.
Generalized q-boson operators associated with the simultaneous creation of several q-bosons are introduced. These operators give rise to a realization of the quantum Weyl-Heisenberg algebra and are applied to the construction of corresponding Holstein-Primakoff realizations of the quantum algebras SU q (2) and SU q (1,1). Coherent states of these algebras are defined in the various ways suggested by the equivalent definitions of the harmonic oscillator coherent states, and some of their properties are studied. Particular attention is devoted to the squeezing properties of the quadrature of the electromagnetic field in these states.
After deriving a general formula for the quantum probability distribution function higher moments, we apply it to the multiphoton squeezed states [the usual Gaussian and the new non-Gaussian Weyl-Heisenberg, SU(2), and SU(1,1)]. The resulting moments are discussed as functions of the photon-number fluctuations. General criteria are considered to determine optimal squeezing properties with respect to photon-number noise. There result interesting generalized uncertainty relations in the form of scaling laws.
A set of deformed k-boson operators are defined and it is shown that the nonlinear transformations from the usual deformed oscillator operators to the deformed k-boson operators form an Abelian group. It is shown that the multiboson operators satisfy quantum Weyl–Heisenberg group. The generalized Holstein–Primakoff realizations in terms of these multibosons operators are used to define the quantum group-theoretic coherent states of SUq(1,1) and SUq(2). The uncertainties ΔX(k′,q) and ΔP(k′,q) with respect to those group-theoretic and generalized coherent states are computed explicitly and the results show that these coherent states are squeezed.
We define k'-photon uncertainties \ensuremath{\Delta}X(k′){X}_{(k\mathcal{'})}X(k′)​,\ensuremath{\Delta}P(k′){P}_{(k\mathcal{'})}P(k′)​ in k-photon states, and show that all such calculations can be correlated using a fractional photon index r\ensuremath{\equiv}k'/k. We exemplify by choosing as k-photon states the coherent states of the k-photon generalized harmonic oscillator, and also the group-theoretic coherent states corresponding to Holstein-Primakoff realizations of SU(2) and SU(1,1). We compute the squeezing obtained in these cases, and show that they have a common limit.