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Physical Review A

Statistical fractional-photon squeezed states

Giacomo Mauro D’Ariano, Nicoletta Sterpi

Published Feb 1, 1989• 15 citations
Publisher

Abstract

We construct a large class of squeezed states realized in terms of density matrices. The moments of the canonical variable q^ and the number operator n^ distributions are analytically evaluated. The new states can be simultaneously squeezed in q^ and n^ to any desired amount and exhibit almost Gaussian shape in the former and strongly sub-Poisson distribution in the latter. The probability distributions can be synthetically characterized in terms of the fractional-photon index introduced by Katriel, Rassetti, and Solomon [Phys. Rev. D 35, 1248 (1987)].

DOI
10.1103/physreva.39.1860

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We construct a large class of squeezed states realized in terms of density matrices. The moments of the canonical variable q^ and the number operator n^ distributions are analytically evaluated. The new states can be simultaneously squeezed in q^ and n^ to any desired amount and exhibit almost Gaussian shape in the former and strongly sub-Poisson distribution in the latter. The probability distributions can be synthetically characterized in terms of the fractional-photon index introduced by Katriel, Rassetti, and Solomon [Phys. Rev. D 35, 1248 (1987)].
A prototype mechanism for producing amplitude-squeezed states, based on the so-called ``photon fractioning'' procedure, is investigated. The coherent state is connected with a quasinumber eigenstate through an intermediate density matrix involving multiphoton processes. The enhancement in the phase fluctuations and the simultaneous reduction of the noise in the photon number result from two competing limits; the ``bright limit,'' which is semiclassical and requires highly excited states, and the photon fractioning limit, which is pure quantum. Asymptotical evaluations of the number probability distribution and of the Q function show that the mixed states are very close to the exact number-phase minimum-uncertainty states.
We construct a new type of photon-added squeezed coherent state generated by repeatedly operating the bosonic creation operator on a new type of squeezed coherent state [Fan H Y and Xiao M 1996 Phys. Lett. A 220 81]. We find that its normalization factor is related to single-variable Hermite polynomials. Furthermore, we investigate its statistical properties, such as Mandel’s Q parameter, photon-number distribution, and Wigner function. The nonclassicality is displayed in terms of the intense oscillation of photon-number distribution and the negativity of the Wigner function.
We construct a new type of photon-added squeezed coherent state generated by repeatedly operating the bosonic creation operator on a new type of squeezed coherent state [Fan H Y and Xiao M 1996 Phys. Lett . A 220 81 ]. We find that its normalization factor is related to single-variable Hermite polynomials. Furthermore, we investigate its statistical properties, such as Mandel's Q parameter, photon-number distribution, and Wigner function. The nonclassicality is displayed in terms of the intense oscillation of photon-number distribution and the negativity of the Wigner function.
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.
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.