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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Squeezing versus photon-number fluctuations

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

Published Oct 15, 1987• 36 citations
Publisher

Abstract

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.

DOI
10.1103/physrevd.36.2399

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We examine the generalized squeezed states defined as eigenstates of a linear combination of the lowering and raising operators a2{\mathrm{a}}^{2}a2 and (${\mathrm{a}}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$)2{)}^{2})2, respectively. This approach is entirely equivalent to the minimum-uncertainty method applied to the amplitude-squared operators. We solve the eigenvalue equation in Glauber's coherent-state representation and find two independent solutions. Their Fock-state expansions, one containing only even and the other only odd number states, reveal a strong nonclassical character. We show that the calculation of the mean photon number is sufficient to obtain the expectation values of interest. Consequently, photon statistics is investigated in both cases by using the generating function of the photon-number distribution. We find the conditions under which the second-order squeezed states display photon antibunching and quadrature squeezing. Also discussed is the preservation of their amplitude-squared squeezing by linear amplification at gains exceeding 2. Analytically, our results are simple formulas in terms of Kummer and Gauss hypergeometric functions that allow straightforward numerical calculations.
A squeezed state of light is usually taken to mean a state in which the fluctuations of one of the quadratures are below the level associated with a coherent or vacuum state. These squeezed fluctuations are phase sensitive in the sense that they vary periodically as the field evolves, but the relationship between squeezing and phase fluctuations is not trivial. While squeezed states can exhibit an arbitrarily low level of phase fluctuations, the phase states, which are eigenstates of the Hermitian optical phase operator are not themselves quadrature squeezed. A phasesqueezed state is one with a lower level of phase fluctuations than a coherent state with the same mean photon number. Phase squeezing is fundamentally different from quadrature squeezing in much the same way that sub-Poissonian photon statistics is different. For this reason optimally squeezed states and states with minimum phase noise will be distinct. We derive the form of the states which minimum phase noise for a given intensity. These have an Airy function behavior for the photon number probability distribution. These phase squeezed states have a far better phase resolution than coherent states of similar intensity, but do not seem to offer a significant reduction of phase fluctuations compared with conventional squeezed states.
We investigate a free monochromatic electromagnetic field which is the superposition of a squeezed thermal radiation and a coherent one. The main tool in our analysis is the characteristic function that has a Gaussian form. We establish an analytic formula for an arbitrary correlation function, as well as its strong-squeezing limit. Besides the usual quasiprobability densities, the coherent-state, number-state, coordinate, and momentum representations of the density operator are derived. We point out the nonclassical oscillations of the photon-number distribution and find its generating function. Collaterally, displaced thermal states and squeezed thermal states are revisited as nontrivial particular cases. We examine finally the squeezing properties of the field using the distribution functions of the quadratures.