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STRONG OSCILLATIONS OF CUMULANTS OF PHOTON DISTRIBUTION FUNCTION IN SLIGHTLY SQUEEZED STATES

Published Jan 1, 1994• 0 citations
OpenAlex

Abstract

The cumulants and factorial moments of photon distribution for squeezed and correlated light are calculated in terms of Chebyshev, Legendre and Laguerre polynomials. The phenomenon of strong oscillations of the ratio of the cumulant to factorial moment is found. Running title: Oscillations of cumulants in squeezed states.

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By use of the squeezed-number-state basis, we study several features of the squeezed thermal states. Higher-order squeezing properties are found to be entirely determined by the second-order ones. The photon-number distribution is expressed in closed form in terms of a Legendre polynomial. We point out that at a critical value of the squeeze parameter, pairwise oscillations of the photon-number distribution set in. Finally, a compact analytical expression of the exact lth-order correlation function is derived and examined.
The primary aim of the present paper is to attract the attention of particle physicists to new developments in studying squeezed and correlated states of the electromagnetic field as well as of those working on the latest topic to new findings about multiplicity distributions in quantum chromodynamics. The new types of nonclassical states used in quantum optics as squeezed states, correlated states, even and odd coherent states (Schrödinger cat states) for one--mode and multimode interaction are reviewed. Their distribution functions are analyzed according to the method used first for multiplicity distributions in high energy particle interactions. The phenomenon of oscillations of particle distribution functions of the squeezed fields is described and confronted to the phenomenon of oscillations of cumulant moments of some distributions for squeezed and correlated field states. Possible extension of the method to fields different from the electromagnetic field (gluons, pions, etc.) is speculated.
We develop closed-form expressions for the moments and cumulants of Gaussian states when measured in the photon-number basis. We express the photon-number moments of a Gaussian state in terms of the loop Hafnian, a function that when applied to a ( 0 , 1 ) -matrix representing the adjacency of a graph, counts the number of its perfect matchings. Similarly, we express the photon-number cumulants in terms of the Montrealer, a newly introduced matrix function that when applied to a ( 0 , 1 ) -matrix counts the number of Hamiltonian cycles of that graph. Based on these graph-theoretic connections, we show that the calculation of photon-number moments and cumulants are #P-hard. Moreover, we provide an exponential time algorithm to calculate Montrealers (and thus cumulants), matching well-known results for Hafnians. We then demonstrate that when a uniformly lossy interferometer is fed in every input with identical single-mode Gaussian states with zero displacement, all the odd-order cumulants but the first one are zero. Finally, we employ the expressions we derive to study the distribution of cumulants up to the fourth order for different input states in a Gaussian boson sampling setup where K identical states are fed into an ℓ -mode interferometer. We analyze the dependence of the cumulants as a function of the type of input state, squeezed, lossy squeezed, squashed, or thermal, and as a function of the number of non-vacuum inputs. We find that thermal states perform much worse than other classical states, such as squashed states, at mimicking the photon-number cumulants of lossy or lossless squeezed states.