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International Journal of Theoretical Physics

Analytical and Phase Space Description of “Near” States

M. R. Bazrafkan, E. Nahvifard

Published Sep 16, 2024• 0 citations
Publisher
DOI
10.1007/s10773-024-05766-w

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The analytic properties of a class of generalized Husimi functions are discussed, with particular reference to the problem of state reconstruction. The class consists of the subset of Wódkiewicz's operational probability distributions for which the filter reference state is a squeezed vacuum state. The fact that the function is analytic means that perfectly precise knowledge of its values over any small region of phase space provides enough information to reconstruct the density matrix. If, however, one only has imprecise knowledge of its values, then the amplification of statistical errors which occurs when one attempts to carry out the continuation seriously limits the amount of information which can be extracted. To take account of this fact a distinction is made between explicate or experimentally accessible information, and information which is only present in implicate or experimentally inaccessible form. It is shown that an explicate description of various aspects of the system can be found localized on various two-real-dimensional surfaces in complexified phase space. In particular, the continuation of the function to the purely imaginary part of complexified phase space provides an explicate description of the Wigner function.
In recent years, strong-field processes such as high-order harmonic generation (HHG) and above-threshold ionization driven by nonclassical states of light have become an increasingly popular field of study. The theoretical modeling of these processes often applies an approximate phase-space expansion of the nonclassical driving field in terms of coherent states, which has been shown to accurately predict the harmonic spectrum. However, its accuracy for the computation of quantum optical observables like the degree of squeezing and photon statistics has not been thoroughly considered. In this work, we introduce this approximative phase-space description and discuss its accuracy, and we find that it mischaracterizes the quantum optical properties of the driving laser by making it an incoherent mixture of classical states. We further show that this error in the driving field description maps onto the light emitted from HHG, as neither sub-Poissonian photon statistics nor quadrature squeezing below vacuum fluctuations can be captured by the approximative phase-space description. Lastly, to benchmark the approximative phase-space description, we consider the quantum HHG from a one-band model, which yields an exact analytical solution. Using the approximative phase-space representation with this specific model, we find a small quantitative error in the quadrature variance of the emitted field that scales with pulse duration and emitter density. Our results show that using this approximative phase-space description can mischaracterize quantum optical observables. Attributing physical meaning to such results should therefore be accompanied by a quantitative analysis of the error.