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Physical Review A

Quantum statistics of displaced Kerr states

Vlasta Peřinová, V. Vrana, A. Lukš, J. Křepelka

Published Mar 1, 1995• 10 citations
Publisher

Abstract

Fluctuations in photon number and quantum phase and in quadratures of the displaced Kerr states have been studied for different values of Kerr evolution parameter. The dynamics of phase distributions, taking into account that of the preferred phase, have resulted in well centered peaks in the phase interval. The role of the preferred phase in the distributions of quadratures has been considered, leading to approximate identification of conjugate quadratures with the number-phase-operator pair. Nonclassical properties have been demonstrated in the case of noise-minimum displaced Kerr states. The necessary displacement parameter has been represented as a center of a crescent-shaped phase region and expressed in terms of the lowest moments of the ${\mathrm{\ensuremath{\Phi}}}_{\mathit{s}\mathit{c}\mathit{r}\mathit{A}}$ quasidistribution.

DOI
10.1103/physreva.51.2499

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Quantum and Semiclassical Optics Journal of the European Optical Society Part B•Apr 1, 1998•2 citations
Fluctuations in photon number and quantum phase and in quadratures of the displaced Kerr states have been studied for different values of Kerr evolution parameter. The dynamics of phase distributions, taking into account that of the preferred phase, have resulted in well centered peaks in the phase interval. The role of the preferred phase in the distributions of quadratures has been considered, leading to approximate identification of conjugate quadratures with the number-phase-operator pair. Nonclassical properties have been demonstrated in the case of noise-minimum displaced Kerr states. The necessary displacement parameter has been represented as a center of a crescent-shaped phase region and expressed in terms of the lowest moments of the \ensuremathΦscrA{\mathrm{\ensuremath{\Phi}}}_{\mathit{s}\mathit{c}\mathit{r}\mathit{A}}\ensuremathΦscrA​ quasidistribution.
We study the statistical and phase properties of the output states of a Mach-Zehnder interferometer with a nonlinear Kerr medium in one of its arms. The combination of nonlinearity and displacement produced by the interferometer generates output states (displaced Kerr states) that, unlike those derived from a Kerr medium alone (Kerr states), do not necessarily retain the photon statistics of the initially coherent field. As the parameter characterizing the nonlinear medium is varied, the statistics of the output field vary from sub-Poissonian to super-Poissonian. The photon-number distributions of these states are rationalized by examining their quasiprobability distributions, which vary from banana-shaped for sub-Poissonian statistics to complicated interference structures for highly super-Poissonian statistics. The number-phase properties of the Kerr and displaced Kerr states are studied for the case of small photon numbers (relevant to cavity quantum electrodynamics) using a methodology based on the Pegg-Barnett formalism. In particular, we determine the range of parameters for which the states are minimum-uncertainty states and whether they are squeezed with respect to photon number and phase, drawing a distinction between a sub-Poissonian nature and number squeezing. Finally, we find that sub-Poissonian statistics can coexist with quadrature squeezing for a range of parameters.
Displaced and squeezed Fock states have recently been thoroughly investigated. Application of these states as the input of a Kerr nonlinear medium reveals many nonclassical effects. We solve this problem analytically and demonstrate results by numerical examples. We introduce a phase distribution for the antinormal quasidistribution ${\mathrm{\ensuremath{\varphi}}}_{\mathit{s}\mathit{c}\mathit{r}\mathit{A}}$ and show that the ``phase'' of the new states can be sharply localized at some values. This happens when revivals appear in the field. If the states are shifted by a reference wave, photon-number distributions clearly detect the structure of ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{scrA}}$. Damping of the new states is briefly discussed.
We construct a distinct class of nonlinear displaced Kerr states by application of the displacement operator upon a state which is prepared by sending the well-known photon-added coherent state through a normal Kerr medium. A sketch for the experimental setup for preparing the state is suggested. We evaluate some statistical properties such as the photon number distribution, Mandel’s Q parameter, Husimi-Q and Wigner functions, and quadrature squeezing for the nonlinear displaced Kerr state and then analyze the nonclassicality in terms of these standard parameters. We reduce the infinite-level problem to a truncated discrete two-level system by using a low Kerr parameter approximation and then convert the generated nonclassicality into bipartite entanglement between the two modes of an output state of a linear optical device.
Using Pegg-Barnett formalism the author has studied the phase properties of displaced number states. It has been found that the phase probability distribution exhibits the same oscillations as the photon number distribution. Using the Hermitian phase operator the fluctuations of the phase of the displaced number states have been studied. The dependence of the fluctuations on the value of the reference phase have been analysed in detail. It has been found that phase of displaced number states exhibit squeezing in the high-intensity limit of the driving field. These states are approximately equivalent to phase states with precisely defined phase. It has been shown that displaced number states are not minimum uncertainty states with respect to the phase and the number operators.
The evolution of a quadrature-squeezed state through a Kerr medium is studied from the point of view of amplitude squeezing. We propose a Mach-Zehnder interferometer configuration, which incorporates a squeeze- state generator and a nonlinear Kerr medium, to generate and subsequently displace these states. The displaced states show amplitude squeezing for a proper choice of the initial quadrature squeezed state, and we study two such cases here in detail. In the limit where the evolution through a Kerr medium results in a macroscopic superposition, an initial phase-squeezed coherent state shows significant amplitude squeezing, and we have shown that the photon-number uncertainty 〈(\ensuremath{\Delta}n^)2{)}^{2})2〉 can be minimized to a value smaller than 〈n^〉2/3{\mathrm{〉}}^{2/3}〉2/3. On the other hand, an initial quadrature-squeezed state, whose direction of squeezing is along the direction of excitation, upon evolving through a Kerr medium for a suitable small duration of time shows substantial amplitude squeezing for an appropriate displacement, and we show that the photon-number uncertainty in this case can be minimized to 〈n^〉1/5{\mathrm{〉}}^{1/5}〉1/5, which is far below the previously achieved limits. \textcopyright{} 1996 The American Physical Society.