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Journal of Modern Optics

Evolution of Coherent States in a Dispersionless Fibre with Saturable Nonlinearity and the Generation of Macroscopic Quantum-superposition States

M. L. Lyra, A.S. Gouveia-Neto

Published Jul 1, 1994• 5 citations
Publisher

Abstract

We study the evolution of a initially coherent state of an electromagnetic field propagating in a Kerr medium with saturable nonlinearity. By using the quantum phase distribution formalism, we analyse the dependence of the output signal phase configuration upon input field amplitude. We observe that the saturation of the nonlinear contribution of the refractive index of the propagation medium introduces interference effects that compromise the observation of macroscopically well distinguishable components of the output state. For input amplitudes much larger than a characteristic saturation amplitude the final state differs from the input state only by an overall phase shift. Possible relevance of the present results in the experimental search of Schrödinger cat-like states using semiconductor-doped glass optical fibres is discussed.

DOI
10.1080/09500349414551311

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We study the evolution of a initially coherent state of an electromagnetic field propagating in a Kerr medium with saturable nonlinearity. By using the quantum phase distribution formalism, we analyse the dependence of the output signal phase configuration upon input field amplitude. We observe that the saturation of the nonlinear contribution of the refractive index of the propagation medium introduces interference effects that compromise the observation of macroscopically well distinguishable components of the output state. For input amplitudes much larger than a characteristic saturation amplitude the final state differs from the input state only by an overall phase shift. Possible relevance of the present results in the experimental search of Schrödinger cat-like states using semiconductor-doped glass optical fibres is discussed.
Intense light beams propagating in a lossless, dispersionless, single-mode optical fiber are subject to the Kerr effect, i.e., to the intensity-dependent refractive index of the fiber’s fused-silica core. Classically, Kerr-effect-induced self-phase modulation (SPM) can be used for spectral broadening of a picosecond pulse for grating-pair pulse compression down to femtosecond duration. Quantum mechanically, Kerr-effect-induced four-wave mixing (FWM) has been used to produce squeezed-state light. We present a quantum propagation theory for a lossless, dispersionless fiber with the Kerr nonlinearity. The theory includes classical SPM and quantum FWM within their regions of validity. It introduces a material time constant for the Kerr interaction, limiting the quantum phase shifts caused by the broadband zero-point fluctuations that accompany any input field, to develop a coarse-grained time multitemporal mode field analysis. Explicit expressions are obtained for the first and the second output-field moments when the fiber’s input field is in an arbitrary Gaussian state. These results are used to obtain homodyne-detection noise spectra, which are employed, in turn, to seek experimentally accessible manifestations of the Kerr time constant.
A two-step interaction scheme involving (2) and (3) nonlinear media is suggested for the generation of Schrödinger cat-like states of a single-mode optical field. In the first step, a weak coherent signal undergoes a self-Kerr phase modulation in a (3) crystal, leading to a Kerr kitten , namely a microscopic superposition of two coherent states with opposite phases. In the second step, such a Kerr kitten enters a (2) crystal and, in turn, plays the role of a quantum seed for stimulated phase-sensitive amplification. The output state in the above-threshold regime consists in a quantum superposition of mesoscopically distinguishable squeezed states, i.e. an optical cat-like state. The whole setup does not rely on conditional measurements, and is robust against decoherence, as only weak signals interact with the Kerr medium.
The generation of Schr\"odinger-cat-like states for quantum fundamental solitons in optical fibers is investigated. It is shown that the generation is possible at proper times in the dynamical evolution of an appropriately chosen initial state. Such a Schr\"odinger-cat-like state has a minimum number of components, three, and the superposition coefficients of components are all real. Mean values of the field and the intensity operators are calculated and the associated quantum interference behavior characteristic of quantum coherent superposition in the Schr\"odinger-cat-like states is revealed. The effect of dissipation on the Schr\"odinger-cat-like states is analyzed using a beam splitter. The severity with which losses tend to destroy interference between components of the input Schr\"odinger-cat-like state is noted.
We consider two copropagating fields in a nonlinear Kerr medium, each with a particular phase and intensity. The Kerr medium possesses an intensity-dependent refractive index and the phase shift of each field thus depends on the intensities of the fields. Classically it is possible to induce an arbitrary phase shift of one field (the signal field) by either increasing the intensity of the other field (the control field) or by increasing the interaction legnth. We show that if the intensity of the control field is low, the phase shift on the signal is limited by the discrete nature of the photon-number distribution in the control field and cannot be increased simply by increasing the interaction length. In general the maximum phase shift of the signal field is \ensuremath{\varphi} if the control field possesses \ensuremath{\varphi} photons. This limit arises as a consequence of quantum recurrence effects.
We consider quantum-mechanically partially polarized light propagating through a Kerr-like medium. Using the usual form of the induced polarization P=A(E\ensuremath{\cdot}E∗{\mathrm{E}}^{\mathrm{*}}E∗)E+B(E\ensuremath{\cdot}E)E∗{\mathrm{E})\mathrm{E}}^{\mathrm{*}}E)E∗, the theory is formulated in terms of an effective Hamiltonian which is quartic in terms of the operators for two orthogonally polarized modes. Exact solutions in closed form for the Heisenberg equations of motion are obtained. These solutions are used to evaluate the physical behavior of various observables as the field propagates through a nonlinear medium. We also present explicit results for the time evolution of the input coherent and Fock states of the field. We show the generation of states that are macroscopic superposition of coherent states. We also find that if the input field is completely polarized, then due to quantum effects the output field becomes partially polarized. This is in contrast to the classical prediction and can have an important bearing on questions like topological phases of light propagating through a nonlinear medium. Numerical results for the energy in each mode, the correlation between two modes, and the higher-order correlations are presented. The input photon statistics is found to make a considerable difference in the dynamics.