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Journal of Optics B Quantum and Semiclassical Optics

Collisions between a small number of photons in a nonlinear Kerr medium

Y. Ben-Aryeh

Published Jan 1, 1999• 2 citations
Publisher

Abstract

The quantum field theory for the one-dimensional Schrödinger equation with attractive delta function potentials is used for treating the dynamics of collisions between a small number of photons in a nonlinear Kerr medium. The present theoretical analysis takes into account both bound states and dispersive continuum states and the general theory is applied, especially for the systems of diphotons and three-photons.

DOI
10.1088/1464-4266/1/2/006

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The quantum field theory for the one-dimensional Schrödinger equation with attractive delta function potentials is used for treating the dynamics of collisions between a small number of photons in a nonlinear Kerr medium. The present theoretical analysis takes into account both bound states and dispersive continuum states and the general theory is applied, especially for the systems of diphotons and three-photons.
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Propagation of light beams in a Fabry-Perot resonator filled with a self-focusing Kerr material is described by the nonlinear Schrodinger equation. If the degrees of freedom are restricted to one transverse dimension through the use of a line focus at the input face, the system possesses classical soliton solutions analogous to those for travelling-wave self-trapped beams. When the classical field theory is quantized, one obtains an effective nonrelativistic many-body field theory in which pairs of photons interact through attractive delta function potentials. The existence of bound states between photons is the microscopic phenomenon that prevents beams from spreading by diffraction. The fundamental bound state consists of two photons, which we call the "diphoton." We consider production of these quasi-particles in a single longitudinal mode Fabry-Perot resonator filled with an alkali vapor. The injected light is strongly attenuated to excite few photons in the cavity and is detuned slightly above the atomic resonance, giving rise to the self-focusing nonlinearity. Detection of the diphoton can be achieved by rejecting the unbound component by using a spatial filter and detecting the remaining signal in coincidence counting. The diphoton signature is a characteristic Lorentzian-squared two-point correlation profile.1
The emergence of multi-photon bound states in quantum nonlinear media is an intriguing phenomenon that has attracted significant theoretical and experimental interest. Such bound states can emerge in quite different settings, such as arrays of quantum emitters coupled to photonic waveguides and Rydberg nonlinear media. The theoretical tools to characterize the bound states are equally diverse, ranging from Bethe ansatz to effective field theories. Here, we propose and utilize a spin-model formulation of quantum atom-light interactions as a general, unified framework to study bound states, and apply this formalism to several distinct systems. We show how to obtain the bound-state dispersion relation within the two-excitation subspace, and also numerically investigate how the properties of higher multi-photon bound states manifest themselves in the propagation of large photon number pulses and spatio-temporal correlations at the output. Interestingly, for all the specific systems studied (chiral and bi-directional waveguides, and Rydberg media), we find that the large-photon number limit always coincides with the well-known semi-classical soliton phenomenon of self-induced transparency or immediate generalizations thereof.
We study the dynamics and quantum characteristics of a single two-level atom interacting with a single mode cavity field undergoing a multi-photon processes in the presence of a nonlinear Kerr-like medium. The wavefunctions of the multi-photon system are obtained when the atom starts in the excited and in the ground state. The atomic inversion, the squeezing of the radiation field and the quasiprobability distribution Q-function of the field are discussed. Numerical results for these characteristics are presented when the atom starts in the excited state and the field mode in a coherent state. The influence of the presence and absence of the number operator and the Kerr medium for the one- and two-photon processes on the evolution of these characteristics are analyzed.
The time-dependent Schrodinger equation for a particle in one dimension in the field of two colliding delta potentials can be solved exactly. Assuming an initial state which is bound about the left delta potential for large negative times, the solution has been used to calculate the probabilities of finding a state bound about one of the potentials long after the collision. These probabilities are oscillating functions of the reciprocal velocity. The ionisation probability goes to zero for high collision velocities and reaches 1 / 2 for small velocities.
Exact solutions to a one-dimensional, time-dependent Schrodinger equation with two colliding delta-function potentials of different strengths are presented. The equation can serve as a model for resonant and quasi-resonant ion-atom collisions. Electrons initially bound to one of the potentials may be transferred to the other potential or to a continuum state and probabilities for these processes are calculated. The model reproduces the damped oscillatory behaviour of the charge-transfer probability seen in quasi-resonant collisions when the collision energy is varied.