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arXiv:1403.5048 · Progress of Theoretical and Experimental Physics

Two-photon paired solitons supported by medium polarization

M. Yoshimura, N. Sasao

Published Jul 1, 2014• 7 citations
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Abstract

We derive for the first time fundamental equations that describe soliton spatial profiles consisting of two-photon mode fields and macroscopic polarization of a medium.Numerical solutions of this basic equation are presented to suggest both single soliton and multiple soliton chains in infinitely long targets, using the example of para-H 2 v = 0 ↔ 1 (E1 forbidden) vibrational parameters.Although the effects of dissipative relaxation are included in the general form for the two-level system, the existence of a static soliton condensate is established.For finite-size targets, we can precisely formulate the profile equation in the framework of a nonlinear eigenvalue problem.Its first iteration provides approximate semi-analytic results under a potential well in the linearized equation; these results have qualitatively similar profiles to the case of an infinitely long target, an important difference being the exponentially decreasing profile near target ends.A large number of weakly interacting solitons correspond to localized portions between adjacent nodes of highly excited bound-state wave functions in a 1D potential well of large size.These soliton condensates are expected to be important in enhancing the signal to the background ratio in the proposed neutrino mass spectroscopy using atoms.. . . . . .

DOI
10.1093/ptep/ptu094

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We derive for the first time fundamental equations that describe soliton spatial profiles consisting of two-photon mode fields and macroscopic polarization of a medium.Numerical solutions of this basic equation are presented to suggest both single soliton and multiple soliton chains in infinitely long targets, using the example of para-H 2 v = 0 ↔ 1 (E1 forbidden) vibrational parameters.Although the effects of dissipative relaxation are included in the general form for the two-level system, the existence of a static soliton condensate is established.For finite-size targets, we can precisely formulate the profile equation in the framework of a nonlinear eigenvalue problem.Its first iteration provides approximate semi-analytic results under a potential well in the linearized equation; these results have qualitatively similar profiles to the case of an infinitely long target, an important difference being the exponentially decreasing profile near target ends.A large number of weakly interacting solitons correspond to localized portions between adjacent nodes of highly excited bound-state wave functions in a 1D potential well of large size.These soliton condensates are expected to be important in enhancing the signal to the background ratio in the proposed neutrino mass spectroscopy using atoms.. . . . . .
The transition regimes of solitons in four-photon resonant processes in the case of two-photon absorption of the fundamental radiation are numerically investigated. The standard system of equations for the amplitudes of probability of finding the system in state with certain energy is used to derive the expression for the induced polarization in the nonlinear medium. As for the equations for the amplitudes of the optical pulses, the general case is considered in which both the amplitudes and phases are space-time dependent. We focus on the finite difference methods and the case of simultaneously propagating solitons at all frequencies of the interacting waves (simultons). The obtained results indicate that upon certain threshold conditions all interacting pulses become the solitons of Lorentzian shape. The numerical analysis has also shown that the soliton amplitudes significantly depend on the ratio between the nonlinear polarizability at the fundamental frequency ω0 and that of combination of ω0 and the trigger-field frequency ω1(2ω0 + ω1). In the second part of the paper, we apply the method of phase planes to show that at typical values of parameters, the solitons are stable.
We develop for dipole-forbidden transition a dynamical theory of two-photon paired superradiance (PSR). This is a cooperative process characterized by two photons emitted back to back with equal energies. By irradiating the trigger laser from two target ends, with its frequency tuned at the half energy between two levels, a macroscopically coherent state of medium and fields dynamically emerges as time evolves, and a large signal of amplified output occurs with a time delay. The basic semiclassical equations in 1+11+11+1 space-time dimensions are derived for the field-plus-medium system to describe the space-time evolution of the entire system and are numerically solved to demonstrate the existence of both explosive and weak PSR phenomena in the presence of relaxation terms. The explosive PSR event terminates accompanying a sudden release of most of the energy stored in the target. Our numerical simulations are performed using the vibrational transition ${X}^{1}{\ensuremath{\Sigma}}_{g}^{+}v=1\ensuremath{\rightarrow}0$ of a para-H2{}_{2}2​ molecule and taking many different excited atom number densities and different initial coherences between the metastable and the ground states. In an example with a number density close to O(1021O({10}^{21}O(1021 cm${}^{\ensuremath{-}3}$) and a high initial coherence, the explosive event terminates several nanoseconds after the trigger irradiation, when the phase relaxation time larger than OOO(10 ns) is taken. After PSR events the system is expected to follow a steady-state solution which is obtained by analytic means and is made of many objects of field condensates endowed with a topological stability.
The problem of light propagation of frequency corresponding to half of the energy difference between a metastable excited state and the ground state of atoms is examined, and solved for coherent medium by analytic means. We demonstrate that the non-linear system of Maxwell-Bloch equation for the effective model of the Λ -type three levels is integrable in the mathematical sense. Analytic solutions thus obtained describe pulse splitting accompanied by compression, indicating a kind of non-linear instability of propagating pulses. The instability is eventually terminated by coherent two photon emission (called paired superradiance or PSR in short). These results are displayed by numerical outputs for visual understanding, as well. It is further shown that the integrable system allows a new class of soliton solutions. Solitons, implying the phenomenon of self-induced transparancy at non-resonant frequencies, are stable against PSR. One of our goals of the present work is construction of a calculable theoretical framework for PSR rates associated with a trigger pulse propagation, which is achieved by combining analytic results with perturbative methods. PSR photon spectrum and its rate ∝(target number density) 2 , along with their time structure, are clarified this way. These results may open a new path for interesting technological applications such as quantum entanglement and for solving the remaining problems of the still mysterious neutrino. Some basic strategy for realistic experiments of PSR detection and soliton production is also outlined.
In the framework of the nonlinear Λ model, propagation of solitons was analysed in atomic vapours and Bose-Einstein condensates. The complicated nonlinear interplay between fast and slow-light solitons in a Λ-type medium was shown to facilitate control of its optical transparency and formation of optical gates. An exact analytical description was given for the deceleration, stopping and revival of slow-light solitons in the experimentally relevant non-adiabatic regime. A stopping slow-light soliton imprints a localized immobile polarization pattern in the medium, which, as explicitly demonstrated here, can be used as a bit of readable optical memory. The whole process can be controlled with the background field and an auxiliary laser field. The latter regulates the signal velocity, while the slow-light soliton can be stopped by switching off the former. The location and shape of the imprinted memory bit were also determined. With few assumptions characteristic of slow light, the Λ model was reduced to a simpler nonlinear model that also describes two-dimensional dilatonic gravity. Exact solutions could now be derived also in the presence of relaxation. Spontaneous decay of the upper atomic level was found to be strongly suppressed, and the spatial form of the decelerating slow-light soliton was preserved, even if the optical relaxation time was much shorter than the typical time scale of the soliton. The effective relaxation coefficient of the slow-light soliton was significantly smaller than that of an arbitrary optical pulse. Such features are obviously of great importance when this kind of system is applied, in practice, to information processing. A number of experimentally observable properties of the solutions reported were found to be in good agreement with recent experimental results, and a few suggestions are also made for future experiments.
The stationary solutions of the Gross-Pitaevskii equation can be divided in two classes: those which reduce, in the limit of vanishing nonlinearity, to the eigenfunctions of the associated Schrödinger equation and those which do not have linear counterpart. Analytical and numerical results support an existence condition for the solutions of the first class in terms of the ratio between their proper frequency and the corresponding linear eigenvalue. For one-dimensional confined systems, we show that solutions without linear counterpart do exist in presence of a multi-well external potential. These solutions, which in the limit of strong nonlinearity have the form of chains of dark or bright solitons located near the extrema of the potential, represent macroscopically excited states of a Bose-Einstein condensate and are in principle experimentally observable.