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Journal of Modern Optics

Time evolution of a quantum soliton in a Kerr medium

Natalia Korolkova, R. Loudon, Gerlinde Gardavsky, M. W. Hamilton, Gerd Leuchs

Published Jul 1, 2001• 12 citations
Publisher

Abstract

The Q-function of the quantum soliton in a fibre is derived in a new form suitable for the assessment of the possibilities for experimental observations of specific quantum-soliton effects. The characteristic effects of soliton evolution are associated with a cubic term in the nonlinear phase angle and appear at distances well beyond the attenuation length (long range). At the accessible distances, the nonlinear dynamics of a fundamental soliton follows essentially the single-mode dynamics (short and middle range). However, for the higher-order solitons, the form of the evolution parameter suggests strong deviations of soliton dynamics from the single-mode with a square of soliton number. The use of enhanced-nonlinearity fibres and higher-order solitons might make the experimental studies of the specific quantum-soliton features more viable.

DOI
10.1080/09500340108232466

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The Q-function of the quantum soliton in a fibre is derived in a new form suitable for the assessment of the possibilities for experimental observations of specific quantum-soliton effects. The characteristic effects of soliton evolution are associated with a cubic term in the nonlinear phase angle and appear at distances well beyond the attenuation length (long range). At the accessible distances, the nonlinear dynamics of a fundamental soliton follows essentially the single-mode dynamics (short and middle range). However, for the higher-order solitons, the form of the evolution parameter suggests strong deviations of soliton dynamics from the single-mode with a square of soliton number. The use of enhanced-nonlinearity fibres and higher-order solitons might make the experimental studies of the specific quantum-soliton features more viable.
We analyse a nonlinear QW model which can be experimentally implemented using the components of the electric field on an optical nonlinear Kerr medium, which translates into a rotation in the coin operator, with an angle which depends (in a nonlinear fashion) on the state of the walker. This simple dependence makes it easy to consider the space-time continuum limit of the evolution equation, which takes the form of a nonlinear Dirac equation. The analysis of this continuum limit allows us, under some approximations, to gain some insight into the nature of soliton structures, which is illustrated by our numerical calculations. These solitons are stable structures whose trajectories can be modulated by choosing the appropriate initial conditions. We have also studied the stability of solitons when they are subject to an additional phase that simulates an external electric field, and also explored if they are formed in higher dimensional spaces.
Summary form only given. Optical solitons in fibres are attractive macroscopical quantum objects, which are particle-like pulses propagating over large distances without distortion due to the balance between Kerr nonlinearity and group velocity dispersion effects. However, in the frame of the quantum theory, they exhibit a complicated internal quantum structure with nontrivial nonlinear dynamics. In the process of the propagation along a fiber, the nonlinear dynamics of secant hyperbolical shaped optical pulses lead to the formation of classically stable soliton pulses with a rich variety of nonclassical properties, such as quantum noise reduction, spectral quantum noise correlations, and quantum non-demolition interactions. We discuss the experimental characterisation of these quantum properties of femtosecond optical solitons at the telecommunication wavelength of 1.5 /spl mu/m experiencing nonlinear-interactions in silica fibres.
Propagation characteristics of optical soliton in a three level atomic medium are analyzed by treating the material medium quantum mechanically, but the electromagnetic wave classically. Both the cubic and quintic components of the nonlinear polarization of the electromagnetic field are considered along with those generated due to the dipole formation of the material. A numerical simulation is carried out with the help of split-step technique. It is observed that the power of the pulse, distance of propagation and degree of dispersion are intimately related. The role of polarization due to the material is duely compensated by keeping higher order dispersive terms. In this connection we have seen that keeping the higher order dispersive terms, up to the eighth order, which is actually the phenomenon of continuum generation, results in a better form of the pulse. In our paper, we have analyzed the effects in both the cases, that is, including and excluding the quintic terms and in each case we have considered the effects of second-order dispersion (β2) as well as the higher order dispersion terms ( 8 2 j j β =
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.
Abstract We present some numerical results for nonlinear quantum walks (NLQWs) studied by the authors analytically (Maeda et al 2018 Discrete Contin. Dyn. Syst. 38 3687–3703; Maeda et al 2018 Quantum Inf. Process. 17 215). It was shown that if the nonlinearity is weak, then the long time behavior of NLQWs are approximated by linear quantum walks. In this paper, we observe the linear decay of NLQWs for range of nonlinearity wider than studied in (Maeda et al 2018 Discrete Contin. Dyn. Syst. 38 3687–3703). In addition, we treat the strong nonlinear regime and show that the solitonic behavior of solutions appears. There are several kinds of soliton solutions and the dynamics becomes complicated. However, we see that there are some special cases so that we can calculate explicit form of solutions. In order to understand the nonlinear dynamics, we systematically study the collision between soliton solutions. We can find a relationship between our model and a nonlinear differential equation.
Quantum walk (QW) provides a versatile tool to study fundamental physics and also to make a variety of practical applications. We here start with the recent idea of nonlinear QW and show that introducing nonlinearity to QW can lead to a wealth of remarkable possibilities, e.g., simulating nonlinear quantum dynamics, thus enhancing the applicability of QW above the existing level for a universal quantum simulator. As an illustration, we show that the dynamics of a nonlinear Dirac particle can be simulated on an optical nonlinear QW platform implemented with a measurement-based-feedforward scheme. The nonlinear evolution induced by the feed-forward introduces a self-coupling mechanism to (otherwise linear) Dirac particles, which accordingly behave as a soliton. We particularly consider two kinds of nonlinear Dirac equations, one with a scalar-type self-coupling (Gross-Neveu model) and the other with a vector-type one (Thirring model), respectively. Using their known stationary solutions, we confirm that our nonlinear QW framework is capable of exhibiting characteristic features of a soliton. Furthermore, we show that the nonlinear QW enables us to observe and control an enhancement and suppression of the ballistic diffusion.