Wave-Packet Dynamics in Chains with Delayed Electronic Nonlinear Response

Abstract

We study the dynamics of one electron wave packet in a chain with a non-adiabatic electronphonon interaction. The electron-phonon coupling is taken into account in the time-dependent Schrödinger equation by a delayed cubic nonlinearity. In the limit of an adiabatic coupling, the self-trapping phenomenon occurs when the nonlinearity parameter exceeds a critical value of the order of the band width. We show that a weaker nonlinearity is required to produce self-trapping in the regime of short delay times. However, this trend is reversed for slow nonlinear responses, resulting in a reentrant phase-diagram. In slowly responding media, self-trapping only takes place for very strong nonlinearities. PACS numbers: 71.30.+h; 73.20.Jc; 05.60.Gg 71.38.Ht The study of the physical mechanisms involved in transport phenomena taking place in nonlinear systems is a fundamental issue in solid state physics. Concerning the electronic transport, nonlinearity arises from the interaction between electrons and lattice vibrations [1,2,3,4,5,6,7]. In this context, the discrete nonlinear Schrödinger equation (DNLSE) effectively describes the influence of lattice vibrations on the electron dynamics. The most important property associated with the DNLSE is the self-trapping which occurs when the nonlinearity parameter exceeds a critical value of the order of the band width [1,2,3]. In this regime, an initially localized electronic wave packet does not spread continuously over the lattice. Therefore, the probability of finding the electron at its initial site remains finite in the long-time limit.In low-dimensional systems, the effect of nonlinearity seems to be dominant over the role played by disorder [8,9,10,11]. Recently, the spreading of an initially localized wave packet in two nonlinear chains with disorder was studied [8]. Considering a discrete nonlinear Schrödinger and quartic Klein-Gordon equations with disorder, it was proved that the second moment and the participation number of the wave packet do not diverge simultaneously [8]. The spreading of an initially localized wave packet in a one-dimensional discrete nonlinear Schrödinger lattice with disorder was also recently studied [9]. It was observed that the Anderson localization is destroyed and a subdiffusive dynamics takes place above a certain critical nonlinearity strength. [9] Moreover, analytical and numerical calculations for a reduced FermiPasta-Ulam chain demonstrated that energy localization does not require more than one conserved quantity [10].From the experimental point of view, the interplay between disorder and nonlinearity was investigated in Ref. [11]. The evolution of linear and nonlinear waves in coupled optical waveguides patterned on an AlGaAs substrate were directly measured. Nonlinear perturbations enhance localization of linear waves while induce delocalization of the nonlinear ones [11]. In the presence of disorder, a transition from ballistic wave packet expansion to exponential localization was observed. Within a more general scope, the study of w...

Fundamental solitons in discrete lattices with a delayed nonlinear response

“…These findings complement the recently published study of effects of the time-delayed nonlinear response on the transition to delocalization of wave packets [18]. Altogether, these results provide for an insight into the dynamics of localized patterns in slowly responding media.…”

“…In this regard, an essential issue is the dynamics of localized modes in media with the non-instantaneous nonlinear response. In particular, it was demonstrated that the relaxation of the nonlinearity has a strong influence on the wave-packet dynamics [17,18].…”

“…The model is formulated in Section II, following Ref. [18], where it was introduced for electron-photon systems featuring a nonadiabatic interaction. The MI is also considered in Section II.…”

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“…Within the context of electronic transport in low-dimensional nonlinear discrete lattices, one of the most known properties is the self-trapping (ST) phenomena. In general lines, ST occurs when the strength of the nonlinearity surpasses a threshold which is of the order the bandwidth for initially fully localized wavepackets [2][3][4][5][6]. In this case, the electron wavepacket remains trapped around the initial position with the probability of finding the electron at its initial position remaining finite in the long-time limit.…”

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