arXiv:1110.3024v1 [cond-mat.dis-nn] 13 Oct 2011 Enhanced transport when Anderson localization is destroyed Yevgeny Krivolapov, 1 Liad Levi, 1 Shmuel Fishman, 1 Mordechai Segev, 1 and Michael Wilkinson 2 1 Physics Department, Technion - Israel Institute of Technology, Haifa 32000, Israel. 2 Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, England We investigate the anomalous transport in optically-induced potentials that are random in both space and time. We find that the time variation destroys Anderson localization, replacing it by transport that is faster than diffusion, which in some cases can be even faster than ballistic. We relate this phenomenon to Chirikov’s theory of overlapping resonances, and find radical differences between the anomalously-enhanced transport in one-dimensional and two-dimensional systems. It is well known that a wavepacket (or a particle) mov- ing in a spatially-disordered time-independent potential can exhibit Anderson localization, and that this is the generic situation in one or two dimensions [1, 2]. At the same time, it is also known that, if the disordered potential is also fluctuating in time, localization is lost and transport is restored. That is, for localization to occur the disordered potential must be ‘frozen’ in time [1]. Consequently, it might naturally be expected that, when Anderson localization is destroyed, transport will become diffusive. Over the years, several different mech- anisms were proposed, for the breakdown of Anderson localization due to temporal fluctuations of the poten- tial. Mott [3] considered the effect of phonons at low temperatures, and argued that this gives rise to a diffu- sive motion termed ‘variable-range hopping’ conductiv- ity. Mott [4] also considered the effects of a weak AC field, and suggested that a resonant interaction domi- nates the low-frequency response. Alternatively, it has been argued that another limiting procedure leads to a distinct type of diffusive response to a low-frequency elec- tric field, termed adiabatic transport [5]. On the other hand, there are reasons to expect that the response to time-dependent potentials may be different than diffu- sion. Specifically, the diffusion constants predicted by such mechanisms sensitively depend upon energy. Also, if the potential is time-dependent, the energies of the particles will not be constant. Hence, if the diffusion co- efficient is a rapidly increasing function of energy, the re- sponse to a time-dependent perturbation can be a super- diffusive. The existence of such ‘anomalous diffusion’ has been demonstrated in classical (‘effective particle’) mod- els with spatially and temporally fluctuating potentials [6–10]. However, experimental evidence for anomalous diffusion is still missing: electron-electron interactions render such phenomena, as well as Anderson localization effects, extremely difficult to observe via charge transport experiments in solids. In fact, Anderson localization is most easily demonstrated in optical and in matter-waves systems, where the potential is induced by light waves. In this Letter, we investigate potentials that are in- duced by light waves, for example in cold atoms [11] and in optically-induced photonic potentials [12, 13]. We find that transport in these temporally-fluctuating and spatially-random potentials is faster than diffusion, and in some cases can be even faster than ballistic. When the potential is made up of a discrete set of momentum components, we establish connections with the Chirikov theory of overlapping resonances, and show how this ap- proach maps into a diffusive approximation in the con- tinuum limit. Finally, we find radical differences between the anomalously-enhanced transport in one-dimensional and two-dimensional systems. The random potentials induced by optical waves in photonic lattices [13] and in atom optics experiments [11, 14], rely on transforming an intensity pattern into an effective potential for the light (the former) or the cold atoms (the latter). Such potentials are naturally de- scribed in terms of the Fourier spectrum of the waves in- ducing them, and their spectral coefficients are assumed to be independent random variables. Physically, the spectral content of optically-induced potentials is rep- resented by a finite number, N , of Fourier coefficients, each representing a plane-wave component at some angle with respect to the optical axis. In typical experiments, this number may be large, hence we shall consider po- tentials with very dense Fourier transform, approximat- ing the limit as N →∞ . The problem is modeled by the Schrödinger equation with such type of potentials, where the initial condition is a localized wave-packet and the question is the rate of spreading of the wave-packet, when Anderson localization is destroyed. In the regime where the wave-packet has already acquired large values of momentum, one expects classical mechanics to provide a reasonable approximation. Therefore, we will explore here the classical dynamics for potentials, which are de- fined by their spectral content [15]. In particular, we consider the classical dynamics of a particle in potentials that are both random in space and in time, emphasizing the spreading of the momentum acquired by the parti- cle, as time evolves. When the potential is made up of a finite number of Fourier components, the theory can be formulated in terms of Chirikov’s resonance overlap criterion [16–18]. In the limit as N →∞ , we show how the Chirikov resonances are related to an expression for a diffusion coefficient D characterizing random changes