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We study the effect of localized modes in lattices of size N with parity-time (PT) symmetry. Such modes are arranged in pairs of quasidegenerate levels with splitting delta approximately exp(-N/xi) where xi is their localization length. The level "evolution" with respect to the PT breaking parameter gamma shows a cascade of bifurcations during which a pair of real levels becomes complex. The spontaneous PT symmetry breaking occurs at gammaPT approximately min{delta}, thus resulting in an exponentially narrow exact PT phase. As N/xi decreases, it becomes more robust with gammaPT approximately 1/N2 and the distribution P(gammaPT) changes from log-normal to semi-Gaussian. Our theory can be tested in the frame of optical lattices.

A random matrix ensemble incorporating both GUE and Poisson level statistics while respecting U(N) invariance is proposed and shown to be equivalent to a system of noninteracting, confined, one dimensional fermions at finite temperature.PACS numbers: 05.45+b, 72.15-Rn, 03.65-w (a) Permanent address:

The intensity pattern generated by a monochromatic point source in a random medium is studied. Thc intensity-intensity correlation function ls calculated and lt ls sh0%n that. the intensity, as a function of coordinate, exhibits large fluctuations (the speckle pattern). The sensitivity of this speckle pattern to small changes in the source frequency is also studied. PACS numbers: 42.20.Ji, 71.55.Jv A wave propagating in a random medium undergoes multiple scattering from the inhomogeneities.The scattered waves interfere with each other and, as a result, a certain intensity pattern is formed. In a random medium, as opposed, e.g. , to a crystal, one would, naively, expect an efficient averaging process and therefore a fairly smooth intensity pattern, with only small intensity fluctuations.Instead, however, one finds a highly irregular pattern, with large intensity changes over short distances. The irregularities are not due to noise. Each microscopic realization of the random medium, i.e. , each sample of the statistical ensemble, displays its own pattern -a "fingerprint" which reflects the specific arrangement of the inhomogeneities (impurities) in that sample. This phenomenon is quite familiar in optics where it is termed "a speckle pattern" and usually refers to an intensity pattern formed on a screen by light reflected from a rough surface. Below, this term is used in a somewhat broader sense and refers to an intensity pattern formed in the bulk of a disordered medium when a wave (electromagnetic, acoustic, or an electron wave) propagates through it.There exists a huge literature on the subject. ' 3 In early work, usually certain assumptions were made directly on the statistics of the scattered light (rather than on the statistical properties of the disordered medium). The "first principles" work, i.e. , that which tries to derive properties of the speckle patterns directly from the wave equation, is mostly limited to smooth inhomogeneities (the wavelength much shorter than the characteristic inhomogeneity size). 2~T he subject of light propagation in random media has been recently given a new boost as a result of a number of experiments. These experiments revealed an enhanced backscattering, in combination with large intensity fluctuations6 and high sensitivity of the speckle pattern to relatively small changes of the source frequency. 7 Similar phenomena exist, and are being extensively studied, in the electron transport in disordered systems. The point is that as long as the sample size is smaller than the inelastic scattering length (the mesoseopic regime), an electron propagates coherently through the entire sample and, thus, takes a "fingerprint" of the specifi, for that sample, impurity arrangement.This manifests itself in various interference phenomena and in extreme sensitivity of the conductance to small changes of various factors. s '2 The purpose of the present work is to calculate some properties of speckle patterns, specifically the intensity correlation function, for a scalar field. Such a fiel...

The review deals with the physics of cold atomic gases in the presence of disorder. The emphasis is on the theoretical developments, although several experiments are also briefly discussed. The review is intended to be pedagogical, explaining the basics and, for some of the topics, presenting rather detailed calculations . arXiv:1112.5736v1 [cond-mat.quant-gas]

Areal density of disorder-induced resonators with a high quality factor, Q ≫ 1, in a film with fluctuating refraction index is calculated theoretically. We demonstrate that for a given kl > 1, where k is the light wave vector, and l is the transport mean free path, when on average the light propagation is diffusive, the likelihood for finding a random resonator increases dramatically with increasing the correlation radius of the disorder. Parameters of most probable resonators as functions of Q and kl are found.

We study the effect of Anderson localization on the expansion of a Bose-Einstein condensate, released from a harmonic trap, in a 3D random potential. We use scaling arguments and the selfconsistent theory of localization to show that the long-time behavior of the condensate density is controlled by a single parameter equal to the ratio of the mobility edge and the chemical potential of the condensate. We find that the two critical exponents of the localization transition determine the evolution of the condensate density in time and space. Recently there has been much interest in the possibility of observing Anderson localization of Bose-Einstein condensates obtained by trapping and cooling bosonic atoms [5,6,7,8]. A Bose-Einstein condensate is characterized by a macroscopic occupation of a single quantum state [9] and hence exhibits quantum, wave-like behavior despite its macroscopic size. Atomic Bose-Einstein condensates subjected to random external (optical) potentials are potentially good candidates for observing Anderson localization of matter waves. Up to now, the experiments have focused on 1D configurations [5], where all single-particle eigenstates are localized. In a typical experiment, the condensate is created in an optical or magneto-optical trap. The trap is then turned off and the condensate is allowed to expand.In this Letter we study the expansion of the BoseEinstein condensate in a 3D random potential. Unlike in 1D, a critical energy (the mobility edge ǫ c ) exists in 3D which separates extended and localized states. An eigenstate is extended (localized) if the corresponding energy is larger (smaller) than ǫ c . When the condensate is released from the trap, the atoms achieve kinetic energies up to the chemical potential µ of the trapped condensate. For weak disorder ǫ c < µ, and a fraction of atoms diffuses away, whereas the remainder is localized, as was pointed out in [7]. We show here that, surprisingly, even for strong disorder ǫ c > µ only a fraction of the condensate will be localized. We study the full dynamics of the condensate expansion by accounting for weak localization at energies ǫ > ǫ c , strong localization at ǫ < ǫ c , and critical behavior around the mobility edge. Our main result is that the effect of disorder on the expansion of the condensate is controlled by a single parameter ǫ c /µ, and that Anderson localization plays an important role even when the chemical potential of the condensate µ is much larger than the mobility edge ǫ c . We show that the behavior of the average condensate densityn(r, t) at large distances r and long times t is governed by the critical exponents ν and s of the localization transition. This could provide a direct way to measure these exponents. The density of the localized part of the condensaten(r, ∞) does not decay exponentially with r, as one could have expected, but follows a power law.Consider a Bose-Einstein condensate of N ≫ 1 atoms of mass m trapped in a 3D spherically-symmetric harmonic potential V ω (r), characterized by the trap freque...

Recently, a macroscopic theory of TV-channel disordered conductors treated the evolution (with the length L) of the probability distribution of the transfer matrix for the full conductor and allowed a theoretical description of the universal conductance fluctuations. Those results are used here to calculate the correlation function between transmission as well as reflection coefficients: In the case L^>W (width of the sample), the former essentially coincides with the one obtained from microscopic perturbative calculations. The latter, on the other hand, is a prediction of the present model. PACS numbers: 72.15.Cz, 02.50.+S Much effort has recently been devoted to the understanding of the intensity fluctuations of waves multiply scattered from disordered media. In quantum electronic transport, multiple scattering leads to anomalously large (the so-called "universal") conductance fluctuations 1 " 9 (UCF): The variance of the dimensionless conductance g (in units of e 2 /h) is of order unity and, to a large extent, does not depend on the size of the sample nor on the degree of disorder. Multiple scattering is equally important in light propagation through a random medium, where it is responsible for various effects in intensity statistics (the speckle pattern). 6 ' 10 " 19 It turns out, 6 ' 10 for instance, that the variance of the transmission coefficient Tb=H a T a b (towards the right, say, when a single mode b is excited on the left) is of the order //TVL, which is again much larger than what a naive statistical consideration would suggest. Here / is the (elastic) mean free path, L the length of the system, and TV the total number of channels.Another interesting fact was pointed out is Refs. 3 and 5: The naive assumption that (in the TV»1 limit) the various transmission factors T a b are statistically independent violates the notion of UCF. It was then suggested in Ref. 5 that lack of correlation between reflection factors R a b might be consistent with UCF. The correlation coefficient Cab,a'b' between T a b and T a 'b' was explicitly evaluated later, 6 by use of diagrammatic perturbation theory. As far as we know, there is no explicit evaluation of the correlation coefficient C §,,ab' between reflection factors.The standard theoretical description of the abovementioned problems, usually based on a perturbative treatment or on numerical simulations, is of a microscopic nature. In Refs. 3 and 7-9, on the other hand, the input to the analysis is the transfer matrix for the full conductor: The approach was thereby named macroscopic in Refs. 8 and 9. The theory of Ref. 8 is based on the properties of flux conservation, time-reversal invariance, and the appropriate combination law when two wires are put together. The distribution associated with systems of very small length is selected on the basis of a maximum-entropy criterion; the combination law then shows that the "evolution" of the distribution with the length L is governed by a Fokker-Planck or diffusion equation in TV dimensions. In Ref. 9 it was shown that ...