Abstracts of talks

The motion of quantum vortex in a two-dimensional spinless superfluid is analyzed within Popov's hydrodynamic description.In the long healing length limit
(where a large number of particles are inside the vortex core) the superfluid dynamics is determined by saddle points of Popov's action, which allow for weak solutions of the Gross-Pitaevskii equation.
The resulting equations are solved for a vortex moving with respect to the superfluid. It is  found that the vortex core is reconstructed in a non-analytic way. The response of the vortex to applied
force produces an anomalously large dipole moment of the vortex and, as a result, the spectrum associated with the vortex motion exhibits narrow resonances lying within the phonon part of the spectrum,
contrary to traditional view.

 *In collaboration with Avraham Klein and Igor Aleiner.

The momentum spectrum of a periodic network (quantum graph) has a band-gap structure. We investigate the relative density of the bands or, equivalently, the probability that a randomly chosen
momentum belongs to the spectrum of the periodic network. We show that this probability exhibits universal properties. More precisely, the probability to be in the spectrum does not depend on the edge
lengths (as long as they are generic) and is also invariant within some classes of graph topologies. Based on a joint work with Gregory Berkolaiko.

Many-body localization is a peculiar dynamical transition between ergodic and non-ergodic phases, which may occur at any temperature and in any dimension. Current theory suggest that for
temperatures below the transition the system is non-ergodic and localized, such that conductivity vanishes at the thermodynamic limit, while for temperatures above the transition the system is thermal
and conductive. In this talk I will present a study of ergodicity and dynamical properties of the many-body localization transition using a combination of non-equilibrium diagrammatic techniques and
numerically exact methods. I will present the dynamical phase-diagram at infinite temperature, and provide evidence of existence of a novel phase, which is ergodic yet has a vanishing dc conductivity

We show that the one-particle density matrix ρ can be used to characterize the interaction-driven many-body localization transition in closed fermionic systems. The natural orbitals (the eigenstates of ρ)
are localized in the many-body localized phase and spread out when one enters the delocalized phase, while the occupation spectrum (the set of eigenvalues of ρ) reveals the distinctive Fock-space structure of the
many-body eigenstates, exhibiting a step-like discontinuity in the localized phase. The associated one-particle occupation entropy is small in the localized phase and large in the delocalized phase, with diverging fluctuations
at the transition. We analyze the inverse participation ratio of the natural orbitals and find that it is independent of system size in the localized phase.

Topological states of matter are distinguished by an invariant that
counts the number of protected subgap states, either bound to a defect
or propagating along a boundary. In a superconductor these are Majorana
fermions, described by a real rather than a complex wave function. The
absence of complex phase factors in the scattering amplitudes
fundamentally modifies the quantum signatures of chaos. We review these
recent developments, with an emphasis on electrical and thermal
transport properties that can probe the Majorana fermions.

Background reading: http://arxiv.org/abs/1407.2131

The quantum phase transition in clean metallic ferromagnets is generically first order, as predicted by theory, with a tricritical point in the phase diagram and tricritical wings in an external magnetic field.
In the presence of quenched disorder the transition is usually continuous, or second order, and in many systems additional disorder effects have been observed that coexist with the critical singularities. This talk will give an
overview of this field and review the status of the theoretical understanding of the experimental observations. 

It is demonstrated that the problem of two Aharonov-Bohm vortices can be solved analitically. The solution consists in  a series of ordinary differential equations which includes the Painlevé III equation. 

Electron-electron interactions are responsible for a correction to the conductance of a diffusive metal, the "Altshuler-Aronov correction". I'll
discuss the counterpart of this correction for a ballistic conductor, in which the electron motion is governed by chaotic classical dynamics. In
the ballistic conductance, the Ehrenfest time enters as an additional time scale that determines the magnitude of quantum interference effects.

For a generic 4D symplectic map we visualize the global organization of regular tori using 3D phase-space slices. The regular 2D tori are shown to be arranged around a skeleton of elliptic 1D tori.
These families of 1D tori show bifurcations and allow for understanding the hierarchy in phase space.

As applications we consider:
(a) power-law trapping,
(b) representation of eigenstates
    of the corresponding quantum map to
    investigate the semiclassical eigenfunction hypothesis,
(c) spectral and eigenvector statistics, and
(d) regular-to-chaotic tunneling.

[1] S. Lange, M. Richter, F. Onken, A. Bäcker, and R. Ketzmerick,
Global structure of regular tori in a generic 4D symplectic map,
Chaos  24, 024409 (2014)
[2] M. Richter, S. Lange, A. Bäcker, and R. Ketzmerick,
 Visualizing classical structures and quantum states of
4D maps by 3D phase-space slices,
      Phys. Rev. E 89 , 022902 (2014).
[3] For videos of 3D phase space slices see:\
http://www.comp-phys.tu-dresden.de/supp/

This is an introductory talk in which I will discuss, in a pedagogical way, the main features of classical and quantum motion of the kicked rotor and its relation to Anderson localization.

Radu Chicireanu,1 Isam Manai,1 Clément Hainaut,1 Jean-Fran�ois Clément,1
Jean Claude Garreau,1 Pascal Szriftgiser,1 and Dominique Delande2
1Laboratoire de Physique des Lasers, Atomes et Molécules,
Université de Lille 1 Sciences et Technologies, CNRS; F-59655 Villeneuve d'Ascq Cedex, France
2Laboratoire Kastler Brossel, UPMC, CNRS, ENS,
Coll�ge de France; 4 Place Jussieu, F-75005 Paris, France
Dimension 2 is expected to be the lower critical dimension for Anderson localization in a time-
reversal-invariant disordered quantum system. Quantitative experimental study of 2D Anderson
localization has been considered for a long time, and still is a real defy. Using an atomic quasiperi-
odically kicked rotor - equivalent to a two-dimensional Anderson-like model - we experimentally
study Anderson localization in dimension 2 and we observe a localized wavefunction dynamics. We
also show that the localization length depends exponentially on the disorder strength and is in quan-
titative aggreement with the predictions of the self-consistent theory of 2D Anderson localization.

In the past 20 years, momentum or trend following strategies have become an established part of the investor toolbox. We introduce a new way of analyzing momentum strategies by looking at the information ratio
(IR, average return divided by standard deviation). We calculate the theoretical IR of a momentum strategy, and show that if momentum is mainly due to the positive autocorrelation in returns, IR as a function of the portfolio
formation period (look-back) is very different from momentum due to the drift (average return). The IR shows that for look-back periods of a few months, the investor is more likely to tap into autocorrelation. However, for
look-back periods closer to 1 year, the investor is more likely to tap into the drift. We compare the historical data to the theoretical IR by constructing stationary periods. The empirical study finds that there are periods/regimes
where the autocorrelation is more important than the drift in explaining the IR (particularly pre-1975) and others where the drift is more important (mostly after 1975). We conclude our study by applying our momentum strategy to
100 plus years of the Dow-Jones Industrial Average. We report damped oscillations on the IR for look-back periods of several years and model such oscilations as a reversal to the mean growth rate. 

This talk is a brief review of quantum and topological properties of the system of charged particles periodically kicked by a 1D spatially periodic potential V(x) perpendicularly to a uniform magnetic field [1-7].
This system is a paradigm of Quantum Chaos and exact special cases of it are the kicked Harper models [2,4]. A general approach to the system [2-7] takes into account all the values of a constant of the motion, the x
coordinate xc of the cyclotron-orbit center. In fact, all these values appear in a generic ensemble of particles and the classical and quantum properties are very sensitive to xc;. In particular,
for some xc = xc*; and for small kick strength κ, the system behaves as if this strength is effectively κ2 [2,6,7]. This implies, e.g., a chaotic diffusion much slower than the already
slow one for other xc values. This phenomenon, which we call "superweak chaos" (SWC), was first observed [2] as a classical fingerprint of quantum antiresonance ( flat quasienergy (QE) band).
The classical (quantum) transition xc → xc*; was later studied in work [6] (work [7]). For rational values of a scaled Planck constant, the QE bands can be characterized by topological
Chern numbers satisfying a Diophantine equation [8], analogous to the one for the quantum Hall effect [9]. Then, the transition xc → xc*; turns out to be, quantally, a topological phase transition.
Recently, we found that SWC emerges in other systems under much more generic conditions.

[1] G.M. Zaslavskii et al., Sov. Phys. JETP  64, 294 (1986).
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I will discuss my close relationship with Dick Prange over a 27 year period as colleagues and friends at Maryland, and our recent research on quantum topological phenomena.

Atomic ensembles have many potential applications in quantum information science. Owing to collective enhancement, working with ensembles at high densities increases the overall efficiency of quantum operations, but at the same time also increases the collision rate and markedly changes the time dynamics of a stored coherence. We study theoretically and experimentally the coherent dynamics of cold atoms under these conditions. A closed form expression for the spectral line shape is derived for discrete fluctuations in terms of the bare spectrum and the Poisson rate constant of collisions, which deviates from the canonical stochastic theory of Kubo [1] and can measure the kernel of velocity changing collisions [2]. We measure a prolongation of the coherence times of optically trapped rubidium atoms as their density increases, a phenomenon we call collisional narrowing in analog to the well-known motional narrowing effect in NMR [3]. We explain under what circumstances collisional narrowing can be transformed into collisional broadening [4].
On account of collisions, conventional echo techniques fail to suppress this dephasing, and multi-pulse dynamical decoupling sequences are required. We demonstrate a 20-fold increase of the coherence time with a sequence of >200 ¹ pulses [5]. We perform quantum process tomography and demonstrate that using the decoupling scheme a dense ensemble with an optical depth of >200 can be used as an atomic memory with coherence times exceeding 3 sec. Further optimization requires utilizing specific features of the collisional bath [6] and control noise [7], which we measure directly. Finally, the spectral system can be mapped on real space anomalous diffusion [8] and optical coherence [9] problems that we also investigate.
Reference
[1] Y. Sagi, R. Pugatch, I. Almog, and N. Davidson, Phys. Rev. Lett. 104, 253003 (2010).
[2] J. Coslovsky et. al., unpublished.
[3] Y. Sagi, I. Almog and N. Davidson, Phys. Rev. Lett. 105, 093001 (2010(.
[4] Y. Sagi, R. et. al, Phys. Rev. A. 83, 043821 (2011).
[5] Y. Sagi, I. Almog and N. Davidson, Phys. Rev. Lett. 105, 053201 (2010).
[6] I. Almog et. al., J. Phys. B: At. Mol. Phys. 44, 154006 (2011).
[7] I. Almog et. al., submitted, arXiv:1303.2045.
[8] Y. Sagi, M. Brook, I. Almog, and N. Davidson, Phys. Rev. Lett. 108, 093002 (2012).
[9] M. Nixon et. al., Nature Photonics 7, 919 (2013).

Because their internal as well as their external degrees of freedom can be very well
controlled, cold atoms make it possible to experimentally study a number of
fundamental physical processes for quantum disordered systems
or few/many-body interacting systems, such as ballistic/diffusive transport,
and phenomena due to quantum interference
between multiple scattering paths, such as
weak localization, coherent back-scattering and strong (a.k.a. Anderson) localization.
In 3D, there is a metal/insulator transition in strong disorder,
which can be studied using cold atoms exposed to a disordered optical potential 
created by a laser speckle. We show that the unusual statistical properties 
of speckle potentials are responsible on the one hand for large deviations from the predictions of the self-consistent theory of localization, and on the other hand to unexpected difficulties in interpreting the experimental results.

We determined with unprecedented accuracy the lowest 900 eigenvalues of two quantum constant-width billiards from resonance spectra measured with flat, superconducting microwave resonators. While the classical dynamics
of the constant-width billiards is unidirectional,a change of the direction of motion is possible in the corresponding quantum system via dynamical tunneling. This becomes manifest in a splitting of the vast majority of resonances
into doublets of nearly degenerate ones. The fluctuation properties of the two respective spectra are demonstrated to coincide with those of a random-matrix model for systems with violated time-reversal invariance and a mixed dynamics.
Furthermore, we investigated tunneling in terms of the splittings of the doublet partners. On the basis of the random-matrix model we derived an analytical expression for the splitting distribution which is generally applicable to systems
exhibiting dynamical tunneling between two regions with (predominantly) chaotic dynamics.

This work was supported by the DFG within the Collaborative Research Centers 634.

Strong field multiple ionization has puzzled scientists, among other things, because of the strong correlations among outgoing electrons: their momenta parallel to the polarization axis of the field are very similar.
We have proposed that the origin of this correlation can be found in the classical dynamics of two electrons escaping from an attracting nucleus. We find that the repulsion between electrons amplifies deviations from a symmetric escape.
This observation can be exploited to describe many of the observed properties, to derive a nonlinear threshold law near the onset of double ionization, and to develop a simple 1+1-dimensional model for effective classical and quantum mechanical
simulations. Quantum simulations of this model show good agreement with the classical model. In the present contribution, we will focus on multiple ionization in molecules, and on triple ionization in  atomic systems.

This is joint work with Lars Bannow and Jan Thiede.

Phys. Rev. A 63, 043414 (2001)
Phys. Rev. A 64, 053401 (2001) 
Europhys. Lett. 56, 651--657 (2001)
J. Phys. B: At. Mol. 36, 3923--3935 (2003)
Phys. Rev. A 71, 033407 (2005)
J. Phys. B 39, 3865 (2006)
Phys. Rev. Lett. 98, 203002 (2007)
Phys. Rev. A 77, 015402 (2008)  
Phys. Rev. A 78, 013419 (2008)

In this talk I will review theories of topological insulating states
of matter in both non-interacting and interacting electron systems.  I
will start with introducing the by now standard theory of topological
band insulators in two and three dimensions. A simple pictorial
illustration/animation of a topological index, characterizing a band
structure, will be provided.  I will also explain the Fu-Kane method
to calculate the Z2 topological index for  time-reversal-invariant
band structures in three dimensions.  Then, I will focus on a new
class of topological material systems - topological Kondo insulators,
which appear as a result of interplay between strong correlations and
spin-orbit interactions.  The Fu-Kane method will be used to show that
hybridization between the conduction electrons and localized
f-electrons in certain heavy fermion compounds gives rise to
interaction-induced topological insulating behavior. Next, I will
briefly mention recent experimental results, which  appear to confirm
our predictions in the Samarium hexaboride compound, where the
long-standing puzzle of the residual low-temperature conductivity has
been shown to originate from topological surface states.  Samarium
hexaboride is perhaps the only material currently available, where
low-temperature transport  is truly surface-dominated and has
essentially no conduction in the bulk. In conclusion, I  will discuss
recent non-perturbative results on strongly-correlated topological
Kondo insulators.

We show in microwave measurements, numerical simulations and analytic theory that the transmission matrix provides the key to a universal description of wave propagation inside random media. The spectral
derivative of the composite phase of transmission eigenchannels is the delay in transmission and gives the integral of intensity over the sample volume of each transmission eigenchannel and its contribution to the density
of states (1,2,3). The ensemble average of the longitudinal profile of energy density of transmission eigenchannels &W_tau(x/L); is given in terms of Dorokhov's auxiliary localization lengths (4), determined from
the corresponding transmission eigenvalues, τ and sample length, &L; (5). Each of these profiles may be expressed as the product of the profile for the perfectly transmitting channel and a function which depends
only upon τ and the ratio of the sample and localization lengths, &L/xi;, &W_tau(x/L)=W_{tau=1}(x/L)S_tau(x/L);. &S_tau(x/L); is the strength of the source term in a generalized diffusion equation incorporating a
position dependent diffusion coefficient &D(x); (6) for the intensity correlator within the medium. These results reveal the rich structure of transmission eigenchannels and provide a measure of control over wave
propagation and energy deposition inside random media.

The research was supported by the National Science Foundation (DMR-1207446) and by the Tsinghua University ISRP.

References 
(1) M. G. Krein, Dokl. Akad. Nauk SSSR  144, 475 (1962).
(2) M. S. Birman and D. R. Yafaev, Algebra i Analiz  4, 1 (1992).
(3) M. Davy, Z. Shi, J. Wang, X. Cheng, and A. Z. Genack, Phys. Rev. Lett. {bf 114}, 033901 (2015).
(4) O. N. Dorokhov, Solid State Commun. 51, 381 (1984).
(5) M. Davy, Z. Shi, J. Park, C. Tian, and A. Z. Genack, Nat. Commun. (To appear in April, 2015).
(6) C. Tian, S. Cheung, and Z. Zhang, Phys. Rev. Lett.  105, 263905 (2010).

Recently, several models where quantum wave functions display multifractal properties have been identified. In the domain of quantum chaos, they correspond to pseudointegrable systems, with properties intermediate between
integrability and chaos. In condensed matter, they include models of electrons in a disordered potential at the Anderson metal-insulator transition. This multifractality leads to particular transport properties and
appear in conjunction with specific types of spectral statistics. In parallel, progress in experimental techniques allow to observe finer and finer properties of the wavefunctions of quantum or wave systems, as well as
to perform experiments with unprecedented control on the dynamics of the systems studied. In this context, this talk will discuss the robustness of
multifractality in presence of small perturbations. We present two scenarios for the breakdown of quantum multifractality under the effect of such perturbations. In the first scenario,
multifractality is preserved unchanged below a certain scale of the quantum fluctuations. In the other scenario, the fluctuations of the wave functions are modified at the different scales and each multifractal dimension smoothly goes
to the ergodic value. We use as generic examples a one-dimensional dynamical system and the three-dimensional Anderson model at the metal-insulator transition, and show that for different types of perturbation the destruction of
multifractal properties can be linked to one of these scenarios depending on the perturbation. Our results thus suggest a universality in the destruction of quantum multifractality. We also discuss the experimental implications.

While many physical phenomena have been described in terms of fractals or multifractals, the application of
these concepts to quantum mechanics is much more recent. Various models displaying self-similar structures at the quantum
level have been identified in the past few years, from the Anderson model of an electron in a disordered potential at criticality
to various quantizations of pseudo-integrable systems intermediate between integrability and chaos. These multifractal properties
appear in connection with particular spectral statistics. I will introduce various ensembles of critical random matrices,
based on Lax matrices of integrable models of interacting particles on a line, and describe a few universal properties of
critical ensembles, notably the connection between multifractal dimensions and eigenvalue statistics, as well as some properties
of extreme value statistics for multifractal states. 

Classical dynamical systems designed so that momentum can only change by multiples of a constant exhibit dynamical behaviors, that mimic the scenario
of quantum dynamical localization; namely, strong inhibition of diffusion, or quadratic energy growth, depending on the arithmetic nature of a
constant. Classical systems of this type may be produced by inserting piecewise linear, continuous potentials in the Standard Map (e.g., the"sawtooth map") but
they also appear  in some polygonal billiards. They  are believed to be "weakly ergodic", and challenge rigorous mathematical analysis. Such purely classical behavior
is in this talk  justified by a phase-space generalization of a well known quantum argument by Fishman, Grempel, and Prange,  whereby the kicked rotor localization was
assimilated to the Anderson localization. 

Correlations play a crucial role when estimating risk in the financial markets. This is so from a systemic viewpoint when trying to assess the stability of the markets, but also from a practical
one when, e.g., optimizing portfolios. However, the financial markets are far away from any kind of equlibrium. They are highly non-stationary, implying that important quantities, in particular the correlations, change in time in a seemingly
random fashion. This poses challenges for the modelling usually not encountered in the more traditional systems of statistical physics. We present two recent results: First we identify distinct quasi-stationary states in the correlation structure which emerge,
exist with some lifetime and then disappear. Second, we put forward a new random matrix ansatz to capture the non-stationarity. As an example we work out multivariate return distributions and compare with data.
At first sight, these two results might appear contradictory, but we argue that this is not so and that they are compatible 

In the framework of semiclassical theory the universal properties of quantum systems with classically chaotic dynamics can be accounted for through correlations between partner periodic orbits with small action differences.
So far, however, the scope of this approach has been mainly limited to systems of a few particles with low-dimensional phase spaces. In the present work we consider N-particle chaotic systems with local homogeneous interactions, where
N is not necessarily small. Based on a model of coupled cat maps we demonstrate emergence of a new mechanism for correlation between periodic orbit actions. In particular, we show the existence of partner orbits which are specific to
many-particle systems. For a sufficiently large N these new partners dominate the spectrum of correlating periodic orbits and seem to be necessary for construction of a consistent many-particle semiclassical theory. 

Random matrix theory provides a phenomenological description of universal behavior of chaotic dynamics. While not capable of giving conditions for universal behavior it is extremely useful when a 'self-averaging' quantity
is evaluated as an average over a suitable ensemble of matrices and an analytical expression is available. In some important cases such analytical expressions can be recovered without ensemble averages by suitable semiclassical technique,
for individual dynamics. Example will be given.

K. Müller, J. Richard, V. Volchkov, V. Denechaud, P. Bouyer, A. Aspect and V. Josse

Ultracold atomic systems in presence of disorder have attracted a lot of interest over the past decade, in particular to study the physics of Anderson localization (AL) in a renewed perspective. Landmark experiments have been demonstrated,
in 1D [1,2] and 3D [3,4,5,6] geometries. However many challenges remain and new ideas have emerged, as for instance the search for original signatures of Anderson localization in momentum space [7].
Here I will describe our progresses along that line where a weak localization effect has been directly observed, i.e. the Coherent Backscattering (CBS) phenomenon [8]. In particular I will report on the recent observation of suppression
and revival of CBS when a controlled dephasing kick is applied to the system [9]. This observation demonstrates a novel and general method, introduced by T. Micklitz and coworkers [10], to study probe phase coherence in disordered systems
by manipulating time reversal symmetry of the experimental time sequence.

References

[1]	J. Billy et al.,, Nature 453, 891 (2008).
[2]	G. Roati et al.,, Nature 453, 895 (208).
[3]	S. Kondov et al., Science 334, 66 (2011).
[4]	F. Jendrzejewki et al., Nat. Phys. 8, 398 (2012).
[5]	S. Semeghini et al. arXiv.1404.3528 (2014).
[6]	C. A. Müller and B. Shapiro, Phys. Rev. Lett. 113, 099601 (2014).
[7]	T. Karpiuk et al., Phys. Rev. Lett. 109, 190601 (2012)..
[8]	F. Jendrzejewski et al., Phys. Rev. Lett. 109, 195302 (2012).
[9]	K. Müller et al., arXiv.1411.1671 (2014).
[10]	T. Micklitz et al., Phys. Rev. B 91, 064203 (2015).

This talk will introduce quantum-classical correspondence for systems
with classically coexisting regular and chaotic dynamics. This will be
visualized for wave packet dynamics and eigenstates in billiards. For
transport in Hamiltonian systems the quantum signatures of dividing
surfaces with holes in a chaotic phase space (so-called partial
barriers) are of great relevance. New results on the localization of
eigenstates will be presented.

By means of a microwave tight-binding analogue experiment of a graphene like lattice, we observe a topological transition between a phase with a pointlike
band gap characteristic of massless Dirac fermions and a gapped phase. By applying a controlled anisotropy on the structure, we investigate the transition directly via
density of states measurements. The wave function associated with each eigenvalue is mapped and reveals new states at the Dirac point, localized on the armchair edges.
We find that with increasing anisotropy, these new states are more and more localized at the edges. We study their existence and the internal
structure of the three main edges: zigzag, bearded, and armchair. We show that the uniaxial strain can be used to manipulate the edge states: a single parameter controls
their existence and their spatial extension into the bulk.

Laser cooling of atoms has made possible to study a large variety of complex quantum systems under optimal conditions. This talk will describe a recent experimental characterization of the Anderson metal-insulator transition,
a quantum phase transition notoriously difficult to observe because it is highly sensitive to decoherence and interactions. I will show how a system of cold atoms subjected to a chaotic dynamics allows to explore this transition in a controlled
way: to measure for the first time its critical exponent (Chabé et al., PRL (2008)), to characterize its critical state (Lemarié et al., PRL (2010)) and to portray its phase diagram (Lopez et al., NJP (2013)).

Periodically driven quantum systems, such as semiconductors subject to light and cold atoms in optical lattices,
provide a novel and versatile platform for realizing topological phenomena. Among these are analogs of topological insulators
and superconductors, attainable in static systems. However, some of these phenomena are unique to the periodically driven case.
I will describe how the interplay between periodic driving, disorder, and interactions gives rise to new steady states exhibiting
robust topological phenomena, with no analogues in static systems. Specifically, I will show that disordered two dimensional
driven systems admit an Òanomalous" phase with chiral edge states that coexist with a fully localized bulk. This phase serves
as a basis for a new topologically protected, far-from-equilibrium transport phenomenon: quantized non-adiabatic charge pumping.
I will make a comparison to interacting one dimensional driven systems, and show that despite the fact that they cannot support
such a phenomenon, they do harbor current carrying states with excessively long life times.

Signatures of Anderson localization in the momentum distribution of a cold atom cloud after a quantum quench are studied. We consider a quasi one-dimensional cloud initially prepared in a well defined momentum state, and
expanding for some time in a disorder speckle potential. Quantum interference generates a peak in the forward scattering amplitude which, unlike the common weak localization backscattering peak, is a signature of emph{strong} Anderson
localization. We present a non-perturbative, and fully time resolved description of the phenomenon, covering the entire diffusion--to--localization crossover. Our results should be observable by present day experiments.

Anderson localization is a universal phenomenon affecting non-interacting quantum particles in a disordered environment.
In three spatial dimensions, theory predicts a quantum phase transition from localization to diffusion at a critical energy, the mobility
edge, which depends on the disorder strength. Although it has been recognized already long ago as a prominent feature of disordered systems,
a complete experimental characterization of the mobility edge is still missing.
I will report the measurement of the mobility edge for ultracold atoms in a disordered potential created by laser speckles.
We are able to control both the disorder strength and the energy of the system, so as to probe the position of the localization
threshold in the disorderÐenergy plane. Our results might allow a direct experimentÐtheory comparison, which is a prerequisite to
study the even more challenging problem of disorder and interactions. 

We investigate the spectral statistics chaotic many-body systems, using a trace formula that expresses the level density of chaotic many-body systems as a smooth term plus a sum over contributions
associated to  solutions of the nonlinear Schr"odinger equation. Our formula applies to bosonic systems with discrete sites, such as the Bose-Hubbard model, in the semiclassical limit as well as in the limit
where the number of particles is taken to infinity. The focus of the talk will be to investigate the two point correlation function of the level density by studying interference between solutions of the
nonlinear Schr"odinger equation. We show that in the limits taken the statistics of fully chaotic many-particle systems becomes universal and agrees with predictions from the Wigner-Dyson ensembles of random
matrix theory. We also discuss the effect of discrete geometric symmetries on this statistics for the example of the Bose-Hubbard model without disorder. The conditions for Wigner-Dyson statistics
involve a gap in the spectrum of the Frobenius-Perron operator, leaving the possibility of different statistics for systems with weaker chaotic properties. (Joint work with Remy Dubertrand)

I will present the observation of topological protection in optics - specifically, a photonic Floquet topological insulator. Topological insulators (TIs) are solid-state materials that are insulators in the bulk, but
conduct electricity along their surfaces - and are intrinsically robust to disorder. In particular, when a surface electron in a TI encounters a defect, it simply goes around it without scattering, always exhibiting Ð quite strikingly Ð
perfect transmission. The structure is an array of coupled helical waveguides (the helicity generates a fictitious circularly-polarized electric field that leads to the TI behavior), and light propagating through it is Ôtopologically protectedÕ
from scattering. Topological protection therefore has the potential to endow photonic devices with quantum Hall-like robustness. 

 Semiclassical methods in Quantum Chaos have been successfully applied to single-particle quantum systems
with corresponding low-dimensional chaotic dynamics. The generalization to interacting many-body systems
remains to be one of the major challenges in this field. I will present two approaches to devise semiclassical
many-particle techniques in the complementary limits of small hbar [1] and  large particle numbers [2,3]. In
particular, I will discuss many-body generalizations of the Gutzwiller-van Vleck propagator and the Gutzwiller
trace formula and show how they allow for investigating coherent backscattering and echo phenomena in Fock
space, arising from many-body quantum interference.

[1] The Weyl expansion for systems of independent identical particles,
    Q. Hummel, J.-D. Urbina, and K. Richter,  J. Phys. A. 47, 01510 (2013).

[2] Coherent Backscattering in Fock Space: a Signature of Quantum Many-Body
    Interference in Interacting Bosonic Systems,
    T. Engl, J. Dujardin, A. ArgŸelles, P. Schlagheck, K. Richter and J.-D. Urbina,, Phys. Rev. Lett. 112, 140403 (2014).

[3] Many-Body Spin Echo
    T. Engl, J.-D. Urbina, and K. Richter, arXive 1409.8065 (2014).

For particles in three or more dimensions are either bosons or fermions.
Restricting particles to the plane the fundamental group of the configuration space is the braid group and a new form of particle statistics corresponding to its abelian representations appears, anyon statistics. Restricting the dimension of
the space further to a quasi-one-dimensional quantum graph opens new forms of statistics determined by the connectivity of the graph.
We develop a full characterization of abelian quantum statistics on graphs. For two connected graphs the statistics are independent of the particle number. On three connected non-planar graphs particles are either bosons or fermions while
in three connected planar graphs they are anyons.
Graphs with more general connectivity exhibit interesting mixtures of these behaviors which we illustrate. This is work with Jon Harrison, Jon Keating, and Jonathan Robbins.

Topological photonic systems provide robust modes whose properties can be well controlled. Photon creation and annihilation processes induce a new class of exploitable symmetries which do not exist in the original electronic context,
but also serve as an extra source of noise. I describe how these concepts can be utilized in two settings: PT-symmetric Lasers [1] and resonator chains with selective amplification of a topologically induced defect mode [2].

[1] H.Schomerus, Quantum Noise and Self-Sustained Radiation of PT-Symmetric Systems, Phys. Rev. Lett. 104, 233601 (2010).
[2] C. Poli, M. Bellec, U.Kuhl, F. Mortessagne, H. Schomerus, arxiv:1407.3703, Nature Communications (in press)

We introduce a number of random matrix models describing the Google matrix G of directed networks. The properties of their spectra and eigenstates are analyzed by numerical matrix diagonalization.
We show that for certain models it is possible to have an algebraic decay of PageRank vector with the exponent similar to real directed networks. At the same time the spectrum has no spectral gap and a 
broad distribution of eigenvalues in the complex plain. The eigenstates of G are characterized by the Anderson transition from localized to delocalized states and a mobility edge curve in the complex plane of eigenvalues.

In quantum chaotic systems with symmetries the energy spectrum decomposes into subspectra that correspond to irreducible representations of the symmetry group. An interesting question then is: what is the correct random matrix
ensemble for describing a particular subspectrum. This depends not only on the behaviour of the system under time reversal, but also on properties of the irreducible representation. In some cases standard representation theory is sufficient
to answer this question, but in general Wigner's theory of corepresentations is required that involves also anti-unitary operators. Applying this theory one can identify relatively simple systems with unusual statistics. 

I shall present a novel approach for the derivation of the spectral joint probability distribution for random ensembles of graph adjacency matrices (or in general of  Bernoulli matrices). It is similar in spirit to Dyson's Brownian
motion approach, but since it deals with discrtete random walks on a finite set of matrices, major modifications are required, and these will be discussed in detail. It will be shown that as the matrix dimension increases, a Fokker-Planck
equation provides an increasingly better approaximation to the joint probability distibution.  

We shall discuss the conjectured spectral phase transition in the Wegner N-orbital model, and justify the localisation side of the  phase diagram (with the sharp dependence of the threshold on
the number of orbitals). We also prove a sharp Wegner estimate for a class of random operators including the Wegner orbital model and Gaussian band matrices. Based on joint works with M. Aizenman, R. Peled, J. Schenker, M. Shamis.

The discovery of integer quantum Hall effect (IQHE) in 1980, a transport quantization phenomenon, heralded a revolution in condensed matter physics. This notwithstanding, IQHE is commonly conceived as being unrelated to chaos
ubiquitous in Nature. Indeed, the salient characteristic of chaos Ð the sensitivity of systemÕs behavior to disturbances Ð is conceptually incompatible with the robustness of transport quantization in IQHE. Moreover, while chaos occurs even
in simple one-body systems, IQHE is known to be a ground-state property of many-electron systems. Surprisingly, we discover in a canonical chaotic one-body system a Planck's quantum-driven phenomenon bearing a firm analogy to IQHE but of
chaotic origin. Our finding indicates that rich topological quantum phenomena can emerge from chaos.

Energy transport is an important theme in natural processes, e.g., chemical reactions and photosynthesis. There is ongoing debate on how the environment influences the efficiency of energy transfer in these systems and to which
extent quantum mechanics plays a role. By interfacing electronically highly excited (Rydberg) atoms with laser light we simulate energy transfer dynamics in a controlled many-body system. In particular, Rydberg atoms experience quantum state
changing interactions similar to Förster processes in complex molecules, offering a model system to study the nature of dipole-mediated energy transport. The extension to multiple interacting excitations could enable elementary realisations
of quantum spin models involving strong and long-range spin-dependent interactions. We report on a new imaging method, which we apply to monitor the migration of electronic excitations with high time and spatial resolution using a background
atomic gas as an amplifier. Through precise control of interactions and the coupling to the environment via the laser fields, we find different mechanisms at work which shed new light on the nature of energy and spin transport in complex quantum
systems.

State-of-the-art experiments with ultracold atoms and molecules allow for an unprecedented control of microscopic degrees of freedom and the bottom-up construction of nano-structures in various regimes (glasses, crystals, superfluids).
! In close collaboration with experiments, we propose to create structures consisting of a few hundred to a few million particles, covering the cross-over from microscopic to macroscopic matter. We focus hereby on both the production of stable many-body
structures, such as static lattices mimicking magnetic materials or coherent solitons in optical lattices [1], as well as on quantum transport problems. The dynamics of ultracold fermions and bosons simulates the transport of electrons in solids, with the
great advantage of unprecedented experimental control and in situ observation possibilities [2]. New forms of current transport, including materials with negative differential resistivity [3], driven by many-body correlation effects have been observed
already. This opens many frontiers for the design of matter and the control of currents [4,5] on the verge between the quantum and the classical world.
!
(1) G. Kordas, S. Wimberger, D. Witthaut, EPL 100, 30007 (2012)
(2) G. Kordas, D. Witthaut, S. Wimberger, Non-equilibrium dynamics in dissipative
Bose-Hubbard chains, Ann. Phys. (Berlin), available online: DOI: 10.1002/andp.
201400189 (2015)
(3) R. Labouvie, B. Santra, S. Heun, S. Wimberger, H. Ott, arXiv:1411.5632 (preprint
2015)
(4) Atom-based analogues to electronic devices, Europhysics News 44, 6, 20 (2013),
highlighting: G. Ivanov, G. Kordas, A. Komnik, S. Wimberger, Eur. Phys. J. B 86,
345 (2013)
(5) Stromuübertragung auf atomarer Ebene, Newspaper article in the Rhein-Neckar-
Zeitung (Heidelberg, Germany), 9/4/2014

Similarly to the probability distribution of energy in physics, the probability distribution of money among the agents in a closed economic system is also expected to follow the exponential Boltzmann-Gibbs law, as a consequence
of entropy maximization.  Analysis of empirical data shows that income distributions in the USA, European Union, and other countries exhibit a well-defined two-class structure.  The majority of the population (about 97%) belongs to the lower
class characterized by the exponential (`thermal') distribution.  The upper class (about 3% of the population) is characterized by the Pareto power-law (`superthermal') distribution, and its share of the total income expands and contracts
dramatically during booms and busts in financial markets.  Globally, data analysis of energy consumption per capita around the world shows decreasing inequality in the last 30 years and convergence toward the exponential probability distribution,
in agreement with the maximal entropy principle. Similar results are found for the global probability distribution of CO2 emissions per capita.  All papers are available at url http://physics.umd.edu/~yakovenk/econophysics/.
For recent coverage in Science magazine, see url http://www.sciencemag.org/content/344/6186/828