Experimental test of universality of the Anderson transition

Quantum diffusion induced by small quantum chaos

It is demonstrated that quantum systems classically exhibiting strong and homogeneous chaos in a bounded region of the phase space can induce a global quantum diffusion. As an ideal model system, a small quantum chaos with finite Hilbert space dimension N weakly coupled with M additional degrees of freedom which is approximated by linear systems is proposed. By twinning the system the diffusion process in the additional modes can be numerically investigated without taking the unbounded diffusion space into account explicitly. Even though N is not very large, diffusion occurs in the additional modes as the coupling strength increases if M≥3. If N is large enough, a definite quantum transition to diffusion takes place through a critical subdiffusion characterized by an anomalous diffusion exponent.

Kardar-Parisi-Zhang Physics in the Density Fluctuations of Localized Two-Dimensional Wave Packets

We identify the key features of Kardar-Parisi-Zhang (KPZ) universality class in the fluctuations of the wave density logarithm in a two-dimensional Anderson localized wave packet. In our numerical analysis, the fluctuations are found to exhibit an algebraic scaling with distance characterized by an exponent of 1/3, and a Tracy-Widom probability distribution of the fluctuations. Additionally, within a directed polymer picture of KPZ physics, we identify the dominant contribution of a directed path to the wave packet density and find that its transverse fluctuations are characterized by a roughness exponent 2/3. Leveraging on this connection with KPZ physics, we verify that an Anderson localized wave packet in 2D exhibits a stretched exponential correction to its well-known exponential localization.

Numerical investigation of localization in two-dimensional quasiperiodic mosaic lattice

A one-dimensional lattice model with mosaic quasiperiodic potential is found to exhibit interesting localization properties, e.g. clear mobility edges (Wanget al2020Phys. Rev. Lett.125196604). We generalize this mosaic quasiperiodic model to a two-dimensional version, and numerically investigate its localization properties: the phase diagram from the fractal dimension of the wavefunction, the statistical and scaling properties of the conductance. Compared with disordered systems, our model shares many common features but also exhibits some different characteristics in the same dimensionality and the same universality class. For example, the sharp peak atg∼0of the critical distribution and the largeglimit of the universal scaling functionβresemble those behaviors of three-dimensional disordered systems.

Localization and delocalization properties in quasi-periodically-driven one-dimensional disordered systems

Localization and delocalization of quantum diffusion in a time-continuous one-dimensional Anderson model perturbed by the quasiperiodic harmonic oscillations of M colors is investigated systematically, which has been partly reported by a preliminary Letter [H. S. Yamada and K. S. Ikeda, Phys. Rev. E 103, L040202 (2021)2470-004510.1103/PhysRevE.103.L040202]. We investigate in detail the localization-delocalization characteristics of the model with respect to three parameters: the disorder strength W, the perturbation strength ε, and the number of colors, M, which plays the similar role of spatial dimension. In particular, attention is focused on the presence of localization-delocalization transition (LDT) and its critical properties. For M≥3 the LDT exists and a normal diffusion is recovered above a critical strength ε, and the characteristics of diffusion dynamics mimic the diffusion process predicted for the stochastically perturbed Anderson model even though M is not large. These results are compared with the results of discrete-time quantum maps, i.e., the Anderson map and the standard map. Further, the features of delocalized dynamics are discussed in comparison with a limit model which has no static disordered part.

Binary Bose-Einstein condensates in a disordered time-dependent potential

We study the non-equilibrium evolution of binary Bose-Einstein condensates in the presence of a weak random potential with Gaussian correlation function using the time-dependent perturbation theory. We apply this theory to construct a closed set of equations that highlight the role of the spectacular interplay between the disorder and the interspecies interactions in the time evolution of the density induced by disorder in each component. It is found that this latter increases with time favoring localization of both species. The time scale at which the theory remains valid depends on the respective system parameters. We show analytically and numerically that such a system supports a steady state that periodically changing during its time propagation. The obtained dynamical corrections indicate that disorder may transform the system into a stationary out-of-equilibrium states. Understanding this time evolution is pivotal for the realization of Floquet condensates.

Protected quantum coherence by gain and loss in a noisy quantum kicked rotor

We study the effects of non-Hermiticity on quantum coherence via a noisy quantum kicked rotor (NQKR). The random noise comes from the fluctuations in kick amplitude at each time. The non-Hermitian driving indicates the imaginary kicking potential, representing the environment-induced atom gain and loss. In the absence of gain and loss, the random noise destroys quantum coherence manifesting dynamical localization, which leads to classical diffusion. Interestingly, in the presence of non-Hermitian kicking potential, the occurrence of dynamical localization is highly sensitive to the gain and loss, manifesting the restoration of quantum coherence. Using the inverse participation ratio arguments, we numerically obtain a phase diagram of the classical diffusion and dynamical localization on the parameter plane of noise amplitude and non-Hermitian driving strength. With the help of analysis on the corresponding quasieigenstates, we achieve insight into dynamical localization, and uncover that the origin of the localization is interference between multiple quasi-eigenstates of the quantum kicked rotor. We further propose an experimental scheme to realize the NQKR in a dissipative cold atomic gas, which paves the way for future experimental investigation of an NQKR and its anomalous non-Hermitian properties.

Interactions and Mobility Edges: Observing the Generalized Aubry-André Model

Using synthetic lattices of laser-coupled atomic momentum modes, we experimentally realize a recently proposed family of nearest-neighbor tight-binding models having quasiperiodic site energy modulation that host an exact mobility edge protected by a duality symmetry. These one-dimensional tight-binding models can be viewed as a generalization of the well-known Aubry-André model, with an energy-dependent self-duality condition that constitutes an analytical mobility edge relation. By adiabatically preparing low and high energy eigenstates of this model system and performing microscopic measurements of their participation ratio, we track the evolution of the mobility edge as the energy-dependent density of states is modified by the model's tuning parameter. Our results show strong deviations from single-particle predictions, consistent with attractive interactions causing both enhanced localization of the lowest energy state due to self-trapping and inhibited localization of high energy states due to screening. This study paves the way for quantitative studies of interaction effects on self-duality induced mobility edges.

Observation of two-dimensional Anderson localisation of ultracold atoms

Anderson localisation -the inhibition of wave propagation in disordered media- is a surprising interference phenomenon which is particularly intriguing in two-dimensional (2D) systems. While an ideal, non-interacting 2D system of infinite size is always localised, the localisation length-scale may be too large to be unambiguously observed in an experiment. In this sense, 2D is a marginal dimension between one-dimension, where all states are strongly localised, and three-dimensions, where a well-defined phase transition between localisation and delocalisation exists as the energy is increased. Here, we report the results of an experiment measuring the 2D transport of ultracold atoms between two reservoirs, which are connected by a channel containing pointlike disorder. The design overcomes many of the technical challenges that have hampered observation of localisation in previous works. We experimentally observe exponential localisation in a 2D ultracold atom system.

Critical phenomena of dynamical delocalization in quantum maps: Standard map and Anderson map

Following the paper exploring the Anderson localization of monochromatically perturbed kicked quantum maps [Phys. Rev. E 97, 012210 (2018)2470-004510.1103/PhysRevE.97.012210], the delocalization-localization transition phenomena in polychromatically perturbed quantum maps (QM) is investigated focusing particularly on the dependency of critical phenomena on the number M of the harmonic perturbations, where M+1=d corresponds to the spatial dimension of the ordinary disordered lattice. The standard map and the Anderson map are treated and compared. As the basis of analysis, we apply the self-consistent theory (SCT) of the localization for our systems, taking a plausible hypothesis on the mean-free-path parameter which worked successfully in the analyses of the monochromatically perturbed QMs. We compare in detail the numerical results with the predictions of the SCT by largely increasing M. The numerically obtained index of critical subdiffusion t^{α} (t:time) agrees well with the prediction of one-parameter scaling theory α=2/(M+1), but the numerically obtained critical exponent of localization length significantly deviates from the SCT prediction. Deviation from the SCT prediction is drastic for the critical perturbation strength of the transition: If M is fixed, then the SCT presents plausible prediction for the parameter dependence of the critical value, but its value is 1/(M-1) times smaller than the SCT prediction, which implies existence of a strong cooperativity of the harmonic perturbations with the main mode.

Impact of Lattice Vibrations on the Dynamics of a Spinor Atom-Optics Kicked Rotor

We investigate the effect of amplitude and phase noise on the dynamics of a discrete-time quantum walk and its related evolution. Our findings underline the robustness of the motion with respect to these noise sources, and can explain the stability of quantum walks that has recently been observed experimentally. This opens the road to measure topological properties of an atom-optics double kicked rotor with an additional internal spin degree of freedom.

Localization Transition for Light Scattering by Cold Atoms in an External Magnetic Field

We establish a localization phase diagram for light in a random three-dimensional (3D) ensemble of motionless two-level atoms with a threefold degenerate upper level, in a strong static magnetic field. Localized modes appear in a narrow spectral band when the number density of atoms ρ exceeds a critical value ρ_{c}≃0.1k_{0}^{3}, where k_{0} is the wave number of light in the free space. A critical exponent of the localization transition taking place upon varying the frequency of light at a constant ρ>ρ_{c} is estimated to be ν=1.57±0.07. This classifies the transition as an Anderson localization transition of 3D orthogonal universality class.

Suppression of Heating in Quantum Spin Clusters under Periodic Driving as a Dynamic Localization Effect

We investigate numerically and analytically the heating process in ergodic clusters of interacting spins 1/2 subjected to periodic pulses of an external magnetic field. Our findings indicate that there is a threshold for the pulse strength below which the heating is suppressed. This threshold decreases with the increase of the cluster size, approaching zero in the thermodynamic limit, yet it should be observable in clusters with fairly large Hilbert spaces. We obtain the above threshold quantitatively as a condition for the breakdown of the golden rule in the second-order perturbation theory. It is caused by the phenomenon of dynamic localization.

Scaling properties of dynamical localization in monochromatically perturbed quantum maps: Standard map and Anderson map

Dynamical localization phenomena of monochromatically perturbed standard map (SM) and Anderson map (AM), which are both identified with a two-dimensional disordered system under suitable conditions, are investigated by the numerical wave-packet propagation. Some phenomenological formula of the dynamical localization length valid for wide range of control parameters are proposed for both SM and AM. For SM the formula completely agree with the experimental formula, and for AM the presence of a new regime of localization is confirmed. These formula can be derived by the self-consistent mean-field theory of Anderson localization on the basis of a new hypothesis for the cut-off length. Transient diffusion in the large limit of the localization length is also discussed.

Return to the Origin as a Probe of Atomic Phase Coherence

We report on the observation of the coherent enhancement of the return probability ["enhanced return to the origin" (ERO)] in a periodically kicked cold-atom gas. By submitting an atomic wave packet to a pulsed, periodically shifted, laser standing wave, we induce an oscillation of ERO in time that is explained in terms of a periodic, reversible dephasing in the weak-localization interference sequences responsible for ERO. Monitoring the temporal decay of ERO, we exploit its quantum-coherent nature to quantify the decoherence rate of the atomic system.

Anderson Transition of Cold Atoms with Synthetic Spin-Orbit Coupling in Two-Dimensional Speckle Potentials

We investigate the metal-insulator transition occurring in two-dimensional (2D) systems of noninteracting atoms in the presence of artificial spin-orbit interactions and a spatially correlated disorder generated by laser speckles. Based on a high order discretization scheme, we calculate the precise position of the mobility edge and verify that the transition belongs to the symplectic universality class. We show that the mobility edge depends strongly on the mixing angle between Rashba and Dresselhaus spin-orbit couplings. For equal couplings a non-power-law divergence is found, signaling the crossing to the orthogonal class, where such a 2D transition is forbidden.

Experimental Observation of Two-Dimensional Anderson Localization with the Atomic Kicked Rotor

Dimension 2 is expected to be the lower critical dimension for Anderson localization in a time-reversal-invariant disordered quantum system. Using an atomic quasiperiodic kicked rotor-equivalent to a two-dimensional Anderson-like model-we experimentally study Anderson localization in dimension 2 and we observe localized wave function dynamics. We also show that the localization length depends exponentially on the disorder strength and anisotropy and is in quantitative agreement with the predictions of the self-consistent theory for the 2D Anderson localization.

Critical phenomena of dynamical delocalization in a quantum Anderson map

Using a quantum map version of the one-dimensional Anderson model, the localization-delocalization transition of quantum diffusion induced by coherent dynamical perturbation is investigated in comparison with the quantum standard map. Existence of critical phenomena, which depends on the number of frequency component M, is demonstrated. Diffusion exponents agree with theoretical prediction for the transition, but the critical exponent of the localization length deviates from it with increase in the M. The critical power ε(c) of the normalized perturbation at the transition point remarkably decreases as ε(c)∼(M-1)(-1).

Mobility edge for cold atoms in laser speckle potentials

Using the transfer-matrix method, we numerically compute the precise position of the mobility edge of atoms exposed to a laser speckle potential and study its dependence versus the disorder strength and correlation function. Our results deviate significantly from previous theoretical estimates using an approximate, self-consistent approach of localization. In particular, we find that the position of the mobility edge in blue-detuned speckles is much lower than in the red-detuned counterpart, pointing out the crucial role played by the asymmetric on-site distribution of speckle patterns.

Interaction effects on dynamical localization in driven helium

Dynamical localization prevents driven atomic systems from fast fragmentation by hampering the excitation process. We present numerical simulations within a collinear model of microwave-driven helium Rydberg atoms and prove that dynamical localization survives the impact of electron-electron interaction, even for doubly excited states in the presence of fast autoionization. We conclude that the effect of electron-electron repulsion on localization can be described by an appropriate rescaling of the atomic level density and of the external field with the strength of the interaction.

How nonlinear interactions challenge the three-dimensional Anderson transition

In disordered systems, our present understanding of the Anderson transition is hampered by the possible presence of interactions between particles. We demonstrate that in boson gases, even weak interactions deeply alter the very nature of the Anderson transition. While there still exists a critical point in the system, below that point a novel phase appears, displaying a new critical exponent, subdiffusive transport, and a breakdown of the one-parameter scaling description of Anderson localization.

Exponential quantum spreading in a class of kicked rotor systems near high-order resonances

Long-lasting exponential quantum spreading was recently found in a simple but very rich dynamical model, namely, an on-resonance double-kicked rotor model [J. Wang, I. Guarneri, G. Casati, and J. B. Gong, Phys. Rev. Lett. 107, 234104 (2011)]. The underlying mechanism, unrelated to the chaotic motion in the classical limit but resting on quasi-integrable motion in a pseudoclassical limit, is identified for one special case. By presenting a detailed study of the same model, this work offers a framework to explain long-lasting exponential quantum spreading under much more general conditions. In particular, we adopt the so-called "spinor" representation to treat the kicked-rotor dynamics under high-order resonance conditions and then exploit the Born-Oppenheimer approximation to understand the dynamical evolution. It is found that the existence of a flat band (or an effectively flat band) is one important feature behind why and how the exponential dynamics emerges. It is also found that a quantitative prediction of the exponential spreading rate based on an interesting and simple pseudoclassical map may be inaccurate. In addition to general interests regarding the question of how exponential behavior in quantum systems may persist for a long time scale, our results should motivate further studies toward a better understanding of high-order resonance behavior in δ-kicked quantum systems.

Kicked-Harper model versus on-resonance double-kicked rotor model: from spectral difference to topological equivalence

Recent studies have established that, in addition to the well-known kicked-Harper model (KHM), an on-resonance double-kicked rotor (ORDKR) model also has Hofstadter's butterfly Floquet spectrum, with strong resemblance to the standard Hofstadter spectrum that is a paradigm in studies of the integer quantum Hall effect. Earlier it was shown that the quasienergy spectra of these two dynamical models (i) can exactly overlap with each other if an effective Planck constant takes irrational multiples of 2π and (ii) will be different if the same parameter takes rational multiples of 2π. This work makes detailed comparisons between these two models, with an effective Planck constant given by 2πM/N, where M and N are coprime and odd integers. It is found that the ORDKR spectrum (with two periodic kicking sequences having the same kick strength) has one flat band and N-1 nonflat bands with the largest bandwidth decaying in a power law as ~K(N+2), where K is a kick strength parameter. The existence of a flat band is strictly proven and the power-law scaling, numerically checked for a number of cases, is also analytically proven for a three-band case. By contrast, the KHM does not have any flat band and its bandwidths scale linearly with K. This is shown to result in dramatic differences in dynamical behavior, such as transient (but extremely long) dynamical localization in ORDKR, which is absent in the KHM. Finally, we show that despite these differences, there exist simple extensions of the KHM and ORDKR model (upon introducing an additional periodic phase parameter) such that the resulting extended KHM and ORDKR model are actually topologically equivalent, i.e., they yield exactly the same Floquet-band Chern numbers and display topological phase transitions at the same kick strengths. A theoretical derivation of this topological equivalence is provided. These results are also of interest to our current understanding of quantum-classical correspondence considering that the KHM and ORDKR model have exactly the same classical limit after a simple canonical transformation.

Localization-delocalization transition of the instantaneous normal modes of liquid water

Despite the fact that the localization-delocalization transition (LDT) widely exists in wave systems, quantitative studies on its critical and multifractal properties are mainly focused on solids. In this work, these properties are investigated on the vibrational motions of liquid water. Simulations of up to 18000 molecules on the flexible extended simple point charge water model provide nearly 10(6) instantaneous normal modes. They are shown to undergo an LDT close to the translational transition and exhibit multifractal fluctuations while approaching the LDT. In combination with finite-size scaling, multifractal analysis predicts the critical frequency Im(ω(c))≈-131.6 cm(-1) for unstable modes at room temperature. The estimated critical exponent ν≈1.60 is close to those of other calculated systems in the same Wigner-Dyson class. At the LDT, the fractal spectrum f(α) and the most probable local vibrational intensity α(mc)≈4.04 coincide with those of the Anderson model, which might be additional universal properties of LDT in more general wave systems. The results extend the validity of the multifractal scaling approach beyond Andersonian systems to a Hessian system.

Quantized adiabatic transport in momentum space

Though topological aspects of energy bands are known to play a key role in quantum transport in solid-state systems, the implications of Floquet band topology for transport in momentum space (i.e., acceleration) have not been explored so far. Using a ratchet accelerator model inspired by existing cold-atom experiments, here we characterize a class of extended Floquet bands of one-dimensional driven quantum systems by Chern numbers, reveal topological phase transitions therein, and theoretically predict the quantization of adiabatic transport in momentum space. Numerical results confirm our theory and indicate the feasibility of experimental studies.

Interacting ultracold bosons in disordered lattices: sensitivity of the dynamics to the initial state

We study the dynamics of a nonlinear one-dimensional disordered system obtained by coupling the Anderson model with the Gross-Pitaevskii equation. We introduce a single quantity globally characterizing the localization of the system. This quantity obeys a scaling law with respect to the width of the initial state, which can be used to characterize the dynamics independently of the initial state.