Atoms in double-delta-kicked periodic potentials: chaos with long-range correlations

Floquet Second-Order Topological Phases in Momentum Space

Higher-order topological phases (HOTPs) are characterized by symmetry-protected bound states at the corners or hinges of the system. In this work, we reveal a momentum-space counterpart of HOTPs in time-periodic driven systems, which are demonstrated in a two-dimensional extension of the quantum double-kicked rotor. The found Floquet HOTPs are protected by chiral symmetry and characterized by a pair of topological invariants, which could take arbitrarily large integer values with the increase of kicking strengths. These topological numbers are shown to be measurable from the chiral dynamics of wave packets. Under open boundary conditions, multiple quartets Floquet corner modes with zero and π quasienergies emerge in the system and coexist with delocalized bulk states at the same quasienergies, forming second-order Floquet topological bound states in the continuum. The number of these corner modes is further counted by the bulk topological invariants according to the relation of bulk-corner correspondence. Our findings thus extend the study of HOTPs to momentum-space lattices and further uncover the richness of HOTPs and corner-localized bound states in continuum in Floquet systems.

Nonmonotonic diffusion rates in an atom-optics Lévy kicked rotor

The dynamics of chaotic Hamiltonian systems such as the kicked rotor continues to guide our understanding of transport and localization processes. The localized states of the quantum kicked rotor decay due to decoherence effects if subjected to noise. The associated quantum diffusion increases monotonically as a function of a parameter characterizing the noise distribution. In this Rapid Communication, for the atom-optics Lévy kicked rotor, the quantum diffusion displays nonmonotonic behavior as a function of a parameter characterizing the Lévy distribution. The optimal diffusion rates are experimentally obtained using an ultracold cloud of rubidium atoms in a pulsed optical lattice. The parameters for optimal diffusion rates are obtained analytically and show a good agreement with our experimental and numerical results. The nonmonotonicity is shown to be a quantum effect that vanishes in the classical limit.

Floquet states of a kicked particle in a singular potential: Exponential and power-law profiles

It is well known that, in the chaotic regime, all the Floquet states of kicked rotor system display an exponential profile resulting from dynamical localization. If the kicked rotor is placed in an additional stationary infinite potential well, its Floquet states display power-law profile. It has also been suggested in general that the Floquet states of periodically kicked systems with singularities in the potential would have power-law profile. In this work, we study the Floquet states of a kicked particle in finite potential barrier. By varying the height of finite potential barrier, the nature of transition in the Floquet state from exponential to power-law decay profile is studied. We map this system to a tight-binding model and show that the nature of decay profile depends on energy band spanned by the Floquet states (in unperturbed basis) relative to the potential height. This property can also be inferred from the statistics of Floquet eigenvalues and eigenvectors. This leads to an unusual scenario in which the level spacing distribution, as a window in to the spectral correlations, is not a unique characteristic for the entire system.

Quantum properties of double kicked systems with classical translational invariance in momentum

Double kicked rotors (DKRs) appear to be the simplest nonintegrable Hamiltonian systems featuring classical translational symmetry in phase space (i.e., in angular momentum) for an infinite set of values (the rational ones) of a parameter η. The experimental realization of quantum DKRs by atom-optics methods motivates the study of the double kicked particle (DKP). The latter reduces, at any fixed value of the conserved quasimomentum βℏ, to a generalized DKR, the "β-DKR." We determine general quantum properties of β-DKRs and DKPs for arbitrary rational η. The quasienergy problem of β-DKRs is shown to be equivalent to the energy eigenvalue problem of a finite strip of coupled lattice chains. Exact connections are then obtained between quasienergy spectra of β-DKRs for all β in a generically infinite set. The general conditions of quantum resonance for β-DKRs are shown to be the simultaneous rationality of η,β, and a scaled Planck constant ℏ(S). For rational ℏ(S) and generic values of β, the quasienergy spectrum is found to have a staggered-ladder structure. Other spectral structures, resembling Hofstadter butterflies, are also found. Finally, we show the existence of particular DKP wave-packets whose quantum dynamics is free, i.e., the evolution frequencies of expectation values in these wave-packets are independent of the nonintegrability. All the results for rational ℏ(S) exhibit unique number-theoretical features involving η,ℏ(S), and β.

Classical and quantum transport in one-dimensional periodically kicked systems

This paper is a brief review of classical and quantum transport phenomena, as well as related spectral properties, exhibited by one-dimensional periodically kicked systems. Two representative and fundamentally different classes of systems will be considered: those satisfying the classical Kolmogorov−Arnol’d−Moser scenario and those that do not. The experimental realization of some of these systems using atom-optics methods will be mentioned.

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.

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.

Long-lasting exponential spreading in periodically driven quantum systems

Using a dynamical model relevant to cold-atom experiments, we show that long-lasting exponential spreading of wave packets in momentum space is possible. Numerical results are explained via a pseudoclassical map, both qualitatively and quantitatively. Possible applications of our findings are also briefly discussed.

Generating a fractal butterfly Floquet spectrum in a class of driven SU(2) systems: eigenstate statistics

The Floquet spectra of a class of driven SU(2) systems have been shown to display butterfly patterns with multifractal properties. The implication of such critical spectral behavior for the Floquet eigenstate statistics is studied in this work. Following the methodologies for understanding the fractal behavior of energy eigenstates of time-independent systems on the Anderson transition point, we analyze the distribution profile, the mean value, and the variance of the logarithm of the inverse participation ratio of the Floquet eigenstates associated with multifractal Floquet spectra. The results show that the Floquet eigenstates also display fractal behavior but with features markedly different from those in time-independent Anderson-transition models. This motivated us to propose random unitary matrix ensemble, called "power-law random banded unitary matrix" ensemble, to illuminate the Floquet eigenstate statistics of critical driven systems. The results based on the proposed random matrix model are consistent with those obtained from our dynamical examples with or without time-reversal symmetry.

Doubly excited ferromagnetic spin chain as a pair of coupled kicked rotors

We show that the dynamics of a doubly excited Heisenberg spin chain, subject to short pulses from a parabolic magnetic field may be analyzed as a pair of quantum kicked rotors. By focusing on the two-magnon dynamics in the kicked XXZ model we investigate how the anisotropy parameter--which controls the strength of the magnon-magnon interaction--changes the nature of the coupling between the two "image" coupled kicked rotors. We investigate quantum state transfer possibilities and show that one may control whether the spin excitations are transmitted together, or separate from each other.

Quantum ratchet accelerator without a bichromatic lattice potential

In a quantum ratchet accelerator system, a linearly increasing directed current can be dynamically generated without using a biased field. Generic quantum ratchet acceleration with full classical chaos [J. B. Gong and P. Brumer, Phys. Rev. Lett. 97, 240602 (2006)] constitutes a new element of quantum chaos and an interesting violation of a sum rule of classical ratchet transport. Here we propose a simple quantum ratchet accelerator model that can also generate linearly increasing quantum current with full classical chaos. This model does not require a bichromatic lattice potential. It is based on a variant of an on-resonance kicked-rotor system, periodically kicked by two optical lattice potentials of the same lattice constant, but with unequal amplitudes and a fixed phase shift between them. The dependence of the ratchet current acceleration rate on the system parameters is studied in detail. The cold-atom version of our quantum ratchet accelerator model should be realizable by introducing slight modifications to current cold-atom experiments.

Fractional variant Planck's over 2pi scaling for quantum kicked rotors without Cantori

Previous studies of quantum delta-kicked rotors have found momentum probability distributions with a typical width (localization length L) characterized by fractional variant Planck's over 2pi scaling; i.e., L approximately variant Planck's over 2pi;{2/3} in regimes and phase-space regions close to "golden-ratio" cantori. In contrast, in typical chaotic regimes, the scaling is integer, L approximately variant Planck's over 2pi;{-1}. Here we consider a generic variant of the kicked rotor, the random-pair-kicked particle, obtained by randomizing the phases every second kick; it has no Kol'mogorov-Arnol'd-Moser mixed-phase-space structures, such as golden-ratio cantori, at all. Our unexpected finding is that, over comparable phase-space regions, it also has fractional scaling, but L approximately variant Planck's over 2pi;{-2/3}. A semiclassical analysis indicates that the variant Planck's over 2pi;{2/3} scaling here is of quantum origin and is not a signature of classical cantori.

Directed deterministic classical transport: symmetry breaking and beyond

We consider transport properties of a double delta -kicked system, in a regime where all the symmetries (spatial and temporal) that could prevent directed transport are removed. We analytically investigate the (nontrivial) behavior of the classical current and diffusion properties and show that the results are in good agreement with numerical computations. The role of dissipation for a meaningful classical ratchet behavior is also discussed.

Classical momentum diffusion in double-delta-kicked particles

We investigate the classical chaotic diffusion of atoms subjected to pairs of closely spaced pulses ("kicks") from standing waves of light (the 2delta-KP ). Recent experimental studies with cold atoms implied an underlying classical diffusion of a type very different from the well-known paradigm of Hamiltonian chaos, the standard map. The kicks in each pair are separated by a small time interval E<<1, which together with the kick strength K, characterizes the transport. Phase space for the 2delta-KP is partitioned into momentum "cells" partially separated by momentum-trapping regions where diffusion is slow. We present here an analytical derivation of the classical diffusion for a 2delta-KP including all important correlations which were used to analyze the experimental data. We find an asymptotic (t-->infinity) regime of "hindered" diffusion: while for the standard map the diffusion rate, for K>>1 , D approximately K(2)/2[1-2J(2)(K)...] oscillates about the uncorrelated rate D(0)=K(2)/2, we find analytically, that the 2delta-KP can equal, but never diffuses faster than, a random walk rate. We argue this is due to the destruction of the important classical "accelerator modes" of the standard map. We analyze the experimental regime 0.1 less or approximately KE less or approximately 1 , where quantum localization lengths L approximately Planck's (-0.75) are affected by fractal cell boundaries. We find an approximate asymptotic diffusion rate D proportional to K(3)E, in correspondence to a D proportional to K(3) regime in the standard map associated with the "golden-ratio" cantori.

Theory of 2 delta-kicked quantum rotors

We examine the quantum dynamics of cold atoms subjected to pairs of closely spaced delta kicks from standing waves of light and find behavior quite unlike the well-studied quantum kicked rotor (QKR). We show that the quantum phase space has a periodic, cellular structure arising from a unitary matrix with oscillating bandwidth. The corresponding eigenstates are exponentially localized, but scale with a fractional power L is less similar to h(-0.75), in contrast to the QKR for which L is less similar to h(-1). The effect of intercell (and intracell) transport is investigated by studying the spectral fluctuations with both periodic as well as "open" boundary conditions.

Entanglement and dynamics of spin chains in periodically pulsed magnetic fields: accelerator modes

We study the dynamics of a single excitation in a Heisenberg spin-chain subjected to a sequence of periodic pulses from an external, parabolic, magnetic field. We show that, for experimentally reasonable parameters, a pair of counterpropagating coherent states is ejected from the center of the chain. We find an illuminating correspondence with the quantum time evolution of the well-known paradigm of quantum chaos, the quantum kicked rotor. From this we can analyze the entanglement production and interpret the ejected coherent states as a manifestation of the so-called "accelerator modes" of a classically chaotic system.

Localization-delocalization transition in a system of quantum kicked rotors

The quantum dynamics of atoms subjected to pairs of closely spaced delta kicks from optical potentials are shown to be quite different from the well-known paradigm of quantum chaos, the single delta-kick system. We find the unitary matrix has a new oscillating band structure corresponding to a cellular structure of phase space and observe a spectral signature of a localization-delocalization transition from one cell to several. We find that the eigenstates have localization lengths which scale with a fractional power L approximately h(-0.75) and obtain a regime of near-linear spectral variances which approximate the "critical statistics" relation summation2(L) approximately or equal to chi(L) approximately 1/2 (1-nu)L, where nu approximately 0.75 is related to the fractal classical phase-space structure. The origin of the nu approximately 0.75 exponent is analyzed.