Quantum Dynamics of Periodic and Limit-Periodic Jacobi and Block Jacobi Matrices with Applications to Some Quantum Many Body Problems

We investigate quantum dynamics with the underlying Hamiltonian being a Jacobi or a block Jacobi matrix with the diagonal and the off-diagonal terms modulated by a periodic or a limit-periodic sequence. In particular, we investigate the transport exponents. In the periodic case we demonstrate ballistic transport, while in the limit-periodic case we discuss various phenomena, such as quasi-ballistic transport and weak dynamical localization. We also present applications to some quantum many body problems. In particular, we establish for the anisotropic XY chain on \({\mathbb{Z}}\) with periodic parameters an explicit strictly positive lower bound for the Lieb–Robinson velocity.     

Decoherence and disorder in quantum walks: from ballistic spread to localization

We investigate the impact of decoherence and static disorder on the dynamics of quantum particles moving in a periodic lattice. Our experiment relies on the photonic implementation of a one-dimensional quantum walk. The pure quantum evolution is characterized by a ballistic spread of a photon's wave packet along 28 steps. By applying controlled time-dependent operations we simulate three different environmental influences on the system, resulting in a fast ballistic spread, a diffusive classical walk, and the first Anderson localization in a discrete quantum walk architecture.     

Localization and recurrence of a quantum walk in a periodic potential on a line

We present a numerical study of a model of quantum walk in a periodic potential on a line. We take the simple view that different potentials have different affects on the way in which the coin state of the walker is changed. For simplicity and definiteness, we assume that the walker＇s coin state is unaffected at sites without the potential, and rotated in an unbiased way according to the Hadamard matrix at sites with the potential. This is the simplest and most natural model of a quantum walk in a periodic potential with two coins. Six generic cases of such quantum walks are studied numerically. It is found that, of the six cases, four cases display significant localization effect where the walker is confined in the neighborhood of the origin for a sufficiently long time. Associated with such a localization effect is the recurrence of the probability of the walker returning to the neighborhood of the origin.     

Manifestation of scarring in a driven system with wave chaos

We consider wave propagation in a model of a deep ocean acoustic wave guide with a periodic range dependence. It is assumed that the wave field is governed by the parabolic equation. Formally the mathematical model of the wave guide coincides with that of a quantum system with time-dependent Hamiltonian. From the analysis of Floquet modes of the wave guide it is shown that there exists a "scarring" effect similar to that observed in quantum systems. It turns out that the segments of an unstable periodic ray trajectory may be distinguished in the spatial distribution of the wave field intensity at a finite wavelength. Besides the scarring effect, it is found that the so-called "stable islands" in the phase space of ray dynamics reveal themselves in the coarse-grained Wigner functions of the Floquet modes.     

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 on Quantum Graphs with Random Vertex Couplings

We consider Schrödinger operators on a class of periodic quantum graphs with randomly distributed Kirchhoff coupling constants at all vertices. We obtain necessary conditions for localization on quantum graphs in terms of finite volume criteria for some energy-dependent discrete Hamiltonians. These conditions hold in the strong disorder limit and at the spectral edges.     

Extension of the GRWP model to translation symmetry breaking processes

This work is concerned with the quantum measurement model proposed by Ghirardi-Rimini-Weber [1] (GRW) in its version presented by Pearle [2], as a stochastic modification of the Schrödinger evolution. These authors conjectured a spontaneous random hitting process which is described by a localization operator, a guassian function acting on microscopic particles which carries two free parameters; the frequency and the localization width ^–1/2. These two parameters can be understood as new constants of nature if the spontaneous localization is considered as a fundamental physical process. We can extend the model by making new conjectures about the possible values of these parameters. In particular, assuming the localization width of order of the atomic distance, the breakdown of the translational symmetry demands that the hitting process must be applied to a non-isolated quantum system. In this way we show that in the present approach the suppression of the coherence of quantum states is mainly due to the dissipation/fluctuation process and not to the random hitting process introduced by the GRW model.     

Low-energy theory of disordered graphene

At low values of external doping, graphene displays a wealth of unconventional transport properties. Perhaps most strikingly, it supports a robust "metallic" regime, with universal conductance of the order of the conductance quantum. We here apply a combination of mean-field and bosonization methods to explore the large scale transport properties of the system. We find that, irrespective of the doping level, disordered graphene is subject to the common mechanisms of Anderson localization. However, at low doping a number of renormalization mechanisms conspire to protect the conductivity of the system, to an extend that strong localization may not be seen even at temperatures much smaller than those underlying present experimental work.     

Periodic locking of chaos in semiconductor lasers with optical feedback

We show how a chaotic system can be locked to emit a periodic waveform belonging to its chaotic attractor. We numerically demonstrate our idea in a system composed of a semiconductor laser driven to chaos by optical feedback from an external cavity. The clue is the injection of an appropriate periodic signal that modulates the phase and amplitude of the intra-cavity radiation, a chaotic analogy of conventional mode-locking. The result is a time process that manifests a chaotic signature embedded in a long-scale periodic train.     

Dynamic localization and the Coulomb blockade in quantum dots under ac pumping

We study conductance through a quantum dot under Coulomb blockade conditions in the presence of an external periodic perturbation. The stationary state is determined by the balance between the heating of the dot electrons by the perturbation and cooling by electron exchange with the cold contacts. We show that the Coulomb blockade peak can have a peculiar shape if heating is affected by dynamic localization, which can be an experimental signature of this effect.     

Local spectral density for a periodically driven system of coupled quantum states with strong imperfection in the unperturbed energies

A random matrix theory approach is applied in order to analyze the localization properties of local spectral density for a generic system of coupled quantum states with strong imperfection in the unperturbed energy levels. The system is excited by an external periodic field, the temporal profile of which is close to monochromatic. The shape of the local spectral density is shown to be well described by the contour obtained from a relevant model of a periodically driven two-state system with irreversible losses to an external thermal bath. The shape width and the inverse participation ratio are determined as functions both of the Rabi frequency and of parameters specifying the localization effect for the generic system considered in the absence of an external field.     

Localization in a quantum spin Hall system

The localization problem of electronic states in a two-dimensional quantum spin Hall system (that is, a symplectic ensemble with topological term) is studied by the transfer matrix method. The phase diagram in the plane of energy and disorder strength is exposed, and demonstrates "levitation" and "pair annihilation" of the domains of extended states analogous to that of the integer quantum Hall system. The critical exponent nu for the divergence of the localization length is estimated as nu congruent with 1.6, which is distinct from both exponents pertaining to the conventional symplectic and the unitary quantum Hall systems. Our analysis strongly suggests a different universality class related to the topology of the pertinent system.     

Dynamical Suppression of Decoherence in Two-Qubit Quantum Memory

In this paper, we have detailedly studied the dynamical suppression of the phase damping for the two-qubit quantum memory of Ising model by the quantum “bang-bang” technique. We find the sequence of periodic radio-frequency pulses repetitively to flip the state of the two-qubit system and quantitatively find that these pulses can be used to effectively suppress the phase damping decoherence of the quantum memory and freeze the system state into its initial state. The general sequence of periodic radio-frequency pulses to suppress the phase damping of multi-qubit of Ising model is also given.     

Semiclassical theory of transport in antidot lattices

Motivated by a recent experiment by Weiss et al. [Phys. Rev. Lett. 70, 4118 (1993)], we present a detailed study of quantum transport in large antidot arrays whose classical dynamics is chaotic. We calculate the longitudinal and Hall conductivities semiclassically starting from the Kubo formula. The leading contribution reproduces the classical conductivity. In addition, we find oscillatory quantum corrections to the classical conductivity which are given in terms of the periodic orbits of the system. These periodic-orbit contributions provide a consistent explanation of the quantum oscillations in the magnetoconductivity observed by Weiss et al. We find that the phase of the oscillations with Fermi energy and magnetic field is given by the classical action of the periodic orbit. The amplitude is determined by the stability and the velocity correlations of the orbit. The amplitude also decreases exponentially with temperature on the scale of the inverse orbit traversal time/T . The Zeeman splitting leads to beating of the amplitude with magnetic field. We also present an analogous semiclassical derivation of Shubnikov-de Haas oscillations where the corresponding classical motion is integrable. We show that the quantum oscillations in antidot lattices and the Shubnikov-de Haas oscillations are closely related. Observation of both effects requires that the elastic and inelastic scattering lengths be larger than the lengths of the relevant periodic orbits. The amplitude of the quantum oscillations in antidot lattices is of a higher power in Planck's constant and hence smaller than that of Shubnikov-de Haas oscillations. In this sense, the quantum oscillations in the conductivity are a sensitive probe of chaos.This paper is dedicated to Prof. H. Wagner on the occasion of his 60th birthday     

Soliton Trains Induced by Adaptive Shaping with Periodic Traps in Four-Level Ultracold Atom Systems

It is well known that an optical trap can be imprinted by a light field in an ultracold-atom system embedded in an optical cavity,and driven by three different coherent fields.Of the three fields coexisting in the optical cavity there is an intense control field that induces a giant Kerr nonlinearity via electromagnetically-induced transparency,and another field that creates a periodic optical grating of strength proportional to the square of the associated Rabi frequency.In this work elliptic-soliton solutions to the nonlinear equation governing the propagation of the probe field are considered,with emphasis on the possible generation of optical soliton trains forming a discrete spectrum with well defined quantum numbers.The problem is treated assuming two distinct types of periodic optical gratings and taking into account the negative and positive signs of detunings(detuning above or below resonance).Results predict that the competition between the self-phase and cross-phase modulation nonlinearities gives rise to a rich family of temporal soliton train modes characterized by distinct quantum numbers.     

Infinite chain of different deltas: A simple model for a quantum wire

We present the exact diagonalization of the Schr?dinger operator corresponding to a periodic potential with N deltas of different couplings, for arbitrary N. This basic structure can repeat itself an infinite number of times. Calculations of band structure can be performed with a high degree of accuracy for an infinite chain and of the correspondent eigenlevels in the case of a random chain. The main physical motivation is to modelate quantum wire band structure and the calculation of the associated density of states. These quantities show the fundamental properties we expect for periodic structures although for low energy the band gaps follow unpredictable patterns. In the case of random chains we find Anderson localization; we analize also the role of the eigenstates in the localization patterns and find clear signals of fractality in the conductance. In spite of the simplicity of the model many of the salient features expected in a quantum wire are well reproduced. Received 24 June 2002 Published online 29 November 2002     

二维Sinai台球系统的量子混沌研究

研究了二维Sinai台球系统的经典与量子的对应关系,运用定态展开法和Gutzwiller的周期轨道理论对Sinai台球系统的态密度经傅里叶变换得到的量子长度谱进行分析,并把量子长度谱中峰的位置与其所对应的经典体系的周期轨道长度做对比,发现两者之间存在很好的对应关系.观察到了一些量子态局域在短周期轨道附近形成量子scarred态或量子superscarred态.还研究了同心与非同心Sinai台球系统的能级最近邻间距分布,发现同心Sinai台球系统是近可积的,非同心Sinai台球系统在θ=3π/8下,随两中心间距离的增加,能级最近邻间距分布将由近可积向维格那分布过渡.     

Numerical finite size scaling approach to many-body localization

We develop a numerical technique to study Anderson localization in interacting electronic systems. The ground state of the disordered system is calculated with quantum Monte Carlo simulations while the localization properties are extracted from the "Thouless conductance" g, i.e., the curvature of the energy with respect to an Aharonov-Bohm flux. We apply our method to polarized electrons in a two-dimensional system of size L. We recover the well-known universal beta(g)=dlogg/dlogL one parameter scaling function without interaction. Upon switching on the interaction, we find that beta(g) is unchanged while the system flows toward the insulating limit. We conclude that polarized electrons in two dimensions stay in an insulating state in the presence of weak to moderate electron-electron correlations.     

Hybrid Quasicrystals,Transport and Localization in Products of Minimal Sets

We consider convex combinations of finite-valued almost periodic sequences (mainly substitution sequences) and put them as potentials of one-dimensional tight-binding models. We prove that these sequences are almost periodic. We call such combinations hybrid quasicrystals and these studies are related to the minimality, under the shift on both coordinates, of the product space of the respective (minimal) hulls. We observe a rich variety of behaviors on the quantum dynamical transport ranging from localization to transport. CRdO thanks the partial support by CNPq     

Many-body energy localization transition in periodically driven systems

According to the second law of thermodynamics the total entropy of a system is increased during almost any dynamical process. The positivity of the specific heat implies that the entropy increase is associated with heating. This is generally true both at the single particle level, like in the Fermi acceleration mechanism of charged particles reflected by magnetic mirrors, and for complex systems in everyday devices. Notable exceptions are known in noninteracting systems of particles moving in periodic potentials. Here the phenomenon of dynamical localization can prevent heating beyond certain threshold. The dynamical localization is known to occur both at classical (Fermi–Ulam model) and at quantum levels (kicked rotor). However, it was believed that driven ergodic systems will always heat without bound. Here, on the contrary, we report strong evidence of dynamical localization transition in both classical and quantum periodically driven ergodic systems in the thermodynamic limit. This phenomenon is reminiscent of many-body localization in energy space.