Quantum Kicked Rotor Research Articles

Quadratic Growth of Out-of-Time-Ordered Correlators in Quantum Kicked Rotor Model.

We investigate both theoretically and numerically the dynamics of out-of-time-ordered correlators (OTOCs) in quantum resonance conditions for a kicked rotor model. We employ various operators to construct OTOCs in order to thoroughly quantify their commutation relation at different times, therefore unveiling the process of quantum scrambling. With the help of quantum resonance condition, we have deduced the exact expressions of quantum states during both forward evolution and time reversal, which enables us to establish the laws governing OTOCs' time dependence. We find interestingly that the OTOCs of different types increase in a quadratic function of time, breaking the freezing of quantum scrambling induced by the dynamical localization under non-resonance condition. The underlying mechanism is discovered, and the possible applications in quantum entanglement are discussed.

Quasilocalization dynamics in a Fibonacci quantum rotor

We analyze the dynamics of a quantum kicked rotor (QKR) driven with a binary Fibonacci sequence of two distinct drive amplitudes. While the dynamics at low drive frequencies is found to be diffusive, a long-lived pre-ergodic regime emerges in the other limit. Further, the dynamics in this pre-ergodic regime can be associated with the onset of a dynamical quasi-localization, similar to the dynamical localization observed in a regular QKR. We establish that this peculiar behavior arises due to the presence of localized eigenstates of an approximately conserved effective Hamiltonian, which drives the evolution at Fibonacci instants. However, the effective Hamiltonian picture does not persist indefinitely and the dynamics eventually becomes ergodic after asymptotically long times.

Enhancement in breaking of time-reversal invariance in the quantum kicked rotor.

We study the breaking of time-reversal invariance (TRI) by the application of a magnetic field in the quantum kicked rotor (QKR), using Izrailev's finite-dimensional model. There is a continuous crossover from TRI to time-reversal noninvariance (TRNI) in the spectral and eigenvector fluctuations of the QKR. We show that the properties of this TRI to TRNI transition depend on α^{2}/N, where α is the chaos parameter of the QKR and N is the dimensionality of the evolution operator matrix. For α^{2}/N≳N, the transition coincides with that in random matrix theory. For α^{2}/N<N, the transition shows a marked deviation from random matrix theory. Further, the speed of this transition as a function of the magnetic field is significantly enhanced as α^{2}/N decreases.

Resonant Quantum Kicked Rotor as A Continuous-Time Quantum Walk

We analytically investigate the analogy between a standard continuous-time quantum walk in one dimension and the evolution of the quantum kicked rotor at quantum resonance conditions. We verify that the obtained probability distributions are equal for a suitable choice of the kick strength of the rotor. We further discuss how to engineer the evolution of the walk for dynamically preparing experimentally relevant states. These states are important for future applications of the atom-optics kicked rotor for the realization of ratchets and quantum search.

Experimental realization of an ideal Floquet disordered system

The atomic Quantum Kicked Rotor (QKR) is an outstanding ‘quantum simulator’ for the exploration of transport in disordered quantum systems. Here we study experimentally the phase-shifted QKR, which we show to display properties close to an ideal disordered quantum system, opening new windows into the study of Anderson physics.

Asymptotic behavior of a rotational population distribution in a molecular quantum-kicked rotor with ideal quantum resonance

We performed a mathematical analysis of the time-dependent dynamics of a quantum-kicked rotor implemented in a diatomic molecule under the condition of ideal quantum resonance. We examined a model system featuring a diatomic molecule in a periodic train of terahertz pulses, regarding the molecule as a rigid rotor with the state-dependent transition moment and including the effect of the magnetic quantum number M. We derived the explicit expression for the asymptotic distribution of a rotational population by making the transition matrix correspondent with a sequence of ultraspherical polynomials. The mathematical results obtained were validated by numerical simulations.

Unconventional quantum criticality in the kicked rotor

The quantum kicked rotor (QKR) driven by $d$ incommensurate frequencies realizes the universality class of $d$-dimensional disordered metals. For $d>3$, the system exhibits an Anderson metal-insulator transition which has been observed within the framework of an atom optics realization. However, the absence of genuine randomness in the QKR reflects in critical phenomena beyond those of the Anderson universality class. Specifically, the system shows strong sensitivity to the algebraic properties of its effective Planck constant $\tilde h \equiv 4\pi /q$. For integer $q$, the system may be in a globally integrable state, in a `super-metallic' configuration characterized by diverging response coefficients, Anderson localized, metallic, or exhibit transitions between these phases. We present numerical data for different $q$-values and effective dimensionalities, with the focus being on parameter configurations which may be accessible to experimental investigations.

Theory of localization and resonance phenomena in the quantum kicked rotor

We present an analytic theory of quantum interference and Anderson localization in the quantum kicked rotor (QKR). The behavior of the system is known to depend sensitively on the value of its effective Planck's constant, . We show here that for rational values of , it bears a similarity to a disordered metallic ring of circumference q and threaded by an Aharonov–Bohm flux. Building on this correspondence, we obtain quantitative results for the time-dependent behavior of the QKR kinetic energy, (this is an observable that sensitively probes the system's localization properties). For values of q smaller than the localization length ξ, we obtain scaling , where Δ=2π/q is the quasi-energy level spacing on the ring. This scaling is indicative of a long time dynamics that is neither localized nor diffusive. For larger values q≫ξ, the functions saturate (up to exponentially small corrections ∼exp(−q/ξ)), thus reflecting essentially localized behavior.

On the Spectrum of the Resonant Quantum Kicked Rotor

It is proven that none of the bands in the quasi-energy spectrum of the Quantum Kicked Rotor is flat at any primitive resonance of any order. Perturbative estimates of bandwidths at small kick strength are established for the case of primitive resonances of prime order. Different bands scale with different powers of the kick strength, due to degeneracies in the spectrum of the free rotor.

Dynamical instability in kicked Bose-Einstein condensates

Bose-Einstein condensates subject to short pulses (``kicks'') from standing waves of light represent a nonlinear analog of the well-known chaos paradigm, the quantum kicked rotor. Previous studies of the onset of dynamical instability (i.e., exponential proliferation of noncondensate particles) suggested that the transition to instability might be associated with a transition to chaos. Here we conclude instead that instability is due to resonant driving of Bogoliubov modes. We investigate the Bogoliubov spectrum for both the quantum kicked rotor (QKR) and a variant, the double kicked rotor (QKR-2). We present an analytical model, valid in the limit of weak impulses which correctly gives the scaling properties of the resonances and yields good agreement with mean-field numerics.

Quantum Chaos with Spin-Chains in Pulsed Magnetic Fields

Recently it was found that the dynamics in a Heisenberg spin-chain subjected to a sequence of periodic pulses from an external, parabolic, magnetic field can have a close correspondence with the quantum kicked rotor (QKR). The QKR is a key paradigm of quantum chaos; it has as its classical limit the well-known Standard Map. It was found that a single spin excitation could be converted into a pair of non-dispersive, counter-propagating spin coherent states equivalent to the accelerator modes of the Standard Map. Here we consider how other types of quantum chaotic systems such as a double-kicked quantum rotor or a quantum rotor with a double-well potential might be realized with spin chains; we discuss the possibilities regarding manipulation of the one-magnon spin waves.

Bohmian quantum mechanical and classical Lyapunov exponents for kicked rotor

Using de Broglie–Bohm approach to quantum theory, we show that the kicked rotor at quantum resonance exhibits quantum chaos for the control parameter K above a threshold. Lyapunov exponents are calculated from the method of Benettin et al. for bounded systems for both the quantum and classical kicked rotor. In the chaotic regime we find stability regions for control parameters equal to even and odd multiples of π, but the quantum regions are only remnants of the classical ones.

Theory of2δ-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.

Ehrenfest time in the weak dynamical localization

The quantum kicked rotor (QKR) is known to exhibit dynamical localization in the space of its angular momentum. The present paper is devoted to the systematic first--principal (without a regularizer) diagrammatic calculations of the weak--localization corrections for QKR. Our particular emphasis is on the Ehrenfest time regime -- the phenomena characteristic for the classical--to--quantum crossover of classically chaotic systems.

Estimates on Green's Functions, Localization and the Quantum Kicked Rotor Model

and phase space L2(1r). It has been conjectured by a number of authors (cf. [FGP], [Bel]) that for typical parameter values for a, b, the wave function t satisfying equations (0.1), (0.2) will be almost periodic in time; hence f(t) (the Fourier transform) remains localized. We will prove this here, assuming the parameter S in (0.2) small. Thus we have: THEOREM. For K stnall and (a,b) o?>tside a set of small rneasure ( > O for K > 0), the following holds. Let 4! = W(t, x) solve (0 1), (0 2) and let to = 4!(0, x) be a suJgciently smooth function on T. Then @ is an alrnost periodic function of timbe, say as an Hl(T)-valued map. In particular supt ll@(t)llHl

Quantum suppression of chaos in the Fermi accelerator

We address the question of how a quantum particle exchanges energy with a moving wall with an imposed smooth, periodic motion law. In the semiclassical region of parameter space, the energy gain of the particle is initially consistent with the classical result. This correspondence is maintained for some time, until quantum localization effects suppress the classical diffusion. We discuss similarities with the well studied case of the Quantum Kicked Rotor.

Field Theory of the Quantum Kicked Rotor.

The quantum kicked rotor is investigated by field theoretical methods. It is shown that the effective theory describing the long wave length physics of the system is precisely the supersymmetric nonlinear sigma-model for quasi one-dimensional metallic wires. This proves that the analogy between chaotic systems with dynamical localization and disordered metals can indeed be exact. The role of symmetries is discussed.

Spectral properties of systems with dynamical localization. I. The local spectrum

The authors study the correlations in the quasi-energy (QE) spectra of systems with dynamical localization, using the quantum kicked rotor (QKR) as a paradigm. The specific spatial structure of the QE eigenstates is taken into account by investigating the local spectrum, which gives each eigenstate an individual weight according to its overlap with some reference state. Two-point correlations in the local spectrum are related by Fourier transform to the time evolution of the probability to stay at the initial state. They devise a scaling theory for this dynamical quantity in the case of the QKR, containing the participation ratio as a single parameter. It implies that the local spectrum is characterized by positive correlations, in contrast to the unbiased spectra in classically chaotic systems with a bounded phase space. This is consistent with recent results on spectral properties of systems with Anderson localization. A scheme for experimental measurements of spectral two-point correlation functions is proposed.