We investigate the quantum-classical transition in the delta-kicked rotor and the attainment of the classical limit in terms of measurement-induced state-localization. It is possible to study the transition by fixing the environmentally induced disturbance at a sufficiently small value, and examining the dynamics as the system is made more macroscopic. When the system action is relatively small, the dynamics is quantum mechanical and when the system action is sufficiently large there is a transition to classical behavior. The dynamics of the rotor in the region of transition, characterized by the late-time momentum diffusion coefficient, can be strikingly different from both the purely quantum and classical results. Remarkably, the early time diffusive behavior of the quantum system, even when different from its classical counterpart, is stabilized by the continuous measurement process. This shows that such measurements can succeed in extracting essentially quantum effects. The transition regime studied in this paper is accessible in ongoing experiments.
Edge localization is a fascinating quantum phenomenon. In this paper, the underlying mechanism generating it is presented analytically and verified numerically for a weakly kicked three-dimensional rotor. Analogy to tight binding model in solid state physics is used. The edge states result of the edge at zero angular momentum of the three-dimensional kicked rotor.
We investigate the quantum kicked rotor in resonance subjected to momentum measurements with a Lévy waiting time distribution. We find that the system has a sub-ballistic behavior. We obtain an analytical expression for the exponent of the power law of the variance as a function of the characteristic parameter of the Lévy distribution and connect this anomalous diffusion with a fractional dynamics.
We study the resonances of the quantum kicked rotor subjected to an excitation that follows an aperiodic Fibonacci prescription. In such a case the secondary resonances show a sub-ballistic behaviour like the quantum walk with the same aperiodic prescription for the coin. The principal resonances maintain the well-known ballistic behaviour. Then the parallelism previusly established between the kicked rotor and the generalized quantum walk is retained only with the secondary resonances.
We develop the Wigner phase space representation of a kicked particle for an arbitrary but periodic kicking potential. We use this formalism to illustrate quantum resonances and anti--resonances.
We show that the dynamics of a doubly-excited 1D Heisenberg ferromagnetic 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 eac…
We develop a system consisting of a quantum kicked rotor with an additional degree of freedom. This models a single two-level atom with internal ground and excited states, and it is characterized by its quantum resonances with ballistic spreading and by the entanglement between the internal and momentum degrees of freedom. These behaviors establish an equivalence between our model and the usual quantum walk on the line.
We present measurements of the mean energy for an atom optics kicked rotor ensemble close to quantum resonance. Oscillations in the mean energy in this regime are are shown to be in agreement with a quasi--classical pendulum approximation. The period of the oscillations is shown to scale with a single variable, which depends on the number of kicks.
We present experimental measurements of the mean energy for the atom optics kicked rotor after just two kicks. The energy is found to deviate from the quasi--linear value for small kicking periods. The observed deviation is explained by recent theoretical results which include the effect of a non--uniform initial momentum distribution, previously applied only to systems using much colder atoms than ours.
We show that some classically chaotic quantum systems uncoupled from noisy environments may generate intrinsic decoherence with all its associated effects. In particular, we have observed time irreversibility and high sensitivity to small perturbations in the initial conditions in a quasiperiodic version of the kicked rotor. The existence of simple quantum systems with intrinsic decoherence clarifies the quantum--classical correspondence in chaotic systems.
We investigate possible ways in which a quantum wavepacket spreads. We show that in a general class of double kicked rotor systems, a wavepacket may undergo superballistic spreading; i.e., its variance increases as the cubic of time. The conditions for the observed superballistic spreading and two related characteristic time scales are studied. Our results suggest that the symmetry of the studied model and whether it is a Kolmogorov-Arnold-Moser system are crucial to its wavepacket spreading behavior. Our study also sheds new light on the exponential wavepacket spreading phenomenon previously …
We experimentally verify the analytical expressions that exist for the diffusion rate in the quantum delta kicked rotor system for small numbers of kicks. We show development of diffusion resonances from two to five kicks, and of multiple resonances for high kick strengths. Furthermore, we show that, in contrast to classical predictions, the results are purely periodic in the kick period, and reproduce the predicted quantum- and diffusion resonances.
We present mean energy measurements for the atom optics kicked rotor as the kicking period tends to zero. A narrow resonance is observed marked by quadratic energy growth, in parallel with a complete freezing of the energy absorption away from the resonance peak. Both phenomena are explained by classical means, taking proper account of the atoms' initial momentum distribution.
We study the quantum kicked rotor in resonance subjected to an unitary noise defined through Kraus operators, we show that this type of decoherence does not, in general, lead to the classical diffusive behavior. We find exact analytical expressions for the density matrix and the variance in the primary resonances. The variance does not loose its ballistic behavior, however the coherence decays as a power law. The secondary resonances are treated numerically, obtaining a power-law decay for the variance and an exponential law decay for the coherence.
We study the resonances of the quantum kicked rotor subjected to an excitation that follows a deterministic time-dependent prescription. For the primary resonances we find an analytical relation between the long-time behavior of the standard deviation and the external kick strength. For the secondary resonances we obtain essentially the same result numerically. Selecting the time sequence of the kick allows to obtain a variety of asymptotic wave-function spreadings: super-ballistic, ballistic, sub-ballistic, diffusive, sub-diffusive and localized.
The influence of an initial momentum on the appearance of "quantum resonances" in the delta-kicked rotor system is explored experimentally. We show that for certain initial momenta, a resonance can be negated entirely, whereas at others a resonance can be made to appear. At a larger number of kicks, all resonances are shown to narrow. We show that as a consequence, the individual "diffraction peaks" may split. We compare our results to numerical simulations as well as analytical theory.
As an extension of our recent work (Results in Physics 12, 147-152 (2019)), we show, both theoretically and numerically, that the breakdown of agreement between the non-relativistic and relativistic quantum dynamical predictions in the non-relativistic regime also occurs for an electron subjected to a time-dependent force.
Considering a kicked rotor coupled to a model heat bath both the classical and quantum entropy productions are calculated exactly. Starting with an initial wave packet, the von Neuman entropy as a function of time is determined from the reduced density matrix while the Liouville evolution of the corresponding Husimi distribution provides us with the classical entropy. It is found that both these entropies agree reasonably satisfying the same asymptotic growth law and more importantly both are proportional to the classical Liapounov exponent.
We present a theoretical and numerical study of the competition between two opposite interference effects, namely interference-induced ballistic transport on one hand, and strong (Anderson) localization on the other. While the former effect allows for resistance free transport, the latter brings the transport to a complete halt. As a model system, we consider the quantum kicked rotor, where strong localization is observed in the discrete momentum coordinate. In this model, we introduce the ballistic transport in the form of a Hadamard quantum walk in that momentum coordinate. The two transport…
We present a perturbative result for the temporal evolution of the fidelity of the quantum kicked rotor, i.e. the overlap of the same initial state evolved with two slightly different kicking strengths, for kicking periods close to a principal quantum resonance. Based on a pendulum approximation we describe the fidelity for rotational orbits in the pseudo-classical phase space of a corresponding classical map. Our results are compared to numerical simulations indicating the range of applicability of our analytical approximation.
The kicked rotor and the kicked top are two paradigms of quantum chaos. The notions of quantum resonance and the pseudoclassical limit, developed in the study of the kicked rotor, have revealed an intriguing and unconventional aspect of classical-quantum correspondence. Here, we show that, by extending these notions to the kicked top, its rich dynamical behavior can be appreciated more thoroughly; of special interest is the entanglement entropy. In particular, the periodic synchronization between systems subject to different kicking strength can be conveniently understood and elaborated from t…
We present experimental observations of diffusion resonances for the quantum kicked rotor with weak decoherence. Cold caesium atoms are subject to a pulsed standing wave of near-resonant light, with spontaneous emission providing environmental coupling. The mean energy as a function of the pulse period is determined during the late-time diffusion regime for a constant probability of spontaneous emission per unit time. Structure in the late-time energy is seen to increase with physical kicking strength. The observed structure is related to Shepelyansky's predictions of the initial quantum diffu…
Quantum resonances in the kicked rotor are characterized by a dramatically increased energy absorption rate, in stark contrast to the momentum localization generally observed. These resonances occur when the scaled Planck's constant hbar=(r/s)*4pi, for any integers r and s. However only the hbar=r*2pi resonances are easily observable. We have observed high-order quantum resonances (s>2) utilizing a sample of low temperature, non-condensed atoms and a pulsed optical standing wave. Resonances are observed for hbar=(r/16)*4pi r=2-6. Quantum numerical simulations suggest that our observation of hi…
We investigate the dynamics of the out-of-time-ordered correlators (OTOCs) via a non-Hermitian extension of the quantum kicked rotor model, where the kicking potential satisfies $\mathcal{PT}$-symmetry. The spontaneous $\cal{PT}$-symmetry breaking emerges when the strength of the imaginary part of the kicking potential exceeds a threshold value. We find, both analytically and numerically, that in the broken phase of $\cal{PT}$ symmetry, the OTOCs rapidly saturate with time evolution. Interestingly, the late-time saturation value scales as a pow-law in the system size. The mechanism of such sca…
We study a system of two coupled kicked rotors, both classically and quantum mechanically, for a wide range of coupling parameters. This was motivated by two published reports, one of which reported quantum localization, while the other reported diffusion. The classical systems are chaotic, and exhibit normal diffusive behavior. In the quantum systems, we found different regimes, depending on the strength of the coupling. For weak coupling, we found quantum localization similar to that exhibited by single kicked rotors. For strong coupling, we found a quasi-diffusive growth of the width of the…
The effect of pulse train noise on the quantum resonance peaks of the Atom Optics Kicked Rotor is investigated experimentally. Quantum resonance peaks in the late time mean energy of the atoms are found to be surprisingly robust against all levels of noise applied to the kicking amplitude, whilst even small levels of noise on the kicking period lead to their destruction. The robustness to amplitude noise of the resonance peak and of the fall--off in mean energy to either side of this peak are explained in terms of the occurence of stable, $ε$--classical dynamics [S. Wimberger, I. Guarneri, and…
We study the ergodicity and mixing of quantum kicked rotor (QKR) with two distinct approaches. In one approach, we use the definitions of quantum ergodicity and mixing recently proposed in [Phys. Rev. E 94, 022150 (2016)], which involve only eigen-energies (Floquet quasi-energies for QKR). In the other approach, we study ergodicity and mixing with quantum Poincarè section, which is plotted with a method that maps a wave function unitarily onto quantum phase space composed of Planck cells. Classical Poincarè section can be recovered with the effective Planck constant gradually diminishing. We d…
Bose-Einstein condensates subject to short pulses (`kicks') from standing waves of light represent a nonlinear analogue of the well-known chaos paradigm, the quantum kicked rotor. Previous studies of the onset of dynamical instability (ie exponential proliferation of non-condensate 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 excitation of Bogoliubov modes for both the quantum kicked rotor (QKR) and a variant, the double kicked …
We numerically investigate the quantum transport in a coupled kicked rotors with the $\mathcal{PT}$-symmetric potential. We find that the spontaneous $\mathcal{PT}$-symmetry breaking of wavefunctions emerges when the amplitude of the imaginary part of the complex potential is beyond a threshold value, which can be modulated by the coupling strength effectively. In the regime of the $\mathcal{PT}$-symmetry breaking, the particles driven by the periodical kicks move unidirectionally in momentum space, indicating the emergence of a directed current. Meanwhile, with increasing the coupling strengt…
We present an investigation into effects exhibited by the two-frequency kicked rotor. Experiments were performed and in addition quantum and classical dynamics were simulated and compared with the experimental results. The experiments involved pulsing the optical standing wave with two pulsing periods of differing frequencies and variable initial phase offset. The ratio of pulsing periods was sampled for rational and irrational values for different experimental runs. In this paper we present these results and examine the measured momentum distributions for the cause of any structures that are …
We present a detailed theory of spectacular semiclassical catastrophes happening during the time evolution of a kicked quantum rotor (Phys.Rev. Lett. {\bf 87}, 163601 (2001)). Both two- and three-dimensional rotational systems are analyzed. It is shown that the wave function of the rotor develops a {\em cusp} at a certain delay after a kick, which results in a sharply focused rotational wave packet. The {\em cusp} is followed by a fold-type catastrophe manifested in the {\em rainbow}-like moving angular singularities. In the three-dimensional case, the rainbows are accompanied by additional si…
We present the results of experiments performed on cold caesium in a pulsed sinusoidal optical potential created by counter-propagating laser beams having a small frequency difference in the laboratory frame. Since the atoms, which have average velocity close to zero in the laboratory frame, have non-zero average velocity in the co-moving frame of the optical potential, we are able to centre the initial velocity distribution of the cloud at an arbitrary point in phase-space. In particular, we demonstrate the use of this technique to place the initial velocity distribution in a region of phase-…
We show that classical-quantum correspondence of center of mass motion in two coupled delta-kicked rotors can be obtained from intrinsic decoherence of the system itself which occurs due to the entanglement of the center of mass motion to the internal degree of freedom without coupling to external environment.
We present a quantum localization phenomenon that exists in periodically kicked 3D rotors, but is absent in the commonly studied 2D ones: edge localization. We show that under the condition of a fractional quantum resonance there are states of the kicked rotor that are strongly localized near the edge of the angular momentum space at $J=0$. These states are analogs of surface states in crystalline solids, and they significantly affect resonant excitation of molecular rotation by laser pulse trains.
We study the resonances of the quantum kicked rotor subjected to an extended initial distribution. For the primary resonances we obtain the dispersion relation for the map of this system. We find an analytical dependence of the statistical moments on the shape of the initial distribution. For the secondary resonances we obtain numerically a similar dependence. This allows us to devise an extended initial condition which produces an average angular momentum pointing in a preset direction which increases with time with a preset ratio.
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.
Quantum systems lose coherence upon interaction with the environment and tend towards classical states. Quantum coherence is known to exponentially decay in time so that macroscopic quantum superpositions are generally unsustainable. In this work, slower than exponential decay of coherences is experimentally realized in an atom-optics kicked rotor system subjected to non-stationary Lévy noise in the applied kick sequence. The slower coherence decay manifests in the form of quantum subdiffusion that can be controlled through the Lévy exponent. The experimental results are in good agreement with…
The double kicked rotor model is a physically realizable extension of the paradigmatic kicked rotor model in the study of quantum chaos. Even before the concept of Floquet topological phases became widely known, the discovery of the Hofstadter butterfly spectrum in the double kicked rotor model [J. Wang and J. Gong, Phys. Rev. A 77, 031405 (2008)] already suggested the importance of periodic driving to the generation of unconventional topological matter. In this work, we explore Floquet topological phases of a double kicked rotor with an extra spin-1/2 degree of freedom. The latter has been ex…
We describe a method allowing transient suppression of decoherence effects on the atom-optics realization of the kicked rotor. The system is prepared in an initial state with a momentum distribution concentrated in an interval much sharper than the Brillouin zone; the measure of the momentum distribution is restricted to this interval of quasimomenta: As most of the atoms undergoing decoherence processes fall outside this detection range and thus are not detected, the measured signal is effectively decoherence-free.
We study the dynamics of Bogoliubov excitations of a Bose-Einstein condensate in the quasiperiodic kicked rotor. In the weakly interacting regime, the condensate is stable and both the condensate and the excitations undergo a phase transition from a quasilocalized to a diffusive regime. The corresponding critical exponents are identical for the condensate and the excitations, and compare very well with the value $ν\approx1.6$ for non-interacting particles.
We investigate the parametric fluctuations in the quantum survival probability of an open version of the delta-kicked rotor model in the deep quantum regime. Spectral arguments [Guarneri I and Terraneo M 2001 Phys. Rev. E vol. 65 015203(R)] predict the existence of parametric fractal fluctuations owing to the strong dynamical localisation of the eigenstates of the kicked rotor. We discuss the possibility of observing such dynamically-induced fractality in the quantum survival probability as a function of the kicking period for the atom-optics realisation of the kicked rotor. The influence of t…
We examine the effect of the initial atomic momentum distribution on the dynamics of the atom-optical realisation of the quantum kicked rotor. The atoms are kicked by a pulsed optical lattice, the periodicity of which implies that quasi-momentum is conserved in the transport problem. We study and compare experimentally and theoretically two resonant limits of the kicked rotor: in the vicinity of the quantum resonances and in the semiclassical limit of vanishing kicking period. It is found that for the same experimental distribution of quasi-momenta, significant deviations from the 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 stationary noise. The associated quantum diffusion increases monotonically as a function of a parameter characterising the noise distribution. In this work, for the Levy kicked atom-optics rotor, it is experimentally shown that by tuning a parameter characterizing the Levy distribution, quantum diffusion displays non-monotonic behaviour. The parameters …
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 wavefunction 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.
We numerically investigate momentum diffusion rates for the pulse kicked rotor across the quantum to classical transition as the dynamics are made more macroscopic by increasing the total system action. For initial and late time rates we observe an enhanced diffusion peak which shifts and scales with changing kick strength, and we also observe distinctive peaks around quantum resonances. Our investigations take place in the context of a system of ultracold atoms which is coupled to its environment via spontaneous emission decoherence, and the effects should be realisable in ongoing experiments…
Dynamical kicking systems possess rich topological structures. In this work, we study Floquet states of matter in a non-Hermitian extension of double kicked rotor model. Under the on-resonance condition, we find various non-Hermitian Floquet topological phases, with each being characterized by a pair of topological winding numbers. A generalized mean chiral displacement is introduced to detect these winding numbers dynamically in two symmetric time frames. Furthermore, by mapping the system to a periodically quenched lattice model, we obtain the topological edge states and unravel the bulk-edg…
We propose an experimental configuration, within an ion trap, by which a quantum mechanical delta-kicked harmonic oscillator could be realized, and investigated. We show how to directly measure the sensitivity of the ion motion to small variations in the external parameters.
We study the dynamics of a Bose-Einstein condensate in the quasiperiodic kicked rotor described by a Gross-Pitaevskii equation with periodic boundary conditions. As the interactions are increased, Bogoliubov excitations appear and deplete the condensate; we characterize this instability by considering the population of the first Bogoliubov mode, and show that it does not prevent, for small enough interaction strengths, the observation of the transition. However, the predicted subdiffusive behavior is not observed in the stable region. For higher interaction strengths, the condensate may be str…
We report a numerical analysis of the Anderson transition in a quantum-chaotic system, the quasiperiodic kicked rotor with three incommensurate frequencies. It is shown that this dynamical system exhibits the same critical phenomena as the truly random 3D-Anderson model. By taking proper account of systematic corrections to one-parameter scaling, the universality of the critical exponent is demonstrated. Our result $ν= 1.59 \pm 0.01$ is in perfect agreement with the value found for the Anderson model.
We observe and study the phenomenon of Anderson localization in a system of true quantum kicked rotors. Nitrogen molecules in a supersonic molecular jet are cooled down to 27~K and are rotationally excited by a periodic train of 24~high-intensity femtosecond pulses. Exponential distribution of the molecular angular momentum - the most unambiguous signature of Anderson localization - is measured directly by means of coherent Raman scattering. We demonstrate the suppressed growth of the molecular rotational energy with the number of laser kicks and study the dependence of the localization length…
We study the dynamics of the many-body atomic kicked rotor with interactions at the mean-field level, governed by the Gross-Pitaevskii equation. We show that dynamical localization is destroyed by the interaction, and replaced by a subdiffusive behavior. In contrast to results previously obtained from a simplified version of the Gross-Pitaevskii equation, the subdiffusive exponent does not appear to be universal. By studying the phase of the mean-field wave function, we propose a new approximation that describes correctly the dynamics at experimentally relevant times close to the start of subd…
We experimentally study a system of quantum kicked rotors - an ensemble of diatomic molecules exposed to a periodic sequence of ultrashort laser pulses. In the regime, where the underlying classical dynamics is chaotic, we investigate the quantum phenomenon of dynamical localization by means of state-resolved coherent Raman spectroscopy. We examine the dependence of the exponentially localized angular momentum distribution and of the total rotational energy on the time period between the pulses and their amplitude. The former parameter is shown to provide control over the localization center, …
We study the mechanisms responsible for quantum diffusion in the quasiperiodic kicked rotor. We report experimental measurements of the diffusion constant on the atomic version of the system and develop a theoretical approach (based on the Floquet theorem) explaining the observations, especially the ``sub-Fourier'' character of the resonances observed in the vicinity of exact periodicity, i.e. the ability of the system to distinguish two neighboring driving frequencies in a time shorter than the inverse of the difference of the two frequencies.
We show that the quantum fidelity is accessible to cold atom experiments for a large class of evolutions in periodical potentials, properly taking into account the experimental initial conditions of the atomic ensemble. We prove analytically that, at the fundamental quantum resonances of the atom-optics kicked rotor, the fidelity saturates at a constant, time-independent value after a small number of kicks. The latter saturation arises from the bulk of the atomic ensemble, whilst for the resonantly accelerated atoms the fidelity is predicted to decay slowly according to a power law.
