Kicked Rotor Model Research Articles

Dynamical localization in a non-Hermitian Floquet synthetic system

We investigate the non-Hermitian effects on quantum diffusion in a kicked rotor model where the complex kicking potential is quasi-periodically modulated in the time domain. The synthetic space with arbitrary dimension can be created by incorporating incommensurate frequencies in the quasi-periodical modulation. In the Hermitian case, strong kicking induces the chaotic diffusion in the four-dimension momentum space characterized by linear growth of mean energy. We find that the quantum coherence in deep non-Hermitian regime can effectively suppress the chaotic diffusion and hence result in the emergence of dynamical localization. Moreover, the extent of dynamical localization is dramatically enhanced by increasing the non-Hermitian parameter. Interestingly, the quasi-energies become complex when the non-Hermitian parameter exceeds a certain threshold value. The quantum state will finally evolve to a quasi-eigenstate for which the imaginary part of its quasi-energy is large most. The exponential localization length decreases with the increase of the non-Hermitian parameter, unveiling the underlying mechanism of the enhancement of the dynamical localization by non-Hermiticity.

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.

A pseudoclassical theory for the wavepacket dynamics of the kicked rotor model

A pseudoclassical theory for the wavepacket dynamics of the kicked rotor model

Scaling laws of the out-of-time-order correlators at the transition to the spontaneous PT -symmetry breaking in a Floquet system

We investigate both numerically and analytically the dynamics of out-of-time-order correlators (OTOCs) in a non-Hermitian kicked rotor model, addressing the scaling laws of the time dependence of OTOCs at the transition to the spontaneous $\mathcal{PT}$-symmetry breaking. In the unbroken phase of $\mathcal{PT}$ symmetry, the OTOCs increase monotonically and eventually saturate with time, demonstrating the freezing of information scrambling. Just beyond the phase transition points, the OTOCs increase in the power laws of time, with the exponent being larger than 2. Interestingly, the quadratic growth of OTOCs with time emerges when the system is far beyond the phase transition points. The above numerical findings are validated by our theoretical analysis, which provides a general framework with important implications for Floquet engineering and information scrambling in chaotic systems.

Scaling laws of out-of-time-order correlators in a non-Hermitian kicked rotor model

We investigate the dynamics of the out-of-time-order correlators (OTOCs) via a non-Hermitian extension of the quantum kicked rotor model, where the kicking potential satisfies PT-symmetry. The spontaneous 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 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 scaling law results from the interplay between the effects of the nonlocal operator in OTOCs and the time reversal induced by non-Hermitian-driven potential.

Diagonalization in a quantum kicked rotor model with non-analytic potential

In this paper we study the lattice quasi-periodic operators with power-law long-range hopping and meromorphic monotone potentials, and diagonalize the operators via a Nash-Moser iteration scheme. As applications, we obtain uniform power-law localization, uniform dynamical localization and Lipschitz continuity of the integrated density of states (IDS) for such operators. Our main motivation comes from investigating quantum suppression of chaos in a quantum kicked rotor model with non-analytical potential.

Dynamical Stability in a Non-Hermitian Kicked Rotor Model

We investigate the quantum irreversibility and quantum diffusion in a non-Hermitian kicked rotor model for which the kicking strength is complex. Our results show that the exponential decay of Loschmidt echo gradually disappears with increasing the strength of the imaginary part of non-Hermitian driven potential, demonstrating the suppress of the exponential instability by non-Hermiticity. The quantum diffusion exhibits the dynamical localization in momentum space, namely, the mean square of momentum increases to saturation with time evolution, which decreases with the increase of the strength of the imaginary part of the kicking. This clearly reveals the enhancement of dynamical localization by non-Hermiticity. We find, both analytically and numerically, that the quantum state are mainly populated on a very few quasieigenstates with significantly large value of the imaginary part of quasienergies. Interestingly, the average value of the inverse participation ratio of quasieigenstates decreases with the increase of the strength of the imaginary part of the kicking potential, which implies that the feature of quasieigenstates determines the stability of wavepacket’s dynamics and the dynamical localization of energy diffusion.

Robust quantum boomerang effect in non-Hermitian systems

Anderson localization is a general phenomenon that applies to a variety of disordered physical systems. Recently, a novel manifestation of Anderson localization for wave packets launched with a finite average velocity was proposed, the Quantum boomerang effect (QBE). This phenomenon predicts that the disorder-averaged center of mass of a particle initially moves ballistically, then makes a U-turn, and finally slowly returns to its initial position. The QBE has been predicted to take place in several Hermitian models with Anderson localization and has been experimentally observed in the paradigmatic quantum kicked rotor model. In this work, we investigate the emergence of the QBE in non-Hermitian systems and clarify the importance of symmetries of the Hamiltonian and the initial state. We generalize the analytical arguments available in the literature and show that even in the case of complex spectrum a boomerang-like behavior can appear in a non-Hermitian system. We confirm our analytical results through a careful numerical investigation of the dynamics for several non-Hermitian models. We find that non-Hermiticity leads to the breakdown of the dynamical relation, though the QBE is preserved. This work opens up new avenues for future investigations in Anderson localized systems. The models studied here may be implemented using cold atoms in optical lattices.

Statistical complexity of the kicked top model considering chaos

Abstract The concept of the statistical complexity is studied to characterize the classical kicked top model which plays important role in the qbit systems and the chaotic properties of the entanglement. This allow us to understand this driven dynamical system by the probability distribution in phase space to make distinguish among the regular, random and structural complexity on finite simulation. We present the dependence of the kicked top and kicked rotor model through the strength excitation in the framework of statistical complexity.

Chaotic dynamics of complex trajectory and its quantum signature* *Project partially supported by the National Natural Science Foundation of China (Grant Nos. 12065009, 11804130, and 11805165) and Zhejiang Provincial Nature Science Foundation, China (Grant No. LY20A050001).

We investigate both the quantum and classical dynamics of a non-Hermitian system via a kicked rotor model with symmetry. For the quantum dynamics, both the mean momentum and mean square of momentum exhibit the staircase growth with time when the system parameter is in the neighborhood of the symmetry breaking point. If the system parameter is much larger than the symmetry breaking point, the accelerator mode results in the directed spreading of the wavepackets as well as the ballistic diffusion in momentum space. For the classical dynamics, the non-Hermitian kicking potential leads to the exponentially-fast increase of classical complex trajectories. As a consequence, the imaginary part of the trajectories exponentially diffuses with time, while the real part exhibits the normal diffusion. Our analytical prediction of the exponential diffusion of imaginary momentum and its breakdown time is in good agreement with numerical results. The quantum signature of the chaotic diffusion of the complex trajectories is reflected by the dynamics of the out-of-time-order correlators (OTOC). In the semiclassical regime, the rate of the exponential increase of the OTOC is equal to that of the exponential diffusion of the complex trajectories.

Non-Hermitian Floquet topological phases in the double-kicked rotor

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-edge correspondence of the non-Hermitian double kicked rotor. These results uncover the richness of Floquet topological states in non-Hermitian dynamical kicking systems.

Quantum-classical correspondence in a nonlinear Gross–Pitaevski system

We investigate the quantum-classical correspondence of the Bose–Einstein condensate (BEC) which is described by the Gross–Pitaevski equation with the periodic, both temporally and spatially, modulation of nonlinear interaction. It is found that this system is equivalent to a generalized kicked rotor (GKR) model. Interestingly, the classical dynamics of the GKR system exhibits the regular-to-chaotic transition as time evolves, and the corresponding energy diffusion is exponentially fast. For the BEC system, the energy diffusion can be predicted by the classical mapping equation of the GKR model. However, the ratio of the energy between the BEC system and the classical GKR model is not unity, even if the effective Planck constant is very small, which is contradictory to our conventional understanding of quantum-classical correspondence. We find that the BEC system is not exactly a quantum counterpart of the classical GKR system. We construct a better quantum counterpart and derive an analytical expression for the exponentially-fast diffusion of energy, which is in perfect agreement with numerical results.

Directed momentum current induced by the PT-symmetric driving.

We investigate the directed momentum current in the quantum kicked rotor model with PT-symmetric deriving potential. For the quantum nonresonance case, the values of quasienergy become complex when the strength of the imaginary part of the kicking potential exceeds a threshold value, which demonstrates the appearance of the spontaneous PT symmetry breaking. In the vicinity of the transition point, the momentum current exhibits a staircase growth with time. Each platform of the momentum current corresponds to the mean momentum of some eigenstates of the Floquet operator whose imaginary parts of the quasienergy are significantly large. Above the transition point, the momentum current increases linearly with time. Interestingly, its acceleration rate exhibits a kind of "quantized" increment with the kicking strength. We propose a modified classical acceleration mode of the kicked rotor model to explain such an intriguing phenomenon. Our theoretical prediction is in good agreement with numerical results.

Floquet topological phases in a spin- 1/2 double kicked rotor

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 experimentally engineered in a quantum kicked rotor recently by loading Rb87 condensates into a periodically pulsed optical lattice. Under the on-resonance condition, the spin-1/2 double kicked rotor admits fruitful topological phases due to the interplay between its external and internal degrees of freedom. Each of these topological phases is characterized by a pair of winding numbers, whose combination predicts the number of topologically protected 0 and \pi-quasienergy edge states in the system. Topological phases with arbitrarily large winding numbers can be easily found by tuning the kicking strength. We discuss an experimental proposal to realize this model in kicked Rb87 condensates, and suggest to detect its topological invariants by measuring the mean chiral displacement in momentum space.

Quantum-classical transition and quantum activation of ratchet currents in the parameter space.

The quantum ratchet current is studied in the parameter space of the dissipative kicked rotor model coupled to a zero-temperature quantum environment. We show that vacuum fluctuations blur the generic isoperiodic stable structures found in the classical case. Such structures tend to survive when a measure of statistical dependence between the quantum and classical currents are displayed in the parameter space. In addition, we show that quantum fluctuations can be used to overcome transport barriers in the phase space. Related quantum ratchet current activation regions are spotted in the parameter space. Results are discussed based on quantum, semiclassical, and classical calculations. While the semiclassical dynamics involves vacuum fluctuations, the classical map is driven by thermal noise.

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.

Nonlinearly-enhanced energy transport in many dimensional quantum chaos

By employing a nonlinear quantum kicked rotor model, we investigate the transport of energy in multidimensional quantum chaos. This problem has profound implications in many fields of science ranging from Anderson localization to time reversal of classical and quantum waves. We begin our analysis with a series of parallel numerical simulations, whose results show an unexpected and anomalous behavior. We tackle the problem by a fully analytical approach characterized by Lie groups and solitons theory, demonstrating the existence of a universal, nonlinearly-enhanced diffusion of the energy in the system, which is entirely sustained by soliton waves. Numerical simulations, performed with different models, show a perfect agreement with universal predictions. A realistic experiment is discussed in two dimensional dipolar Bose-Einstein-Condensates (BEC). Besides the obvious implications at the fundamental level, our results show that solitons can form the building block for the realization of new systems for the enhanced transport of matter.

Semiclassical approach to the quantum Loschmidt echo in deep quantum regions: From validity to breakdown

Semiclassical results are usually expected to be valid in the semiclassical regime. An interesting question is, in models in which appropriate effective Planck constants can be introduced, to what extent will a semiclassical prediction stay valid when the effective Planck constant is increased? In this paper, we numerically study this problem, focusing on semiclassical predictions for the decay of the quantum Loschmidt echo in deep quantum regions. Our numerical simulations, carried out in the chaotic regime in the sawtooth model and in the kicked rotator model and also in the critical region of a one-dimensional Ising chain in transverse field, show that the semiclassical predictions may work even in deep quantum regions, particularly for perturbation strength in the so-called Fermi-Golden-Rule regime.

Calculation of Anderson Localization Length by Using Generalized Quantum Kicked Rotator Analytically

We use the generalized quantum kicked rotator model and its relation with the Anderson model. We calculate localization length analytically. For this reason, we consider a one-dimensional rotator model for a special potential whose an impulse is applied at equal time intervals, T. We obtain time evolution of the wave function between two successive impulses by an evolution matrix. We change this model to the tight-binding model for a particle on a one-dimensional lattice. At special case the wave function is localized, and then we derive the Anderson localization length for the system analytically.

Quantum control of ultra-cold atoms: uncovering a novel connection between two paradigms of quantum nonlinear dynamics

Controlling the translational motion of cold atoms using optical lattice potentials is of both theoretical and experimental interest. By designing two on-resonance time sequences of kicking optical lattice potentials, a novel connection between two paradigms of nonlinear mapping systems, i.e. the kicked rotor model and the kicked Harper model, is established. In particular, it is shown that Hofstadter's butterfly quasi-energy spectrum in periodically driven quantum systems may soon be realized experimentally, with the effective Planck constant tunable by varying the time delay between two sequences of control fields. Extensions of this study are also discussed. The results are intended to open up a new generation of cold-atom experiments of quantum nonlinear dynamics.