Quasienergy States Research Articles

Comparative investigation between complex energy and transfer matrix techniques for quasi-energy states in heterostructure materials

A comparative analysis is performed to study the resonant tunneling through symmetric multiple double-barrier resonant tunneling heterostructure materials composed of nanoscale semiconductors ZnSe/BeTe, AlAs/GaAs, InGaAs/AlInAs, and AlGaAs/GaAs. Two techniques are applied in the present investigation for calculating the quasi-resonant energy states and their associated lifetimes in nanoscale double-barrier quantum well heterostructures. The first technique is based on the complex energy solution of the time-dependent Schrödinger equation with the time–energy uncertainty condition. The second one is employing the transfer matrix method. The quasi-resonant energies, lifetimes, and frequencies in nanoscale double-barrier quantum well heterostructures are calculated by both methods and compared. The validity and accuracy of both techniques are tested and compared extensively with various current numerical methods. Excellent agreements are obtained between our results and the available experimental and theoretical values. It is also found that the complex energy technique is recommended when handling the quasi-energy states and quasi-lifetimes of novel nanoscale devices.

Quasienergy operators and generalized squeezed states for systems of trapped ions

Collective many-body dynamics for time-dependent quantum Hamiltonians is investigated for a dynamical system that exhibits multiple degrees of freedom, in this case a combined (Paul and Penning) trap. Quantum stability is characterized by a discrete quasienergy spectrum, while the quasienergy states are symplectic coherent states. We introduce the generators of the Lie algebra of the symplectic group SL(2,R), which we use to build the coherent states (CS) associated to the system under investigation. The trapped ion is treated as a harmonic oscillator (HO) to which we associate the quantum Hamilton function. We obtain the kinetic and potential energy operators as functions of the Lie algebra generators and supply the expressions for the classical coordinate, momentum, kinetic and potential energy, along with the total energy. Moreover, we also infer the dispersions for the coordinate and momentum, together with the asymmetry and the flatness parameter for the distribution. The system interaction with laser radiation is also examined for a system of identical two-level atoms. The Hamilton function for the Dicke model is derived. The optical system is modelled as a HO (trapped ion) that undergoes interaction with an external laser field and we use it to engineer a squeezed state of the electromagnetic (EM) field. We consider coherent and squeezed states associated to both ion dynamics and to the EM field. The approach used enables one to build CS in a compact and smart manner by use of the group theory.

Autonomous quantum error correction in a four-photon Kerr parametric oscillator

Autonomous quantum error correction has gained considerable attention to avoid complicated measurements and feedback. Despite its simplicity compared with the conventional measurement-based quantum error correction, it is still a far from practical technique because of significant hardware overhead. We propose an autonomous quantum error correction scheme for a rotational symmetric bosonic code in a four-photon Kerr parametric oscillator. Our scheme is the simplest possible error correction scheme that can surpass the break-even point—it requires only a single continuous microwave tone. We also introduce an unconditional reset scheme that requires one more continuous microwave tone in addition to that for the error correction. The key properties underlying this simplicity are protected quasienergy states of a four-photon Kerr parametric oscillator and the degeneracy in its quasienergy level structure. These properties eliminate the need for state-by-state correction in the Fock basis. Our schemes greatly reduce the complexity of autonomous quantum error correction and thus may accelerate the use of the bosonic code for practical quantum computation.

Effective Hamiltonians for almost-periodically driven quantum systems* *Contribution to the special collection ‘Koopman methods in classical and quantum-classical mechanics’

We present an effective Hamiltonian theory available for some quasi-periodically driven quantum systems which does not need the knowledge of the Fourier frequencies of the control signal. It could also be available for some chaotically driven quantum systems. It is based on the Koopman approach which generalizes the Floquet approach used with periodically driven systems. We show the properties of the quasi-energy states (eigenvectors of the effective Hamiltonian) as quasi-recurrent states of the quantum system.

Charging Quantum Batteries with a General Harmonic Driving Field

AbstractA general harmonic driving field was considered for improving the charging efficiency of the quantum battery system. Charge saturation was used to describe the charging efficiency, where the charging mode is divided into saturated and unsaturated. The relationships between the time‐dependent charge saturation and parameters of the general driving field were evaluated both analytically and numerically. The Floquet theorem was used to express time‐dependent charge saturation with the quasienergy and Floquet states of the system. The analytical and numerical results were used to identify the best parameter values for optimizing the charging efficiency.

Enhanced excitation of a driven bistable system induced by spectrum degeneracy

The non-equilibrium statistics and kinetics of a simple bistable system (resonantly driven nonlinear oscillator coupled to reservoir) have been investigated by means of master equation for the density matrix and quasiclassical Fokker-Planck equation in quasienergy space. We found out that the system's statistical and kinetic properties drastically change when the quasienergy states become nearly degenerate and the occupation of the most excited state is strongly enhanced. It has been revealed that in nearly degenerate case a new critical quasienergy parameter emerges. Below the critical quasienergy value the eigenstates are superpositions of the quasiclassical states from different phase space regions, while above this value the eigenstates correspond to only one particular region of the phase space. We have also generalized Keldysh theory for ionization of atoms in the electromagnetic field for bistable systems. It has been demonstrated that Keldysh parameter in bistability region is large when pumping intensity is smaller than the critical value. It has been shown by direct calculations that multi-photon transition amplitude coincides with the tunneling amplitude. So, multi-photon transitions and tunneling between the regions of the phase space are just the same effects. We also demonstrated that for bistable systems the Keldysh parameter logarithmically depends on the external field amplitude.

Effects of tunneling and multiphoton transitions on squeezed-state generation in bistable driven systems

Bistability of nonlinear resonantly driven oscillator in the presence of external noise is analyzed by means of classical Fokker-Planck equation in quasienergy space with account for tunneling effects and by quantum master equation in quasienergy states representation. Two time scales responsible for different stages of bistable system relaxation have been obtained. We found out that the slow relaxation rate caused by fluctuation--induced transitions between different stable states can be enhanced by several orders due to tunneling effects. It was also revealed that tunneling between nearly degenerate quasienergy states and resonant multiphoton transitions between the genuine eigenstates of the nonlinear oscillator are just the similar effects. It was demonstrated that the quasienergy states in the bistability region corresponding to higher amplitude are squeezed. The degree of squeezing is determined by the ratio between nonlinearity and detuning, so the uncertainty of one quadrature can be considerably smaller than the quantum limit. It was found out that tunneling effects can enhance the generation of output oscillator squeezed states. It is shown that 1D Fokker--Planck equation is a quasiclassical limit of a quantum master equation.

Quantum tunneling effect on switching rates of bistable driven system

The influence of quantum tunneling effects on the bistability of nonlinear resonantly driven oscillators in the presence of external noise is analyzed by means of the generalized Fokker–Planck equation in quasi-energy space. Two time scales responsible for different stages of bistable system relaxation have been revealed. It was found that quantum tunneling effects could strongly change the switching rates between different stable states and considerably enhance occupation of the stable state with higher amplitude. It was also revealed that tunneling between nearly degenerate quasi-energy states and resonant multi-photon transitions between the genuine eigenstates of the nonlinear oscillator are just similar effects.

Schrödinger–Koopman quasienergy states of quantum systems driven by classical flow

We study the properties of the quasienergy states of a quantum system driven by a classical dynamical system. The quasienergies are defined in the same manner as in light-matter interaction but where the Floquet approach is generalized by the use of the Koopman approach of dynamical systems. We show how the properties of the classical flow (fixed and cyclic points, ergodicity, chaos) influence the driven quantum system. This approach of Schrödinger–Koopman quasienergies can be applied to quantum control, quantum information in presence of noises, and dynamics of mixed classical-quantum systems. We treat the example of a kicked spin ensemble where the kick modulation is governed by discrete classical flows as the Arnold’s cat map and the Chirikov standard map.

Topological properties of adiabatically varied Floquet systems.

Energy or quasienergy (QE) band spectra depending on two parameters may have a nontrivial topological characterization by Chern integers. Band spectra of one-dimensional (1D) systems that are spanned by just one parameter, a Bloch phase, are topologically trivial. Recently, an ensemble of 1D Floquet (time-periodic) systems, double-kicked rotors (DKRs) that are classically nonintegrable and depend on an external parameter, has been studied. It was shown that a QE band spanned by both the Bloch phase and the external parameter is characterized by a Chern integer. The latter determines the change in the mean angular momentum of a state in the band when the external parameter is adiabatically varied by a natural period. We show here, under conditions much more general than in previous works, that the ensemble of DKRs for all values of the external parameter corresponds to a 1D double-kicked particle (DKP) having translational invariance in the position-momentum phase plane. This DKP can be characterized by a second Chern integer, which is shown to be connected with the integer above for the DKR ensemble. This connection is expressed by a Diophantine equation (DE), which we derive. The DE, involving the band degeneracies of the DKR ensemble and of the DKP system, determines the allowed values of the DKR-ensemble integer. In particular, this integer is generically nonzero, showing the general topological nontriviality of the DKR ensemble.

Preparing quasienergy states on demand: A parametric oscillator

We study a nonlinear oscillator, which is parametrically driven at a frequency close to twice its eigenfrequency. By judiciously choosing the frequency detuning and linearly increasing the driving amplitude, one can prepare any even quasienergy state starting from the oscillator ground state. Such state preparation is effectively adiabatic. We find the Wigner distribution of the prepared states. For a different choice of the frequency detuning, the adiabaticity breaks down, which allows one to prepare on demand a superposition of quasienergy states using Landau-Zener-type transitions. We find the characteristic spectrum of the transient radiation emitted by the oscillator after it has been prepared in a given quasienergy state.

Edge-state transport in Floquet topological insulators

Floquet topological insulators are systems in which the topology emerges out of equilibrium when a time periodic perturbation is applied. In these systems one can define quasi-energy states which replace the quilibrium stationary states. The system exhibits its non-trivial topology by developing edge localized quasi-energy states which lie in a gap of the quasi energy spectrum. These states represent a non-equilibrium analogue of the topologically protected edge-states in equilibrium topological insulators. In equilibrium these edge-states lead to very specific transport properties, in particular the two-terminal conductivity of these systems is $2e^2/h$. Here we explore the transport properties of the edge-states in a Floquet topological insulator. In stark contrast to the equilibrium result, we find that the two terminal conductivity of these edge states is significantly different from $2e^2/h$. This fact notwithstanding, we find that for certain external potential strengths the conductivity is smaller than $2e^2/h$ and robust to the effects of disorder and smooth changes to the Hamiltonian's parameters. This robustness is reminiscent of the robustness found in equilibrium topological insulators. We provide an intuitive understanding of the reduction of the conductivity in terms of scattering by photons. This leads us to consider a previously proposed Floquet sum rule which recovers the equilibrium value of $2e^2/h$ for the conductivity when edge states are present. We show that this sum rule holds in our system using both numerical and analytic techniques.

Observation of Floquet States in a Strongly Driven Artificial Atom.

We present experiments on the driven dynamics of a two-level superconducting artificial atom. The driving strength reaches 4.78GHz, significantly exceeding the transition frequency of 2.288GHz. The observed dynamics is described in terms of quasienergies and quasienergy states, in agreement with Floquet theory. In addition, we observe the role of pulse shaping in the dynamics, as determined by nonadiabatic transitions between Floquet states, and we implement subnanosecond single-qubit operations. These results pave the way to quantum control using strong driving with applications in quantum technologies.

Charge and spin edge currents in two-dimensional Floquet topological superconductors

A time periodic driving on a topologically trivial system induces edge modes and topological properties. In this work we consider triplet and singlet superconductors subject to periodic variations of the chemical potential, spin-orbit coupling, and magnetization in both topologically trivial and nontrivial phases, and study their influence on the charge and spin currents that propagate along the edges of the two-dimensional system, for moderate to large driving frequencies. Currents associated with the edge modes are induced in the trivial phases and enhanced in the topological phases. In some cases there is a sign reversal of the currents as a consequence of the periodic driving. The edge states associated with the finite quasienergy states at the edge of the Floquet zone are in general robust, while the stability of the zero quasienergy states depends on the parameters. Also, the spin polarization of the Floquet spectrum quasienergies is strong as for the unperturbed topological phases. It is found that in some cases the unperturbed edge states are immersed in a continuum of states due to the perturbation, particularly if the driving frequency is not large enough. However, their contribution to the edge currents and spin polarization is still significant.

Dramatic enhancement of high-order-harmonic generation via the quantum interference of dipole-transition paths

We present a theoretical study of the high-order-harmonic generation (HHG) from coherent superposition states of a three-level system driven by monochromatic cw laser fields. The HHG spectrum can be enhanced by about ${10}^{5}$ times via tuning the relative phase between the states which are initially prepared. Even if the ionization effects are taken into account, the enhancement factors are still up to about ${10}^{2}$ times. The enhancement is interpreted as the quantum constructive interference of different dipole transitions between quasienergy states. To realize the interference, the intensity of the driving laser field is chosen such that frequencies of the harmonics generated from the different dipole-transition paths, which are dependent on the ac Stark shifted energy separation of quasienergy states, are equal to each other. Moreover, the case studies of HHG spectra from a coherent superposition state of a three-level system show that the enhancement of harmonics is closely related to the harmonic strengths of the dipole transitions, as well as the parity of quasienergy states which are involved in the dipole-transition paths. The reduction of enhancement factors due to ionization effects is also qualitatively analyzed.

Critical quasienergy states in driven many-body systems

We discuss singularities in the spectrum of driven many-body spin systems. In contrast to undriven models, the driving allows us to control the geometry of the quasienergy landscape. As a consequence, one can engineer singularities in the density of quasienergy states by tuning an external control. We show that the density of levels exhibits logarithmic divergences at the saddle points, while jumps are due to local minima of the quasienergy landscape. We discuss the characteristic signatures of these divergences in observables like the magnetization, which should be measurable with current technology.

Quasi-energy of single quantum particles and a Bose-Einstein condensate in a dynamical trap

The quasi-energy states have been found analytically for single quantum particles and an atomic Bose-Einstein condensate in a trap with periodically oscillating walls with a small modulation depth. A resonance is shown to exist as the modulation frequency approaches the difference of the frequencies corresponding to the levels in the unperturbed problem. Quasi-energy splitting and, accordingly, beats with a periodic population exchange between two levels in resonance have been found in the resonant case. Bistability of the response to trap size modulation, when the sustenance (depending on the initial conditions) of various quasi-energy states is possible under the same conditions, has been found for a Bose-Einstein condensate under resonance conditions.

Quantum fluctuations in modulated nonlinear oscillators

We use a modulated oscillator to study quantum fluctuations far from thermal equilibrium. A simple but important nonequilibrium effect that we discuss first is quantum heating, where quantum fluctuations lead to a finite-width distribution of a resonantly modulated oscillator over its quasienergy (Floquet) states. We also discuss the recent observation of quantum heating. We analyze large rare fluctuations responsible for the tail of the quasienergy distribution and switching between metastable states of forced vibrations. We find the most probable paths followed by the quasienergy in rare events, and in particular in switching. Along with the switching rates, such paths are observable characteristics of quantum fluctuations. As we show, they can change discontinuously once the detailed balance condition is broken. A different kind of quantum heating occurs where oscillators are modulated nonresonantly. Nonresonant modulation can also cause oscillator cooling. We discuss different microscopic mechanisms of these effects.

Coherent destruction of tunneling with optimally designed polychromatic external field

A suitably designed polychromatic field with a very low field strength and low frequency (∼10-5 atomic unit) can bring about coherent destruction of tunneling (CDT) in a symmetric double well system. It is analyzed that in the presence of an external perturbation the difference of energy between the two lowest quasi-energy states may increase or decrease depending on the spatial and temporal nature of the perturbation. We have designed sets of polychromatic fields both spatially symmetric and antisymmetric, which cause CDT in symmetric double well system. A stochastic optimizer (Simulated Annealing) has been used to design such a polychromatic field periodic in time. Both spatial symmetry preserving or symmetry breaking perturbations may cause CDT for a symmetric double well potential.

Generation and coherent control of even-order harmonics driven by intense frequency-comb and cavity-mode fields inside a fsEC

We present a theoretical investigation of the multiphoton resonance dynamics driven by intense frequency-comb and cavity-mode fields inside a femtosecond enhancement cavity (fsEC). The many-mode Floquet theorem is employed to provide a nonperturbative and exact treatment of the interaction between a quantum system and laser fields. The quasienergy structure driven by the frequency-comb laser field is modified by coupling the cavity-mode field and the multiphoton resonance processes between modified quasienergy states, resulting in the generation of even-order harmonics. The high-order harmonic generation (HHG) from a two-level system driven by the laser fields can be coherently controlled by tuning the laser parameters. In particular, the tuning intensity of the cavity-mode field allows one to coherently control the HHG power spectra without changing the absolute positions of comb frequencies.