Floquet Eigenstates Research Articles

Emergent strong zero mode through local Floquet engineering

Periodically driven quantum systems host exotic phenomena that often do not have any analog in undriven systems. Floquet prethermalization and dynamical freezing of certain observables, via the emergence of conservation laws, are realized by controlling the drive frequency. Recent experimental developments in synthetic quantum matter, such as superconducting qubits and cold atoms, have opened avenues for implementing local Floquet engineering which can achieve spatially modulated quantum control of states. Here, we uncover the novel memory effects of local periodic driving in a nonintegrable spin-half staggered Heisenberg chain. For a boundary-driven protocol at the dynamical freezing frequency, we show the formation of an approximate strong zero mode, a prethermal quasilocal operator, due to the emergence of a discrete global Z2 symmetry. This is captured by constructing an accurate effective Floquet Hamiltonian using a higher-order partially resummed Floquet-Magnus expansion. The lifetime of the boundary spin can be exponentially enhanced by enlarging the set of suitably chosen driven sites. We demonstrate that in the asymptotic limit, achieved by increasing the number of driven sites, a strong zero mode emerges, where the lifetime of the boundary spin grows exponentially with system size. The nonlocal processes in the Floquet Hamiltonian play a pivotal role in the total freezing of the boundary spin in the thermodynamic limit. The novel dynamics of the boundary spin is accompanied by a rich structure of entanglement in the Floquet eigenstates where specific bipartitions yield an area-law scaling while the entanglement for random bipartitions scales as a volume-law. Our work addresses the long-standing question of the existence of a strong zero mode in a nonintegrable model and elucidates the complex nature of thermalization in locally driven systems. Published by the American Physical Society 2024

Signatures of a sampling quantum advantage in driven quantum many-body systems

A crucial milestone in the field of quantum simulation and computation is to demonstrate that a quantum device can perform a computation task that is classically intractable. A key question is to identify setups that can achieve such goal within current technologies. In this work, we provide formal evidence that sampling bit-strings from a periodic evolution of a unitary drawn from the circular orthogonal ensemble (COE) cannot be efficiently simulated with classical computers. As the statistical properties of COE coincide with a large class of driven analog quantum systems thanks to the Floquet eigenstate thermalization hypothesis, our results indicate the possibility that those driven systems could constitute practical candidates for a sampling quantum advantage. To further support this, we give numerical examples of driven disordered Ising chains and 1D driven Bose–Hubbard model.

Periodically driven model with quasiperiodic potential and staggered hopping amplitudes: Engineering of mobility gaps and multifractal states

In this study, we examine whether periodic driving of a model with a quasiperiodic potential can generate interesting Floquet phases that have no counterparts in the static model. Specifically, we consider the Aubry-Andr\'e model, which is a one-dimensional time-independent model with an on-site quasiperiodic potential ${V}_{0}$ and a nearest-neighbor hopping amplitude that is taken to have a staggered form. We add a uniform hopping amplitude that varies in time either sinusoidally or as a square pulse with a frequency $\ensuremath{\omega}$. Unlike the static Aubry-Andr\'e model, which has a simple phase diagram with only two phases (only extended or only localized states), we find that the driven model has four possible phases for the Floquet eigenstates: a phase with gapless quasienergy bands and only extended states, a phase with multiple mobility gaps separating different quasienergy bands, a mixed phase with coexisting extended, multifractal, and localized states, and a phase with only localized states. The multifractal states have generalized inverse participation ratios that scale with the system size with exponents that are different from the values for both extended and localized states. In addition, we observe intricate reentrant transitions between the different kinds of states when $\ensuremath{\omega}$ and ${V}_{0}$ are varied. The appearance of such transitions is confirmed by the behavior of the Shannon entropy. Many of our numerical results can be understood from an analytic Floquet Hamiltonian derived using a Floquet perturbation theory that uses the inverse of the driving amplitude as the perturbation parameter. In the limit of high frequency and large driving amplitude, we find that the Floquet quasienergies match the energies of the undriven system, but the Floquet eigenstates are much more extended. We also study the spreading of a one-particle wave packet, and we find that it is always ballistic but the ballistic velocity varies significantly with the system parameters, sometimes showing a nonmonotonic dependence on ${V}_{0}$ that does not occur in the static model. Finally, we compare the results for the driven model, which has a static staggered hopping amplitude, with a model that has a static uniform hopping amplitude, and we find some significant differences between the two cases. All of our results are found to be independent of the driving protocol, either sinusoidal or square pulse. We conclude that the interplay of the quasiperiodic potential and driving produces a rich phase diagram that does not appear in the static model.

Periodically driven Rydberg chains with staggered detuning

We study the stroboscopic dynamics of a periodically driven finite Rydberg chain with staggered ($\Delta$) and time-dependent uniform ($\lambda(t)$) detuning terms using exact diagonalization (ED). We show that at intermediate drive frequencies ($\omega_D$), the presence of a finite $\Delta$ results in violation of the eigenstate thermalization hypothesis (ETH) via clustering of Floquet eigenstates. Such clustering is lost at special commensurate drive frequencies for which $\hbar \omega_d=n \Delta$ ($n \in Z$) leading to restoration of ergodicity. The violation of ETH in these driven finite-sized chains is also evident from the dynamical freezing displayed by the density-density correlation function at specific $\omega_D$. Such a correlator exhibits stable oscillations with perfect revivals when driven close to the freezing frequencies for initial all spin-down ($|0\rangle$) or Neel ($|{\mathbb Z}_2\rangle$, with up-spins on even sites) states. The amplitudes of these oscillations vanish at the freezing frequencies and reduces upon increasing $\Delta$; their frequencies, however, remains pinned to $\Delta/\hbar$ in the large $\Delta$ limit. In contrast, for the $|{\bar {\mathbb Z}_2}\rangle$ (time-reversed partner of $|{\mathbb Z}_2\rangle$) initial state, we find complete absence of such oscillations leading to freezing for a range of $\omega_D$; this range increases with $\Delta$. We also study the properties of quantum many-body scars in the Floquet spectrum of the model as a function of $\Delta$ and show the existence of novel mid-spectrum scars at large $\Delta$. We supplement our numerical results with those from an analytic Floquet Hamiltonian computed using Floquet perturbation theory (FPT) and also provide a semi-analytic computation of the quantum scar states within a forward scattering approximation (FSA).

Missing quantum number of Floquet states

We reformulate the Floquet theory for periodically driven quantum systems following a perfect analogy with the proof of Bloch theorem. We observe that the current standard method for calculating the Floquet eigenstates by the quasi-energy alone is incomplete and unstable, and pinpoint an overlooked quantum number, the average energy. This new quantum number resolves many shortcomings of the Floquet method stemming from the quasi-energy degeneracy issues, particularly in the continuum limit. Using the average energy quantum number we get properties similar to those of the static energy, including a unique lower-bounded ordering of the Floquet states, from which we define a ground state, and a variational method for calculating the Floquet states. This is a first step towards reformulating Floquet first-principles methods, that have long been thought to be incompatible due to the limitations of the quasi-energy.

Signatures of multifractality in a periodically driven interacting Aubry-André model

We study the many-body localization (MBL) transition of Floquet eigenstates in a driven, interacting fermionic chain with an incommensurate Aubry-Andr\'e potential and a time-periodic hopping amplitude as a function of the drive frequency ${\ensuremath{\omega}}_{D}$ using exact diagonalization (ED). We find that the nature of the Floquet eigenstates change from ergodic to Floquet-MBL with increasing frequency; moreover, for a significant range of intermediate ${\ensuremath{\omega}}_{D}$, the Floquet eigenstates exhibit nontrivial fractal dimensions. We find a possible transition from the ergodic to this multifractal phase followed by a gradual crossover to the MBL phase as the drive frequency is increased. We also study the fermion autocorrelation function, entanglement entropy, normalized participation ratio (NPR), fermion transport, and the inverse participation ratio (IPR) as a function of ${\ensuremath{\omega}}_{D}$. We show that the autocorrelation, fermion transport, and NPR display qualitatively different characteristics (compared to their behavior in the ergodic and MBL regions) for the range of ${\ensuremath{\omega}}_{D}$ which supports multifractal eigenstates. In contrast, the entanglement growth in this frequency range tend to have similar features as in the MBL regime; its rate of growth is controlled by ${\ensuremath{\omega}}_{D}$. Our analysis thus indicates that the multifractal nature of Floquet-MBL eigenstates can be detected by studying autocorrelation function and fermionic transport of these driven chains. We support our numerical results with a semianalytic expression of the Floquet Hamiltonian obtained using Floquet perturbation theory (FPT) and discuss possible experiments which can test our predictions.

Time molecules with periodically driven interacting qubits

We provide numerical evidence for a temporal quantum-mechanical interference phenomenon: time molecules (TMs). A variety of such stroboscopic states are observed in the dynamics of two interacting qubits subject to a periodic sequence of π-pulses with the period T. The TMs appear periodically in time and have a large duration, δt TM ≫ T. All TMs are characterized by almost zero value of the total polarization and a strong enhancement of the entanglement entropy S up to the maximum value of S ≃ ln 2 indicating the presence of corresponding Bell state. Moreover, the TMs demonstrate a stroboscopic switching between the two maximally entangled Bell states and a slow leakage into other eigenstates. The TMs are generated by the commensurability of the Floquet eigenvalues and the presence of maximally entangled Floquet eigenstates. The TMs remain stable with detuned system parameters and with an increased number of qubits. In particular, we observed the TMs in the dynamics of three interacting qubits, and these TMs show a stroboscopic switching between the four Greenberger–Horne–Zeilinger states. The TMs can be observed in microwave experiments with an array of superconducting qubits.

Topological and dynamical features of periodically driven spin ladders

Studies of periodically driven one-dimensional many-body systems have advanced our understanding of complex systems and stimulated promising developments in quantum simulation. It is hence of interest to go one step further, by investigating the topological and dynamical aspects of periodically driven spin ladders as clean quasi-one-dimensional systems with spin-spin interaction in the rung direction. Specifically, we find that such systems display subharmonic magnetization dynamics reminiscent to that of discrete time crystals (DTCs) at finite system sizes. Through the use of generalized Jordan-Wigner transformation, this feature can be attributed to the presence of corner Majorana $\ensuremath{\pi}$ modes (MPMs), which are of topological origin, in the systems' equivalent Majorana lattice. Special emphasis is placed on how the coupling in the rung direction of the ladder prevents degeneracy from occurring between states differing by a single spin excitation, thus preserving the MPM-induced $\ensuremath{\pi}/T$ quasienergy spacing of the Floquet eigenstates in the presence of parameter imperfection. This feature, which is absent in their strict one-dimensional counterparts, may yield fascinating consequences in future studies of higher dimensional Floquet many-body systems.

Mobility edge and multifractality in a periodically driven Aubry-André model

We study the localization-delocalization transition of Floquet eigenstates in a driven fermionic chain with an incommensurate Aubry-Andr\'e potential and a hopping amplitude which is varied periodically in time. Our analysis shows the presence of a mobility edge separating single-particle delocalized states from localized and multifractal states in the Floquet spectrum. Such a mobility edge does not have any counterpart in the static Aubry-Andr\'e model and exists for a range of drive frequencies near the critical frequency at which the transition occurs. The presence of the mobility edge is shown to leave a distinct imprint on fermion transport in the driven chain; it also influences the Shannon entropy and the survival probability of the fermions at long times. In addition, we find the presence of CAT (linear superposition of localized single particle states) states in the Floquet spectrum with weights centered around a few nearby sites of the chain. This is shown to be tied to the flattening of Floquet bands over a range of quasienergies. We support our numerical studies with a semianalytic expression for the Floquet Hamiltonian (${H}_{F}$) computed within a Floquet perturbation theory. The eigenspectra of the perturbative ${H}_{F}$ so obtained exhibit qualitatively identical properties to the exact eigenstates of ${H}_{F}$ obtained numerically. Our results thus constitute an analytic expression of a ${H}_{F}$ whose spectrum supports multifractal and cat states. We suggest experiments which can test our theory.

Many-body scar state intrinsic to periodically driven system

The violation of the Floquet version of eigenstate thermalization hypothesis is systematically discussed with realistic Hamiltonians. Our model is based on the PXP type interactions without disorder. We exactly prove the existence of many-body scar states in the Floquet eigenstates, by showing the explicit expressions of the wave functions. Using the underlying physical mechanism, various driven Hamiltonians with Floquet-scar states can be systematically engineered.

Extended nonergodic regime and spin subdiffusion in disordered SU(2)-symmetric Floquet systems

We explore thermalization and quantum dynamics in a one-dimensional disordered SU(2)-symmetric Floquet model, where a many-body localized phase is prohibited by the non-abelian symmetry. Despite the absence of localization, we find an extended nonergodic regime at strong disorder where the system exhibits nonthermal behaviors. In the strong disorder regime, the level spacing statistics exhibit neither a Wigner-Dyson nor a Poisson distribution, and the spectral form factor does not show a linear-in-time growth at early times characteristic of random matrix theory. The average entanglement entropy of the Floquet eigenstates is subthermal, although violating an area-law scaling with system sizes. We further compute the expectation value of local observables and find strong deviations from the eigenstate thermalization hypothesis. The infinite temperature spin autocorrelation function decays at long times as $t^{-\beta}$ with $\beta < 0.5$, indicating subdiffusive transport at strong disorders.

Time crystals in the driven transverse field Ising model under quasiperiodic modulation

We investigate the transverse field Ising model subject to a two-step periodic driving protocol and quasiperiodic modulation of the Ising couplings. Analytical results on the phase boundaries associated with Majorana edge modes and numerical results on the localization of single-particle excitations are presented. The implication of a region with fully localized domain-wall-like excitations in the parameter space is eigenstate order and exact spectral pairing of Floquet eigenstates, based on which we conclude the existence of time crystals. We also examine various correlation functions of the time crystal phase numerically, in support of its existence.

Time-crystalline phases and period-doubling oscillations in one-dimensional Floquet topological insulators

In this work, we reported a ubiquitous presence of topological Floquet time crystal (TFTC) in one-dimensional periodically-driven systems. The rigidity and realization of spontaneous discrete time-translation symmetry (DTS) breaking in our model require necessarily coexistence of anomalous topological invariants (0 modes and $\pi$ modes), instead of the presence of disorders or many-body localization. We found that in a particular frequency range of the underlying drive, the anomalous Floquet phase coexistence between zero and pi modes can produce the period-doubling (2T, two cycles of the drive) that breaks the spontaneously, leading to the subharmonic response ($\omega/2$, half the drive frequency). The rigid period-oscillation is topologically-protected against perturbations due to both non-trivially opening of 0 and $\pi$-gaps in the quasienergy spectrum, thus, as a result, can be viewed as a specific "Rabi oscillation" between two Floquet eigenstates with certain quasienergy splitting $\pi/T$. Our modeling of the time-crystalline 'ground state' can be easily realized in experimental platforms such as topological photonics and ultracold fields. Also, our work can bring significant interests to explore topological phase transition in Floquet systems and to bridge the gap between Floquet topological insulators and photonics, and period-doubled time crystals.

Computing Floquet Hamiltonians with symmetries

Unitary matrices arise in many ways in physics, in particular as a time evolution operator. For a periodically driven system one frequently wishes to compute a Floquet Hamilonian that should be a Hermitian operator $H$ such that $e^{-iTH}=U(T)$ where $U(T)$ is the time evolution operator at time corresponding the period of the system. That is, we want $H$ to be equal to $-i$ times a matrix logarithm of $U(T)$. If the system has a symmetry, such as time reversal symmetry, one can expect $H$ to have a symmetry beyond being Hermitian. We discuss here practical numerical algorithms on computing matrix logarithms that have certain symmetries which can be used to compute Floquet Hamiltonians that have appropriate symmetries. Along the way, we prove some results on how a symmetry in the Floquet operator $U(T)$ can lead to a symmetry in a basis of Floquet eigenstates.

Quantized large-bias current in the anomalous Floquet-Anderson insulator

We study two-terminal transport through two-dimensional periodically driven systems in which all bulk Floquet eigenstates are localized by disorder. We focus on the Anomalous Floquet-Anderson Insulator (AFAI) phase, a topologically-nontrivial phase within this class, which hosts topologically protected chiral edge modes coexisting with its fully localized bulk. We show that the unique properties of the AFAI yield remarkable far-from-equilibrium transport signatures: for a large bias between leads, a quantized amount of charge is transported through the system each driving period. Upon increasing the bias, the chiral Floquet edge mode connecting source to drain becomes fully occupied and the current rapidly approaches its quantized value.

Formation and suppression of nonthermal statistics in peridically driven quantum Ising models

In classic statistical physics, an isolated system corresponds to a constant energy shell in the phase space, which can be described by the microcanonical ensemble. While, for an isolated quantum system, the conventional treatment is to subject the system to a narrow energy window in the Hilbert space instead of the energy shell in classical phase space, and then confine the participating eigen states of system wave function in the narrow window, so that the microcanonical ensemble can be recovered in the framework of quantum mechanics. Apart from the traditional theory, there is a more self-consistent description for the isolated quantum system, that is, the quantum microcanonical (QMC) ensemble. The QMC ensemble abandons the narrow energy window assumption, and allows all the eigen states to contribute to the system wave function on condition that the system average energy is fixed at a given value. At the same time, the total occupation probability of these eigen states is conserved to unity. The most probable probability distribution obtained in the Hilbert space for an isolated quantum system according to the constraints specified above is called the QMC statistics. There is a clear difference between the QMC distribution and the traditional Gibbs distribution having an exponential form. Through the external periodic drives, an isolated quantum system may produce the QMC distribution, which is a consequence of the interplay between internal origins and external drives. In this paper, we investigate the conditions for the formation and suppression of QMC distribution by using the exact diagonalization method based on the one-dimensional Ising model. We start with the one-dimensional Ising model and focus on three different cases of periodic drives: systems under vertical (along the &lt;i&gt;z&lt;/i&gt; axis), horizontal (along the &lt;i&gt;x&lt;/i&gt; axis), horizontal magnetic field together with random internal (along the &lt;i&gt;y&lt;/i&gt; axis) magnetic field. For all these three cases, the external magnetic fields are set to be ordinary rectangular pulses and the Gibbs distributions are taken as the initial states. We then study the evolutions and their asymptotic tendencies to the QMC distributions of the eigen state occupation probability under the effect of external periodic magnetic field. The results show that under the vertical magnetic field, the eigen state occupation probability does not change, and the system cannot produce the QMC distribution; under the horizontal magnetic field, the system tends to display a QMC distribution, but only partly; under horizontal and random internal magnetic fields at the same time, the transition to QMC distribution can be fully realized, and finally the system is almost completely thermalized. In order to clarify the different behaviors of the Ising model in the three cases, we also calculate the information entropy of the eigen state of Floquet operator in the eigen representation of the unperturbed Hamiltonian. We find that as the information entropy of the Floquet eigen state increases, the convergence to the QMC distribution in the Hilbert space is improved. We also notice that the mechanism for the emergence of QMC distribution is closely related to the thermalization effect of the isolated quantum system. Our analyses show that when the magnetic field is vertical, it cannot trigger the thermalization of the system. When the magnetic field is horizontal, the system becomes partly, but not completely, thermalized. When we add a horizontal periodic magnetic field and a random internal magnetic field at the same time, the system can be completely thermalized to infinite temperature. Thus, the asymptotic behavior towards the QMC statistics is a reflection of the fact that the isolated quantum system is thermalizable under periodic drives.

Dynamics of quasiperiodically driven spin systems

We study the stroboscopic dynamics of a spin-S object subjected to δ-function kicks in the transverse magnetic field which is generated following the Fibonacci sequence. The corresponding classical Hamiltonian map is constructed in the large spin limit, S→∞. On evolving such a map for large kicking strength and time period, the phase space appears to be chaotic; interestingly, however, the geodesic distance increases linearly with the stroboscopic time implying that the Lyapunov exponent is zero. We derive the Sutherland invariant for the underlying SO(3) matrix governing the dynamics of classical spin variables and study the orbits for weak kicking strength. For the quantum dynamics, we observe that although the phase coherence of a state is retained throughout the time evolution, the fluctuations in the mean values of the spin operators exhibit fractality which is also present in the Floquet eigenstates. Interestingly, the presence of an interaction with another spin results in an ergodic dynamics leading to infinite temperature thermalization.

Steady states of interacting Floquet insulators

Floquet engineering offers tantalizing opportunities for controlling the dynamics of quantum many body systems and realizing new nonequilibrium phases of matter. However, this approach faces a major challenge: generic interacting Floquet systems absorb energy from the drive, leading to uncontrolled heating which washes away the sought after behavior. How to achieve and control a non-trivial nonequilibrium steady state is therefore of crucial importance. In this work, we study the dynamics of an interacting one-dimensional periodically-driven electronic system coupled to a phonon heat bath. Using the Floquet-Boltzmann equation (FBE) we show that the electronic populations of the Floquet eigenstates can be controlled by the dissipation. We find the regime in which the steady state features an insulatorlike filling of the Floquet bands, with a low density of additional excitations. Furthermore, we develop a simple rate equation model for the steady state excitation density that captures the behavior obtained from the numerical solution of the FBE over a wide range of parameters.

Floquet-theoretical formulation and analysis of high-order harmonic generation in solids

By using the Floquet eigenstates, we derive a formula to calculate the high-order harmonic components of the electric current (HHC) in the setup where a monochromatic laser field is turned on at some time. On the basis of this formulation, we study the HHC spectrum of electrons on a one-dimensional chain with a staggered potential to study the effect of multiple sites in the unit cell such as the systems with charge density wave order. With the help of the solution for the Floquet eigenstates, we analytically show that two plateaus of different origins emerge in the HHC spectrum. The widths of these plateaus are both proportional to the field amplitude, but inversely proportional to the laser frequency and its square, respectively. We also show numerically that multistep plateaus appear when both the field amplitude and the staggered potential are strong.

Nonreciprocal Optical Dissipation Based on Direction-Dependent Rabi Splitting

We consider a ring resonator with two resonant modes undergoing dynamic modulation. Using the Floquet analysis, we show that with a proper choice of modulation profile, such a resonator can achieve a direction-dependent Rabi splitting of its Floquet eigenstates, which results in direction-dependent dissipation and nonreciprocal transmission for waves. We numerically demonstrate this nonreciprocal effect and show that a realistic refractive index modulation of the ring can result in significant transmission in one direction and near complete dissipation in.