We research the properties of Floquet prethermal says in two-dimensional Mott-insulating Hubbard clusters under consistent optical excitation. With exact-diagonalization simulations, we show that Floquet prethermal says arise not just off resonance, yet also for resonant excitation, gave a small area amplitude. In the resonant situation, the long-lived quasi-stationary Floquet says are identified by Rabi oscillations of observables such as double occupation and kinetic power. At stronger fields, thermalization to unlimited temperature is observed. We administer explanations to these results by implies of time-dependent perturbation concept. The primary findings are substantiated by a finite-dimension analysis.

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Coherent optical manipulation of issue is a growing area of research because of the advancement of intense laser sources. Photoinduced phase transitions in many-body systems are currently feasible with non-thermal procedures without loss of quantum jiyuushikan.org1,2,3,4,5,6. For instance, current mid-infrared pump-probe experiments revealed exciting phenomena such as light-induced superconductivity7,8,9, ultrafast structural transitions10,11, and also metasteady charge ordering12, which are driven by brief optical excitation of phonons. More straight manipulation of digital says have the right to be realized through the so-called Floquet engineering by continuous optical excitation13,14,15. Notable success in this direction are the prediction and realization of dynamical localization16,17,18 and also topological band structures19,20,21,22.

In the high-frequency limit, Floquet concept gives an excellent summary of low-power phenomena in terms of effective static Hamiltonians. In this limit, heating is a fairly minor effect also in interacting units, considering that the drive frequency is amethod from any characteristic absorption energy of the system23,24,25. Therefore, long-lived quasi-stationary claims, which appear to be “thermalized”, are realized till the much later on heating time scales26,27; such states are termed as Floquet prethermal claims (FPSs). On the various other hand also, when the drive frequency is close to an energy scale of a generic interacting device, or its submultiples, heating is intended due to possible photon absorption procedures. In isolated units, this leads to thermalization to infinite temperature28. Thus, resonant excitation provides the evaluation by the Floquet image more complex.

As an different to the high-frequency limit, Floquet prethermalization is likewise observed in systems cshed to integrability, in which quasi-integrals of motion constrain the dynamics for finite however long times29,30,31,32,33,34,35,36,37,38,39,40,41,42. In certain, units in the Mott-insulating phase of the infinite-dimensional Hubbard models were shown to screen exceptionally long-lived prethermal says, also for frequencies cshed to resonance36,41. Further examination of these long-lived quasi-secure says is of great prominence to advance the Floquet design protocols for generic frequencies. To this finish, we usage a realistic finite lattice geomeattempt and investigate the stability and also controllcapability of possible FPSs.

In this occupational, we examine Floquet prethermalization in two-dimensional (2D) Hubbard clusters under consistent optical excitation. The setup can be realized in solid-state devices such as quantum dot arrays under laser fields43,44,45,46 or in ultracold atoms in shaking optical lattices26,27,47. Starting from the Mott-insulating state, we calculate the thrust time development by exact diagonalization. We find that Floquet prethermal says arise also at frequencies resonant through absorption-top energies, offered a little area amplitude. Remarkably, these long-lived quasi-secure says show Rabi oscillations of observables such as double occupation and kinetic energy. The spectral density reflects that the mechanism oscillates between the ground state and the one-photon excited state. For stronger excitation, the system goes to the infinite-temperature state, in basic, and the spectral thickness is spread over many type of excited states. We elucidate the beginning of the Rabi oscillations of the FPSs via the help of time-dependent perturbation concept. A finite-size evaluation shows that the phenomena are robust for miscellaneous lattice geometries as much as fourteenager sites, and possibly for bigger systems.


Two-dimensional cluster geometries. In this work-related, we think about mostly the cluster through (L=10) sites with a regular boundary (left), while various other geometries (right) are also provided for the finite-dimension evaluation.

$$eginaligned eginaligned H(t)&= H_J (t) + U amount _i n_iuparrow n_idownarrowhead ,\ H_J(t)&= - sum _mathinner langle i,j angle , sigma J_0 e^-ifracehbar int _varvecR_j^varvecR_i dvarvecr cdot varvecA(t) c^dagger _i sigma c_jsigma + message H.c.\&= - sum _mathinner langle i,j angle , sigma J_0 e^-i fracehbar varvecl_ij cdot varvecA(t) c^dagger _isigma c_jsigma + message H.c., endaligned endaligned$$

where (c_i sigma ) and (c^dagger _i sigma ) are annihilation and development operators at website i with spin (sigma), and (varvecl_ij = varvecR_i - varvecR_j) the bond vector connecting sites i and also j at positions (varvecR_i) and also (varvecR_j). Oppowebsite spin densities (n_i sigma = c^dagger _isigma c_i sigma ) are topic to a neighborhood Hubbard repulsion U. We take (hbar = c = e = 1), and usage (J_0 = 1) and also (U = 6), which ensures that the initial ground state is a Mott insulating state. In the adhering to, we emphasis on a two-dimensional cluster through (L=10) sites via a routine boundary condition (Fig. 1)48. Results hold qualitatively unchanged for other geometries and also sizes as shown below. Optical excitation is induced by electric fields (varvecE(t) = - partial varvecA(t)/partial t) alengthy the x-axis, wbelow the vector potential (varvecA(t)) is switched on as

$$eginaligned eginaligned varvecA(t)&= A(t) hatx = A_0 cos (omega _d t) f(t) hatx, \ f(t)&= {left{ eginarrayll exp left< - 4log (2) left( fract- au _0 au ight) ^2 ight> & t

The create of the excitation is encouraged by experiments through ultracold atoms, where driving alengthy one axis is typically used26,27. Our envelop feature f(t) is preferred for computational convenience, while a continuous drive amplitude after a linear ramp is offered in Refs.26,27. We have additionally checked that fields alengthy the diagonal direction ((hatx+haty)) lug comparable outcomes.

The initial ground state is calculated by the conventional Lanczos method48,49,50,51, and the subsequent time-evolution is imposed by the Krylov-space approach through time step (dt= 0.02) and Krylov measurement (M = 20)52,53,54,55. For each time action, we use the midallude Hamiltonian, while a greater order Magnus growth is also possible56.

To characterize the time-progressed states, we calculate double occupation (d(t) = amount _i mathinner langle n_iuparrow n_idownarrow angle /L) and kinetic power (E_message kin(t) = mathinner langle H_J(t) angle /L). For ultracold atoms in an optical lattice, these quantities can be measured by radio-frequency spectroscopy and also time-of-flight measurements, respectively26,27,57. Because time development is unitary, isolated devices do not thermalize as a whole. For a huge enough system dimension, however, neighborhood observables perform thermalize, as explained by the eigenstate-thermalization hypothesis28. In propelled isolated units, this leads towards the infinite-temperature state, which is identified by (d = 0.25) and (E_ ext kin=0). We further present two quantities to achieve a in-depth picture of the moment evolution. One is the spectral thickness of the waveattribute (mathinner ),

$$eginaligned S(t,omega ) = sum _n |mathinner langle psi (t) angle |^2 delta (omega -epsilon _n), endaligned$$

wright here (mathinner n angle ) and also (epsilon _n) are the eigenstates and eigenenergies (via respect to the ground state) of the zero-area Hamiltonian. Such a spectral decomplace deserve to be approximately calculated by the Lanczos method. The other amount is efficient dimension39 or inverse participation ratio58,

$$eginaligned kappa (t) = frac1mathinner langle n angle , endaligned$$

which quantifies just how many kind of of the power eigenclaims are populated. For instance, in the ground state, the spectral density is peaked at zero energy and also (kappa =1), whereas if the system oscillates in between two levels we have actually (kappa approx 2) and two peaks in the spectral density. The infinite-temperature limit might be identified as the spreading of the wavefeature over all the claims of the Hilbert room (mathbb H) and also hence (kappa =dim mathbb H). We note that Loschmidt amplitudes have the right to be used to calculate the spectral density as well59,60.


Linear absorption spectrum (alpha (omega )) in Eq. (5) for the two-dimensional instance via (L=10) sites and also (U/J_0 = 6).

Now we revolve to the two-dimensional cluster with (L=10). As a basis to translate the time-dependent phenomena, let us initially look at the direct absorption spectrum50,61

$$eginaligned alpha (omega ) = - frac1pi L^2 mathfrak Imathinner I_x frac1omega - H_0 + epsilon _0 I_x mathinner 0 angle , endaligned$$

wbelow (I_x) is the full present operator along the x-axis, namely along the direction of the optical field . (H_0) is the unperturbed zero-area Hamiltonian in Eq. (1). In Fig. 2, we plot (alpha (omega )) for (U/J_0 = 6) via small broadening aspect (eta = 1/L). Tright here are 3 absorption peaks roughly U at (omega = 4.52, 5.67), and also 6.6, which reexisting transitions to the one-photon excited says.

Double occupation


Average double occupation (overlined(t)) as a function of drive frequency (omega _d) and amplitude (A_0) in the 2D Hubbard cluster ((L=10)). Time average is taken in between (t = 30) and 600. The one- and two-photon resonances are suggested by the dashed and also dash-dotted lines, respectively.


Time advancement of double occupation at the one-photon resonance (omega _d = 4.52) (a), at the two-photon resonance (omega _d=2.4) (b), and also off resonance (c,d) in the 2D Hubbard cluster ((L=10)). (e,f) Amplitude dependence of the Rabi frequency (Omega _message R) at resonance.

First, we comment on the average double occupation after the optical fields are turned on. Here the onset of the excitation is set to be ( au _0 = 10) through a ramp of width ( au = 0.1); we have actually checked that the stable claims are not sensitive to the particular value of ( au). This additionally indicates that the phase of the drive is not decisive for the secure claims. The time-averaged worths are plotted as a duty of drive frequency (omega _d) and also amplitude (A_0) in Fig. 3.

As supposed, we find solid absorption at frequencies resonant through the three peaks in the optical absorption spectrum (omega _dapprox 4.52), 5.67, and 6.63. We alert, but, that also at these resonance problems, heating is still moderate for small area amplitudes. We uncover added two-photon resonances at around half the frequencies of one-photon resonances. These resonances stem from second-order processes not caught by the linear absorption spectrum, Eq. (5), and involve says through about the very same parity under reflection, as we comment on listed below in even more detail. At strong fields, the double occupation saturates to the infinite-temperature value, (d = 0.25), at all frequencies. The appearance of the infinite-temperature state is in comparison to the two-site model, wright here pure oscillatory actions are observed as a result of the tiny measurement of the Hilbert room (watch the supplementary material). We likewise note that tbelow is no resonance at frequencies commensurate to U, i.e., (omega _d = U/n) with (n in mathbb N), in comparison to36, which provides an boundless dimensional lattice and a strong area amplitude. The discrepancy is probably due to the different lattice geometries or the impact of finite temperatures.

In Fig. 4a we plot, for miscellaneous field amplitudes, the moment advancement of double occupation at the one-photon resonance (omega _d=4.52). Comparable outcomes are acquired at the other one-photon resonances. Regardless of the resonance problem, for weak areas, we uncover FPSs wbelow the double occupation oscillates around a consistent value approximately (t=2000) (not shown in the figure). The oscillations have the right to be considered as Rabi oscillations, given that their frequencies, (Omega _ ext R), boost approximately lialmost through drive amplitude (A_0) . Below, we explain the linear dependence by time-dependent perturbation theory. Such a FPS through persistent oscillations is among our main findings. In contrast, for solid areas, the oscillations are on optimal of a boosting value saturating about the infinite-temperature limit. Kinetic power viewpoints to zero in this limit (view the supplementary material). We have actually confirmed that the very same qualitative habits holds as much as (L=14) sites. As the mechanism size grows, among the one-photon excited states, which has actually excitonic jiyuushikan.org because of the spin-polaron formation, becomes prominent62,63,64. Therefore, we also suppose that larger systems show Rabi oscillations by using a resonant light to this state.

Comparable oscillations are uncovered at the two-photon resonance (omega _d=2.40) for weak excitation . In comparison to the one-photon resonance, the Rabi frequency counts nonlialmost on the field amplitude , because of the two-photon absorption process. We find weak dependence of (Omega _R) on (A_0) at little field amplitudes, which is maybe because of the slight deviation of (omega _d) from the resonance. In enhancement to the slow Rabi oscillations, tright here exist fast oscillations via frequency (omega approx 4.5), which presumably comes from the systematic superplace of the ground state and also the excited says through absorption-peak energies. At large area amplitudes, heating and also thermalization to boundless temperature occur.

Ameans from the resonances, Rabi oscillations disappear and also 2 distinctive actions arise instead. At weak areas and low frequencies , the double occupation rises nearly livirtually in time. We may take into consideration this linear production of double occupation as an analog of the DC response65,66. On the other hand, at weak areas and high frequencies , FPSs without Rabi oscillations appear as in36,41. Their steady-state values depfinish on the drive amplitude (A_0).

As displayed in the supplemental material, at resonances, we find monotonic growth of the double occupation and the kinetic energy as the drive amplitude boosts. Thus, the complete energy additionally boosts monotonically until the device reaches the infinite-temperature state. This amplitude dependence contrasts to a propelled two-level device, wbelow absorption saturation occurs for intermediate and huge drive amplitudes67. The distinction originates from the lack of spontaneous emission and the easily accessible multi-photon excited states in our design.

Spectral density


Spectral densities (S(t, omega )) for frequency (omega _d = 4.3) and amplitude (A_0 = 0.03) (a), (A_0 = 0.06) (b), and (A_0 = 0.3) (c) at times multiples of the drive duration (T_d = 2pi /omega _d). The delta attributes are approximated by Lorentzian functions. For (A_0 = 0.03), the approximate expression, Eq. (10), is also plotted (solid line). Effective dimensions (kappa) calculated from each spectrum are likewise displayed.

In order to deepen our knowledge of the Rabi oscillations and also the dependence on the drive frequency and amplitude, we think about the spectral density (S(t, omega )) of the time-evolved wavefunction, Eq. (3), and the reliable measurement (kappa), Eq. (4).

Figure 5 reflects the spectral density (S(t, omega )) at times multiple of the drive duration (T_d=2pi /omega _d) for drive frequency cshed to the one-photon resonance (omega _d=4.3). At small amplitude (A_0=0.03) the weight oscillates largely between the ground state and the one-photon excited state. Hence, the efficient dimension (kappa) continues to be listed below 2. During the oscillations, the mechanism repeatedly absorb a photon and release it through stimulated emission68. Due to the fact that our model treats only the classic light areas, spontaneous emission does not happen. Therefore, the spontaneous decay of the population in the excited claims is not observed. Increasing to (A_0=0.06) , weight on the two-photon excited states (omega approx 2 omega _d) becomes not negligible, while still the weight is localized on a couple of states, i.e., (kappa) stays below 3. Finally, at strong field (A_0=0.30) , the spectral thickness spreads over a broad energy range due to multiple-photon absorption and (kappa > 10), definition that the mechanism has actually reached the infinite-temperature state, i.e., the double occupation and also the kinetic energy reaches the infinite-temperature values. We note that as a result of the limited variety of Krylov dimensions (below (M=50)), the reliable dimension calculated by the Lanczos strategy cannot recreate the mass infinite-temperature actions, (kappa ightarrowhead dim mathbb H).

The spectrum density demonstprices the discrete jiyuushikan.org of the multi-photon excited states and also progressive filling of these high-energy states via increasing drive amplitudes, which resembles the state-filling effect in semiconductor quantum dots. In the latter, the discrete power levels are developed by strong spatial confinement of electrons and also stronger photoexcitation leads to larger populaces in the high-power levels69,70.

Time-dependent perturbation theory

Here we talk about the physical origin of the Rabi oscillations at the resonant excitation and the amplitude dependence utilizing time-dependent perturbation concept. For this purpose, we define the unperturbed zero-area Hamiltonian (H_0) and also the time-dependent perturbation,

$$eginaligned eginaligned V(t)&= H(t) - H_0 \&= - amount _mathinner langle i,j angle , sigma J_0 left( e^-i fracehbar varvecl_ij cdot varvecA(t) - 1 ight) c^dagger _isigma c_jsigma + message h.c.\&= - I_x sin - K_x left 1 - cos ight , endaligned endaligned$$
$$eginaligned I_x = - isum _i, sigma J_0 c^dagger _isigma c_i-hatx, sigma + ext h.c., qquad K_x = - sum _i, sigma J_0 c^dagger _isigma c_i-hatx, sigma + message h.c.. endaligned$$

which represents the paramagnetic and diamagnetic contributions. Applying first-order time-dependent perturbation theory to the ground state (mathinner 0 angle ), we discover the shift amplitude to an additional eigenstate of the unperturbed Hamiltonian (mathinner n angle ) as

$$eginaligned eginaligned mathinner langle psi (t) angle &simeq -i int _0^t mathinner langle V(t) angle e^i epsilon _n t" dt" \&simeq i mathinner langle n angle int _0^t A(t") e^i epsilon _n t" dt" + fraci2 mathinner langle n angle int _0^t ^2 e^i epsilon _n t" dt". endaligned endaligned$$

The first term defines the one-photon absorption procedure. If the matrix element (mathinner langle I_x angle ) is peaked at one or a few eigenstates through equivalent energies, the form has actually the same structure to the two-level system, and offers Rabi oscillations under consistent excitation (A(t) = A_0 cos (omega _d t)) through (omega _d simeq epsilon _n)68,71. From the straight absorption spectrum, Fig. 2a, we check out that this matrix facet has actually discrete peaks, which then describes the oboffered Rabi oscillations. Within the rotating wave approximation, the Rabi frequency is offered by (Omega _R = sqrt(omega _d - epsilon _n)^2 + D^2) through (D propto A_0 mathinner langle I_x angle ). The second term, in addition to the second-order terms, defines two-photon excitation, which gives rise to the nonlinear amplitude dependence (sim A_0^2) and also to the boost of efficient measurement by spreading the weight of the wavefeature to more states . These terms are responsible for the two-photon resonance at (omega _d simeq epsilon _n/2).

At the lowest order in A(t) (i.e., retaining only the paramagnetic term), the spectral thickness is uncovered to be

$$eginaligned S(t, omega ) simeq |mathinner langle 0 angle |^2 delta (omega )+ amount _n >0 |mathinner langle I_x angle |^2 left| int _0^t A(t") e^i epsilon _n t" dt" ight| ^2delta (omega - epsilon _n). endaligned$$

Note that the ground state occupation is calculated as (|mathinner langle psi (t) angle |^2 = 1- amount _n>0 |mathinner langle n angle |^2) to avoid the use of second-order perturbation. In Fig. 5a, the approximate expression, Eq. (10), is compared with the spectral thickness obtained by the Lanczos approach at a weak amplitude, which well reproduces the Rabi oscillations. On the various other hand also, for resonant excitation, the change probcapability becomes also huge and also provides the perturbative expression invalid. For bigger amplitudes, greater orders in perturbation theory are compelled.

Expressions comparable to Eq. (9) can be obtained for various other forms of time-dependent perturbation. An broadly stupassed away instance is the driven-interaction protocol41,58,

For a weak interactivity (J_0/U gtrsim 1), the matrix aspect (mathinner langle n angle ) is not peaked nor the excited states are degeneprice, and also we execute not suppose Rabi oscillations. However, in the limit of (J_0/ U ll 1), the excited says are nearly degeneprice, and the lower and upper Hubbard bands develop an efficient two-level mechanism causing Rabi oscillations58.

Up to this suggest, we have elaborated on a two-dimensional cluster via (L=10) sites. The vital question is if the Floquet prethermal says that we find for the (L=10) cluster endure for larger devices. For instance, a number of studies show that the crucial drive amplitude to reach the infinite-temperature state vanishes in the thermodynamic limit for different models38,40. In order to inspect the robustness of the outcomes acquired over, right here we simulate miscellaneous sizes of one-dimensional lattices (chains), ladders, and other two-dimensional clusters (see Fig. 1). For each lattice, the drive frequency is taken to be at the lowest one-photon excitation peak in the straight absorption spectrum. In the inset of Fig. 6, we show an instance of time advancement of double occupation for the (L=14) ladder. We confirm that the Rabi oscillations appear for weak drive amplitudes, while the system viewpoints to the infinite-temperature state for bigger amplitudes.

Critical field amplitude (omega _d A_0^*) to reach the infinite-temperature state for various lattice geometries. The error bars show the regions wbelow the double occupation does not reach a secure state within the simulation time (t le 1000). Tbelow is no solid size dependence for (L ge 10). The incollection mirrors time advancement of double occupation for the (L=14) ladder.

We estimate the crucial area amplitudes that sepaprice the prethermal routine (d(t) and also the infinite-temperature regimen (d(t) approx 0.25), based on the time-averaged double occupation close to the end of the simulations (t sim 1000). For comparichild of various lattices, we convert the drive amplitude (A_0) to the area amplitude (E_0 = omega _d A_0). The acquired instrumental area amplitudes for assorted lattices are plotted in Fig. 6. For (L , because of the limited available absorption processes, the crucial area amplitudes are rather big compared to (J_0). This is constant via the two-site model (check out the supplementary material). For larger devices (Lge 10), the crucial field amplitudes are of the order of (J_0). However, tbelow is no methodical decrease as the system dimension grows as Refs.38,40, and also thus we mean that the results are valid for larger units.

In this work-related, we have actually elaborated on the properties of prethermal Floquet says in optically excited Hubbard clusters.We have actually demonstrated that the prethermal says exist also at resonance, as far as the drive strength is weak. In particular, the Floquet prethermal states at resonance involve Rabi oscillations, whose frequency scales linearly with the area amplitude at one-photon resonances, and also with the square amplitude at two-photon resonances. Experimentally, these outcomes can be tested by using ultracold atoms or quantum dot arrays. We have actually elucidated the beginning of the Rabi oscillations by time-dependent perturbation theory. A finite-size evaluation has shown the robustness of the main outcomes.

The observation of Rabi oscillations in this model suggests feasible systematic manipulation of quantum many-body claims. Rabi-favor oscillations are intended to be a basic phenomenon for resonantly thrust Hubbard-choose clusters; for instance, comparable oscillations are found after short pulses72,73. Considering the systematic jiyuushikan.org of Rabi oscillations, the device can be unified via various forms of drive pulses and provided for optical control of associated quantum says. We expect that presenting dissipation or spontaneous emission68,74 to our design will certainly weaken the Rabi oscillations while stabilizing the prethermal Floquet states, and it is an amazing open question to investigate their impacts on the final secure states.


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This occupational is sustained by Research Foundation for Opto-Science and Technology and also by Georg H. Endress Foundation. The authors acunderstanding assistance by the state of Baden–Württemberg with bwHCOMPUTER and also the Germale Research Foundation (DFG) through give no INST 40/575-1 FUGG (JUSTUS 2 cluster). We thank S. Stumper for constructive discussions on the manuscript.


Open Access funding permitted and organized by Projekt DEAL.

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Institute of Physics, University of Freiburg, Hermann-Herder-Str. 3, 79104, Freiburg, Germany

Junichi Okamoto

EUCOR Centre for Quantum Science and also Quantum Computing, College of Freiburg, Hermann-Herder-Str. 3, 79104, Freiburg, Germany

Junichi Okamoto

Max Planck Institute for the Physics of Complex Equipment, Nöthnitzer Straße 38, 01187, Dresden, Germany

Francesco Peronaci


J.O. planned the job, enforced the simulations, and analyzed the data. F.P. examined the results. Both authors debated the outcomes and wrote the manuscript.

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Correspondence to Junichi Okamoto.

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Okamoto, J., Peronaci, F. Floquet prethermalization and Rabi oscillations in optically excited Hubbard clusters. Sci Rep 11, 17994 (2021). https://doi.org/10.1038/s41598-021-97104-x

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Received: 12 May 2021

Accepted: 18 August 2021

Published: 09 September 2021

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