The optimal frequency window for Floquet engineering in optical lattices

The concept of Floquet engineering is to subject a quantum system to time-periodic driving in such a way that it acquires interesting novel properties. It has successfully been employed in atomic quantum gases in driven optical lattices. Typically, Floquet engineering is based on two approximations. On the one hand, it is assuming that resonant excitations to high-lying states above some energy gap are suppressed for sufficiently low driving frequencies, so that the system can be described within some low-energy subspace (e.g. spanned by the lowest Bloch band of a lattice). On the other hand, the driving frequency is also assumed to still be large compared to the typical energy scale of this low-energy subspace, so that it does not resonantly create excitations within this space. Eventually, on some time scale τ , deviations from these approximations will make themselves felt as unwanted heating. Floquet engineering, thus, requires a window of driving frequencies, where both types of heating processes are suppressed on the experimentally relevant time scale. In this paper, we theoretically investigate the existence of such an optimal frequency window, using the example of interacting bosons in a shaken optical lattice. We find that the maximum value of τ , measured in the experimentally relevant unit of the tunneling time, increases with the lattice depth.


I. INTRODUCTION
The idea of Floquet engineering is to subject a quantum system to time-periodic driving in such a way that it acquires interesting novel properties that are difficult to achieve by other means. This concept has been applied very successfully applied to systems of atomic quantum gases in optical lattices [1]. The fact that these systems are extremely clean, well isolated from their environment, and highly tunable also in a time-dependent fashion makes them an ideal platform for studying coherent many-body dynamics. Examples for Floquet engineering in optical lattices include, among others, dynamic localization [2,3], photon-assisted tunneling [4][5][6][7][8], the control of an interaction-induced quantum phase transition [9,10], the creation of kinetic frustration [11,12], artificial magnetic fields [13][14][15][16][17][18][19][20][21][22] and topological band structures [23][24][25].
A simple explanation of the basic concept underlying Floquet engineering is often given by considering the onecycle time-evolution operator where T denotes time ordering andĤ(t) =Ĥ(t + T ) a timer-periodic Hamiltonian. The fact that this operator is unitary allows one, at least formally, to express it in terms of an hermitian operatorĤ F that is called called Floquet Hamiltonian, This effective time-independent HamiltonianĤ F governs the time evolution of the system, when it is monitored stroboscopically in integer steps of the driving period. Thus, naively one can expect that the driven system behaves as some effective non-driven system described by the HamiltonianĤ F . Note that the above reasoning applies to small quantum systems only. The situation in many-body systems is more complex. Here the eigenstates ofĤ F will typically be superpositions of states having very different energies. This is a consequence of resonant coupling (in a large system resonances will be ubiquitous). The lack of energy conservation, which is reflected in such resonant coupling, suggests that in the thermodynamic limit the system approaches an infinite-temperature-like state, so that in the sense of eigenstate thermalization the eigenstates ofĤ F represent an infinite-temperature ensemble [26,27]. From this point of view, the Floquet Hamiltonian does not seem to be a suitable object for engineering interesting novel system properties.
However, while the very definition of a Floquet Hamil-tonianĤ F via Eq. (2) as a sufficient starting point for Floquet engineering is too optimistic, the conclusion suggested at the end of the previous paragraph is too pessimistic. In fact, Floquet engineering can be a useful concept also in many-body quantum systems. This fact is related to the observation that in some parameter regimes the time scale τ associated with the detrimental resonance effects that are responsible for heating can become rather long. Thus, on times shorter than τ , we might be able to engineer and study interesting drivinginduced phsyics, before eventually heating sets in. In such a regime one can derive an approximate effective HamiltonianĤ eff that captures the impact of the periodic drive apart from detrimental heating effects. The standard strategy employed for this purpose involves two steps [9]. The first step is given by a low-frequency approximation. In condensed-matter physics often low-energy approximations are made. For example, higher-lying orbital states spanning Bloch bands of a lattice above a band gap are neglected, when deriving Hubbard-type tight-binding models, or doublon-holon excitations above the charge gap of a Mott insulator are eliminated, in order to derive spin Hamiltonians. While in static systems, such lowenergy approximations are often very well justified, this is not the case in periodically driven systems. Here resonant excitations to the neglected excited states can occur, where the drive provides one or several energy quanta ω. Such processes contribute to the aforementioned detrimental heating. However, when the driving frequency is low compared to the gap, they can be slow. By estimating the associated heating rate [28][29][30][31][32][33][34][35], we might be able to argue that we can neglect higher-lying states on the time scale of an experiment.
The second step is given by a high-frequency approximation. Let us assume that according to the first step we are able to neglect, say, higher-lying Bloch bands, so that we can describe our system by a Hubbard Hamiltonian acting in the lowest band of a lattice. Now, the periodic drive can still resonantly create excitations within this low-energy subspace. This form of heating can be reduced considerably by considering driving frequencies that are sufficiently large, so that absorbing an energy quantum of ω corresponds to a slow high-order process in which several elementary excitations are created at once [36,37]. If this is the case, we can employ a rotating-wave approximation and describe the system by the time-averaged low-energy Hamiltonian (or compute also further corrections using a high-frequency expansion [37][38][39][40][41]). In this way, we arrive at an approximate effective HamiltonianĤ eff that describes the dynamics of our system on time scales before driving-induced heating sets in.
The two steps outlined above require that there is a window of suitable driving frequencies that are both low compared to the relevant energy gap separating the lowenergy subspace from higher-lying states and large compared to the energy scales governing this low-energy subspace. In this paper, we investigate the question, whether such an optimal frequency window exists, using the experimentally relevant example of repulsively interacting bosonic atoms in a periodically shaken one-dimensional optical lattice. For this purpose, we compute the time evolution of a small two-band Bose-Hubbard model by means of exact numercial time integration.

II. SYSTEM AND MODEL
We consider a system of ultracold bosonic atoms in a one-dimensional optical lattice potential Here the laser wave number k L defines the recoil energy E R = 2 k 2 L /(2m) with atom mass m, corresponding to the kinetic energy required to localize a particle on the length of a lattice constant a = π/k L . Typical recoil energies take values of a few kHz. The deep confining potential V ⊥ (y, z) m 2 ω 2 ⊥ (y 2 + z 2 ) shall reduce the dynamics to one spatial dimension via a large transverse excitation gap ω ⊥ that freezes the particles in the lowest transverse single-particle state.
The system shall be driven periodically in time by the homogenous sinusoidal force pointing in the lattice direction e x , It is characterized by the driving strength K, corresponding to the amplitude of the potential offset between neighboring lattice sites, and the angular driving frequency ω, which defines also the driving period T = 2π/ω. Such a force can be realized as an inertial force by shaking the lattice back and forth in x direction.
In the absence of periodic forcing, experiments performed in the regime of deep lattices, V 0 /E R 5, at the typical ultracold quantum gas temperatures are described accurately by the single-band Bose Hubbard model [42] Here the index denotes the lattice sites in ascending order and the label s indicates the lowest Bloch band (to be distinguished from the first excited band, labeled by p, which is considered below). Moreover,b † α ,b α , andn α =b † α b α denote the creation, annihilation and number operator for a boson in a Wannier state of band α on site . Nearest-neighbor tunneling is described by the parameter J s and on-site interactions by the Hubbard parameter U s .
While in a non-driven system, a description in the lowenergy subspace of the s band is well justified, this assumption is not as clear in a system that is driven periodically. Even if the driving frequency is small compared to the band gap separating the s band from the first excited p band, states of excited bands might still be populated via multiphoton excitations corresponding to either single-particle processes [29,34] or two-particle scattering [35]. If periodic driving is used to control the physics of the lowest band, such excitation processes must be viewed as unwanted heating. In order to estimate this effect, below we will also take into account the first excited band, which for the undriven lattice is captured by the Hamiltonian and coupled to the s band via the interband interaction term Here ∆ denotes the orbital energy required to excite a particle to a Wannier state of the p band and J p and U p describe nearest-neighbor tunneling and on-site interactions in this p band, respectively. The interactions between s and p states are quantified by the on-site parameters U sp . If the energy scales of the periodic force, ω and K, remain below ∆, the bands of the undriven problem, s and p, provide a useful basis also for the description of the driven system (see the supplemental material of Ref. [35], where also the case of ω ∆ is discussed). Assuming this regime, we project the potential −r ·F (t) induced by the force to the lowest two bands and obtain the driving term of the Hamiltonian: (8) where η is the dipole matrix element between two Wannier states of the s and the p band on the same lattice site in units of the lattice constant.
The total Hamiltonian to be used for our analysis is now given bŷ The number of independent parameters that describe this model is reduced considerably by noticing that J s /E R , J p /E R , ∆/E R , and η are determined completely by the dimensionless lattice depth V 0 /E R . Moreover, the interaction parameters U s , U p , and U sp , which also depend on V 0 /E R , share the very same (linear) dependence on both the s-wave scattering length a s and the transverse confinement ω ⊥ , so that the interactions can be characterized by the strength U s as well as by the lattice depthV 0 /E R , which determines the ratios U p /U s and U sp /U s . Thus, taking J s and /J s as the units for energy and time, respectively, the undriven model is characterized by V 0 /E R and U s /J s as well as by the number of particles per site, N/L. The periodic driving is, furthermore, characterized by the dimensionless diving strength K/J s and angular frequency ω/J s . The dependence of the model parameters on the lattice depth V 0 /E R , obtained from band-structure calculations, is shown in Fig. 1.

III. SINGLE-BAND AND HIGH-FREQUENCY APPROXIMATION
So far most schemes of Floquet engineering in optical lattices (such as, for example, the control of the bosonic Mott transition [9,10], the implementation of kinetic frustration [11,12], the creation of artificial gauge fields 0.01 Tight-binding parameters for the lowest two bands of the one-dimensional optical cosine lattice with respect to the lattice depth V0/ER. [13,[15][16][17][18][19]22], and the realization of Floquet topological insulators [23][24][25]) are based on two approximations [9]: a low-frequency approximation with respect to orbital degrees of freedom and a high-frequency approximation with respect to processes occuring in the lowest band described by H s . The low-frequeny single-band approximation is based on the assumption that the driving frequency and amplitude remain low enough to ensure that the system remains in the subspace spanned by the lowest (s-type) Wannier-like orbital at each lattice site. It roughly requires driving frequencies ω ∆ (10) and driving amplitudes K that remaining below a threshold value K th below which multi-photon transitions are expected to be suppressed exponentially with the photon number ∆/ ω [34]. It leads to a description of the system in terms of a tight-binding model with a single orbital state per lattice site, which in our case is given by the single-band model The high-frequency approximation is based on the assumption that the driving frequency is still large compared to the energy scales J s and U s governing the lowenergy model (11), Under these conditions we can approximate the effective time-independent Hamiltonian describing the time evolution of the periodically driven system using a highfrequency expansion [9,37,40,41]. For that purpose, we first perform a gauge transformation with the timeperiodic unitary operator with θ(t) = K/( ω) sin(ωt) , which integrates out the driving term. The transformed HamiltonianĤ = U †Ĥ sbÛ − iÛ †U readŝ The fact that it possesses typical matrix elements that are small compared to ω even for large K ∼ ω justifies the high-frequency expansion also for strong driving. Its leading order is given by the rotating-wave approximation, where the system is described by the time-averaged Hamiltonian Here the effective tunneling matrix element acquired a dependence on the scaled driving amplitude K/ ω described by a Bessel function J n . All in all, the time evolution of the system's state |ψ(t) is approximately described by In particular, we expect for integers n, when monitoring the dynamics stroboscopically in steps of the driving period at those times t = nT , for whichÛ (nT ) = 1. Higher orders of the high-frequency expansion will provide relative corrections of the order of J s / ω to the evolution governed byĤ eff [37,40]. The single-band high-frequency approximation, leading to a description of the system's dynamics in terms of the approximate effective Hamiltonian (15), requires that there is a window of driving frequencies for which both conditions (10) and (12) are fulfilled. Since with increasing lattice depth V 0 /E R both J s decreases rapidly and ∆ increases moderately (see Fig. 1), while the interaction parameter U s can be made small by tuning the swave scattering length using a Feshback resonance, such a window will open for sufficiently large V 0 /E R . However, even within such a frequency window heating will not be suppressed completely and eventually make itself felt on some time scale τ . This heating time τ has to be compared to the typical duration of an experiment, which will be given by some fixed multiple of the tunneling time /J s . The tunneling time, in turn, increases exponentially with the lattice depth: asymptotically for deep lattices one finds ln(J s /E R ) −2 V 0 /E R [43] (see also Fig. 1). Thus, in order to take into account also this latter effect, in the following we will investigate the behavior of the dimensionless heating time τ J s / . In doing so, we have to keep in mind that there will also be background heating (resulting from noise, three-body collisions, or scattering with background particles), which happens on some time scale τ 0 . Assuming τ 0 ∼ 1s (∼ 10s), requiring τ 0 /J s , and noting that E R ∼ 2π · 3kHz for typical experiments, we can see from Fig. 1 that the lattice depth is limited to values V 0 /E R 15 (20).

IV. INTRABAND HEATING
Let us first investigate the validity of the highfrequency approximation, before considering also heating due to the coupling to the first excited band. For this purpose we consider the following quench scenario. We assume that the system is prepared in the ground state of the undriven Hamiltonian (5), when at time t = 0 the driving amplitude is switched on abruptly to a finite value K. We integrate the time evolution of the system described by the time-dependent single-band Hamiltonian H sb (t) [Eq. (11)] and compare it to the approximate solution |ψ eff n [Eq. (18)] obtained from the time-averaged single-band HamiltonianĤ eff . For that purpose we consider a small system of N = 6 particles on L = 10 lattice sites, for which we can integrate the time evolution exactly.
In order to monitor the deviation between the exact time evolution and the dynamics predicted by the rotating-wave approximation, we consider the expectation value which corresponds to the mean occupation of the singleparticle state with quasimomentum zero in the s band. The difference between the exact expectation value and the one obtained within the rotating-wave approximation taken at times t = nT with integer n will serve us as an indicator for the validity of the approximations made. While for the results presented in this section, n 0 (t) refers to the dynamics computed from the dynamics generated by the time-dependent single-band Hamiltonian (11), in the following section n 0 (t) will correspond to the dynamics of the full driven two-band model (9). In Fig. 2 we plot ∆n 0 (t) for a strong quench to a large driving amplitude K/ ω = 4 (the other parameters are specified in the caption). For this value the effective tunneling parameter changes its sign, J eff s ≈ −0.4J, so that the quench is significant also on the level of the rotatingwave approximation. We can see that ∆n 0 (t) shows an irregular oscillatory behavior, with a linearly growing envelop. We define the heating time τ as the time at which |∆n 0 (t)| exceeds the value ∆n cut = 0.2 for the first time, so that |∆n 0 (t)| < ∆n cut ∀t < τ.
Note that τ gives only and estimate for the time scale on which heating starts to play a role. The value of ∆n cut is somewhat arbitrary. It is chosen to be much smaller than the initial occupation of the zero momentum state, which is of the order of N , while it is also smaller than (and of the order of) the filling factor N/L = 0.6 corresponding to the mean occupation of each momentum state. The linear spreading of the envelop of ∆n 0 (t) implies that altering ∆n cut by a factor of order one will simply alter the heating time τ by roughly the same factor. Note also that the typical deviations |∆n 0 (t)| at time t = τ are smaller than ∆n cut = 0.2, since in most cases ∆n cut is reached the first time during the time evolution when an extreme fluctuation of |∆n 0 (t)| occurs. In Fig. 3, we plot the heating time τ J s / versus the driving frequency ω/J for two different values of the interaction strength, U/J s = 1 and U/J s = 5 (the other parameters are specified in the caption). We see that the heating time is considerably reduced for the larger value of the interactions. Moreover, an exponential dependence of the heating time on the driving frequency can be observed. This agrees with the expectation for heating processes based on a perturbative argument. Namely, one can argue that the order of the process of absorbing an energy quantum ω, corresponding to the number of elemantary excitations (quasiparticles) that have to be collectively excited, will grow like a power of ω and that the corresponding matrix element will be suppressed exponentially with the order [36]. Such an exponential suppression of heating with respect to the driving frequency has recently been proven for spin systems having a finite local energy bound [44,45]. However. these proves do not apply to the bosonic Hubbard model considered here, which in principle allows for macroscopic site occupations.

V. INTRABAND AND INTERBAND HEATING
The exponential increase of the heating time with respect to the driving frequency visible in Fig. 3 is an artifact of the single-band description of the driven lattice system. Namely, for sufficiently large driving frequencies unwanted excitations to higher-lying orbital states (spanning excited Bloch bands) will become the dominant heating effect. In order to take into account this effect, we will include also the coupling to the first excited Bloch band (the p-band). For this purpose we consider the two-band Hamiltonian (9) and monitor the heating time τ defined in the same way as in the previous section. In an experiment, of course, also further bands above the p band will play a role. But the coupling to the first excited band is most dominant, because with respect to the lowest band it both is energetically closest and possesses the largest coupling matrix elements. Therefore, the characteristic time scale for interband heating processes is determined by transitions to the p band. Higherlying bands can still make themselves felt, e.g. in the precise shape of resonance lines (as discussed in Ref. [29]). However, such effects are not crucial for the present analysis, which is interested in time scales only.
In Fig. 4 we plot the heating time τ versus the driving frequency for a system of N = 4 particles on L = 8 sites (corresponding to 16 single-particle states) with lattice depth V 0 /J = 14. For strong driving, K/ ω = 4, and two different interaction strengths, U s /J s = 1 and U s /J s = 5, we compare the heating times obtained from the single-band model (11) (open circles) to those obtained from the two-band model (9) (filled circles). As expected, we can observe that the coupling to the p band does not influence the heating time for low frequencies, while it becomes dominant for larger driving frequencies.
For the two-band model the interplay between intraband and interband heating gives rise to a maximum of the heating time, τ opt , at some optimal intermediate driving frequency ω opt . For the larger interaction strength τ opt is lower and occurs at a larger frequency.
To study the impact of interactions in more detail, we compare the frequency-dependent heating times for various interaction strengths U s /J s in Fig. 5. The inset shows the maximum heating time τ opt (diamonds, right axis) and the corresponding optimal driving frequency ω opt (circles, left axis) versus U s /J s . We observe a significant reduction of τ opt combined with an upshift of ω opt , when increasing the interaction strength U s /J s up to values of about 3. Both the shift of ω opt and the noticeable reduction of τ opt for the single-band model (Fig. 3) suggest that increasing the interactions mainly enhances intraband heating, so that intraband heating becomes dominant for larger ω. For values of U s /J s that are larger than 3 both τ opt and ω opt approximately saturate. We attribute this saturation to the reaching of the hard-core-boson limit within the lowest band, which is expected for U s /|J eff s | ≈ 2.5(U s /J s ) 1. In Fig. 6 we depict the lowest-band zeroquasimomentum occupation n 0 (t) in units of its initial value n 0 (0) after a time of t ≈ 40 /J s , which is sufficiently long for interesting experiments. It is  Here n0(t) is defined in Eq. (19). We assumed a recoil energy of ER = 3.332π kHz, a typical value for an experiment with 8 7Rb atoms, for which the chose time span corresponds to tJs/ ≈ 40 tunneling times. The driving strength corresponds to an effective tunneling matrix element of J eff s ≈ 0.5Js.
plotted versus the interaction strength U s /J s and the driving frequency ω/J s , where the low-frequency regime is shown in the left panel, while results for higher driving frequencies are given in the right panel.
In the underlying simulations, we have considered a lattice depth of V 0 /E R = 10 and a driving strength of K/ ω = 1.5, which is smaller than the one used previously and does not induce a sign change of the effective tunneling matrix element (16), J eff s ≈ 0.51J s . The latter implies that on the level of the effective Hamiltonian, the quench induced when switching on shows the optimal (maximum) heating time τopt (diamonds) and the corresponding optimal frequency ωopt versus V0/ER. the driving, does not correspond to an inversion of the effective dispersion relation, but rather to a reduction of the band width by a factor of one half. On the level of the time-averaged Hamiltonian (15), this rather mild quench will not induce a large amount of energy to the system, so that the occupation n 0 (t)/n 0 (0) will retain a rather large value also during the dynamics following the quench. Thus, a significant reduction of n 0 (t)/n 0 (0) indicates heating. Note also that (for fixed K/ ω) the ideal dynamics generated byĤ eff , and thus also n 0 (t)/n 0 (0), is independent of the driving frequency.
In Fig. 6, we can observe a significant reduction of n 0 (t)/n 0 (0) that indicates heating in various regimes. In the regime of weak interactions U s /J s 1, heating is visible both for too low frequencies, when ω ∼ J s , as well as for too high frequencies, when ω ∼ ∆ (with ∆/J s ∼ 250 for the given lattice depth). When the interband intaractions U s become larger than the interband tunneling J s , low frequency heating sets in already for larger ω, in accordance with condition (10). At the same time, we can also observe that interband heating at large frequencies is enhanced in the presence of interactions. For the chosen lattice depth of V 0 /E R = 10, we observe that strong interactions U s /J s 1 lead to significant heating for any frequency.
The dependence of the heating time τ J s / on the lattice depth V 0 /E R is investigated in detail in Fig. 7, where we plot τ J s / versus ω/J s for various values of V 0 /J s and for rather strong interactions U s /J s = 5 and driving K/ ω = 4. The inset shows τ opt J s / and ω opt /J s versus V 0 /E R . We can observe that both τ opt J s / and ω opt /J s increase with the lattice depth. The main figure shows that this behavior is associated with a significant reduction of heating for large ω/J s . Let us discuss this behavior in more detail.
First, we can notice that the intraband dynamics, de-scribed by the single-band Hamiltonian (11) and measured in the natural unit of the tunneling time /J s , is determined by the dimensionless ratios U s /J s , ω/J s , and K/ ω, which we kept fixed in our simulations when increasing the lattice depth V 0 /E R . This choice of fixed parameters is natural from the point of view of quantum simulation, where we wish to engineer the properties of the lowest band described by the approximate effective Hamiltonian (15). It explains why for small ω/J s , for which interband coupling is negligible, the dimensionless heating time τ J s / is hardly influenced by the lattice depth. This can be seen from the fact that all curves in Fig. 7 agree up to the point (∼ ω opt /J s ), where τ J s / starts to be reduced by interband processes.
In turn, we can observe in Fig. 7 that the interband heating, which is responsible for the reduction of τ J s / at large frequencies, is significantly reduced with increasing lattice depth. This behavior is not obvious, since it results from the interplay of various effects. We can first note that with increasing lattice depth V 0 /E R the band separation ∆/E R increases, whereas the interband coupling parameter η decreases (Fig. 1). Both effects tend to reduce interband heating. An additional and much stronger reduction of interband heating will, however, result from the exponential suppression of the tunneling parameter J s with respect to the lattice depth V 0 /E R (Fig. 1). Namely, since we keep the dimensionless ratio ω/J s fixed (for the reason explained in the previous paragraph), the number of photons (i.e. energy quanta ω) needed to overcome the band separation ∆, will increase exponentially with the lattice depth. At the same time, the rate for interband transitions will decrease exponetially with n ph [29,34]. Besides these effects, which lead to a very strong suppression of interband heating, a strong enhancement of the role of interband heating results from the exponential increase of the tunneling time /J s with V 0 /E R in the units of which τ is measured. However, the results presented in Fig. 7 clearly show that the former effects win over the latter one, so that all in all for large driving frequencies τ J s / is reduced significantly when the lattice depth V 0 /E R is increased. Therefore, we can also see a clear increase of the maximum heating time τ opt J s / (as well as an upshift of ω opt /J s ), when the lattice depth is increased. The data shown in Fig. 7 indicate that in terms of the natural unit of time, the s-band tunneling time /J s relevant for quantum simulation, we can increase the heating time associated with the periodic drive as much as we want. However, as we discussed already above, this possibility is limited by the increase of the tunneling time with respect to the time scale τ 0 associated with other heating processes (such as three-body collisions, scattering with background particles, noise, etc.). We estimated above that for a typical experimental value of τ 0 = 1s (10s), the lattice depth should not be larger than about shows the optimal (maximum) heating time τopt (diamonds) and the corresponding optimal frequency ωopt versus K/ ω. (20). Note that, our results imply that any reduction of background heating processes can be used to significantly reduce also driving induced heating by making the lattice deeper. This is one of the main results of this paper.
Let us, finally, also have a look at the dependence of the heating time on the drivng strengths. In Fig. 8 we plot τ J s / versus K/ ω for a system with V 0 /E R = 14 and U s /J s = 5. We focus on values of K/ ω that are interesting for Floquet engineering (i.e. that are large enough to achieve a significant modification of J eff s and not much larger than required for tuning J eff s to negative values). For the smallest considered driving strength of K/ ω = 1.5 a narrow window of frequencies is found for which the heating time takes large values of more than 300 tunneling times. This window disappears for stronger driving. Note that we do not find a simple monotonous decrease of the heating time with respect to the driving strength. This must be attributed to the fact that the finite-frequency components ∝ e imωt of the timedependent Hamiltonian (14) [as well as those of the corresponding gauge-transformed time-dependent two-band Hamiltonian], which describe heating processes beyond the rotating-wave approximation (15) where the system exchanges m energy quanta with the drive ω, involve Bessel-function expressions J m (K/ ω) that depend in a non-monotonous way on the driving strength K/ ω.

VI. CONCLUSIONS
In summary, we have investigated the conditions for Flouqet engineering in optical lattices. In particular we were interested in the existence of a frequency window where both low-frequency intraband heating and highfrequency interband heating is suppressed on a time scale τ that is large compared to the tunneling time. Considering the concrete example of a small one-dimensional system of interacting bosons in a shaken optical lattice, we presented numerical results that show that such a frequency window exists for sufficiently deep lattices. The maximum ratio of heating and tunneling time, τ opt J s / , (which is found for an optimal intermediate driving frequency ω opt /J s ) is found to increase with the lattice depth. This result, which is not obvious since also the tunneling time increases exponentially with the lattice depth, implies that we can reduce driving-induced heating, by simply ramping up the lattice depth. However, we have pointed out that the increase of the tunneling time in this limit, limits this strategies to lattice depths where the tunneling time is still much smaller than the time scale τ 0 for non-driving-induced background heating. Thus, the larger the time scale for such background heating, the more we can reduce also driving induced heating.
We have also found that ramping up the interaction strengths, driving-induced heating is significantly enhanced, until a saturation value is reached roughly when the ratio U s /J s reaches values of 3. This saturation is a promising result regarding the possibility to use Floquet engineering for the preparation of strongly correlated states of matter such as fractional Chern insulators [46][47][48][49].
An interesting direction for future work concerns the role of disorder. It has been argued that many-body localization can protect the driven system against unwanted heating associated with deviations from the highfrequency approximation [50,51]. Roughly speaking, within the localization length, the system is not able to create excitations of a sufficiently large energy ω. This mechanism is crucial also for the stabilization of discrete time crystals [52][53][54][55]. However, disorder-induced localization cannot be expected to protect the system also against heating associated with deviations from the low-frequency approximation. Unwanted resonant multiphoton excitations to states above the gap can and will still occur. It is an important question, in how far the corresponding heating rates are influenced by disorderinduced localization.