Floquet Perturbation Theory: Formalism and Application to Low-Frequency Limit

We develop a low-frequency perturbation theory in the extended Floquet Hilbert space of a periodically driven quantum systems, which puts the high- and low-frequency approximations to the Floquet theory on the same footing. It captures adiabatic perturbation theories recently discussed in the literature as well as diabatic deviation due to Floquet resonances. For illustration, we apply our Floquet perturbation theory to a driven two-level system as in the Schwinger-Rabi and the Landau-Zener-St\"uckelberg-Majorana models. We reproduce some known expressions for transition probabilities in a simple and systematic way and clarify and extend their regime of applicability. We then apply the theory to a periodically-driven system of fermions on the lattice and obtain the spectral properties and the low-frequency dynamics of the system.

Analytically, the appearance of these novel features is usually understood within a high-frequency approximation, e.g. the rotating-wave approximation, Floquet-Magnus expansion, and Brillouin-Wigner theory [24][25][26][27][28]. These approximations often break down as frequency is lowered below the typical energy scale of the static system, such as the bandwidth or an equilibrium insulating gap. Though in certain cases other perturbative schemes, such as the Schrieffer-Wolff theory [29], provide valuable insight away from the high-frequency regime, understanding the low-frequency behavior of these novel phases remains challenging. In the opposite limit of vanishingly small frequency, one may expect the dynamics be governed by adiabatic evolution. Perturbative methods to account for diabatic correction to this adiabatic evolution have been developed [30][31][32][33][34]. However, the connection between these methods and the Floquet theory used for higher frequencies is not clear.
In this paper, we develop a systematic perturbation theory based on the Floquet theorem within the extended Floquet Hilbert space furnishing the steady states of a periodically driven quantum system [35,36]. Our approach is general and works whenever an operator in the Floquet Hamiltonian describing the dynamics of the system in the extended Floquet Hilbert space can be taken to be small. Indeed, we show how this Floquet perturbation theory leads to perturbative expansion both in the high-and the low-frequency limits. In both cases, we reproduce previous results in a compact and efficient way and show how higher-order terms is worked out systematically. Moreover, using this formalism we expand the applicability of these results and show when deviations are expected. In the low-frequency limit, we clarify the deviations from adiabatic evolution near quasienergy resonances [38] that lead to Rabi oscillations. Finally, using our Floquet perturbation theory, we study a system of non-interacting fermions moving on a driven one-dimensional lattice at low frequency [15,39]. We derive the Floquet spectrum and show when the low-frequency limit does and does not approach the adiabatic evolution.
We note that in the low-frequency limit the periodicity assumed in the Floquet theory is not a real restriction for reproducing the results of the adiabatic perturbation theory for a general drive. Basically, in this limit one can think of any drive as one big cycle of a periodic drive and find the desired evolution at any time mid cycle. The additional periodic structure in the Floquet theory is important only if when one wishes to study the Floquet spectra of an actual periodic drive. One may call this low-frequency Floquet perturbation theory the "Floquet adiabatic perturbation theory;" however, this term is already used in the literature [40] to describe the evolution of a driven system when a parameter of the drive is slowly varied. To avoid confusion, we do not use this terminology.
The paper is organized as follows. In Section 2, the Floquet perturbation theory is developed within the extended Floquet Hilbert space and used to derive high-and low-frequency series expansions of the Floquet spectrum. Formal aspects of the theory are presented in Appendix A. In Section 3, we illustrate the formalism by applying it to transition probabilities in a driven two-level system, described separately by the Rabi-Schwinger and the Landau-Zener-Stückelberg-Majorana models. In Section 4, we develop the degenerate low-frequency Floquet perturbation theory and demonstrate its application near quasienergy degeneracies in the low-frequency regime of the Landau-Zener model as well as the driven Su-Schrieffer-Heeger model of non-interacting fermions moving on a one-dimensional lattice. We conclude with a summary and outlook in Section 5. Some technical details of our calculations are given in Appendix B.

Floquet Theory and the Extended Floquet Hilbert Space
Floquet theorem is the statement that the solution to a differential equation with periodic coefficients can be written as a phase factor multiplied by a periodic function. A direct consequence of this statement in the condensed matter setting is the Bloch theorem for the solution to the Schrödinger equation in the presence of a spatially periodic potential due to a lattice. In our discussion, we reserve the Floquet theorem for a system with parameters that are periodic in time, t. The details of the Floquet theory formalism are presented in Appendix A; here, we provide a summary.
For a HamiltonianĤ(t) =Ĥ(t + T ) with period T = 2π/Ω, Floquet theorem states that the time-dependent Shrödinger equation i d dt |ψ(t) =Ĥ(t)|ψ(t) takes steady-state solutions of the form where the quasienergy α is a conserved quantity and the periodic Floquet mode |φ α (t) = |φ α (t + T ) satisfies the Floquet Schrödinger equation The |φ α (t) form a time-dependent orthonormal basis for the Hilbert space H and can be viewed as the eigenstates of the effective HamiltonianĤ eff (t) ≡Ĥ(t) − i d dt with eigenvalues belonging to the Floquet zone, α ∈ [−Ω/2, Ω/2]. Using Floquet theorem, the evolution operator with t 0 < t and Texp the time-ordered exponential, can be decomposed aŝ where define, respectively, the Floquet HamiltonianĤ F and the micromotion operatorΦ(t). The structure we have described above can be formalized in terms of an extended Floquet Hilbert space F = H ⊗I , where the auxiliary space I is the space of bounded periodic function over [0, T ) [36]. We denote the states in H , I , and F respectively by |· , |·), and |· and the operators acting on each respective space asÔ, " O, andÔ. The space I is spanned by a continuous orthonormal basis {|t)}, 0 ≤ t < T , where " I is the identity operator in I . The auxiliary space I is also spanned by the orthonormal Fourier basis satisfying (n|m) = δ nm , n∈Z |n)(n| = " I. We note |t) = n∈Z e inΩt |n).
A loop in H , i.e. a cyclic time-dependent state |φ(T ) = |φ(0) , can be lifted to a loop in F given by |φ t := |φ(t) |t). Associated with any loop |φ t ∈ F is the center of the loop, We define the Fourier-integral and the time-derivative operators, such that for a loop |φ t ∈ F , is the n-th Fourier integral and is the center of the time-derivative of the loop. Then, the Floquet Schrödinger Eq. (2) can be written in F as Note, however, that the set {|φ α } of solutions to Eq. (14) is not large enough to furnish a complete basis for F . Noting that [Ĥ,μ n ] = 0 and we can write instead where αn ≡ α − nΩ and |φ αn ≡μ n |φ α . Indeed,μ n is the ladder operator forĤ −Ẑ, mapping the solution |φ α with quasienergy α to |φ αn with quasienergy α −nΩ. Now, the solutions |φ αn to Eq. (24) provides a full basis for F .

High-Frequency Expansion
As an example, we derive a high-frequency expansion using the Floquet perturbation theory. Assuming the frequency is larger than the typical quasienergy, we shall take the unperturbed HamiltonianĤ 0 = 0 andV =Ĥ. Therefore, α(0) = 0, and |φ α(0) can be obtained from lifting an arbitrary time-independent set |φ α in H to F . After some algebra we find Thus, following the first-order degenerate perturbation theory, |φ α is chosen as the eigenstate ofĤ (0) , and the the quasienergies are obtained by diagonalizinĝ This is indeed the same expression obtained using other high-frequency expansions, such as Floquet-Magnus expansion and Brillouin-Wigner perturbation theory [25][26][27].

Low-Frequency Expansion
A perturbative expansion at low frequencies can be obtained by rescaling time to τ = Ωt and noting that the periodicity of the HamiltonianĤ(τ ) is maintained when translating τ → τ + 2π. The Floquet Schrödinger equation in rescaled units read One may now attempt a perturbative expansion at low frequencies taking −ΩẐ as the perturbation operator. However, there is a subtlety that must be addressed: the Floquet perturbation theory we developed in the previous section takesẐ as part of the unperturbed Hamiltonian. This is necessary to ensure that the eigenvalues of the unperturbed operator have the same modular structure as the final quasienergies. By, takingĤ as the unperturbed operator without includingẐ, the eigenvalues of the unperturbed Hamiltonian will no longer be modular. Indeed, the eigenstates ofĤ are nothing but the eigenstates of the instantaneous HamiltonianĤ(τ ) lifted to F : where |ψ ατ = |ψ α (τ ) |τ ), andĤ(τ ) |ψ α (τ ) = E α (τ ) |ψ α (τ ) , and E ατ = E α (τ ). The eigenvalues E ατ ofĤ are, therefore, not modular, unlike the eigenvalues αn ofĤ − ΩẐ. Therefore, in order to use perturbation theory to build the spectrum ofĤ − ΩẐ as a power series over the spectrum ofĤ, we need to amend our Floquet perturbation theory to ensure we obtain a modular spectrum. This can be done by using the general relationship between the Floquet spectrum and the center of loops in F found by lifting the loop of Floquet modes in H . We first use perturbation theory to find |ψ ατ (i) to the desired order i. The modular spectrum is then found by taking the Fourier transform of this loop in F , wheredτ ≡ dτ /(2π). This defines the proper eigenstate ofĤ − ΩẐ with a modular eigenvalue αn(i) = α(i) − nΩ, and the quasienergy Note that for i ≥ 1 the first term vanishes. This equation follows from the Floquet Schrödinger equation and noting that φ α(0) |φ α = 2π 0 ψ ατ (0) |ψ ατ dτ = 1. Explicitly, for the first few terms we find, and There is one final loose end we now address: there is a gauge freedom in the choice of the instantaneous basis |ψ ατ that we need to fix. Explicitly, Thus, fixing the gauge by setting up to a constant, we obtain This is the zeroth-order Floque Schrödinger equation. Therefore, we must indeed choose |φ α(0) = |κ α . This gauge-fixing was previously used by Martiskainen and Moiseyev [37]; however, they only justified its use numerically by showing it improves the accuracy of the perturbative expansion. Here, we see that this gauge must be fixed for consistency of the adiabatic solution as the zeroth order term in the general low-frequency Floquet perturbation theory.
We shall now consider a more general periodic Hamiltonian is a vector, whose direction, n, as well as its magnitude, d, change periodically in time. We denote the average 2π 0 d(τ )dτ ≡d. Then, the Schrödinger equation is not in general exactly solvable. Transforming to the rotating frame, for example, will not produce a time-independent Hamiltonian any more. A high-frequency expansion can be developed in the rotating frame; however, these expansion fail at low enough frequency. Instead, we shall use the Floquet perturbation theory based on the instantaneous spectrum given by eigenstates |ψ ± (τ ) = e ∓iΛ(τ ) ±f ∓ (θ(τ )/2)e −iϕ(τ ) f ± (θ(τ )/2) and eigenvalues E ± (τ ) = ±d(τ ), where θ and ϕ are the polar angles of n and the gauge So, The first-order correction to the Floquet steady state reads

Driven Landau-Zener Model
Consider the Hamiltonian where f (τ ) is a periodic function satisfying f (0) = −f (π) = 1 and a, b > 0. Half way during the cycle, the first term switches sign, thus realizing the usual situation in the Landau-Zener model for large frequencies. In our general notation, . The minimum gap in the instantaneous spectrum is b obtained when f vanishes. We take a b. For b = 0, the exact solution for the evolution operator is Thus, the Floquet spectrum is given by the quasienergies ±(0) = ±aF (2π)/(2π) = ±af (0) and Floquet steady states we expect the operator bσ y to be "small' compared tô H 0 − ΩẐ with H 0 (τ ) = af (τ )σ x ; thus, we may use the Floquet perturbation theory to find corrections to the Floquet spectrum: where g ± n (a/Ω) = 2π 0 e −inτ e ±i(a/Ω)Λ(τ )d τ . Note that [g ± n (a/Ω)] * = g ∓ −n (a/Ω). Thus, starting from an initial state |φ ±(0) (0) , the probability of transitioning to state |φ ∓(0) (τ ) at time τ is, where the approximation is to the second-order in b. This yields, after some algebra, This expression can also be obtained [42,43] directly from the Schrödinger equation written in the adiabatic basis, Starting with initial state |φ ±(0) (0) and assuming A ± (τ ) Our derivation using the more systematic Floquet perturbation theory, apart from being an application of the formalism, shows that this result is valid not only when b/Ω 1, but also when b a, even near for resonant b ≈ Ω/2 a. In the latter case, the usual rotating-wave approximation for large frequencies fails; however, the highly oscillating phase factor suppresses higher order corrections.
In Figs. 1 and 2, we compare P − (τ ) obtained by the exact numerical solution with the Floquet perturbation theory for f (τ ) = cos(τ ) as well as the sawtooth-linear function Panels (a) and (b) in each figure show two choices of parameters at or near resonance b/Ω = 0.5. We note that while agreement is good in the first half of the cycle, it becomes less reliable in the second half. In fact, the end-of-cycle behavior of the Floquet perturbation theory is sensitive to the choice of parameters: for small final values of P − (τ ) around τ = 2π the agreement is reasonably good, but for certain larger final values, as in Fig. 2(b), the Floquet perturbation result becomes less reliable. One way to understand these variations is to compare Floquet perturbation theory and exact results at half-and full-cycle. In panels (c) and (d) of each figure we compare our results for τ = π and τ = 2π, respectively, at a fixed ratio b/a 1 as a/Ω is varied. For the cosine ramp, the Floquet perturbation theory gives, where J 0 and H 0 are, respectively, Bessel and Struve functions. For the linear ramp, we x 0 f ± (πy 2 /2)dy are Fresnel integrals. We note that for large a/Ω 1, the prefactor approaches 1/2 and this reproduces the classic Landau-Zener-Stückelberg-Majorana formula [44][45][46][47], However, the perturbative result starts to fail for larger a/Ω and fixed b/Ω since the magnitude of the exponent in P LZ becomes large. We show P LZ in panel (c) of each figure. We note that while the classic result is in very good agreement with the exact result, it misses the oscillations as a function of a/Ω. By contrast, perturbative result captures these oscillations very well.
We also see the sporadic nature of the agreement between Floquet perturbation theory and the exact result in the second half-cycle. As seen in panels (d) of each figure, the final value P − (2π) shows large oscillations as a function of a/Ω and vanishes periodically. The Floquet perturbation theory result captures these oscillations and, in particular, the zeros of P − (2π) remarkably well. This feature is similar to the coherent destruction of transitions discussed in a periodically driven double-well potential [48][49][50][51].
For larger values of a/Ω going beyond its applicability, the Floquet perturbation theory overshoots the amplitude of the oscillations, eventually giving unphysical values larger than unity. This is due to the non-unitary nature of perturbative expansion of the micromotion operator. We expect that a more controlled expansion that respects the unitarity of the micromotion operator, similar to the Floquet-Magnus expansion, should resolve this problem.

Degenerate Low-Frequency Floquet Perturbation Theory
In the preceding discussion we have assumed the quasienergy spectrum is nondegenerate. We shall now address the case where this is not true; as we will see this has interesting consequences for the dynamics.

Formalism
First, let us say a few words about the different conventions for the range of quasienergies, which we formally restricted to the first Floquet zone [−Ω/2, Ω/2]. This choice is entirely arbitrary, of course, and it may be more suitable in some problems to make other choices. Shifting a quasienergy αn → αn − mΩ maps |φ αn →μ m |φ αn = |φ αn+m . Doing so for all or just a subset of quasienergies has no physical effect. For example, in Eq. (17) all such shifts can be absorbed into a shift of the summation variable n. Now, in Floquet perturbation theory, even if the unperturbed quasienergies α(0) are in the first Floquet zone, the corrections, Eq. (19a), may not be. Thus, in this section, we shall relax this condition.
Degeneracies arise when two or more quasienergies r α , labeled by r, coincide when shifted by integer multiples of frequency, n r . For accidental or symmetry related degeneracies, n r = 0, and one might as well restrict the unperturbed quasienergies to the first Floquet zone from the outset. However, for low-frequency Floquet perturbation theory, where even the unperturbed quasienergies need to be calculated according to Eq. (28a), it is more convenient to allow unperturbed quasienergies take values outside first Floquet zone. In this case, we shall assume the shifts n r are unique. (Of course, one may always set one of them to zero.) Thus, r α − n r Ω = mΩ/2 for some integer m. This is an unusual situation in textbook perturbation theory, since the zeroth-order quantities are resonant up to first order in the small quantity Ω. Nevertheless, we may still proceed by trying to find the proper superpositions of degenerate states that resolve the degeneracy. Define with coefficients c rs α to be determined by solving (Ĥ − ΩẐ) |χ r α = (mΩ/2 + r α(1) ) |χ r α + O(Ω 2 ).
This yields, where W α is a matrix with diagonal elements W rr α = r α − n r Ω − mΩ/2, and the offdiagonal elements, where we have used φ r α (τ )|φ s α (τ ) = 0 for r = s. The eigenvalue set of equations in (50) constitute the first-order degenerate low-frequency Floquet perturbation theory.

Floquet Resonances in Landau-Zener Model
Let us revisit the driven Landau-Zener model, Eq. (41), in the low-frequency Floquet perturbation theory. Here, we shall take f (τ ) = cos τ . The instantaneous eigenvalues are E ± (τ ) = ± √ a 2 cos 2 τ + b 2 ; so, the lowest-order quasienergies are where E(x) and K(x) are complete elliptic integrals. (The first-order correction to quasienergy vanishes.) For Ω/b 1, one can always find quasienergy degeneracies by tuning a/b and sufficiently large shifts n. However, we note quasienergy degeneracies also occur for Ω/b 1, so the associated dynamics is not restricted to (very) low frequencies.
In Fig. 3, we show the transition probability and the expectation value of energy at two such quasienergy degeneracies, one at a lower frequency Ω/b = 0.75 and the other at a much higher frequency Ω/b = 2.45 with, respectively, n = 3 and n = 1. In both cases, we see that the low-frequency Floquet perturbation theory works well. This is remarkable for these frequencies are not particularly low compared to the minimum gap of 2b. In fact, in the second case, the frequency Ω/b = 2.45 is larger than the gap, which is why it was chosen to obtain the lowest value of n = 1. In this case, numerically calculated quasienergies are far from Floquet zone edges and simply eyeballing their evolution with a/b does not hint at an avoid quasienergy degeneracy in the adiabatic approximation. Nevertheless, the lowest-order degenerate low-frequency Floquet perturbation theory produces a reasonably accurate result.
A few remarks are in order. The low frequency oscillations observed in Fig. 3 are manifestations of Rabi-like oscillations between resonant statesμ n |φ +(0) and |φ −(0) ; at degeneracy, the frequency of these oscillations is |z n |Ω. These oscillations indicate a breakdown of adiabaticity [38]: the system starting in the adiabatic state |φ −(0) would transition out fully over 1/(4|z n |) cycles. However, one must be quite careful about statements of adiabatic breakdown at low frequencies. For smooth drive protocols, the splitting |z n | is typically exponentially small at low frequencies, meaning that a non-adiabatic transition would take exponentially long times. For example, in Appendix B we show that for Ω/b 1, a quasienergy degeneracy of the lowest shift order is obtained for a ∼ Ω with n ∼ bΩ and the quasienergy splitting vanishes as (Ω/b) 2b/Ω . Thus, the transition time out of the adiabatic evolution diverges as a factorially large number ∼ e −2(b/Ω) log(b/Ω) . This trend is seen in Fig. 3. The period of Rabi oscillations is multiplied by a factor of about six when going from a principal resonance at n = 1 and Ω/b = 2.45 to the next resonance at n = 3 and Ω = 0.7. Using low-frequency Floquet perturbation theory, we found that for the first resonance in the driven Landau-Zener model with n = 7 and Ω/b = 0.29, this period rises up to more than 2 × 10 5 cycles.
We have shown here that the celebrated adiabatic approximation is the lowest order of the more general low-frequency Floquet perturbation theory, which is built on the Floquet Green's function. The latter correctly accounts for corrections to the adiabatic approximation and, in particular, its potential breakdown due to Rabi oscillations at quasienergy degeneracies. We must note, too, that not all quasienergy resonances at low frequency exhibit Rabi oscillations. The low-frequency Floquet perturbation theory shows when adiabatic evolution is preserved due to protected quasienergy crossings. For example, in our driven Landau-Zener model, z n = 0 for even n. At such crossings, Rabi oscillations become infinitely long, regardless of frequency, restoring adiabatic evolution in the degenerate subspace.

Driven Su-Schrieffer-Heeger Model at Low Frequency
As a second application, we apply the (degenerate) low-frequency Floquet perturbation theory to a driven lattice model of non-interacting fermions, namely the Su-Schrieffer-Heeger (SSH) model. Both the static [52,53] and driven [15,39] versions of this model show distinct topological phases that are distinguished by the appearance of protected bound states at the edges of the lattice with open boundary conditions or a topological winding number in the Brillouin zone for a system with periodic boundary conditions.
For a given Ω < 4w there are a set of points ±k ℘ (Ω) in the Brillouin zone where quasienergies become degenerate: k ℘ − = k ℘ + − n k ℘ Ω, with 4|δ|/Ω ≤ n k ℘ ≤ 4w/Ω . Near these points we must employ degenerate low-frequency Floquet perturbation theory. Expand k± = ±(n k ℘ Ω/2 + ∆ ℘ k ) and |χ ℘± k = c ± k+μ n k ℘ |φ k+(0) + c ± k− |φ k−(0) , to determines the gap opening at the degeneracy point as ℘± k(1) = ± ∆ ℘ k 2 + Ω 2 |z ℘ k | 2 and the solutions |χ ℘± k in a fashion similar to Eqs. (4.2). In Fig. 4 we compare the spectral measures obtained from the adiabatic approximation and the low-frequency Floquet perturbation theory with the numerically exact solution for a smooth drive protocol δ(τ ) =δ + δ 0 cos τ . The quasienergies found using degenerate low-frequency Floquet perturbation theory match the exact solution remarkably well. The infidelity of an approximate solution, |φ(τ ) , is defined as 1 − F, where F(τ ) = | φ e (τ )|φ(τ ) | 2 and |φ e (τ ) is the exact solution. Near a quasienergy degeneracy point, the infidelity of the adiabatic approximation increases dramatically as the frequency increases. The infidelity of the solution obtained using the degenerate low-frequency Floquet perturbation theory, on the other hand, is not only small relative to the adiabatic approximation, but it also remains small on the absolute scale even for larger frequencies. This demonstrates the consistency and accuracy of the Floquet perturbation theory.
As the frequency is lowered, the order n k ℘ increases as ∼ 1/Ω. For a smooth drive protocol, the adiabatic limit is obtained as the off-diagonal element z ℘ k vanishes.
For example, for the sinusoidal drive protocol z ℘ k vanishes exponentially similar to the Floquet resonances in the Landau-Zener model. However, for a drive protocol with sharp features, such as a step-wise protocol, the approach to the adiabatic limit may be slower or even violated. In Fig. 5 we show the dependence of z k,n on k and the order n of the quasienergy degeneracy. Indeed, as shown in Fig. 5(a) and the inset, for the sinusoidal drive protocol, z k,n vanishes with n in an exponential manner. However, for step-wise protocol, shown in Fig. 5(b), z k,n approaches a limiting value as n increases that depends on the parity of n.

Summary and Outlook
We have formulated the Floquet perturbation theory that recasts time-dependent perturbation theory of a periodically driven quantum system in terms of a timeindependent perturbation theory in the extended Floquet Hilbert space of the periodic operators. This formalism is transparent and strips away cumbersome book-keeping that is required to track the time-evolution in the original Hilbert space. We showed how both high-frequency and low-frequency expansions of the dynamics can be obtained using this formalism.
While we reproduced some results already reported in the literature, for example in two-level systems, our approach allowed us to clarify and extend the range of applicability of these results. Additionally, using the formalism in this paper one can readily obtain the full-cycle dynamics not usually accessible in traditional approaches. For example, the usual treatment of the Landau-Zener model assumes an infinite duration for the transition. However, the actual transition takes a finite time [43,54]. In experimentally relevant situations the drive may take a comparable time as the time needed for the transition. In these cases, the Floquet perturbation theory is useful and provides a more detailed description of the dynamics within the drive cycle.
In the low-frequency limit, we obtained a systematic and compact derivation of the adiabatic perturbation theory. Moreover, in this formalism, the occurrence of quasienergy degeneracies that cause diabatic deviations via Rabi oscillations is easily and accurately captured using a degenerate low-frequency Floquet perturbation theory. We saw that for typical, smooth drive protocols, such as sinusoidal, the approach to the adiabatic limit is exponential since the matrix element leading to Rabi oscillations vanishes (super-)exponentially in 1/Ω. However, for drive protocols with sharp features, such as step-wise, this matrix element may approach an asymptotic value, thus invalidating the adiabatic approximation. In such scenarios, the Floquet perturbation theory is essential for describing the correct dynamics of the system at low frequencies.
We briefly mention some interesting problems for future application of our work. On the technical side, it would be useful to improve the Floquet perturbation theory for the expansion of the Floquet modes by ensuring the unitarity of the micromotion operator, perhaps along the lines of the Floquet-Magnus expansion [25,26]. This would, for example, improve the accuracy of the calculated transition probabilities in the driven Landau-Zener model and avoid divergence for slow drive. Other interesting problems for which the Floquet perturbation theory might be useful are the fate and structure of Floquet topological [15] and Floquet many-body localized [18,20] phases at low frequencies.

Appendix A.2. Micromotion
In the above decomposition, we chose to set the initial time t 0 = 0. This choice is arbitrary; we could choose any other time within a cycle 0 ≤ t 0 < T . In general, for t > t 0 we have:Û The two-time micromotion operator produces the periodic evolution of Floquet modes, Φ(t, t 0 ) |φ α (t 0 ) = |φ α (t) . We note that, Thus,Ĥ F ,Ĥ F (t 0 ), andĤ eff (t) are all unitarily equivalent. Especially, the eigenvalues of H F (t 0 ) do not depend on the initial time. However, when certain approximate methods are used to findĤ F (t 0 ) it turns out the eigenvalues acquire a spurious dependence on t 0 . One way to avoid this problem is by using the decompositionΦ(t, The dependence on t 0 is then entirely accounted for in the micromotion operator. We also note that |φ = n∈Z |φ (n) |n), where the Fourier components Similarly, a two-parameter periodic family of operatorsÂ(t, t ) : H → H witĥ A(t + T, t ) =Â(t, t + T ) =Â(t, t ) is lifted to an operatorÂ : F → F defined aŝ Two special cases arise whenÂ is diagonal in either I or H . As an example of the former case,Â(t, t ) =Â(t) T " δ(t − t ) andÂ (n,n ) =Â (n−n ) = As a simple case of the latter case, we takeÂ to be the identity in H , so thatÂ(t, t ) is just a complex-valued function A(t, t ) of its arguments. As an important example, we choose A(t, t ) = e −inΩt " δ(t − t ) =: " µ n (t, t ) to find where |(dφ/dt) t is defined as the loop d dt |φ(t) lifted to F . In particular, φ t |Ẑ |φ t = φ (t)| i ∂ ∂t |φ(t) T δ(t − t ). (A.30) We also note that, A loop of periodic family of bases |φ α (t) for H can be lifted to a basis |φ αt for F , since it may be easily seen φ αt |φ βt = δ αβ T δ(t − t ), (A.32) x sin τ e imτ 1 + x 2 cos 2 τd τ = i m ( 1 + 1/x 2 − 1/x) |m| , (B.5) for odd m and the fact that the integral vanishes for even m, we find z n = 0 for even n and, for odd n = 2q + 1, where y = 1 + (b/a) 2 − (b/a) ∈ [0, 1], and J q,p (x) = 1 2 [J q−p (2x) + J q+p+1 (2x)]. For small Ω/b 1, the smallest shift is obtained when a/b 1 to be q ∼ b/Ω. In this limit, y ≈ a/2b. Assuming a/Ω 1, we find Λ 0 = a 2 /(8bΩ) 1. The biggest contribution to z 2q+1 would not come from the smallest power of y, since this is multiplied by J q (2Λ 0 ), which is suppressed by the factorial q!. Instead, it comes from the term with J 0 , that is as discussed in Sec. 4.2.