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Prethermal quasiconserved observables in Floquet quantum systems

Prethermalization, by introducing emergent quasiconserved observables, plays a crucial role in protecting Floquet many-body phases over exponentially long time, while the ultimate fate of such quasiconserved operators can signal thermalization to infinite temperature. To elucidate the properties of prethermal quasiconservation in many-body Floquet systems, here we systematically analyze infinite temperature correlations between observables. We numerically show that the late-time behavior of the autocorrelations unambiguously distinguishes quasiconserved observables from non-conserved ones, allowing to single out a set of linearly-independent quasiconserved observables. By investigating two Floquet spin models, we identify two different mechanism underlying the quasi-conservation law. First, we numerically verify energy quasiconservation when the driving frequency is large, so that the system dynamics is approximately described by a static prethermal Hamiltonian. More interestingly, under moderate driving frequency, another quasiconserved observable can still persist if the Floquet driving contains a large global rotation. We show theoretically how to calculate this conserved observable and provide numerical verification. Having systematically identified all quasiconserved observables, we can finally investigate their behavior in the infinite-time limit and thermodynamic limit, using autocorrelations obtained from both numerical simulation and experiments in solid state nuclear magnetic resonance systems.

Prethermalization, by introducing emergent quasiconserved observables, plays a crucial role in protecting periodically driven (Floquet) many-body phases over exponentially long time, while the ultimate fate of such quasiconserved operators can signal thermalization to infinite temperature. To elucidate the properties of prethermal quasiconservation in many-body Floquet systems, here we systematically analyze infinite temperature correlations between observables. We numerically show that the late-time behavior of the autocorrelations unambiguously distinguishes quasiconserved observables from non-conserved ones, allowing to single out a set of linearly-independent quasiconserved observables. By investigating two Floquet spin models, we identify two different mechanism underlying the quasiconservation law. First, we numerically verify energy quasiconservation when the driving frequency is large, so that the system dynamics is approximately described by a static prethermal Hamiltonian. More interestingly, under moderate driving frequency, another quasiconserved observable can still persist if the Floquet driving contains a large global rotation. We show theoretically how to calculate this conserved observable and provide numerical verification. Having systematically identified all quasiconserved observables, we can finally investigate their behavior in the infinite-time limit and thermodynamic limit, using autocorrelations obtained from both numerical simulation and experiments in solid state nuclear magnetic resonance systems.
While the connection to an effective time-independent Hamiltonian is appealing, the active drive leads to energy absorption by the Floquet many-body system, which is then expected to heat up to infinite temperature. The heating is detrimental to any quantum application, as no local quantum information is retained and all interesting phenomena mentioned above disappear [43][44][45]. It has been shown theoretically [46][47][48][49][50] and experimentally [51,52] that even when the system heats up, the thermalization time can be exponentially long in the drive parameters (typically the frequency of a rapid drive). Then, a long-lived prethermal quasi-equilibrium is established, that allows exploiting the engineered Floquet Hamiltonian for quantum simulation [53][54][55]. The emergent symmetries and conserved observables in the prethermal state distinguish it from the fully thermalized state, and underpin the existence of novel Floquet phases [34,35,50]. Even more surprisingly, some numerical studies have shown that the emergent conserved observables might not display thermalizing behavior even in the infinite-time limit [53][54][55][56]. Many-body localization [32,[57][58][59][60][61][62][63], dynamic localization [53,55,64], and some fine-tuned driving protocols [54,56,65] provide a way to escape the thermalization fate, which could also be absent in finite-size systems. Indeed, distinguishing the long-lived prethermal state from an eventual thermal state is challenging. Numerical studies are bound to finite-size (and often small) systems, while experiments can only probe finite times, before the external environment induces thermal relaxation.
Here we tackle this problem by a numerical and experimental study of two Floquet models in spin chains, namely the kicked dipolar model (KDM) and the alternating dipolar model (ADM). While most studies on spin chain dynamics have focused on evolution of pure states, here we propose to study Floquet prethermalization using infinite temperature correlations. This metric provides information about quasisconserved observables across the whole spectrum and serves as a direct measurable quantity in nuclear magnetic resonance (NMR) experiments. In Sec. II we show that the existence of long-lived quasi-arXiv:2005.11150v1 [quant-ph] 21 May 2020 conserved observables can be unambiguously identified using late-time behavior of the correlations, based on which we provide a method to systematically search for all linearly-independent local quasiconserved quantities. Then we provide both numerical and analytical tools to investigate such prethermal conserved observables and their origins. We first show that the prethermal Hamiltonian H pre obtained from the Magnus expansion under rapid drive yields a quasiconserved observable in each model in Sec. III A. We further show in Sec. III B that when the driving Hamiltonian contains a large global rotation, the Floquet propagator can induce an additional conserved observable, as shown by going beyond the usual Magnus expansion. With all the quasiconserved observables at hand, we investigate in Sec. IV whether they exist in the thermodynamic limit and infinite-time limit, by looking at the dependence of autocorrelations on system size (numerically) and on time (experimentally). Both methods indicate quasiconserved observables vanish and the system thermalizes to infinite temperature.

A. Hamiltonians and Correlations
In this paper we use the Trotter-Suzuki scheme for the driving protocol, where the time-dependent Hamiltonian is piecewise constant in one driving period. However, our results are general for any form of periodic driving. The evolution of the system we study is given by the unitary propagator in one period U F = e −iH2τ e −iH1τ , where in each period we consider the system to be under the Hamiltonian H 1 for a time τ , and then under H 2 for another duration τ . Motivated by NMR experiments, we consider two models of an L-site spin-1/2 chain: the kicked dipolar model (KDM), where  Here 3 is the dipolar interaction operator in an arbitrary direction set by α (α = x, y, z), where S j α are spin-1/2 operators of the j-th spin (j = 1, · · · , L) and S j = (S j x , S j y , S j z ) T . As shown in Ref. [35], the 1/r 3 interaction is sufficiently short range in 1D to yield no qualitative difference with respect to the nearest-neighbor interaction, thus for simplicity in numerical and analytical studies we only keep the nearest-neighbor interaction unless explicitly mentioned. Z = j S j z is the collective magnetization operator along z-axis, and below we will also use X = j S j x , Y = j S j y . J and h are the strength of the dipolar interaction and the collective z-field respectively, and we fix h = J throughout the paper. In numerics we assume periodic boundary conditions.
To investigate quasi-conservation properties we use infinite-temperature correlations as our metric, where U t is the unitary evolution during time t, O and O are observables, and the norm is defined as O ≡ √ TrO 2 . In the following we drop the subscript β = 0 for simplicity. Figure 1 shows numerical simulations of some exemplary correlations, the magnetization along three axes O = O = Z, X, Y in KDM (the qualitative behavior is general for other observables and models.) The autocorrelations of X and Y display oscillations around 0 and damping, which originate from the z-field and the dipolar interaction, respectively. Instead, Z(t)Z exhibits a more interesting behavior. For small Jτ , it quickly equilibrates at a nonzero value independent of L, and it remains constant afterwards. For relatively large Jτ , there is a slow decay of Z(t)Z toward a final value that decreases with increasing L. We thus expect the final value to be zero in the thermodynamic limit, corresponding to an infinite-temperature final state. Indeed, the observable Z displays the defining characteristics of what we deem a quasiconserved observable in the prethermal regime: the autocorrelation of a quasiconserved observable is nonzero in the prethermal regime, but goes to zero in the fully thermalized state. In simulations, autocorrelations of quasiconserved observables still have nonzero value at infinite time due to the small system size (e.g. Z(t)Z in Fig. 1), while for non-conserved observables autocorrelations are zero (e.g. X(t)X in Fig. 1). These distince behaviors serve as a direct metric to identify quasiconserved observables. As any observable that overlaps with a quasiconserved observable would have non-zero infinite-time autocorrelation, we want to find a linearly independent, orthogonal set of eigen-quasiconserved observables.

B. Eigen-quasiconserved Observables
We design a systematic procedure to search for the set of eigen-quasiconserved observables, We note that eigenvectors {E µ } of the Floquet (super)propagatorÛ F form an orthogonal vector basis for the space of operators (herê However, this operator basis is in general highly nonlocal, and thus not practical. We then want to find a small, local set of observables that approximate the exact eigen-observables, and have non-zero eigenvalues, that is, are quasiconserved. We start from a basis set {O (α) } of Hermitian observables that are translationally invariant sums of local operators: Here (α) ≡ (α 1 , · · · , α r ) with α k ∈ {x, y, z, 0}, where S j 0 denotes the identity matrix operating on the j-th spin. By imposing α 1 , α r = 0, we say O (α) is of range r: each term in O (α) acts non-trivially on at most r neighboring spins. Since the number of operators is exponentially large in system size, we restrict our search to the operator subspace spanned by O (α) whose range r ≤ r c , which are local and thus experimentally relevant. Starting from an orthonormal operator basis {O µ } of this subspace (with O µ O ν = δ µν ) we construct a matrix from all pair correlations, Λ µν = O µ (∞)O ν . The matrix Λ is the projection of the infinite-time propagatorÛ F (t → ∞) onto the r c -local subspace. The diagonalization of Λ yields the local eigen-observables E k , and eigenvalues λ k , satisfying E k (∞)E l = λ k δ kl . Note that since Λ is not ensured to be unitary, its eigenvalues do not have unit amplitude, λ k ≤ 1. We note that the larger the λ k , the better E k approximates an exactly conserved observable. The correlations O(∞)O between any two observables whose locality is bounded by r c can be directly derived by decomposing the observables onto the E µ basis We apply this systematic procedure to the two models under consideration. time-independent system. Even for small Trotter steps, most eigenvalues are already small, and decrease when increasing system size. However, a few eigenvalues are large, and show little dependence on system size. This last group comprises the eigenvalues associated with the eigen-quasiconserved observables that govern the nontrivial dynamics at long times. Based on these results, we find that there are two eigen-quasiconserved observables for KDM, E for KDM is close to D z [red curves in Fig. 2(a)]. We can thus more carefully analyze these quasiconserved observables and describe analytically their origin in the limit of small τ in the next section. Even so, we remark that there is an interesting regime at intermediate τ , where E are still large, but they deviate from their static (τ → 0) counterparts. This indicates that the quasiconserved observables truly arise from the Floquet dynamics, and are not simply a remnant of the approximated, static Hamiltonian.

A. Prethermal Hamiltonian
It is intuitive to expect that a quasiconserved observable might emerge from energy conservation. Indeed, one can always regard the Floquet evolution as arising from an effective static Hamiltonian by setting U F = e −iτ H F for some Hermitian operator H F . However, in general this Hamiltonian is highly non-local and thus it is not associated to a local quasi-conserved observable. Still, when the driving frequency is large compared to local energy scales (here J, h), the stroboscopic dynamics is given by a time-independent local prethermal Hamiltonian H pre plus a small correction δH(t) [46,48], which may be nonlocal. It is this prethermal Hamiltonian H pre that can be associated with a local quasiconserved observable. H pre can be obtained from the Floquet-Magnus expansion [66,67] truncated at an optimal order m * : where the zeroth order term is the average Hamiltonian Ω 0 = H = 1/τ τ 0 H(t)dt and higher order terms Ω m involve m nested commutators. Then, for spin chains with nearest-neighbor couplings the range of Ω m grows linearly with m.
The truncation m * is crucial not only to keep the prethermal Hamiltonian local, but also because the series in Eq. 3 diverges for a generic many-body system [48]. The time-dependent correction δH is however exponentially small in 1/Jτ , leading to an exponentially long time t pre for the system to heat up. Thus, for t < t pre , the system effectively prethermalizes to the state e −βHpre where β is determined by the initial state energy, making H pre an eigen-quasiconserved observable. Although one should investigate the prethermalization process by studying the dynamics of an infinitely large system at long times approaching infinity, numerically we can only tackle small system sizes, so we take a different approach -we set the time to infinity, and study how the observable correlations change when increasing system size. The validity of this approach relies on the fact that for a system size L < m * the term δH does not appear in the expan-sion, making O 1 = H pre exactly conserved even at infinite time for sufficiently small τ . From a physics point of view, this means that the energy 2π /τ is larger than the many-body bandwidth (∼ JL), and thus the system cannot absorb energy from the drive if it is faster than 1/JL. Since the zeroth order term of H pre is H, the autocorrelation of H pre provides a bound for that of H, leading to bounded Trotter error in the Trotter-Suzuki scheme [53].
As further verification, we calculate numerically the Floquet Magnus expansion, Eq. (3), up to m = 10 and evaluate not only the convergence of the expansion, but also operator conservation. For the first metric, we plot Ω m in Fig. 3(a) and (d) for the two models studied. We find that, up to the computationally accessible order, the norm of Ω m decays exponentially, indicating that H pre converges when τ is small. From the slopes in Fig. 3(a) and (d), we get radii of convergence Jτ ≈ 3 for both models. Still, the expansion convergence does not guarantee the resulting H pre is a quasiconserved observable. In Fig. 3(b) and (e), we compute the long-time infidelity (1 − H pre (∞)H pre ) by truncating the expansion in Eq. (3) at increasing orders. When Jτ is small, the autocorrelation exponentially approaches 1 with increasing order, suggesting that the optimal truncation order m * should be larger than our largest accessible order here, or even absent in the system size we study. Instead, for larger Jτ , the correlation stops converging at some order; for even larger Jτ (Jτ = 1 for example) the correlation is almost zero for all orders. Therefore, even within the radius of convergence Jτ ≈ 3, H pre from Eq. 3 may fail to be quasiconserved. We plot the infinite-time correlation H pre (∞)H pre versus Jτ in Fig. 3(c) and (f) and show how it changes with system size (here H pre is evaluated to 7 th order). The drop of H pre (∞)H pre with increasing system size is evident for Jτ 1.2 in both models, suggesting that for the system size we explore the effective Hamiltonian picture fails in the above parameter space. Note that in the L → ∞ limit the correlations are expected to be zero for any τ > 0 as will be discussed in Sec. IV.

B. Emergent dipolar order
To search for additional conserved observables in KDM we develop a method inspired by the existence of discrete time-translation symmetry-protected phases in prethermal Floquet systems [50]. Similar results have been obtained for the static Hamiltonian H = hZ + JD y associated with the (zero-order) KDM. For this model, it has been shown that the polarization Z is quasiconserved, and does not reach its thermal equilibrium value until a time exponentially long in h/J [49,50,68], even if according to ETH the system should thermalize.
Since the average Hamiltonian picture breaks down when increasing τ and we see from Fig. 2(a) that the other observable is conserved for even larger τ , we must go beyond the static case, and work directly in the Floquet system. This kind of system was first studied in [50], where they further focused on the case h = π to identify a prethermal Floquet time crystal. Here we generalize their analysis to obtain the novel quasiconserved observable for any h.
We transform the Floquet operator by going to a rotated frame as and demand [Z, D] = 0. By appropriately choosing S, D, it will be shown that δH is exponentially small in [69]. Therefore, for small τ and large enough ratio h/J 0.5 [68], the operator D approximately commutes with the Floquet unitary in the rotated frame, making D pre = e −S De S a prethermal quasiconserved observable in the original frame. We emphasize that the right-hand side of Eq. 4 still describes a Floquet system, therefore we derived the quasiconservation without first transforming to a static Hamiltonian. Note that Z pre = e −S Ze S is quasiconserved in the same sense as D pre . However, whereas D pre , is orthogonal to H pre to zeroth order, Z pre ≈ H pre − D pre and it cannot thus be considered an eigen-quasiconserved observable. Now we give the details of finding the desired S, D. We first write the transformation Eq. 4 in an equivalent form e i hZτ e S e −i hZτ e −i 2 H1τ e −S = e −iτ (D+δH) , Here we assume that J/h and hτ are small parameters whose magnitude are of the same order marked by , and do perturbation in 1. After expanding the operators, D = D 1 + 2 D 2 + · · · , S = S 1 + 2 S 2 + · · · , one can collect terms that are of order j on both sides of Eq. 5, and get a series of equations indexed by j. In practice we do not calculate exponentials directly but use the Magnus expansion of the left-hand side. The j-th order is given by where h j only contains −ihZτ , −iH 1 τ and S j with j < j − 1. The first few orders can be written explicitly, , while higher orders can be found recursively. Assuming all orders S j with j < j −1 are known (which is trivially true for j = 2), we determine S j−1 from Eq. 6 by requiring [S j−1 , −ihZτ ] to cancel the terms in h j that do not commute with Z. Similar to the prethermal Hamiltonian Eq. 3, the expansion in generally diverges and should be truncated at some order, leading to the exponentially small nonlocal residual δH, see, e.g. Ref. [46,50].
Here we explain in detail how to obtain S j−1 from Eq. 6 by taking advantage of the special structure of the field operator Z. We first decompose h j = q=0,±1,··· h jq such that [Z, h jq ] = qh jq (h jq are called the q-th quantum coherence of Z [70][71][72]). This decomposition is only possible when the dominant part of the Hamiltonian has equally spaced eigenvalues, such as for the collective rotation H (K) 2 = JZ in our case. Equation 6 is then satisfied by setting −iτ D j = h j0 and S j−1 = i q =0 h jq /(hqτ ). We note that S is a sufficiently local operator, r(S j ) = j, for KDM with nearest-neighbor interaction.
When τ is small, the S j operators are dominated by the (J/h) j term. Therefore, in the τ → 0 limit, the quasiconserved observable found here for the Floquet model reduces to the prethermal quasiconserved observable of the static Hamiltonian H (K) [50,68], where the expansion is a series of J/h and δH ≈ exp(−O(h/J)). In this regime, , and the expansion converges for h/J 0.5 (up to truncation at exponentially large order) as shown in Ref. [68] (Note that here we used h/J = 1). Instead, for relatively larger hτ , the S j operators are dominated by (hτ ) j and δH ≈ exp(−O(1/hτ )), in agreement with the exponentially slow Floquet heating.
We numerically evaluate the convergence properties of D pre in the KDM [ Fig. 4(a)], using the metrics discussed in the previous section, convergence of the order-by-order expansion terms and infinite-time autocorrelation. We find that the series converges up to order 7 in the hτ regime we are interested in. The infinite-time autocorrelation is close to 1 at small τ , as shown in Fig. 4(b) and (c), confirming that the local truncation of D pre (as obtained by the first few orders) gives rise to quasiconserved  . Comparing these results to the prethermal Hamiltonian shown in Fig. 3(b) and (c), we find that (i) the normalized autocorrelation of D pre converges to 1 in a larger parameter range (Jτ 1.6 for D pre and Jτ 1 for H pre ), (ii) the autocorrelation shows a significant drop at Jτ 1.8 for D pre and Jτ 1.2 for H pre , with a steeper drop when L is increased from 8 to 12. Both facts suggest that D pre is more robust than H pre , in agreement with the experimental results presented in Ref. [51]. This provides evidence that it is possible to realize novel Floquet phases beyond the effective Hamiltonian picture.

IV. TOWARD INFINITE TEMPERATURE: EXPERIMENTAL AND NUMERICAL SIGNATURES
Although it is generally believed that Floquet manybody systems should heat up to infinite temperature, some numerical works [53][54][55][56] have found signs of nonthermal behavior in various models. Here we provide evidence of thermalization in the long-time and thermodynamic limit, using numerics and experiments in a NMR quantum simulator [51,68,70], respectively. In simulations, we can access the infinite-time limit using exact diagonalization, but only for small system sizes. Conversely, the system size in NMR experiments is large enough to achieve the thermodynamic limit, but the evolution time cannot be too long due to hardware limitation. Still, by looking at the dynamics for increasingly longer times (experimentally) and larger system sizes (numerically), we can extract insight on the final fate of the Floquet systems.
The experimental system is a single crystal of fluorapatite (FAp) [74]. We study the dynamics of 19 F spin-1/2 using NMR techniques. Although the sample is 3D, 19 F form quasi-1D structure because the interaction within the chain is ∼40 times larger than the interaction between different chains [75][76][77]. Average chain length is estimated to be > 50 and the coherence time of the 19 F spins is T 1 ≈ 0.8s. The sample is placed in 7 T magnetic field where the Zeeman interaction dominates, thus reducing the 19 F spins interaction to the secular dipolar Hamiltonian H = J 0 D z with J 0 = −29.7 krad/s (we define z as the magnetic field direction). While the corresponding 1D, nearest-neighbor XXZ Hamiltonian is integrable [78][79][80], the experimental 1/r 3 Hamiltonian can lead to diffusive [81,82] and chaotic behavior [83] in 3D.
In the presence of a transverse field, the system is known to show a quantum phase transition [84]. We use 16 RF pulses [51,68,70,85] to engineer the natural Hamiltonian into H = JD x with tunable J. This enables varying the Floquet steps by tuning J, while keeping τ fixed. Then, experimental imperfections such as decoherence and pulse errors remain the same, and we can faithfully quantify the Floquet heating rate. The initial state is a high-temperature thermal state with small thermal polarization in the magnetic field direction ρ(0) ≈ (1 − Z)/2 L with ≈ 10 −5 , and the observable is the collective magnetization along x-axis O = X. As the identity part does not change under unitary evolution and does not contribute to signal, it is convenient to consider only the deviation from the identity δρ(0) = Z, which can be rotated to a desired observable O . Therefore, the NMR signal is equivalent to an infinite- We experimentally study the heating rates of the quasiconserved observables and their scaling with Floquet period, to reveal the prethermal phase and investigate the eventual heating to infinite temperature. In Fig. (5) we show results for ADM (the two quasiconserved observable in KDM show similar behavior as reported elsewhere [51].) To study the autocorrelation of H pre = H + O(τ ) in ADM, we measure the average Hamiltonian    In (a-b), error bars are determined from the noise in the free induction decay (see SM [73] for details on the experimental scheme).
thermalizes to the prethermal Hamiltonian. When more Floquet periods are applied, the autocorrelation of D z slowly decays from its prethermal value. The decay rate in the prethermalization regime is shown in Fig. 5(b), and can be fitted to an exponential function in 1/(Jτ ) on top of a constant background decay (which is due to experimental imperfections, see SM [73] for more details.) By normalizing the data to the data collected under the fastest drive (Jτ = 0.35), the background decay is cancelled, and the resulting dynamics only arises from the coherent evolution, as shown in Fig. 5(c). For given n, the normalized correlation decreases when increasing Jτ , because H pre = H + O(Jτ ) thus H that we measure has less overlap with the true quasiconserved observable H pre for larger Jτ . The overall drop of the curves when increasing n is instead an indicator of Floquet heating.
To better quantify the final thermalization process, we define a critical value J c such that when Jτ > J c τ the system is thermalized, at a given number n of periods in the thermodynamic limit, or for a system size L at infinite time. Studying the scaling of J c as a function of n (experimentally) and L (numerically) provides hints on the long-time, thermodynamic limits.
We numerically obtain the autocorrelations O(∞)O as a function of Jτ , using exact diagonalization. In Fig. 6(a)  consider the exact dipolar interaction instead of truncating to nearest neighbors.) Note that both observables in KDM show a non-monotonic behavior. They appear to be quasiconserved until Jτ = 1; the decrease in overlap is however interrupted by a revival at Jτ = 1.6. This is because H (K) and D z are approximation of H pre and D pre to leading order. Thus H (K) (D z ) still has a small overlap with D pre (H pre ), giving rise to a second plateau at Jτ ≈ 1.6 (Jτ ≈ 1). The experimentally measured autocorrelations of quasiconserved observables in KDM can be find in [51]. For both experiments and simulations we then find J c τ from the point where the curves drop below a threshold value of 0.5 (any other reasonable choice would not qualitatively change the results). We linearly interpolate between data points to get J c τ for every quasiconserved observable and plot the J c τ in Fig. 6(b) and (c). The decrease of numerically calculated J c τ with L in Fig. 6(b) indicates that even the correlations of quasiconserved observables decay to zero as the system thermalizes to infinite temperature, suggesting this non-thermalizing behavior should not persist to the thermodynamic limit. Similar result is also observed from experimentally measured J c τ as shown in Fig. 6(c) [87]. Note that although J c τ for H (K) (n)H shows only a moderate dependence on n [ Fig. 6(c)], its decay is still larger than experimental uncertainties.

V. CONCLUSION
As Floquet driving is a promising avenue for quantum simulation, it is crucial to evaluate its robustness, the existence of a long-lived prethermal phase, and the eventual thermalization to infinite temperature. Investigating Floquet heating, which breaks the prethermal regime, is particularly challenging, not only because of inherent limitations in numerical and experimental studies, but also because of the challenge to properly identifying all quasiconserved observables in the complex, many-body driven dynamics.
Here we tackle both these issues by combining analytical, numerical and experimental tools. First, we provide a systematic strategy to find local, eigen-quasiconserved observables in the prethermal regime using infinitetemperature correlations. By systematically searching over local operators, we find that counter-intuitive quasiconserved observables might emerge, as we identify two eigen-quasiconserved observables: the first, not surprisingly is associate with energy, H pre , under sufficient fast drive; in addition, we find another quasiconserved observable, D pre , for the KDM in the presence of a large driving field. We then use numerical and experimental evidence to obtain insight into the inaccessible thermodynamic limit and long-time regime, to show that autocorrelations of quasiconserved observables indeed decrease toward zero due to Floquet heating, suggesting the Floquet system approaches the infinite temperature state.
Our results not only provide a metric to study thermalization in driven quantum systems, but also open intriguing perspectives into the existence of quasiconserved observables other than the energy. It is an open ques-tion when they emerge and how they interact with each other. A better understanding of quasiconserved observables would benefit understanding of heating in closed driven systems, and designing robust protocol to slow down thermalization.
In the main text we measured the Floquet heating for a periodic, Hamiltonian switching scheme. While it would be easy to change the period by increasing the time between switches, this would lead to experiments performed with different total times or a different number of control operations. In turns, this can introduce variable amount of decoherence and relaxation effects, and of control errors. Instead, we kept the time for one Floquet period constant and used Hamiltonian engineering to vary the Hamiltonian strength in order to vary the Floquet driving frequency.
One of the assumptions in our work is that the background decay rate does not change much with driving frequency (compared to the change in Floquet heating rate). In this section, we provide experimental evidence for this assertion. When changing driving frequency, we are changing (i) the effective strength J of the engineered dipolar interaction JD y and (ii) the kicking angle in the kicked dipolar model by a phase shift (see VII C). As phase shift angles are usually very accurately implemented in NMR experiments, we focus on the engineered dipolar interaction, which is obtained by Floquet engineering itself, as explained in VII C. To quantify how good is the engineered JD y , we measure Y (n)Y and D y (n)D y under the engineered Hamiltonian JD y , without kicking field nor direction alternation, as shown in Fig. 7.
Note that the maximum difference between the decay rate of D y (n)D y over the range of Jτ considered is ∼ 0.003, much smaller than the Floquet heating rate in the main text. A quantitative analysis is challenging because the specific form of error terms is unknown, and JD y is an interacting Hamiltonian thus error accumulation is intractable. Here we use some simple arguments to argue that variations in the background decay with Jτ have little to no influence on our results. First, we note that while in the main text we are interested in the decay of the autocorrelation of H pre and D pre , here with H = JDy we can only discuss the decay of D y and Y , since other observables that are not conserved display very fast decay which is not informative. For example, in the main text we measure D z , which thermalizes even under the ideal D y and thus we cannot distinguish thermalization from decay due to experimental imperfections in the engineered dipolar Hamiltonian D y . Still, as D z and D y overlap, if the background decay of D z had a significant change with Jτ , it would be reflected in D y , which is not observed. Therefore, we expect the change in the background decay rate for D z (n)D z to be small as well.
Here we can only probe the background decay rate of Y , while in the main text we are interested in the longitudinal magnetization, Z, that appears in H (K) (n)H (K) [see Fig. 6(c)]. The transverse magnetization decay rate is, however, a upper bound for Z, since in NMR experiments Z is usually more robust against errors than Y due to the large magnetic field in z-axis that suppresses decoherence and experimental errors that do not conserve the total Zeeman energy (we note that we typically do not explicitly write the Zeeman energy in the Hamiltonians as we work in the rotating frame). Even if the variation in the background decay for Z were as large as what observed for Y in these experiments (∼ 0.009), it would still be still small compared with Floquet (see inset of Fig. 7). In addition, in the kicked dipolar model, we can consider D y as being subjected to rotations along Z that further cancel out the error terms in the engineered JD y that do not conserve Z. As a result, the decay rate of Y due to the engineered D y is larger, by about a factor of 2, than the baseline decay of H The system used in the experiment was a single crystal of fluorapatite (FAp). Fluorapatite is a hexagonal mineral with space group P 6 3 /m, with the 19 F spin-1/2 nuclei forming linear chains along the c-axis. Each fluorine spin in the chain is surrounded by three 31 P spin-1/2 nuclei. We used a natural crystal, from which we cut a sample of approximate dimensions 3 mm×3 mm×2 mm. The sample is placed at room temperature inside an NMR superconducting magnet producing a uniform B = 7 T field. The total Hamiltonian of the system is given by The first two terms represent the Zeeman interactions of the F(S) and P(s) spins, respectively, with frequencies ω F = γ F B ≈ (2π)282.37 MHz and ω P = γ P B = (2π)121.51 MHz, where γ F/P are the gyromagnetic ratios. The other three terms represent the natural magnetic dipole-dipole interaction among the spins, given generally by where r ij is the vector between the ij spin pair. Because of the much larger Zeeman interaction, we can truncate the dipolar Hamiltonian to its energy-conserving part (secular Hamiltonian). We then obtain the homonuclear Hamiltonians H P = 1 2 λ<κ J P κλ (2s λ z s κ z − s λ x s κ x − s λ y s κ y ) (10) and the heteronuclear interaction between the F and P spins, with J jk = γ j γ k 1−3 cos(θ jk ) 2 | r jk | 3 , where θ jk is the angle between the vector r jk and the magnetic field z-axis.
The maximum values of the couplings (for the closest spins) are given respectively by J F = −32.76 krad s −1 , J P = 1.20 krad s −1 and J F P = 6.12 krad s −1 .
The dynamics of this complex many-body system can be mapped to a much simpler, quasi-1D system. First, we note that when the crystal is oriented with its c-axis parallel to the external magnetic field the coupling of fluorine spins to the closest off-chain fluorine spin is ≈ 40 times weaker, while in-chain, next-nearest neighbor couplings are 8 times weaker. Previous studies on these crystals have indeed observed dynamics consistent with spin chain models, and the system has been proposed as solid-state realizations of quantum wires [75,77,88]. This approximation of the experimental system to a 1D, short-range system, although not perfect has been shown to reliably describe experiments for relevant time-scales [76,89]. The approximation breaks down at longer times, with a convergence of various effects: long-range in-chain and cross-chain couplings, as well as pulse errors in the sequences used for Hamiltonian engineering. In addition, the system also undergoes spin relaxation, although on a much longer time-scale (T 1 = 0.8 s for our sample).

B. Error analysis
In experiments, we want to measure the correlation δρ(t)O , where δρ(t) = U (t)δρ(0)U (t) is the nontrivial part of the density matrix evolved under a pulse-control sequence for a time t. Instead of just performing a single measurement after the sequence, we continuously monitor the free evolution of δρ(t) under the natural Hamiltonian H dip , from t to t + t FID . The measured signal is called in NMR free induction decay (FID) and a typical FID trace is shown in Fig. 9). This signal trace allows us to extract not only the amplitude of the correlation (from the first data point) but also its uncertainty. We take the standard deviation of the last 20 data points in the FID as the uncertainty of the δρ(t)O . This uncertainty is used with linear error propagation to obtain the error bars of all the quantities analyzed in the main text.

C. Hamiltonian Engineering
In the main text we focused on the Floquet heating (Trotter error) for a periodic alternating scheme, switching between two Hamiltonians. In order to avoid longer times and/or different numbers of control operations when changing the Trotter step (Floquet period), we engineered Hamiltonians of variable strengths. Then, !" x y y x x y y x x y y x x y y x !# 2!" !# 2!" !# 2!" !# 2!" !# 2!" !# 2!" !# 2!" !# !" Note that thanks to the ability of inverting the sign of the Hamiltonian, the scheme amounts to measuring out-of-time order correlations.