# Counterdiabatic Driving for Periodically Driven Systems

Paul M. Schindler<EMAIL_ADDRESS>Max Planck Institute for the Physics of
Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany  Marin Bukov
<EMAIL_ADDRESS>Max Planck Institute for the Physics of Complex Systems,
Nöthnitzer Str. 38, 01187 Dresden, Germany

###### Abstract

Periodically driven systems have emerged as a useful technique to engineer the
properties of quantum systems, and are in the process of being developed into
a standard toolbox for quantum simulation. An outstanding challenge that
leaves this toolbox incomplete is the manipulation of the states dressed by
strong periodic drives. The state-of-the-art in Floquet control is the
adiabatic change of parameters. Yet, this requires long protocols conflicting
with the limited coherence times in experiments. To achieve fast control of
nonequilibrium quantum matter, we generalize the notion of variational
counterdiabatic driving away from equilibrium focusing on Floquet systems. We
propose two approaches to find local approximations to the adiabatic gauge
potential for the effective Floquet Hamiltonian, based on the inverse-
frequency expansion and a non-perturbative variational principle. They enable
transitionless driving of Floquet eigenstates far away from the adiabatic
regime. We discuss applications to two-level, Floquet band, and interacting
periodically-driven models. In particular, we demonstrate that the developed
methods allow us to capture non-perturbative photon resonances and obtain
high-fidelity protocols that respect experimental limitations like the
locality of the accessible control terms. Our work lays the foundations for a
quantum control theory of nonequilibrium systems.

##  I Introduction

Figure 1:  Sketch: Floquet State Manipulation and Floquet counterdiabatic
driving. Periodic driving (blue) is used to engineer properties of the
system—here exemplified by the Kapitza pendulum where the otherwise unstable
fixed point at $\theta=\pm\pi$ (see potential left) is stabilized by the
driving (see other two potentials). To prepare the target Floquet state
$\ket{\psi_{\mathrm{target}}}$ from the initial state
$\ket{\psi_{\mathrm{initial}}}$, additional control (yellow) has to be added.
Control in Floquet systems in general takes one of three forms (see box left):
$(i)$ amplitude ramps, $(ii)$ frequency chirps, and $(iii)$ change of the
static part of the Hamiltonian. State manipulation without additional
counterterms (unassisted) away from the adiabatic limit fails. However, using
Floquet counterdiabatic driving (FCD) we can obtain fast transitionless
protocols.

Over the last decade, periodic (Floquet) drives have emerged as a useful
technique to engineer the properties of quantum systems [1, 2, 3, 4] in a
variety of platforms ranging from condensed matter systems [4] to quantum
simulators [3, 5, 6]. The periodicity of the drive allows us to describe the
dynamics of Floquet systems by an effective time-independent (a.k.a. Floquet)
Hamiltonian at times integer multiple of the driving period [7, 8, 2]. These
effective Hamiltonians can have properties drastically different from those of
the non-driven system. Such nonequilibrium behavior is exploited in a Floquet
engineering approach [1, 2, 3] to realize Hamiltonians otherwise hardly
implementable in the lab: most notably, they can exhibit artificial gauge
fields [9, 10, 11, 5] and topological lattice models [12, 13, 14, 15, 16, 17,
18, 19, 20], but can also give rise to dynamically stabilized [21, 22, 23, 24]
and localized [25, 26, 27] matter. Moreover, certain periodically driven
systems display nonequilibrium phases of matter without static counterparts,
e.g., discrete time crystals [28, 29, 30, 31, 32, 33] or anomalous topological
insulators [34, 35, 36, 37, 38, 39]. In many body systems the lack of energy
conservation due to the periodic drive typically leads the system to heat to a
featureless infinite temperature state at sufficiently late times [40, 41].
However, Floquet heating is exponentially suppressed in the driving frequency
[41, 42, 43, 44]. Thus, a parametrically long-lived and metastable prethermal
plateau forms whose properties can be designed using Floquet engineering. Due
to their versatility and accessibility, techniques based on strong periodic
drives are developed into a standard toolbox for quantum simulation.

However, engineering a local effective Hamiltonian that governs the system
within a stable long-lived prethermal plateau, is necessary yet insufficient
for a self-contained quantum simulation toolbox. An essential prerequisite for
investigating the physics of the effective Hamiltonian is the ability to
prepare and probe its eigenstates. At present, manipulating the system’s state
under the presence of a strong periodic drive remains an outstanding
challenge. More generally, control of nonequilibrium quantum matter remains
largely unexplored. Yet, developing such a theory is imperative to complete
Floquet engineering into a self-contained quantum simulation toolbox.

The existence of a time-independent Floquet Hamiltonian may suggest borrowing
concepts from equilibrium quantum control. However, as we show in this work,
the physics of state manipulation is richer and more intricate for genuinely
nonequilibrium systems. Unlike their equilibrium counterparts, the exact
effective Hamiltonian for generic Floquet systems is a non-local operator, and
its exact analytic form is known only in a handful of cases. This poses
considerable challenges for developing analytical control techniques.
Additionally, the control terms explicitly break the time-periodicity of the
Hamiltonian which renders Floquet’s theorem – the cornerstone of Floquet
engineering – not directly applicable, further limiting analytical progress.
Indeed, if applied carelessly, extra control terms can become resonant with
the periodic drive which will inevitably reduce the lifetime of the prethermal
plateau. For all these reasons, developing theories to control Floquet systems
is a difficult yet important problem, whose solution has the potential to
advance quantum simulation.

The state-of-the-art approach to control periodically driven systems is the
adiabatic change of parameters. However, the adiabatic limit for the (local)
effective Hamiltonian does not exist since photon absorption gaps in the
quasi-energy spectrum need to be passed diabatically while the co-existing
conventional Landau-Zener gaps adiabatically [45, 46, 47]. In generic many-
body systems, these two kinds of gaps appear indistinguishable (cf. Fig. 2);
thus, manipulating state population dynamics becomes intractable. Moreover, in
practice, adiabatic state preparation requires slow protocols to suppress
excitations due to diabatic transitions. This stands in contrast to the
limited coherence timescales in present-day quantum simulators [48, 49] that
require fast processes to avoid decoherence.

In this work, we lay the foundations of a quantum control theory away from
equilibrium, by focusing on periodically driven systems. We address the
problems inherent to adiabatic Floquet control schemes by generalizing the
concept of transitionless driving [50, 51, 52, 53, 54, 55, 56, 57, 58, 59] to
Floquet systems. This theory of Floquet counterdiabatic driving (FCD)[Fig. 1]
yields transitionless control protocols that transfer population between
Floquet eigenstates in finite time. This is achieved by introducing additional
terms which counteract diabatic transitions.

In particular, we introduce two complementary approaches. The first, based on
inverse frequency expansions (IFE), is tailored to Floquet engineering in the
high-frequency regime. As a perturbative method, it is unable to resolve
photon resonances in the quasi-energy spectrum. We demonstrate how this
deficiency can be turned into a useful engineering feature: passing resonances
diabatically can suppress undesired excitation channels. Second, we present a
non-perturbative variational approach based on a least-action principle.
Provided with a suitable ansatz, this method suppresses diabatic excitations
for all avoided crossings in the quasi-energy spectrum – including the non-
perturbative photon resonances.

We illustrate our theory on three paradigmatic and genuinely nonequilibrium
control problems without static counterparts. First, we compare the two
approaches using the pedagogical example of a driven two-level system. We
demonstrate that both methods lead to transitionless driving between quasi-
energy states for conventional resonances. In addition, we show that choosing
either the IFE or variational approach allows us to toggle whether photon
resonances are passed diabatically or adiabatically. This property can add
extra versatility to the Floquet engineering toolbox. Second, we apply FCD
driving to the experimentally highly relevant case of Floquet state
manipulation in an ultracold atom quantum simulator. We demonstrate that,
taking experimental constraints into account, our method leads to a
significant reduction in diabatic transitions. Finally, we use our theory to
enhance the many-body fidelity of state population transfer in a non-
integrable Ising model. In particular, we showcase how periodic drives open up
new ways to aid and improve state preparation even in interacting static
systems.

##  II Theory of Counterdiabatic Driving for Floquet Systems

Figure 2:  Sketch: Comparison of local Floquet counterdiabatic driving
methods. (A) Comparison of IFE and variational procedure to obtain the
(local)-FCD protocol. For IFE a transformation to the perturbative Floquet
frame and back is needed, while the variational method operates on the level
of the lab frame. (B) Action of the two methods on avoided level crossings in
the quasi-energy spectrum. $(i)$ For a standard resonance—where both quasi-
energies and non-driven or perturbative energies are close in energy—both
l-FCD approaches follow the adiabatic path. $(ii)$ For a photon
resonance—where perturbative energy levels which are multiples of the driving
frequency $\omega$ apart hybridize—the variational method still follows the
adiabatic path. However, the IFE method leads to a diabatic transition,
violating adiabaticity but possibly suppressing heating.

Consider a periodically driven Hamiltonian
$\mathcal{H}_{\lambda}(t)=\mathcal{H}_{\lambda}(t+T)$ with period $T$ and
control parameter $\lambda$. Common examples include switching on a periodic
drive by ramping up its amplitude, changing the drive frequency, or another
parameter in the static part of the Hamiltonian, cf. Fig. 1. For every fixed
value of $\lambda$, the stroboscopic evolution ($t=nT$ $n\in\mathbb{Z}$) is
generated by a time-independent Floquet Hamiltonian
$\mathcal{H}_{F\\!,\lambda}$,

$e^{-iT\mathcal{H}_{F\\!,\lambda}}=\mathcal{T}e^{-i\int_{0}^{T}\mathcal{H}_{\lambda}(t)\mathrm{d}t}\,.$
(1)

More generally, there exists a rotating (Floquet) frame in which the dynamics
(both stroboscopic and non-stroboscopic) are exactly described by the Floquet
Hamiltonian, i.e.,

$\mathcal{H}_{F\\!,\lambda}=\mathcal{P}_{\\!\lambda}^{\dagger}(t)\mathcal{H}_{\lambda}(t)\mathcal{P}_{\\!\lambda}(t)-i\mathcal{P}_{\\!\lambda}^{\dagger}(t)\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{P}_{\\!\lambda}(t)\,.$
(2)

This frame is defined by the micromotion operator
$\mathcal{P}_{\\!\lambda}\pqty{t}=\mathcal{P}_{\\!\lambda}\pqty{t+T}\equiv\exp(-i\mathcal{K}_{\lambda}(t))$
generated by the kick operator $\mathcal{K}_{\lambda}(t)$.

Our goal is to achieve transitionless driving between eigenstates of the
Floquet Hamiltonian $\mathcal{H}_{F\\!,\lambda}$ (a.k.a., Floquet states) upon
varying the control parameter $\lambda=\lambda\pqty{t}$ from some initial
value $\lambda_{\mathrm{i}}$ to a final value $\lambda_{\mathrm{f}}$ in a
finite time $T_{\mathrm{ramp}}$ [Fig. 1]. The difficulty comes from the
observation that the time-dependence of the control parameter need not be
periodic in general. Hence, formally it breaks the periodicity of the
Hamiltonian, i.e.,
$\mathcal{H}_{\lambda(t+T)}(t+T)\neq\mathcal{H}_{\lambda(t)}(t)$.

The adiabatic theorem [60, 61] guarantees transitionless (adiabatic) driving
for gapped states only in the limit of infinitely slow variations, i.e., as
$\dot{\lambda}\to 0$ ($T_{\mathrm{ramp}}\to\infty$). In the absence of quasi-
energy level-crossings along the protocol, this also holds for Floquet systems
[45]. However, the strict adiabatic limit is inaccessible in practice. As a
result, changing the control parameter faster than the inverse quasi-energy
gap to nearby levels unavoidably leads to sizeable diabatic excitations. In
non-driven systems, such diabatic excitations can be exactly suppressed with
the help of counterdiabatic (CD) driving. By adding additional counterterms to
the Hamiltonian, exact CD driving allows us to follow the adiabatically-
connected instantaneous state of $\mathcal{H}_{\lambda}$ at all times during
the application of the protocol.

In this paper, we generalize the theory of CD driving to periodically driven
systems. In other words, we answer the question as to which counterterms need
to be added to the controlled lab-frame Hamiltonian $\mathcal{H}_{\lambda}(t)$
to achieve transitionless driving in the eigenstates of the instantaneous
effective Floquet Hamiltonian $\mathcal{H}_{F\\!,\lambda}$.

###  II.1  Floquet Adiabaticity and Floquet Counterdiabatic Driving

To suppress diabatic excitations, we first identify the cause for diabatic
transitions. This requires generalizing the concept of the adiabatic gauge
potential (AGP) $\mathcal{A}_{\lambda}(t)$ [50, 51, 58] to periodically driven
systems and define the Floquet adiabatic gauge potential (FAGP). To this end,
we revise the transformation to the Floquet frame that removes the micromotion
dynamics, to include the time-dependence of the protocol $\lambda(t)$ [45]

$\tilde{\mathcal{H}}_{\lambda}=\mathcal{H}_{F\\!,\lambda}-\dot{\lambda}\tilde{\mathcal{A}}_{\mathcal{P}\\!,\lambda}.$
(3)

where we take into account the dependence of $\mathcal{P}_{\\!\lambda}(t)$ on
the control parameter $\lambda$, and
$\tilde{\mathcal{A}}_{\mathcal{P}\\!,\lambda}(t)=i\mathcal{P}_{\\!\lambda}^{\dagger}\partial_{\lambda}\mathcal{P}_{\\!\lambda}$.
To make the cause for excitations explicit, we now perform a second
transformation to a co-moving frame where the Floquet Hamiltonian
$\mathcal{H}_{F\\!,\lambda}$ is diagonal. Noting the implicit dependence on
time of the instantaneous change-of-basis operator $V_{\lambda}$ via the
protocol $\lambda(t)$, we find

$\tilde{\tilde{\mathcal{H}}}_{\lambda}=\tilde{\mathcal{H}}_{F\\!,\lambda}-\dot{\lambda}\pqty{\tilde{\tilde{\mathcal{A}}}_{F\\!,\lambda}+\tilde{\tilde{\mathcal{A}}}_{\mathcal{P}\\!,\lambda}}\,,$
(4)

where
$\tilde{\mathcal{H}}_{F\\!,\lambda}=V_{\lambda}^{\dagger}\mathcal{H}_{F\\!,\lambda}V_{\lambda}=\sum_{n}\ket{n}E_{{F},n}(\lambda)\bra{n}$
is a diagonal matrix containing the instantaneous Floquet (quasi-)energies
$E_{{F},n}$, and
$\tilde{\tilde{\mathcal{A}}}_{F\\!,\lambda}=iV_{\lambda}^{\dagger}\partial_{\lambda}V_{\lambda}$.

In this second frame, the Hamiltonian $\tilde{\mathcal{H}}_{F\\!,\lambda}$ is
diagonal at all times, and therefore all diabatic transitions are necessarily
caused by the off-diagonal Floquet adiabatic gauge potential (FAGP)
$\tilde{\tilde{\mathcal{A}}}=\tilde{\tilde{\mathcal{A}}}_{F\\!,\lambda}+\tilde{\tilde{\mathcal{A}}}_{\mathcal{P}\\!,\lambda}$.
The two contributions in the FAGP correspond to the change in the eigenbasis
of the Floquet Hamiltonian ($\mathcal{A}_{F}$) and the change of Floquet frame
($\mathcal{A}_{\mathcal{P}}$). Therefore, we can suppress all diabatic
transitions in the presence of the periodic drive, by adding the gauge
potential term. In the original lab frame, this leads us to the
counterdiabatic Hamiltonian

$\displaystyle\mathcal{H}_{\mathrm{CD}\\!,\lambda}(t)$
$\displaystyle=\mathcal{H}_{\lambda}(t)+\dot{\lambda}\mathcal{A}_{\lambda}(t),\,$
(5) $\displaystyle\mathcal{A}_{\lambda}(t)$
$\displaystyle=i\partial_{\lambda}(\mathcal{P}_{\\!\lambda}(t)V_{\lambda})V_{\lambda}^{\dagger}\mathcal{P}_{\\!\lambda}^{\dagger}(t).$

Hence, Eq. (5) provides the sought-after Floquet counterdiabatic protocol. We
emphasize that the structure of the operator $\mathcal{A}_{\lambda}$ is
independent of the specific form of the protocol $\lambda(t)$ 111We note that
the FAGP is independent of the form of the protocol only if the frequency is
constant during the protocol, see Sec. III.. Therefore, in the adiabatic limit
$\dot{\lambda}\to 0$, all diabatic transitions vanish since
$\dot{\lambda}\mathcal{A}_{\lambda}\to 0$.

###  II.2  Approximate Floquet Counterdiabatic Driving

Notice that the definition of the FCD protocol (5) requires (i) solving the
Floquet problem, and (ii) fully diagonalizing the Floquet Hamiltonian for all
parameter values $\lambda(t)$. These are notoriously hard tasks, as obtaining
the full spectrum in many-body systems is exponentially hard in the system
size, while finding closed-form expressions for the Floquet Hamiltonian is
challenging even in few-level systems due to time-ordering, let alone in
generic many-body systems. Thus, computing an exact FAGP using Eq. (5) is out
of reach for almost all models in practice. Moreover, even if one could obtain
the exact FAGP, it often contains long-range or multi-body interactions [58]
which are challenging to implement in an actual experiment [63]. Therefore,
the relevant question for experimental applications concerns finding an
optimal local approximation to the exact CD Hamiltonian (5). We refer to this
procedure as local Floquet counterdiabatic driving (l-FCD).

Below, we present two distinct yet viable approaches for local FCD. First, we
consider an inverse frequency expansion method (IFE), computing
$\mathcal{A}_{\mathcal{P}}$ and $\mathcal{A}_{F}$ separately. As an
alternative, we then derive a least-action principle for the FAGP which allows
for a direct approximation of the exact FAGP using a variational principle
(variational method).

####  II.2.1 Inverse Frequency Expansion

A key observation for deriving the IFE approach is the absence of the periodic
time-dependence in the Floquet frame, cf. Eq. (3). Therefore, provided we have
access to (some approximation to) the Floquet Hamiltonian
$\mathcal{H}_{F\\!,\lambda}$ and its micromotion operator
$\mathcal{P}_{\\!\lambda}(t)$, one can apply techniques from static CD driving
to find a local approximation $\mathcal{X}_{F\\!,\lambda}$ to the exact gauge
potential $\mathcal{A}_{F,\lambda}$.

In particular, we can use local counterdiabatic driving [64]. That is, using a
parametrized ansatz, $\mathcal{X}_{F\\!,\lambda}$, for the FAGP in the Floquet
frame, the optimal parameters are obtained from the CD variational principle
$\delta_{\mathcal{X}_{F\\!,\lambda}}S_{F}=0$ with action

$\displaystyle S_{F}[\mathcal{X}_{F\\!,\lambda}]$
$\displaystyle=\Tr\left(\mathcal{G}_{F}^{2}\pqty{\mathcal{X}_{F\\!,\lambda}}\right),$
(6) $\displaystyle\mathcal{G}_{F}\pqty{\mathcal{X}_{F\\!,\lambda}}$
$\displaystyle=i\commutator{\mathcal{H}_{F\\!,\lambda}}{\mathcal{X}_{F\\!,\lambda}}-\partial_{\lambda}\mathcal{H}_{F\\!,\lambda}\,.$

In general, we do not have access to the exact Floquet Hamiltonian
$\mathcal{H}_{F\\!,\lambda}$ and micromotion operator
$\mathcal{P}_{\\!\lambda}$. However, in Floquet engineering one is mostly
interested in large driving frequencies compared to local energy scales. In
this regime, a perturbative solution in terms of an inverse-frequency
expansion can be obtained to a desired order $n$:

$\displaystyle\mathcal{H}_{F\\!,\lambda}$
$\displaystyle=\mathcal{H}_{F\\!,\lambda}^{(n)}+O(\omega^{-n-1})=\sum_{j=0}^{n}\omega^{-j}\mathcal{H}_{F\\!,\lambda}^{[j]}+O(\omega^{-n-1})\,,$
(7) $\displaystyle\mathcal{K}_{F\\!,\lambda}$
$\displaystyle=\mathcal{K}_{F\\!,\lambda}^{(n)}+O(\omega^{-n-1})=\sum_{j=0}^{n}\omega^{-j}\mathcal{K}_{F\\!,\lambda}^{[j]}+O(\omega^{-n-1})\,,$

e.g., using a van Vleck [65, 2, 66], Floquet-Magnus [67, 68, 69, 1] or
Brillouin-Wigner [70, 71] expansion.

The final approximate FAGP $\mathcal{X}$ that should be implemented in the lab
frame, is then obtained by transforming $\mathcal{X}_{F}$ back to the lab
frame and adding the additional Floquet-frame contribution
$\mathcal{A}_{\mathcal{P}\\!,\lambda}$:

$\mathcal{X}^{(n)}_{\lambda}(t)=\mathcal{P}^{(n)}_{\\!\lambda}\mathcal{X}^{(n)}_{F\\!,\lambda}\mathcal{P}^{(n)\dagger}_{\\!\lambda}+\mathcal{A}^{(n)}_{\mathcal{P}\\!,\lambda}\,.$
(8)

Here, $\mathcal{X}^{(n)}_{F,\lambda}$ denotes the solution to Eq. (6) computed
with respect to the $n$’th order Floquet Hamiltonian $\mathcal{H}^{(n)}_{F}$.
Using an inverse frequency expansion for Eq. (8) and recalling that
$\mathcal{K}_{F\\!,\lambda}^{[0]}=0$ 222For strongly coupled drives, this can
always be achieved by an additional transformation to a rotating frame., we
find for the lowest two orders in the inverse frequency:

$\displaystyle\mathcal{X}_{\lambda}^{(0)}(t)$
$\displaystyle=\mathcal{X}^{(0)}_{F\\!,\lambda}$ (9)
$\displaystyle\mathcal{X}_{\lambda}^{(1)}(t)$
$\displaystyle=\mathcal{X}^{(1)}_{F\\!,\lambda}+\omega^{-1}\pqty{i\commutator{\mathcal{K}_{F\\!,\lambda}^{[1]}(t)}{\mathcal{X}^{(1)}_{F\\!,\lambda}}-\partial_{\lambda}\mathcal{K}_{F\\!,\lambda}^{[1]}(t)}\,,$

where $\mathcal{X}^{(0)}_{F\\!,\lambda}$ and
$\mathcal{X}^{(1)}_{F\\!,\lambda}$ are obtained from minimizing the action (6)
for $\mathcal{H}_{F}^{(0)}$ and $\mathcal{H}_{F}^{(1)}$, respectively. For
convenience, we provide analytic expressions up to third order in App. B.1.

Note that, the IFE approach to the Floquet adiabatic gauge potential, is bound
to the validity of the inverse frequency expansion which breaks down whenever
photon resonances occur [65, 73, 74]. This is particularly problematic for
interacting systems.

Moreover, the IFE may also be problematic from a practical perspective. To see
why, recall that the main purpose for introducing the variational ansatz Eq.
(6) is the ability to select the operator structure of
$\mathcal{X}_{F,\lambda}$, and hence to incorporate experimental constraints
such as locality. However, within the IFE approach, the variational principle
is applied in the Floquet frame. Thus, the extra additive contribution from
the transformation $\mathcal{P}_{\\!\lambda}^{(n)}$ back to the lab frame –
$\mathcal{A}_{\mathcal{P},\lambda}^{(n)}$ in Eq. (8) – is fixed and cannot be
shaped. Therefore, there exists no obvious way to obtain experimentally
implementable local approximations to this additional contribution within the
IFE approach.

Nonetheless, we emphasize that the IFE method can still be useful for many
Floquet engineering studies that are designed to work in the high-frequency
regime, where the $\mathcal{H}_{F\\!,\lambda}^{(n)}$ is the object of
interest. Here, both $\mathcal{H}_{F\\!,\lambda}^{(n)}$ and
$\mathcal{K}^{(n)}_{F\\!,\lambda}$ are already known, making the application
of IFE straightforward and hence allowing for an improved state manipulation
at a low computational cost.

Moreover, the incapability of the IFE to capture photon resonances can even be
advantageous. In particular, in Floquet engineering applications governed by a
perturbative Floquet Hamiltonian $\mathcal{H}_{F}^{(n)}$, state preparation
protocols should ideally follow the eigenstates of the approximate Floquet
Hamiltonian $\mathcal{H}_{F}^{(n)}$ rather than the exact $\mathcal{H}_{F}$.
Since photon resonances are not captured by the perturbative Floquet
Hamiltonian, the resulting IFE CD protocol will ignore any photon-induced
hybridization gaps and traverse them diabatically, as if they are not present.
This can help avoid undesired hybridization of quasi-energy eigenstates and
some forms of heating in few-particle and weakly-interacting or integrable
systems. Indeed, as we show in Sec. IV.1, using the IFE protocol and a
suitably short ramp duration allows us to pass through the spectral gaps of
the approximate $\mathcal{H}_{F}^{(n)}$ adiabatically, while traversing photon
resonances diabatically.

####  II.2.2 Variational Procedure

Considering the shortcomings of the IFE-approach it would be advantageous to
have a variational principle akin to Eq. (6) for the lab frame FAGP without
relying on the inverse frequency expansion or any additional change-of-frame
transformations.

Using the definition of $\mathcal{H}_{F}$ in Eq. (2) and the action for
$\mathcal{A}_{F}$ (6), we now state the following variational principle for
$\mathcal{A}_{\lambda}$:

$\displaystyle S\bqty{\mathcal{X}_{\lambda}}$
$\displaystyle=\int_{0}^{T}\Tr(\mathcal{G}^{2}(\mathcal{X}_{\lambda}(t)))\,\mathrm{d}t,$
(10) $\displaystyle\mathcal{G}\pqty{\mathcal{X}_{\lambda}}$
$\displaystyle=i\commutator{\mathcal{H}_{\lambda}(t)}{\mathcal{X}_{\lambda}(t)}+\partial_{t}\mathcal{X}_{\lambda}(t)-\partial_{\lambda}\mathcal{H}_{\lambda}(t)\,,$

where the integral over time and the partial time derivative are evaluated at
fixed $\lambda$. To see where Eq. (10) comes from intuitively, notice that
plugging the definition of $\mathcal{H}_{F\\!,\lambda}$ (2) into the action
(6), we obtain

$\mathcal{P}_{\lambda}^{\dagger}\mathcal{G}\mathcal{P}_{\lambda}=i\commutator{\mathcal{H}_{\lambda}(t)}{\mathcal{X}_{F}(t)+\mathcal{A}_{\mathcal{P}}}+\partial_{t}\pqty{\mathcal{X}_{F}(t)+\mathcal{A}_{\mathcal{P}}}+\partial_{\lambda}\mathcal{H}_{\lambda}(t)\,,$

where the explicit time dependence corresponds to the periodic time dependence
only. Replacing $\mathcal{X}_{F}(t)+\mathcal{A}_{\mathcal{P}}$ by an ansatz
operator $\mathcal{X}$, taking the trace, and integrating the action over one
period $T$ we arrive at Eq. (10). A rigorous detailed derivation using the
extended Hilbert space is given in App. B.2.

Equation (10) is the desired Floquet variational principle, allowing for the
determination of an approximate FAGP using only lab frame quantities. Notably,
the ansatz $\mathcal{X}_{\lambda}(t)$ for the variational FAGP carries an
explicit periodic time dependence. Therefore, a complete basis to expand
$\mathcal{X}_{\lambda}(t)$ in must, in addition to all operators acting on the
Hilbert space, also include a complete set of periodic functions. Hence, in
general, $\mathcal{X}_{\lambda}(t)$ contains infinitely many terms arising
from the Fourier harmonics of the periodic time dependence.

However, in practical applications, it is often sufficient to truncate the
infinite sum to a finite-dimensional subset including only a finite number
$N_{h}$ of Fourier harmonics. Then, we can choose an ansatz with $N_{h}$
harmonics and $N_{O}$ local operators

$\mathcal{X}_{\lambda}(t)=\sum_{m=1}^{N_{O}}\sum_{\ell=-N_{h}}^{N_{h}}\chi_{\ell
m}e^{i\ell\omega t}\mathcal{O}_{m}\,,$ (11)

where $\chi_{\ell m}$ are the variational parameters. The operators
$\mathcal{O}_{m}$ are chosen to reflect any external constraints, e.g.,
locality or accessibility in the lab. In order to close the truncated algebra
under multiplication we ignore any Fourier harmonics $e^{i\ell\omega t}$ with
$\absolutevalue{\ell}>N_{h}$ resulting from a product of two time-dependent
functions.

For the variational principle, we have to truncate $\mathcal{G}$ to the given
number of harmonics $N_{h}$ before computing the action $S$. This leads to the
truncated variational principle

$\displaystyle S^{(N_{h})}\bqty{\mathcal{X}_{\lambda}}$ $\displaystyle=$
$\displaystyle\int_{0}^{T}\Tr(\mathcal{G}^{(N_{h})}\pqty{\mathcal{X}_{\lambda}(t)}^{2})\mathrm{d}t$
(12) $\displaystyle\mathcal{G}^{(N_{h})}\pqty{\mathcal{X}_{\lambda}}$
$\displaystyle=$ $\displaystyle
i\commutator{\mathcal{H}_{\lambda}(t)}{\mathcal{X}(t)}+\partial_{t}\mathcal{X}_{\lambda}(t)-\partial_{\lambda}\mathcal{H}_{\lambda}(t)\big{|}_{N_{h}}\,,$

which is equivalent to Eq. (10) if a complete ansatz ($N_{h}\to\infty$) is
considered. A detailed algorithm to compute an approximate variational FAGP
numerically is found in App. E.

Note that the action (10) is quadratic in the gauge potential such that the
minimization is convex and guaranteed to converge to a global optimum. Let us
re-iterate that the variational principle, Eq. (10), is a fully non-
perturbative method that allows for the direct determination of a local
approximate FAGP in the lab frame. As such, it overcomes all of the
shortcomings of the IFE approach, at the expense of dealing with an infinite-
dimensional variational space. However, as we will see below, in practical
applications, a truncation to a finite number of harmonics proves sufficient,
see Eqs. (11) and (12).

##  III Counter-Diabatic Frequency Modulation

The frequency modulation ($\lambda=\omega$), often referred to as chirping, is
commonly considered for state-manipulation in experimental setups [75, 38, 76]
as it can lead to drastic changes in the system’s response. However, we stress
that the Floquet adiabaticity of chirps requires extra caution as it leads to
a change in the driving period: $T=T(t)$. This becomes evident when
considering the micromotion contribution to the adiabatic gauge potential,
$\mathcal{A}_{\mathcal{P}}=-i\mathcal{P}_{\\!\omega}\partial_{\omega}\mathcal{P}_{\\!\omega}^{\dagger}$.
In general, the micromotion operator $\mathcal{P}_{\\!\omega}(t)$ need not
reduce to the identity $\mathcal{P}_{\\!\omega}(t)\not\equiv\mathds{1}$ at any
time $t\neq nT$($n\in\mathbb{Z}$), and we can write

$\mathcal{P}_{\\!\omega}(t)=\sum_{\ell}\mathcal{P}_{\ell,\omega}e^{i\ell\omega
t}\,.$

Hence, $\norm{\mathcal{A}_{\mathcal{P}}}\sim t$ as $t{\to}\infty$, since
$\partial_{\omega}e^{i\ell\omega t}=t\times i\ell e^{i\ell\omega t}$. This
leads to a non-zero contribution
$\dot{\lambda}\norm{\mathcal{A}_{\mathcal{P}}}\propto\dot{\lambda}t{\not\to}0$
in the FAGP even in the adiabatic limit, where $\dot{\lambda}{\to}0$ and
$T_{\mathrm{ramp}}{\to}\infty$. Therefore, even in the adiabatic limit, the
evolution does not follow the instantaneous eigenstates of the Floquet
Hamiltonian (3). This suggests, that the infinite slow evolution follows the
eigenstates of a different effective Hamiltonian. In fact, one can convince
oneself that including the explicitly time-dependent term –
$\mathcal{P}^{\dagger}_{\\!\omega}\pqty{\sum_{\ell}\mathcal{P}_{\ell,\omega}\partial_{\omega}e^{i\ell\omega
t}}{\subset}\mathcal{P}^{\dagger}_{\\!\omega}\partial_{\omega}\mathcal{P}_{\\!\omega}$
– in the Floquet Hamiltonian rather than the FAGP, adiabaticity is restored,
i.e., $\dot{\lambda}\norm{\mathcal{A}_{\mathcal{P}}}{\to}0$ as
$\dot{\lambda}{\to}0$ and $T_{\mathrm{ramp}}{\to}\infty$.

In practice, this can be achieved by replacing the time-modulated frequency
$\omega$ by the instantaneous frequency $\nu$:

$\omega(t)\to\nu(t)=\derivative{t}\pqty{\omega(t)t}=\omega(t)+t\dot{\omega}(t)\,$
(13)

in the Floquet frame Eq. (3).

The instantaneous frequency $\nu$ being the relevant frequency for the Floquet
problem instead of the time-modulated frequency $\omega$ can have unexpected
and counterintuitive consequences. To illustrate this, consider a linear ramp
$\omega(t)=\omega_{0}+(\omega_{1}-\omega_{0})t/T_{\text{ramp}}$: the adiabatic
protocol follows the instantaneous frequency $\nu$, i.e., the system will
evolve following the instantaneous Floquet eigenstate corresponding to
$\nu(t)=\omega_{0}+2\times(\omega_{1}-\omega_{0})t/T_{\text{ramp}}$, cf. Eq.
(13). This comes in stark contrast to the naïve expectation that the system
will trace the Floquet eigenstates corresponding to
$\omega(t)=\omega_{0}+(\omega_{1}-\omega_{0})t/T_{\text{ramp}}$ [note the
missing factor of $2$ compared to $\nu(t)$], Sec. IV.2 for more details.

As a consequence, the variational principle (10) also needs to be adjusted
accordingly (see App. D.1)

$\mathcal{G}_{\\!\omega}\pqty{\mathcal{X}_{\omega}}=i\commutator{\mathcal{H}_{\omega}(t)}{\mathcal{X}_{\omega}(t)}+\partial_{t}\mathcal{X}_{\omega}(t)-\frac{\dot{\nu}}{\nu}\mathcal{H}_{\omega}(t)-\dot{\omega}\partial_{\omega}\mathcal{H}_{\omega}(t)$
(14)

where the partial derivative $\partial_{\omega}$ acts only on explicitly
$\omega$-dependent terms and does not act on the Fourier harmonics
$e^{i\ell\omega t}$ 333The fact that $\partial_{\omega}$ in Eq. (14) does not
act on $e^{i\ell\omega t}$ motivates us to introduce the phase variable
$\phi=\omega t$. Then, $e^{i\ell\omega t}\to e^{i\ell\phi}$ is no longer
explicitly $\omega$-dependent and $\nu=\dot{\phi}$ appears naturally as the
correct frequency. We refer the interested reader to App. B.2 for all
details.. The l-FCD Hamiltonian for chirps then reads as

$\mathcal{H}_{\mathrm{l{-}FCD}}=\mathcal{H}+\mathcal{X}_{\omega}\,,$ (15)

where any $\dot{\omega}$ and $\dot{\nu}$ dependence is already implicitly
included in $\mathcal{X}_{\omega}$.

##  IV Applications

To demonstrate the usefulness of Floquet CD driving, we now apply the
techniques we developed in Sec. II to three examples of increasing complexity.
The control setups we consider include resonant drives and frequency chirps,
chosen to emphasize the out-of-equilibrium character of the controlled system.
Their lack of static counterparts brings about genuinely nonequilibrium
effects absent in static control problems.

We first discuss a periodically-driven two-level system exposed to an
additional external control field. While the IFE approach gives a good
approximation to the FAGP in the off-resonant high-frequency regime, it breaks
down for resonant control drives. The variational approach, on the other hand,
is capable of finding accurate Floquet gauge potentials even on resonance. We
then move on to apply the variational technique to prepare the state of a
Floquet-engineered fermionic band model by speeding up a frequency chirp. We
place a particular emphasis on experimental constraints and discuss how these
can be incorporated to compute a sufficiently local approximation to the exact
FAGP. Finally, we show how Floquet control can be deployed to open up
nonequilibrium control channels in a static non-integrable many-body spin
chain, thus enhancing equilibrium control techniques. In particular, we find
that l-FCD driving can aid the preparation of many-body ground states
belonging to different quantum phases of matter even when all accessible
equilibrium control channels require crossing the critical point.

In all examples, we quantify the advantage brought in by l-FCD driving by
computing the fidelity. We use either the instantaneous fidelity

$F(t)=\absolutevalue{\innerproduct{\psi_{0,\lambda(t)}}{\psi(t)}}^{2}$ (16)

during the protocol between the time-evolved state $\ket{\psi(t)}$ and the
instantaneous Floquet eigenstate $\ket{\psi_{0,\lambda(t)}}$ [cf. App. A], or
the final fidelity at the end of the ramp, $F(T_{\text{ramp}})$. In general,
the closer the fidelity to unity, the better the FAGP approximation.

###  IV.1 Linearly polarized Two-Level System

Figure 3:  FCD for linearly driven two-level system (17) in high-frequency (A)
and one-photon resonance (B) regime. The level-splitting $\lambda$ is varied
linearly within $\lambda\in\bqty{-5g,\,5g}$ and
$\lambda\in\bqty{0.9\omega,\,1.1\omega}$, respectively. $(i)$ Numerically
exact instantaneous Floquet energies $E_{F}$ as a function of time $t$. $(ii)$
Instantaneous fidelity $F(t)$ for unassisted (gray), IFE (red circles) and
variational protocol (blue boxes). Inset: Deviation of fidelity from unity,
note the logarithmic scale. $(iii)$ $x$ (blue, solid) and $y$ (orange, dashed)
component of the variational gauge potential $\mathcal{X}$, see Eqs. (20) and
(22), (the $z$ component always vanishes). In high-frequency regime (A) the
IFE method successfully suppresses diabatic transitions but fails in the
presence of photon resonances (B). The variational method suppresses diabatic
transitions in all scenarios. We use a linear ramp
$\lambda(t)=\lambda_{\mathrm{i}}+(\lambda_{\mathrm{f}}-\lambda_{\mathrm{i}})t/T_{\mathrm{ramp}}$
for the control parameter
$\lambda\in\bqty{\lambda_{\mathrm{i}},\lambda_{\mathrm{f}}}$. Other parameters
are $\omega/g=100$, $A/g=2.5$, $T_{\mathrm{ramp}}g=0.5$.

A hallmark feature of Floquet systems is the existence of photon resonances in
the quasi-energy spectrum. To build intuition about the methods introduced in
Sec. II, we investigate a pedagogical example of a linearly driven two-level
system 444cf. App. C for the exactly solvable circularly driven 2LS. We are
primarily interested in comparing the behavior of the IFE and variational
approaches to the FAGP in the regime of photon resonances where the IFE method
is expected to break down.

Consider the Hamiltonian of the linearly polarized two-level system

$\mathcal{H}_{\lambda}\pqty{t}=\lambda\,S^{z}+\bqty{g+A\cos(\omega
t)}\,S^{x}\,,$ (17)

with level-splitting $\lambda$, level-hybridization $g$, drive amplitude
$A$(${=}2.5\,g$) and drive frequency $\omega=2\pi/T$(${=}100\,g$). Despite its
simple Hilbert-space structure, there exists no known closed-form expression
for the Floquet Hamiltonian.

high-frequency Regime. Let us first focus on the high-frequency regime. In
particular, we consider a linear ramp of the level-splitting $\lambda$

$\lambda(t)=(\lambda_{\mathrm{f}}-\lambda_{\mathrm{i}})\frac{t}{T_{\text{ramp}}}+\lambda_{\mathrm{i}}$
(18)

from $\lambda_{\mathrm{i}}=-5g$ to $\lambda_{\mathrm{f}}=5g$ with an duration
$T_{\mathrm{ramp}}=1/(2g)$. Such control setups arise naturally when trying to
manipulate the behavior of Floquet-engineered systems using external controls,
e.g., to demonstrate the presence of a non-zero Berry curvature in a Floquet
topological band [14, 79], or to probe the metastability of dynamically
stabilized matter [80]. Like in these situations, we assume that the system is
initiated in a Floquet eigenstate before the ramp $\lambda(t)$ is turned on.

Note that, due to the invariance of the quasi-energies with respect to a shift
by $n\omega$ ($n\in\mathbb{Z}$) the Floquet Hamiltonian does not have a well-
defined ground state. Therefore, we use as an initial state the Floquet
eigenstate which has a larger overlap with the ground state of the non-driven
Hamiltonian $\lambda_{\mathrm{i}}\,S^{z}$.

Since the frequency is large compared to all other parameters, $\omega\ll
g,A,\lambda$, the system is well described by the lowest order Floquet
Hamiltonian (see App. F.1)

$\mathcal{H}_{F}^{(0)}=\lambda S^{z}+gS^{x},\qquad\mathcal{K}_{F}^{(0)}=0\,,$
(19)

The Floquet energies exhibit an avoided crossing around $\lambda=0$, see Fig
3A(i). In fact, in the absence of any counterterms (referred henceforth as
unassisted control, cf. Fig. 1), the protocol leads to an almost complete loss
of fidelity, see grey line in Fig. 3A(ii).

Note that, the lowest-order Floquet Hamiltonian (19) describes a standard
Landau-Zener crossing [81, 82, 83, 84] as a function of $\lambda$. Hence, the
corresponding approximate FAGP for (19) takes the well-known form [58]

$\mathcal{A}_{\lambda}^{(0)}(t)=-\frac{g}{\lambda^{2}+g^{2}}S^{y}\,,$ (20)

where, in addition, we also used that to lowest order
$\mathcal{A}_{\lambda}^{(0)}(t)=\mathcal{A}_{F,\lambda}^{(0)}$, following Eq.
(9). In contrast to the unassisted protocol, the IFE FCD protocol
significantly suppresses diabatic excitations leading to high fidelity
throughout the entire protocol duration, see red circles in Fig 3A(ii).

For comparison, we can also consider a variational FAGP. To this end, we use
the ansatz

$\mathcal{X}(t)=\bqty{y_{0}+y_{1}\cos(\omega t)}S^{y}+x_{1}\sin(\omega
t)S^{x}+z_{1}\sin(\omega t)S^{z}\,,$ (21)

with variational parameters $\Bqty{y_{0},x_{1},y_{1},z_{1}}$. Since real
symmetric Hamiltonians are diagonalized by orthogonal transformations, the
generating AGP can be chosen imaginary-valued [58]. To generalize this
observation to Floquet systems for the ansatz above, note that $\cos(\omega
t)\propto e^{i\omega t}+e^{-i\omega t}$ can be considered a real-valued, and
$\sin(\omega t)\propto i(e^{i\omega t}-e^{-i\omega t})$ an imaginary-valued
function [see App. B.2 for the details]. Similar to the IFE-FAGP, the
variational FAGP fully counteracts the diabatic transitions, see blue squares
in Fig 3A(ii).

To reconcile the IFE and variational approaches, we apply a high-frequency
expansion within the variational principle (12). This allows us to
analytically derive the form of the variational FAGP shown in Fig 3A(iii). As
expected, the latter coincides with Eq. (20) up to $O(\omega^{-1})$
corrections (see App. F.1).

Photon Resonance Regime. Let us now turn to the one-photon resonance regime at
$\lambda\approx\omega$. In particular, we consider the linear ramp from Eq.
(18) with $\lambda_{\mathrm{i}}=0.9\omega$ and
$\lambda_{\mathrm{f}}=1.1\omega$.

In this regime, the energy levels appear far separated [Fig. 3B(i)], such that
identifying an avoided level crossing is not immediately obvious. However, the
energy separation is almost resonant with the external drive
$E_{F,1}-E_{F,2}\approx\omega$, leading to a hybridization of the states and
an avoided crossing at around $\lambda\approx\omega$, see Fig. 3B(i).

Since the ramp duration is small compared to the inverse photon absorption
gap, the Landau-Zener condition for adiabatic passage is not satisfied, and
the unassisted drive creates excitations. However, this Floquet resonance is
not captured by the high-frequency expansion 555Resonant drives can also be
dealt with directly in some cases, using an additional change-of-frame
transformation [73, 2]., and, in addition, the IFE CD protocol fails as well,
see Fig. 3B(ii). The variational approach, on the other hand, fully captures
this non-perturbative resonance and allows for transitionless driving through
the avoided crossing, see Fig. 3B(ii). In fact, using the ansatz (21) and
taking a large frequency limit in the variational minimization
($\omega{\approx}\lambda{\gg}A,g$), we find (see App. F.1)

$\mathcal{X}_{\lambda}(t)=\frac{2A}{A^{2}+4(\lambda-\omega)^{2}}\bqty{\cos(\omega
t)S^{y}-\sin(\omega t)S^{x}}+O(\omega^{-1})\,,$ (22)

The structure of the variational FAGP is shown in Fig. 3B(iii).

This simple model already illustrates the potential advantages of the
variational over the IFE method. Indeed, using a suitable ansatz, the
variational method allows us to capture both perturbative and non-perturbative
effects in the quasi-energy spectrum and the Floquet eigenstates. We expect
this technique to be particularly useful in atomic physics where high-
precision few-level control can be sped up using FCD drives.

For the purposes of Floquet engineering governed by the approximate Floquet
Hamiltonian, however, following the adiabatically connected quasi-energy
manifold in the presence of photon-resonances can sometimes be undesirable,
[see discussion of Fig. 2]. In such cases, one can use the IFE protocol and a
short $T_{\mathrm{ramp}}$ to traverse standard quasi-energy gaps
adiabatically, while passing through photon-resonances diabatically.
Therefore, depending on the control problem at hand, either the IFE or the
variational approach is preferable.

###  IV.2 Fermionic Band Model

Figure 4:  Fermionic Floquet band model: (A) Schematic of setup with $s$
(blue) and $p$ (red) band energies $\epsilon_{s,p}$, intra-band hoppings
$J^{1,2,3}_{s,p}$ and drive-induced inter-band hoppings $\eta^{0,1}_{sp}$. (B)
In this work we replace the linear drive (left panel) by a cubic drive (right
panel), see Eq. (25), to mitigate the error when considering the time-
modulated frequency $\omega(t)$ (gray dashed) instead of the instantaneous
frequency $\nu(t)$ (black solid). (C) Quasi-energy bands at the beginning
$(i)$ and end $(ii)$ of the frequency ramp as a function of quasi-momentum,
color indicates overlap with $s$ (blue) and $p$ (red) band of the non-driven
model. During the state preparation the two bands become $s$ and $p$ band
character.

The experimental realization of topological band models [18] represents a
milestone achievement of ultracold atom quantum simulators. Due to atoms being
neurally charged, implementing topological Hamiltonians in ultracold atom
platforms, such as the Harper-Hofstadter [12, 79], Haldane [14], or Shackley
[75] model, would be unthinkable without the help of Floquet engineering.
However, studying a topologically non-trivial state in an experiment requires
a high-fidelity state preparation procedure. Therefore, (l-)FCD can play an
important role in enhancing those state preparation schemes at a moderate-to-
low cost.

Our methods are straightforward to apply to generic Floquet-engineered
(topological) band systems. We illustrate this using the setup from a recent
experiment demonstrating Floquet topological pumping in a system of one-
dimensional non-interacting fermions [75]. The system is effectively described
by the free fermion Hamiltonian in momentum space

$\mathcal{H}_{\lambda}(t)=\sum_{q=1}^{N_{f}}\boldsymbol{\Psi}_{q}^{\dagger}\cdot\boldsymbol{h}_{\lambda}(q,t)\cdot\boldsymbol{\Psi}_{q},$
(23)

where $N_{f}$ is the number of fermions and
$\boldsymbol{\Psi}_{q}^{\dagger}=(c_{p,q}^{\dagger},\,c_{s,q}^{\dagger})$ with
$c_{xq}^{\dagger}$ is the fermionic creation operator at quasi-momentum $q$ in
the $x{=}s,\,p$ Bloch band. The Bloch Hamiltonian is given by

$\displaystyle\boldsymbol{h}_{\lambda}(q,t)=$
$\displaystyle\pqty{\epsilon_{+}+\sum_{j=1}^{3}J_{+}^{j}\cos(jqa-
jA)}\;\mathds{1}$ (24) $\displaystyle-
aF(t)\pqty{\eta^{0}_{sp}+2\eta_{sp}^{1}\cos(qa-A)}\;\sigma^{x}$
$\displaystyle+\pqty{\epsilon_{-}-2\sum_{j=1}^{3}J_{-}^{j}\cos(jqa-
jA)}\;\sigma^{z}\,,$

with the external time-periodic driving force $F(t){=}\omega K_{1}\cos(\omega
t){+}2\omega K_{2}\cos(2\omega t+\varphi)$, and the Peierls phase
$A(t){=}K_{1}\sin(\omega t){+}2K_{2}\sin(2\omega t+\varphi)$; lattice spacing
constant is $a$; the mean and difference between the $s$ an $p$-band in
energies is $2\epsilon_{\pm}=\epsilon_{p}\pm\epsilon_{s}$; the intra-band
hopping is $2J_{\pm}^{j}=J_{p}^{j}\pm J_{s}^{j}$, and the inter-band hoppings
are denoted by $\eta_{sp}^{0,1}$ 666For the specific numerical values of the
different quantities see Tab. 1 in App. F.2., see sketch in Fig. 4A.

Depending on the values of the control parameters
$(K_{1},K_{2},\omega,\varphi)$ the Floquet-Bloch bands of Eq. (23) can be
either topological or trivial [75]. The two-tone drive breaks the symmetry
protecting the topological phase, thus opening up an adiabatic path between
the trivial and topological phases. In the experiment, the initial state is
the completely filled $s$-band of the static system $K_{1}{=}K_{2}{=}0$ and
belongs to the trivial phase. During the state preparation protocol, first the
amplitude $K_{2}$, then the frequency $\omega$, and eventually the second
amplitude $K_{1}$ are linearly ramped to prepare the initial state of the
Floquet pump. We note in passing that, only during the topological Floquet
pump is the symmetry restored and the state becomes topologically non-trivial.

The theory of FCD driving is applicable to all steps of the state preparation
protocol, and can even speed up the subsequent Floquet pump itself. However,
the overall bottleneck for the entire state preparation sequence is set by the
smallest gap encountered during the frequency ramp of the three-step state
preparation protocol. Moreover, for frequency ramps, the size of the Floquet
zone varies with time which may lead to the appearance of photon absorption
resonances during the drive. Unlike amplitude (i.e., coupling) ramps,
frequency chirps have no equilibrium analogue. Therefore, from a theoretical
standpoint, they represent the most challenging and interesting of the three
protocol steps. For these reasons, we set $\lambda(t)=\omega(t)$ and focus
exclusively on the frequency chirp.

In the experiment, a linear frequency ramp was considered [75]. However, since
this linear frequency chirp overshoots the instantaneous frequency
$\nu_{\mathrm{f}}\neq\omega_{\mathrm{f}}$, see Fig. 4B, and hence leads to a
potentially wrong final state in the adiabatic limit [see Sec. III], we
consider instead a cubic chirp. The cubic ramp with initial value
$\omega(t_{\mathrm{i}})=\omega_{\mathrm{i}}$ and final value
$\omega(t_{\mathrm{f}})=\omega_{\mathrm{f}}$ is defined as

$\omega(t)=\omega_{\mathrm{i}}+\pqty{\omega_{\mathrm{f}}-\omega_{\mathrm{i}}}\pqty{\frac{t-t_{\mathrm{i}}}{t_{\mathrm{f}}-t_{\mathrm{i}}}}^{3}\,.$
(25)

Since $\omega(t_{\mathrm{f}})=\nu(t_{\mathrm{f}})$, using the cubic chirp
guarantees that the final state corresponds to an eigenstate of the Floquet
Hamiltonian at the correct drive frequency, cf. Fig. 4B.

During the state preparation protocol the quasi-energy bands hybridize such
that both Floquet bands attain admixtures of the original $s$ and $p$ bands,
see Fig. 4. The hybridization of the two bands results in avoided level
crossings for the different quasi-momenta modes, with gaps $\Delta_{q}$.
Therefore, performing the unassisted state preparation protocol in a finite
duration $T_{\mathrm{ramp}}$ leads to a significant loss in fidelity $F(q,t)$,
for all quasi-momenta where the ramp duration is small compared to the inverse
gap, $T_{\mathrm{ramp}}\not{\gg}1/\Delta_{q}$, see Fig. 5A(i). Hence, in the
following, we consider FCD-assisted state preparations. As the Floquet
engineering effect in this example relies on photon resonances, we deploy the
variational approach.

Figure 5:  Fermionic Floquet band model FCD assisted state preparation: (A)
Momentum resolved instantaneous Infidelity $1{-}F(q,t)$ for unassisted $(i)$,
exact FCD $(ii)$ and l-FCD $(iii)$ for $N_{q}=2$ protocol for
$T_{\mathrm{ramp}}=1\,\mathrm{ms}$. (B) Amplitude $\norm{\mathcal{X}(q,t)}$ of
corresponding FAGP potential for FCD $(i)$ and l-FCD $(ii)$. (C) Final
infidelity $1-F$ as a function of $T_{\mathrm{ramp}}$ for unassisted (cyan
diamonds), and l-FCD protocols (purple squares) for $N_{q}=1,2,3$ (light to
dark). Inset shows zoom into the region around the $1\,\%$ infidelity
threshold (gray line) and ramp times needed for the different protocols to
pass the threshold. variational method suppresses all diabatic transitions but
requires experimentally infeasible highly non-local counter term.
Approximating counterterms with local terms still yields a boost in fidelity
compared to using no counter term. Parameters: We use the cubic chirp Eq. (25)
with $\omega_{\mathrm{i}}{=}2\pi{\times}5\,\mathrm{kHz}$ and
$\omega_{\mathrm{f}}{=}2\pi{\times}7.5\,\mathrm{kHz}$, $N_{f}{=}500$ fermions
and $N_{h}{=}32$ Fourier harmonics.

Note that in order for the system to evolve according to the dynamics of the
Hamiltonian (23) once the state preparation procedure is over, the FAGP must
vanish at the end of the protocol. This property is also guaranteed by using
the cubic ramp, since
$\dot{\nu}(t_{\mathrm{f}})=\dot{\omega}(t_{\mathrm{f}})=0$.

To compute the FAGP, recall that the Hamiltonian in momentum space reduces to
a collection of independent two-level systems, one for each quasimomentum $q$,
cf. Eq. (24). Therefore, we can leverage the techniques developed in the two-
level system example, Sec. IV.1, to compute the FAGP for each fixed momentum
$q$ individually. To be precise, we choose an ansatz for the adiabatic gauge
potential of the form

$\mathcal{X}_{q;\lambda}(t)=\sum_{\alpha=x,y,z}\sum_{\ell=0}^{N_{h}}\pqty{\chi^{\alpha,\ell}_{q;\lambda}e^{i\ell\omega
t}}\;\sigma^{\alpha}\,,$ (26)

with variational parameters $\chi^{\alpha,\ell}_{q;\lambda}$ which are smooth
functions of $(q,\,\lambda)$. The variational parameters are obtained
numerically using the action (14).The amplitude, $\norm{\mathcal{X}(q,t)}$, of
the resulting variational FAGP is shown in Fig 5B(i). Note that the ansatz
(26) cannot reproduce the Floquet gauge potential exactly, due to the finite
number $N_{h}$ of Fourier harmonics kept. Still, as seen in Fig. 5A(ii), the
associated FCD protocol leads to almost perfect transitionless driving.

The FAGP is localized in quasi-momentum space [see Fig 5B(i)], involving many
lattice harmonics, $e^{ijqa}$, $j=0,\dots,N_{f}-1$. Since the $j$’th lattice
harmonic $e^{ijqa}$ corresponds to a $j$’th-neighbor coupling in real space,
the exact Floquet adiabatic gauge potential is delocalized in real space.
Therefore, to implement the exact FCD protocol in an experiment, precise
control over all individual $j$’th neighbor couplings is required. At the
moment, to the best of our knowledge, engineering such long-range couplings is
out of reach for most cold atomic experimental platforms [although arbitrary-
range hopping is feasible using the techniques developed for synthetic
dimensions [87, 88, 89, 90].].

Fortunately, the versatile character of the variational FAGP approach allows
us to build any experimental restrictions directly into the ansatz, so long as
one is willing to tolerate a certain fidelity degradation. Frequent
experimental constraints reflect the accessibility of control channels or the
locality of available control terms. For example, depending on the
experimental platform, the imaginary-valued terms $\propto\sigma^{y}$ required
for the gauge potential, may be difficult to implement. However, rotating to a
moving frame we can replace any $\propto\sigma^{y}$ term using $\sigma^{x}$
and $\sigma^{z}$ terms [64]. Another interesting way to generate the gauge
potential terms is using Floquet engineering [91].

To alleviate the problem of locality in our system, we consider a local
approximation (l-FCD) to the exact FCD protocol restricting the control to a
limited number, $N_{q}$ ($j=0,\dots,N_{q}$), of lattice harmonics. This
approximation may be implemented by considering a real space ansatz for the
variational FAGP and then solving the variational principle in Eq. (10) for
the many-particle Hamiltonian (23) instead of considering each momentum
individually. However, since we already have access to a nonlocal gauge
potential that does the job, we can simply truncate the exact result to the
given number of lattice harmonics $N_{q}$.

As anticipated, while the l-FCD counterterms take a simpler form in quasi-
momentum space they cannot suppress all excitations leading to finite
infidelity, see Fig. 5B(ii) and Fig. 5A(iii). Nevertheless, compared to the
unassisted protocol, the local FCD still leads to a notable increase in
fidelity, see Fig. 5C. In particular, considering a $99\,\%$ fidelity
threshold the l-FCD protocols lead to a $5$-$20\,\%$ reduction in ramp-time
depending on the amount of added lattice harmonics $N_{q}=1$-$3$,
respectively. These numbers are, however, strongly model-dependent and should
not be considered as a generic reference point for the speedup offered by FCD
driving, as we will see in Sec. IV.3.

In general, Floquet CD driving, like the conventional static CD driving, may
require the implementation of experimentally infeasible long-range
counterterms. However, as demonstrated in the experimentally relevant example
above, local approximations can still lead to a notable and experimentally
desirable increase in fidelity compared to unassisted protocols with only
moderate control requirements.

###  IV.3 Periodically Driven Many-Body System

Figure 6:  Local Floquet counterdiabatic driving for circularly driven spin
chain. (A) Sketch of circularly driven spin chain, see Eq. (28). (B) Rotating
frame fields in $\tilde{h}_{x}$-$\tilde{h}_{z}$ plane. The non-driven protocol
(gray dashed) ramps through the quantum critical point (QCP, red) at
$(\tilde{h}_{x},\tilde{h}_{z})=(J,0)$. The driven protocol (blue solid) yields
an effective non-zero $\tilde{h}_{z}$ component allowing to circumvent the
QCP. (C) Time-dependence of state-preparation schemes. (i) Variational
parameter $\chi(t)$ (30) computed using the algorithm detailed in App. E. (ii)
Instantaneous Fidelities $F\pqty{t}$ for non-driven (gray circles), driven
(blue squares) and l-FCD assisted driven protocols (turquoises triangles). (D)
Final fidelity for protocols from C for different ramp times
$T_{\mathrm{ramp}}$ (i) and different system sizes $L$ (ii). The Floquet
engineered finite $h_{z}$ component leads to a notable increase in fidelity
and enhanced scaling with ramp time and system size. Around
$JT_{\mathrm{ramp}}=5$ the l-FCD protocol enables state-preparation close to
unit fidelity almost independent of system size. Parameters: We use
$A(t)/J=\bqty{10-9.5\times\lambda\pqty{t}}$ and
$\omega(t)=\bqty{\omega_{\mathrm{max}}\times\sin(\pi\lambda\pqty{t})}$ where
$\omega_{\mathrm{max}}=0.2\,J$ and $\lambda\pqty{t}$ is a cubic ramp, see Eq.
(25), from $\lambda_{\mathrm{i}}=0$ to $\lambda_{\mathrm{f}}=1$. Further,
$L=24$ and $JT_{\mathrm{ramp}}=5$, if not stated otherwise. We add a small
symmetry breaking field $h_{z}/J{=}10^{-3}$ to single out one of the
degenerate states in the FM phase.

An important problem in quantum many-body control is the preparation of ground
states of an ordered phase starting from a ground state in the disordered
phase. This is a notoriously hard task since by definition there exists no
adiabatic path in the thermodynamic limit connecting the two phases [92, 93,
94, 95] as long as the protecting symmetry is left intact. Therefore, in
practice one either makes use of the finite size gap which vanishes in the
thermodynamic limit and thus leads to divergent unassisted protocol durations,
or one needs to introduce additional control channels to break the protecting
symmetry explicitly.

However, in some cases such symmetry breaking controls cannot be directly
implemented, for example if the system is intrinsically symmetric. As a way
out, nonequilibrium periodic drives have been proposed as a useful tool to
engineer the implementation of additional terms assisting the state
preparation protocol [96, 97, 98, 99, 91].

Here, we show that FCD driving belongs to this list, providing a proof-of-
principle demonstration of how periodic driving assisted by l-FCD can be used
to improve state preparation protocols for static many-body systems. We
consider as a toy model the transverse field Ising (TFI) chain of $L$ spins,

$\mathcal{H}_{\mathrm{TFI}}=\sum_{\ell=1}^{L}\bqty{J\sigma_{\ell}^{z}\sigma_{\ell+1}^{z}+h_{x}\sigma_{\ell}^{x}+h_{y}\sigma_{\ell}^{y}}\,,$
(27)

with ferromagnetic interactions $J{<}0$, and two transverse fields $h_{x}$ and
$h_{y}$; we use periodic boundary conditions. Independent of the ratio
$h_{x}/h_{y}$, the system undergoes a quantum phase transition at
$h{\equiv}\sqrt{h_{x}^{2}+h_{y}^{2}}{=}J$ from a disordered paramagnetic (PM)
phase ($h{>}|J|$) to an ordered ferromagnetic (FM) phase ($h{<}|J|$), where
the energy gap closes. Therefore, in the static system and in the
thermodynamic limit, there exists no adiabatic path connecting the two phases.

In order to explicitly break the underlying $\mathbb{Z}_{2}$ symmetry, access,
e.g., to a parallel $z$-component field
($\propto\sum_{\ell}\sigma^{z}_{\ell}$) is needed. Notice that the
instantaneous Hamiltonian preserves the $\mathbb{Z}_{2}$ symmetry exactly for
all parameter values. Hence, the PM and FM phases are strictly disconnected in
equilibrium. In the following, we make the plausible assumption that we can
only access terms already present in the Hamiltonian (27) for control
purposes. We will now demonstrate how one can use nonequilibrium periodic
driving to engineer an adiabatic path connecting the two phases using only the
$h_{x}$ and $h_{y}$ field components.

Consider the circularly driven TFI model

$\mathcal{H}(t;A,\omega)=\sum_{j=1}^{L}\bqty{J\sigma_{j}^{z}\sigma_{j+1}^{z}+A\pqty{\cos(\omega
t)\sigma_{j}^{x}+\sin(\omega t)\sigma_{j}^{y}}}\,,$ (28)

where we periodically modulate the $h_{x}$ and $h_{y}$ components with
amplitude $A{=}A(t)$ and frequency $\omega{=}\omega(t)$, see also sketch in
Fig. 6A. We stress that the time dependence explicitly breaks the
integrability of the system, and brings out its intrinsically many-body
character. Indeed, using the reference frame transformation
${V(t)=\exp(-i\omega(t)t\sum_{n}\sigma_{n}^{z}/2)}$, Eq. (28) is mapped to the
non-integrable mixed field Ising model (MFI)

$\mathcal{H}_{\mathrm{MFI}}(t;A,\omega)=\sum_{j=1}^{L}\bqty{J\sigma_{j}^{z}\sigma_{j+1}^{z}+\tilde{h}_{x}\sigma_{j}^{x}+\tilde{h}_{z}\sigma_{j}^{z}}\,,$
(29)

with transverse field $\tilde{h}_{x}(t){=}A(t)$ and longitudinal field
$\tilde{h}_{z}(t){=}-\nu(t)/2$, and instantaneous drive frequency
$\nu(t)=\partial_{t}(\omega(t)t)$ (see Eq. (13). We deliberately choose the
circular drive in order to avoid potential heating processes present
ubiquitously in generic periodically driven many-body systems.

The key insight behind applying the circular drive is that it leads to an
effective $z$-field which opens up a new parameter space direction to
circumvent the quantum critical point, see Fig. 6B. However, we stress that
having access to a $z$-field control is only necessary but not sufficient for
high-fidelity many-body state manipulation. Indeed, a naïve application of the
circular protocol does not directly lead to a significant increase in fidelity
and enhanced scaling in the ramp time $T_{\mathrm{ramp}}$ and system size $L$,
compared to the $\omega(t){\equiv}0$ protocol, see Fig. 6C and D,
respectively.

In order to improve the performance of the protocol, we now apply FCD driving.
Notice that the exact AGP in a generic many-body system requires the
implementation of highly non-local both long-range and multi-body interactions
including Pauli strings $\sigma_{j_{1}}\sigma_{j_{2}}\dots\sigma_{j_{s}}$ of
arbitrary length $s=1,\dots,L$ [58, 64]. The multi-body terms appear as a
direct consequence of interactions and are also featured in the FAGP. As a
result, obtaining the exact FAGP is computationally expensive. In addition,
implementing such highly non-local multi-body terms is experimentally
infeasible 777In fact, the non-local terms become more dominant in the
proximity of the QCP. Thus, local CD is expected to fail when applied to the
static system which inevitably comes close to the QCP..

To comply with these constraints, we exploit again the variational character
of l-FCD driving: we focus on CD protocols that require only local single-
particle counter-drives built from the accessible $h_{x}$ and $h_{y}$ fields
already present in Eq. (27). In particular, we choose the variational ansatz

$\mathcal{X}(t)=\chi(A,\omega)\sum_{j=1}^{L}\bqty{\cos(\omega
t)\sigma_{j}^{y}-\sin(\omega t)\sigma_{j}^{x}}\,,$ (30)

with a single variational parameter $\chi$, see Fig. 6C(i). The ansatz (30)
gives rise to an $y$-field $\tilde{h}_{y}{=}\chi$ in the rotating frame CD
Hamiltonian (29).

Since we are only interested in transitionless driving of the groundstate, we
can further replace the trace $\Tr(\cdot)$ in the variational principle (10)
by the expectation value $\expectationvalue{\cdot}_{\psi_{0}(t)}$ [58] with
respect to the instantaneous Floquet groundstate $\ket{\psi_{0}(t)}$. In
general, the notion of Floquet ground states is ill-defined as quasi-energies
$E_{F}$ are only defined up to integer multiples of the driving frequency
$E_{F}\hat{=}E_{F}+n\omega$, $n\in\mathbb{Z}$. Usually, in such cases
$\ket{\psi_{0}(t)}$ may represent the manifold of adiabatically connected
target Floquet eigenstates. Here, however, we can use the exact mapping to the
MFI model to define the instantaneous Floquet ground states using the
instantaneous ground state of the static MFI model (29). We emphasize that
using the groundstate expectation value instead of the trace leads to an
enhanced state preparation protocol, see App. F.3, but is computationally more
demanding as it requires the computation of the instantaneous Floquet
groundstate at each step of the protocol.

Remarkably, using the numerically computed l-FCD assisted protocol (see Fig.
6C(i)) leads to an increase of fidelity by almost one order of magnitude
compared to the unassisted driven protocol, see Fig. 6C. Notably, for
protocols as short as $JT_{\mathrm{ramp}}{=}5$, the l-FCD assisted protocol
leads to almost unit fidelity for a wide range of system sizes $L{=}8{-}24$,
see Fig. 6D(ii). Let us comment, that already few Floquet cycles are
sufficient to yield a nonequilibrium-based enhancement. This emphasizes the
advantage of following the Floquet states even in the absence of periodic
drives in the original control setup.

In conclusion, we demonstrated that combining l-FCD driving with ideas from
Floquet theory enables the engineering of more efficient state preparation
schemes without any additional control requirements. More generally, using
nonequilibrium drives allows us to open up new effective control directions
via suitably engineered change-of-reference transformations. In particular, we
have shown that l-FCD is also applicable in nonintegrable quantum many-body
systems.

##  V Discussion & Outlook

###  V.1 Summary

Periodic driving has emerged as a powerful tool to engineer complex
Hamiltonians in quantum simulation. However, to complete this set of ideas
into a self-contained toolbox, requires us to develop a technique for high-
precision control of the engineered Hamiltonians that allows, among others,
for high-fidelity state manipulation. The current state of the art in Floquet
control is adiabatic manipulation of parameters. Still, adiabatic control
requires long protocols, which appears at odds with the limited coherence
times in experiments that benefit from short preparation protocols. To achieve
fast control of nonequilibrium quantum matter we consider a conceptually
different approach that aims at suppressing diabatic transitions by adding
counterterms to the lab-frame Hamiltonian.

We introduced two complementary approaches to (local) Floquet counterdiabatic
driving. They enable the computation of controlled approximations to the exact
Floquet adiabatic gauge potential used to generate optimal transitionless
driving protocols. Both methods rely only on the structure of the Hamiltonian,
i.e., do not require access to eigenvalues or eigenvectors. The first, based
on an inverse frequency expansion, is tailored to Floquet engineering
applications in the high-frequency regime. However, as it inherits issues from
the inverse frequency expansion, this method breaks down in the presence of
photon resonances. Therefore, we developed a second, non-perturbative,
variational procedure based on a least-action principle in the lab-frame which
overcomes the problems of the perturbative high-frequency approach.

Using the example of a linearly driven two-level system, we demonstrated that
the variational method works in the genuinely nonequilibrium regime of
resonant drives. Moreover, by choosing which of the inverse frequency
expansion or variational methods to use, we can engineer whether the system
should pass through photon resonances diabatically or adiabatically. The
control framework we developed in this example can be directly extended to
yield fast and efficient control and population transfer between isolated
states in few-level systems, such as atoms and molecules, exposed to a
periodic drive.

In a second application, we sped up a frequency-modulation drive to prepare
the ground state of a Floquet fermionic band insulator. Such frequency chirps
represent an example of nonequilibrium protocols without equilibrium control
counterparts. We used this example to illustrate the versatility of our method
to obtain short high-fidelity protocols that respect experimental limitations.
In particular, we showed how to avoid the implementation of experimentally
inaccessible non-local terms present in the exact Floquet adiabatic gauge
potential.

Finally, we applied local Floquet counterdiabatic driving to transfer the
population from a disordered to an ordered ground state in a non-integrable
Ising spin chain. This example illustrates a conceptually new approach to
enhancing state preparation in static systems. Using periodic drives we can
construct a nonequilibrium adiabatic path that connects two states of matter,
even when these states are not connected by adiabatic paths in the original
non-driven Hamiltonian. This can be useful, e.g., in quantum annealing where
phase transitions in the parameter landscape pose significant limitations to
the performance of solving optimization problems [101, 102]. In summary,
Floquet counterdiabatic driving can significantly enhance state preparation
without additional hardware requirements.

###  V.2 Outlook

Besides having immediate applications in quantum simulation and quantum
computing, our work opens up a couple of new exciting directions. First, there
are certain straightforward extensions of our nonequilibrium control methods
to more complex systems. In particular, the algebraic structure, underlying
the exactly solvable Floquet toy model in App. C, can be generalized to
discover new classes of analytically solvable Floquet models. For realistic
models, where this dynamical integrability is broken, expressive matrix-
product operator ansätze may be useful to approximate the adiabatic gauge
potential beyond system sizes amenable to exact diagonalization.

Second, our work paves the way for a more general control theory for
nonequilibrium systems. Specifically, the variational procedure can be used as
a starting point to generalize more sophisticated shortcuts to adiabaticity
techniques [103, 59] to periodically driven systems, such as counterdiabatic-
optimized local driving [104] or a second periodic drive may even be
implemented to help engineer the Floquet adiabatic gauge potential [91]. In
fact, our theory also applies to classical periodically driven systems [58,
105], and can be extended to other nonequilibrium systems described by local
effective Hamiltonians, such as random multipolar [106] and quasi-periodic
driving [107].

In addition, we expect our work to provide new insights into the study of
thermalization in nonequilibrium systems. On one hand, the adiabatic gauge
potential can be used to detect chaos and integrability breaking [108, 109].
On the other hand, an important open problem is the manipulation of prethermal
states of matter where additional control fields may interfere with and
compromise the stability of the prethermal plateau.

Finally, let us re-emphasize that the non-perturbative variational principle
enables the study of Floquet systems beyond the restrictions inherent to high-
frequency expansions. For example, the resulting non-perturbative variational
adiabatic gauge potential allows for computing linear response functions [58]
and detecting photon resonances, without needing to compute the exact non-
local Floquet Hamiltonian (see also App. F.2.2). Furthermore, the relation
between the gauge potential and the quantum geometric tensor opens up new ways
to investigate quantum geometry [58] away from equilibrium. In addition, as
the variational gauge potential contains information about the entire non-
perturbative quasi-energy spectrum, the Floquet variational principle we
developed could even provide a way to obtain a variational Floquet
Hamiltonian.

## Acknowledgements

We thank P. W. Claeys, M. Kolodrubetz, and A. Polkovnikov for insightful
discussions. Funded by the European Union (ERC, QuSimCtrl, 101113633). Views
and opinions expressed are however those of the authors only and do not
necessarily reflect those of the European Union or the European Research
Council Executive Agency. Neither the European Union nor the granting
authority can be held responsible for them. The simulations make use of the
QuSpin Python package [110, 111]. Simulations were performed on the MPI PKS
HPC cluster.

##  Appendix A  Instantaneous Floquet eigenstates

Notice that, to evaluate the instantaneous fidelity (16),
$F(t)=\absolutevalue{\innerproduct{\psi_{0,\lambda(t)}}{\psi(t)}}^{2}$, we
have to compute the ‘instantaneous’ Floquet eigenstate
$\ket{\psi_{0,\lambda(t)}}$. However, the notion of an instantaneous Floquet
eigenstate is ill-defined. Instead, there exists an entire family of Floquet
eigenstates $\Bqty{\ket{\psi_{0,\lambda}(\phi)}}_{\phi\in\bqty{0,\,2\pi}}$
connected by the micromotion operator $\mathcal{P}_{\lambda}(\phi)$.

Therefore, the question arises which state, or which phase $\phi$, yields the
correct representative. Considering a trivial protocol, where all parameters
are fixed, it becomes evident from the Floquet theorem (3) that
$\phi(t)=\omega t+\phi_{0}$ is the correct choice for the phase. In fact, the
same relation holds true for frequency modulations as well, which can be
motivated by considering the integrated phase
$\phi(t)-\phi_{0}=\int_{0}^{t}\\!\nu(s)\mathrm{d}s=\omega(t)t$ with respect to
the instantaneous frequency $\nu(t)$.

To compute the instantaneous Floquet eigenstates for a phase $\phi=\omega t$,
we first evaluate the Floquet Hamiltonian $\mathcal{H}_{F}[t]$ via

$e^{-i\mathcal{H}_{F}[t]T}=\mathcal{T}e^{-i\int_{t}^{t+T}\mathcal{H}_{\lambda(t)}\\!\pqty{s}\mathrm{d}s}\,,$
(31)

and then diagonalizing it.

##  Appendix B  Additional Theoretical Results

###  B.1 Explicit Exppressions for Higher Order Inverse Frequency Expansion
Approach

In this section, we report on higher order expressions for the inverse
frequency expansion (IFE) (9). Let us recall the inverse frequency expansion
for the kick operator and Floquet Hamiltonian (7)

$\displaystyle\mathcal{H}_{F\\!,\lambda}^{(n)}$
$\displaystyle=\sum_{j=0}^{n}\omega^{-n}\mathcal{H}_{F\\!,\lambda}^{[n]}=\mathcal{H}_{F\\!,\lambda}+O(\omega^{-n-1})\,,$
(32) $\displaystyle\mathcal{K}_{\lambda}^{(n)}$
$\displaystyle=\sum_{j=0}^{n}\omega^{-n}\mathcal{K}_{\lambda}^{[n]}=\mathcal{K}_{\lambda}+O(\omega^{-n-1})\,,$

as well as the general structure of the transformation

$\mathcal{X}_{\lambda}(t)=\mathcal{P}_{\\!\lambda}\mathcal{X}_{F\\!,\lambda}\mathcal{P}_{\\!\lambda}^{\dagger}+i\pqty{\partial_{\lambda}\mathcal{P}_{\\!\lambda}}\mathcal{P}_{\\!\lambda}^{\dagger}\,.$
(33)

Using again $K^{[0]}_{\lambda}=0$ and that $\mathcal{X}_{\lambda}^{(n)}$ is
the (approximate) AGP for $\mathcal{H}_{\lambda}^{(n)}\pqty{\lambda}$, the IFE
terms up to the third order read

$\mathcal{X}_{\lambda}^{(0)}(t)=\mathcal{X}_{\lambda}^{(0)}\,,$ (34a)
$\mathcal{X}_{\lambda}^{(1)}(t)=\mathcal{X}_{\lambda}^{(1)}+\omega^{-1}\pqty{i\commutator{\mathcal{K}_{\lambda}^{[1]}(t)}{\mathcal{X}_{\lambda}^{(1)}}-\partial_{\lambda}\mathcal{K}_{\lambda}^{[1]}(t)}\,,$
(34b) $\displaystyle\mathcal{X}_{\lambda}^{(2)}(t)$
$\displaystyle=\mathcal{X}_{\lambda}^{(2)}-\partial_{\lambda}\mathcal{K}_{\lambda}^{(2)}(t)$
(34c) $\displaystyle\hskip
2.84544pt+\commutator{\mathcal{K}_{\lambda}^{(2)}(t)}{\mathcal{X}_{\lambda}^{(2)}}-\frac{i}{2}\commutator{\mathcal{K}_{\lambda}^{(1)}(t)}{\partial_{\lambda}\mathcal{K}_{\lambda}^{(1)}(t)}$
$\displaystyle\hskip
2.84544pt+\commutator{\commutator{\mathcal{K}_{\lambda}^{(1)}(t)}{\mathcal{X}_{\lambda}^{(2)}}}{\mathcal{K}_{\lambda}^{(1)}(t)}\,,$
and $\displaystyle\mathcal{X}_{\lambda}^{(3)}(t)$
$\displaystyle=\mathcal{X}_{\lambda}^{(3)}-\partial_{\lambda}\mathcal{K}_{\lambda}^{(3)}(t)$
(34d) $\displaystyle\hskip
2.84544pt+\commutator{\mathcal{K}_{\lambda}^{(3)}(t)}{\mathcal{X}_{\lambda}^{(3)}}$
$\displaystyle\hskip
2.84544pt-\frac{i}{2}\commutator{\mathcal{K}_{\lambda}^{(2)}(t)}{\partial_{\lambda}\mathcal{K}_{\lambda}^{(2)}(t)}$
$\displaystyle\hskip
2.84544pt+\commutator{\commutator{\mathcal{K}_{\lambda}^{(2)}(t)}{\mathcal{X}_{\lambda}^{(3)}}}{\mathcal{K}_{\lambda}^{(2)}(t)}$
$\displaystyle\hskip
2.84544pt-\frac{1}{6}\commutator{\commutator{\mathcal{K}_{\lambda}^{(1)}(t)}{\partial_{\lambda}\mathcal{K}_{\lambda}^{(1)}}}{\mathcal{K}_{\lambda}^{(1)}(t)}$
$\displaystyle\hskip
2.84544pt-\frac{i}{6}\commutator{\commutator{\commutator{\mathcal{K}_{\lambda}^{(1)}(t)}{\mathcal{X}_{\lambda}^{(3)}}}{\mathcal{K}_{\lambda}^{(1)}(t)}}{\mathcal{K}_{\lambda}^{(1)}(t)}\,,$
where for brevity the third order expression also includes higher order terms
in
$\commutator{\mathcal{K}_{\lambda}^{(2)}(t)}{\partial_{\lambda}\mathcal{K}_{\lambda}^{(2)}(t)}$
and
$\commutator{\commutator{\mathcal{K}_{\lambda}^{(2)}(t)}{\mathcal{X}_{\lambda}^{(3)}}}{\mathcal{K}_{\lambda}^{(2)}(t)}$.

Let us emphasize that a similar expansion of $\mathcal{X}_{\lambda}^{(n)}$ in
powers of $\omega^{-1}$ does not exist in general, i.e., the difference
between $\mathcal{X}_{\lambda}^{(n)}$ and $\mathcal{X}_{\lambda}^{(n-1)}$
might be of higher order than $O(\omega^{-n})$. Indeed, if the $n$’th order
correction opens a gap then the FAGP generally scales as $\omega^{n}$ as it is
proportional to $1/\mathrm{gap}$ close to an avoided gap closing. A common
scenario where this can occur is the presence of quasi-conservation laws.
There, the low order Floquet Hamiltonians may preserve a symmetry but higher
order terms explicitly break the symmetry, possibly lifting degenerate energy
levels.

###  B.2 Alternative Derivation of Variational Method using extended Floquet
Hilbert Space

In this section we provide an additional derivation of the variational method
using the so-called extended Floquet Hilbert space approach also referred to
as two-time treatment [66].

##### Floquet Theorem for States

To this end, we will first recap the derivation of the extended Floquet
Hilbert space approach. In the following, we denote the periodic time-
dependence by $\phi=\omega t$ in order to avoid confusion between periodic
time-dependence and the time-dependence associated with the change of
parameter $\lambda=\lambda(t)$. Using this convention Eq. (3) reads

$\mathcal{H}_{F,\lambda}=\mathcal{P}_{\\!\lambda}^{\dagger}(\phi)\mathcal{H}_{\lambda}(\phi)\mathcal{P}_{\\!\lambda}(\phi)-i\dot{\phi}\mathcal{P}_{\\!\lambda}(\phi)^{\dagger}\derivative{\phi}\mathcal{P}_{\\!\lambda}(\phi)\,.$
(35)

In the following, we will suppress the $\lambda$ index for all operators.

As $\mathcal{H}_{F}$ is time-independent and hermitian there exist a complete
set of orthonormal eigenstates $\ket{\epsilon_{n}^{F}}$ such that
$\mathcal{H}_{F}\ket{\epsilon_{n}^{F}}=\epsilon_{n}^{F}\ket{\epsilon_{n}^{F}}$.
Transforming back to the lab frame the time-dependent Schrödinger equation
(TDSE)

$\pqty{i\derivative{t}-\mathcal{H}(\phi)}\ket{\psi(t)}=0\,,$ (36)

is therefore solved by the so-called Floquet states

$\ket{\psi_{n}(t)}=e^{-i\epsilon_{n}^{F}t}\ket{u_{n}(\phi)}\,,$ (37)

with the periodic Floquet functions
$\ket{u_{n}(\phi)}=\mathcal{P}(\phi)\ket{\epsilon_{n}^{F}}$. Therefore, any
solution of the TDSE can be written as

$\ket{\psi(t)}=\sum_{n}c_{n}e^{-i\epsilon_{n}^{F}t}\ket{u_{n}(\phi)}\,,$ (38)

with complex expansion coefficients $c_{n}=\innerproduct{\psi(0)}{u_{n}(0)}$.

Note that there are two distinct time-dependences in $\ket{\psi_{n}(t)}$, a
periodic time-dependence inherited from the micromotion and the phase
accumulation $e^{-i\epsilon_{n}^{F}t}$ as known from usual static systems.
Therefore, it is natural to split the time-derivative $\derivative{t}$ in the
TDSE into two contributions
$\derivative{t}=\partial_{t}+\derivative{\phi}{t}\partial_{\phi}$ such that

$i\partial_{t}\ket{\psi(t)}=\pqty{i\nu\partial_{\phi}+\mathcal{H}(t)}\ket{\psi(t)}\,,$
(39)

where we used the instantaneous frequency
$\nu=\dot{\phi}=\omega+\dot{\omega}t$ and
$\ket{\psi(t)}=\sum_{n,m}c_{nm}(t)e^{-in\phi}\ket{m}$ is considered to be
expanded in Fourier modes $\Bqty{e^{-in\phi}}_{n\in\mathbb{Z}}$ as well as
some physical basis $\Bqty{\ket{m}}$.

##### Extended Floquet Hilbert Space

This Fourier expansion of the state motivates the introduction of an extended
Floquet Hilbert space where the periodic time dependence is lifted to its own
degree of freedom. Then, the periodic functions
$\Bqty{e^{in\phi}}_{n\in\mathbb{z}}$ form an orthogonal basis of the Hilbert
space of periodic smooth functions $\mathscr{L}_{\circ}$ with inner product

$\left(\left.n\right|m\right)=\int_{0}^{2\pi}e^{-in\phi}e^{im\phi}\frac{\differential{\phi}}{2\pi}$
(40)

where we introduced the bra-ket notation $e^{in\phi}\to\left|n\right)$.

Likewise, expanding the Hamiltonian in Fourier modes
$\mathcal{H}(t)=\sum_{n}\mathcal{H}_{n}e^{-in\phi}$ we can promote the product
$e^{-i\phi n}\cdot$ to an operator acting on $\mathscr{L}_{\circ}$, i.e.,
$\mathcal{E}_{n}\hat{=}e^{in\phi}:\mathscr{L}_{\circ}\to\mathscr{L}_{\circ}$
with $\mathcal{E}_{n}\left|m\right)=\left|n+m\right)$. Eventually, we can also
express the derivative $\partial_{\phi}$ as an operator on the Hilbert space
$\mathscr{L}_{\circ}$, in fact from $-i\partial_{\phi}e^{in\phi}=ne^{in\phi}$
it follows that the derivative acts as a number operator on the space of all
harmonics
$\mathcal{N}\hat{=}-i\partial_{\phi}:\mathscr{L}_{\circ}\to\mathscr{L}_{\circ}$
with $\mathcal{N}\left|n\right)=n\left|n\right)$.

The extended Floquet Hilbert space
$\mathscr{F}=\mathscr{H}\otimes\mathscr{L}_{\circ}$ is the combination of the
physical Hilbert space $\mathscr{H}$ and $\mathscr{L}_{\circ}$. Elements in
this Hilbert space are denoted by
$\left|\left.\Xi\right>\right>\in\mathscr{F}$ and can be expanded as
$\left|\left.\Xi\right>\right>=\sum_{n,m}c_{nm}\ket{n}\otimes\left|m\right)$.

Therefore, we can rewrite Eq. (39) as

$i\partial_{t}\left|\left.\psi\right>\right>=\mathcal{Q}\left|\left.\psi\right>\right>\,,$
(41)

with the time-independent quasi-energy operator
$\mathcal{Q}=\nu\mathcal{N}+\sum\mathcal{H}_{n}\mathcal{E}_{n}$. Then, Eq.
(41) can be solved by a separation Ansatz
$\left|\left.\psi(t)\right>\right>=e^{-iqt}\left|\left.\Phi\right>\right>$,
with eigenfunctions $\left|\left.\Phi\right>\right>$ which satisfy
$\mathcal{Q}\left|\left.\Phi\right>\right>=q\left|\left.\Phi\right>\right>$
with quasi-energies $q$. Therefore, the quasi-energy operator $\mathcal{Q}$ is
a time-independent object and contains all information about the Floquet
problem, i.e., diagonalizing it amounts to computing the Floquet functions
$\ket{u_{n}}$ and quasi-energies $\epsilon_{n}^{F}$ in Eq. (37). Note that,
using the extended Floquet Hilbert space the instantaneous frequency
$\nu=\dot{\phi}$ appears naturally as the frequency of interest.

Before we continue deriving the variational ansatz a short comment about the
extended Floquet Hilbert space is in order. Note that, since
$\left|\left.\Phi\right>\right>$ is an eigenfunction of $\mathcal{Q}$ with
eigenvalue $q$, also $\mathcal{E}_{n}\left|\left.\Phi\right>\right>$ is an
eigenfunction of $\mathcal{Q}$ with eigenvalue $q-n\omega$. However,
projecting
$\left|\left.\psi(t)\right>\right>=e^{-iqt}\left|\left.\Phi\right>\right>$ and
$\left|\left.\psi^{\prime}(t)\right>\right>=e^{-i(q-n\omega)t}\mathcal{E}_{n}\left|\left.\Phi\right>\right>$
back to the physical Hilbert space the two states are in fact the same
$\ket{\psi(t)}{=}\ket{\psi^{\prime}(t)}$. Hence, $\mathscr{F}$ contains
infinitely many copies of the physical problem each separated by $n\omega$.

##### Floquet Counterdiabatic Driving

Reintroducing the control parameter $\lambda$ in Eq. (41) we have

$i\partial_{t}\left|\left.\psi(t)\right>\right>=\mathcal{Q}_{\lambda}\left|\left.\psi(t)\right>\right>$

with

$\mathcal{Q}_{\lambda}=\nu(\lambda)\mathcal{N}+\sum_{n}\mathcal{H}_{n}(\lambda)\mathcal{E}_{n}\,.$
(42)

Note that formally Floquet counterdiabatic driving is equivalent to usual
static counterdiabatic driving if one replaces the Hamiltonian by the quasi-
energy operator $\mathcal{Q}_{\lambda}$. In fact, as $\mathcal{Q}_{\lambda}$
is hermitian there exists for each fixed $\lambda$ a unitary transformation
$\mathcal{V}_{\lambda}$ such that
$\tilde{\mathcal{Q}}_{\lambda}{=}\mathcal{V}_{\lambda}^{\dagger}\mathcal{Q}_{\lambda}\mathcal{V}_{\lambda}=\nu\mathcal{N}+\tilde{\mathcal{H}}$
is diagonal. Thus, in the moving frame
$\left|\left.\tilde{\psi}(t)\right>\right>=\mathcal{V}_{\lambda(t)}^{\dagger}\left|\left.\psi(t)\right>\right>$
the TDSE reads

$\pqty{i\partial_{t}+\tilde{\mathcal{Q}}_{\lambda}-\dot{\lambda}\tilde{\mathcal{A}}_{\lambda}}\left|\left.\tilde{\psi}(t)\right>\right>=0\,,$
(43)

with the Floquet adiabatic gauge potential (FAGP)
$\tilde{\mathcal{A}}_{\lambda}=i\mathcal{V}_{\lambda}^{\dagger}\partial_{\lambda}\mathcal{V}_{\lambda}$.
As $\tilde{\mathcal{Q}}$ is diagonal all transitions are driven by the FAGP
$\tilde{\mathcal{A}}_{\lambda}$, such that the Floquet counterdiabatic quasi-
energy operator reads

$\mathcal{Q}_{\mathrm{CD}}=\mathcal{Q}_{\lambda}+\dot{\lambda}\mathcal{A}_{\lambda}\,.$
(44)

The Floquet theorem in the extended Hilbert space guarantees that, for fixed
$\lambda$, $\mathcal{Q}_{\lambda}$ can be diagonalized by a unitary
$\mathcal{V}_{\lambda}$ which corresponds to a time-periodic operator [65].
Therefore, also $\mathcal{A}_{\lambda}(\phi)$ is a time-periodic operator, for
fixed $\lambda$.

The FAGP satisfies the usual identities also satisfied by standard AGPs, see
e.g., Ref. [58], upon using the correct extended Floquet Hilbert space
analoga,

$\displaystyle\pqty{\mathcal{A}_{\lambda}}_{nm}$
$\displaystyle=i\left<\innerproduct{n_{\lambda}}{\partial_{\lambda}m_{\lambda}}\right>\,,$
(45a) $\displaystyle\pqty{\mathcal{A}_{\lambda}}_{nm}$
$\displaystyle=i\frac{\left<\left<n_{\lambda}\right|\right.\partial_{\lambda}\mathcal{Q}_{\lambda}\left|\left.m_{\lambda}\right>\right>}{q_{n}(\lambda)-q_{m}(\lambda)}\,,$
(45b) $\displaystyle\commutator{\mathcal{A}_{\lambda}}{\mathcal{Q}_{\lambda}}$
$\displaystyle=i\partial_{\lambda}\mathcal{Q}_{\lambda}+i\mathcal{M}_{\lambda}\,,$
(45c) $\displaystyle 0$
$\displaystyle=\commutator{\mathcal{Q}_{\lambda}}{i\partial_{\lambda}\mathcal{Q}_{\lambda}-\commutator{\mathcal{A}}{\mathcal{Q}_{\lambda}}}\,,$
(45d)

where $\left|\left.n_{\lambda}\right>\right>$ is the eigenstate of
$\mathcal{Q}_{\lambda}$ with eigenvalue $q_{n}(\lambda)$ and
$\mathcal{M}_{\lambda}=-\sum_{n}\left|\left.n_{\lambda}\right>\right>\partial_{\lambda}q_{\lambda}\left<\left<m_{\lambda}\right|\right.$
is the generalized force operator.

Let us briefly sketch the proofs of Eqs. (45) which are equivalent to the
usual AGP case (c.f. [58]). Equation (45)a is a direct result of the
definition of
$\mathcal{A}=i\mathcal{V}_{\lambda}\partial_{\lambda}\mathcal{V}^{\dagger}_{\lambda}$.
Likewise b follows directly from a using
$\mathcal{Q}_{\lambda}\left|\left.n_{\lambda}\right>\right>=q_{n}(\lambda)\left|\left.n_{\lambda}\right>\right>$,
with c being the operator expression of b. Eventually, Eq. (45)d follows from
Eq. (45)c using that
$\commutator{\mathcal{Q}_{\lambda}}{\mathcal{M}_{\lambda}}=0$, which in turn
is a direct result of the two operators sharing the same eigenstates.

##### Variational Principle

Equations (45) can be used to derive the variational principle in complete
analogy to the static case(see e.g. [58]). For completeness, we provide a
brief derivation. To this end, let us introduce

$\mathcal{G}_{\lambda}(\mathcal{X})=i\commutator{\mathcal{Q}_{\lambda}}{\mathcal{X}_{\lambda}}-\partial_{\lambda}\mathcal{Q}_{\lambda}$
(46)

where $\mathcal{X}_{\lambda}$ is an hermitian test operator. Then, Eq. (45)c
can be written as
$\mathcal{G}_{\lambda}(\mathcal{A}_{\lambda})=\mathcal{M}_{\lambda}$.
Therefore, the FAGP is the unique operator minimizing the Frobenius distance

$\displaystyle D^{2}_{\lambda}(\mathcal{X}_{\lambda})$
$\displaystyle=\Tr_{\mathscr{F}}(\bqty{\mathcal{G}_{\lambda}(\mathcal{X}_{\lambda})-\mathcal{M}_{\lambda}}^{2})$
(47)
$\displaystyle=\Tr_{\mathscr{F}}(\mathcal{G}^{2}_{\lambda}(\mathcal{X}_{\lambda}))-2\Tr_{\mathscr{F}}(\mathcal{G}_{\lambda}(\mathcal{X}_{\lambda})\mathcal{M}_{\lambda})+\Tr_{\mathscr{F}}(\mathcal{M}^{2}_{\lambda})\,.$

Note that, for periodic operators that we consider, the trace over the
extended Floquet Hilbert space can be written as
$\Tr_{\mathscr{F}}(\cdot)=\Tr_{\mathscr{H}}(\left(0\right|\cdot\left|0\right))$,
where $\Tr_{\mathscr{L}_{\circ}}(\mathds{1}_{\mathscr{L}_{\circ}})=1$.

Using the cyclicity of the trace it is easy to show that
$\Tr_{\mathscr{F}}(\commutator{\mathcal{X}}{\mathcal{Q}}\mathcal{M})=0$ and
thus

$D^{2}_{\lambda}(\mathcal{X})=\Tr_{\mathscr{F}}(\mathcal{G}^{2}_{\lambda}(\mathcal{X}))+\Tr_{\mathscr{F}}(\mathcal{M}^{2}_{\lambda})-2\Tr_{\mathscr{F}}(\partial_{\lambda}\mathcal{Q}_{\lambda}\mathcal{M}_{\lambda})\,.$
(48)

As the two last terms in Eq. (48) are independent of $\mathcal{X}$ minimizing
$D^{2}_{\lambda}(\mathcal{X})$ is equivalent to minimizing the action

$S[\mathcal{X}]=\Tr_{\mathscr{F}}(\mathcal{G}^{2}_{\lambda}(\mathcal{X}))\,.$
(49)

Equation (49) can be projected to the physical Hilbert space by using
$\Tr_{\mathscr{F}}(\cdot)\to\int_{0}^{2\pi}\Tr_{\mathscr{H}}(\cdot)\differential{\phi}/(2\pi)$
and
$i\commutator{\mathcal{X}_{\lambda}}{\mathcal{N}}\to\partial_{\phi}\mathcal{X}_{\lambda}$
to read

$\displaystyle S[\mathcal{X}_{\lambda}]$
$\displaystyle=\int_{0}^{2\pi}\Tr_{\mathscr{H}}(\mathcal{G}^{2}_{\lambda}(\mathcal{X}_{\lambda}))\frac{\differential{\phi}}{2\pi}\,,$
(50) $\displaystyle\mathcal{G}_{\lambda}(\mathcal{X}_{\lambda})$
$\displaystyle=i\commutator{\mathcal{H}}{\mathcal{X}_{\lambda}}+\nu\partial_{\phi}\mathcal{X}_{\lambda}-\partial_{\lambda}\mathcal{H}\,,$

which is the same as Eq. (10) reported on in the main text, up to an
irrelevant multiplicative factor.

##  Appendix C Exactly Solvable Model

In this section, we report on the analytically solvable circularly driven two-
level system

$\mathcal{H}_{\circ}(t)=\Delta S^{z}+A\bqty{\cos(\omega t)S^{x}+\sin(\omega
t)S^{y}}\,,$ (51)

with level splitting $\Delta$, driving amplitude $A$ and driving frequency
$\omega$. This model is the non-interacting limiting case of the circularly
driven Ising model (28) considered in the main text. Likewise, it can be
mapped to a static problem using the frame transformation
$V=\exp(-i\phi(t)S^{z})$ with $\phi(t)=\omega t$. This leads to

$\widetilde{\mathcal{H}}=(\Delta-\omega)S^{z}+AS^{x}\,,$ (52)

where we used that $\dot{\phi}=\omega$ for fixed frequency. Note that Eq. (52)
is not yet the Floquet Hamiltonian, as its spectrum is not folded into a
window $\omega$ of energies [2]. However, since this folding is trivial, i.e.,
no photon resonances occur, we can skip this step for further consideration.
So far, the analysis is general and applies to any parameter in Eq. (51). In
the following, we focus on a constant in time frequency
$\derivative{t}\omega=0$. The case of a time-dependent frequency, so-called
chirp, is studied in detail in Sec. D.3.

In the rotating frame, the Hamiltonian (52) can be diagonalized via
$U(\alpha)=\exp(-i\alpha S^{y})$, where $\tan(\alpha)=A/(\Delta-\omega)$.
Therefore, the FAGPs $\mathcal{A}_{\Delta}$ and $\mathcal{A}_{A}$ for
$\tilde{\mathcal{H}}_{\circ}$ upon changing $\Delta$ and $A$, respectively,
read

$\displaystyle\mathcal{A}_{\Delta}(t)$
$\displaystyle=\frac{-A}{(\Delta-\omega)^{2}+A^{2}}\pqty{\cos(\omega
t)S^{y}-\sin(\omega t)S^{x}}\,,$ (53) $\displaystyle\mathcal{A}_{A}(t)$
$\displaystyle=\frac{\Delta-\omega}{(\Delta-\omega)^{2}+A^{2}}\pqty{\cos(\omega
t)S^{y}-\sin(\omega t)S^{x}}\,.$

Notice that there is no additional contribution due to
$V^{\dagger}\partial_{\lambda}V$ as $V$ is independent of both $A$ and
$\Delta$.

We can also derive the FAGP in the extended Floquet Hilbert space. This allows
us to gain more insights into the system and the mechanism allowing for an
exact solution. To this end, let us consider the quasi-energy operator

$\mathcal{Q}_{\circ}=\omega\mathcal{N}+\Delta
S^{z}+\frac{A}{2}S^{+}\mathcal{E}^{+}+\frac{A}{2}S^{-}\mathcal{E}^{-}\,,$ (54)

where $\mathcal{E}^{\pm}\hat{=}e^{\pm i\omega t}$. Additionally, we introduced
the spin-$\frac{1}{2}$ raising and lowering operators $S^{\pm}=(S^{x}\mp
iS^{y})$ which fulfill $\commutator{S^{z}}{S^{\pm}}=\pm iS^{\pm}$.

Note that, we can define generalized raising and lowering operators
$\Sigma^{\pm}=S^{\pm}\mathcal{E}^{\pm}$ with associated pseudo-spin operators
$\Sigma^{x}=(\Sigma^{+}+\Sigma^{-})/2$,
$\Sigma^{y}=(\Sigma^{+}-\Sigma^{-})/2i$ and $\Sigma^{z}=S^{z}$. These
operators also fulfill the usual spin-$\frac{1}{2}$ algebra

$\displaystyle\commutator{\Sigma^{z}}{\Sigma^{\pm}}$ $\displaystyle=\pm
i\Sigma^{\pm}\,,$ (55) $\displaystyle\commutator{\Sigma^{+}}{\Sigma^{-}}$
$\displaystyle=2\Sigma^{z}\,,$
$\displaystyle\commutator{\Sigma^{\alpha}}{\Sigma^{\beta}}$
$\displaystyle=i\epsilon_{\alpha\beta\gamma}\Sigma^{\gamma}\,,$

with $\alpha,\beta,\gamma\in\Bqty{x,y,z}$ and the fully anti-symmetric Levi-
Civita symbol $\epsilon$. Additionally, we find

$\displaystyle i\commutator{\mathcal{N}}{\Sigma^{\pm}}$
$\displaystyle=\pm\Sigma^{\pm}\,,$ (56) $\displaystyle
i\commutator{\mathcal{N}}{\Sigma^{\alpha}}$
$\displaystyle=-i\epsilon_{z\alpha\gamma}\Sigma^{\gamma}\,.$

Therefore, the set of operators
$\Bqty{\mathcal{N},\,\Sigma^{x},\,\Sigma^{y},\,\Sigma^{z}}$ form a closed
algebra akin to the standard Pauli-algebra, with the number operator
$\mathcal{N}$ taking a similar role as $\Sigma^{z}$.

Eventually, let us consider the action of the adjoint representation of
$\exp(-i\alpha\Sigma^{y})$

$\displaystyle\exp(i\alpha\Sigma^{y})\Sigma^{z}\exp(-i\alpha\Sigma^{y})$
$\displaystyle=\cos(\alpha)\Sigma^{z}-\sin(\alpha)\Sigma^{x}\,,$ (57)
$\displaystyle\exp(i\alpha\Sigma^{y})\Sigma^{x}\exp(-i\alpha\Sigma^{y})$
$\displaystyle=\cos(\alpha)\Sigma^{x}+\sin(\alpha)\Sigma^{z}\,,$
$\displaystyle\exp(i\alpha\Sigma^{y})\mathcal{N}\exp(-i\alpha\Sigma^{y})$
$\displaystyle=\cos(\alpha)\mathcal{N}+\sin(\alpha)\Sigma^{x}\,,$

which follows immediately from Eqs. (55) and (56).

Using the generalized raising and lowering operators, we can write the quasi-
energy operator (54) as

$\mathcal{Q}_{\circ}=\omega\mathcal{N}+\Delta\Sigma^{z}+A\Sigma^{x}\,.$ (58)

Then, Eq. (57) implies that $\mathcal{W}=\exp(-i\alpha\Sigma^{y})$
diagonalizes the quasi-energy operator for $\tan\alpha=(\Delta-\omega)/g$.
Hence, the adiabatic gauge potentials in the extended Floquet Hilbert space
read

$\displaystyle\mathcal{A}_{\Delta}$
$\displaystyle=\frac{-A}{(\Delta-\omega)^{2}+A^{2}}\Sigma^{y}\,,$ (59)
$\displaystyle\mathcal{A}_{A}$
$\displaystyle=\frac{\Delta-\omega}{(\Delta-\omega)^{2}+A^{2}}\Sigma^{y}\,,$

which is equivalent to Eqs. (53) upon projecting to the physical Hilbert
space.

In summary, the circularly driven two-level system is analytically solvable
due to the closed algebra generated by the quasi-energy operator in the
extended Floquet Hilbert space. This insight might be generalized to obtain
more complex, analytically tractable, Floquet systems, i.e., by ensuring a
closure relation among the operators appearing in the quasi-energy operator.

##  Appendix D  Details on Frequency Ramps

In this section, we use the extended Floquet Hilbert space approach to
rigorously derive the variational action (14) for frequency ramps. Moreover,
we illustrate our findings on the exactly solvable circularly driven two-level
system.

###  D.1 Floquet Counterdiabatic Driving for Chirps

First, we give a rigorous derivation of the variational action (14) for chirps
using the extended Floquet Hilbert space approach. In Sec. B.2, we used the
instantaneous frequency throughout such that the quasi-energy operator (42),

$\mathcal{Q}_{\lambda}=\nu_{\lambda}\mathcal{N}+\sum_{n}\mathcal{H}_{n,\lambda}\mathcal{E}_{n}\,.$

forms a perfect starting point. Note that, using the quasi-energy operator
$\mathcal{Q}_{\lambda}$ (42) we find for a chirp, $\lambda=\nu$, that
$\partial_{\nu}\mathcal{Q}_{\lambda}=\mathcal{N}$ appears in the variational
principle (49). The appearance of the number operator in the variational
principle is not a fundamental problem. However, the practical computation of
the variational FAGP becomes more difficult due to the non-periodicity and
unboundedness of $\mathcal{N}$.

An important observation is that changing the quasi-energy operator by a
multiplicative (possibly time-dependent) factor
$\mathcal{Q}\to\mathcal{Q}/f(\lambda)$ leaves the eigenstates unchanged,
hence, does not change the FAGP. Therefore, we can instead consider the re-
scaled (dimensionless) quasi-energy operator

$q_{\lambda}=\frac{\mathcal{Q}_{\lambda}}{\nu_{\lambda}}=\mathcal{N}+\nu_{\lambda}^{-1}\sum_{n}\mathcal{H}_{n,\lambda}\mathcal{E}_{n}\,.$
(60)

Then, the number operator no longer appears in the derivative
$\partial_{\nu}q_{\lambda}=-\nu_{\lambda}^{-2}\sum_{n}\mathcal{H}_{n,\lambda}\mathcal{E}_{n}$,
making the variational principle easier to solve.

However, note that since the instantaneous frequency $\nu$ explicitly depends
on $\dot{\omega}$ and time $t$, the FAGP attains an explicit time dependence.
This is in contrast to the mere implicit time dependency it carries via
$\lambda=\lambda(t)$ if the frequency is fixed. Therefore, it is convenient to
choose $\lambda=t$ as the fundamental protocol parameter. Then, combining
everything we arrive at the action of form Eq. (49) with

$\displaystyle S[\mathcal{X}_{\lambda}]$
$\displaystyle=\int_{0}^{2\pi}\Tr_{\mathscr{H}}(\mathcal{G}^{2}_{\lambda}(\mathcal{X}))\differential{\phi}\,,$
(61) $\displaystyle\mathcal{G}_{\omega}(\mathcal{X}_{\lambda})$
$\displaystyle=i\commutator{\mathcal{H}}{\mathcal{X}_{\lambda}}+\nu\partial_{\phi}\mathcal{X}_{\lambda}-\frac{\dot{\nu}}{\nu}\mathcal{H}-\dot{\lambda}\partial_{\lambda}\mathcal{H}\,,$

where $\partial_{\lambda}$ acts on any protocol parameter that is not related
to the periodicity of the drive $\phi$. A common scenario where the frequency
appears both to describe the periodicity and a non-periodic parameter is
strong driving. There, the amplitude $A$ of a drive scales with the frequency
$A\propto\omega$.

###  D.2  Adjusted Floquet Counterdiabatic Driving for Chirps

The action (14) describes the FAGP which leads to transitionless driving along
the adiabatic path. Let us recall that for a chirp this adiabatic path is
described by the instantaneous frequency $\nu(t)$ instead of the time-
dependent frequency $\omega(t)$, see Sec. III.

Here, we demonstrate that using an adjusted action we can obtain
transitionless driving along the instantaneous Floquet states described by the
change in the time-dependent frequency $\omega(t)$; not the instantaneous
frequency $\nu(t)$.

To this end, let us recall that in Sec. III, we concluded that parts of the
‘naive’ FAGPT contribution
$\mathcal{A}_{\mathcal{P}}=-i\mathcal{P}_{\omega}\partial_{\omega}\mathcal{P}_{\omega}^{\dagger}$
must be included into the Floquet Hamiltonian. This procedure restores
adiabaticity, i.e., $\dot{\lambda}\norm{\mathcal{A}_{\mathcal{P}}}\to 0$ in
the adiabatic limit, but requires the change $\nu(t)\to\omega(t)$. However, if
we insist on using the entire expression
$\mathcal{A}_{\mathcal{P}}=-i\mathcal{P}_{\omega}\partial_{\omega}\mathcal{P}_{\omega}^{\dagger}$
as part of the FAGP, the state protocol will follow the time-dependent
frequency $\omega(t)$. This comes at the expense that the counter term will
not vanish in the adiabatic limit, see also example in Sec D.3.

In order to derive the variational principle for this case one simply retraces
the steps from Eq. (36) up to Eq. (49) setting explicitly $\nu=\omega$, i.e.,
assuming $\derivative{t}=\omega\derivative{\phi}$. However, in order to
correct for the fact that the phase $\phi=\omega t$ depends on $\omega$, one
eventually has to use $\derivative{\omega}=\partial_{\omega}+t\partial_{\phi}$
such that the variational principle reads

$\displaystyle S^{\prime}[\mathcal{X}_{\omega}]$
$\displaystyle=\int_{0}^{2\pi}\Tr_{\mathscr{H}}(\mathcal{G}^{2}_{\omega}(\mathcal{X}_{\omega}))\differential{\phi}\,,$
(62) $\displaystyle\mathcal{G}^{\prime}_{\omega}(\mathcal{X}_{\omega})$
$\displaystyle=i\commutator{\mathcal{H}}{\mathcal{X}_{\omega}}+\omega\partial_{\phi}\mathcal{X}_{\omega}-\partial_{\omega}\mathcal{H}-t\partial_{\phi}\mathcal{H}\,.$

###  D.3 Example: Circularly Driven two-level System

Figure 7:  Protocol Dependence for Chirp. Unassisted state preparation for
smooth (A) and linear (B) frequency ramp in a circularly driven two-level
system (51). (i) Driving protocol for time-dependent frequency $\omega(t)$
(gray solid) and instantaneous frequency $\nu(t)$ (blue dashed). (ii)
$z$-polarization of time evolved state (blue) as a function of the ramp time
$T_{\mathrm{ramp}}$. The constant black line corresponds to $z$-polarization
of the target state. (iii) Fidelity of time evolved state at the end of the
protocol with respect to the Floquet eigenstate at instantaneous frequency,
$\nu{=}\dot{\omega t}$; for B) also with respect to $\omega$ (inset). The
adiabatic path does not follow the time-dependent frequency $\omega$-ramp but
rather the instantaneous frequency $\nu$, which is different for $t\neq 0$ for
a linear ramp but coincides with the expected result for the smooth ramp at
the beginning and end of the protocol, since
$\dot{\omega}(t=T_{\mathrm{ramp}}{=})0$. We used $g=1$ and $\Delta=10$,
$\omega_{\mathrm{i}}=20$ and $\omega_{\mathrm{f}}=10$; the smooth ramp is
given by
$\omega(t)=\omega_{\mathrm{i}}+(\omega_{\mathrm{f}}-\omega_{\mathrm{i}})\sin^{2}(x\pi/2)$
with $x=(t-t_{\mathrm{i}})/T_{\mathrm{ramp}}$.  Figure 8:  Floquet
Counterdiabatic Driving for Chirp in Circularly Driven Two-level System. Main
Panel. Fidelity as a function of ramp time $T_{\mathrm{ramp}}$ for unassisted
protocol (blue dots) and CD drive (black line), c.f. Eq. (64). Inset (i)
Amplitude of counterdiabatic term measured as the norm of the operator
$\norm{\dot{\lambda}\mathcal{A}_{\lambda}}^{2}=\absolutevalue{\dot{\lambda}}{\Tr(\mathcal{A}_{\lambda}^{2})}$
where $\dot{\lambda}=\dot{\nu}$ or $\dot{\omega}$ for $\mathcal{A}$ (black
solid) or $\mathcal{A}^{\prime}$ (green dashed), respectively. Inset (ii)
$z$-polarization as a function time for $T_{\mathrm{ramp}}=A$ for CD
protocols, colors as in $(i)$. Floquet counterdiabatic driving leads to
transitionless driving along the adiabatic path. Adjusted FAGP,
$\mathcal{A}^{\prime}$ allows for following instantaneous states along
$\omega$ at the expense of additional finite terms for all ramp times.
Parameters are as in Fig. 7.

To illustrate the theoretical consideration above, let us consider again the
exactly solvable circularly driven two-level system (51), c.f. Sec. C,

$\mathcal{H}_{\circ}=\Delta S^{z}+A\left(\cos(\omega t)S^{x}+\sin(\omega
t)S^{y}\right)\,.$

As detailed in Sec. C, we can use a unitary transformation,
$V=\exp(-i\phi(t)S^{z})$, to map Eq. (51) to a static system

$\widetilde{\mathcal{H}}=(\Delta-\omega)S^{z}+AS^{x}\,.$

Let us consider ramping the frequency from some initial value
$\omega(0)=\omega_{\mathrm{i}}$ to a final value
$\omega(T_{\mathrm{ramp}})=\omega_{\mathrm{f}}$. Then, we would naively expect
that the $\dot{\phi}$ contribution which deviates from $\omega$ should be part
of the adiabatic gauge potential, i.e., $\dot{\omega}\mathcal{A}\supset
t\dot{\omega}S^{z}$. However, as shown in Sec. III, the $t\dot{\omega}S^{z}$
contribution does not vanish in the adiabatic limit therefore it cannot be
part of the adiabatic gauge potential.

##### Adiabatic Limit

To restore the adiabatic limit, $t\dot{\omega}S^{z}$ should be considered a
relevant perturbation regardless of the protocol and hence should be included
in the Floquet Hamiltonian

$\widetilde{\mathcal{H}}=(\Delta-\nu(t))S^{z}+AS^{x}\,.$ (63)

Note that, while in the adiabatic limit $\dot{\omega}\to 0$ the effective
frequency does not approach the actual frequency at all times
$\nu\not\to\omega$, however, $\dot{\nu}\to 0$. Let us further emphasize that
the frequency ramp leads to a change in the frequency appearing in the quasi-
energy operator (42), but does not affect the period of the oscillations,
i.e., the micromotion operator $V=\exp(i\omega(t)tS^{z})$ remains unchanged.

For sufficiently smooth protocols that approach their final value with a
vanishing slope $\dot{\omega}(T_{\mathrm{ramp}})=0$ the two frequencies agree
at the final point $\omega(T_{\mathrm{ramp}})=\nu(T_{\mathrm{ramp}})$; thus,
the final states also agree. However, this may still lead to an increase in
the adiabaticity time scale: in general the maximal slope of $\nu$,
$\max_{t}\absolutevalue{\dot{\nu}}$, will be larger than the maximal slope of
$\omega$, $\max_{t}\absolutevalue{\dot{\omega}}$. This is a direct result of,
$\nu$ being a non-convex function in time whenever
$\dot{\omega}(T_{\mathrm{ramp}})=0$. If, however,
$\dot{\omega}(T_{\mathrm{ramp}})\neq 0$ as is the case for a linear frequency
ramp, then also $\omega(T_{\mathrm{ramp}})\neq\nu(T_{\mathrm{ramp}})$, and the
prepared state might differ from the targeted state.

In Fig. 7, we demonstrate the problem of a linear frequency ramp (B) in
contrast to a smooth ramp (A). Let us consider, that we want to prepare the
$x$-polarized state—or more precisely the Floquet state which circles around
the $z$-axis in the $xy$-plane—starting from the (almost) $z$-polarized state.
To this end, we choose a frequency chirp with $(\Delta-\omega_{\mathrm{i}})\gg
A$ and $(\Delta-\omega_{\mathrm{f}})=0$. This leads to
$\widetilde{\mathcal{H}}(t=t_{\mathrm{f}})=AS^{x}$ if the state follows the
time-dependent frequency $\omega(t)$. In Fig. 7(i), we display the
corresponding frequency ramps $\omega=\omega(t)$ and the effective frequency
$\nu(t)$ experienced by the system.

We first estimate the performance of the protocol on a physical observable.
Therefore, we consider the $z$-magnetization, $\expectationvalue{\sigma^{z}}$,
which should vanish for the target $x$-polarized state. As the
$z$-magnetization is unaffected by the micromotion operator
$\mathcal{P}\propto\sigma^{z}$ a finite $z$-polarization indicates failure of
the state preparation procedure. With increasing ramp time
$T_{\mathrm{ramp}}{\to}\infty$ the $z$-magnetization saturates for both ramps
indicating that the adiabatic limit has been reached, see Fig. 7(ii). However,
for the linear ramp, Fig. 7B(ii), the $z$-magnetization attains a non-zero
value ($\expectationvalue{\sigma^{z}}\to 1$) suggesting that the protocol
failed to prepare the target state.

In addition, we consider the fidelity $F$ with respect to the Floquet
eigenstate following the instantaneous frequency $\nu$, see Fig 7(iii). We
find for both ramps, linear and smooth, that the fidelity approaches unity for
large ramp time $T_{\mathrm{ramp}}$; confirming that the adiabatic state
follows $\nu(t)$ and not $\omega(t)$. For the smooth ramp, we find that the
adiabatic limit is reached only at much larger ramp times compared to the
linear ramp. However, the final state in the adiabatic limit agrees with the
target state.

##### Counterdiabatic Driving

As discussed in Secs. D.1 and D.2 there are two ways to perform
counterdiabatic driving in the case of chirps. One can either follow the
adiabatic path described by instantaneous eigenstates following $\nu(t)$,
i.e., eigenstates of (63). Alternatively, one can follow the naively expected
path which corresponds to instantaneous eigenstates with respect to
$\omega(t)$, i.e., eigenstates of Eq. (52).

For both cases, we can readily write down the counterdiabatic term using the
results from Sec. C:

$\displaystyle\dot{\nu}\mathcal{A}_{\omega}(\nu)$
$\displaystyle=\frac{A\dot{\nu}}{(\Delta-\nu)^{2}+A^{2}}\pqty{\cos(\phi)S^{y}-\sin(\phi)S^{x}}\,,$
(64a) $\displaystyle\dot{\omega}\mathcal{A}^{\prime}_{\omega}(\omega)$
$\displaystyle=\dot{\omega}\mathcal{A}_{\omega}(\omega)+\dot{\omega}tS^{z}\,,$
(64b)

where $\mathcal{A}^{\prime}_{\omega}$ denotes the alternative case. From Eq.
(64)b it is clear that the contribution
$\dot{\omega}\mathcal{A}^{\prime}_{\omega}$ will remain finite in the
adiabatic limit $\dot{\omega}\to 0$.

In Fig. 8 the effect of the two FCD protocols is depicted for the linear ramp.
As expected, the FCD protocols lead to unit fidelity regardless of the ramp
time; fidelity is measured with respect to the according targeted ‘adiabatic’
path. However, in agreement with the quantum speed limit the strength of the
counter term strongly depends on the ramp time $T_{\mathrm{ramp}}$, see Fig.
8(i). While the usual counter term, $\mathcal{A}_{\omega}$, follows the
adiabatic curve described in Fig. 7 using the adjusted counter term
$\mathcal{A}_{\omega}^{\prime}$ allows for state preparation following the
naive path $\omega$, see Fig. 8(ii). However, following $\omega(t)$ instead of
$\nu(t)$ comes at the expense that the alternative FAGP does not vanish in the
adiabatic limit.

##  Appendix E Details of the Numerical Algorithm

Computing the (approximate) variational adiabatic gauge potential
$\mathcal{X}$, cf. Eq. (10), analytically quickly becomes infeasible for a
large number of variational parameters. Therefore, most variational FAGPS
presented in the main text are computed numerically.

In this section, we describe the algorithm that we use to compute the
variational Floquet adiabatic gauge potential numerically and comment on some
straightforward extensions of our algorithm.

##### General framework for solving variational principle

Let us start by recalling the equations that need to be solved. The
variational FAGP minimizes the action

$\displaystyle S\bqty{\mathcal{X}_{\lambda}}$
$\displaystyle=\int_{0}^{2\pi}\Tr(\mathcal{G}^{2}(\mathcal{X}_{\lambda}(\phi)))\mathrm{d}\phi$
$\displaystyle\mathcal{G}$
$\displaystyle=i\commutator{\mathcal{H}_{\lambda}(\phi)}{\mathcal{X}_{\lambda}(\phi)}+\partial_{\phi}\mathcal{X}_{\lambda}(\phi)+\partial_{\lambda}\mathcal{H}_{\lambda}(\phi)\,,$

for each parameter value $\lambda$, see also Eq. (10) in the main text.

In a numerical approach, we can not compute the variational FAGP for all
continuous values of $\lambda$ in the interval
$\lambda\in\bqty{\lambda_{i},\,\lambda_{f}}$ but rather need to restrict to a
discrete subset. Moreover, it would be computationally too demanding to
compute the variational FAGP on the fly while computing the time evolution,
i.e., while solving the time-dependent Schrödinger equation (TDSE). Therefore,
we pre-compute the variational FAGP $\mathcal{X}(\lambda_{n})$ for a given set
of discrete parameters $\lambda_{n}\in\bqty{\lambda_{i},\,\lambda_{f}}$,
$n=1,\dots,N$. Then, we interpolate the results,
$\mathcal{X}(\lambda_{n})\to\mathcal{X}_{\lambda}^{\bqty{N}}$, to obtain a
continuous function, which is then used to solve the TDSE. To ensure that our
results are sufficiently converged we assert that the error between the
interpolation $\mathcal{X}_{\lambda}^{\bqty{N}}(\lambda_{n^{\prime}})$ and the
exact result, evaluated on a set of unseen points
$\lambda_{n^{\prime}}{\neq}\lambda_{n}$, is below some threshold value
$\epsilon$,
$\norm{\pqty{\mathcal{X}_{\lambda}^{\bqty{N}}-\mathcal{X}_{\lambda}}(\lambda_{n^{\prime}})}\leq\epsilon$.
If the error exceeds the threshold value we increase the number of points used
for interpolation and repeat the procedure until the error is below threshold.
For simplicity, we draw points for the evaluation equidistantly in the
interval $\bqty{\lambda_{i},\,\lambda_{f}}$. More sophisticated sampling
techniques, like importance sampling or Gaussian process regression, may lead
to a significant reduction in computational resources.

Let us now consider how to obtain the variational FAGP for a fixed value
$\lambda$. To this end, we first need to make an ansatz for the variational
FAGP. Throughout, we will consider an ansatz given by an operator expansion of
the form

$\hat{\mathcal{X}}_{\lambda}=\sum_{n}x_{n\\!,\lambda}\hat{O}_{n}\pqty{\phi}\,,$
(65)

with variational parameters $x_{n\\!,\lambda}$ and linearly independent
operators $\hat{O}_{n}\pqty{\phi}$ which may carry a periodic time-dependency.
Later, we consider more concrete examples of operators but in general, one may
consider operators that satisfy experimental constraints. In particular, in
most experimental setups there are strict constraints on the physical
operators which can be implemented but there are hardly any constraints on
their time-dependency. Therefore, a separation into time and physical
operators might be useful, i.e.,
$\hat{O}_{n}(\phi)\equiv\hat{O}_{n=(kl)}(\phi)=\hat{O}_{k}f_{l}(\phi)$.

Note that, since $\mathcal{G}$ is linear in $\mathcal{X}_{\lambda}$ and $S$
quadratic in $\mathcal{G}$, the action is quadratic in the variational
parameters, leading to a convex optimization problem. Moreover, plugging Eq.
(65) into $\nabla_{\boldsymbol{x}}S=0$ we find that the optimal parameters are
given by

$\boldsymbol{M}\boldsymbol{x}=\boldsymbol{b}\,,$ (66)

with

$\displaystyle M_{jk}(\lambda)$
$\displaystyle=\int\mathrm{d}\phi\Tr(g_{j\\!,\lambda}(\phi)g_{k\\!,\lambda}(\phi))$
(67) $\displaystyle b_{j}(\lambda)$
$\displaystyle=\int\mathrm{d}\phi\Tr(g_{j\\!,\lambda}(\phi)\partial_{\lambda}\mathcal{H}_{\lambda}(\phi)])$
$\displaystyle g_{j\\!,\lambda}(\phi)$
$\displaystyle=i\commutator{\mathcal{H}_{\lambda}\pqty{\phi}}{O_{j}(\phi)}+\partial_{\phi}O_{j}(\phi)\,.$

Notice that, for a given Hamiltonian computing the matrix $\boldsymbol{M}$ and
vector $\boldsymbol{b}$ can be done easily using an appropriate ansatz and
exploiting the underlying Hilbert space algebra. Therefore, computing the
variational FAGP at a fixed parameter $\lambda$ only amounts to solving the
linear system of equation, Eq. (66); thus only scales in the number of
variational parameters but not in the Hilbert space dimension. However, for a
generic system, the exact FAGP is expected to be non-local. Thus, to obtain
the exact FAGP variationally the number of variational parameters must be
comparable to the Hilbert space dimension.

##### Basis Representation

If one is interested in only one particular Hamiltonian and a single
variational ansatz with few parameters, we suggest to compute the quantities
in Eq. (67) (semi-)analytically for the specific operators and solve the
corresponding system of linear equations analytically or numerically. However,
if multiple Hamiltonians or ansätze with the same underlying Hilbert space
need to be solved a different approach might be beneficial. As we show below,
one can avoid computing the linear system (66) for each individual one by
exploiting the linear structure of Eq. (66).

To this end, let us consider an operator basis $\Bqty{\mathcal{B}_{n}}_{n}$ of
the extended Floquet Hilbert space
$\mathscr{F}^{2}=\mathscr{H}^{2}\otimes\mathscr{L}_{\circ}^{2}$, with Hilbert
space of physical operators $\mathscr{H}^{2}$ and the Hilbert space of
operators acting on periodic smooth functions $\mathscr{L}_{\circ}^{2}$. For
example, for a spin system of $L$ spins a complete basis is given by the
operators $\sigma^{\boldsymbol{\mu}}\cdot e^{in\omega t}$, where
$n\in\mathbb{Z}$ and $\boldsymbol{\mu}=\pqty{\mu_{1},\dots\mu_{L}}$ with
$\mu_{j}=0,x,y,z$ and Pauli-strings
$\sigma^{\boldsymbol{\mu}}=\sigma^{\mu_{1}}\otimes\dots\sigma^{\mu_{L}}$.

A key insight which we exploit for our derivation is that the commutator
${}_{n}\hat{\mathcal{C}}\equiv
i\commutator{\mathcal{B}_{n}}{\cdot}:\mathscr{F}^{2}\to\mathscr{F}^{2}$ and
derivative
$\hat{\mathcal{N}}\equiv\partial_{\phi}\pqty{\cdot}\equiv\commutator{\mathcal{N}}{\cdot}:\mathscr{F}^{2}\to\mathscr{F}^{2}$
are linear (super-)operators acting on the Hilbert space of operators. Thus,
we can expand them in the operator basis
${}_{n}\hat{\mathcal{C}}(\mathcal{B}_{k})=i\commutator{{}_{n}\hat{\mathcal{C}}}{\mathcal{B}_{k}}=\sum_{k}{}_{n}\hat{\mathcal{C}}_{kl}\,\mathcal{B}_{l}$
and
$\hat{\mathcal{N}}(\mathcal{B}_{k})=\partial_{\phi}\mathcal{B}_{k}=\sum_{k}\hat{\mathcal{N}}_{kl}\mathcal{B}_{l}$
with matrix entries ${}_{n}\hat{\mathcal{C}}_{kl}$ and
$\hat{\mathcal{N}}_{nm}$.

Note that, computing the matrix elements ${}_{n}C_{kl}$ and $N_{nm}$ may seem
like a significant overhead. However, depending on the choice of basis they
are directly related to the structure factor of the operator Lie-algebra,
hence involve low to no computational overhead.

Using the operator basis, $\Bqty{\mathcal{B}_{n}}_{n}$, we can express any
Hamiltonian $\mathcal{H}$ and its derivative $\partial_{\lambda}\mathcal{H}$
as a linear combination of basis elements:
$\mathcal{H}=\sum_{n}h_{n}\mathcal{B}_{n}$ and
$\partial_{\lambda}\mathcal{H}=\sum_{n}\partial_{\lambda}h_{n}\mathcal{B}_{n}$
with sets of real numbers $h_{n}$ and $\partial h_{n}$, respectively.
Likewise, we can also choose a variational ansatz of the form
$\mathcal{X}=\sum_{n\in\mathbb{Y}}x_{n}\mathcal{B}_{n}$, which in general only
includes a subset of the operator basis
$\Bqty{\mathcal{B}_{n}}_{n\in\mathbb{Y}}\subset\Bqty{\mathcal{B}_{n}}_{n}$.
Then, we can express the operator $\mathcal{G}$ (10) similarly using the
basis:

$\displaystyle\mathcal{G}(\mathcal{X})$
$\displaystyle=\sum_{j\in\mathbb{Y}}x_{j}\pqty{\hat{\mathcal{N}}(\mathcal{B}_{j}))+\sum_{m}h_{m}\,{}_{m}\hat{\mathcal{C}}(\mathcal{B}_{j})}+\sum_{n}\partial_{\lambda}h_{n}\mathcal{B}_{n}$
$\displaystyle=\sum_{n}\bqty{\sum_{j\in\mathbb{Y}}\sum_{m}\pqty{x_{j}h_{m}\,{}_{m}C_{jn}}+\partial_{\lambda}h_{n}}\mathcal{B}_{n}\,.$

Thus, the action, Eq. (10), can be expressed as

$\displaystyle S\bqty{\boldsymbol{x}}$
$\displaystyle=\sum_{j,k\in\mathbb{Y}}x_{j}x_{k}\sum_{n,m}S_{jn}\eta_{nm}S_{mk}$
(68)
$\displaystyle\vspace{10pt}+2\sum_{j\in\mathbb{Y}}x_{j}\sum_{n,m}S_{jn}\eta_{nm}\partial_{\lambda}h_{m}+\mathrm{const.}$
$\displaystyle=\boldsymbol{x}^{T}\boldsymbol{S}^{T}\boldsymbol{\eta}\boldsymbol{S}\boldsymbol{x}+2\boldsymbol{x}^{T}\boldsymbol{S}^{T}\boldsymbol{\eta}\partial_{\lambda}\boldsymbol{h}+\mathrm{const.}$

where we defined the metric
$\eta_{nm}{=}\int\mathrm{d}\phi\Tr(\mathcal{B}_{n}\mathcal{B}_{m})$ and matrix
$\boldsymbol{S}$ with entries
$S_{nm}{\equiv}(\sum_{p}h_{p}\,{}_{p}\hat{\mathcal{C}}_{nm})+\hat{\mathcal{N}}_{nm}$.
Let us emphasize that, in general, $\boldsymbol{S}$ is a rectangular matrix:
considering the matrix entries $S_{nm}$, then, $n$ runs over all indices in
the basis, $\Bqty{\mathcal{B}_{n}}_{n}$, but $m\in\mathbb{Y}$ only. In
addition, if the basis is orthonormal the metric is diagonal
$\eta_{nm}=\delta_{nm}$.

Using the superoperator expressions has the advantage that all model and
ansatz dependence enters the variational action (68) simply by changing the
dimension and linear coefficients of the matrix $\boldsymbol{S}$ and vector
$\partial_{\lambda}\boldsymbol{h}$. Therefore, considering a multitude of
ansätze or models can be done at low cost, justifying the possible overhead
caused by computing the superoperators ${}_{n}\hat{\mathcal{C}}$ and
$\hat{\mathcal{N}}$.

Note that, if $\boldsymbol{S}$ is invertible Eq. (66) simplifies to

$\boldsymbol{S}\boldsymbol{x}=\partial_{\lambda}\boldsymbol{h}\,.$ (69)

In general, however, the variational ansatz $\mathcal{X}$ does not contain all
elements of the basis $\Bqty{B_{n}}_{n}$ such that $\boldsymbol{S}$ is
rectangular and hence non-invertible. Therefore, an obvious approximation
scheme is to truncate $\boldsymbol{S}$ to a square matrix, i.e., neglecting
all operators $\mathcal{B}_{m}$ in
$\mathcal{G}=\sum_{m}\mathcal{G}_{m}\mathcal{B}_{m}$ which do not appear in
$\mathcal{X}$. This also naturally leads to a truncation in the number of
harmonics as discussed in the main text. Such a truncation is, however, not
strictly necessary, and in general, the matrix $\boldsymbol{M}$ and vector
$\boldsymbol{b}$ in Eq. (66) are obtained from

$\displaystyle\boldsymbol{M}$
$\displaystyle=\boldsymbol{S}^{T}\boldsymbol{\eta}\boldsymbol{S}$ (70)
$\displaystyle\boldsymbol{b}$
$\displaystyle=\boldsymbol{S}^{T}\boldsymbol{\eta}\partial_{\lambda}\boldsymbol{h}\,.$

##  Appendix F Additional Material for examples

In this section, we provide additional material for the examples studied in
Sec. IV of the main text.

Figure 9:  Performance of high-frequency expansion in linearly driven two-
level system in the large frequency regime. Left Panel: Exact (solid line) and
lowest order perturbative (circles) Floquet quasi-energies. Right Panel: Error
in lowest order perturbative Floquet energies compared to exact energies. The
lowest order perturbative Floquet Hamiltonian accurately describes the exact
Floquet quasi-energies. Parameters are as in Fig. 3.

###  F.1  Linearly Driven Two-Level System

####  F.1.1 High-frequency Expansion

In this section, we briefly support the claim that the lowest order high-
frequency expansion (19),

$H_{F}^{(0)}=\lambda S^{z}+gS^{x}\,,$

is sufficient to describe the linearly driven two-level system (17) in the
high-frequency regime.

To this end, we compare the numerically computed spectrum of the exact Floquet
Hamiltonian with the spectrum obtained from the lowest order high-frequency
expansion (19), see Fig. 9. We find that the exact and high-frequency Floquet
quasi-energies agree up to the fourth digit, suggesting that the lowest order
Floquet Hamiltonian is sufficient to describe the system at high accuracy.

####  F.1.2 Analytical Derivation of Variational FAGP

In this subsection, we derive in detail the analytical variational FAGP
reported on in the main text Sec. IV.1.

To this end, let us consider the ansatz Eq. (21) and plug it into the action
Eq. (12), resulting in

$S[\boldsymbol{X}]=\boldsymbol{X}\cdot\boldsymbol{M}\cdot\boldsymbol{X}-2\boldsymbol{X}^{T}\boldsymbol{b}+\mathrm{const.}\,.$
(71)

Hence, minimizing the action $\delta S=0$ amounts to solving the linear system
of equations

$\boldsymbol{M}\cdot\boldsymbol{X}=\boldsymbol{b}\,,$ (72)

where $\boldsymbol{X}^{T}=(y_{0},\,x_{1},\,y_{1},\,z_{1})$,
$\boldsymbol{b}^{T}=\left(2g,\,0,\,A,\,0\right)$ and

$\boldsymbol{M}=\begin{pmatrix}2\lambda^{2}+2g^{2}+4A^{2}&0&3Ag&-2A\omega\\\
0&\omega^{2}+\lambda^{2}&2\lambda\omega&-\lambda g\\\
3Ag&2\lambda\omega&g^{2}+\omega^{2}+\lambda^{2}+A^{2}/2&-2g\omega\\\
-2A\omega&-\lambda g&-2g\omega&\omega^{2}+\lambda^{2}\end{pmatrix}\,.$ (73)

While inverting the $4\times 4$ matrix may be performed analytically using,
e.g., a Gaussian elimination procedure, the general expression is complicated,
and not much insight can be gained from it. Also, in general, a larger number
of variational parameters may be considered which makes an analytical
treatment infeasible.

Therefore, we focus on deriving an expression for the variational parameters
for the two regimes considered in the main text, i.e., the high-frequency (I)
and one-photon resonance (II) regime.

##### (I) High-frequency regime

In the high-frequency regime $\omega$ is the largest energy scale,
$\omega\gg\lambda,\,g,\,A$, such that it is favourable to remove the
$\propto\omega,\,\omega^{2}$ contributions in $\boldsymbol{M}$. This is
achieved by considering the re-scaling
$\tilde{X}^{T}=(y_{0},\,x_{1}\omega,\,y_{1}\omega,\,z_{1}\omega)$ and
$\tilde{b}^{T}=(2g,0,A/\omega,0)$ leading to

$\tilde{\boldsymbol{M}}=\begin{pmatrix}2\lambda^{2}+2g^{2}+4A^{2}&0&3\frac{Ag}{\omega}&-2\frac{A}{\omega}\\\
0&1+\frac{\lambda^{2}}{\omega^{2}}&2\frac{\lambda}{\omega}&-\frac{\lambda
g}{\omega^{2}}\\\
3\frac{Ag}{\omega^{2}}&2\frac{\lambda}{\omega}&1+\frac{g^{2}+\lambda^{2}+A^{2}/2}{\omega^{2}}&-2\frac{g}{\omega}\\\
-2\frac{A}{\omega}&-\frac{\lambda
g}{\omega^{2}}&-2\frac{g}{\omega}&1+\frac{\lambda^{2}}{\omega^{2}}\end{pmatrix}\,,$

taking the limit
$\frac{A}{\omega},\,\frac{g}{\omega},\,\frac{\lambda}{\omega}\to 0$ the linear
system Eq. (72) reduces to

$\begin{pmatrix}2\lambda^{2}+2g^{2}+4A^{2}&0&0&-2A\\\ 0&1&0&0\\\ 0&0&1&0\\\
-2A&0&0&1\\\ \end{pmatrix}\cdot\tilde{\boldsymbol{X}}=\begin{pmatrix}2g\\\
0\\\ 0\\\ 0\end{pmatrix}\,,$

which is solved by

$\displaystyle y_{0}$ $\displaystyle=\frac{g}{\lambda^{2}+g^{2}}$
$\displaystyle x_{1}$ $\displaystyle=0$ $\displaystyle y_{1}$
$\displaystyle=0$ $\displaystyle z_{1}$
$\displaystyle=\frac{A}{\omega}\frac{g}{\lambda^{2}+g^{2}}\,,$

which is equivalent to the IFE approach Eq. (20) up to $O(\omega^{0})$.

##### (II) One-photon resonance regime

In the one-photon resonance regime, we have $\lambda\approx\omega$, such that
both $\lambda$ and $\omega$ are large scales. Therefore, in order to get rid
of the large scales a re-scaling of the parameters
$\tilde{X}^{T}=(y_{0}\omega,\,x_{1}\omega,\,y_{1}\omega,\,z_{1}\omega)$ and
$\tilde{b}^{T}=(2g/\omega,0,A/\omega,0)$ is needed. Performing the re-scaling
and a large frequency expansion $\frac{\lambda}{\omega}\to 1$ and
$\frac{A}{\omega},\,\frac{g}{\omega}\to 0$ in Eq. (73), however, leads to the
linear system

$\begin{pmatrix}1&0&0&0\\\ 0&1&1&0\\\ 0&1&1&0\\\ 0&0&0&1\\\
\end{pmatrix}\cdot\tilde{\boldsymbol{X}}=\begin{pmatrix}0\\\ 0\\\ 0\\\
0\end{pmatrix}\,.$

Naively one might assume that the ideal parameters are just given by the
trivial choice of parameters, i.e., $\tilde{\boldsymbol{X}}_{0}=0$. However,
as the matrix is singular also $\tilde{\boldsymbol{X}}_{1}=c\pqty{0,1,-1,0}$
with arbitrary $c$ is a solution. Thus, the solution to the variational
principle is ill-defined.

To understand whether $\tilde{\boldsymbol{X}}_{0}$ or
$\tilde{\boldsymbol{X}}_{1}$ is the correct choice of parameters, we consider
expanding around these solutions. To this end, we represent the matrix (73) in
the basis
$\Bqty{\hat{\boldsymbol{e}}_{1},\pqty{\hat{\boldsymbol{e}}_{2}+\hat{\boldsymbol{e}}_{3}}/\sqrt{2},\pqty{\hat{\boldsymbol{e}}_{2}-\hat{\boldsymbol{e}}_{3}}/\sqrt{2},\hat{\boldsymbol{e}}}_{4}$,
where $\hat{\boldsymbol{e}}_{i}$ is the $i$’th unit vector. This leads to the
modified linear systems of equations

$\boldsymbol{M}^{\prime}\cdot\boldsymbol{X}^{\prime}=\boldsymbol{b}^{\prime}\,,$
(74)

with $\boldsymbol{X}^{\prime}=(y_{0},\chi,\eta,z_{0})$,
$\boldsymbol{b}^{\prime}=\pqty{g,-A,A,0}$, and

$\boldsymbol{M}^{\prime}=\begin{pmatrix}2\lambda^{2}+2g^{2}+4A^{2}&3Ag&3Ag&-2A\omega\\\
3Ag&2\Omega_{+}^{2}+G^{2}&G^{2}&-g\omega-g\Omega_{+}\\\
3Ag&G^{2}&2\Omega_{-}^{2}+G^{2}&-g\omega-g\Omega_{-}\\\
-2A\omega&-g\omega-g\Omega_{+}&-g\omega-g\Omega_{-}&\omega^{2}+\lambda^{2}\end{pmatrix}\,,$
(75)

where we defined $\Omega_{\pm}=\lambda\pm\omega$ and $G^{2}=g^{2}+A^{2}/2$.
Note that, by definition of the one-photon regime $\omega\approx\lambda$, such
that $\Omega_{-}=(\lambda-\omega)$ neither scales with $\omega$ nor $\lambda$.
Thus, the $M^{\prime}_{\eta\eta}$ index in Eq. (75) is a finite, non-
diverging, contribution even if $\lambda,\omega\to\infty$ is considered.
Therefore, to get rid of the diverging energy scales when taking the infinite
frequency limit $\lambda\approx\omega\to\infty$ we should choose the re-
scaling
$\boldsymbol{X}^{\prime}\to\tilde{\boldsymbol{X}}^{\prime}=\pqty{y_{0}\omega,\chi\omega,\eta,z_{0}\omega}$.
Performing this re-scaling and taking the large frequency limit in Eq. (75) we
eventually arrive at

$\begin{pmatrix}2&0&0&0\\\ 0&2\frac{\Omega_{+}^{2}}{\omega^{2}}&0&0\\\
0&0&2\Omega_{-}^{2}+G^{2}&-g\\\
0&0&-g&1\end{pmatrix}\cdot\tilde{\boldsymbol{X}}^{\prime}=\begin{pmatrix}0\\\
0\\\ A\\\ 0\end{pmatrix}\,,$

which is solved by

$\displaystyle\boldsymbol{X}^{\prime}$
$\displaystyle=\pqty{0,0,\eta,\frac{g}{\omega}\eta}^{T}\,,$
$\displaystyle\eta$ $\displaystyle=\frac{2A}{4(\lambda-\omega)^{2}+A^{2}}\,.$

Transforming back to the original basis leads to

$\displaystyle y_{0}$ $\displaystyle=0$ $\displaystyle x_{1}$
$\displaystyle=-\frac{2A}{4(\lambda-\omega)^{2}+A^{2}}$ $\displaystyle y_{1}$
$\displaystyle=+\frac{2A}{4(\lambda-\omega)^{2}+A^{2}}$ $\displaystyle z_{1}$
$\displaystyle=0+O(\omega^{-1})\,,$

which is exactly Eq. (22) in the main text up to $O(\omega^{-1})$ corrections.

###  F.2  Floquet Topological Pump Experiment

####  F.2.1  Derivation of the Bloch Hamiltonian

In this section, for completeness, we derive the Bloch Hamiltonian, Eq. (24),
from the real space lab frame Hamiltonian.

Following Ref. [75], the fermionic Hamiltonian in the lab frame and real space
is described by

$H_{\mathrm{lab}}(t)=\frac{\hat{p}^{2}}{2m}+V(\hat{x}-x_{0}(t))\,,$ (76)

with potential energy $V(x)=V_{0}\cos^{2}(\pi x/a)$ and lattice constant $a$.
The lattice position is driven by the two-tone frequency drive

$x_{0}(t)=\frac{c_{\mathrm{exp}}}{\omega}\pqty{K_{1}\cos(\phi)+\frac{1}{2}K_{2}\cos(2\phi+\varphi)}\,,$
(77)

with phase shift $\varphi$ and time-dependent phase $\phi=\omega t$.
Transforming to the co-moving frame of the shaken lattice, the Hamiltonian
reads

$\hat{H}(t)=\frac{\hat{p}^{2}}{2m}+V(\hat{x})-F(t)\hat{x}\,,$ (78)

with inertial force $F(t)=-m\ddot{x}_{0}(t)$.

Using the tight-binding picture, the second quantized operators read