# Photon pumping in a weakly-driven quantum cavity-spin system

Christina Psaroudaki  Department of Physics and Institute for Quantum
Information and Matter, California Institute of Technology, Pasadena, CA
91125, USA  Institute for Theoretical Physics, University of Cologne, D-50937
Cologne, Germany  Gil Refael  Department of Physics and Institute for Quantum
Information and Matter, California Institute of Technology, Pasadena, CA
91125, USA  Walter Burke Institute for Theoretical Physics, California
Institute of Technology, Pasadena, CA 91125, USA

###### Abstract

We investigate the photon pumping effect in a topological model consisting of
a periodically driven spin-1/2 coupled to a quantum cavity mode out of the
adiabatic limit. In the strong-drive adiabatic limit, a quantized frequency
conversion of photons is expected as the temporal analog of the Hall current.
We numerically establish a novel photon pumping phenomenon in the
experimentally accessible nonadiabatic driving regime for a broad region of
the parameter space. The photon frequency conversion efficiency exhibits
strong fluctuations and high efficiency that can reach up $80\%$ of the
quantized value for commensurate frequency combinations. We link the pumping
properties to the delocalization of the corresponding Floquet states which
display multifractal behavior as the result of hybridization between localized
and delocalized sectors. Finally we demonstrate that the quantum coherence
properties of the initial state are preserved during the frequency conversion
process in both the strong and ultra-weak-drive limit.

##  I Introduction

The development of efficient frequency conversion mechanisms is a process with
various technological applications, relevant for quantum communications and
quantum computing Lauk  _et al._ (2020). Photonic channels in the IR-band
appear to be very attractive for the long-distance transmission of a quantum
state, providing low-loss transmission O’Brien  _et al._ (2009) based on
quantum-compatible storage and repeater protocols Briegel  _et al._ (1998);
Chanelière  _et al._ (2006); Duan  _et al._ (2001). On the other hand, the
elementary quantum processors and memories based on atoms Boozer  _et al._
(2007); Chanelière  _et al._ (2005); Olmschenk  _et al._ (2009), spins in
quantum dots Gerardot  _et al._ (2008), or superconducting qubits Devoret and
Schoelkopf (2013), operate in different frequency regions. Thus, the
realization of networks connecting disparate quantum systems Kimble (2008);
Tanzilli  _et al._ (2005) requires the development of quantum interfaces,
capable of bridging the frequency gap Huang and Kumar (1992); Kumar (1990);
Mirhosseini  _et al._ (2020). The bidirectional transfer of quantum
information relies on mechanisms that shift a quantum state of light from its
original frequency band to a desired one, while preserving all other quantum
properties Andrews  _et al._ (2014); Rakher  _et al._ (2010).

Figure 1: Frequency conversion platform. A single spin (purple) is coupled to
a cavity mode of frequency $\omega_{1}$ (red) and driven by an external
periodic magnetic field of frequency $\omega_{2}$ (yellow).

Temporal analogs of topological models can provide a new platform for the
efficient transfer of photons from one electromagnetic mode to another. In
such systems, novel types of topological order are realized by subjecting a
trivial quantum system to periodic driving Kitagawa  _et al._ (2010); Lindner
_et al._ (2011); Potter  _et al._ (2016); Choi  _et al._ (2017); Peng and
Refael (2018). In particular, driving a spin-1/2 with two elliptically
polarized periodic waves of incommensurate frequencies generates the dynamical
analog of a 2D topological insulator, where the Hall current corresponds to a
quantized pumping of energy between the two sources Martin  _et al._ (2017);
Crowley  _et al._ (2019). Recently, this topological frequency conversion
effect has been experimentally realized using a single-qubit IBM device Malz
and Smith (2020). Analogously, a periodically driven magnetic particle coupled
to a cavity induces a quantized energy transfer from the external field to the
cavity mode Nathan  _et al._ (2019). Such a quantized frequency conversion
occurs as long as the system is in the near-adiabatic limit realized for
sufficiently strong drives, when the instantaneous energy gap is large
compared to the photon frequencies. Nevertheless, a rough estimation of the
single-spin coupling rate to a cavity modeAbe  _et al._ (2011) suggests that
the strong-drive requirement for a quantized energy flow is experimentally
challenging.

In this work, we investigate the photon pumping effect in a topological model
consisted of a periodically driven spin-1/2 coupled to a quantum cavity mode
out of the adiabatic limit. The pumping properties of such a non-equilibrium
system are described as the delocalization of the corresponding Floquet states
in the synthetic photon-number states along the direction of the drive
frequencies. To describe this phenomenon we employ a statistical analysis of
the Floquet modes using the participation ratio approach, in analogy with
electronic transport in the context of Anderson localizationAnderson (1958);
Wegner (1980). We thus find echos of Anderson Localization in a dynamics
problem. In addition, similarly to Thouless topological charge pumping
Thouless (1983), the quantized energy transfer is not expected to remain
robust to nonadiabatic effects. Through a numerical estimation of the energy
transfer between the two modes, we establish a novel pumping phenomenon that
persists in the nonadiabatic driving regime for a broad region of the
parameter space of the model.

We find that the photon pumping efficiency in the weak-driving regime exhibits
strong fluctuations. For $\eta\leq\omega_{i}$, with $\eta$ the driving
amplitude and $\omega_{i}$ the lowest frequency, the strong fluctuations
emerge due to the nonadiabatic driving conditions, before the pumping rate
saturates to the quantized value for sufficiently strong drives
$\eta\approx\eta_{\scalebox{0.6}{ad}}$, when adiabaticity is restored. These
fluctuations are stronger for rationally-related frequencies, with the
conversion efficiency of the order of $80\%$ of the quantized value for
$\eta\approx 30\%\omega_{1}$, while in the same limit irrationally-related
frequencies have a vanishing conversion efficiency. In the nonadiabatic limit
and around the topological boundary, we demonstrate that a finite pumping
phenomenon persists in the trivial phase, but is less effective as we move
further away from the boundary. This energy pumping is characterized by a
strong sensitivity to the initial phase of the drive, although we can always
specify extended regions of the parameter space with efficiency that exceeds
$80\%$ of the quantized value.

A finite energy pumping signals the delocalization of the corresponding
quasienergy states in the space of photon-number in the cavity. To link the
pumping properties of the model to the delocalization of the Floquet states,
we follow a statistical analysis of the Floquet modes, closely related to
electronic transport in the context of Anderson localization Anderson (1958);
Wegner (1980), applied here to a frequency-space tight-binding model. This is
achieved by calculating the inverse participation ratio and extract its
scaling exponent with respect to the number of photon states. Our findings
corroborate the results of the photon pumping efficiency, namely in the
nonadiabatic regime commensurate frequencies have larger delocalization
exponents and a finite transport of states in the nontopological regime.
Perhaps the most interesting finding is that the Floquet eigenstates also
display fractal behavior for any finite $\eta$ as the result of hybridization
between the localized and delocalized sectors, induced by the periodic drive.
Note that the behavior we explore here is the complement of the non-chaotic
frequenecy-locking behavior in the same system investigated in Ref. Nathan
_et al._ , 2020.

In addition, we study whether the quantum coherence properties of the initial
state are preserved during the frequency conversion process, by evaluating the
phase probability of the time-evolved state. During the adiabatic pumping,
quantum coherence is preserved and the phase distribution is well-approximated
by a Gaussian curve, although the phase undergoes a diffusion process.
Surprisingly, in the ultra-weak-drive limit $\eta\ll\omega$, the pumped photon
state is described by a central Gaussian peak, suggesting the existence of a
state with a well defined phase. As $\eta$ is increased, additional secondary
peaks and phase fluctuations are generated and quantum coherence is
suppressed.

The structure of the paper is as follows. In Sec. II we discuss the temporal
topological models in the strong-drive regime and evaluate the quantized
energy pumping for different model parameters. In Sec. III we calculate the
energy flow in the weak-drive regime and identify regimes of efficient
frequency conversion by exploring a broad parameter space. Such an energy flow
is understood in terms of the delocalization of Floquet states, presented in
Sec. IV, while the coherence properties of the pumped state are investigated
in Sec. V. The experimental implications of our work are discussed in Sec. VI.
Finally, our main conclusions are summarized in Sec. VII.

##  II Quantized Energy Transfer

The purpose of this section is to present the topological properties of an
externally driven spin, coupled to a dynamical quantum mechanical cavity mode
and discuss the quantized conversion of energy in the topologically nontrivial
parameter regime. The quantum system is described by a Hamiltonian of the
form,

$\displaystyle\mathcal{H}(t)=\omega_{1}\hat{n}-\eta\hat{\mathbf{B}}\cdot\hat{\boldsymbol{\sigma}}\,,$
(1)

where $\hat{\boldsymbol{\sigma}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ and
$\hat{\mathbf{B}}=(\hat{B}_{x},\hat{B}_{y},\hat{B}_{z})$, with
$\hat{B}_{x}=B_{d}\sin(\omega_{2}t+\phi)$,
$\hat{B}_{y}=B_{0}(\hat{a}-\hat{a}^{\dagger})/2i$ and
$\hat{B}_{z}=B_{m}-B_{d}\cos(\omega_{2}t+\phi)-B_{0}(\hat{a}+\hat{a}^{\dagger})/2$.
Here $\hat{a}$ denotes the photon annihilation operator of the cavity mode
with frequency $\omega_{1}$ and amplitude $B_{0}$,
$\hat{n}\equiv\hat{a}^{\dagger}\hat{a}$ is the photon number operator, $B_{m}$
represents a static field along the $z$ direction and $B_{d}$ is the amplitude
of the external periodic field of frequency $\omega_{2}$ and phase $\phi$. We
also use $\hbar=1$ throughout.

A rough estimation of the single-spin coupling rate to a cavity mode is given
by $\eta_{0}=\eta B_{0}=g\mu_{B}\sqrt{\mu_{0}\omega_{c}/2\hbar V_{c}}$, where
$V_{c}$ is the cavity mode volume and $\omega_{c}$ the cavity frequency,
further enhanced to $\eta_{s}=\eta_{0}\sqrt{N_{s}}$ by utilizing an ensemble
of $N_{s}$ noninteracting spins Abe  _et al._ (2011). For a microwave cavity
mode with $\omega_{c}=10$ GHz, a small $V_{c}=10^{-12}$ m3 gives
$\eta_{0}=143$ Hz or $\eta=0.32$ GHz for $N_{s}=5\times 10^{12}$ spins. The
single-spin cavity strength has been experimentally measured using a X-band
resonator to $\eta_{0}=0.38$ Hz and $\eta=37.4$ MHz for $V_{c}=2\times
10^{-10}$ m3 and $N_{s}=7.8\times 10^{14}$ Abe  _et al._ (2011), suggesting
that the strong-drive requirement for a quantized energy flow,
$\eta\gg\omega_{c}$, is experimentally challenging.

The model of Eq. (1) is regarded as a one dimensional semi-infinite tight-
binding model, with $n$, the number of photons, being the lattice site. The
quantized transfer of energy for incommensurate frequencies occurs in the
near-adiabatic limit, when the frequencies $\omega_{i}$ are smaller than the
instantaneous energy. Thus when $\eta\mbox{min}|B_{c}-B_{\pm}|\gg\omega_{i}$,
with $B_{\pm}=|B_{m}\pm B_{d}|$ and $B_{c}=B_{0}\sqrt{n}$ the effective
amplitude of the electromagnetic mode in a cavity on $n$ photons.
Additionally, the system is in its topological regime when
$B_{c}\in[B_{-},B_{+}]$. When the spin is aligned with the instantaneous
field, and under the requirements specified above, the topological effect
emerges as an increase in the photon number at the quantized rate

$\displaystyle n_{Q}=\frac{\omega_{2}C}{2\pi}\,,$ (2)

together with an energy transfer at the rate
$dE/dt=\omega_{1}\omega_{2}C/2\pi$ with $E=\omega_{1}\langle n\rangle$. We
note that although the system can be chosen initially to be in its topological
regime, since $B_{c}=B_{0}\sqrt{n}$ (with $n$ here a dynamical quantity), we
expect that at later times the condition $B_{c}\in[B_{-},B_{+}]$ breaks down.
Thus, an increase of the photon number with a quantized rate can take place as
long as $n\in[B_{-}^{2}/B_{0},B_{+}^{2}/B_{0}]$.

Figure 2: a) Frequency conversion efficiency $\bar{n}_{F}$ in the strong-drive
regime $\eta=2$, between commensurate frequencies $\omega_{1}/\omega_{2}=2/3$,
as a function of the amplitude of the external drive $B_{d}$ and the Zeeman
field $B_{m}$. We choose $N_{\mbox{\tiny ph}}=1$, $B_{0}=1$, $\omega_{2}=1$,
and $\phi=0$. b)-(c) Time evolution of $\langle n(t)\rangle$ for
$B_{m}=8=B_{d}$ and b) incommensurate frequencies
$\omega_{1}/\omega_{2}=(\sqrt{5}-1)/2$ with unity efficiency $\bar{n}_{F}=1$,
and c) commensurate frequencies $\omega_{1}/\omega_{2}=2/3$ and efficiency
$\bar{n}_{F}\approx 1$. In both cases, the number of photon in the cavity
increases at the quantized rate $n_{Q}$ depicted with the black solid line
from the value $N_{\mbox{\tiny ph}}=1$ until it reaches the topological
boundary $(B_{m}+B_{d})^{2}/B_{0}=256$ illustrated with black dashed line at
approximately time $t/T=(B_{m}+B_{d})^{2}/B_{0}-N_{\mbox{\tiny ph}}=255$.
Figure 3: Frequency conversion efficiency $\bar{n}_{F}$ as a function of the
effective drive amplitude $\eta_{\mbox{\tiny eff}}=\eta\sqrt{N_{\mbox{\tiny
ph}}}$, for a cavity with initially $N_{\mbox{\tiny ph}}=20$ photons, and a
choice of $B_{0}=1$, $B_{m}=8=B_{d}$, $\phi=0$, and
$\omega_{1}/\omega_{2}=(\sqrt{5}-1)/2$ (blue line) or
$\omega_{1}/\omega_{2}=2/3$ (red line). Due to the nonadiabatic effects, the
efficiency fluctuates before it converges to unity for sufficiently strong
drives $\eta_{\mbox{\tiny ad}}\approx 10$. Fluctuations are stronger for
commensurate frequencies, with an efficient pumping effect $\bar{n}_{F}\approx
0.8$ in the deep nonadiabatic regime with $\eta_{\mbox{\tiny eff}}$ as small
as $3\%\times\eta_{\mbox{\tiny ad}}$.

To address the frequency conversion in the weak-drive or commensurate
frequency case, where no quantization should be expected, we numerically
calculateWeinberg and Bukov (2017) the expectation value of the photon number
operator as $\langle\hat{n}(t)\rangle=\langle\Psi(t)|\hat{n}|\Psi(t)\rangle$,
with $|\Psi(t)\rangle$ the time-evolved state
$|\Psi(t)\rangle=U(t)|\Psi_{0}\rangle$ and
$U(t)=\mathcal{T}\exp[-i\int_{0}^{t}\mathcal{H}(t^{\prime})dt^{\prime}]$ the
time evolution operator Weinberg and Bukov (2017). The Hamiltonian of Eq. (1)
assumes a quantized light-matter interaction with eigenstates of the form
$|n,s\rangle$, where $n$ is the photon and $s$ the spin index. At time $t=0$,
the system is prepared such that the spin is in its ground state
$|\downarrow\rangle$, aligned with the instantaneous magnetic field, and the
photon mode in a coherent state with mean number of photons
$N_{\scalebox{0.6}{ph}}$,
$|\Psi_{0}\rangle=|N_{\scalebox{0.6}{ph}},\downarrow\rangle$. With this
condition, $\langle\hat{n}(t)\rangle$ is an increasing function over some
finite amount of time, while the initial condition
$|\Psi_{0}\rangle=|N_{\scalebox{0.6}{ph}},\uparrow\rangle$ leads to a
decreasing photon number (not considered here).

To evaluate the efficiency of the frequency conversion we use the quantized
photon increase rate $n_{Q}$ of Eq. (2) as a reference, and study the time-
averaged photon number expectation value,

$\displaystyle\bar{n}(t_{\scalebox{0.6}{max}})=\frac{2}{n_{Q}t_{\scalebox{0.6}{max}}^{2}}\int_{0}^{t_{\scalebox{0.6}{max}}}dt~{}(\langle\hat{n}(t)\rangle-
N_{\scalebox{0.6}{ph}})\,,$ (3)

where we used $\langle\hat{n}(t=0)\rangle=N_{\scalebox{0.6}{ph}}$. Further, we
focus on the quantity $\bar{n}_{F}=\bar{n}(t_{\scalebox{0.6}{top}})$, where
$t_{\scalebox{0.6}{top}}=TB_{+}^{2}/B_{0}-N_{\scalebox{0.6}{ph}}$ and
$T=2\pi/\omega_{2}$ the period of the drive. Under this choice, we study the
study the frequency conversion effect for
$\langle\hat{n}(t)\rangle\in[B_{-}^{2}/B_{0},B_{+}^{2}/B_{0}]$. Thus, for unit
efficiency $\bar{n}_{F}=1$, the number of photons in the cavity increases from
the value $N_{\scalebox{0.6}{ph}}$ up to
$N_{\scalebox{0.6}{ph}}+B_{+}^{2}/B_{0}$, at the quantized rate $n_{Q}$, a
process that takes place within a $t_{\scalebox{0.6}{top}}$ amount of time.

The main features of the quantized frequency conversion are depicted in Fig.
2, plotted in the strong-drive regime $\eta=2$ and a choice of
$N_{\scalebox{0.6}{ph}}=1$, $B_{0}=1$, $\omega_{2}=1$ and $\phi=0$. Figs. 2-b)
and c) suggest that in this limit, the number of photons in the cavity
increases at the quantized rate $n_{Q}$, from the initial value
$N_{\scalebox{0.6}{ph}}$ up to the upper topological boundary
$B_{+}^{2}/B_{0}$ at time $t_{\scalebox{0.6}{top}}$. Choosing $B_{m}=8=B_{d}$,
we find that incommensurate frequencies $\omega_{1}/\omega_{2}=(\sqrt{5}-1)/2$
exhibit unit efficiency $\bar{n}_{F}=1$, while commensurate frequencies
$\omega_{1}/\omega_{2}=2/3$ can be as efficient with $\bar{n}_{F}\approx 1$.
Finally, the colored surface of Fig. 2-a) represents the conversion efficiency
$\bar{n}_{F}$-dependence on parameters $B_{m}$ and $B_{d}$ for the
commensurate case $\omega_{1}/\omega_{2}=2/3$. White solid lines indicate the
boundaries of the topological regime. Simple inspection reveals that there is
a broad parameter range beyond the topological regime for which $\bar{n}_{F}$
reaches and even exceeds the unit efficiency.

Below we demonstrate that within this parameter region and for the
experimentally accessible weak-drive limit, it is possible to transfer photons
from the external mode to the cavity mode, starting from the quantum
mechanical few-photon limit, with high efficiency that can reach up to
$\bar{n}_{F}=0.8$.

##  III Nonadiabatic Pumping

With this preparation, we turn to the main task of this paper and calculate
the frequency conversion efficiency in the weak-drive regime,
$\eta\leq\omega_{i}$. In analogy with the quantization of the charge transport
upon a cyclic adiabatic driving of a band insulating system, known as Thouless
topological pumping Thouless (1983), the quantized energy transfer is not
expected to remain robust in the face of nonadiabatic effects Privitera  _et
al._ (2018). In the adiabatic limit discussed in Sec. II-B, the non-quantized
pumping power for commensurate frequencies is understood in terms of a partial
sampling of the Berry curvature, contrary to the incommensurate case with a
robust quantized pumping proportional to $C=1$. Thus, for an efficient pumping
beyond the topological regime, it appears promising to tune the photon
frequencies to a rational ratio. At time $t=0$, the system is in the state
$|\Psi_{0}=|N_{\scalebox{0.6}{ph}},\downarrow\rangle$.

Figure 4: Frequency conversion efficiency $\bar{n}_{\mbox{\tiny max}}$ around
the topological transition (blue line), and approaching irrationally-related
frequencies (red line). In both cases we use $N_{\mbox{\tiny ph}}=20$,
$B_{0}=1$ and $B_{d}=8$. The blue line depicts the conversion efficiency as we
vary $B_{m}$ from the topological to the trivial phase for
$\omega_{1}/\omega_{2}=2/3$ and a topological phase boundary at $B_{m}=12.5$.
The red line depicts the conversion efficiency for $B_{m}=8$ and rationally-
related frequencies of the form $\omega_{1}/\omega_{2}=q/p$, with increasing
$\sqrt{p^{2}+q^{2}}$. Figure 5: The colored surface represents the conversion
efficiency $\bar{n}_{F}$ as a function of the initial phase of the external
periodic drive $\phi$ and the drive amplitude $\eta$, with $B_{0}=1$,
$\omega_{1}/\omega_{2}=2/3$, $N_{\mbox{\tiny ph}}=100$ and $B_{m}=20=B_{d}$.
We observe that $\bar{n}_{F}$ is highly sensitive with respect to both $\eta$
and $\phi$.

In Fig. 3 we plot the frequency conversion efficiency $\bar{n}_{F}$ as a
function of the effective drive amplitude
$\eta_{\scalebox{0.6}{eff}}=\eta\sqrt{N_{\mbox{\tiny ph}}}$, for a cavity with
initially $N_{\scalebox{0.6}{ph}}=20$ photons in the topological regime with
$B_{0}=1$ and $B_{m}=8=B_{d}$. We examine both rational
$\omega_{1}/\omega_{2}=2/3$ (red line) and irrational
$\omega_{1}/\omega_{2}=(\sqrt{5}-1)/2$ frequency combinations in a broad
region of the driving amplitude including the nonadiabatic (NA)
$\eta_{\scalebox{0.6}{eff}}\leq\omega_{i}$ (red shaded area), the intermediate
(IM)
$\omega_{i}\lessapprox\eta_{\scalebox{0.6}{eff}}\leq\eta_{\scalebox{0.6}{ad}}$
(blue shaded area), and the adiabatic regime (AD) (gray shaded are) for
$\eta_{\scalebox{0.6}{eff}}\geq\eta_{\scalebox{0.6}{ad}}$. Fig. 3 allows for
two major observations. First, for both frequency combinations, due to the
nonadiabatic effects, $\bar{n}_{F}$ strongly fluctuates before it converges to
unity for sufficiently strong drives $\eta_{\scalebox{0.6}{ad}}\approx 10$. In
addition, in the nonadiabatic regime depicted in detail in the inset of Fig.
3, fluctuations are stronger for rational frequencies, with efficient pumping
effects of the order of $\bar{n}_{F}\approx 0.8$ at
$\eta_{\scalebox{0.6}{eff}}$ that could be as small as $30\%\omega_{1}$. For
intermediate amplitudes $2\lessapprox\eta_{\scalebox{0.6}{eff}}\lessapprox 6$,
thus for $\eta_{\scalebox{0.6}{eff}}\approx
20-60\%\times\eta_{\scalebox{0.6}{ad}}$, there is still a finite photon
pumping $0.5\lessapprox\bar{n}_{F}\lessapprox 0.8$, a behavior that is
independent of the frequency combination. To justify the choice of the field
amplitude $B_{m}=8=B_{d}$, we note for the parameters summarized in Fig. 2-3,
the frequency conversion effect takes place within a sufficiently long time
interval $t_{\scalebox{0.6}{top}}=1071$, during which the cavity photon number
is increased by $\Delta
n=\langle\hat{n}(t_{\scalebox{0.6}{top}})\rangle-\langle\hat{n}(0)\rangle=256$
photons. In physical units, $t_{\scalebox{0.6}{top}}=1~{}\mu$s for
$\omega_{1}=1$ GHz, while both $B_{m}$ and $B_{d}$ are in the mT regime.

In the nonadiabatic regime $\eta_{\scalebox{0.6}{eff}}\leq\omega_{i}$,
irrational frequencies display vanishing efficiency $\bar{n}_{F}=0$. Such a
striking contrast of the nonadiabatic pumping between commensurate and
incommensurate frequencies is understood as follows. The energy pumping in the
adiabatic regime is generated by engineering synthetic dimensions, where each
lattice dimension equals the number of irrationally-related drive frequencies.
When the two frequencies have a rational ratio, $\omega_{1}/\omega_{2}=q/p$
with $p,q$ coprime integers, the two-dimensional lattice is compactified into
a cylinder of circumference $\sqrt{p^{2}+q^{2}}$ Martin  _et al._ (2017).
Thus, in the limit of $p,q\gg 1$, one expects the pumping to be insensitive to
the details of the frequency ratio and obtain its universal value which
derives from the underlying topology. In turn, the topological effect cannot
apply at weak drive. This is supported by the results summarized in Fig. 4
(red line), where we plot $\bar{n}_{\scalebox{0.6}{max}}$ in the nonadiabatic
regime for $N_{\scalebox{0.6}{ph}}=20$, $B_{0}=1$, $B_{m}=B_{d}=8$, and
various frequency combinations. Here $\bar{n}_{\scalebox{0.6}{max}}$ is
defined as the maximum value of $\bar{n}_{F}$ in the nonadiabatic regime,
$\bar{n}_{\scalebox{0.6}{max}}=\mbox{max}\\{\bar{n}_{F}(\eta_{\scalebox{0.6}{eff}}):\eta_{\scalebox{0.6}{eff}}\in[0,1]\\}$.
It becomes apparent that as $p,q$ increase, the nonadiabatic pumping effect
vanishes. The full fluctuating behavior of
$\bar{n}_{F}(\eta_{\scalebox{0.6}{eff}})$ is presented in Fig. 12.

Figure 6: Inverse participation ratio exponent $\alpha_{2}$ as a function of
the driving amplitude $\eta$ for $\omega_{1}/\omega_{2}=2/3$ and $B_{0}=1$. We
use three different parameter combinations, two in the topological regime
$B_{m}=10=B_{d}$ (red line) and $B_{m}=4=B_{d}$ (blue line), and one in the
trivial regime $B_{m}=4$, $B_{d}=10$. The inset depicts $\alpha_{2}(\eta)$ for
$B_{0}=1$, $B_{m}=10=B_{d}$, and $\omega_{1}/\omega_{2}=2/3$ (red line) or
$\omega_{1}/\omega_{2}=2/(\sqrt{5}-1)$ (blue line).

Let us now focus on the conversion efficiency around the topological
transition, as illustrated in Fig. 4 (blue line). We present
$\bar{n}_{\scalebox{0.6}{max}}$ as a function of $B_{m}$ for $B_{0}=1$,
$B_{d}=8$, $\omega_{1}/\omega_{2}=2/3$, and a topological phase boundary at
$B_{m}=12.47$. The overall picture suggested is that a finite $\bar{n}_{F}$
persists in the nontopological regime but with intensity that vanishes as we
go further away from the topological boundary. The full fluctuating behavior
of $\bar{n}_{F}(\eta_{\scalebox{0.6}{eff}})$ is presented in Fig. 13.

To complete the description we must also examine the sensitivity of
$\bar{n}_{F}$ to the initial phase of the external periodic drive $\phi$, due
to dephasing of the time evolved state $|\Psi(t)\rangle$. The main features
are depicted in Fig. 5, where we plot $\bar{n}_{F}$ as a function of both
$\eta$ and $\phi$ and a choice of $B_{0}=1$, $\omega_{1}/\omega_{2}=2/3$,
$N_{\scalebox{0.6}{ph}}=100$ and $B_{m}=20=B_{d}$. We observe that although
$\bar{n}_{F}$ is highly sensitive with respect to both $\eta$ and $\phi$, we
can specify extended regions of the parameter space with efficient pumping
$\bar{n}_{F}\geq 0.8$. This sensitivity on $\phi$ is a remnant of an analogous
$\phi$-dependence of $\bar{n}_{F}$ in the adiabatic regime, where the pumping
effect is proportional to the integral of the Berry curvature along a path
selected by $\phi$, further enhanced here by the nonadiabatic driving
conditions. The sensitivity of $\bar{n}_{F}$ on the initial conditions
motivates studies on the statistical behavior of the pumping effect for an
ensemble of random Hamiltonians. The distribution of the energy pumping
efficiency along with the Floquet level statistics is explored in a subsequent
publication Psaroudaki and Refael, 2021.

##  IV Floquet Localization

A finite energy pumping in a frequency-space tight-binding model signals the
delocalization of the corresponding quasienergy states along the direction of
the drive frequencies, that now play the role of an effective electric field.
In this section we link the pumping properties of the quantum cavity-spin
model to the delocalization properties of its Floquet modes. In the case of a
simple periodic drive with frequency $\omega_{2}$, the time evolution of an
arbitrary initial state has the formShirley (1965)

$\displaystyle|\Psi(t)\rangle=\sum_{j}c_{j}e^{-i\varepsilon_{j}t}|u_{j}(t)\rangle\,,$
(4)

where $\varepsilon_{j}$ are the quasienergies and $c_{j}$ are time-independent
coefficients. The Floquet states $|u_{j}(t)\rangle$ diagonalize the single-
period time evolution operator
$U(T)=\mathcal{T}\exp[-i\int_{0}^{T}\mathcal{H}(t^{\prime})dt^{\prime}]$,
where $T=2\pi/\omega_{2}$ is the period of the drive.

Figure 7: Inverse participation ratio exponent $\alpha_{2}$ for $B_{0}=1$,
$B_{m}=10=B_{d}$, $\eta=0.16$, $\omega_{2}=1$ and various frequency
combinations $\omega_{1}$. We note that $\alpha_{2}$ is maximized for
commensurate frequencies.

Based on an expansion of the zero time Floquet states in the photon basis
$|u_{j}(0)\rangle=\sum_{n}b_{j}^{n}|n\rangle$, the generalized inverse
participation ratio (IPR) is defined as Romito  _et al._ (2018)

$\displaystyle\mathcal{R}_{q}=\frac{1}{N}\sum_{j,n=0}^{N-1}|b_{j}^{n}|^{2q}=\frac{1}{N^{\alpha_{q}}}\,,$
(5)

where $N$ is both the number of the Floquet modes and the number of photons in
the cavity, while for convenience we first focus on $\mathcal{R}_{2}$. If a
Floquet state is delocalized among many photon states, then each of them
contributes $b_{j}^{n}\approx 1/\sqrt{N}$ and $\mathcal{R}_{2}\approx 1/N$ for
$N\gg 1$ ($\alpha_{2}=1$). In the opposite limit, a localized Floquet state in
the photon lattice will give $\mathcal{R}_{2}=1$ ($\alpha_{2}=0$). For a given
set of parameters and driving amplitude $\eta$, we calculate $\mathcal{R}_{2}$
for $N\in[10,200]$ and extract the exponent $\alpha_{2}$ by a numerical fit.

In Fig. 6 we plot the exponent $\alpha_{2}$ for the model (1) as a function of
the driving amplitude $\eta$ and examine its behavior as the system
transitions from the topological to the trivial regime. In the topological
class, Floquet states are delocalized for any finite $\eta$ with $\alpha_{2}$
exhibiting strong fluctuations in the $0\leq\eta\leq 0.2$ regime around
$\alpha_{2}\approx 0.8$, before it asymptotically approaches
$\alpha\rightarrow 1$ for larger $\eta$. In the trivial class and for large
$\eta$, $\alpha_{2}\rightarrow\alpha_{\scalebox{0.6}{as}}$ with
$\alpha_{\scalebox{0.6}{as}}<1$, indicating a hybridization between the
localized and delocalized sectors. The inset depicts the delocalization of
Floquet states in the topological regime $B_{0}=1$ and $B_{m}=10=B_{d}$ for
commensurate $\omega_{1}/\omega_{2}=2/3$ (red line) and incommensurate
$\omega_{1}/\omega_{2}=2/(\sqrt{5}-1)$ (blue line) frequency combinations. As
expected, in the nonadiabatic regime commensurate frequencies have larger
delocalization exponents, that is translated into large photon pumping in the
cavity. This is further supported by the results summarized in Fig. 7 for
$B_{0}=1$, $B_{m}=10=B_{d}$, $\eta=0.16$ and various frequency combinations
$\omega_{1}$. We note that $\alpha_{2}$ is significantly enhanced for
commensurate frequency combinations, corroborating the results of Sec. III.

Figure 8: Inverse participation ratio exponent $\alpha_{q}$ as a function of
the fractal dimension $q$ for $B_{0}=1$, $B_{m}=10=B_{d}$,
$\omega_{1}/\omega_{2}=2/3$ and three driving amplitudes $\eta=0.01,0.1$, and
$0.5$. The black dashed line denotes the expected $\alpha=q-1$ behavior for
delocalized states.

Perhaps the most interesting finding is that in all cases the scaling exponent
of $\mathcal{R}_{2}$ reflects a hybridization between the localized and
delocalized sectors, that is a sign of Floquet multifractality. The presence
of multifractal states is associated with the critical spectral statistics of
a system exactly on the metal- insulator transition point Hofstadter (1976) or
the Anderson transition point Evers and Mirlin (2008); Mirlin  _et al._
(1996). To analyze the multifractal behavior of the Floquet eigenstates of the
driven quantum-cavity system, we examine the generalized IPR of Eq. (5) as a
function of the fractal dimension $q$. For localized states it holds
$\alpha_{q}=0$, for delocalized $\alpha_{q}=q-1$, and any other combination
signals multifractionality. The overall picture suggested by Fig. 8, where we
plot $\alpha_{q}$ for three values of the drive $\eta=0.01,0.1,0.5$, is that
multifractality of the Floquet states is present for any $\eta>0$ and Floquet
states become increasingly delocalized with increasing $\eta$. This surprising
result motivates further studies on the criticality of the Floquet eigenstate
statistics Wang and Gong (2009) but is beyond the scope of the present paper
and we leave it for the future.

##  V Quantum Coherence

In this section we study the coherent properties of the pumped time-evolved
photon state $|\Psi(t)\rangle$ to examine whether the phase coherence of the
initial state is preserved during the frequency conversion process, a
necessary condition for a quantum information transfer Tanzilli  _et al._
(2005). In the context of quantum optics, a number of probability
distributions can be employed to investigate the properties of quantum states.
Here we focus on the Pegg-Barnett phase probability distribution based on a
construction of a Hermitian phase operator to study whether outgoing states
have a well defined phase Pegg and Barnett (1989). We introduce a complete set
of orthonormal phase states Pegg and Barnett (1989)

$\displaystyle|\theta_{m}\rangle=\frac{1}{N}\sum_{n,s}e^{in\theta_{m}}|n,s\rangle\,,$
(6)

where $\theta_{m}=2\pi m/N$, $m=0,\cdots,N-1$, $|n,s\rangle$ are eigenstates
of the coupled photon-spin system and $N$ is the total number of photon
states. The phase probability distribution
$P(\theta)=|\langle\theta|\Psi\rangle|^{2}$ contains important information on
the phase properties of a general state $|\Psi\rangle$. Here we initially
prepare the system at a coherent photon state $N_{\scalebox{0.6}{ph}}$ and the
spin aligned with the magnetic field,
$|\Psi_{0}\rangle=|N_{\scalebox{0.6}{ph}},\downarrow\rangle$ and let it evolve
under the driven spin-cavity Hamiltonian $\mathcal{H}(t)$ of Eq. (1).

Figure 9: Quantum coherence properties of the pumped photon state in the
adiabatic limit $\eta=10/\sqrt{N_{\mbox{\tiny ph}}}$. a) Time evolution of the
phase distribution $P(\theta)$ for $B_{0}=1$, $B_{m}=8=B_{d}$ (topological
phase), $\omega_{1}/\omega_{2}=2/3$, and $N_{\mbox{\tiny ph}}=20$. $P(\theta)$
follows a Gaussian distribution with a standard deviation $\sigma_{\theta}$ an
increasing function over time. b), c) Snapshots of the phase distribution
$P(\theta)$ at several time incidents between $t=0$ and $t=100$. d) Time
evolution of $\langle n(t)\rangle$ with $\bar{n}_{F}=1$. e) Time evolution of
$\sigma_{\theta}$, well approximated by
$\sigma_{\theta}(t)=(c_{0}+c_{1}t/T)/\sqrt{\langle\hat{n}(t)\rangle}$, with
$c_{0}=0.65$ and $c_{1}=0.068$. Figure 10: Quantum coherence properties of the
pumped photon state in the nonadiabatic limit $\eta=0.3/\sqrt{N_{\mbox{\tiny
ph}}}$. a) Time evolution of the phase distribution $P(\theta)$ for $B_{0}=1$,
$B_{m}=8=B_{d}$ (topological phase), $\omega_{1}/\omega_{2}=2/3$, and
$N_{\mbox{\tiny ph}}=20$. b), c) Snapshots of the phase distribution
$P(\theta)$ at several time incidents between $t=0$ and $t=100$. d) Time
evolution of $\langle n(t)\rangle$ with $\bar{n}_{F}\approx 0.8$. e) Time
evolution of $\sigma_{\theta}$ of the central Gaussiann peak.

The quantum coherence properties of the pumped photon state $|\Psi(t)\rangle$
in the adiabatic limit are summarized in Fig. 9, for parameters in the
topological phase $B_{0}=1$, $B_{m}=8=B_{d}$, $\omega_{2}=1$, commensurate
frequencies $\omega_{1}/\omega_{2}=2/3$, and strong drive
$\eta=10/\sqrt{N_{\scalebox{0.6}{ph}}}$. Panel a) depicts the time evolution
of the Pegg-Barnett phase distribution $P(\theta)$ for
$N_{\scalebox{0.6}{ph}}=20$. As expected quantum coherence is maintained and
$P(\theta)$ is described by a Gaussian peak at all times [see Fig. 9-b) and
c)], although the standard deviation $\sigma_{\theta}$ of the distribution
increases over time [see Fig. 9-e)]. The photon number $\langle n(t)\rangle$
increases over time at the quantized rate $n_{Q}$ [see Fig. 9-d)]. The
standard deviation $\sigma_{\theta}$ is well approximated by
$\sigma_{\theta}(t)=(c_{0}+c_{1}t/T)/\sqrt{\langle n(t)\rangle}$ with
$c_{0}=0.65$, $c_{1}=0.068$, and $T=2\pi/\omega_{2}$.

The broadening of the Gaussian curve suggests that the phase of the pumped
state undergoes a diffusion process, evident in the time evolution of both the
mean amplitude $\langle\hat{a}(t)\rangle$ and $\langle
e^{i\hat{\Phi}_{\theta}}(t)\rangle$. Here $\hat{a}$ is the photon annihilation
operator and
$\hat{\Phi}_{\theta}=\sum_{m}\theta_{m}|\theta_{m}\rangle\langle\theta_{m}|$
is the phase operator. In Fig. 11 we present
$\langle\hat{a}(t)\rangle/\langle\hat{n}(t)\rangle$ (upper panel) and extract
the rate of amplitude decay as $\Gamma_{n}=1.7\times 10^{-3}$, a direct
measure of the phase diffusion rate Walls and Milburn (1995). The lower panel
depicts $|\langle
e^{i\hat{\Phi}_{\theta}}(t)\rangle|=e^{-\frac{1}{2}\sigma^{2}_{\theta}(t)}\approx
e^{-\Gamma_{\theta}t}$ with $\Gamma_{\theta}=c_{1}^{2}/2T=3.66\times 10^{-4}$.
Thus, the outgoing pumped state is described by a diffusion coefficient
$\Gamma_{\theta}\approx 3.66~{}\times 10^{-4}\times\omega_{1}$, a quantity
that is equivalent to a single-mode laser linewidth. Typically, narrow-band
sources are preferable for quantum information applications as they provide
access to atomic transitions.

The quantum coherence properties of the pumped photon state $|\Psi(t)\rangle$
in the nonadiabatic limit are summarized in Fig. 10, for parameters in the
topological phase $B_{0}=1$, $B_{m}=8=B_{d}$, commensurate frequencies
$\omega_{1}/\omega_{2}=2/3$, and ultra-weak drive
$\eta=0.3/\sqrt{N_{\scalebox{0.6}{ph}}}$. Panel a) depicts the time evolution
of the Pegg-Barnett phase distribution $P(\theta)$ for
$N_{\scalebox{0.6}{ph}}=20$. $P(\theta)$ is still described by a central
Gaussian peak [see Fig. 9 b) and c)], although additional peaks develop due to
the nonadiabatic driving conditions. Thus, the standard deviation
$\sigma_{\theta}$, plotted in Fig. 10-e), increases/decreases over time due to
the appearance/disappearance of secondary peaks. For all cases plotted, the
photon number in the cavity increases over time with frequency conversion
efficiency $\bar{n}_{F}\approx 0.8$. As we further increase $\eta$ additional
phase fluctuations are generated and phase coherence is suppressed. A summary
of this behavior is depicted in Fig. 14 of the Appendix, where we also
illustrate how phase coherence is restored as the system approaches the
strong-drive adiabatic limit.

In summary, the frequency conversion in the adiabatic limit preserves the
quantum coherence properties of the initial state, although the state is
characterized by a diffusion coefficient. For nonadiabatic driving conditions
and ultra-low drive, $P(\theta)$ is still described by a central Gaussian
peak, suggesting that the pumped quantum state has a well defined phase. As we
further increase the driving amplitude, additional secondary peaks and phase
fluctuations are generated and quantum coherence is suppressed.

Figure 11: Time evolution of the mean amplitude
$\langle\hat{a}(t)\rangle/\langle\hat{n}(t)\rangle$ (upper panel) and $\langle
e^{i\hat{\Phi}_{\theta}}(t)\rangle$ (lower panel) for
$\eta=10/\sqrt{N_{\mbox{\tiny ph}}}$, $B_{0}=1$, $B_{m}=8=B_{d}$,
$\omega_{1}/\omega_{2}=2/3$, and $N_{\mbox{\tiny ph}}=20$. The rate of the
amplitude decay is $\Gamma_{n}=0.011$ for the former and
$\Gamma_{\theta}=0.0023$ for the later. We note that on top of the exponential
behavior, $\langle\hat{a}(t)\rangle/\langle\hat{n}(t)\rangle$ oscillates at a
frequency $\omega_{1}=2/3$.

##  VI Experimental Implications

Here we discuss whether the nonadiabatic frequency conversion discussed in the
previous section is experimentally feasible. To give an estimate in physical
units under the choice of parameters summarized in Fig. 3, thus for $B_{0}=1$
and $B_{d}=8=B_{m}$, we first note that the adiabatic requirement holds as
long as
$\eta\geq\hbar\omega_{1}\eta_{\scalebox{0.6}{ad}}/\sqrt{N_{\scalebox{0.6}{ph}}}\approx
10\hbar\omega_{1}/\sqrt{N_{\scalebox{0.6}{ph}}}$. Therefore, in the few-photon
limit, starting with a state of $N_{\scalebox{0.6}{ph}}=20$ coherent photons,
the adiabatic requirement is met for a spin-photon coupling of the order of
$\eta/h\omega_{1}=0.18$ at $B_{d}=B_{m}=100$ mT, and grows to
$\eta/h\omega_{1}=0.8$ in the fully quantum limit with
$N_{\scalebox{0.6}{ph}}=1$ and $B_{d}=B_{m}=450$ mT. In the opposite many-
photons limit, starting with a cavity of $N_{\scalebox{0.6}{ph}}=200$ photons,
the adiabatic limit corresponds to $\eta/h\omega_{1}=0.06$ at $32$ mT, and for
$N_{\scalebox{0.6}{ph}}=400$ we find $\eta/h\omega_{1}=0.04$ at $23$ mT. On
the contrary, a nonadiabatic pumping is possible for even smaller values of
the spin-photon coupling of approximately
$\eta=0.3\hbar\omega_{1}/\sqrt{N_{\scalebox{0.6}{ph}}}$. This translates to
$\eta/h\omega_{1}=5.3\times 10^{-3}$ for $N_{\scalebox{0.6}{ph}}=20$ at $3$
mT, and $\eta/h\omega_{1}=2.4\times 10^{-2}$ for $N_{\scalebox{0.6}{ph}}=1$ at
$14$ mT.

The direct interaction of a single spin with the cavity magnetic field is
exceedingly small, with $\eta_{0}/h\omega_{1}=10^{-9}$. A generalization is to
consider of an ensemble of $N_{s}$ identical two-level systems resonantly
interacting with a single electromagnetic field, a model analyzed by Tavis and
Cummings Tavis and Cummings (1968). In the scope of this model, the collective
spin-photon coupling is $\eta=\sqrt{N_{s}}\eta_{0}$, where $\eta_{0}$ is the
coupling strength of each individual spin, a scaling that has been
experimentally verified, while a similar $\sqrt{N_{s}}$ behavior prevails also
for the magnon-photon coupling in magnetically interacting systemsHuebl  _et
al._ (2013); Tabuchi  _et al._ (2014); Zhang  _et al._ (2014); Maier-Flaig
_et al._ (2016); Kostylev  _et al._ (2016). The coupling of large electron
spin ensembles to microwave cavity photons has been experimentally found to be
of the order of $\eta/h\omega_{1}=5\times 10^{-4}$ ($\omega_{1}=9.7$ GHz) for
a large ensemble of $N_{s}=10^{16}$ spins in a 3D microwave cavityAbe  _et
al._ (2011), and of the order of $\eta/h\omega_{1}=4\times 10^{-3}$ for
$N_{s}=10^{12}$ spins in a superconducting cavityKubo  _et al._ (2010);
Schuster  _et al._ (2010) ($\omega_{1}=2.87$ GHz at $40$ mK).

Several approaches have been explored in order to enhance the spin-photon
coupling. Recently, it has been demonstrated that by reducing the cavity
effective volume at superconducting nanoconstrictions, the microwave magnetic
field is enhanced and results strong spin-photon couplings of the order of
$\eta/h\omega=1.4\times 10^{-3}$ for $N_{s}=10^{8}$ spins in a microwave
cavity with an average of $5\times 10^{5}$ thermal photonsGimeno  _et al._
(2020) ($\omega_{1}=1.4$ GHz at $44$ mK). A different approach is to consider
a number of exchange-coupled spins, where the coupling strength is reported to
be at least one order of magnitude larger as compared to noninteracting spins,
i.e. $\eta/h\omega_{1}=7.6\times 10^{-2}$ Huebl  _et al._ (2013)
($\omega_{1}=5.6$ GHz at $50$ mK). In these systems, microwave cavity photons
interact coherently with the collective spin excitations in ferromagnetic
crystals Kostylev  _et al._ (2016), such as the ferromagnetic insulator
yttrium iron garnet (YIG). A strong coupling is achieved when the cavity and
the magnetostatic mode are on resonance, leading to $\eta/h\omega_{1}\approx
3\times 10^{-3}$ Tabuchi  _et al._ (2014) ($\omega_{1}=10.6$ GHz at $10$ mK).
The resonance condition justifies the single photon mode assumption in the
model of Eq. 1, as higher-energy modes are off-resonant and can be neglected.
This is further supported by the results of Ref. Nathan  _et al._ , 2019 in a
classical multi-mode cavity, where the authors demonstrated that the energies
of higher-energy modes, effectively suppressed by tuning their dissipation
rates, decay fast to zero. In view of the increasing interest on the coherent
interaction between magnons and microwave or optical photons, we believe our
work could serve as a basis for studies on photon frequency conversion in a
system of $N_{s}$ interacting spins, also motivated by recent results on the
enhancement of frequency conversion in a system of two interacting spins
Körber  _et al._ (2020).

Nevertheless, in the above studies the spin-phonon coupling has been explored
away from the quantum single-spin to single-photon limit. A promising platform
for the realization of the Tavis-Cummings model is the use of two-state atoms
coupled to a resonant cavity mode with a large reported single-qubit to
single-photon coupling of the order of $\eta_{0}/h\omega_{1}=1.3\times
10^{-2}$ Fink  _et al._ (2009)$\omega_{1}=6.7$ GHz at $20$ mK). The coupling
of single electron spins in silicon quantum dots to single microwave photons
has been reported to be of the order of $\eta_{0}/h\omega_{1}=2-7\times
10^{-3}$, with an average thermal photon number in the resonator well below
1Mi  _et al._ (2017, 2018); Landig  _et al._ (2018) ($\omega_{1}=7.7$ GHz at
$10-30$ mK). Finally, at room temperature, the magnon to microwave photon
coupling is of the order of $\eta/h\omega_{1}=1.4-3.3\times 10^{-3}$ Zhang
_et al._ (2014); Maier-Flaig  _et al._ (2016) ($\omega_{1}=7.9$ GHz), while
the spin-photon coupling in a molecular crystal is $\eta/h\omega_{1}=1.2\times
10^{-3}$ Breeze  _et al._ (2017) ($\omega_{1}=1.45$ GHz). From the above
considerations it becomes apparent that the predicted frequency conversion
effect in the nonadiabatic limit is experimentally feasible using a large
ensemble of spins and is within experimental reach for the quantum few-photon
limit.

##  VII Discussion

In this work, we consider the transfer of energy in a periodically driven
spin-1/2 coupled to a quantum cavity mode out of the adiabatic limit. We
establish a novel pumping phenomenon that persists in the weak-drive regime
and examine its efficiency for a broad range of the parameter space. We
demonstrate that the frequency conversion efficiency exhibits strong
fluctuations due to the nonadiabatic effects before it saturates to unity for
sufficiently strong drives, $\eta_{\scalebox{0.6}{ad}}\approx 10\omega_{i}$.
It is more efficient for rationally-related frequencies and can reach up to
$80\%$ of the quantized value for $\eta=0.3\omega_{i}$. Emphasis is put on the
magnitude of the pumping effect in the nontopological phase of the model. A
finite frequency conversion efficiency persists in the trivial phase, but is
less effective as we move further away from the topological boundary.

A finite-energy pumping signals the delocalization of the corresponding
Floquet states, which are found to display a multifractal behavior, due to the
hybridization between localized and delocalized sectors. The presence of
multifractal states, associated with critical spectral statistics, motivates
future studies on the criticality of the Floquet eigenstate statistics.
Finally, we show that in the adiabatic limit, during the frequency conversion
process, the quantum coherent properties of the initial state are preserved,
but the phase undergoes a diffusion process. For ultra-low drive, the pumped
quantum state has still a well defined phase but as we further increase the
driving amplitude, additional secondary peaks and phase fluctuations are
generated and quantum coherence is suppressed.

Experimental systems are subject to dissipation mechanisms that need to be
incorporated for the construction of a realistic model (although we don’t
expect a qualitative change of our results). In the adiabatic driving limit
and within a Markov-Lindblad framework, the inclusion of external noise and
dissipation stabilizes the conversion effect as the system approaches a steady
state with a quantized number of emitted photons per driving period Nathan
_et al._ (2019). In the nonadiabatic limit, it was recently demonstrated that
the introduction of a non-Hermitian tailored time-periodic dissipation
restores the topological transport quantization of Thouless pumps in plasmonic
waveguide arrays, emphasizing the uniqueness of Floquet topological systems
Fedorova  _et al._ (2020).

###### Acknowledgements.

We are grateful to Ivar Martin and Frederik Nathan for useful discussions.
C.P. has received funding from the European Union’s Horizon 2020 research and
innovation program under the Marie Sklodowska-Curie grant agreement No 839004.
We are also grateful to the U.S. Department of Energy, Office of Science,
Basic Energy Sciences under Award de-sc0019166. GR is also grateful to the NSF
DMR grant number 1839271. NSF and DOE supported GR’s time commitment to the
project in equal shares. NSF provided partial support to C.P. This work was
performed in part at Aspen Center for Physics, which is supported by National
Science Foundation grant PHY-1607611. G.R. is also grateful for support from
the Simons Foundation and the Packard Foundation.

Figure 12: Frequency conversion efficiency $\bar{n}_{F}$ as a function of the
effective drive amplitude $\eta_{\mbox{\tiny eff}}=\eta\sqrt{N_{\mbox{\tiny
ph}}}$, for a cavity with initially $N_{\mbox{\tiny ph}}=20$ photons in the
topological class $B_{0}=1$, and $B_{m}=8=B_{d}$. Four different commensurate
frequency combinations are presented of the form $\omega_{1}/\omega_{2}=q/p$,
with increasing $\sqrt{p^{2}+q^{2}}$. When $q,p\gg 1$, the nonadiabatic photon
pumping vanishes. Figure 13: Frequency conversion efficiency $\bar{n}_{F}$ as
a function of the effective drive amplitude $\eta_{\mbox{\tiny
eff}}=\eta\sqrt{N_{\mbox{\tiny ph}}}$, for a cavity with initially
$N_{\mbox{\tiny ph}}=20$ photons, $B_{0}=1$, $B_{d}=8$, and
$\omega_{1}/\omega_{2}=2/3$. Four different Zeeman fields are presented, two
in the topological phase, $B_{m}=10,12$ and two in the trivial phase
$B_{m}=14,16$. The topological phase boundary is at $B_{m}=12.47$. Figure 14:
Time evolution of the phase distribution $P(\theta)$ for parameters ranging
between the ultra-weak-drive limit $\eta=0.067$ up to the strong-drive limit
$\eta=2.236$. We use $B_{0}=1$, $B_{m}=8$, $B_{d}=8$, $N_{\mbox{\tiny
ph}}=20$, and $\omega_{1}/\omega_{2}=2/3$. In the ultra-weak limit
$\eta\ll\omega_{i}$ (left panel), $P(\theta)$ is described by a central
Gaussian peak, suggesting that the phase is preserved during the photon
pumping effect. As $\eta$ is increased further, additional secondary peaks and
fluctuations are generated, which suppress quantum coherence. As the system
approaches the adiabatic limit $\eta\gg\omega_{i}$ (right panel), coherence is
restored and $P(\theta)$ is well described by a single Gaussian curve. In all
depicted cases there is a finite photon pumping effect with
$0.6\leq\bar{n}_{F}\leq 1$.

##  Appendix A Appendix

In this Appendix we provide further details on the frequency conversion
effect. In Fig. 12 we plot the dependence of the frequency conversion
efficiency $\bar{n}_{F}$ on $\eta_{\mbox{\tiny eff}}$, for a cavity with
initially $N_{\mbox{\tiny ph}}=20$ photons in the topological class $B_{0}=1$,
$B_{m}=8=B_{d}$, and four different commensurate frequency combinations of the
form $\omega_{1}/\omega_{2}=q/p$. We note that as $q,p\gg 1$, the nonadiabatic
pumping effect vanishes, and the overall behavior resembles the one for
irrationally-related frequencies.

Moreover, we are interested in exploring whether the pumping effects persists
for parameters outside the topological regime. We therefore plot $\bar{n}_{F}$
as a function of $\eta_{\mbox{\tiny eff}}$ for a cavity with initially
$N_{\mbox{\tiny ph}}=20$ photons, $B_{0}=1$, $B_{d}=8$ and
$\omega_{1}/\omega_{2}=2/3$ and four different values of the Zeeman field
amplitude. The topological phase boundary is at $B_{m}=12.5$. Although the
pumping effects persists in the trivial phase of the model, the intensity
diminishes as we go further away from the boundary.

Finally, for completeness we present the phase distribution $P(\theta)$
calculated in Sec. V for parameters ranging between the ultra-weak-drive up to
the strong-drive limit. In Fig. 14 we present $P(\theta)$ for seven values of
$\eta$, $B_{0}=1$, $B_{m}=8$, $B_{d}=8$, $N_{\mbox{\tiny ph}}=20$, and
$\omega_{1}/\omega_{2}=2/3$. Phase coherence is preserved in the ultra-weak
limit $\eta\ll\omega_{i}$, and $P(\theta)$ is described by a central Gaussian
peak. As $\eta$ is increased, additional secondary peaks and phase
fluctuations are generated and the pumped states is no longer characterized by
a well defined phase. Phase coherence is restored in the strong-drive
adiabatic limit, where $P(\theta)$ is a Gaussian curve at all times.

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