text:
\section{Introduction}
In recent years, periodic driving has been explored
as a way to create desirable properties in
otherwise ordinary systems~\cite{Dalibard_2011,Eckardt_2017,Cooper_2019,Oka_2019_review,Rudner_2019_review,Harper_2019_review}.
In addition to inducing exotic phases that already exist in equilibrium~\cite{Oka_2009,Lindner_2011,Jotzu_2014}, periodic driving can also induce exotic phenomena with no equilibrium counterpart~\cite{Kitagawa2010, Jiang2011,Rudner_2013, Titum_2016, Khemani_2016, Else_2016b, Martin_2017,Nathan_2019,Crowley, Kolodrubetz_2018, Nathan_2019b, ChandranSondhi,Fidkowski,Roy,Fidkowski2,TC-potter,Else2,Keeling,Gong,sondhi1,Huang_2018,Efetov,Peer,Mathey, Yao-Demler, Ueda, AMRey}.
These predictions inspired a wide range of experiments, leading to the realization and observation of
 new drive-induced phenomena, such as Floquet time crystals, and anomalous Floquet insulators~\cite{Broome2010,Sacha_2015,Hu2015,vonKeyserlingk_2016,Choi_2017, Zhang_2017, Maczewsky2017, Mukherjee2017, Moessner_2017,Ho_2017,Russomano_2017,McIver_2019,Lukin1,Wintersperger2020}.
\begin{figure}[!ht]
\includegraphics[width=0.99\columnwidth]{Figure_1.pdf}
\caption{(a)
We study a  2-level system (orange), coupled to a single cavity mode (green) while driven periodically with  frequency $\Omega$ (blue).
(b) Quasienergy locking: while the quasienergies of the system (gray, red, and blue lines)
are effectively uniformly distributed between $0$ and $\Omega$, in certain finite parameter regimes
the spacings between certain subsets of levels (red and blue)  are exponentially close to
$\Omega /q$ for some integer $q$
(here we illustrate this clustering for $q=3$).
(c) Number of period-3 quasienergy-locked levels
in the model, obtained from numerical exact evolution and diagonalization of the system's Floquet operator, as function of cavity frequency $\omega_{\rm c}$ and qubit-cavity coupling $\eta$ (see Sec.~\ref{sec:model} for model details, Sec.~\ref{sec:numerics} for details of the simulation, {and Appendix~\ref{app:higher_mode_locking} for similar plots for other frequency locking ratios)}.
The inset shows a histogram of quasienergy level spacings in the frequency-locked regime (parameters indicated by the cross  in main panel).
}
\label{fig:Fig1}
\end{figure}

In this work, we consider another class of such
driving-induced phenomena:  quantum frequency locking~\cite{Holthaus_1994,Sacha_2015,Zhang_2017,Sacha_2017,Huang_2018,Giergiel_2018,Pizzi_2019,Matus_2019}.
Quantum frequency locking arises when a quantum system with an intrinsic characteristic frequency $\omega_{\rm c}$ is driven at a frequency $\Omega = 2\pi/T$ close to (but not necessarily equal to) a rational multiple $q/r$ of the intrinsic frequency $\omega_{\rm c}$.
In this case, the system can  respond by robust oscillations with period locked {\it exactly}  to an integer  multiple of the driving period,  $q T$.

Here we propose a new and accessible realization of quantum frequency locking. Namely, we consider a periodically-driven qubit coupled to a quantized electromagnetic cavity (Fig.~\ref{fig:Fig1}a).
When the qubit is driven close to resonance with a rational multiple  of the cavity's resonance frequency, the cavity mode oscillates with frequency  locked to $r\Omega/q$.
This phenomenon has a wide range of interesting implications and uses, which we explore in this paper:
in particular,  it implies the formation of characteristic  subsets of quasienergy levels which are separated by
$\Omega /q$, up to exponentially small corrections (see sketch in Fig.~\ref{fig:Fig1}b)~\cite{Holthaus_1994,Zhang_2017}; we term this related phenomenon ``quasienergy locking.''
Quantum frequency locking   is, moreover, a robust effect, which does not require  fine-tuning~\cite{frequency_tuning}. It persists both for weak and strong qubit-cavity coupling, and for finite ranges of the driving frequency  (see Fig.~\ref{fig:Fig1}c).

Quantum frequency locking has previously been considered in various platforms and
settings, including ultracold atoms interacting with a vibrating mirror~\cite{Sacha_2015,Matus_2019}, spin chains or clock-models~\cite{Russomano_2017,Surace_2019,Pizzi_2019}, parametrically driven chains of electromagnetic cavities \cite{huang2020non}, and  nonlinear oscillators~\cite{Holthaus_1994,Guo_2013,Zhang_2017}.
Our
proposal provides a new and complimentary realization that can capitalize on recent advances in control of  few-level quantum systems in Rydberg atoms, quantum dots, and superconducting qubits interacting with microwave cavities~\cite{Gross_2017,Burkard_2020,Clerk_2020}.
In this way our results provide a direct path for realizing quantum frequency locking on readily-available experimental platforms.
In addition to being of fundamental interest, the robust coherent oscillations of the cavity mode can moreover
be used as a frequency converter to generate a coherent signal at a frequency different from the drive (by a factor $r/q$).

Quantum frequency locking has a well-established classical counterpart, known as Arnold Tongues~\cite{Rasband_1990,strogatz, Peer, Armour, Gabrielse}.
Quantum mechanics, however, introduces several new aspects to this effect:
the wave-packets of a  system do not spread in phase space when observed stroboscopically~\cite{Buchleitner_2002, huang2020non}.
Moreover,  as explained above, the robust period-multiplied oscillations implies a nontrivial ordering of the quasienergy spectrum in the system (see Fig.~\ref{fig:Fig1}b).
The Wigner functions of the corresponding Floquet eigenstates  exhibit a remarkably rich structure (see Fig.~\ref{fig:contours}cd below).

The nontrivial organization of the quasienergy spectrum mentioned above is a signature of the breakdown of discrete time-translation symmetry~\cite{Holthaus_1994,Sacha_2015,Bruder,Zhang_2017}.
Thus frequency-locked quantum systems also present examples of ``Floquet time-crystals''~\cite{Khemani_2016,Else_2016b},
and
demonstrate how time-translation symmetry breaking (in a broader sense than defined in Ref.~\onlinecite{Else_2016b}) may be realized in few-body quantum systems [see also Refs.~\onlinecite{Sacha_2015,Zhang_2017}]; see Sec.~\ref{sec:quasienergy_locking} for further discussion.

In what follows we introduce the qubit-cavity model we study
in Sec.~\ref{sec:model}.
In Sec.~\ref{sec:theory}, we perform an approximate semiclassical analysis of the model  to describe how   quantum frequency locking  emerges, and identify the conditions under which it occurs.
We discuss the implications of quantum frequency locking for the quasienergy spectrum of the system in \ref{sec:phase_locking}, before confirming our approach numerically in Sec.~\ref{sec:numerics}.
Here we also discuss how quantum frequency locking may be utilized for frequency conversion (Sec.~\ref{sec:signatures}).
We conclude with a discussion in Sec.~\ref{sec:discussion}.
Technical details of our analysis are provided in the Appendices.

\section{model}
\label{sec:model}
The system we consider consists of a two-level system, such as a qubit, quantum dot, or a spin-1/2 magnetic moment, coupled linearly to a quantized  electromagnetic cavity mode, and to a periodic drive (see  Fig.~\ref{fig:Fig1}a).
Without loss of generality, we  refer to the two-level system simply as a spin below.

The Hamiltonian of the  system is given by
\begin{equation}
\hat H(t) = \hat H_{\rm c}
 +\hat  H_{\rm s}(t).
\label{eq:hamiltonian_1}
\end{equation}
Here $\hat H_{\rm c}$ and $\hat H_{\rm s}(t)$ denote the  Hamiltonian  of the cavity and spin, respectively, defined such that  $\hat H_{\rm s}(t)$      includes the spin-cavity coupling;  this term [and hence also $\hat H(t)$] depends on time $t$.
The cavity Hamiltonian is simply given by $\hat H_{\rm c} = \omega_{\rm c} \hat b^\dagger \hat b$, where $\omega_{\rm c}$ and $\hat b$ denote  the frequency and bosonic annihilation operator of the cavity mode, respectively (here and below, we work in units where $\hbar = 1$).
The Hamiltonian of the spin, $\hat H_{\rm s}(t)$, consists of three parts: a static (Zeeman) part, $\hat H_0$,   a term coupling the spin to a  time-dependent  driving field, $\hat V_{\rm dr}(t)$, and a term  coupling the spin
to the cavity field, $\hat H_{\rm sc}$:
\begin{equation}
\hat H_{\rm s}(t) = \hat H_0 +\hat  V_{\rm dr}(t) +\hat  H_{\rm sc}.
\end{equation}
The drive encoded in $\hat V_{\rm dr}(t)$ has $T$-periodic time-dependence  (angular frequency $\Omega \equiv{2\pi}/{T}$): $\hat V_{\rm dr}(t)= \hat V_{\rm dr}(t+T)$.

We do not expect frequency-locking to depend on the specific details of $\hat H_0$, $\hat V_{\rm dr}(t)$ and $\hat H_{\rm sc}$. For concreteness, however, we use the forms:
\begin{eqnarray}
\hat H_0 &=& \eta \hat \sigma_x B_0,\\
\hat V_{\rm dr} (t)&=& \eta A_d [\sin(\Omega t)\hat  \sigma_x + \cos(\Omega t)\hat \sigma_z],\\
\hat H_{\rm sc} &=& \eta ( \hat b \hat \sigma^+ + \hat b^\dagger\hat  \sigma^-)\label{eq:j_sc_def}.
\end{eqnarray}
Here   $\eta$ parametrizes the spin's coupling to
the external (Zeeman and driving) fields and to the cavity field, $\hat \sigma_x,\hat \sigma_y,\hat \sigma_z$   denote the Pauli matrices acting on the  spin, with $\hat \sigma^{\pm} \equiv \frac{1}{2} (\hat \sigma_x \pm i \hat \sigma_y)$, and $B_0$ and $A_d$ are dimensionless numbers denoting  the effective Zeeman field strength and driving amplitude, respectively. This model was shown to support topological frequency conversion  in Refs.~\onlinecite{Martin_2017,Nathan_2019, Crowley}.

The  cavity mode is described by a harmonic oscillator, and can thus
 conveniently be represented using the dimensionless position and momentum operators \cite{x_p_normalization}  $ \hat x \equiv \frac{1}{2} (\hat b+\hat b^\dagger)$ and $ \hat p  \equiv\frac{1}{2i}(\hat b-\hat b^\dagger)$.
In terms of these  operators,  the Hamiltonian of the full system is given by
\begin{equation}
\hat H(t) = \frac{\omega_c}{2}(\hat  x^2+ \hat p^2) + \eta \vec b(\hat  x ,\hat  p ,t)\cdot \vec{\hat  S},
\label{eq:Ht}
\end{equation}
where  $\vec{\hat S} = (\hat \sigma_x,\hat \sigma_y,\hat \sigma_z)$ denotes the
{effective} spin operator {describing the qubit}, while
\begin{equation}
\vec b(x,p,t)= (B_0-A_d\sin\Omega t -x,p,A_d\cos \Omega t )
\label{eq:h_def}
\end{equation}
can be seen as the effective Zeeman  field acting on the spin, as a function of the cavity mode's position and momentum, and  time.

As mentioned in the introduction, the model above can be realized in various ways. Most appealing perhaps are realizations using superconducting qubits~\cite{manucharyan2009fluxonium, PhysRevX.9.041041, Clerk_2020} and nitrogen-vacancy (NV) centers~\cite{nv, PhysRevLett.113.197601},  as well as atoms in optical cavities (see, e.g., Refs.~\onlinecite{Schleier-Smith,Lev-Cavity}).
We expect that our following discussion generalizes to cavities with multiple modes~\cite{Nathan_2019}.

\section{Semiclassical picture of Frequency locking}
\label{sec:theory}

When the  driving field of the in model of  Sec.~\ref{sec:model} has an off-resonant frequency, $\Omega$, close   to a rational multiple $q/r$ of the cavity eigenfrequency $\omega_{\rm c}$  (where $q$ and $r$ are integers), there exists finite regions of phase space where  the  cavity mode responds to the drive with coherent oscillations whose frequency is  locked {\it exactly} to $r\Omega/q$.
We refer to this phenomenon as {quantum frequency locking}.
In this section, we demonstrate from heuristic semiclassical arguments how quantum frequency locking arises in the model.
We confirm our approach using numerical simulations in  Sec.~\ref{sec:numerics}.

Our first step towards deriving frequency locking
is to transform the cavity mode's degrees of freedom, $(\hat x,\hat p)$, to a  frame
rotating with frequency $\tilde \Omega \equiv r\Omega/q$ (as in previous studies of quantum frequency locking; see, e.g., Refs.~\cite{Guo_2013,Zhang_2017}).
In this rotating frame, the cavity mode becomes much slower than the driving period and the spin's dynamics.
In Sec.~\ref{sec:effective_hamiltonian}, we identify conditions under which this separation of time scales allows us to effectively integrate out the spin and the driving field.
This
results in a time-independent semiclassical  effective  Hamiltonian that governs the evolution of the cavity mode in the rotating frame:
\begin{equation}
\mathcal H
{\rm eff}
(x,p) = \frac{\delta \omega}{2} \left(x^2+p^2\right) + \varepsilon (x,p),
\label{eq:heff}
\end{equation}
where $x$ and $p$ denote the semiclassical position and momentum variables of the cavity mode (in the rotating frame), and $\delta \omega\equiv \omega_c- \tilde \Omega$ is the cavity detuning from $\tilde \Omega$.
The potential $\varepsilon (x,p)$ in Eq. (\ref{eq:heff}) results from  integrating out the spin and the driving field, and plays a central role in our analysis.
We  identify two distinct parameter regimes where the above separation of timescale occurs, namely,   the large-$\eta$ {\it adiabatic} regime, and the small-$\eta$ {\it Floquet} regime (see Sec.~\ref{sec:effective_hamiltonian} for discussion of these regimes).
Both regimes support quantum frequency locking, but result    in two distinct expressions for the potential $\varepsilon (x,p)$.

Frequency locking can naturally be understood  by
inspecting the effective Hamiltonian $\mathcal H_{\rm eff}(x,p)$:  in Fig.~\ref{fig:contours}ab we plot $\mathcal H_{\rm eff}(x,p)$
for the two regimes  where  quantum frequency locking occurs  with period $3$.
In both cases $\mathcal H_{\rm eff}(x,p)$ has local extrema at nonzero values of $x$ and $p$.
When the  cavity mode is initialized at one of these extrema, its phase space  location  in the rotating frame  remains stationary.
As a result, in the original ``lab'' frame, the cavity mode will  oscillate with frequency $\Omega/3$.

The picture above also explains the robustness of  frequency locking.
 Even if the cavity is initialized near (but not precisely at) the extremum of $\mathcal H_{\rm eff}$, its location in the rotating frame will remain confined near the extremum at all times.
Thus,  frequency locking can be achieved  with only moderate requirements for control over initial conditions; for instance, in Fig.~\ref{fig:contours}, it will occur with significant probability as long as the displacement amplitude of the cavity mode, $\sqrt{x^2+p^2}$, is less than $30$ {in the dimensionless units we have adopted [see text above  Eq.~\eqref{eq:Ht}].
The corresponding motion in the lab frame must remain close to this point each time three driving periods have passed. As a result, the frequency spectrum of the cavity mode's motion features a sharp, well-defined peak at $\Omega/3$.
The extrema moreover cannot be removed by weak perturbations, implying that frequency locking persists in finite parameter ranges.

In Appendix~\ref{app:photon_lattice}  we present a different, complementary perspective on quantum frequency locking, based on the dynamics in the combined Fock space of the oscillator and driving field.
The   approach presented there can in principle be  used to study quantum frequency locking for any driven finite-dimensional quantum system coupled to a quantized cavity mode, in the limit of small nonlinearity and detuning.

\begin{figure}
\includegraphics[width=0.99\columnwidth]{Figure3_v1.png}
\caption{(a) and (b): Constant (quasi)energy contours of the effective  Hamiltonian for the cavity mode in the frame where phase space rotates with frequency $\Omega/3$, $\mathcal H_{\rm eff}(x,p)$, [see Eq.~\eqref{eq:heff}],
Panels (a) and (b) depict two parameter sets where period-$3$ quantum frequency locking occurs, within the adiabatic and Floquet regimes, respectively; see Sec.~\ref{sec:effective_hamiltonian} for parameters.
(c) and (d): Wigner functions of quasienergy-locked Floquet eigenstates, using the same parameters as in panels (a) and (b), respectively.
The contours of the corresponding effective Hamiltonians from panels (a) and (b) shown as grey lines.
{\it Inset, panel} (c):  Wigner function for  an equal-weight linear combination of the $3$ related   Floquet eigenstates of which one is depicted in (c) (see Sec.~\ref{sec:phase_locking}). }
\label{fig:contours}
\end{figure}

\subsection{Effective cavity mode  Hamiltonian}
\label{sec:effective_hamiltonian}
We now obtain the effective Hamiltonian in Eq.~\eqref{eq:heff}.
To this end,  we consider the dynamics of the system in the rotating frame that was described above.
The Hamiltonian of the system in this rotating frame is given by $\tilde H(t) = \hat U^\dagger_0(t) \big[\hat H(t)- \tilde \Omega \hat b^\dagger\hat  b\big] \hat U_0(t) ,
\label{eq:rf_transf}
$
where the unitary operator $\hat U_0(t)  \equiv e^{-i \tilde \Omega \hat b^\dagger\hat  b t}$   generates the transformation to the rotating frame:
 the Schr\"odinger equation in the lab frame is solved by $|\psi(t)\rangle = \hat U_0(t)|\tilde \psi(t)\rangle$, where $\partial _t |\tilde \psi(t)\rangle =  -i \tilde H(t)|\tilde \psi(t)\rangle$.
Noting that $\hat U_0(t)$ only acts nontrivially on $\hat H_{\rm sc}$, we find
\begin{equation}
\tilde H(t)   = \frac{\delta \omega}{2} (\hat  x ^2 + \hat  p ^2) + \eta \vec{h}(\hat  x,\hat  p,t) \cdot \vec{\hat S},
\label{eq:rotating_frame_hamiltonian}
\end{equation}
where $\vec h$ is obtained from $\vec b$ in Eqs.~(\ref{eq:Ht}) and (\ref{eq:h_def}) after rotating the oscillator phase space  by $\tilde \Omega t$:
$\vec h ( x, p,t)= \vec b\big(x \cos\tilde \Omega t +   p \sin\tilde \Omega t, p \cos\tilde \Omega t -   x \sin\tilde \Omega t,t\big)$.
Note that the  Hamiltonian $\tilde H(t)$ in Eq.~\eqref{eq:rotating_frame_hamiltonian} describes a periodically driven system with {extended} period $\tilde T = qT$, through the explicit time-dependence of $\vec h(x,p,t)$ above (recall that $\tilde{\Omega} = r\Omega/q$).

To derive the effective semiclassical Hamiltonian for the cavity mode,
$\mathcal H_{\rm eff}(x,p)$, we consider the  equations of motion generated by $\tilde H(t)$ for the Heisenberg picture operators  $\hat x(t)$,
$\hat p(t)$, and $\vec{\hat S}(t)$:
\begin{align}
\partial _t \hat x(t)&= \delta \omega \, \hat p(t)+ \eta \vec v_p(t)
\cdot \vec{\hat S},
\label{eq:x_eom}\\
\partial _t \hat p(t) &=  -\delta \omega\, \hat x(t) - \eta  \vec v_x(t) \cdot \vec{\hat S} ,
\label{eq:p_eom}\\
\partial _t{\vec{\hat S}}(t) &=  \eta \vec h[\hat x(t),\hat p(t),t] \times \vec {\hat S}.
\label{eq:s_eom}
\end{align}
where $\vec v_x(t)$ and $\vec v_p(t)$ are  vectors with unit norm:
$\vec v_x(t)\equiv (-\cos\tilde\Omega t,\sin\tilde \Omega t,0)$,  $\vec v_p(t)\equiv (\sin{\tilde{\Omega}} t,\cos {\tilde{\Omega}} t,0)$.
We apply a semiclassical approximation to the equations of motion above, by assuming that the cavity mode's location in phase space is relatively well-defined at all times:
in Eq.~\eqref{eq:s_eom}, we  approximate $\vec h(\hat x(t) , \hat p(t) , t ) \approx \vec h(x(t),p(t),t)$, where $x(t)\equiv \langle \hat x(t)\rangle$, and $p(t)\equiv \langle \hat p(t)\rangle$ denote the expectation values of the position and momentum operators.
We expect this
approximation to be justified when the characteristic  scales in phase space of
variations in $\vec h(x,p,t)$ are   large compared to the scale of quantum fluctuations $\Delta x,\Delta p \sim 1$.

The approximation above reduces  Eq.~\eqref{eq:s_eom} to a Bloch-equation with a time-dependent field $\vec h(x(t),p(t),t)$.
By moreover taking the expectation values on both sides of Eqs.~\eqref{eq:x_eom}-\eqref{eq:s_eom},  we then obtain three coupled  equations of motion for the (semi)classical variables $x(t),p(t),$ and $\vec S(t)\equiv \langle \vec S(t)\rangle$: namely Eqs.~\eqref{eq:x_eom}-\eqref{eq:s_eom} with the operators $\hat x(t)$, $\hat p(t) $, and $\vec{\hat S}(t)$  replaced by their expectation values  $ x(t)$, $p(t) $, and $\vec{ S}(t)$.
These equations of motion are generated by the time-dependent classical Hamiltonian $\tilde {\mathcal H}(t)$, given by
\begin{equation}
\tilde{\mathcal H}(t)   = \frac{\delta \omega}{2} (  x ^2 +   p ^2) + \eta \vec{h}(  x,  p,t) \cdot \vec{ S}.
\label{eq:classical_rotating_frame_hamiltonian}
\end{equation}
Note that the dynamics of the cavity mode, $(x(t),p(t))$, has characteristic frequencies $\delta \omega, \eta$, which can be much smaller than $\tilde\Omega$ (which is
on the same order as the resonance frequency of the oscillator) and the characteristic frequencies of the spin's dynamics.
Below, we identify two parameter regimes where this separation of timescales allows us to effectively eliminate the spin, $\vec S(t)$, and obtain the static effective Hamiltonian for the cavity mode in Eq.~\eqref{eq:heff}.

\subsubsection{Adiabatic regime}
\label{sec:adiabatic}
The simplest ``adiabatic'' regime occurs for large $\eta$. In this regime  the direction of the instantaneous Zeeman field, $\vec h(x(t),p(t),t)$, changes adiabatically with respect to the (fast) Larmor precession of the spin, which has frequency $\sim\eta |\vec h(x,p,t)|$ [see Eq.~\eqref{eq:classical_rotating_frame_hamiltonian}].
In this case, the equations of motion [Eq.~\eqref{eq:s_eom} with $\hat x, \hat p$, and $\vec{\hat S}$ substituted by their expectation values $x$ and $p$] has the two distinct solutions: $\vec S(t) \approx \pm \vec h(x,p,t)/|\vec h(x,p,t)|$~\cite{spin_initialization}.
With these solutions for $\vec S$, Eq.~\eqref{eq:classical_rotating_frame_hamiltonian} becomes
\begin{equation}
\tilde{\mathcal{H}}^{\pm}_{\rm cav}(x,p,t) = \frac{\delta\omega }{2} (x^2 + p^2)  \pm \eta  |\vec h(x,p,t)|.\label{heffa}
\end{equation}

Similarly to quantum systems, the stroboscopic time-evolution generated by
$\tilde {\mathcal H}^{\pm}_{\rm cav}(x,p,t)$   (i.e.,  time-evolution at integer multiples of $\tilde T$) is equivalent to that generated by some time-independent effective classical Hamiltonian~\cite{Oteo_1991}.
When the cavity mode in the rotating frame oscillates  slowly compared to
$\tilde \Omega$ (see below for more detailed conditions),
this effective Hamiltonian
is well-approximated  by the time-average of $\tilde{\mathcal H}_{\rm cav}^{\pm}(x,p,t)$; i.e., by $\mathcal H_{\rm eff}(x,p)$ in Eq.~\eqref{eq:heff}, with
\begin{equation}
\varepsilon (x,p) =  \pm \frac{\eta}{{\tilde{T}}} \int_0^{{\tilde{T}}}\!\!\! {\rm d}t\, |\vec h(x,p,t)|,
\label{eq:heff_expression_adiabatic}
\end{equation}
where the sign ($\pm$) depends on the initial alignment of the spin~\cite{spin_initialization}.
Eq.~\eqref{eq:heff_expression_adiabatic} can be obtained using a Magnus expansion of the evolution operator generated by the system's Liouvillian (see Ref.~\cite{Oteo_1991} for details).

The considerations above show that $\mathcal H_{\rm eff} (x,p)$  in Eqs.~\eqref{eq:heff}~and~(\ref{eq:heff_expression_adiabatic})  describes the system when the Larmor precession frequency of the spin, $\eta|\vec h(x,p,t)|$ is much larger than the driving frequency, which in turn should be much larger than the characteristic frequency of the cavity mode in the rotating frame.
The latter is given by the {\it renormalized} frequency detuning $\delta \omega'(x,p)$, given  by the radial gradient of $\mathcal H_{\rm eff} (x,p)$, divided by the amplitude of the cavity mode, $A_{\rm cav} = \sqrt{x^2 + p^2}$.
Hence the adiabatic regime arises when $\delta \omega'(x,p) \ll \Omega\ll\eta|\vec h(x,p,t)|$  (here we used that $\Omega$ and $\tilde \Omega$ have the same order of magnitude).
This condition  is satisfied in the vicinity of the extrema of $\mathcal H_{\rm eff}(x,p)$, as long as these  occur in regions of phase space where $\eta |\vec h(x,p,t)|\gg \Omega$.
As an illustration, in Fig.~\ref{fig:contours}a, we plot the constant (quasi)energy contours of $\mathcal H_{\rm eff}(x,p)$ in the adiabatic regime, obtained from direct numerical evaluation of Eqs.~\eqref{eq:heff}~and~\eqref{eq:heff_expression_adiabatic}.
We use the      parameters  $A_d= 15$, $B_0=7$, $\omega_c =0.34 \Omega$  and $\eta =0.56 \Omega $.
These parameters are indicated by the cross in Fig.~\ref{fig:Fig1}c, and fall within the adiabatic regime.
We expect $\mathcal H_{\rm eff}(x,p)$ to describe the system accurately near the three minima located at radius $A_{\rm cav}\approx 24$, where its gradient  (and hence $\delta \omega'$) vanishes.
In Sec.~\ref{sec:numerics} (see also Fig.~\ref{fig:Fig1}c),  we confirm that these local minima indeed lead to quantum frequency locking at these parameters, as explained above.

\subsubsection{Floquet regime}
\label{sec:floquet}
The Floquet parameter regime for frequency locking occurs not when the instantaneous Hamiltonian changes adiabatically, but rather when the  effective {\it Floquet}  Hamiltonian of the spin (with $x$ and $p$ held fixed) changes adiabatically.

To obtain the effective Hamiltonian $\mathcal H_{\rm eff}(x,p)$ in the Floquet regime, we consider the dynamics resulting from  Eq.~\eqref{eq:s_eom} with  $x$ and $p$ held fixed.
In this case,  the time-periodicity of $\vec h(x,p,t)$ implies that all solutions $\vec S(t)$ to   Eq.~\eqref{eq:s_eom} satisfy $\vec S([n+1]\tilde T) = R_0(x,p) \vec S(n\tilde T)$,  for some fixed three-dimensional orthogonal matrix $R_0(x,p)$ with unit determinant.
Like any orthogonal matrix with unit determinant, $R_0(x,p)$ can be expressed as a rotation about some axis $\vec a(x,p)$ by some angle $\theta(x, p)$ between $0$ and $\pi$ (note that the required interval for $\theta$  fixes the direction of $\vec a$).
As a result, for fixed $x$ and $p$, there exists a   time-periodic solution to the Bloch equation in Eq.~\eqref{eq:s_eom}  (up to a constant scale factor), $\vec{S}(t) = \vec n_0(x,p,t)$, in which $\vec n_0(x,p,0)=\vec a (x,p)$ is parallel to the net rotation axis, and $\vec n_0(x,p,t)$ evolves according to Eq.~\eqref{eq:s_eom}.
Thus, for fixed $x$ and $p$, we  identify $\mathcal H_{\rm eff}^{{\rm spin}} (x,p)= \theta (x,p) \vec a(x,p) \cdot \vec S$ as the effective Hamiltonian of the spin (see Appendix~\ref{app:floquet_locking} for further details).

When $x$ and $p$ are not fixed, but the evolution of effective precession axis $\vec a(x,p)$ (due to the motion of $x$ and $p$)  evolves slowly relative to the energy gap
of  $H_{\rm eff}^{{\rm spin}}$, $\delta \varepsilon (x,p) \equiv \min(\theta (x,p),2\pi-2\theta(x,p))/\tilde T$ (see Appendix~\ref{app:floquet_locking}), the stroboscopic motion of the spin closely follows stroboscopic motion resulting from the adiabatically changing Hamiltonian $H_{\rm eff}^{{\rm spin}}(x(t),p(t))$~\cite{Weinberg_2017}.
As a result, if initially aligned or anti-aligned with $\vec a(x(0),p(0))$, the spin's evolution at later (stroboscopic) times will satisfy $\vec S(n\tilde T) \approx \pm \vec a(x(n\tilde T),p(n\tilde T))$, where the sign depends on the initial alignment.
In Appendix~\ref{app:floquet_locking}, we substitute this solution into Eq.~\eqref{eq:classical_rotating_frame_hamiltonian} and take the time-average, making use of our assumption that the cavity mode is effectively stationary within the driving period $\tilde T$~\cite{Oteo_1991}.
Doing this, we find that  the cavity mode evolves according to the  effective Hamiltonian  in Eq.~\eqref{eq:heff}, with
\begin{equation}
\varepsilon (x,p) = \pm  \theta (x,p)/2\tilde T.
\label{eq:heff_expression_floquet}
\end{equation}
Here the sign depends on the initial alignment of the spin with
$\vec a(x,p)$~\cite{spin_initialization}.
The angle $\theta(x,p)$ can be  straightforwardly calculated for the system by exact time-evolution of the Bloch equation for the spin in Eq.~\eqref{eq:s_eom}.

The Floquet regime arises  when the dynamics of the cavity mode occur
on a much longer time-scale  than the driving period $\tilde T$, and when the change of the effective axis of rotation, $\vec a(x,p)$ (due to the motion of $x$ and $p$) is slow compared to the effective Larmor frequency $\theta(x,p)/\tilde T$.
In Appendix~\ref{app:floquet_locking}, we show that these conditions are satisfied when  $\eta \ll \theta(x,p)/\tilde T$ and $\delta\omega \ll \tilde \Omega/A_{\rm cav}$, where $A_{\rm cav} =\sqrt{x^2+p^2}$ denotes the amplitude of the cavity field.
Note that, since $\tilde T > T$, and $\theta(x,p)\leq \pi$, the Floquet regime requires $\eta \ll \Omega$.
Thus the Floquet regime arises in the limit of small spin-cavity coupling, $\eta$, and detuning, $\delta \omega$ (since our semiclassical approximation  requires $A_{\rm cav} \gg 1$ for quantum fluctuations not to play a role).

As an illustration, In Fig.~\ref{fig:contours}b, we plot the contours of $\mathcal H_{\rm eff}(x,p)$ for antialigned spin  (i.e., with $\vec S = -\vec a$) using the parameters $A_d =15, B_0 = 7, \omega_c =\Omega/3$  and $\eta =0.048\Omega $.
Since $\delta \omega =0$,  the conditions for Floquet locking outlined above imply that $\mathcal H_{\rm eff}$  accurately describes the dynamics  of the cavity mode whenever $ H_{\rm eff}(x,p) \gg \eta$.

\section{Quasienergy locking and  symmetry breaking}
\label{sec:phase_locking}
\label{sec:quasienergy_locking}
In Sec.~\ref{sec:theory}, we   identified the conditions for quantum frequency locking: namely, it arises for parameters in the adiabatic or Floquet regimes where
 the effective   Hamiltonian
 $\mathcal H_{\rm eff}(x,p)$ [Eq.~\eqref{eq:heff}] has extrema at  nonzero amplitude in phase space.
While our  treatment in Sec.~\ref{sec:theory} in principle also  applies to  fully classical systems (indeed, our results were derived using semiclassical arguments), in this section we demonstrate some novel aspects of the quantum mechanical version
 of the phenomenon. In particular, we show that

period-$q$ quantum  systems feature
  characteristic multiplets of  quasienergy levels that differ by rational fractions of the drive frequency, $\Omega/q$, up to exponentially suppressed corrections [see Eq.~\eqref{eq:qel} below].
We term this phenomenon ``quasienergy locking''.
It was also identified in earlier work~\cite{Holthaus_1994,Sacha_2015,Else_2016b,Zhang_2017}, and
can be seen as  the defining feature of quantum frequency locking.

Quasienergy locking can be seen as a breakdown of the discrete time translation symmetry which is  inherently present in the
driven
system~\cite{Sacha_2015,Else_2016b,Zhang_2017}:
 as we show below,
linear combinations of  quasienergy-locked Floquet eigenstates define a family of nearly stationary states of the system's  evolution that break the discrete time translation symmetry of the drive (up to exponentially long times).

In the remainder of this section we  review the defining features of quasienergy locking (Sec.~\ref{sec:qel_def}), and subsequently show how it arises the qubit-cavity model (Sec.~\ref{sec:qel_derivation}).
We finally discuss
 how quasienergy locking can be understood as a breakdown of discrete time translation symmetry (Sec.~\ref{sec:ttsb}).
To highlight physical aspects of the phenomenon, we provide our arguments on a heuristic level, while  a   more rigorous (but  technical) treatment  is given in Appendix.~\ref{app:qe_locking}.

\subsection{Quasienergy locking}
\label{sec:qel_def}
Quasienergy locking is a phenomenon that arises in  the quasienergy spectrum of
 periodically driven quantum systems~\cite{Holthaus_1994,Sacha_2015,Else_2016b,Zhang_2017}.
In such systems the quasienergies $\{\varepsilon _n\}$ define
the eigenvalues of the system's time-evolution operator over one period (known as the Floquet  operator),  $\hat U(T) |\psi_n \rangle = e^{-i\varepsilon _n T}|\psi_n\rangle$, where $\hat U(t) \equiv Te^{-i\int_0^t dt'\hat  H(t')}$ denotes the system's time-evolution operator and $\mathcal T$ denotes the time-ordering operation.
The corresponding  eigenstates $\{|\psi_n\rangle\}$, termed Floquet eigenstates, hence form a complete basis of states that are mapped to themselves after each driving period $T$, up to a unitary phase $e^{-i\varepsilon _nT}$.
Quasienergy thus plays a role analogous to energy for the evolution of periodically driven quantum systems:
the evolution at  integer  multiples of the driving period, $k$, can  be resolved   as
$
|\psi(kT)\rangle = \sum_n c_n e^{-i\varepsilon _n kT}|\psi_n\rangle,
$
where
the coefficients
$\{c_n\}$ are determined from the initial conditions~\cite{floquet_propagation}.
However, note that each $\varepsilon _n$ is only defined modulo $\Omega$.

The quasienergies of a generic periodically driven quantum system
 are naturally distributed uniformly  between $0$ and $\Omega$.
However,
when period-$q$ quantum frequency locking  arises, the spectrum features
 characteristic multiplets of
quasienergy levels $\varepsilon _1 ,\ldots \varepsilon _q$ that  differ by  $\Omega/q$,
  up to exponentially suppressed corrections:
\begin{equation}
\varepsilon _\ell = \varepsilon  + \ell \Omega/q +\mathcal O(\delta \varepsilon ),
\label{eq:qel}
\end{equation}
where $\varepsilon$ generally differs from multiplet to multiplet.
Here $\delta \varepsilon $ is a quasienergy scale that can be many times
 smaller than the  average quasienergy level spacing in the system, such that the feature above would not occur by coincidence  (see, e.g., the inset in Fig.~\ref{fig:Fig1}c and Sec.~\ref{sec:numerics}).
For the driven qubit-cavity system we consider in this work, we show below that
 $\delta \varepsilon \sim e^{-d/\xi}$, where $d$ denotes the separation  in phase space between the extrema of the effective Hamiltonian
 $\mathcal H_{\rm eff}(x,p)$, while $\xi\sim 1$ denotes the scale of quantum fluctuations.
We term the  formation of these multiplets ``quasienergy locking''.

The  Floquet eigenstates associated with  each multiplet of  locked quasienergy levels,  $|\psi_1\rangle,\ldots |\psi_q\rangle$, have interesting features of  their own.
Specifically, they take the form
\begin{equation}
|\psi_\ell\rangle = \frac{1}{\sqrt{q}}\sum_{k=1}^q e^{-2\pi i \ell k /q}|\chi_k\rangle,
\label{eq:qel_fes_form}
\end{equation}
where, for the model
we consider,
 each state $|\chi_k\rangle$  has support only near a particular extremum of $\mathcal H_{\rm eff}(x,p)$ in phase space  (see Appendix~\ref{app:qe_locking} and Sec.~\ref{sec:qel_derivation} below).
Note from Eq.~\eqref{eq:qel_fes_form}  that, for each $k$, $|\chi_k\rangle \equiv \frac{1}{\sqrt{q}}\sum_{k} e^{2\pi i \ell k /q}|\psi_k\rangle$.
One can thus verify that  the states $|\chi_1\rangle \ldots |\chi_k\rangle$  are  orthogonal, and mapped to each other under evolution by one driving period $T$, up to a phase and an exponentially suppressed correction:
\begin{equation}
\hat U(T) |\chi_k\rangle = e^{-i\varepsilon T} |\chi_{k+1}\rangle  + \mathcal O (\delta \varepsilon  T),
\label{eq:chi_relation}
\end{equation}
with $|\chi_{q+1}\rangle \equiv |\chi_q\rangle$~\cite{other_families}.

\subsection{Derivation of quasienergy locking for the driven qubit-cavity system}
\label{sec:qel_derivation}
Having reviewed the defining features of quasienergy locking, we now   show how it  emerges in the driven qubit-cavity system that we consider in this work.
We provide our arguments on a heuristic basis, by analyzing the dynamics of the system in the rotating frame introduced in Sec.~\ref{sec:effective_hamiltonian}.
For simplicity we consider  the   limit
 where the
{evolution} of the system in this
frame
is  fully captured by the effective semiclassical  Hamiltonian from Sec.~\ref{sec:theory}, $\mathcal H_{\rm eff}(x,p)$ (see Sec.~\ref{sec:effective_hamiltonian} for specific conditions).
In Appendix~\ref{app:floquet_locking} we provide a more  rigorous line of arguments that
 holds in the Floquet and adiabatic limits whenever $\mathcal H_{\rm eff}(x,p)$ features extrema with surrounding ``potential wells''  that are much larger than the scale of quantum fluctuations, $\xi \sim 1$ (see below for definition of potential wells).

To derive quasienergy locking
we investigate  the properties of the driven qubit-cavity system's Floquet eigenstates and quasienergies, $\{| \psi_n\rangle\}$ and $\{\varepsilon _n\}$.
As a first step, we note that the former are identical to the Floquet eigenstates in the rotating frame (see Sec.~\ref{sec:effective_hamiltonian}), $\{|\tilde \psi_n\rangle\}$,
while each quasienergy
 $\varepsilon _n$ is identical to  the corresponding quasienergy  in the rotating frame, $\tilde \varepsilon _n$, modulo $\tilde \Omega \equiv \Omega / q$~\cite{qe_correspondence}.
To obtain
$\{|\tilde \psi_n\rangle\}$ and $\{\tilde\varepsilon _n\}$, we
we recall  our assumption that  $\mathcal H_{\rm eff}(x,p)$ fully captures the system's dynamics in the rotating frame.
Hence we expect the stroboscopic evolution of the quantized cavity mode (in the rotating frame) to be generated by the   effective quantum  Hamiltonian $\mathcal H_{\rm eff}(\hat x,\hat p)$.
We moreover expect the spin to be locked to the effective axis of precession as a function of $ x$ and $ p$, as explained in Sec.~\ref{sec:theory}.
For simplicity, we therefore  neglect the spin in the following.
Through the above correspondence between the  rotating and lab frames, in the idealized limit we consider here, the Floquet eigenstates of the system in the lab frame are thus
given by  the eigenstates of $\mathcal H_{\rm eff}(\hat x,\hat p)$, while the quasienergies are given by the corresponding eigenvalues, up to integer multiples of $\tilde \Omega$.

To obtain the eigenstates and eigenvalues of  $\mathcal H_{\rm eff}(\hat x,\hat p)$,
 we consider the structure of $\mathcal H_{\rm eff}(\hat x,\hat p)$ in the frequency-locked regime.
 We  recall
 that frequency locking arises when $\mathcal H_{\rm eff}(x,p)$ has extrema at nonzero displacement amplitude.
These extrema are surrounded by {classical} trajectories [i.e., contours of $\mathcal H_{\rm eff}(x,p)$]
 which encircle and remain close to their respective fixed points at all times.
We refer to each such extremum, along  with its surrounding neighborhood that contains these encircling trajectories (out to the separatrices beyond which the trajectories encircle other fixed points) as a potential well.
Note that  $\mathcal H_{\rm eff}(x,p)$ has a built-in symmetry of discrete rotation  by   $2\pi/q$ in phase space, as is evident in the numerical examples plotted in Fig.~\ref{fig:contours}ab, where $q=3$ (see also Appendix~\ref{app:qe_locking}).
This symmetry, which is generated by $\hat U_0(T)$,
guarantees that each potential well of $\mathcal H_{\rm eff}(x,p)$  forms part of a ring of $q$  wells which are mapped to each other through rotations by  $2\pi/q$ in phase space.
We refer to these  wells as wells $1\ldots q$ in the following, such that well $k$ is mapped to well $k+1\, (\mod q)$ a  through phase space rotation by $2\pi /q$.

We  expect
 $\mathcal H_{\rm eff}(\hat x, \hat p)$   to support
{approximate} eigenstates   that are confined within the potential wells of $\mathcal H_{\rm eff}(x,p)$, whose wave functions  decay with the distance from the well, $r$, as $\mathcal O(e^{-r/\xi})$.
For $k=1,\ldots q$, we let $|\phi_k\rangle$
  denote such a ``bound'' eigenstate of $\mathcal{H}_{\rm eff}(\hat x , \hat p)$ when  restricting phase space to well $k$, such that $|\phi_k\rangle$ and $|\phi_{k+1}\rangle$ are related through phase space rotation by $2\pi /q$; i.e., we restrict phase space to a  region located within a distance $r_0$ from well $k$, for some $0<r_0<d/2$, where $d$ denotes the distance  in phase space between adjacent wells.
  We let $\varepsilon $ denote the corresponding eigenvalue of $H_{\rm eff}(\hat x ,\hat p)$; this takes  the same value for all $k$ due to the discrete rotation symmetry.
Below we find that the bound states $\{|\phi_k\rangle\}$ are identical to the
states  $\{|\chi_k\rangle\}$ whose linear combinations give
a quasienergy-locked multiplet of  Floquet eigenstates  as in Eq.~\eqref{eq:qel_fes_form}.
Note, however, that this identification is only exact in the idealized limit we consider here, where  $ {\mathcal H}_{\rm eff}(\hat x, \hat p)$ fully captures the stroboscopic evolution in the rotating frame.
Beyond this limit,  $\{|\chi_k\rangle\}$
deviate from  $\{|\phi_k\rangle\}$ by  nonzero, but small, corrections due to, e.g., nonadiabatic corrections and quantum fluctuations.
In Appendix~\ref{app:qe_locking}, we provide a more rigorous way to identify the states $|\chi_n\rangle$ that includes such corrections.

  Due to the exponentially decaying wavefunction of $|\phi_k\rangle$ outside well $k$, when all of phase space is included, each $|\phi_k\rangle$
  remains an approximate eigenstate of $\mathcal{H}_{\rm eff}(\hat x, \hat p )$,  up to an exponentially small correction
 $\mathcal{H}_{\rm eff}(\hat x, \hat p )|\phi_k\rangle = \varepsilon |\phi_k\rangle + \mathcal O(e^{-r_0/\xi})$.
 To zeroth order in  $\lambda  =e^{-r_0/\xi}$ (i.e., in the classical limit $\xi \to 0$), each $|\phi_k\rangle$ is thus an exact eigenstate of $\mathcal{H}_{\rm eff}(\hat x , \hat p)$, with eigenvalue $\varepsilon $.
 In the limit of small but nonzero $\xi$, the corresponding eigenstates of $\mathcal{H}_{\rm eff}(\hat x ,\hat p)$, $|\psi_1\rangle,\ldots|\psi_q\rangle$,

 can be obtained through
  zeroth-order degenerate perturbation theory, and thus can be expressed as linear combinations of the degenerate ``unperturbed'' states $|\phi_k\rangle$.
To identify these linear combinations, we note that the discrete rotation  symmetry  $[\hat U_0(T),\mathcal H_{\rm eff}(\hat x,\hat p)]=0$  requires the  eigenstates of $\mathcal H_{\rm eff}(\hat x,\hat p)$ to also be eigenstates of $\hat U_0(T)$.
Since $\hat U_0(T) |\phi_k\rangle  = |\phi_{k+1}\rangle$,
$|\psi_1\rangle,\ldots |\psi_q\rangle$ are thus given by Eq.~\eqref{eq:qel_fes_form}, with $|\chi_n \rangle = |\phi_n\rangle$.
Using that $\langle \phi_k|\mathcal{H}_{\rm eff}(\hat x, \hat p)|\phi_{k'}\rangle \lesssim e^{-d/\xi}$, we find that the corresponding eigenvalues of $\mathcal{H}_{\rm eff}(\hat x ,\hat p)$,  $\tilde \varepsilon _1,\ldots \tilde \varepsilon _q$, are all given by $\varepsilon $, up to corrections of order $e^{-d/\xi}$.
We conclude that for
 each ring of potential wells  of $\mathcal{H}_{\rm eff}(x,p)$ (if such wells are present),
  the qubit-cavity system
  supports one or more families of  Floquet eigenstates, whose form is given in Eq.~\eqref{eq:qel_fes_form}, where, for each $k$, $|\chi_k\rangle$ has support only in well $k$ of the ring~\cite{more_families} .
The corresponding  quasienergies are given by the same value $\varepsilon $, up to integer multiples of $\Omega /q$, and corrections of order $e^{-d/\xi}$.

In Appendix~\ref{app:qe_locking}, we provide a  more rigorous
 derivation of quasienergy locking, which holds beyond the idealized limit considered here.
 There we confirm that in the  regime, the qubit-cavity system supports families of near-degenerate eigenstates of the form in Eq.~\eqref{eq:qel_fes_form}, where $|\chi_k\rangle$ has support only in well $k$.
However,   $|\chi_k\rangle$ and $|\chi_{k+1}\rangle$ are not  related by  exact phase space rotation by $2\pi/q$, but rather through Eq.~\eqref{eq:chi_relation}~\cite{approximate_rotation}.
It follows that the corresponding quasienergies take the form in Eq.~\eqref{eq:qel}.
This was what we wanted to show in this subsection.

\subsection{Time translation symmetry breaking}
\label{sec:ttsb}\label{sec:qel_signatures}
Here we review how quasienergy locking can be seen as a realization of time-translation symmetry breaking~\cite{Sacha_2015,Else_2016b,Zhang_2017}.
To see this,  note from Eq.~\eqref{eq:chi_relation} that the states $\{|\chi_k\rangle\}$ are taken onto themselves after evolution by the extended period $\tilde T = qT$, up to a phase, and an exponentially suppressed correction.
Each $|\chi_{k}\rangle$ hence is a  (nearly) stationary state of the system's time-evolution that breaks the original discrete time-translation symmetry by $T$.
In contrast to the exact Floquet eigenstates, $\{|\psi_\ell\rangle\}$, which are
 superpositions of states characterized by distinct values of the oscillator phase (i.e, ``{Schr\"{o}dinger} cat'' states), each symmetry-breaking state $|\chi_k\rangle$  has a well-defined phase, and hence corresponds to a semiclassical ``non-cat'' state of the cavity mode.
In the sense above, the driven qubit-cavity system can hence be seen as supporting  steady states  that break  discrete time translation symmetry.

The symmetry-breaking states $\{|\chi_k\rangle\}$  remain stationary states of the system's time-evolution up to the duration of  confinement within the potential wells of $\mathcal H_{\rm eff}$,  $\tau~\sim e^{{d/\xi}}$.
In contrast, the ``coherence time'' of the symmetry-breaking steady states in many-body Floquet time crystals scales exponentially with the  size of the system due to many-body nature of the states, and hence is infinite in the thermodynamic limit~\cite{Khemani_2016,Else_2016b}.
For the qubit-cavity system we consider, {although there is no thermodynamic limit}, $\tau$ still scales exponentially with the system parameters, and thus can be very large compared to the other timescales of the system.
In this sense, we can still regard time-translation symmetry to be  broken in practice.
Note that we consider a more general notion of time-translation symmetry breaking than defined in Ref~\cite{Else_2016b}: namely, we only require some, but not  all,  steady states of the system to break the discrete time-translation symmetry of the system~\cite{Sacha_2015,Zhang_2017}.

\section{Numerical results\label{sec:numerics}}

Here we support our
discussion
with numerical simulations.
We simulate the qubit-cavity model in Sec.~\ref{sec:model} by computing the complete Floquet operator of the system
using direct time-evolution.
We then obtain the quasienergy spectrum and Floquet eigenstates through exact diagonalization.
In these simulations we  truncate  the Hilbert space of the cavity to the first $650$ photon-number eigenstates  (resulting in  Hilbert space  dimension $1300$), and discretize the  Hamiltonian's continuous time-dependence within one period into  $300$ evenly spaced intervals.

Throughout our simulations, we fix the dimensionless Zeeman field component to be $B_0=7$, and the driving amplitude  $A_d=15$, while we vary the qubit-cavity coupling $\eta$ and the cavity resonance frequency
$\omega_{\rm c}$ (see Sec.~\ref{sec:model}).

\subsection{Detection of quantum frequency locking}
\label{sec:locking_detection}
For each choice of the parameters $\eta$ and $\omega_{\rm c}$ we probed, we detected the presence of quantum frequency locking from the quasienergy spectrum of the system, $\{\varepsilon _n\}$.
We begin by sorting the quasienergy level spacings for the system,
$\Delta \varepsilon _{mn} \equiv \varepsilon _{m}-\varepsilon _{n}$ for all $1300\times 1299$  pairs of quasienergy levels where $m\neq n$, into a histogram of $10^5$ bins evenly spaced in the interval between $0$ and $\Omega$ (we consider  the value of each level spacing $\Delta \varepsilon _{mn}$  modulo $\Omega$).
For a generic distribution of quasienergy levels, we  expect the   number of level pairs  $N(\Delta \varepsilon )$  falling into the  bin at level splitting $\Delta \varepsilon $
to be given by $ 1300^2 / 10^{5} \approx 17$.
However, when period-$q$ quantum frequency locking is present, we expect an anomalously high number of level spacings to fall into the bin where $\Delta \varepsilon  = \Omega /q$, c.f.~the discussion in Sec.~\ref{sec:phase_locking}.

To illustrate this, in the inset in Fig.~\ref{fig:Fig1}c, we plot $N(\Delta \varepsilon )$ for
$\eta  = 0.56 \Omega$ and $\omega_{\rm c} = 0.34 \Omega$ (indicated by cross in main panel).
These parameters bring the system into the adiabatic regime; we previously plotted the $q=3$ effective cavity Hamiltonian for this choice of parameters in Fig.~\ref{fig:contours}a (see Sec.~\ref{sec:effective_hamiltonian}).
From the arguments of Sec.~\ref{sec:theory}, the  local extrema of $\mathcal H_{\rm eff}(x,p)$, which are clearly present in Fig.~\ref{fig:contours}a, should give rise  to period-$3$ frequency locking.
The data in the inset of  Fig.~\ref{fig:Fig1}c confirms this:
while $N(\Delta \varepsilon )$ is of order $\sim 17$ for almost all bins in Fig.~\ref{fig:Fig1}c, the spectrum features an anomalously high number of level pairs ($\sim 100$)
whose splitting falls into the bin at $\Delta \varepsilon =\Omega /3$.
From the discussion in Sec.~\ref{sec:phase_locking}, this  is a clear indication of period-$3$ quantum frequency locking.
We expect the model supports  approximately $100-17 \sim 85$ frequency-locked triplets of  Floquet eigenstates of the form in Eq.~\eqref{eq:triplet}.

As the above paragraph demonstrates, we may use the histogram peak-height
$N(\Delta \varepsilon =\Omega/q)$  to estimate the number of period-$q$ frequency-locked Floquet eigenstates  in the system.
In the main panel of Fig.~\ref{fig:Fig1}c, we plot this  number for $q=3$ as a function of $\omega_{\rm c}$ and  $\eta$.
As is evident in Fig.~\ref{fig:Fig1}c, the model supports a large number of period-$3$ frequency-locked Floquet eigenstates in a finite region of parameter space, arising both for weak and strong detuning $\delta \omega = {\omega_{\rm c}}-\Omega/3$ and qubit-cavity coupling $\eta$.

The data in Fig.~\ref{fig:Fig1}c show clear signatures of the two distinct regimes of quantum frequency locking we identified in Sec.~\ref{sec:effective_hamiltonian}.
Focusing on the peak  that emerges from $\omega_{\rm c} = \Omega/3$, for  $\eta\ll \Omega$,  quantum frequency locking occurs when $\omega_{\rm c}\approx \Omega /3$.
However,  for $\eta\gtrsim\Omega/2$, the $\omega_{\rm c}$-interval in which quantum frequency locking occurs splits into two linearly-diverging branches.
This  point   marks the crossover from the Floquet  (lower branch) to the adiabatic regime (upper branches).
Specifically, in the adiabatic regime, after  a simultaneous  rescaling of $\eta$ and $\delta \omega$ by the same positive factor $\lambda$, $\mathcal H_{\rm eff}(x,p)$ is mapped to $\lambda\mathcal H_{\rm eff}(x,p)$ [see Eqs.~\eqref{eq:heff}~and~\eqref{eq:classical_rotating_frame_hamiltonian}].
Moreover a sign reversal of $\delta \omega$ maps $\mathcal H_{\rm eff}(x,p)$ for aligned spin into $-\mathcal H_{\rm eff}(x,p)$ for anti-aligned spin, and vice versa.
Thus, in the adiabatic  regime, $\mathcal H_{\rm eff}(x,p)$  features the same structure of local extrema and  potential wells along the lines $ \delta \omega = \pm \kappa \eta$ for some proportionality factor $\kappa$, and hence,  in this regime, quantum frequency locking should occur along these two lines in parameter space.
This structure of  linearly-diverging branches is clearly evident in Fig.~\ref{fig:Fig1}c.
In contrast, the Floquet  regime only arises when $\eta < \Omega $, and for small values of $\delta \omega$ (see Sec.~\ref{sec:floquet}).
Thus, %
the Floquet  regime gives rise to a single branch  at
$\delta \omega \sim 0, \eta \ll \Omega$.

\subsection{Structure of Floquet eigenstates}
Next, we sought to verify that the frequency-locked Floquet eigenstates have the structure we predicted in Sec.~\ref{sec:phase_locking}: we expect each triplet of  frequency-locked Floquet eigenstates,  $|\psi_n^1\rangle$, $|\psi_n^2\rangle$, and $|\psi^3_n\rangle$ (with corresponding quasienergies $\varepsilon_n + \ell \Omega /3$) to be of   the form in Eq.~\eqref{eq:floquet_state_multiplet_final} (for $q=3$),
where $|\chi_n^k\rangle$  has support only within a particular ``potential well'' of the effective cavity mode Hamiltonian, $\mathcal H_{\rm eff}(x,p)$.

To confirm
the hypothesized structure above, we  obtained the Floquet eigenstates of the model for the  parameter set  used  in  Fig.~\ref{fig:contours}a (see Sec.~\ref{sec:theory} for parameters; note that these were also used for the inset in Fig.~\ref{fig:Fig1}c), where the system exhibits quantum frequency locking in  the adiabatic regime.
We computed the Wigner function $W(x,p)$ for each Floquet eigenstate $|\psi_n\rangle$,
using  the reduced density matrix  of the cavity  $\rho^{n}_{\rm cav}= {\rm Tr}_S [|\psi_n\rangle\langle\psi_n|]$, where ${\rm Tr}_S$ denotes the partial trace over the Hilbert space of the spin.
Fig.~\ref{fig:contours}c shows the Wigner function of a frequency-locked  Floquet eigenstate of the system (i.e.,  one out of the
many Floquet eigenstates whose  quasienergies differ by an exact multiple of $\Omega/3$ from two other quasienergies in the system).
The Wigner function
in Fig.~\ref{fig:contours}c shows a highly structured pattern, and has  support only in $3$ separate regions of phase space that coincide with
the potential wells
of  $\mathcal H_{\rm eff}(x,p)$ in panel (a),  (shown as grey lines in Fig.~\ref{fig:contours}c), consistent with Sec.~\ref{sec:phase_locking}.

Next, we identified the   two other Floquet eigenstates of the triplet in which $|\psi_n\rangle =|\psi_n^3\rangle$ formed a part, $|\psi_n^1\rangle$ and $|\psi_n^2\rangle$ (i.e., we identified the two Floquet eigenstates of the system whose   quasienergies differ  by  $\pm\Omega/3$ from the quasienergy of the eigenstate $|\psi_n^3\rangle$, up to a correction many orders of magnitude smaller than the level spacing of the quasienergy spectrum).
The  Wigner functions of these two  states are nearly identical to the Wigner function  in Fig.~\ref{fig:contours}c, and are not shown here.
According to the hypothesis of Eq.~\eqref{eq:floquet_state_multiplet_final} there exists a gauge choice for the Floquet eigenstates $|\psi_n^1\rangle,\ldots|\psi_n^3\rangle$  such that, for each $k$, $|\chi^k_n\rangle \equiv \frac{1}{\sqrt{3}}\sum_{\ell=1}^3e^{2\pi i k\ell/3}|\psi_n^\ell\rangle$  only has support only in well $k$ of $\mathcal H_{\rm eff}$.
In the inset of Fig.~\ref{fig:contours}c, we show the Wigner function for such a linear combination  (with $k=3$).
In agreement with the discussion in Sec.~\ref{sec:phase_locking},
this Wigner function is only nonzero in a single potential well  of $H_{\rm eff}$ [namely near $(x,p)= (20,0)$].
We confirmed numerically (data not shown here) that with the same gauge choice for the states $|\psi_n^1\rangle,\ldots |\psi_n^3\rangle$,
the two other choices of $k$ led to the  Wigner function of the resulting state $|\chi_k\rangle$ having support in the two other potential wells of $H_{\rm eff}$.
Thus we confirmed that the triplet of Floquet eigenstates has the structure in Eq.~\eqref{eq:floquet_state_multiplet_final}.

We also considered the Wigner functions of frequency-locked Floquet eigenstates in the  Floquet regime ($\eta \ll \Omega, \delta \omega \sim 0$).
Fig.~\ref{fig:contours}d shows the Wigner function of such a frequency-locked Floquet eigenstate of the model, for  parameters  $ \omega_{\rm c} =\Omega/3$ and $\eta =0.048\Omega $, which puts the system in the Floquet regime, and were also used in Fig.~\ref{fig:contours}b.
The  Wigner function
exhibits a very similar structure as in the adiabatic regime: there exist three separate regions where the it is nonzero and smoothly varying that coincide with the potential wells  of the effective Hamiltonian of the system,  $\mathcal H_{\rm eff}(x,p)$, (Fig.~\ref{fig:contours}b).
At the edges of its peaks, the Wigner function   exhibits oscillations from positive to negative with nodal lines parallel to the  the contours of  $\mathcal H_{\rm eff}(x,p)$,  hence strongly supporting the discussion in Sec.~\ref{sec:floquet}.

Note that each  ring of potential wells of $\mathcal H_{\rm eff}(x,p)$
can support several triplets of quasienergy-locked  Floquet eigenstates.
Moreover, for the (Floquet regime) parameters used in Fig.~\ref{fig:contours}bd,  $\mathcal H_{\rm eff}(x,p)$ features multiple rings of potential wells that can each support its own
families of frequency-locked Floquet eigenstates.
We confirm this in Appendix~\ref{app:other_wigners} where   we plot the Wigner functions for  two additional triplets of frequency-locked Floquet eigenstates that both have support in the same well; this well  is different from the one where the Floquet eigenstate   in Fig.~\ref{fig:contours}d has support.

\subsection{Observable signatures and frequency conversion}\label{sec:signatures}

\begin{figure}
\includegraphics[width=0.99\columnwidth]{Time_Evolution_Data_v1.png}
\caption{
Observable signatures of quantum frequency locking.
(a) Frequency spectrum of the cavity field, $\langle \hat x(t)\rangle $, for the system when initialized within (green), and outside (orange) the frequency locking regime,  respectively.
See main text for parameters and further details.
Note that  the green curve is vertically offset by $1$.
(b) Zoom-in of panel (a), in the vicinity of  $\omega = \Omega/3$ (indicated by vertical dashed line). }
\label{fig:dynamics}
\end{figure}
As a final goal for our numerical simulations, we explored  the observable signatures of quantum frequency locking in the system, and their possible applications for frequency conversion.
To this end, we  considered the dynamics of the
observable $\langle \hat x(t)\rangle$, which, depending on the exact realization of the model,
 for instance can measure a component of the electromagnetic field in the cavity  (see Sec.~\ref{sec:model} for definition).
Using the parameters $\omega = 0.34 \Omega$, $\eta = 0.56 \Omega$
(also used in Figs.~\ref{fig:Fig1}c~and~\ref{fig:contours}ac),
we computed   the time-evolution of the system after initializing the cavity mode in a coherent state with phase $0$ and displacement amplitude either $20$ or $10$, corresponding to locations $(x_0,p_0)=(20,0)$ and $(x_0,p_0)=(10,0)$  in phase-space.
For both initializations we  initialized the spin in the state $|\hspace{-3.2pt}\downarrow\rangle$, anti-aligned with the initial effective magnetic field $\vec b(x_0,p_0,0)$.
From the  resulting effective cavity Hamiltonian  of the system shown in  Fig.~\ref{fig:contours}a,
we expect  these two initializations to place the system inside and outside the frequency locking regime, respectively.

In Fig.~\ref{fig:dynamics}a, we show the  dimensionless Fourier transform of $\langle \hat x(t)\rangle$ (absolute value), $|\tilde x(\omega)|$, for the two initializations above, while Fig.~\ref{fig:dynamics}b shows a close-up of the spectrum in the vicinity of $\omega_c= \Omega /3$~\cite{fourier_transform}.
In the frequency-locked regime, $|\tilde x(\omega)|$ features  an extremely sharp peak of magnitude $\sim 10$ at $\omega = \Omega/3$.
The two side-peaks visible in Fig.~\ref{fig:dynamics}b arise from the slow  orbit
of the cavity wave-packet  around the local minimum of $\mathcal H_{\rm eff}(x,p)$ (see Sec.~\ref{sec:effective_hamiltonian}); their offset from the main peak defines the oscillation frequency of this motion.
As is evident in Figs.~\ref{fig:dynamics}cd,  in the  regime the system has a clear, measurable subharmonic response to the driving.
In contrast, outside the frequency-locked regime,  $|\tilde x(\omega)|$  shows a broad feature around the same value, but no well-defined peak.

When weakly coupled to an external environment (such as an electromagnetic waveguide), it may be possible to extract an output signal whose frequency spectrum shares the spectrum of $\langle x(t)\rangle$, and hence exhibits well-defined coherent oscillations at  frequency $\Omega/q$ which is evident in Fig.~\ref{fig:dynamics} .
In this way, the  qubit-cavity system can potentially be exploited for frequency conversion.

\section{Discussion}
\label{sec:discussion}

The discovery of Floquet  time crystals sparked  a broader investigation  of discrete time-translation symmetry breaking.
This work shows how such symmetry breaking can emerge as quantum frequency locking in a periodically driven spin-cavity system.
When frequency-locked, the system exhibits well-defined  oscillations  with extended  period ${\tilde{T}} = qT$,  where $T$ denotes the driving period, and $q$ is an integer.
Quantum frequency locking moreover has remarkable consequences for the  quasienergy spectrum of a system: a large number of multiplets of  Floquet eigenstates emerge whose  quasi-energy differences are exponentially close to $n \Omega/q$ for  $n=1,\ldots,q$.
Using a semiclassical phase-space approach, we  identify two mechanisms for  frequency locking, which allow it to occur in a wide region of parameter space.
Quantum frequency locking  hence  does not require  fine-tuning, and can be reached through appropriately  {controlled but not fine-tuned} initialization of the cavity mode, for a finite range of detuning $\delta \omega =\omega_c -r\Omega/q$, and for both weak and strong qubit-cavity coupling $\eta$.

The frequency locking exhibited by the qubit-cavity system  is of fundamentally different nature than, e.g., time-crystalline behavior in  spin chains (see, e.g., Refs.~\onlinecite{Khemani_2016,Else_2016b}).
In the latter setting, time-translation symmetry breaking is also manifested in a large degeneracy of period-doubled Floquet eigenstates.
However, for these systems, period multiplication  emerges from the many-body nature of the system, and  each quasienergy level in the system forms a part of a quasienergy-locked multiplet.
In contrast,  for the qubit-cavity system,  only a finite (nonzero) number of quasienergy levels form multiplets.

We expect that the nontrivial fixed points of the stroboscopic motion generated by the semiclassical effective Hamiltonian $\mathcal H_{\rm eff}(x,p)$  remain stable in the presence of weak dissipation in the cavity, as would be the case if the radiation is allowed to leak out.
In this case  the frequency locking effect  could be used for extracting  an output signal whose frequency  is given by  a rational fraction of the drive, thus achieving frequency conversion.
This offers an interesting direction for future studies.

The driven spin-cavity system we considered
is perhaps one of the simplest  systems that exhibits quantum frequency locking.
This generic  class of models
can describe a diverse range of settings and physical systems, such as, e.g.,  Rydberg atoms in optical cavities and qubits in contact with microwave modes.
 Due to the simplicity of the model, and the many suitable experimental platforms, we  expect that the qubit-cavity model  forms a convenient and versatile platform for studying the breakdown of discrete time-translation symmetry.
At  strong coupling $\eta$, frequency locking  moreover coexists with the topological energy-pumping regime that was analyzed in Ref.~\onlinecite{Nathan_2019}.
{Thus, the relatively simple and experimentally accessible model of a driven qubit-cavity system supports several  distinct, highly nontrivial non-equilibrium phenomena.
The simplicity of the platform, and the interplay of these nontrivial phenomena makes the driven qubit-cavity system   an interesting subject for future experimental and theoretical studies.}

{\it Acknowledgements ---}
IM was supported by the Materials Sciences and Engineering Division, Basic Energy Sciences, Office of Science, U.S. Dept. of Energy.
FN and MR are grateful to the Villum Foundation and the European Research Council (ERC)
under the European Union Horizon 2020 Research and
Innovation Programme (Grant Agreement No. 678862) for support. GR is grateful for NSF DMR grant number 1839271.
GR is also grateful to the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award  DE-SC0019166. NSF and DOE supported GR's time commitment to the project in equal shares.

redpajama_set_name: RedPajamaArXiv


text:
About Us News & Updates News Sign up to Strictly Get Dancing
Sign up to Strictly Get Dancing
Twinkle your toes on the dance floor this autumn as Martin House takes its hugely popular Strictly Get Dancing competition to the musicals.
Martin House is searching for couples who fancy themselves as the next Fred Astaire and Ginger Rogers to take part in Strictly Get Dancing at the Musicals to sign up now.
The event, which has a new venue at Aspire in Leeds, takes place on Saturday 9th November, but couples need to sign up before September 22nd, before rehearsals begin.
Sara Cracknell, events fundraiser at Martin House, said: "We've all probably secretly dreamed of recreating a classic movie dance – whether it's a classic Fred and Ginger number or a modern film like La La Land.
"Strictly Get Dancing at the Musicals is your chance to make that dream a reality – while raising money for Martin House at the same time."
Couples who sign up to Strictly Get Dancing will attend six lessons at the Danceology studios in Wetherby, where dance teacher Jean Harvey will put them through their paces to learn a Latin and a ballroom dance, with a movie musical theme.
They also need to raise a minimum of £500 in sponsorship, before they perform on the dancefloor at Aspire in front of their family and friends.
The winners will be chosen by a panel of judges which will include ITV Calendar presenter Christine Talbot and Northern Ballet's Laura Hatton, while Christine's co-presenter Duncan Woods will host the evening.
Sara added: "It's a real once-in-a-lifetime opportunity to be a star for an evening while raising money to help children and young people in your area."
It costs £50 per couple to sign up for Strictly Get Dancing at the Musicals, to find out more or to register, email events@martinhouse.org.uk or go to www.martinhouse.org.uk/strictlygetdancing.

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Calumet City Plumbing Company, Inc. was founded in 1967 as a service and repair contractor. During our history, we have steadily grown into the quality firm we have become today. We recognize the need to provide competent 24 hour emergency repairs as well as new construction and alterations. We realize the need to offer a single solution for all your utility repairs, extension, and new installations.
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text:
August 20, 2016 August 20, 2016 ~ latskojerry
It has been rather nice for the past few seasons to have my two favorite teams, the San Francisco Giants and Pittsburgh Pirates, contend for their respective division titles and participate in the post season. The Pirates had an especially long drought period before emerging as contenders in 2013 and many of us who have been around since J. Fred Muggs had as many viewers as the Kardashians recall the woeful days of Kiner's Korner and perennial last place finishes before the happy World Champion days of 1960, 1971, and 1979. The Giants have been, for the most part, flourishing ever since they moved to the Phone Booth in 2000 but there were many years of suffering prior to that in the Bay Area because even when they were winning one had to endure Candlestick Park to see them in person and they almost moved to Florida.
It was good while it lasted but this year it looks like neither team has what it takes to make the playoffs. Oh, sure, the Giants are in second place and there are still many games to play but they have played at a .323 winning percentage since the All Star break after boasting a surprising 57-33 record that had many of us fooled. The Pirates have perked up lately, on the other hand, going 28-17 since June 24. The Cubs faltered a bit over a month ago, but that has been followed by another winning surge that has just about clinched the division title already. The Cardinals and Pirates still have decent teams but will probably wear themselves out just trying to be wild card entries.
The Pirates shot themselves in the foot in the off season. Whether or not they were deliberately conceding to Chicago, the winter additions of David Freese and Matt Joyce did not scare people around the league. The trade of Neil Walker was only going to make sense if Jon Niese became an 18 game winner, but he was 8-6 with a 4.91 ERA for the Bucs. Francisco Liriano went from being a good five inning pitcher to being a mediocre at best one, going 6-11 with a 5.46 ERA. Juan Nicasio has seldom been good. Starting pitching is really all that Pittsburgh needs and there is hope for the future. Jameson Taillon has looked good and others with promise appear on the horizon. The outfield remains perhaps the overall best in baseball, especially defensively. McCutchen obviously needs his dreads back but seems to be snapping out of a year long batting funk. Marte and Polanco both strike out a lot but are otherwise superb. The Bucs were hurt badly by the loss to injury of so many games by Francisco Cervelli and Chris Stewart and catcher is, after all, a vitally important position. The infield defense is not terrible but does not rate with the best either. Jaso and Freese were not acquired for their gold glove potential.
Meanwhile, the Giants' sterling infield defense took a hit with the late July trade of third baseman Matt Duffy to Tampa Bay, where he is evidently going back to playing shortstop. The Giants received a mediocre left handed starter, Matt Moore, in return and acquired a lefty reliever, Will Smith, from Milwaukee as well in an attempt to shore up a wobbly pitching staff. Smith gets the mediocre tag as well and San Francisco will have to hope that Eduardo Nunez, a journeyman infielder who reminds no one of Brooks Robinson or Omar Vizquel, is highly productive offensively. He was an All Star for the Twins, but every team gets an All Star.
Don't get me wrong, either or both of these teams could end up in the 2016 playoffs somehow and I'll be happy if they do. There are proven veteran winners on both squads. As for the World Series, that would be a bit of a reach. For the future, a fresh injection of youthful enthusiasm may be needed if the Giants and Pirates are to regain badass status.
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text:
Bombi Bitt och jag är en roman från 1932 av Fritiof Nilsson Piraten. Romanen har beskrivits som en svensk Tom Sawyer.

Beskrivning och bakgrund
Bombi Bitt och jag är en roman med inslag av humor, ett slags självbiografi av Piraten i unga dagar. Mycket är visserligen fantasi, men det berättas på ett sannolikt och medryckande sätt. Handlingen är till stora delar förlagd i skånska miljöer (centrerade kring Vollsjö) i början av 1900-talet.

Bokens inledning är berömd:

Piraten-boken har inspirerats av Mark Twains fabulerande pojkäventyr, och den har liknats både vid Tom Sawyer och Huckleberry Finn. Titelns "Jag" – Bombi Bitts tre år yngre följeslagare Eli, är Piratens alter ego. Namnet är hämtat från en pojke hos Zacharias Topelius. Samhället Vollsjö har i boken fått det fiktiva namnet Tosterup (ej att förväxla med byn Tosterup i Tomelilla kommun), och miljöerna i boken är hämtade från olika bygder i sydöstra Skåne där Piraten eller andra i hans släkt levat.

Mycket av "barndomsskildringarna" i boken är mer eller mindre fria fabuleringar. Patron Esping och Vricklund, liksom pigan Boel, Bombi Bitt och "Elis" föräldrar, kan ha verkliga förebilder, medan annat är mindre verklighetsförankrat. Nilsson Piratens egen far var dock stins i Vollsjö (alltså "Tosterup").

Filmatiseringar
Nilsson Piratens bok har vid minst två tillfällen bearbetats till produktioner med skådespelare. 1936 hade långfilmen Bombi Bitt och jag premiär. 1968 sändes den fyra avsnitt långa TV-serien Bombi Bitt och jag på svensk TV. I denna TV-produktion hade en 17-årig Stellan Skarsgård huvudrollen; i 1936 års film spelades den enligt boken 14-årige Bombi Bitt av en då 27-årig Sture Lagerwall.

Referenser

Skönlitteratur 1932
Svenska romaner som blivit film
Skåne i skönlitteratur
Svenskspråkig skönlitteratur
Verk av Fritiof Nilsson Piraten

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Catherine Leggitt is the author of the Christine Sterling Mysteries, PAYNE & MISERY, THE DUNN DEAL, and PARRISH THE THOUGHT published by Ellechor Publishing. A native Californian, Catherine raised two daughters, taught school, and cared for aging parents before retirement drove her to the keyboard. These days Catherine crafts suspenseful and convoluted plots—exploring God's mysteries through fiction.
The Her is gone, the princess has vamoosed? My turn is coming. Could someone please find her? Thanks for such a dramatic turn of events, Catherine. This is fun!
Gone as in vanished or gone as in dead?