# Quantum metrology with Bloch Oscillations in Floquet phase space

Keye Zhang<EMAIL_ADDRESS>Quantum Institute for Light and Atoms,
State Key Laboratory of Precision Spectroscopy, School of Physics and
Electronic Science, East China Normal University, Shanghai 200241, China
Weijie Liang  Quantum Institute for Light and Atoms, State Key Laboratory of
Precision Spectroscopy, School of Physics and Electronic Science, East China
Normal University, Shanghai 200241, China  Pierre Meystre  Department of
Physics and College of Optical Sciences, University of Arizona, Tucson, AZ
85721, USA.  Weiping Zhang<EMAIL_ADDRESS>School of Physics and Astronomy,
and Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240,
China  Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan, Shanxi 030006, China  Shanghai Research Center for Quantum Sciences,
Shanghai 201315, China

###### Abstract

Quantum particles performing Bloch oscillations in a spatially periodic
potential can be used as a very accurate detector of constant forces. We find
that the similar oscillations that can appear in the Floquet phase space of a
quantum particle subjected a periodic temporal driving, even in the absence of
periodic lattice potential, can likewise be exploited as detectors. Compared
with their spatial Bloch analog, however, the Floquet-Bloch oscillations
provide significant added flexibility and open the way to a broad range of
precision metrology applications. We illustrate this property on the examples
of a tachometer and a magnetometer.

_Introduction_ – It is well known that quantum particles confined in periodic
potentials and subjected to a constant force do not accelerate uniformly in
real space. Rather, they undergo Bloch oscillations, which have been observed
in a variety of systems, including ultracold atom gases trapped on optical
lattices Dahan1996 ; Wilkinson1996 ; Gustavsson2008  and optical waves in
waveguide arrays Pertsch1999 ; Morandotti1999 . Since the Bloch oscillations
frequency depends only on the lattice spacing and the applied external force
it provides an accurate tool for the measurement of weak forces and the
precise determination of fundamental constants Poli2011 ; Biraben2004 ;
Biraben2006 ; Biraben2011 .

There is much interest in extending the study of these systems to situations
where they are subjected in addition to a time-periodic forcing Arlinghaus2011
; Kudo2011 . This includes in particular the study of ‘super-Bloch
oscillations’, which can result in linear transport in a lattice Haller2010
and spatial oscillations of the system even in the absence of a spatial
lattice Junk2020 . More generally, time-periodic forcing is now exploited in
the emerging area of ‘Floquet engineering’, a powerful tool to control and
modify quantum systems, including for instance the discovery of new out-of-
equilibrum phases, see e.g. Moessner2017 ; Weitenberg2021 , in particular the
Floquet time crystal phase Else2016 ; Sacha2018a . Applications of these
techniques to enhance the precision of quantum metrology have also been
proposed, see e.g. Lukin2020 ; Fiderer2018 ; Jiang2021 .

The present work extends the Bloch measurement approach to the Bloch-like
oscillations that appear in the Floquet phase space of a system subjected to a
periodic temporal drive rather than a spatially periodic potential. Just like
in the case of Bloch oscillations, the measurement of constant forces applied
to systems undergoing such oscillations reduces to frequency measurements,
with the remarkable precision with which they can be carried out. But as we
shall show, the transformation of the Hamiltonian of the ‘measuring apparatus’
to the phase representation brings to the fore two key new features of this
approach, as compared to the familiar Bloch oscillations approach: It is not
limited to the measurement of constant forces, and it is applicable to a
variety of physical observables. We will illustrate these points on the
specific examples of the realization of a tachometer and a magnetometer.

_Bloch Dynamics in Floquet phase space_ – We consider a system described by
the one-dimensional Hamiltonian

$H=H_{0}+H_{1}=\frac{p^{2}}{2m}+U(x)+V(x)\cos\omega t\,,$ (1)

where $H_{0}$ is characterized by a conservative and non-periodic potential
$U(x)$, and $H_{1}=V(x)\cos\omega t$ is weak single-frequency perturbation
oscillating at the frequency $\omega$.

There are two common approaches to reveal the lattice structure of this system
in Floquet phase space. The first one is based on a perturbative approach
combined with a specific time-dependent canonical transformation
Buchleitner2002 , while the second one relies on the use of Floquet quantum
states and quasi-energy theory Grifoni1998 ; Holthaus2016 . We adopt the first
method here, as it is particularly convenient to discuss the various
measurement applications that we have in mind.

Consider first the classical version of the unperturbed Hamiltonian $H_{0}$.
For a given system energy $H_{0}=E_{0}$ it is always possible to introduce
conjugate action-angle variables $(J,\Theta)$ with

$J_{0}=\oint p\,{\rm d}x=\oint\sqrt{2m[E_{0}-U(x)]}{\rm d}x\,,$ (2)

where the integral is over one period of motion, so that the transformed
Hamiltonian $H_{0}$ depends on the action variable $J$ only, and the
associated angle variable $\Theta$ evolves at a constant angular frequency
$\Omega$ in the range $[0,2\pi)$, with Hamilton equations of motion

$\dot{J_{0}}=-\frac{\partial
H_{0}(J_{0})}{\partial\Theta}=0\,\,\,\,;\,\,\,\dot{\Theta}=\frac{\partial
H_{0}(J_{0})}{\partial J_{0}}=\Omega\,.$ (3)

In terms of these action-angle variables, the perturbation potential $V(x)$ in
$H_{1}=V(x)\cos\omega t$ is resolved into multiple components as a Fourier
series in the action angle $\Theta$,

$V(x)\to V(J,\Theta)=\sum_{\ell=-\infty}^{+\infty}V_{\ell}(J)e^{{\rm
i}\ell\Theta}\,,$ (4)

where $V_{\ell}(J)$ are the Fourier components of $V(x)$ evaluated along the
unperturbed classical trajectory of energy $E_{0}$.

If the frequency $\omega$ of the perturbation is a multiple of the unperturbed
angular frequency $\Omega$, $\omega=n\Omega$, all components in that sum will
oscillate rapidly and average out to zero, except for the near-resonant term
$V_{n}\cos[n(\Theta-\Omega t)]$, so that one can invoke a secular
approximation. Following Ref. Buchleitner2002  we then carry out a second
time-dependent canonical coordinate transformation involving the slowly
varying variable $\vartheta=\Theta-\Omega t$, with the result that to the
lowest order in the perturbation the Hamiltonian $H$ reduces to the time
independent form SM

$H(P,\vartheta)=H_{0}+H_{1}\approx\frac{P^{2}}{2M}+V_{n}\cos n\vartheta\,.$
(5)

Here $P\equiv J-J_{0}$ is the deviation $\delta J$ of the action variable from
its unperturbed form and can be thought of as acting as an ‘effective
momentum’; and $M\equiv(\partial^{2}H_{0}/\partial J_{0}^{2})^{-1}$, can be
interpreted as playing the role of an ‘effective mass’ DM . The Hamiltonian
$H(P,\vartheta)$ is then formally identical to the Hamiltonian of a particle
in a periodic potential, except that the lattice periodicity is now in the
slowly varying angle variable $\vartheta$.

We now quantize $H(P,\vartheta)$ by promoting $P$ to an operator,
$P\rightarrow-{\rm i}\hbar\partial_{\vartheta}$. Its eigenfunctions are Bloch-
like functions of the form Sacha2018

$\varphi_{q}(\vartheta)=U_{q}(\vartheta)e^{{\rm i}q\vartheta}\,,$ (6)

where $U_{q}(\vartheta)$ has the periodicity $2\pi/n$ of the potential
$V_{n}\cos n\vartheta$, $U_{q}(\vartheta+2\pi/n)=U_{q}(\vartheta)$. In the
first Brillouin zone the effective ‘quasi-momenta’ $q$ associated with the
successive energy bands are dimensionless integers in the range $[0,n)$, with
dispersion relations $\varepsilon_{q}$. A general state of the system is
therefore of the form

$\psi(\vartheta,t)=\sum_{q=0}^{n-1}c_{q}(0)\varphi_{q}(\vartheta)e^{-{\rm
i}\varepsilon_{q}t/\hbar}\,.$ (7)

where $c_{q}(0)$ are the initial probability amplitudes of the Bloch-like
functions, $c_{q}(0)=\int\psi(\vartheta,0)\varphi_{q}^{*}(\vartheta)\,{\rm
d}\vartheta$.

_Measurements by Floquet-Bloch oscillations_ – We now discuss the use of
systems undergoing such Floquet-Bloch oscillations as ‘measuring apparatus’
that allow under a broad range of conditions for the determination of forces
via frequency measurements. We focus here on the case of a weak force $f$
linearly coupled to $\vartheta$ in Floquet space via the interaction footnote

$V_{\rm probe}=f\vartheta\,.$ (8)

As a result the ‘quasi-momentum’ $q$ will undergo the evolution

$q(t)=q(0)-ft/\hbar\,,$ (9)

and the state (7) becomes, assuming that $V_{\rm probe}$ is weak enough that
it does not induce interband transitions Gluck2002 ,

$\psi(\vartheta,t)=\sum_{q=0}^{n-1}c_{q}(0)U_{q(t)}(\vartheta)e^{{\rm
i}q(t)\vartheta}e^{-\frac{\rm i}{\hbar}\int_{0}^{t}\varepsilon_{q(t)}{\rm
d}t}\,.$ (10)

The group velocity of this wavepacket is
$v_{g}\sim\partial\varepsilon(q)/\partial q$, and its ‘effective mass’
$m_{g}\sim[\partial^{2}\varepsilon(q)/\partial q^{2}]^{-1}$, oscillating
between positive and negative values. As a result, the wave packet moves back
and forth along $\vartheta$ in Floquet phase space at the Bloch period

$T_{\rm B}=\frac{\hbar n}{|f|}\,,$ (11)

the time it takes $q(t)$ to propagate from $q(0)$ to $q(0)+n$ across the
Brillouin zone.

Returning to the original $(P,\Theta)$ picture, we find with
$\vartheta=\Theta-\Omega t$ and $\omega=n\Omega$ that $\Theta(t+T_{\rm
B})-\Theta(t)=\Omega T_{\rm B}=\hbar\omega/|f|$. Therefore after each Bloch
period $\psi(\Theta,t)$ accumulates a drift $\hbar\omega/f$ in the angle
variable $\Theta$,

$\psi(\Theta,t)=\psi\big{(}\Theta+\frac{\hbar\omega}{f},t+T_{\rm
B}\big{)}=\psi\big{(}\Theta+\frac{\hbar\omega}{f},t+\frac{\hbar
n}{f}\big{)}\,,$ (12)

as illustrated in Fig. 1. This is the central result of this paper. It shows
that it is possible to exploit either the period of oscillations or the phase
shift of $\psi(\Theta,t)$ to determine $f$.

Figure 1: Floquet-Bloch oscillations and drift of the probability density
$|\psi(\Theta,t)|^{2}$ for an initial Gaussian wave packet, with a driving
$f=4\hbar\omega/\pi$, $V_{n=100}=50\hbar\omega$, and
$\hbar^{2}n^{2}/(2|M|)=100\hbar\omega$.

_Implementation_ – One simple way to realize potentials of the form (8) is by
using a train of $\delta$-kicks separated by an interval $T_{\rm
D}=2\pi/\Omega$. This follows from the fact that we have then

$\displaystyle V_{\rm probe}$ $\displaystyle=$ $\displaystyle
f\Theta\sum_{\ell=-\infty}^{\infty}T_{\rm D}\,\delta(t-lT_{\rm D})$ (13)
$\displaystyle=$ $\displaystyle f\left[\pi-\sum_{k=1}^{\infty}\frac{2\sin
k\Theta}{k}\right]\sum_{\ell=-\infty}^{\infty}e^{{\rm i}\ell\Omega t}\approx
f\vartheta\,,$

where the first summation in the second line is the Fourier series of $\Theta$
for $0<\Theta<2\pi$, the second summation is the Fourier series of the Dirac
comb function, and the final approximate equality holds under the secular
approximation.

Importantly, since the relationship between the phase coordinate $\Theta$ and
the physical coordinates $(x,p)$ depends on the explicit form of $H_{0}$ it
follows that the Hamiltonian $V_{\rm probe}$ can describe a number of physical
processes, in contrast with the situation with Bloch oscillations in a spatial
lattice. Floquet-Bloch oscillations in phase lattice can therefore be
exploited for the measurement of a broader variety of observables by an
appropriate design of the conservative potential $U(x)$ in $H_{0}$. We
illustrate this point on the examples of a square well potential and of a
triangular well.

_Infinite square well_ – For a particle of mass $m$ in a one-dimensional
infinite square well potential of width $L$ we have

$U(x)=\begin{cases}0&\text{for}\;0<x<L\\\
\infty&\text{otherwise}\,,\end{cases}\ $ (14)

and the corresponding action-angle transformation between the position $x$ and
$\Theta$ is linear,

$x=L\left|\pi-\Theta\right|/\pi\,.$ (15)

The amplitude of the particle momentum is constant, but its direction depends
on $\Theta$,

$p=-{\rm sign}(\sin\Theta)mL\Omega/\pi\,,$ (16)

where the angular frequency is $\Omega=\sqrt{2\pi^{2}E_{0}/mL^{2}}$ with
$E_{0}$ the initial energy and $m$ the mass of the particle SM .

The linear relationship (15) between $\Theta$ and $x$ indicates that in
physical coordinates $V_{\rm probe}(x,p)$ should be linear in position. It
will be linear in $\vartheta$ as well if it consists of a periodic train of
$\delta$-functions, as we have seen. For the infinite square well this would
be a regular sequence of $\delta$-kicks of alternating directions and
separated by a time $T_{\rm D}=2mL/|p|=\sqrt{2m/E_{0}}L$, where we have used
Eq. (16) and the explicit form of $\Omega$. For instance, if the direction of
the kicks is always opposite to the particle motion then

$V_{\rm probe}(x,p)={\rm
sign}(p)F_{0}\sum_{l=-\infty}^{+\infty}\delta(t-lT_{\rm D})x\,,$ (17)

resulting in the required action-angle variables interaction

$V_{\rm probe}(\Theta)\approx f(\Theta-\pi)\sum_{\ell=-\infty}^{\infty}T_{\rm
D}\,\delta(t-lT_{\rm D})\approx f(\vartheta-\pi)\,.$ (18)

Such a potential can be approximated by a periodic sequence of square pulses
$F(t)$ of amplitude $F_{0}$ and duration $\tau\ll T_{\rm D}$.

Combining the transformation $(\ref{xTheta})$ and the approximation (13), such
a sequence of short square pulses gives $f=LF_{0}\tau/\pi T_{\rm D}$. Since
the $\pi$ shift in $\vartheta$ doesn’t affect the Floquet dynamics the
amplitude of the drive force can be obtained from the Floquet-Bloch period as

$F_{0}=\frac{\hbar
n\pi}{\tau}\sqrt{\frac{2m}{E_{0}}}\left(\frac{1}{T_{B}}\right)\,,$ (19)

where we have used Eq. (11). This shows that $F_{0}$ can be determined by a
direct measurement of the frequency $1/T_{\rm B}$ of the Bloch oscillations of
the mean particle position.

_Triangular well_ – We now turn to the situation of a one-dimensional
triangular well

$U(x)=\begin{cases}\eta x&{\rm for}\;x\geq 0\\\ \infty&{\rm
for}\;x<0\,,\end{cases}$ (20)

in which case the position and momentum $x$ and $p$ are related to the action-
angle variables by

$\displaystyle x$ $\displaystyle=$
$\displaystyle\eta(2\pi\Theta-\Theta^{2})/2m\Omega^{2},\,$ $\displaystyle p$
$\displaystyle=$ $\displaystyle\eta(\pi-\Theta)/\Omega\,,$ (21)

where the angular frequency is $\Omega=\eta\pi/\sqrt{2mE_{0}}$ Buchleitner2002
; SM . Importantly, it is now the momentum that is linear in $\Theta$,
suggesting that this potential might find applications in situations where the
probe $V_{\rm probe}$ depends linearly on the momentum $p$ rather than $x$. We
present two examples that exploit such a situation, the first one resulting in
the realization of a tachometer as shown in Fig. 2, and the second in a
magnetometer.

Figure 2: Tachometer based on Floquet-Bloch oscillations.

Consider then first the Hamiltonian of a particle of mass $m$, expressed in a
rotating frame with angular velocity $\vec{w}$. In the presence of the
triangular well $U(x)$ and of a small periodic Floquet perturbation
$V(x)\cos\omega t$ it is given by

$H(\vec{r})=\frac{\vec{p}^{2}}{2m}-\vec{w}\cdot(\vec{r}\times\vec{p})+U(x)+V(x)\cos\omega
t\,.$ (22)

For $\vec{w}$ along the $z$-axis this expression simplifies to

$H_{x}=\frac{p_{x}^{2}}{2m}+U(x)+V(x)\cos\omega t+w_{z}(yp_{x}-p_{y}x)\,.$
(23)

When the position of the particle along $y$ is confined between $0$ and
$y_{0}$ by a periodic sequence of square pulses $g(t)$ of period $T_{\rm D}$
and duration $\tau\ll T_{\rm D}$, approximated by a Dirac comb,

$y(t)=y_{0}\sum_{l=-\infty}^{+\infty}g(t-lT_{{\rm D}})\approx
y_{0}\tau\sum_{l=-\infty}^{+\infty}\delta(t-lT_{{\rm D}})\,,$ (24)

the last term in $H_{x}$ can be interpreted as a probe potential $V_{\rm
probe}=f\vartheta,$ with $f=-w_{z}\eta y_{0}\tau/\pi$ SM . As a result the
system operates as a tachometer, with the Bloch frequency $1/T_{\rm B}$
providing a direct measure of the angular velocity $w_{z}$ via

$w_{z}=\frac{\hbar n}{2\eta y_{0}\tau}\left(\frac{2\pi}{T_{\rm B}}\right)\,.$
(25)

Instead of $x$ in the former case, the transform Eq. (21) justifies that
$T_{\rm B}$ is obtained from the time evolution of the momentum $p_{x}$.

As a concrete example consider a tachometer aimed at measuring angular
velocities of the order of those of the fastest demonstrated spinning objects,
with $w_{z}\sim 10^{9}$rad/s Reimann2018 ; Ahn2018 ; Ahn2020 . For atomic
scale particles of mass $m\sim 10^{-27}$kg whose motion can be controlled at
the nanoscale level, $y_{0}\sim 10^{-9}$m and $\tau\sim 10^{-9}$s, a
perturbation frequency $\omega\sim 100\Omega\sim 10^{10}$Hz, and $\eta\sim
10^{-17}$N, chosen to be much larger than the centripetal force
$mw_{z}^{2}y_{0}$ to ensure that $V_{\rm probe}$ is the weakest perturbation,
we find from Eq. (25) that the Floquet-Bloch period is $T_{\rm B}\sim
10^{-6}$s, which is much shorter than the typical spatial Bloch periods
demonstrated for instance in ultracold atoms ($10^{-3}$s) Dahan1996 . Such
short periods present the considerable advantage of requiring modest demands
on the quantum coherence time of observed objects – we recall that the period
of usual Bloch oscillations of electrons on solid-state lattices is typically
much longer than their quantum coherence time, so that their demonstration has
been limited so far to artificial superlattices Feldmann1992 ; Leo1992  and to
ultracold atoms in optical lattices Dahan1996 . We also note that the use of
the proposed tachometer could possibly be extended to higher angular
frequencies when arranging for $T_{\rm B}$ to still be of the same order of
magnitude, but this would place more severe constraints on the control of
$y_{0}$ and on $\tau$.

Exploiting the formal analogy between inertial and electromagnetic forces
implied by the Larmor’s theorem immediately leads to the possibility of
developing a magnetometer based on the same formal measurement mechanism. As
discussed in Refs. Coisson1973 ; Semon1981 ; Sivardiere1983 , the Hamiltonian
of a particle of mass $m$ and charge $Q$ in a magnetic field $\vec{B}$ can be
obtained by the substitution $\vec{w}\rightarrow Q\vec{B}/2m$ in Eq. (22). For
the case of a uniform magnetic field of amplitude $B_{z}$ along the $z$
direction and considering the same triangular potential and Floquet
perturbation as in the previous example, the Floquet-Bloch driving force
becomes then $f=-QB_{z}\eta y_{0}\tau/2\pi m$ SM . In complete analogy with
Eq. (25) the magnetic field strength is then obtained from $T_{\rm B}$ as

$B_{z}=\frac{m\hbar n}{\eta y_{0}\tau Q}\left(\frac{2\pi}{T_{\rm
B}}\right)\,.$ (26)

For $B_{z}\sim 10^{-18}$T, the highest measurement precision obtained by SQUID
magnetometers Fagaly2006 , a particle with charge at the single-electron
level, $Q\sim 10^{-19}$C, and an atomic scale mass, the resulting weak driving
force $f$ results in a long period $T_{\rm B}$ and hence challenging quantum
coherence time requirements. To obtain $T_{\rm B}\sim 10^{-3}$s with a
perturbation frequency $\omega\sim 10^{6}$Hz and $\eta\sim 10^{-8}$N, the
motion of the particle along the $y$-axis must be controlled at the micron
scale, with $y_{0}\sim 10^{-6}$m and $\tau\sim 10^{-6}$s.

_Other potentials_ – In addition to the examples considered so far, Floquet-
Bloch-oscillation-based measurement schemes can be extended to other
potentials as well, even to some unusual or singular potentials, by following
the same general approach. Examples include the inverse potential $V_{\rm
probe}=-a/x$ corresponding to Coulomb or gravitational force, the logarithmic
potential $V_{\rm probe}=a\log(x)$, and the square-root potential $V_{\rm
probe}=a\sqrt{x}$ found in charged particles’ bounded motion for electrons
Gesztesy1978  and quarks Bose1979 ; Freitas1978 . Their amplitudes $a$, which
are important in a number of problems in nuclear physics and relativistic
quantum mechanics, could in principle be determined by that method. In
general, however, the action-angle transformation results in a form of the
angle variable $\Theta(x,p)$ that is a complicated function of both $x$ and
$p$. Still, we show below that an appropriate design of the trapping potential
$U(x)$ in $H_{0}$ can result in an approximate angle variable $\Theta(x)$ of
the required form to achieve the desired measurement.

From the equations $m\partial_{t}^{2}x=-\partial_{x}U(x)$ and $\Theta=\Omega
t$ we have that

$U(x)=-m\int\frac{\partial^{2}x}{\partial
t^{2}}dx=-m\Omega^{2}\int\frac{\partial^{2}x}{\partial\Theta^{2}}dx.$ (27)

Keeping in mind that $\Theta\in[0,2\pi)$ imposes in general an additional
constraint on of $x$, to ensure that $U(x)$ is a conservative potential, this
equation can be used to find the potential $U(x)$ that results in any desired
relationship between $\Theta$ and $x$,

Consider for example the potential $V_{\rm probe}(x)=-a/x$. A determination of
its amplitude $a$ from the Floquet-Bloch period $T_{\rm B}$ requires now an
action-angle transformation such that $\left|\Theta-\pi\right|=b/x$, where $b$
is some positive constant. From Eq. (27) we must then have
$U(x)=-m\Omega^{2}x^{4}/(2b^{2})$ which can be realized by a repulsive
interaction. In addition, the range restriction $b/\pi<x<L$ with $L\gg b/\pi$
must be imposed on $U(x)$ to match the range of $\Theta$. It follows that
$U(x)$ must be a trapping potential of the form

$U(x)=\begin{cases}-\frac{m\Omega^{2}x^{4}}{2b^{2}}&{\rm
for}\;\frac{b}{\pi}<x<L\,,\\\ \infty&\text{otherwise}\,.\end{cases}$ (28)

This is a one-dimensional infinite well with a negative quartic bottom
profile. The potential energy of the particle is negative, but its kinetic
energy is of course positive. If it dominates the total energy, then the
Floquet dynamics of the particle approaches the previously discussed situation
of one-dimensional infinite square well, which applies as we have seen to the
measurement of spatially invariant forces. A precise measurement of the
amplitude of $V_{\rm probe}(x)$ requires therefore that the quartic potential
energy dominates the total energy, so that $|E_{0}|\ll
m\Omega^{2}b^{2}/(2\pi^{4})$. A periodic driving of the gravitational-like
potential along the lines of Eq. (18) allows then the measurement of $a$
through the Bloch-Floquet period $T_{\rm B}$ SM ,

$a=\frac{\hbar nb}{\tau\Omega}\left(\frac{2\pi}{T_{\rm B}}\right).$ (29)

_Conclusion and outlook_ – In this work we have exploited Floquet engineering
techniques to extend the concept of Bloch oscillations-based precision
measurements to the use of Floquet-Bloch oscillations. Specifically, we showed
that when a quantum system trapped by some potential $U({\bf r})$, the
addition of a weak single-frequency perturbation, combined with a sequence of
$\delta$-kicks, results in the onset the Bloch-like oscillations in its
Floquet phase space. These oscillations present significant advantages and
flexibility over traditional Bloch oscillations for precision measurements, as
a proper design of $U({\bf r})$ and kicking interactions permit to exploit
them for the measurement of a broad variety of observables, including rotation
rates, magnetic fields, and even the strengths of singular potentials.

The present paper concentrated on single-particle physics. Floquet-Bloch
oscillations are vulnerable to dissipation-induced decoherence, and this
imposes limitations of the range of accessible Floquet-Bloch frequencies. A
possible way to circumvent this problem might involve replacing the single-
particle periodic temporal system by Floquet time crystals Else2016 , which
can withstand dissipative environments Gong2018 ; Zhu2019 ; Hemmerich2021 . We
will show in future work how similar measurement schemes as discussed here can
be developed in such systems for properly modulated many-body interactions.

###### Acknowledgements.

_Acknowledgements_ – We acknowledge enlightening discussions with Lu Zhou.
Work was supported by the National Natural Science Foundation of China (Grants
No. 11974116 and No. 11654005), the National Key Research and Development
Program of China (Grant No. 2016YFA0302001), the Fundamental Research Funds
for the Central Universities, the Shanghai Municipal Science and Technology
Major Project under Grant No. 2019SHZDZX01, the Chinese National Youth Talent
Support Program, and the Shanghai Talent Program.

## References

  * (1) S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Q. Niu, and M. G. Raizen, Observation of Wannier-Stark ladders in an accelerating optical potential, Phys. Rev. Lett. 76, 4512 (1996).
  * (2) M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon, Bloch oscillations of atoms in an optical potential, Phys. Rev. Lett. 76, 4508 (1996).
  * (3) M. Gustavsson, E. Haller, M. J. Mark, J. G. Danzl, G. Rojas-Kopeinig, and H.-C. Nägerl, Control of interaction-induced dephasing of Bloch oscillations, Phys. Rev. Lett. 100, 080404 (2008).
  * (4) T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, Optical Bloch oscillations in temperature tuned waveguide arrays, Phys. Rev. Lett. 83, 4752 (1999).
  * (5) R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, Experimental observation of linear and nonlinear optical Bloch oscillations, Phys. Rev. Lett. 83, 4756 (1999).
  * (6) N. Poli, F.-Y. Wang, M. G. Tarallo, A. Alberti, M. Prevedelli, and G. M. Tino, Precision measurement of gravity with cold atoms in an optical lattice and comparison with a classical gravimeter, Phys. Rev. Lett. 106, 038501 (2011).
  * (7) R. Battesti, P. Cladé, S. Guellati-Khélifa, C. Schwob, B. Grémaud, F. Nez, L. Julien, and F. Biraben,, Bloch oscillations of ultracold atoms: A tool for a metrological determination of $h/m_{\rm Rb}$, Phys. Rev. Lett. 92, 253001 (2004).
  * (8) P. Cladé, E. de Mirandes, M. Cadoret, S. Guellati-Khélifa, C. Schwob, F. Nez, L. Julien, and F. Biraben, Determination of the fine structure constant based on Bloch oscillations of ultracold atoms in a vertical optical lattice, Phys. Rev. Lett. 96, 033001 (2006).
  * (9) R. Bouchendira, P. Cladé, S. Guellati-Khélifa, F. Nez, and F. Biraben, New determination of the fine structure constant and test of quantum electrodynamics, Phys. Rev. Lett. 106, 080801 (2011).
  * (10) S. Arlinghaus and M. Holthaus, Generalized acceleration theorem for spatio-temporal Bloch waves, Phys. Rev. B 84, 054301 (2011).
  * (11) K. Kudo and T. S. Monteiro, Theoretical analysis of super-Bloch oscillations, Phys. Rev. A 83, 053627 (2011).
  * (12) E. Haller, R. Hart, M. J. Mark, J. G. Danzl, L. Reichsöllner, and H. Nägerl, Inducing transport in a dissipation-free lattice with super-Bloch oscillations, Phys. Rev. Lett. 104, 200403 (2010).
  * (13) V. Junk, P. Reck, C. Gorini, and K. Richter, Floquet oscillations in periodically driven Dirac systems, Phys. Rev. B 101, 134302 (2020).
  * (14) R. Moessner and S. Sondhi, Equilibration and order in quantum Floquet matter, Nat. Phys. 13, 424 (2017).
  * (15) C. Weitenberg and J. Simonet, Tailoring quantum gases by Floquet engineering, Nat. Phys. 10.1038/s41567-021-01316-x (2021).
  * (16) D. V. Else, B. Bauer, and C. Nayak, Floquet time crystals, Phys. Rev. Lett. 117, 090402 (2016).
  * (17) K. Sacha and J. Zakrzewski, Time crystals: a review, Rep. Prog. Phys. 81, 016401 (2018).
  * (18) H. Zhou, J. Choi, S. Choi, R. Landig, A. M. Douglas, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, P. Cappellaro, H. S. Knowles, H. Park, and M. D. Lukin, Quantum metrology with strongly interacting spin systems, Phys. Rev. X 10, 031003 (2020).
  * (19) L. J. Fiderer and D. Braun, Quantum metrology with quantum chaotic sensors, Nat .Commun. 9, 1351 (2018).
  * (20) M. Jiang, H. Su, Z. Wu, X. Peng, and D. Budker, Floquet maser, Sci. Adv. 7, eabe0719 (2021).
  * (21) A. Buchleitner, D. Delande, and J. Zakrzewski, Non-dispersive wave packets in periodically driven quantum systems, Phys. Rep. 368, 409 (2002).
  * (22) M. Grifoni and P. Hänggi, Driven quantum tunneling, Phys. Rep. 304, 229 (1998).
  * (23) M. Holthaus, Floquet engineering with quasienergy bands of periodically driven optical lattices, J. Phys. B: At. Mol. Opt. Phys. 49, 013001 (2016).
  * (24) See the Supplemental Material for the details of the derivations.
  * (25) Because the Floquet phase space is based on the angle variable $P$ has units of an angular momentum and $M$ of a moment of inertia.
  * (26) These implicitly time-dependent Bloch-like functions correspond to the Floquet eigenstates within the secular approximation, see K. Giergiel, A. Miroszewski, and K. Sacha, Time crystal platform: From quasicrystal structures in time to systems with exotic interactions, Phys. Rev. Lett. 120, 140401 (2018).
  * (27) As we shall see, this does not imply that we limit ourselves to linear interactions is physical space.
  * (28) M. Glück, A. R. Kolovsky, and H. J. Korsch, Wannier-Stark resonances in optical and semiconductor superlattices, Phys. Rep. 366, 103 (2002).
  * (29) R. Reimann, M. Doderer, E. Hebestreit, R. Diehl, M. Frimmer, D. Windey, F. Tebbenjohanns, and L. Novotny, GHz rotation of an optically trapped nanoparticle in vacuum, Phys. Rev. Lett. 121, 033602 (2018).
  * (30) J. Ahn, Z. Xu, J. Bang, Y. Deng, T. M. Hoang, Q. Han, R. Ma, and T. Li, Optically levitated nanodumbbell torsion balance and GHz nanomechanical rotor, Phys. Rev. Lett. 121, 033603 (2018).
  * (31) J. Ahn, Z. Xu, J. Bang, P. Ju, X. Gao, and T. Li, Ultrasensitive torque detection with an optically levitated nanorotor, Nat. Nanotechnol. 15, 89 (2020).
  * (32) J. Feldmann, K. Leo, J. Shah, D. A. B. Miller, J. E. Cunningham, T. Meier, G. von Plessen, A. Schulze, P. Thomas, and S. Schmitt-Rink, Optical investigation of Bloch oscillations in a semiconductor superlattice, Phys. Rev. B 46, 7252 (1992).
  * (33) K. Leo, P. H. Bolivar, F. Brüggemann, R. Schwedler, and K. Köhler, Observation of Bloch oscillations in a semiconductor superlattice, Solid State Commun. 84, 943 (1992).
  * (34) R. Coisson, On the vector potential of Coriolis forces, Am. J. Phys. 41, 585 (1973).
  * (35) M. D. Semon and G. M. Schmieg, Note on the analogy between inertial and electromagnetic forces, Am. J. Phys. 49, 689 (1981).
  * (36) J. Sivardiere, On the analogy between inertial and electromagnetic forces, Eur. J. Phys. 4, 162 (1983).
  * (37) R. L. Fagaly, Superconducting quantum interference device instruments and applications, Rev. Sci. Instrum 77, 101101 (2006).
  * (38) F. Gesztesy and L. Pittner, Electrons in logarithmic potentials I. Solution of the Schrodinger equation, J. Phys. A: Math. Gen. 11, 679 (1978).
  * (39) H. J. W. Müller-Kirsten and S. K. Bose, Solution of the wave equation for the logarithmic potential with application to particle spectroscopy, J. Math. Phys. 20, 2471 (1979).
  * (40) H. Freitas de Carvalho, R. Chanda, and A. Brandão D’Oliveira, The «square-root» potential and the charmonium. Lett. Nuovo Cimento 22, 679 (1978).
  * (41) Z. Gong, R. Hamazaki, and M. Ueda, Discrete time-crystalline order in cavity and circuit QED systems, Phys. Rev. Lett. 120, 040404 (2018).
  * (42) B. Zhu, J. Marino, N. Y. Yao, M. D. Lukin, and E. A. Demler, Dicke time crystals in driven-dissipative quantum many-body systems, New J. Phys. 21, 073028 (2019).
  * (43) H. Keßler, P. Kongkhambut, C. Georges, L. Mathey, J. G. Cosme, and A. Hemmerich, Observation of a dissipative time crystal, Phys. Rev. Lett. 127, 043602 (2021).

Supplemental Materials

##  I Theoretical model

The total Hamiltonian of our model can be divided into four parts,

$H=H_{0}+H_{1}+V_{\rm probe},$ (S1)

where the dominant part

$H_{0}=\frac{p^{2}}{2m}+U(x),$ (S2)

consists of a kinetic energy term and a conservative and time-independent
potential energy term $U(x)$ which, as we will show below, generates a rotor
in the Floquet phase space. The weaker part $H_{1}$ is a time-dependent
perturbation oscillating at a single-frequency $\omega$ and builds a periodic
potential in the Floquet phase space. The weakest term $V_{\rm probe}$
accounts for the time-dependent coupling of the system with the generalized
force to be measured.

###  I.1 Floquet phase space

Without $H_{1,2}$ the energy of the system is conserved, and its trajectories
in the physical phase space $(x,p)$ are closed loops corresponding to periodic
motion. It is then always possible to find a set of canonically conjugate
phase space coordinates adapted to the dynamics of the system where the
Hamilton’s equations of the system take the exceedingly simple form

$\displaystyle\dot{J}$ $\displaystyle=-\frac{\partial
H_{0}}{\partial\Theta}=0\Longrightarrow H_{0}=H_{0}(J),$ (S3)
$\displaystyle\dot{\Theta}$ $\displaystyle=\frac{\partial H_{0}}{\partial
J}=\Omega\Longrightarrow\Theta=\Omega t+\Theta_{0}.$ (S4)

The transformed $H_{0}$ depends on the action variable, traditionally written
$J$, alone, and $J\geq 0$. The conjugate variable is the angle variable
$\Theta$, which evolves in a range $[0,2\pi)$ with a constant angular velocity
$\Omega$, corresponding to the period of the motion $T_{\rm D}=2\pi/\Omega.$
The explicit transforms corresponding to two specific Hamiltonian of interest
in this paper are given in section I.2.

When the system is subjected in addition to a single-frequency perturbation
$H_{1}=V(x)\cos\omega t$ in the physical coordinates, the total Hamiltonian
can be expressed in terms of these action-angle coordinates

$H_{0}+H_{1}=H_{0}(J)+\sum_{\ell=-\infty}^{+\infty}V_{\ell}(J)e^{i\ell\Theta}\cos\omega
t,$ (S5)

where since $\Theta$ is $2\pi$-periodic we have expanded strength of the
perturbation $V(J,\theta)$ by the Fourier series

$V_{\ell}(J)=\frac{1}{2\pi}\int_{0}^{2\pi}V(x)e^{-i\ell\Theta}d\Theta.$ (S6)

For a small perturbation, the phase space trajectories of the perturbed
dynamics will remain close to the unperturbed ones, that is
$\Theta\approx\Omega t$ and $\ell\Theta-\omega t\approx(\ell\Omega-\omega)t$.
If the frequency of the perturbation is such that

$\omega\approx n\Omega,\>n=1,2,3\cdots,$ (S7)

we can take the secular approximation since one then expects that all terms in
the summation oscillate rapidly and average out to zero except the near-
resonant term $V_{n}\cos(n\Theta-\omega t)$. We then perform a time-dependent
canonical transform of coordinates involving the slowly varying variable

$\displaystyle\vartheta$ $\displaystyle=\Theta-\frac{\omega t}{n},$ (S8)

so that the effective secular Hamiltonian becomes a time-independent one in
the new phase space $(I,\vartheta)$,

$H_{0}+H_{1}\approx H_{0}(J)+V_{n}(J)\cos n\vartheta-\frac{\omega J}{n},$ (S9)

where the last term, $-\omega J/n$ is the contribution of the canonical
momentum conjugate to time. Because this Hamiltonian is nothing but the
classical analog of the quantum Floquet Hamiltonian we call the extended phase
space $(J,\vartheta)$ the Floquet phase space below.

For small $V_{n}$, it is reasonable to perform a power expansion of the
unperturbed Hamiltonian in the vicinity of the resonant action $J_{0}$,

$H_{0}+H_{1}\approx H_{0}(J_{0})-\frac{\omega
J_{0}}{n}+\left(\left.\frac{\partial H_{0}}{\partial
J}\right|_{J=J_{0}}-\frac{\omega}{n}\right)(J-J_{0})+\frac{1}{2}\left.\frac{\partial^{2}H_{0}}{\partial
J^{2}}\right|_{J=J_{0}}(J-J_{0})^{2}+V_{n}(J_{0})\cos n\vartheta,$ (S10)

where the constant zero-order term can be removed, and the first-order term is
zero due to that $J_{0}$ is the action corresponding precisely to the resonant
condition $\omega=n\Omega$. If we define a new ‘conjugate momentum’

$P=J-J_{0},$ (S11)

and the ‘effective mass’

$M=(\partial^{2}H_{0}/\partial J^{2}_{0})^{-1},$ (S12)

the effective Hamiltonian takes the same form as one of a particle in lattice
potential,

$H_{0}+H_{1}\approx\frac{P^{2}}{2M}+V_{n}(J_{0})\cos n\vartheta.$ (S13)

For convenience we remove the subscript of the action $J_{0}$ in the main
text. Besides, the units of $P$ and $M$ are actually same as angular momentum
and moment of inertia, respectively because the Floquet phase space is based
on the angle variable.

###  I.2 Action-angle transform

If $H_{0}$ dominates over all other terms in the Hamiltonian and $H_{1,2}$ can
be ignored the motion of the system is periodic, and the action-angle
transform is realized by an integration over one period of the motion,

$J_{0}=\frac{1}{2\pi}\oint
pdx=\frac{1}{\pi}\int_{x_{min}}^{x_{max}}\sqrt{2m(E_{0}-U(x))}dx,$ (S14)

where $E_{0}$ is the energy of the system and $x_{max,min}$ represent two peak
positions of the periodic motion in the physical coordinates. The angular
velocity is given by

$\Omega=\frac{\partial H_{0}}{\partial J_{0}},$ (S15)

and the relation between $\Theta$ and the physical coordinates is given by the
integral

$\Theta-\Theta_{0}=\frac{\partial}{\partial J_{0}}\int_{x_{0}}^{x}pdx,$ (S16)

where $\Theta_{0}$ and $x_{0}$ represent their initial values.

For example, if the conservative potential in $H_{0}$ is a 1D infinite square
well,

$U(x)=\begin{cases}0,&0<x<L\\\ \infty,&\text{otherwise}\end{cases},$ (S17)

from Eqs. (S12, S14, S15) we obtain

$\displaystyle J_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{0}^{L}\sqrt{2mE_{0}}=\frac{L}{\pi}\sqrt{2mE_{0}},$
(S18) $\displaystyle\Omega$ $\displaystyle=$ $\displaystyle\frac{\partial
E_{0}}{\partial J_{0}}=\frac{\pi^{2}J_{0}}{mL^{2}},$ (S19) $\displaystyle M$
$\displaystyle=$ $\displaystyle\left(\frac{\partial^{2}E_{0}}{\partial
J_{0}^{2}}\right)^{-1}=\frac{mL^{2}}{\pi^{2}}.$ (S20)

The integral (S16) gives

$\Theta-\Theta_{0}=\frac{\partial}{\partial
J_{0}}\int_{0}^{x}\sqrt{2mE_{0}}dx=\frac{\pi}{L}xn.$ (S21)

Further taking into account that $x\in(0,L)$ and $\Theta\in(0,2\pi)$, we then
obtain

$\displaystyle x$ $\displaystyle=$
$\displaystyle\frac{L}{\pi}\left|\pi-\Theta\right|,$ (S22) $\displaystyle p$
$\displaystyle=$ $\displaystyle m\dot{x}=-\sqrt{2mE_{0}}\
\rm{sign}(\sin\Theta),$ (S23)

which also gives

$\pi-\Theta={\rm sign}(p)\frac{\pi x}{L}.$ (S24)

Similarly, if the conservative potential is one-dimensional triangular well

$U(x)=\begin{cases}\eta x,&x\geq 0\\\ \infty,&x<0\end{cases},$ (S25)

we obtain

$\displaystyle J_{0}$ $\displaystyle=$
$\displaystyle\frac{(2E_{0})^{3/2}\sqrt{m}}{3\eta\pi},$ (S26)
$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\frac{\eta\pi}{\sqrt{2mE_{0}}},$ (S27) $\displaystyle M$
$\displaystyle=$ $\displaystyle-\frac{4mE_{0}^{2}}{\eta^{2}\pi^{2}},$ (S28)
$\displaystyle x$ $\displaystyle=$
$\displaystyle\frac{E_{0}}{\eta}-\frac{\eta(\pi-\Theta)^{2}}{2m\Omega^{2}},$
(S29) $\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{\eta}{\Omega}(\pi-\Theta).$ (S30)

As a final example, if the conservative potential is an infinite one-
dimensional well with a quartic bottom profile,

$U(x)=\begin{cases}-\frac{m\Omega_{0}^{2}x^{4}}{2b^{2}},&\frac{b}{\pi}<x<L\\\
\infty,&\text{otherwise}\end{cases},$ (S31)

where the boundary $L\gg b/\pi>0$, for a small initial energy, $|E_{0}|\ll
m\Omega_{0}^{2}b^{2}/(2\pi^{4})$, we can obtain the approximate expressions,

$\displaystyle J_{0}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2mE_{0}}}{\pi}\int_{b/\pi}^{L}\sqrt{1+\frac{m\Omega_{0}^{2}x^{4}}{2b^{2}E_{0}}}$
(S32) $\displaystyle\approx$ $\displaystyle\frac{m\Omega_{0}L^{3}}{3\pi
b}+\frac{E_{0}}{\Omega_{0}}-\frac{\pi^{4}E_{0}^{2}}{10mb^{2}\Omega_{0}^{3}},$
$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle\frac{\partial
E_{0}}{\partial J_{0}}\approx\Omega_{0},$ (S33) $\displaystyle M$
$\displaystyle=$ $\displaystyle\left(\frac{\partial^{2}E_{0}}{\partial
J_{0}^{2}}\right)^{-1}\approx\frac{5mb^{2}}{\pi^{4}}-\frac{2E_{0}}{\Omega_{0}^{2}},$
(S34) $\displaystyle x$ $\displaystyle\approx$
$\displaystyle\frac{b}{|\pi-\Theta|+b/L},$ (S35) $\displaystyle p$
$\displaystyle\approx$ $\displaystyle{\rm
sign}(\sin\Theta)\sqrt{2mE_{0}+\frac{m^{2}\Omega_{0}^{2}b^{2}}{(\Theta-\pi)^{4}+b^{4}/L^{4}}}.$
(S36)

It follows that in the case of a probe potential of the form $V_{\rm
probe}(x,t)=-a/x$, the inverse $x$-dependence results in a linear angular
dependence in the phase space,

$-\frac{a}{x}\approx\frac{a}{L}-\frac{a}{b}|\pi-\Theta|.$ (S38)

##  II Measurement schemes

###  II.1 Tachometer

The Hamiltonian for a particle of mass $m$ in a rotating frame of the angular
velocity $\vec{w}$ is

$H(\vec{r})=\frac{1}{2m}(\vec{p}-m\vec{w}\times\vec{r})^{2}-\frac{m}{2}(\vec{w}\times\vec{r})^{2}.$
(S39)

If $\vec{w}$ is along the z-axis, with the triangular well $U(x)$ and the
Floquet perturbation $V(x)\cos\omega t$, the effective Hamiltonian for the
particle in the $x$ dimension is

$H_{x}=\frac{p_{x}^{2}}{2m}+U(x)+V(x)\cos\omega t+w_{z}yp_{x}-w_{z}xp_{y},$
(S40)

When the position of the particle along $y$ is modulated by a periodic
sequence approximately in a Dirac-comb form,

$y(t)\approx y_{0}\tau\sum_{l=-\infty}^{+\infty}\delta(t-lT_{{\rm
D}})=\frac{y_{0}\tau}{T_{\rm D}}\sum_{l=-\infty}^{+\infty}e^{il\Omega t},$
(S41)

the last two term in $H_{x}$ can be interpreted as a probe potential $V_{\rm
probe}$ in the Floquet phase space. For the first term $w_{z}yp_{x}$,
considering the transform Eq. (S30) and applying the secular approximation

$w_{z}yp_{x}=\frac{w_{z}y_{0}\tau}{T_{\rm
D}}\sum_{l=-\infty}^{+\infty}e^{il\Omega
t}\frac{\eta}{\Omega}\sum_{k=1}^{\infty}\frac{2\sin
k\Theta}{k}\approx\frac{w_{z}y_{0}\tau\eta}{2\pi}(\pi-\vartheta).$ (S42)

For the second term $-w_{z}xp_{y}$, in the rotating frame
$p_{y}=m\dot{y}+mw_{z}x$ and then
$-w_{z}xp_{y}=-w_{z}x\dot{y}-mw_{z}^{2}x^{2}$, but we neglect the potential
$-mw_{z}^{2}x^{2}$ for its weaker effect compared with $U(x)$, considering the
transform Eq. (S29) and applying the secular approximation, it gives a same
contribution as term $w_{z}yp_{x}$,

$\displaystyle-w_{z}xp_{y}$ $\displaystyle\approx$
$\displaystyle-\frac{w_{z}y_{0}\tau}{T_{\rm
D}}\sum_{l=-\infty}^{+\infty}il\Omega e^{il\Omega
t}\frac{\eta}{2m\Omega^{2}}(\frac{8\pi^{2}}{3}-\sum_{k=1}^{\infty}\frac{4\cos
k\Theta}{k^{2}})$ (S43) $\displaystyle\approx$
$\displaystyle\frac{w_{z}y_{0}\tau\eta}{2\pi}(\pi-\vartheta).$

Then the probe potential

$V_{\rm probe}=w_{z}(yp_{x}-xp_{y})\approx f(\vartheta-\pi),$ (S44)

with

$f=-\frac{w_{z}\eta y_{0}\tau}{\pi},$ (S45)

and $w_{z}$ can be obtained from the Floquet-Bloch period $T_{\rm B}=\hbar
n/|f|$ via

$w_{z}=\frac{\hbar n\pi}{T_{\rm B}\eta y_{0}\tau}.$ (S46)

###  II.2 Magnetometer

The Hamiltonian for a charged particle of mass $m$ and charge $Q$ in an
electromagnetic field with vector and scalar potentials $\vec{A}$ and $\phi$
is

$H=\frac{(\vec{p}-Q\vec{A})^{2}}{2m}+Q\phi+U(x)+V(x)\cos\omega t.$ (S47)

If the electromagnetic field is a uniform magnetic field $\vec{B}$ along the
$z$ direction, one has in the symmetry gauge $\nabla\cdot\vec{A}=0$ and
$\vec{A}=-\vec{r}\times\vec{B}/2$,

$Q\vec{A}=QB_{z}(x\vec{e}_{y}-y\vec{e}_{x})/2,$ (S48)

and the Hamiltonian $H$ for $x$ dimension simplifies to

$H_{x}=\frac{p_{x}^{2}}{2m}+U(x)+V(x)\cos\omega
t+\frac{QB_{z}}{2m}(yp_{x}-xp_{y})+\frac{Q^{2}B_{z}^{2}x^{2}}{8m}.$ (S49)

After neglecting the last potential that is much weaker than $U(x)$, this
Hamiltonian is same as the one for tachometer in Eq. (S40) except for the
replacement $w_{z}\rightarrow QB_{z}/2m$. Similarly, the Floquet-Bloch driving
force

$f=-\frac{QB_{z}\eta y_{0}\tau}{2\pi m},$ (S50)

and

$B_{z}=\frac{2\pi m\hbar n}{\eta y_{0}\tau QT_{\rm B}}.$ (S51)

We note that the Floquet-Bloch driving force $f$ and the period $T_{\rm B}$
are independent on the electromagnetic gauge. For instance in the Landau gauge

$Q\vec{A}=-QB_{z}y\vec{e}_{x},$ (S52)

and then the probe potential

$V_{\rm probe}=\frac{QB_{z}}{m}yp_{x},$ (S53)

which is different from the case in the symmetry Coulomb gauge, but the double
coefficient $QB_{z}/m$ results a same value of $f$.