# Studying and applying magnetic dressing with a Bell and Bloom magnetometer

Giuseppe Bevilacqua  Valerio Biancalana  Dept. of Information Engineering and
Mathematics - DIISM, University of Siena – Via Roma 56, 53100 Siena, Italy
Yordanka Dancheva  Dept. of Physical Sciences, Earth and Environment - DSFTA,
University of Siena – Via Roma 56, 53100 Siena, Italy; currently at Aerospazio
Tecnologie Rapolano T. (SI) - Italy  Antonio Vigilante  Department of Physics
and Astronomy, University College London, Gower Street,
London WC1E 6BT, United Kingdom<EMAIL_ADDRESS>
###### Abstract

The magnetic dressing phenomenon occurs when spins precessing in a static
field (holding field) are subject to an additional, strong, alternating field.
It is usually studied when such extra field is homogeneous and oscillates in
one direction. We study the dynamics of spins under dressing condition in two
unusual configurations. In the first instance, an inhomogeneous dressing field
produces space dependent dressing phenomenon, which helps to operate the
magnetometer in strongly inhomogeneous static field. In the second instance,
beside the usual configuration with static and the strong orthogonal
oscillating magnetic fields, we add a secondary oscillating field, which is
perpendicular to both. The system shows novel and interesting features that
are accurately explained and modelled theoretically. Possible applications of
these novel features are briefly discussed.

##  1 Introduction

The physics of a precessing magnetization in a static magnetic field is well-
known since the studies performed by Joseph Larmor at the end of the 19th
century. The phenomenology of precessing systems is enriched when additional
time-dependent fields are introduced. In many applications like magnetic
resonance (MR) experiments an oscillating transverse field (which is usually
much weaker than the static one) is applied resonantly to the Larmor one
$\Omega_{\mathrm{L}}/2\pi$. In this case the resonant condition helps in
describing the system dynamics in the rotating wave approximation: the
oscillating field is seen as a superposition of two counter-rotating
components, one of which appears to be static in the reference frame rotating
at $\Omega_{\mathrm{L}}$ around the holding field: it causes a slow precession
in that rotating frame.

The presence of a transverse (off-resonant) alternating field arouse interest
also in the complementary regime, when its strength exceeds the static one.
This condition is technically unfeasible in conventional NMR at Tesla level,
while it is more easily accessible in atomic physics experiments, where the MR
is commonly studied in much weaker fields, e.g. at microtesla level.

The seminal work studying precessing spins subjected to a strong, off-
resonant, transverse field (dressing field) dates back to the late Sixties,
when S.Haroche and co-workers [1] introduced the concept of magnetic dressing.
That contribution started a vivid activity around the subject [2, 3, 4, 5]
Recently, the magnetic dressing was studied also with cold atoms and
condensates [6, 7]. The magnetic dressing of atoms offers a powerful tool in
quantum control experiments [6, 8] and high resolution magnetometry [9].

Our group devoted research activity to characterize the magnetic dressing
phenomenon occurring when an atomic sample is subject to an anharmonic
dressing field [10]. More recently, the potential of applying a non-uniform
dressing field to counteract the effects of static field inhomogeneities was
demonstrated [11]. This has important implications in NMR imaging (MRI) in the
ultra low field regime [12].

As a general feature, the mentioned works studied the magnetic dressing in
configurations where the precession around the holding field is modified by
one time dependent field oriented along one direction in the plane
perpendicular to the holding one (the precession plane).

In a more recent work [13], we studied theoretically and experimentally new
features that emerge when the design of the dressing field uses both the
dimensions of the precession plane. We addressed the case when such time-
dependent two-dimensional field is periodic in time. The periodicity is
obtained by superposing two perpendicular field components, which oscillate
harmonically with various amplitudes and with frequencies that are equal or in
(small) integer ratio $p$.

The system is studied by means of a model that produces analytical results
thanks to a perturbative approach. This approach requires that one of the
time-dependent components is much larger than both the static field and the
other oscillating term. Despite its weakness, the latter (denominated tuning
field) plays an important role.

This paper is organized as follows: in Sec.2, we describe the model and derive
the expression for the effective precession frequency in terms of the dressing
parameters (amplitudes, frequencies and relative phase of the dressing and
tuning fields); in Sec.3, we describe a practical application of magnetic
dressing, which makes a Bell and Bloom atomic magnetometer suited to operate
also in the presence of a strong field gradient. In Sec.4, we report
experimental results obtained in the tuning-dressing configuration, and
consider possible applications where our findings may be of interests. An
appendix deepens some technical aspects of the calculations on which the model
is based.

##  2 Model

The evolution of atomic magnetization in an external field is commonly
described using the Bloch equations. In this work, the considered atomic
sample is a buffered Cs cell at the core of a Bell & Bloom magnetometer [14].
The presence of a weak pump radiation synchronously modulated at the effective
Larmor frequency compensates the weak relaxation phenomena and enables the use
of a model that determines the steady-state solution on the basis of the
simpler Larmor equation.

Consider the Larmor equation
$\mathbf{\dot{M}}=\gamma\mathbf{B}\times\mathbf{M}$ for the atomic
magnetization $\mathbf{M}$ in presence of a magnetic field of the form

$\mathbf{B}=B_{1}\cos(\Omega t)\mathbf{\hat{x}}+B_{2}\cos(p\Omega
t+\phi)\mathbf{\hat{y}}+B_{3}\mathbf{\hat{z}}$ (1)

with $p$ an integer. We refer to $B_{3}$ as static field, to $B_{1}$ as
dressing field and to $B_{2}$ as tuning field, respectively. Using the
adimensional time $\tau=\Omega t$ one obtains explicitly ($\omega_{i}=\gamma
B_{i}$)

$\frac{d\,\mathbf{M}}{d\,\tau}=\left[\frac{\omega_{1}}{\Omega}\cos(\tau)\,A_{1}+\frac{\omega_{2}}{\Omega}\cos(p\tau+\phi)\,A_{2}+\frac{\omega_{3}}{\Omega}A_{3}\right]\mathbf{M},$
(2)

where the three-dimensional matrices $A_{i}$ are antisymmetric with only two
elements different from zero i.e. $(A_{1})_{2,3}=-(A_{1})_{3,2}=-1$. $A_{2}$
and $A_{3}$ are obtained from cyclic permutations of the indices: see eqs.16
in the appendix.

Assuming that $\xi=\omega_{1}/\Omega\gg\omega_{2}/\Omega,\omega_{3}/\Omega$
the perturbation theory let factorize the time evolution operator $U(\tau)$
($\mathbf{M}(\tau)\equiv U(\tau)\mathbf{M}(0)$) in the interaction
representation as

$U(\tau)=\exp{\left[\xi\
\sin\tau\,A_{1}\right]}\,U_{I}(\tau)=\begin{pmatrix}1&0&0\\\
0&\cos\varphi&-\sin\varphi\\\
0&\sin\varphi&\cos\varphi\end{pmatrix}U_{I}(\tau),$ (3)

where $\varphi=\xi\ \sin\tau$, obtaining a dynamical equation for
$U_{I}(\tau)$: $dU_{I}/d\tau$ is given by

$\left[\left(\frac{\omega_{3}}{\Omega}\sin\varphi+\frac{\omega_{2}}{\Omega}\cos\varphi\,\cos(p\tau+\phi)\right)A_{2}+\left(\frac{\omega_{3}}{\Omega}\cos\varphi-\frac{\omega_{2}}{\Omega}\sin\varphi\,\cos(p\tau+\phi)\right)A_{3}\right]U_{I}\equiv\epsilon
A(\tau)U_{I},$

where the parameter $\epsilon$ is used to identify the various orders of
perturbation theory and to label the corresponding terms.

The matrix $A(\tau)$ is periodic $A(\tau+2\pi)=A(\tau)$ so we can use the
Floquet theorem and write

$U_{I}(\tau)=\e^{\Lambda(\tau)}\,\e^{\tau\,F}$ (4)

with $\Lambda(0)=0$ and $\Lambda(\tau+2\pi)=\Lambda(\tau)$, and the Floquet
matrix $F$ is time-independent. Next, we expand $U_{I}$ following the Floquet-
Magnus [15] expansion

$\Lambda=\epsilon\Lambda_{1}+\epsilon^{2}\Lambda_{2}+\cdots\,,\,\,\,\,F=\epsilon
F_{1}+\epsilon^{2}F_{2}+\cdots$ (5)

where the first terms are obtained from the relations

$F_{1}=\frac{1}{2\pi}\int_{0}^{2\pi}A(\tau)\mathrm{d}\tau\,,\,\,\,\,\Lambda_{1}(\tau)=\int_{0}^{\tau}A(\tau^{\prime})\mathrm{d}\tau^{\prime}-\tau
F_{1}.$ (6)

Using the relation involving the Bessel functions $J_{n}$

$\e^{i\,z\sin\theta}=\sum_{n=-\infty}^{+\infty}J_{n}(z)\,\e^{i\,n\,\theta}$
(7)

one finds

$\displaystyle\int_{0}^{\tau}\cos(\varphi(\tau^{\prime}))\mathrm{d}\tau^{\prime}$
$\displaystyle=J_{0}(\xi)\tau+f_{1}(\tau)$ (8a)
$\displaystyle\int_{0}^{\tau}\sin(\varphi(\tau^{\prime}))\mathrm{d}\tau^{\prime}$
$\displaystyle=f_{2}(\tau)$ (8b)
$\displaystyle\int_{0}^{\tau}\cos(\varphi(\tau^{\prime}))\cos(p\tau^{\prime}+\phi)\mathrm{d}\tau^{\prime}$
$\displaystyle=\frac{1+(-1)^{p}}{2}\tau J_{p}(\xi)\cos\phi+f_{3}(\tau)$ (8c)
$\displaystyle\int_{0}^{\tau}\sin(\varphi(\tau^{\prime}))\cos(p\tau^{\prime}+\phi)\mathrm{d}\tau^{\prime}$
$\displaystyle=\frac{-1+(-1)^{p}}{2}\tau J_{p}(\xi)\sin\phi+f_{4}(\tau).$ (8d)

The functions $f_{i}(\tau)$ are reported in the appendix (see eqs. 19 \- 22).

Using the eq. (8) one finds

$F_{1}=\begin{cases}\frac{\omega_{3}}{\Omega}J_{0}(\xi)\,A_{3}+\frac{\omega_{2}}{\Omega}J_{p}(\xi)\cos\phi\,A_{2}&p\,\mathrm{even}\\\
\left[\frac{\omega_{3}}{\Omega}J_{0}(\xi)+\frac{\omega_{2}}{\Omega}J_{p}(\xi)\sin\phi\right]A_{3}&p\,\mathrm{odd}\end{cases}$
(9)

and

$\Lambda_{1}=\left(\frac{\omega_{3}}{\Omega}f_{2}(\tau)+\frac{\omega_{2}}{\Omega}f_{3}(\tau)\right)A_{2}+\left(\frac{\omega_{2}}{\Omega}f_{1}(\tau)-\frac{\omega_{2}}{\Omega}f_{4}(\tau)\right)A_{3}$
(10)

Using the eq. (9) (see appendix for details) we can calculate the dressed
Larmor frequency measured in the experiment for the component of the
magnetization parallel to the dressing field:

$\Omega_{\mathrm{L}}=\begin{cases}\sqrt{\omega_{3}^{2}J_{0}^{2}(\xi)+\omega_{2}^{2}J_{p}^{2}(\xi)\
\cos^{2}\phi}&p\;\mathrm{even}\\\
\left|\omega_{3}J_{0}(\xi)+\omega_{2}J_{p}(\xi)\
\sin\phi\right|&p\;\mathrm{odd}\end{cases}$ (11)

The eq. (11) is at the focus of this paper. It includes also the well known
and simpler case of a single dressing field: for zero values of the tuning
field (i.e. $\omega_{2}=0$) it gives
$\Omega_{\mathrm{L}}=|J_{0}(\xi)\omega_{3}|$. The latter is the instance
discussed in Sec.3, where only the dressing field is applied, with the
peculiarity of a position-dependent parameter: $\xi=\xi(x)$. The more general
case, with the presence of both the dressing and the tuning fields, is
discussed in Sec.4.

##  3 Atomic resonance enhancement with inhomogenous dressing

An interesting magnetometric application of the dressing phenomenon is based
on applying a position-dependent dressing (and/or tuning) field to compensate
atomic MR broadening caused by inhomogeneities of the static field. Unless
counteracted, the broadening induced by strongly inhomogeneous static field
would severely deteriorate the performance of the magnetometer.

Details of the magnetometric experimental apparatus can be found elsewhere
[16]. Beside the field control system described in ref. [16], here several
additional coils enable the application of the time dependent (dressing and
tuning) fields. In particular, the stronger dressing field is applied with a
solenoidal coil surrounding the sensor and the weaker tuning field is applied
by means of an Helmholtz pair. Alternatively, when an inhomogenous dressing
field is required, it is produced by a small source (a coil wound on a hollow-
cylinder ferrite, with the laser beams passing across the hole) placed in the
proximity of the atomic cell.

As discussed in Refs.[11, 12], the application of inhomogenous dressing field
paves the way to detect NMR imaging signals in loco, that is with an atomic
sensor operating in the same place where the NMR sample is. In fact, the
frequency encoding technique used in MRI requires the application of static
gradients which would destroy the atomic resonance.

The basic principle of operation can be summarized as follows: let the NMR
sample and the sensor atom be in a static field $(B_{3}(x))$ with an intensity
dependent on the $x$ co-ordinate , thus
$\omega_{3}=\omega_{3}(x)=\gamma(B_{3}(0)+Gx)$, where $G=\partial
B_{3}/\partial x$ is the gradient applied to the frequency encoding purpose,
which in our case is up to about 150 nT/cm. Over the centimetric size of the
atomic sample, such a gradient would broaden the atomic MR from its original
few-Hz width up to kHz level, smashing it completely and preventing any
magnetometric measurement.

Figure 1: 1-D imaging of a structured (four disks) water sample contained in
the plastic cartridge shown in the picture (b). The plots (a) represent MRI
profiles, as obtained when the sample occupies four different positions
$x_{s}$. Both the structure details and the positions (actual $x_{s}$ values
are reported the right-side y-axes) are determined with submillimetric
precision.

The narrow width of the atomic magnetic resonance can be restored by applying
tailored tuning and/or dressing fields. We will consider here the case of
single field dressing, where the gradient $G$ makes $\Omega_{\mathrm{L}}$
position-dependent according to

$\Omega_{\mathrm{L}}(x)=|J_{0}(\xi(x))\omega_{3}(x)|,$ (12)

where we have considered a position dependent dressing field, making the
dressing parameter $\xi$ dependent on $x$, as well.

An appropriate inhomogeneity of $\xi$ makes possible to achieve the condition
$\partial\Omega_{L}/\partial x=0$, under which the atomic resonance width is
restored. It is worth noting that the nuclear precession is unaffected by the
dressing field. The much smaller gyromagnetic factor, makes their dressing
parameter vanishing and the effects of $B_{1}$ negligible: the MRI frequency
encoding based on $G$ is preserved.

In this configuration, $B_{1}$ is inhomogeneous, and, in a dipole
approximation, at a distance $x$ from the cell center, the dressing field
$B_{x}$ is

$B_{x}(x,t)=\frac{\mu_{0}}{2\pi}\frac{m(t)}{(x_{0}+x)^{3}}=B_{1}(x)\cos(\Omega
t),$ (13)

where $\mu_{0}$ is the vacuum permittivity, $m(t)=m_{0}\cos(\Omega t)$ is the
oscillating dipole momentum, and $x_{0}$ is the distance of the dipole from
the cell center. Taking now into account the dependence on $x$ of both the
static and the dressing fields, accordingly with the eq.11, the dressed
angular frequency results

$\Omega_{L}(x)=\frac{\gamma_{\mathrm{Cs}}}{2\pi}\left(B_{0}+Gx\right)J_{0}\left(\frac{\gamma_{\mathrm{Cs}}B_{1}(x)}{\Omega}\right),$
(14)

or, in a first-order Taylor approximation,

$\begin{split}\Omega_{\mathrm{L}}(x)=\Omega_{\mathrm{L}}(0)+\Omega_{\mathrm{L}}^{\prime}(0)\,x+O(x^{2})\approx\frac{\gamma_{\mathrm{Cs}}}{2\pi}\left(B_{0}J_{0}(\alpha)+\left[\frac{3B_{0}\alpha
J_{1}(\alpha)}{x_{0}}+GJ_{0}(\alpha)\right]x\right),\\\ \end{split}$

where $\alpha=(\mu_{0}/2\pi)(\gamma_{\mathrm{Cs}}m_{0})/(\Omega x_{0}^{3})$.
In conclusion, the condition for compensating the effect of the gradient $G$
is found to be:

$-3\frac{B_{0}}{x_{0}}\frac{\alpha J_{1}(\alpha)}{J_{0}(\alpha)}=G.$ (15)

This shows that with an appropriate choice of $\alpha$, the width of the
atomic magnetic resonance is substantially restored, as to recover the
magnetometer performance to a level making it suited to detect weak MRI
signals.

The Figure 1 shows MRI profiles obtained with this methodology. The frequency
encoding is here obtained with a field gradient $G\approx 100$nT/cm. The
presence of $G$ broadens the atomic MR from its original 25 Hz up to about 800
Hz. Its width is then restored down to 35 Hz by inhomogeneous dressing.

##  4 Tuning-dressing experiment

The Eq.11 expresses the dependence of dressed angular frequency
$\Omega_{\mathrm{L}}$ on the experimental parameters (strengths, relative
phase and frequencies of the tuning and dressing fields). The predicted
behaviour is verified using the mentioned Bell and Bloom magnetometer.
Measurements are made with $p=1,2,3$, and $\Omega_{\mathrm{L}}$ is recorded as
a function of $\omega_{1}$ and $\phi$.

(a)

(b)

Figure 2: (a) $\Omega_{\mathrm{L}}$ as a function of the relative phase $\phi$
between the fields $B_{1}$ and $B_{2}$. In black dots the experimental
measurements, in blue the theoretical predictions. Upper, middle and bottom
panels refer respectively to the cases with $p=1,2,3$, in which $\xi$ has been
fixed to $1.38,3.83,1.54$. (b) $\Omega_{\mathrm{L}}$ as a function of the
Bessel functions argument $\xi$. In black dots the experimental measurements,
in blue the theoretical predictions. Upper, middle and bottom panels refer
respectively to the cases with $p=1,2,3$. In the cases with an odd $p$ the
phase between $B_{1}$ and $B_{2}$ is fixed to $\phi=\pi/2$, when $p$ is even
the phase $\phi$ is set to zero. In each plot, $\xi$ changes varying the value
of $B_{1}$, which ranges from zero to $\approx 15\ \mu$T.

Figure 2 shows comparisons of measured and calculated values of
$\Omega_{\mathrm{L}}$ as a function of $\phi$ for given $\xi$ and as a
function of $\xi$ for given $\phi$, respectively. The data shown in Figure
2(b) are measured with the $\phi$ values that maximize the tuning effect. i.e.
is $\phi=\pi/2$ for the odd $p$ values and $\phi=0$ for the even one.

There is an excellent accordance between the experimental data and the
theoretical prediction. In particular, both the dependence of the relative
phase $\phi$ and on the pertinent $J_{p}$ (with $p=1,2,3$) Bessel functions
are perfectly verified by the experiment. Minor deviations can be attributed
to experimental imperfections, mainly to not-exact perpendicularity of the
applied fields $B_{1}$ and $B_{2}$.

The developed model and its excellent correspondence with the experimental
results demonstrates the possibility to enhance the control of spins evolution
by means of the described tuning-dressing field arrangement.

This enhancement opens up to a variety of new experimental configurations in
which the new set of parameters ($B_{2}$, $p$ and $\phi$) add to $B_{1}$ and
$\Omega$ to make new handles available to finely control the atomic
magnetization dynamics. Compared to the known cases of harmonic [1] or
anharmonic [10] dressing field oscillating along one direction, noticeably
here the dressed frequency $\Omega_{\mathrm{L}}$ (eq.11) may exceed
$\omega_{3}$.

The tuning-dressing scheme makes possible to chose a parameter set to achieve
an arbitrary $d\Omega_{\mathrm{L}}/d\xi$ in conjunction with an arbitrary (to
some extent) value of $\Omega_{\mathrm{L}}$. For instance, it is possible to
produce a condition of critical dressing (equalization of precession
frequencies of different species) [17, 9] with no first-order dependence on
$\xi$, so to attenuate the detrimental effects caused by $B_{1}$
inhomogeneities, which constitute a severe limiting problem in high-resolution
experiments [9].

Similarly, it is possible to fulfill the condition of a large
$d\Omega_{\mathrm{L}}/d\xi$ avoiding the constraint of a strong
$\Omega_{\mathrm{L}}$ reduction. The latter may help in applications like that
described in Sec.3, where $\Omega_{\mathrm{L}}$ is made deliberately position-
dependent by means of a spatially inhomogeneous $\xi$. In addition, for that
kind of application, it is worth noting that the presented scheme makes it
possible to render $\Omega_{\mathrm{L}}$ space dependent by means of an
inhomogeneity of the field $B_{2}$, which is of easier implementation and
control, being $B_{2}\ll B_{1}$.

Other applications where the tuning-dressing scheme may offer important
potentials is suggested by the dependence on $\phi$. As recently reported
[18], an emerging application of highly sensitive magnetometers concerns the
detection of targets made of weakly conductive materials. In that case, the
typical setup is based on a radio-frequency magnetometer, where the target
modifies the amplitude or the phase of a (resonant) radio-frequency field
driving the magnetometer. Alternative setups could be developed, where the
target modifies the field $B_{2}$, whose frequency is not required to match
the atomic resonance. In this case, provided that $J_{p}(\xi)$ is large (e.g.
$\xi\approx 1.84$ in the case of $p=1$), the system would have a large
response to any variation of either the amplitude (if $\phi=\pm\pi/2$) or the
phase (if $\phi=0,\pi$) of $B_{2}$ caused by eddy currents induced in the
target.

##  Appendix A Appendix

The explicit expressions for the matrices in the main text are

$\displaystyle A_{1}=\begin{pmatrix}0&0&0\\\ 0&0&-1\\\
0&1&0\end{pmatrix}\,,\,\,A_{2}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
-1&0&0\end{pmatrix}\,,\,\,A_{3}=\begin{pmatrix}0&1&0\\\ -1&0&0\\\
0&0&0\end{pmatrix}$ (16)

Defining $L_{j}=iA_{j}$ and taking the base that diagonalizes $L_{3}$ one
obtains the familiar form for the angular momentum operators acting on the
$|L=1,M_{L}\rangle$ states. This demonstrates that the algebra generated by
the $A_{i}$ matrices is the same of the quantum angular momentum operators.
Moreover, it is straightforward to demonstrate that the quantum mean value
$\langle\gamma\mathbf{L}\rangle$ satisfies the same classical Larmor equation.

The action of a general antisymmetric matrix
$W=a_{1}A_{1}+a_{2}A_{2}+a_{3}A_{3}$ on a given vector $\mathbf{v}$ is the
cross-product

$W\mathbf{v}=\mathbf{a}\times\mathbf{v}\qquad\mathbf{a}\equiv(a_{1},a_{2},a_{3}).$
(17)

Using the Cayley-Hamilton theorem, the needed matrix exponentials can be
analytically evaluated. Let write $\mathbf{a}=\theta\mathbf{\hat{a}}$ with
$\mathbf{\hat{a}}\cdot\mathbf{\hat{a}}=1$ then

$\e^{W}\mathbf{v}=\sum_{n=0}^{+\infty}\frac{W^{n}}{n!}\mathbf{v}=\mathbf{v}+\sin\theta\,\mathbf{\hat{a}}\times\mathbf{v}+(1-\cos\theta)\,\mathbf{\hat{a}}\times(\mathbf{\hat{a}}\times\mathbf{v}).$
(18)

The auxiliary $f_{i}$ functions introduced in the main text are defined as

$\displaystyle f_{1}(\tau)$
$\displaystyle=\sum_{n=1}^{\infty}\frac{J_{2n}(\xi)}{n}\sin(2n\tau)$ (19)
$\displaystyle f_{2}(\tau)$
$\displaystyle=4\sum_{n=0}^{\infty}\frac{J_{2n+1}(\xi)}{2n+1}\sin^{2}((n+1/2)\tau)$
(20) $\displaystyle f_{3}(\tau)$ $\displaystyle=\Re(g(\tau))$ (21)
$\displaystyle f_{4}(\tau)$ $\displaystyle=\Im(g(\tau))$ (22)

where

$g(\tau)=\e^{i\phi}\sum_{n\neq-p}\frac{J_{n}(\xi)}{i(n+p)}\left(\e^{i(n+p)\tau}-1\right)+\e^{-i\phi}\sum_{n\neq
p}\frac{J_{n}(\xi)}{i(n-p)}\left(\e^{i(n-p)\tau}-1\right)$ (23)

These functions have a limited and oscillating behaviour and are needed in the
evaluation of $\e^{\Lambda_{1}(\tau)}$. One can see by inspection that
$\e^{\Lambda_{1}}\approx\mathbbmss{1}$ is a good approximation.

The initial condition appropriate for the experiment is
$\mathbf{M}(0)\propto(1,0,0)$ and the quantity monitored is $M_{x}(\tau)$. So
applying eq. (18) to evaluate $\e^{\tau F_{1}}$, and then eq. (3), one finds

$M_{x}(t)\propto\cos(\Omega_{\mathrm{L}}\;t).$ (24)

## References

## References

  * [1] Haroche S, Cohen-Tannoudji C, Audoin C and Schermann J P 1970 Phys. Rev. Lett. 24 861–864
  * [2] Yabuzaki T, Tsukada N and Ogawa T 1972 Journal of the Physical Society of Japan 32 1069–1077
  * [3] Kunitomo M and Hashi T 1972 Physics Letters A 40 75–76
  * [4] Ito H, Ito T and Yabuzaki T 1994 Journal of the Physical Society of Japan 63 1337
  * [5] Holthaus M 2001 Phys. Rev. Applied 64 011601
  * [6] Gerbier F, Widera A, Fölling S, Mandel O and Bloch I 2006 Phys. Rev. A 73(4) 041602 URL https://link.aps.org/doi/10.1103/PhysRevA.73.041602
  * [7] Hofferberth S, Fischer B, Schumm T, Schmiedmayer J and Lesanovsky I 2007 Physical Review A - Atomic, Molecular, and Optical Physics 76 013401
  * [8] Pervishko A A, Kibis O V, Morina S and Shelykh I A 2015 Phys. Rev. B 92(20) 205403 URL https://link.aps.org/doi/10.1103/PhysRevB.92.205403
  * [9] Swank C M, Webb E K, Liu X and Filippone B W 2018 Phys. Rev. A 98(5) 053414 URL https://link.aps.org/doi/10.1103/PhysRevA.98.053414
  * [10] Bevilacqua G, Biancalana V, Dancheva Y and Moi L 2012 Phys. Rev. A 85 042510 (Preprint 1112.1309)
  * [11] Bevilacqua G, Biancalana V, Dancheva Y and Vigilante A 2019 Phys. Rev. Applied 11(2) 024049 URL https://link.aps.org/doi/10.1103/PhysRevApplied.11.024049
  * [12] Bevilacqua G, Biancalana V, Dancheva Y and Vigilante A 2019 Appl.Phys.Lett. 115 174102 (Preprint https://arxiv.org/abs/1908.01283) URL https://doi.org/10.1063/1.5123653
  * [13] Bevilacqua G, Biancalana V, Vigilante A, Zanon-Willette T and Arimondo E 2020 Phys. Rev. Lett. 125(9) 093203 URL https://link.aps.org/doi/10.1103/PhysRevLett.125.093203
  * [14] Bell W E and Bloom A L 1957 Phys. Rev. 107(6) 1559–1565 URL https://link.aps.org/doi/10.1103/PhysRev.107.1559
  * [15] Blanes S, Casas F, Oteo J and Ros J 2009 Physics Reports 470 151 – 238 ISSN 0370-1573 URL http://www.sciencedirect.com/science/article/pii/S0370157308004092
  * [16] Bevilacqua G, Biancalana V, Chessa P and Dancheva Y 2016 Applied Physics B 122 103 ISSN 1432-0649 URL http://dx.doi.org/10.1007/s00340-016-6375-2
  * [17] Golub R and Lamoreaux S K 1994 Physics Reports 237 1 – 62 ISSN 0370-1573 URL http://www.sciencedirect.com/science/article/pii/0370157394900841
  * [18] Marmugi L, Deans C and Renzoni F 2019 Applied Physics Letters 115 083503 (Preprint https://doi.org/10.1063/1.5116811) URL https://doi.org/10.1063/1.5116811