# Multi-state interferometric measurement of nonlinear AC Stark shift

Junnosuke Takai<EMAIL_ADDRESS>Department of Physics,
Gakushuin University, Tokyo 171-8588, Japan  Kosuke Shibata  Department of
Physics, Gakushuin University, Tokyo 171-8588, Japan  Naota Sekiguchi
Department of Electrical and Electronic Engineering, Tokyo Institute of
Technology, Tokyo 152-8552, Japan  Takuya Hirano  Department of Physics,
Gakushuin University, Tokyo 171-8588, Japan

###### Abstract

We demonstrate measurement of quadratic AC Stark shifts between Zeeman
sublevels in an 87Rb Bose–Einstein condensate using a multi-state atomic
interferometer. The interferometer can detect a quadratic shift without being
affected by relatively large state-independent shifts, thereby improving the
measurement precision. We measure quadratic shifts in the total spin $F=2$
state due to the light being near-resonant to the D1 line. The agreement
between the measured and theoretical detuning dependences of the quadratic
shifts confirms the validity of the measurement. We also present results on
the suppression of nonlinear spin evolution using near-resonant dual-color
light pulses with opposite quadratic shifts.

##  I Introduction

Atom–field interactions often cause shifts in atomic energy levels, referred
to as AC Stark shifts in the semi-classical treatment. These shifts often
depend on the atomic spin state. Linear and quadratic energy shifts (light
shifts) of Zeeman sublevels can arise from spherical tensor operators of,
respectively, rank-1 (vector) and 2 (tensor) in the irreducible decomposition
of the interaction Hamiltonian [1, 2]. Quadratic shifts enable advanced
quantum state manipulation such as dynamical spin control [3] and
nuclear–electronic spin entanglement [4]. Quadratic shifts have recently been
used to generate the Schrödinger’s kitten state in cold Dy atoms of large spin
$J=8$ [5]. On the other hand, quadratic shifts are often detrimental for
precise measurements, such as in atomic clocks. Even a small energy shift can
be a dominant uncertainty in state-of-the-art precise measurements. Quadratic
shifts are also harmful in spin detection via Faraday rotation [6, 7]. In a
Faraday rotation measurement, near-resonant light gives a large signal but may
also change the atomic spin state through nonlinear spin evolution due to
quadratic shifts [7, 8, 2].

Accurate measurement of quadratic or tensor shifts is important for building a
sound basis for quantum control as well as for precise measurements. The
tensor shifts in clock transitions in alkali atoms have been measured using
the Ramsey method with a hot vapor [9] and a cold-atom fountain [10]. Tensor
shifts in cold lanthanide atoms in the ground state have been determined using
Kapitza–Dirac diffraction from a pulsed standing wave [11], trap frequency
measurement [12, 13] and modulation spectroscopy in an optical lattice [14].
In these measurements [9, 10, 11, 12, 13, 14], the tensor shift is
distinguished from other shifts based on the difference in the frequency
dependence of the scalar and tensor shifts and/or the polarization dependence
of the tensor shift. Tensor shift measurements distinguished in this way tend
to be uncertain due to technical issues, such as imperfect polarization
control at the atomic position. It is difficult to precisely determine a
tensor shift much smaller than a state-independent scalar shift. This is the
case for most atom experiments, although lanthanide atoms can have large
tensor polarizability at specific light frequencies [11, 12, 13, 14].

In the present study, we demonstrate the detection of quadratic light shifts
using a multi-state atomic interferometer [15, 16] in a Bose–Einstein
condensate (BEC) of 87Rb atoms. This scheme is insensitive to state-
independent light shifts and directly measures the quadratic light shift,
thereby realizing a sensitive measurement by avoiding the uncertainty in
distinguishing the tensor shift from other shifts. Direct tensor shift
detection is also advantageous in that we can measure the shift in a
particular experimental configuration without needing to change light
frequencies or polarization. Furthermore, as we can measure the tensor shift
without relying on a priori theoretical knowledge including light frequency
and polarization dependence of the shift, the measurement may be used for
checking the validity of a theory. Conversely, by verifying that the measured
frequency dependence of the quadratic shift is consistent with the theory, the
validity of the measurement scheme is confirmed. We confirm the validity of
our measurement in this manner. We also demonstrate suppression of nonlinear
spin evolution using dual-color light pulses near the D1 transition, with the
light frequencies and powers chosen on the basis of the light shift
measurement to null the net quadratic shift.

The paper is organized as follows. In Sec. II we present our experimental
method and setup. The experimental results are described in Sec. III. We
conclude the paper in Sec. IV.

Figure 1: (color online) (a) Experimental configuration. The BEC in a coherent
spin state exhibits Larmor precession about the $z$ axis along the bias
magnetic field, $B$. A linearly polarized 795-nm light pulse propagates along
the $x$ direction through the BEC. (b) Time sequence for quadratic shift
detection. We use $\pi/2$-$\pi$-$\pi/2$ RF pulses to construct an atomic
interferometer. A train of $n$ light pulses is applied between the $\pi$ pulse
and last $\pi/2$ pulse. (c) Typical TOF images of a BEC measured after the
detection sequence. The field of view is $2.1$ mm $\times$ $0.46$ mm. The
upper and lower panels show TOF images without light pulses and with a train
of $n=9$ pulses of $P=5.71$ mW and $\Delta/2\pi=-840$ MHz, respectively. The
black rectangles in the lower panel indicate the analysis regions for counting
the atom number in each magnetic sublevel. The regions indicated by the red
dotted boxes are used for counting atoms scattered by the superradiance in the
$m_{z}=1$ state.

##  II Experimental method and setup

We produce a BEC of typically $3\times 10^{5}$ 87Rb atoms in a vacuum glass
cell [17]. The BEC is trapped in a crossed optical trap. The trap is composed
of an axial beam at a wavelength of 852 nm and a radial beam at 976 nm. The
axial and radial beam waists are approximately 30 $\mu$m and 70 $\mu$m,
respectively. The axial and radial trap frequencies are measured to be
$2\pi\times(123,16)$ Hz. A bias magnetic field $B$ of 15 $\mu$T is applied
along the axial beam along the $z$ axis. We initially prepare the atoms in the
$|F,m_{z}\rangle=|2,2\rangle$ state, where $F$ is the quantum number for the
total angular momentum of the atoms in the ground state and $m_{z}$ denotes
the magnetic sublevel.

The experimental configuration for the light shift measurement is shown in
Fig. 1(a). We measure the quadratic light shift due to a light pulse near-
resonant to the D1 line ($\lambda=795$ nm), which propagates along the $x$
direction. The D1 light is generated by a distributed feedback (DFB) laser and
is frequency offset-locked to a master external cavity diode laser (ECDL). The
frequency of the ECDL is stabilized to the $F=2\rightarrow F^{\prime}=1$
resonance line, where $F^{\prime}$ represents the total angular momentum for
the excited 2P1/2 state. The beam power is controlled by an acousto-optic
modulator (AOM). The beam is directed to the atoms through an optical fiber
after the AOM. The beam is almost collimated before the atom cell and has a
Gaussian profile with a $1/e^{2}$-radius of $\omega_{0}=0.75$ mm. We control
the polarization state at the atomic position using a half-wave plate (HWP)
and a quarter wave plate (QWP) before the atom cell. Just after the cell we
adjust the polarization of the light to be linearly polarized. The angle
between the polarization plane and the direction of the magnetic field,
$\theta$, is adjusted to 54.7∘, at which the nonlinear spin evolution of the
precessing atoms is averaged over a Larmor period if the light is continuous
[7, 17].

Figure 2: (color online) Power dependence of magnetization, $m$, for
$\Delta/2\pi$ of (a) $-840$ MHz, (b) 240 MHz, (c) 340 MHz, (d) 440 MHz, (e)
540 MHz and (f) 640 MHz. The red symbols (circles and triangles) represent the
measured magnetization. The circle data points are used for fitting (see text
for details) and the blue solid lines are the fitting curves. The green dashed
line is the simulated magnetization including the spontaneous emission effect.
The calculation and simulation are performed for a single pulse of $\tau=667$
ns.

We perform atomic interferometry with the time sequence depicted in Fig. 1(b).
In this sequence, we use the spin echo method to suppress the influence from
low-frequency fluctuations of the bias magnetic field. We apply light pulses
between the middle $\pi$ pulse and last $\pi/2$ rf pulse. Each magnetic
sublevel experiences an AC Stark shift and acquires a phase shift during the
light pulse. Different phase shifts between the magnetic sublevels result in a
change in the population of each sublevel after the last $\pi/2$ rf pulse. We
measure the sublevel populations by absorption imaging along the $x$ axis
after a time-of-flight of $20.6$ ms with Stern–Gerlach spin separation (see
Fig. 1(c)).

The AC Stark shift due to light near-resonant to the D1 line is derived from
the light-shift Hamiltonian [2]:

$\displaystyle\hat{H}_{\mathrm{shift}}$ $\displaystyle=$
$\displaystyle\sum_{F^{\prime}}\frac{\hbar\Omega_{0}^{2}}{4\Delta_{FF^{\prime}}}\\{C_{FF^{\prime}}^{(0)}|\vec{\varepsilon}|^{2}+\mathrm{i}C_{FF^{\prime}}^{(1)}(\vec{\varepsilon}^{\,*}\times\vec{\varepsilon})\cdot\hat{\bm{F}}$
(1) $\displaystyle+$ $\displaystyle
C_{FF^{\prime}}^{(2)}(|\vec{\varepsilon}\cdot\hat{\bm{F}}|^{2}-\frac{1}{3}\hat{\bm{F}}^{2}|\vec{\varepsilon}|^{2})\\},$

where $\Delta_{FF^{\prime}}$ is the amount of light detuning from the
transition frequency between the $F$ and $F^{\prime}$ states,
$C_{FF^{\prime}}^{(k)}$ is a rank-$k$ tensor coefficient representing the
angular momentum dependence [2] and $\vec{\varepsilon}$ is the polarization
vector for the light. $\Omega_{0}$ is defined by

$\Omega_{0}=\frac{\langle P_{1/2}||d||S_{1/2}\rangle E}{\hbar},$ (2)

where $\langle P_{1/2}||d||S_{1/2}\rangle=2.537\times
10^{-29}\,\mathrm{C}\cdot\mathrm{m}$ [18] is the reduced matrix element for
the D1 dipole transition and $E$ is the field amplitude. $\Omega_{0}$ can be
expressed using the beam power, $P$, as

$\Omega_{0}=\frac{\langle
P_{1/2}||d||S_{1/2}\rangle}{\hbar}\sqrt{\frac{4P}{\pi
c\varepsilon_{0}\omega_{0}^{2}}}\equiv\sqrt{\eta P},$ (3)

where $c$ is the speed of light and $\varepsilon_{0}$ is the electric
constant. Hereafter we consider linearly polarized light, which introduces no
vector shift and produces a state-dependent shift solely through the tensor
component. The state-dependent Hamiltonian can be written as

$\hat{H}_{\mathrm{depend}}=\frac{\hbar\Omega_{0}^{2}}{4\Delta_{\mathrm{HFS}}}\chi(\hat{F_{y}}\mathrm{cos}\theta+\hat{F_{z}}\mathrm{sin}\theta)^{2},$
(4)

where $\Delta_{\mathrm{HFS}}=2\pi\times 814.5$ MHz is the hyperfine splitting
between the $F^{\prime}=1$ and $F^{\prime}=2$ states and

$\chi=\sum_{F^{\prime}}\frac{C_{FF^{\prime}}^{(2)}\Delta_{\mathrm{HFS}}}{\Delta_{FF^{\prime}}}$
(5)

represents the dependence of the coupling strength on the light frequency. We
refer to $\chi$ as the coupling coefficient. In an experiment using the ground
$F=2$ state, $\chi$ depends on the laser frequency as

$\chi=\frac{\Delta_{\mathrm{HFS}}}{12}\left(\frac{1}{\Delta}-\frac{1}{\Delta-\Delta_{\mathrm{HFS}}}\right),$
(6)

where $\Delta\equiv\Delta_{21}$ (see Fig. 3(b)). Hereafter, we consider
detuning with respect to the transition frequency between the $F=2$ and
$F^{\prime}=1$ states.

The spin dynamics for $\hat{H}_{\mathrm{depend}}$ can be easily described if
the quantization axis is selected to be along the direction of the
polarization axis, $z^{\prime}$. The state before sending the light pulse,
$|\psi\rangle=\sum_{m_{z^{\prime}}}\beta_{m_{z^{\prime}}}|m_{z^{\prime}}\rangle$,
with $\beta_{m_{z^{\prime}}}$ being the probability amplitude for each
sublevel $m_{z^{\prime}}$ in this frame, evolves under the influence of a
rectangular pulse of width $\tau$ and power $P$ into

$|\psi^{\prime}\rangle=\sum_{m_{z^{\prime}}}\beta_{m_{z^{\prime}}}e^{\mathrm{i}\chi{m_{z^{\prime}}}^{2}\xi
P\tau}|m_{z^{\prime}}\rangle,$ (7)

where

$\xi=\frac{\eta}{4\Delta_{\mathrm{HFS}}}.$ (8)

Here we consider evolution due to a single pulse for simplicity. The extension
to the multi-pulse case is straightforward. We assume that the pulse width,
$\tau$, is sufficiently shorter than the period of Larmor precession and
neglect evolution due to the magnetic field during the pulses. In Eq. (7) we
omit the global (state-independent) phase shift, which has no relevance to the
population change in the magnetic sublevels. If the spin evolution during the
pulse is purely caused by the light shift, the state after the last $\pi/2$
pulse in the $m_{z}$ basis is written using the Wigner D-matrix [16],
$D^{j}(\alpha,\beta,\gamma)$, as

$\displaystyle|\psi_{\mathrm{end}}\rangle$ $\displaystyle=$ $\displaystyle
D^{2}(0,-\frac{\pi}{2},0)D^{2}(0,0,-\phi)D^{2}(-\theta,0,0)^{\dagger}$ (9)
$\displaystyle AD^{2}(-\theta,0,0)D^{2}(0,0,\phi)|\psi_{0}\rangle,$

where $A=\mathrm{diag}(e^{4\mathrm{i}\chi\xi P\tau},e^{\mathrm{i}\chi\xi
P\tau},1,e^{\mathrm{i}\chi\xi P\tau},e^{4\mathrm{i}\chi\xi P\tau})$ represents
the time development by light pulses, $\phi$ is the spin angle in the $x$–$y$
plane with respect to the $y$ axis at the starting time of the light pulse and
$|\psi_{0}\rangle=(-1/4,\mathrm{i}/2,\sqrt{6}/4,-\mathrm{i}/2,-1/4)^{T}$
represents the coherent spin state (CSS) along the $y$ axis ($\phi=0$) as the
basis of the $z$ axis. Here $D^{j}(\alpha,\beta,\gamma)$ is defined in terms
of the Euler angles $(\alpha,\beta,\gamma)$ around the $z$–$y$–$z$ axes as

$D_{q^{\prime}q}^{j}(\alpha,\beta,\gamma)=\langle
jq^{\prime}|\hat{R}(\alpha,\beta,\gamma)|jq\rangle,$ (10)

where
$\hat{R}(\alpha,\beta,\gamma)=e^{-\mathrm{i}\alpha\hat{j}_{z}}e^{-\mathrm{i}\beta\hat{j}_{y}}e^{-\mathrm{i}\gamma\hat{j}_{z}}$
is the rotational operator [19]. We calculate the magnetization,
$m=\langle\psi_{\mathrm{end}}|\hat{F}_{z}|\psi_{\mathrm{end}}\rangle$, using
Eq. (9). The calculated $m$ is a function of $\chi\xi P\tau$, but its explicit
expression is too long to show here. We note $m=2\mathrm{cos}^{3}(\chi\xi
P\tau)$ if $\theta=0$ [20].

##  III Results

###  III.1 Measurement of quadratic light shifts

We first detect quadratic light shifts produced by a single pulse of
$\Delta/(2\pi)=-840$ MHz, which induces less light-assisted collisional atom
losses [17]. The pulse has an almost rectangular shape and its length is fixed
to $\tau=667\,\mathrm{ns}$. The pulse is applied $0.125\,\mathrm{ms}$ after
the $\pi$ pulse, when the spin orientation is along the $x$ axis
($\phi=\pi/2$). We confirm the spin direction by spin-sensitive phase contrast
imaging [17].

We observe changes in the sublevel population using a Stern–Gerlach
measurement, as shown in Fig. 1(c). We experimentally obtain the magnetization
using

$m=\frac{\sum_{i}iN_{i}}{N_{\mathrm{tot}}},$ (11)

where $N_{i}$ is the number of atoms in the $|F,m_{z}=i\rangle$ state
$(i=-2,-1,0,1,2)$ after the read-out pulse and
$N_{\mathrm{tot}}=\sum_{i}N_{i}$ is the total number of atoms. The
magnetization is plotted as a function of the pulse power in Fig. 2 (a). We
fit the data using $(1-\delta
p)\langle\psi_{\mathrm{end}}|\hat{F}_{z}|\psi_{\mathrm{end}}\rangle$, where
$\delta p$ is introduced to account for experimental imperfect state
preparation and control. In the fitting (blue line in Fig.2 (a)), we use Eq.
(9) with $\theta=54.7^{\circ}$ and $\phi=90^{\circ}$, in correspondence with
the experiment. The fitting gives $\chi=-0.0404(8)$. The value in parentheses
denotes the standard deviation of $\chi$ calculated from three sets of data.

Next, we measure the quadratic light shifts for other light frequencies with
positive detunings $\Delta/(2\pi)=+\\{240,340,440,540,640\\}$ MHz. The results
for each detuning are depicted in Figs. 2(b)–2(f). We observe large changes in
$m$. For these positive detunings, $\chi$ is much larger than that for
$\Delta/(2\pi)=-840$ MHz and the observed large change in $m$ is reasonable.
The spontaneous emission rates also become large for these detunings and
optical pumping due to spontaneous emission may also result in spin change. To
evaluate the contribution of spontaneous emissions, we numerically calculate
the dynamics of the magnetization using an atomic master equation including
spontaneous emission [21]. A theoretically predicted value of the coupling
coefficient, $\chi_{\textrm{theo}}$, is used in the simulation. We plot the
simulation results in Fig. 2 (green dotted lines). The numerical simulation
indicates that the spontaneous emissions are not negligible for beam powers
larger than approximately 5 mW, which we were not able to investigate in
detail due to the limited available beam power. In the fitting shown by blue
lines in Fig. 2(b)–(f), to obtain experimental values of the coupling
coefficient, $\chi_{\textrm{exp}}$, we use data points at beam powers for
which the magnetization obtained by the simulation differs from the matrix
calculation by less than 0.06 to lessen the effect of spontaneous emissions on
the estimation of $\chi$. The fitted values of $\chi_{\textrm{exp}}$ are shown
in Table 1.

The frequency dependence of $\chi_{\textrm{exp}}$ is consistent with the
theoretical curve given by Eq. (6), as shown in Fig. 3(a). Note that
interferometric detection does not reveal the sign of $\chi$. We determine the
sign of $\chi$ at each $\Delta$ in Fig. 3(a) to coincide with the
theoretically determined sign. We also note the exact determination of $\chi$
requires precise calibration of the beam intensity at the atom position. It is
more appropriate to analyze the ratio between $\chi_{\textrm{exp}}$ and
$\chi_{\textrm{theo}}$ to estimate the validity of the measurement. This ratio
is shown in Table 1. The sample standard deviation of the ratios, which
represents the precision of the measurement, is $0.02$ (2%).

Figure 3: (color online) (a) Dependence of the coupling coefficient $\chi$ on
detuning. The red circles represent the coefficients obtained from the
experiments. The error bar for each circle represents the standard deviation
over three experimental runs. The blue solid line is the theoretical curve
given by Eq. (6). (b) Energy diagram for 87Rb relevant to the experiment.

###  III.2 Suppression of nonlinear spin evolution

We also demonstrate suppression of spin evolution due to a quadratic shift by
using near-resonant dual-color light pulses. From Eq. (6), we can see that
$\chi$ is positive if $0<\Delta<\Delta_{\mathrm{HFS}}$ and negative otherwise.
If we combine negative ($\Delta<0$) and positive
($0<\Delta<\Delta_{\mathrm{HFS}}$) detuned light with a power ratio of
$p_{\mathrm{neg}}/p_{\mathrm{pos}}=\chi_{\mathrm{pos}}/\chi_{\mathrm{neg}}$,
the quadratic light shift should vanish and nonlinear spin evolution should be
suppressed. We use the DFB laser used in the above experiment at a fixed
detuning of $\Delta_{-}/(2\pi)=-840$ MHz. We prepare another ECDL for
quadratic shift compensation. The ECDL is frequency offset-locked to the
master ECDL, and its detuning is set to
$\Delta_{+}/(2\pi)=+\\{240,340,440,540,640\\}$ MHz. The two laser beams are
mixed at a non-polarizing beam splitter (NPBS) before the AOM for power
control. We adjust the ratio of the DFB laser power to the ECDL power using a
HWP and PBS before the NPBS. The ratios used in the experiment,
$r_{\mathrm{exp}}$, are listed in Table. 1. The values of $r_{\mathrm{exp}}$
are close to the theoretical optimal ratios,

$r_{\mathrm{theo}}=\left(\frac{1}{\Delta_{+}}-\frac{1}{\Delta_{+}-\Delta_{\mathrm{HFS}}}\right)/\left(\frac{1}{\Delta_{-}}-\frac{1}{\Delta_{-}-\Delta_{\mathrm{HFS}}}\right).$
(12)

Table 1: Measured and theoretical coupling coefficients and power ratios for quadratic shift cancellation. $\frac{\Delta}{2\pi}$ (MHz) | $\chi_{\mathrm{exp}}$ | $\chi_{\mathrm{exp}}/\chi_{\mathrm{theo}}$ | $r_{\mathrm{exp}}$ | $r_{\mathrm{theo}}$ | $r_{\mathrm{exp}}/r_{\mathrm{theo}}$
---|---|---|---|---|---
$-840$ | $-0.0404$ | 1.016 | $-$ | $-$ | $-$
240 | 0.411 | 1.025 | 10.17 | 10.08 | 1.01
340 | 0.356 | 1.039 | 8.81 | 8.61 | 1.02
440 | 0.351 | 1.045 | 8.68 | 8.43 | 1.03
540 | 0.379 | 1.017 | 9.39 | 9.37 | 1.00
640 | 0.486 | 0.982 | 12.02 | 12.44 | 0.97
Figure 4: (color online) Magnetization with and without compensation as
function of negative-detuned beam power. (a) Magnetization for single pulse.
The blue circles represent the magnetization without compensation (with only
the negative-detuned light). The red triangle data are taken with the
compensation light with $\Delta_{+}/(2\pi)=+640$ MHz added. (b) Magnetization
for $n=9$ pulses. The magnetization values with compensation light of
$\Delta_{+}/(2\pi)=$ 240, 340, 440, 540, 640 MHz are plotted by pink crosses,
light blue inverted triangles, black squares, green diamonds and red
triangles, respectively. The blue circles represent the magnetization without
compensation. The solid lines are the magnetization by the numerical
simulation including spontaneous emissions.  Figure 5: (color online)
Proportion of atoms kicked by superradiance for (a) $m_{z}=+1$ and (b)
$m_{z}=+2$ as function of negative-detuned beam power.

We observe that the change in magnetization for the pulse mixture is much less
than that for the single negative-detuned pulse. We show the results for
$\Delta_{+}/(2\pi)=+640$ MHz light compensation and without compensation in
Fig. 4(a). We apply $n=9$ pulses to observe the changes in magnetization more
clearly. We set the interval between pulses to $9.5\,\mu$s, equal to the
period of Larmor precession, in order to apply all pulses when the spin is
directed to the probe propagation direction. The results with compensation
light pulses with $\Delta_{+}/(2\pi)=$ 240, 340, 440, 540, 640 MHz are plotted
in Fig. 4(b).

The magnetization changes with increasing power even with compensation. This
change is considered to be due to spin relaxation by optical pumping. We also
observe that the size of the change depends on the detuning of the
compensation beam, $\Delta_{+}$. The closer the frequency is to the
$F=2\rightarrow F^{\prime}=1$ resonance, the greater the magnetization change.
The frequency dependence can be understood as follows. When we take the $x$
axis as the quantization axis, the linearly polarized beam couples to the
atoms via the circular component, as shown in Fig. 6. The coupling strength to
the $F^{\prime}=1$ state is three times stronger than that to the
$F^{\prime}=2$ state. The increase in the magnetization change as $\Delta$
approaches 0 is related to this difference in coupling strength. The frequency
dependence of the magnetization change is confirmed by a numerical simulation.
The simulated change is minimum at $\Delta_{+}/(2\pi)=$ 540 MHz. This is
consistent with the simple estimation of the relaxation rate given by the
product of the scattering rate (approximately proportional to
$3/\Delta_{21}^{2}+1/\Delta_{22}^{2}$) and the beam power required for the
quadratic shift compensation ($\propto 1/\Delta_{21}-1/\Delta_{22}$).

Figure 6: (color online) Coupling strength between
$|F,m_{F}\rangle=|2,2\rangle$ state and excited states. The $x$ axis is taken
as the quantization axis here. The blue arrows represent the allowed
transitions. The number next to each arrow is the dipole matrix element,
expressed as multiples of $\langle P_{1/2}||d||S_{1/2}\rangle$.

The magnetization change in the experiment is larger than that in the
simulation including the spontaneous emissions, especially for detuning from
340 to 540 MHz. A possible cause of this difference is the superradiance [22,
23, 24]. In fact, we observe atoms kicked by superradiant scattering when the
beam is strong. We evaluate the proportion of the number of kicked atoms,
$N_{\mathrm{sup}}$, to the total number of atoms, $N_{\mathrm{tot}}$, as a
measure of superradiant scattering. In the analysis, we count the number of
kicked atoms in the regions adjacent to the zero moment component (see the
lower panel of Fig. 1(c)) as $N_{\textrm{sup}}$. We plot
$N_{\mathrm{sup}}/N_{\mathrm{tot}}$ for the magnetic sublevels of $m_{z}=+1$
and $+2$ in Fig. 5. The data in Fig. 5 is taken with $n=9$ pulses. We consider
that superradiance enhances the spin relaxation rate and may increase the
magnetization change. We estimate from Fig. 5(a) that the scattering rate is
enhanced by up to approximately $30\%$. The Raman superradiance [25] to the
$F=1$ state should also contribute to the scattering rate enhancement. We also
measure the atom number loss, which changes the population among the magnetic
sublevels and causes spin relaxation. However, we do not observe a significant
dependence of the loss rates on $\Delta_{+}$. The larger than expected
magnetization change might be partly due to the enhancement of spontaneous
emission near the critical temperature for Bose–Einstein condensation [26].

##  IV Conclusions and outlook

We have demonstrated the successful measurement of quadratic light shifts
using an atom interferometer. The measured dependence of the shift due to
detuning is consistent with the theoretical prediction. In addition, we
suppressed the influence of nonlinear light shifts by using dual-color light
pulses. Dual-color light is applicable to high-precision measurement in a
probe, especially for BEC magnetometers [17]. A dual-color probe is useful not
only for improving the signal-to-noise ratio in spin measurements by
increasing the probe strength, but also for preventing disturbances when
creating a spin squeezed state via quantum nondemolition measurements [27, 28,
29, 30, 31].

Further improvements of this measurement scheme are feasible. Reducing
technical noises during absorption imaging and spin manipulation by rf pulses
will lead to higher sensitivity. The bias magnetic field may produce a slight
difference between the experimental data and the theoretical prediction
calculated by the Wigner D-matrix and interaction Hamiltonian, in which we
have neglected the spin evolution due to the magnetic field. Therefore, to
improve the estimation accuracy for the coupling coefficients, it may be
effective to use a low magnetic field. These improvements may make it possible
to estimate physical quantities such as transition matrix elements and
hyperfine splittings.

###### Acknowledgements.

This work was supported by the MEXT Quantum Leap Flagship Program (MEXT
Q-LEAP) grant number JPMXS0118070326 and JSPS KAKENHI grant number JP19K14635.

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