# Quantum lock-in detection of a vector light shift

Kosuke Shibata<EMAIL_ADDRESS>Naota Sekiguchi  Takuya Hirano
Department of Physics, Gakushuin University, Tokyo, Japan

###### Abstract

We demonstrate detection of a vector light shift (VLS) using the quantum lock-
in method. The method offers precise and accurate VLS measurement without
being affected by real magnetic field fluctuations. We detect a VLS on a
Bose–Einstein condensate (BEC) of 87Rb atoms caused by an optical trap beam
with a resolution less than 1 Hz. We also demonstrate elimination of a VLS by
controlling the beam polarization to realize a long coherence time of a
transversally polarized $F$ = 2 BEC. Quantum lock-in VLS detection should find
wide application, including the study of spinor BECs, electric-dipole moment
searches, and precise magnetometry.

##  I Introduction

The a.c. Stark shift or light shift plays significant roles in atomic physics.
One example is the optical trap Grimm et al. (2000), which has been
extensively used in cold atom experiments and has been the subject of
intriguing and important research, including low-dimensional Görlitz et al.
(2001) and uniform gases Gaunt et al. (2013), and atoms in an optical lattice
with applications to quantum simulation Bloch et al. (2008) and atomic clocks
Derevianko and Katori (2011); Katori (2011). It has also enabled the study of
multi-component gases and, in particular, spinor Bose–Einstein condensates
(BECs) Stamper-Kurn and Ueda (2013).

The light shift has vector and tensor components and hence is state-dependent
in general Deutsch and Jessen (1998); Geremia et al. (2006); Deutsch and
Jessen (2010). The state dependence has been exploited for realizing state-
selective transport Mandel et al. (2003a, b) and confinement Heinz et al.
(2020). However, a state-dependent shift is often undesirable for situations
in which well-controlled spin evolution is required. Escaping from a vector
light shift (VLS), which is equivalent to a fictitious magnetic field, has
been an important issue in precise measurements, such as the search for an
atomic electric-dipole moment Romalis and Fortson (1999) and exotic spin-
dependent interactions Jackson Kimball et al. (2017a). Reducing the VLS is
also important in atomic magnetometers, in which the VLS introduces systematic
errors. The quantum noise associated with the light shift due to the probe
field ultimately limits the sensitivity Fleischhauer et al. (2000).

The VLS restricts the potential use of optically trapped atoms for
magnetically sensitive experiments. While its effect can be diminished by
applying a bias magnetic field in a direction orthogonal to the wavevector,
the VLS can still be a significant noise source in precise measurements
Romalis and Fortson (1999). It is necessary to reduce the VLS when an ultralow
magnetic field is required. In addition, the relative direction cannot be
chosen satisfactorily in some situations, such as in 3D optical lattice
experiments.

In order to eliminate the VLS caused by optical trapping beams, the light
polarization should be precisely controlled, because the VLS is proportional
to the intensity of a circularly polarized component Grimm et al. (2000);
Deutsch and Jessen (1998); Geremia et al. (2006); Deutsch and Jessen (2010).
However, it is a formidable task to precisely extinguish the circular
component at the atom position located in a vacuum cell. Polarization
measurements and control outside the cell do not assure the degree of linear
polarization due to the stress-induced birefringence of the vacuum windows
Jellison (1999).

Therefore, a sensitive and robust polarization measurement method using atoms
themselves as a probe is important. Most effective polarization measurements
are accomplished by using atoms themselves as a probe. Polarization
measurements with an atomic gas have been performed with various methods
including Larmor precession measurement Zhu et al. (2013), precise microwave
spectroscopy Steffen et al. (2013), and frequency modulation nonlinear
magneto-optical rotation Jackson Kimball et al. (2017b). Differential Ramsey
interferometry has been developed for spinor condensates Wood et al. (2016).
Polarization measurements by fluorescence detection have been recently
demonstrated for ions Yuan et al. (2019).

In this paper, we demonstrate VLS detection by applying the quantum lock-in
method Kotler et al. (2011); de Lange et al. (2011). The measurement is immune
to environmental magnetic field noise, and thus achieves excellent precision
and accuracy. We detect a VLS induced by an optical trap beam on a BEC of 87Rb
atoms with a resolution less than 1 Hz. This detection method is feasible to
implement and should have wide applications in various research areas involved
with optical fields.

The paper is organized as follows. In Sec. II, our experimental method and
setup are presented. The experimental results are described in Sec. III. We
discuss the applications and potential performance of the quantum lock-in VLS
detection in Sec. IV. We conclude the paper in Sec. V.

##  II Experimental method and setup

We produce a BEC in a vacuum glass cell. A BEC of $3\times 10^{5}$ atoms in
the hyperfine spin $F=2$ state is trapped in a crossed optical trap. The trap
consists of an axial beam at the wavelength of 850 nm and a radial beam at
1064 nm. The axial and radial beam waists are $\approx$ 30 $\mu$m and 70
$\mu$m, respectively. A magnetic bias field $B$ of 15 $\mu$T is applied along
the axial beam to define the quantization axis, as shown in Fig. 1(a). The
atoms are initially in the $|F,m_{F}\rangle=|2,2\rangle$ state, where $m_{F}$
denotes the magnetic sublevel. The ellipticity of the axial beam at the atomic
position is controlled with a quarter waveplate (QWP) in the VLS measurement
described below. The QWP is located between a polarization beam splitter for
polarization cleaning and the cell. The angle of the QWP is adjusted with a
precise manual rotation stage. The minimum scale of the rotation stage is 0.28
mrad.

Figure 1: (color online) (a) Experimental configuration. A BEC is trapped in
the axial trap beam along the $z$ axis and the radial trap beam along the $x$
axis (not shown). (b) Typical TOF image of a BEC measured after rf pulses for
the detection. The spin components ($m_{F}=-2,-1,0,1,2$) are spatially
resolved by the Stern–Gerlach method. (c) Time sequence for the quantum lock-
in detection of a VLS. The beam power, $P(t)$, is modulated with a frequency
$\omega_{m}$. The phase of the spin vector evolves with an angular frequency
of $\omega(t)$. The accumulated phase, $\Phi=\int_{0}^{T}\omega(t)dt$, is
finally measured.

The time sequence for the quantum lock-in detection of a VLS is shown in Fig.
1(c). The lock-in technique enables enhanced sensitivity at the modulation
frequency while reducing the effect of unwanted noise. We measure a VLS
induced by the axial optical trap beam with multiple rf pulses. The trap beam
power, $P(t)$, is modulated with a frequency $\omega_{m}$ during the pulse
application as

$P(t)=P_{0}\left(1+p\sin(\omega_{m}t)\right)\equiv
P_{0}+P_{1}\sin(\omega_{m}t),$ (1)

where $P_{0}$ is the mean power and $p$ is the modulation index. $P_{1}$ can
be negative by changing the modulation phase by $\pi$. $\omega_{m}$ is set to
be sufficiently higher than twice the trapping frequency to avoid parametric
heating of the atoms. The modulation generates an a.c. fictitious magnetic
field to be measured, given by

$B_{\mathrm{fic}}=-\frac{1}{4}\alpha^{(\mathrm{1})}\mathcal{C}I_{1}\sin(\omega_{m}t)\equiv
B_{1}\sin(\omega_{m}t),$ (2)

where $\alpha^{(\mathrm{1})}$ is the a.c. vector polarizability, $\mathcal{C}$
is the degree of the circularity and $I_{1}$ is the beam intensity
corresponding to $P_{1}$.

The pulse set consists of an initial $\pi/2$ pulse at $t=0$, an odd number
($N$) of $\pi$-pulses, and a readout $\pi/2$ pulse. The pulses are equally
spaced by $\Delta T$. The spacing satisfies $\omega_{m}$$=\pi/\Delta T$ so
that the evolved phase due to the fictitious field is constructively
accumulated. The relative phase, $\Delta\varphi$, between the initial and
read-out pulses is set to $\pi/2$ for maximum sensitivity to small changes in
the accumulated phase, $\Phi$. $\Phi$ is explicitly given by

$\Phi=\frac{2}{\pi}\frac{g_{F}\mu_{B}B_{1}}{\hbar}T\equiv\frac{2}{\pi}\omega_{1}T,$
(3)

where $g_{F}$ is the Landé g-factor, $\mu_{B}$ is the Bohr magneton, $\hbar$
is the reduced Planck constant and $T=(N+1)\Delta T$ is the phase accumulation
time. $\omega_{1}/(2\pi)$ represents the VLS corresponding to $B_{1}$ in units
of frequency.

The read-out pulse converts $\Phi$ into the magnetization, $m$, as

$m\equiv\frac{\sum_{i}iN_{i}}{N_{\mathrm{tot}}}=\mathcal{V}F\sin\Phi,$ (4)

where $N_{i}$ is the atom number in the $|F,m_{F}=i\rangle$ state
$(i=-2,-1,0,1,2)$ after the read-out pulse, $N_{\mathrm{tot}}=\sum_{i}N_{i}$
is the total atom number, and $\mathcal{V}$ is the visibility. $\mathcal{V}$
is ideally $1$, but in practice it is less than $1$ due to magnetic field
noise Kotler et al. (2011). Imperfections in the initial state preparation and
spin manipulation also decrease $\mathcal{V}$. The magnetization is measured
by standard absorption imaging after a time-of-flight with Stern–Gerlach spin
separation (see Fig. 1(b)).

##  III Results

Figure 2: (color online) Detection of the VLS. The error bars represent the
sample standard deviation. The red solid line is the fitting curve by
$\mathcal{V}F\sin(ap)$. The right axis represents $B_{1}$. It should be noted
that the right axis scale is not linear since $B_{1}$ is proportional to
$\arcsin(\frac{m}{\mathcal{V}F})$.

We first confirm the validity of the detection scheme. We perform a lock-in
detection with $\omega_{m}=2\pi\times 2$ kHz ($\Delta T=0.25$ ms) and $N=27$,
and hence $T=7$ ms. $P_{0}$ is fixed to 11 mW. The change in $m$ is observed
as $p$ is varied. The result is plotted in Fig. 2. Here, the angle of the QWP
axis, $\theta$, is approximately 4∘ apart from the optimal angle,
$\theta^{*}$, minimizing the VLS. The experimental determination of
$\theta^{*}$ is described below. $m$ is well fitted by a sinusoidal function
$\mathcal{V}F\sin(ap)$, indicating the VLS was successfully detected. The
visibility in this detection setting is found to be $\mathcal{V}=0.746(42)$
from an independent measurement with no modulation ($p=0$) where
$\Delta\varphi$ is scanned.

The detection is used to minimize the VLS. We control the VLS by changing
$\theta$ with $p$ fixed to 0.32. The $\theta$-dependence of $m$ is shown in
Fig. 3(a). Because $\mathcal{C}\approx\sin 2(\theta-\theta^{*})\equiv\sin
2\Delta\theta$ when the birefringence in the optical path is small Wood et al.
(2016) and $|\Delta\theta|\ll 1$, we fit $m$ by
$\mathcal{V}F\sin(\beta_{1}(\theta-\theta^{*}))$. The fit gives
$\beta_{1}=6.2(2)$, which is in reasonable agreement with the calculation.
$\theta^{*}$ is found to be -6.6(1.1) mrad. The VLS resolution is evaluated as
$\delta\omega=\beta_{1}\delta\theta^{*}/T=2\pi\times 0.16$ Hz, where
$\delta\theta^{*}$ is the uncertainty in the $\theta^{*}$ estimation.

Figure 3: (color online) Polarization dependence of the signal. (a)
Measurement result with $T$ = 7 ms. (b) Measurement result with $T$ = 28.2 ms.
The blue circles and red squares represent $m_{+}$ and $m_{-}$, respectively.
(c) $\Delta m$ as a function of $\theta$. $\Delta B_{1}$ is the difference
between the fictitious magnetic fields for positive and negative $P_{1}$. The
solid lines in (a) and (c) are the fitting curves.

We perform a fine estimation of $\theta^{*}$ by extending $T$ to 27.2 ms and
applying a larger modulation. In this experiment, $\omega_{m}$ and $N$ are
$2\pi\times 625$ Hz and $33$, respectively. We measure $m$ for $P_{1}=\pm 13$
mW, referred to as $m_{\pm}$, respectively. In finding $\theta^{*}$, we use
$\Delta m=m_{+}-m_{-}$ to cancel the offset due to the background field and
the systematic error in the spin measurement. The results are shown in Figs.
3(b) and (c). $\Delta m$ is fitted by
$4\mathcal{V}F\sin(\beta_{2}(\theta-\theta^{*}))$, giving $\beta_{2}=117(16)$
and $\theta^{*}=0.06$ mrad with $\delta\theta^{*}$ = 0.40 mrad. The angle
resolution is improved 2.8 times.

We observe a larger variance in $m$ in the experiments for the fine
$\theta^{*}$ estimation. The standard deviation of $\Delta m$ is on average
0.64, while that for the reference data without modulation is $0.09$.
Therefore, a further improvement by a factor of at least 7 is possible,
because $\Delta m$ should ideally be independent of $T$ and the modulation
strength. We ascribe the increased variance to the actual variation of the
vector shift over the experimental runs, caused by beam polarization
fluctuation. The result of the sensitive detection implies that the beam
circularity varies with the standard deviation of approximately $3\times
10^{-3}$. On the other hand, from an independent experiment, we expect that
the retardance of the QWP should vary by several mrad due to the temperature
change in our experimental room (within $\approx$ 0.6 K with a period of
around 20 minutes).

The BEC is subject to a fictitious magnetic field gradient without the VLS
cancellation, because it is located at the shoulder of the optical trap beam
due to gravity sag. While the observed fictitious magnetic field is small, the
gradient in the fictitious field can be on the order of 100 $\mu$T/m. The
gradient displaces the trap potential for each spin state other than the
$m_{F}$ = $0$ state, thereby driving the spin dependent motion. We observe an
actual motion in a transversally-spin-polarized BEC in the hyperfine spin
$F=2$ state, prepared after the initial $\pi/2$ pulse. We plot the vertical
displacement of the spin components in the TOF image, which reflects the
momentum, in Figs. 4(a)–(d). The direction of the motion inverts depending on
the sign of $\Delta\theta$ and the motion becomes small at
$\Delta\theta\approx 0$. These observations indicate that the motion is
induced by the fictitious magnetic field.

Figure 4: (color online) Effects of the fictitious magnetic field on a
transversally polarized BEC. (a)–(d) Vertical displacement of the center of
mass in the TOF image. The panels show the data for $\Delta\theta$ = (-1,
+0.02, +1, +4) degrees, respectively. The solid and dashed lines are guides
for the eyes. (e)–(h) Population evolution corresponding to (a)–(d).

The fictitious magnetic field gradient also causes nonlinear spin evolution
and thus a population change, as does the real magnetic field gradient Eto et
al. (2014). The initial polarized atomic spin state breaks due to the spin
mixing seeded by the nonlinear spin evolution. We show
$p_{0}=N_{0}/N_{\mathrm{tot}}$, $p_{1}=(N_{-1}+N_{+1})/(2N_{\mathrm{tot}})$,
and $p_{2}=(N_{-2}+N_{+2})/(2N_{\mathrm{tot}})$ in Figs. 4(e)–(h). The
population changes are observed at an earlier time ($t<100$ ms) except for the
case $\Delta\theta\approx 0$. These changes can be attributed to the
fictitious magnetic field gradient. The faster population change for
$\Delta\theta=4^{\circ}$ is consistent with a qualitative estimation of the
characteristic time for the change of $t_{*}\propto b^{-2/3}$, where $b$ is
the magnetic field gradient Eto et al. (2014). A slow population change, which
occurs regardless of $\Delta\theta$, is caused by a residual axial magnetic
field gradient, $\partial B_{z}/\partial z$. The existence of the axial
gradient in these data is confirmed by the fact that the spin components
separate in the axial direction at later times.

Figure 5: (color online) Atom number losses. The solid line is an exponential
fit to the data with the optimized field gradient. The inset shows the
population evolution for the optimized case. The dotted lines are the
prediction curves of the mean-field driven evolution without including the
inelastic losses Kronjäger et al. (2008).

We next observed the change in the atom loss rate. Figure 5 shows the
evolution of the atom numbers, corresponding to the data in Figs. 4(b) and
(d). The decay is faster when $\Delta=4^{\circ}$ than for $\Delta\approx
0^{\circ}$. In the latter case, the decay rate starts to increase from around
$t$ = 150 ms, where the population changes occur (see Fig. 4(f)). No increase
in the loss rate is observed at the later time when we optimize $\theta$ and
reduce the axial real magnetic field gradient. The $1/e$ time for the
optimized condition is found to be 742 (31) ms. The change of the loss rate
can be understood from the property of the inelastic collisions in the $F=2$
state Tojo et al. (2009). Note that the inelastic collisional loss in the
polarized state is inhibited due to the restriction of the angular momentum
conservation. The break of the polarized state due to the field gradient
results in rapid atom losses.

However, the loss still occurs for the optimized condition. Although the
remaining loss may be due to the residual field inhomogeneity, it is
associated with the spin mixing induced by the quadratic Zeeman energy
Kronjäger et al. (2006, 2008). The model presented in Kronjäger et al. (2006,
2008), however, needs to be modified to explain the observed population
conservation, shown in the inset of Fig.5. According to Kronjäger et al.
(2008), the population evolution in the limit of small quadratic Zeeman
energy, $q$, is approximately given by

$\displaystyle p_{0}=$
$\displaystyle\frac{3}{8}\left[1+\frac{q}{2g_{1}n}(1-\cos(4g_{1}nt/\hbar))\right],$
(5) $\displaystyle p_{1}=$ $\displaystyle\frac{1}{4},$ (6) $\displaystyle
p_{2}=$
$\displaystyle\frac{1}{16}\left[1-\frac{3q}{2g_{1}n}(1-\cos(4g_{1}nt/\hbar))\right],$
(7)

where $g_{1}=\frac{4\pi\hbar^{2}}{m}\frac{a_{4}-a_{2}}{7}$ is the interaction
strength with $a_{\mathcal{F}}$ being the $s$-wave scattering length for the
collisional channel of the total angular momentum $\mathcal{F}$ and $n$ is the
mean atomic density. Following these equations, $p_{0}$ and $p_{2}$ would
undergo oscillations, which is not in agreement with the observed experimental
result. We therefore attribute the population conservation to polarization
purification by inelastic collisional losses Eto et al. (2019). It should be
noted that the observed population conservation contrasts with the case of the
$F=1$ state, in which the magnitude of the polarization modulates Jasperse et
al. (2017).

##  IV Discussion

The quantum lock-in VLS detection is of practical use in cold atom
experiments. It can be used for evaluating the degree of circular polarization
of an optical trap beam at the atomic position, as we have shown. As the
vacuum window birefringence introduces a maximum ellipticity of $10^{-2}$ or
$10^{-1}$ Steffen et al. (2013), a beam with no special care taken with
respect to the in vacuo polarization may generate a fictitious field of
several nT or a VLS of tens of Hz, even with a shallow trap for ultracold atom
experiments. Quantum lock-in detection is sensitive enough to ensure better
linear polarization at the atomic position and therefore will greatly improve
the magnetic conditions in cold atom experiments. The sensitivity is
sufficient to suppress the VLS below the requirements for magnetically
sensitive experiments, including studies of spinor BECs. Although a
homogeneous linear Zeeman shift does not affect the spinor physics due to spin
conservation Stamper-Kurn and Ketterle (2001), a magnetic field gradient below
several $\mu$T/m is typically required to prevent magnetic polarization and
observe the intrinsic magnetic ground state Stenger et al. (1998) or dynamics.
The sub-Hz VLS resolution of quantum lock-in detection meets this challenging
demand.

Reducing the VLS is also important for precise measurements. In addition to a
direct energy shift, an inhomogeneous fictitious field is also detrimental to
measurement accuracy Cates et al. (1988). VLS reduction leads to a long
coherence time, which is a mandatory requirement for highly sensitive
measurements. We have constructed a precise BEC magnetometer using a
transversally polarized $F=2$ BEC with a long coherence time, realized using
VLS elimination as we have shown. The detail of the $F=2$ BEC magnetometer
will be presented elsewhere Sekiguchi et al. (2020).

We finally discuss the sensitivity limitations. The sensitivity of the quantum
lock-in detection is essentially the same as that of a Ramsey interferometer
with an equal phase accumulation time. As the atom shot noise is dominant over
the photon shot noise in typical absorption imaging, the standard quantum
limit in the VLS measurement is given by Giovannetti et al. (2004, 2006)

$\delta\omega=\frac{1}{T\sqrt{N_{\mathrm{tot}}}}.$ (8)

Here we replace the factor $\frac{2}{\pi}$ in Eq. (3) due to the sinusoidal
modulation with the maximal value of 1, which is realized with a rectangular
waveform modulation. Substituting $N_{\mathrm{tot}}=3\times 10^{5}$ and $T$ =
$30$ ms into Eq. (8), we obtain $\delta\omega=2\pi\times 10$ mHz. This is
equivalent to a single shot field sensitivity of $\sim$ 1 pT.

##  V Conclusions

We have demonstrated precise detection of a VLS due to an optical trap using
the quantum lock-in method. We have applied the detection to eliminating the
VLS, to observe the extension of the lifetime of transversally polarized $F=2$
BEC. The attained resolution of sub Hz is sufficient to suppress the VLS below
the required level for magnetically sensitive research, including the study of
spinor BECs. Although our demonstration was performed with a BEC, the scope of
the detection method is not limited to cold atom gases; the proposed method
can be applied to spin systems such as trapped ions and diamond NV centers,
where coherent spin control is possible.

###### 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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