# Experimental study of tune-out wavelengths for spin-dependent optical
lattice in 87Rb Bose-Einstein condensation

Kai Wen  State Key Laboratory of Quantum Optics and Quantum Optics Devices,
Institute of Opto-electronics, Shanxi University, Taiyuan, Shanxi 030006,
People’s Republic of China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, People’s Republic of China
Zengming Meng<EMAIL_ADDRESS> State Key Laboratory of Quantum Optics
and Quantum Optics Devices,
Institute of Opto-electronics, Shanxi University, Taiyuan, Shanxi 030006,
People’s Republic of China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, People’s Republic of China
Liangwei Wang  State Key Laboratory of Quantum Optics and Quantum Optics
Devices,
Institute of Opto-electronics, Shanxi University, Taiyuan, Shanxi 030006,
People’s Republic of China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, People’s Republic of China
Liangchao Chen  State Key Laboratory of Quantum Optics and Quantum Optics
Devices,
Institute of Opto-electronics, Shanxi University, Taiyuan, Shanxi 030006,
People’s Republic of China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, People’s Republic of China
Lianghui Huang  State Key Laboratory of Quantum Optics and Quantum Optics
Devices,
Institute of Opto-electronics, Shanxi University, Taiyuan, Shanxi 030006,
People’s Republic of China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, People’s Republic of China  Pengjun
Wang  State Key Laboratory of Quantum Optics and Quantum Optics Devices,
Institute of Opto-electronics, Shanxi University, Taiyuan, Shanxi 030006,
People’s Republic of China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, People’s Republic of China  Jing
Zhang<EMAIL_ADDRESS>
<EMAIL_ADDRESS>State Key Laboratory of Quantum Optics and Quantum Optics
Devices,
Institute of Opto-electronics, Shanxi University, Taiyuan, Shanxi 030006,
People’s Republic of China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, People’s Republic of China

###### Abstract

We study the periodic potential of one-dimensional optical lattice originated
from scalar shift and vector shift by manipulating the lattice polarizations.
The ac Stark shift of optical lattice is measured by Kapitza-Dirac scattering
of 87Rb Bose-Einstein condensate and the characteristics of spin-dependent
optical lattice are presented by scanning the lattice wavelength between the
D1 and D2 lines. At the same time, tune-out wavelengths that ac Stark shift
cancels can be probed by optical lattice. We give the tune-out wavelengths in
more general cases of balancing the contributions of both the scalar and
vector shift. Our results provide a clear interpretation for spin-dependent
optical lattice and tune-out wavelengths, and help to design it by choosing
the appropriate lattice wavelength.

###### pacs:

34.20.Cf, 67.85.Hj, 03.75.Lm

##  I Introduction

Optical lattice for ultracold atoms has become an increasingly important
technology in many-body physics Bloch  _et al._ (2008), quantum simulation,
quantum computation, quantum information storage and high precision
measurements Aidelsburger  _et al._ (2013); Miyake  _et al._ (2013); Chiu  _et
al._ (2018); Schnorrberger  _et al._ (2009); Martin  _et al._ (2013). When
neutral atoms are trapped in periodic potentials produced by standing wave of
light fields, the trapping potentials of various atomic internal states are
manipulated by lattice polarizations, which is called the spin-dependent
optical lattice Mandel  _et al._ (2003); Duan  _et al._ (2003), which bring
more complicated geometry for ultracold atoms such as spin-dependent hexagonal
lattice Soltan-Panahi  _et al._ (2011), spin-dependent optical superlattice
Yang  _et al._ (2017), and have been used to study many interesting phenomena
such as controlled coherent transport Ke  _et al._ (2018); Mandel  _et al._
(2003), spinor BEC Ostrovskaya and Kivshar (2004), spin-orbit coupling and
artificial gauge fields Ye  _et al._ (2018); Grusdt  _et al._ (2017),
spontaneous emission of matter waves Krinner  _et al._ (2018), twisted-bilayer
optical potentials González-Tudela and Cirac (2019).

Tune-out wavelengths that ac Stark shift cancels was initially introduced in
species-specific optical manipulation LeBlanc and Thywissen (2007) and can be
useful for optical Feshbach resonances Clark  _et al._ (2015) and atomic
interferometer Trubko  _et al._ (2015). Since tune-out wavelengths are
independent of the light intensity Arora  _et al._ (2011); Jiang  _et al._
(2013); Cheng  _et al._ (2013), it can be precisely measured by various
methods Holmgren  _et al._ (2012); Henson  _et al._ (2015); Leonard  _et al._
(2015); Schmidt  _et al._ (2016); Adam and Charles (2016); Trubko  _et al._
(2017); Copenhaver  _et al._ (2019); Décamps  _et al._ (2020). In general,
tune-out wavelength is utilized accurately only for the scalar shift by
cancelling and neglecting the vector and tensor contribution as much as
possible Holmgren  _et al._ (2012); Henson  _et al._ (2015); Leonard  _et al._
(2015); Trubko  _et al._ (2017). In this paper, we investigate the tune-out
wavelengths in more general cases of considering the contributions from both
the scalar and vector shift. The ac Stark shift of optical lattice is measured
by Kapitza-Dirac scattering which diffracts BEC into a number of high momentum
states and the characteristics of spin-dependent optical lattice are
investigated by scanning the lattice wavelength between the D1 and D2 lines.
Kapitza-Dirac scattering becomes a standard tool and shows many applications
in calibrating the lattice depth Ovchinnikov  _et al._ (1999); Cahn  _et al._
(1997); Denschlag  _et al._ (2002); Beswick  _et al._ (2019); Chen  _et al._
(2010), detecting the lattice structure Viebahn  _et al._ (2019); Wen  _et
al._ (2020), performing high-resolution spectroscopy Stenger  _et al._ (1999)
and metrology Gupta  _et al._ (2002); Campbell  _et al._ (2005). The periodic
potential originated from scalar shift and vector shift is manipulated by
controlling the lattice polarizations, which is used to generate spin-
dependent optical lattice. We can design the special spin-dependent optical
lattice with the help of tune-out wavelengths.

##  II theory

###  II.1 AC Stark shift

As we know, ac Stark effect is the result of an interaction between atoms and
a classical light field. Here the total ac Stark shift for alkali-metal atoms
in ground state interacting with a far-off-resonance light field can be
expressed in terms of its scalar, vector, tensor components Beloy (2009);
Lundblad  _et al._ (2010); Le Kien  _et al._ (2013); Becher  _et al._ (2018);
Tsyganok  _et al._ (2019)

$\displaystyle\Delta U=$ $\displaystyle\Delta
U\left(F,{{m}_{F}};\omega\right)$ (1) $\displaystyle=$
$\displaystyle-A[\alpha_{~{}}^{\left(0\right)}\left(\omega\right)+{{\alpha}^{\left(1\right)}}\left(\omega\right)\left(\xi\hat{e}_{k}\cdot\hat{e}_{B}\right)\frac{{m}_{F}}{F}$
$\displaystyle+{{\alpha}^{\left(2\right)}}\left(\omega\right)\frac{3\cos^{2}\theta-1}{2}\frac{3{m}_{F}^{2}-F(F+1)}{F(2F-1)}],$

where ${{\alpha}^{\left(0,1,2\right)}}\left(\omega\right)$ are the scalar,
vector and tensor polarizabilities respectively. $F$ is the total atomic
angular momentum, $m_{F}$ is magnetic quantum number, $A$ is the laser field
intensity with $A=2\epsilon_{0}c\left|E\right|^{2}$, $\omega$ and $E$ are the
frequency and amplitude of optical field,
$\hat{e}_{k}\cdot\hat{e}_{B}=\left|{e}_{k}\right|\left|{e}_{B}\right|\cos(\phi)$,
$\hat{e}_{k}$ and $\hat{e}_{B}$ are unit vectors along the light wave-vector
and magnetic field quantization axis respectively, $\phi$ is the intersection
angle between $\hat{e}_{k}$ and $\hat{e}_{B}$, $\theta$ is the intersection
angle between the linearly polarized component of light field and
$\hat{e}_{B}$. This formula comes from the first non-vanishing term (the
second order) of a perturbation development. Note that the range of values of
light ellipticity is $\xi\in[-1,1]$, $\xi=\pm 1$ denotes left and right
circular polarization. The left and right elliptical polarization is defined
in terms of the light wave-vector. Scalar shift can be interpreted as a spin-
independent light shift. Vector shift acts like an effective magnetic field to
generate the linear Zeeman splitting (light shift proportional to $m_{F}$),
which depends on the ellipticity of the light and the intersection angle
between the laser beam wave vector and magnetic field quantization axis
$\hat{e}_{B}$. Tensor shift is proportional to $m^{2}_{F}$. For alkali-metal
atoms in ground state, the tensor shift can vanish once light detuning
$\delta$ exceed the hyperfine splitting $\Delta_{HF}$. Because the ground
state is $J=1/2$, which induces the tensor shift coefficient
${{\alpha}^{\left(2\right)}}=0$ Becher  _et al._ (2018); Steck (2007);
Rosenbusch  _et al._ (2009). In this work, we consider a far detuning of
$\delta\gg\Delta_{HF}$, so that the ac Stark shift only includes two terms of
the scalar $\alpha^{(0)}$ and vector shift $\alpha^{(1)}$.

For a linear polarized light beam($\xi=0$), vector shift vanishes, but scalar
shift keeps. For circular polarization light, the left and right circular
polarization can change the sign of vector shift to be positive or negative.
Therefore, the different ac Stark shift of two spin states can be cancelled by
controlling the ellipticity, or tuning the angle between $\hat{e}_{k}$ and
$\hat{e}_{B}$ (even changing the strength of the external bias magnetic field
when considering high order contribution Lundblad  _et al._ (2010); Derevianko
(2010); Yang  _et al._ (2016)), which is an important technique for the atomic
clock and qubit for quantum computation.

###  II.2 Scalar and vector shift

For the first excited state of alkali-metal atoms, the fine structure induces
the spectral lines of the D1 (the $5^{2}S_{1/2}\rightarrow 5^{2}P_{1/2}$
transition) and D2 (the $5^{2}S_{1/2}\rightarrow 5^{2}P_{3/2}$ transition)
lines. Because the D1 and D2 lines of the first excited state are larger
detuned than the the excited-state hyperfine splitting, the coefficients of
the scalar and vector shifts in Eq. (1) are expressed as Corwin  _et al._
(1999); Park  _et al._ (2001); Mckay and Demarco (2010); Cao  _et al._ (2012)

$\displaystyle{{{\alpha}^{\left(0\right)}}\left(\omega\right)\approx-\frac{\pi{{c}^{2}}{{\Gamma}_{{{D2}}}}}{2\omega^{3}_{0}}\left(\frac{2}{{{\delta}_{{{D2}}}}}+\frac{1}{{{\delta}_{{{D1}}}}}\right)},$
(2)
$\displaystyle{{\alpha}^{\left(1\right)}}\left(\omega\right)\approx-\frac{\pi{{c}^{2}}{{\Gamma}_{{{D2}}}}}{2\omega^{3}_{0}}\left(\frac{1}{{{\delta}_{{{D2}}}}}-\frac{1}{{{\delta}_{{{D1}}}}}\right){g}_{F}{F},$
$\displaystyle{{\alpha}^{\left(2\right)}}\left(\omega\right)\approx 0,$

where ${{\Gamma}_{{{D}_{2}}}}$ is the decay rate of the excited state for
$D_{2}$ line, ${{\delta}_{{{D}_{1}}}}={{\omega}_{~{}}}-{{\omega}_{D1}}$,
${{\delta}_{{{D}_{2}}}}={{\omega}_{~{}}}-{{\omega}_{D2}}$. ${{g}_{F}}$ is the
gyromagnetic ratio

$\displaystyle g_{F}=g_{J}\left[\frac{F(F+1)+J(J+1)-I(I+1)}{2F(F+1)}\right],$
(3) $\displaystyle g_{J}=1+\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)},$

where $S$ is the spin angular momentum, $L$ is the orbital angular momentum,
$J$ is the total electronic angular momentum, $I$ is the total nuclear angular
momentum. For the ground states $5^{2}S_{1/2}$ of 87Rb atoms, $g_{J}=2$,
$g_{F}=1/2$ for $F=2$, $g_{F}=-1/2$ for $F=1$. Here, we study ${}^{87}Rb$
atoms and present the coefficients of the scalar and vector shifts as the
function of wavelength in Fig. 1. The resonant wavelengths of the D1 and D2
lines of ${}^{87}Rb$ atom are ${\lambda}_{D1}=794.98$ nm and
${\lambda}_{D2}=780.24$ nm respectively. Obviously $\alpha^{(0)}$ has a
crossed zero point at $\lambda=790.005$ nm, and $\alpha^{(1)}$ always is
negative between the D1 and D2 lines. When the wavelength of light is far red-
detuned or blue-detuned by an amount larger than the fine structure splitting
of the excited states, the vector shift approaches zero. Here, we study the
tune-out wavelengths in more general cases of considering the contributions
from both the scalar and vector shift. The tune-out wavelengths of the ground
hyperfine states are given in Table 1 with $\phi=0$, $\xi=0,\pm 1$.

Figure 1: (Color online) The coefficients of the scalar and vector shifts as
the function of wavelength for ${}^{87}Rb$ atom in $|F=2,m_{F}=2\rangle$
state. $\alpha^{(0)}$ has a crossed zero point at $\lambda$ = 790.020 nm. When
the wavelength of light is far red-detuned from D1,
$\alpha_{~{}}^{\left(0\right)}\gg\alpha_{~{}}^{\left(1\right)}\approx 0$.
Table 1: Tune-out wavelengths of the ground states with $\phi=0$ in the
$5^{2}S_{1/2}-5^{2}P_{1/2,}$ 3/2 states of 87Rb.

Polarization | $|F,m_{F}\rangle$ |  ${{\lambda_{zero}}}$(nm)
---|---|---
Calc. use Eq. (3) | Calc. in Ref LeBlanc and Thywissen (2007) | Other Calc. | Expt.
| $|2,2\rangle$ | 790.005 | 790.04 |  | 790.01850(9) Schmidt  _et al._ (2016)
| $|2,1\rangle$ | 790.005 | 790.04 | 790.032439(35) Leonard  _et al._ (2015) |
$\xi=0$ | $|2,0\rangle$ | 790.005 | 790.03 | 790.034(7) Arora  _et al._ (2011) | 790.032388(32) Leonard  _et al._ (2015)
| $|2,-1\rangle$ | 790.005 | 790.04 | 790.032602(193) Wang  _et al._ (2016) |
| $|2,-2\rangle$ | 790.005 | 790.04 |  |
| $|1,1\rangle$ | 790.005 | 790.04 |  | 789.85(1) Catani  _et al._ (2009)
| $|1,0\rangle$ | 790.005 | 790.04 | 790.018187(193) Wang  _et al._ (2016) | 790.018(2) Lamporesi  _et al._ (2010), 790.020(25) *
| $|1,-1\rangle$ | 790.005 | 790.04 |  | 790.01858(23) Schmidt  _et al._ (2016)
| $|2,2\rangle$ | none | none |  | none *
| $|2,1\rangle$ | 792.484 | 792.52 |  |
| $|2,0\rangle$ | 790.005 | 790.06 |  |
$\xi=1$ | $|2,-1\rangle$ | 787.541 | 787.59 |  |
| $|2,-2\rangle$ | 785.093 | 785.14 |  |
| $|1,1\rangle$ | 787.541 | 787.59 |  | 787.590(31) *
| $|1,0\rangle$ | 790.005 | 790.06 |  | 790.020(25) *
| $|1,-1\rangle$ | 792.484 | 792.53 |  |
| $|2,2\rangle$ | 785.093 | 785.14 | 785.11516 Wang (2017) | 785.146(12) *
| $|2,1\rangle$ | 787.541 | 787.59 |  |
| $|2,0\rangle$ | 790.005 | 790.06 |  |
$\xi=-1$ | $|2,-1\rangle$ | 792.484 | 792.52 |  |
| $|2,-2\rangle$ | none | none | none Wang (2017) |
| $|1,1\rangle$ | 792.484 | 792.53 |  |  $\approx$792.4 Schmidt  _et al._ (2016), 792.462(22) *
| $|1,0\rangle$ | 790.005 | 790.06 |  | 790.020(25) *
| $|1,-1\rangle$ | 787.541 | 787.59 |  |  $\approx$787.620 Schmidt  _et al._ (2016)

  * *

Our experimental measurements.

In order to measure the tune-out wavelengths, we employ a one-dimensional (1D)
optical lattice along the external bias magnetic field ($\phi=0$) with
different polarization configurations. Here, the two laser beams have the same
intensity with
$A=2\epsilon_{0}c\left|E_{1}\right|^{2}=2\epsilon_{0}c\left|E_{2}\right|^{2}$.
For case 1, two laser beams with the same linear polarization counter-
propagate along the z axis. Because of the parallel polarized beams, it can
produce the spatial intensity modulation to form a 1D optical lattice,

$\begin{split}\Delta
U_{L1}\left(F,{{m}_{F}};\omega\right)=&-4A{\alpha}^{\left(0\right)}\cos^{2}kz.\\\
\end{split}$ (4)

The optical lattice potential for this case only originates from the scalar
shift. Therefor it can be used for measuring the tune-out wavelength for the
scalar shift. For case 2, two counter-propagated laser beams have linear
orthogonal polarization (lin$\perp$lin polarization configuration). The
orthogonally polarized beams can not produce a spatial intensity modulation.
In contrast, it can produce the ellipticity modulation of polarization in
space. This optical lattice potential is called Sisyphus optical potential,
which has been used for Sisyphus cooling Dalibard and Cohen-Tannoudji (1989).
The periodic potential is given by

$\begin{split}\Delta
U_{L2}\left(F,{{m}_{F}};\omega\right)=&-2A[{\alpha}^{\left(0\right)}+{\alpha}^{\left(1\right)}\xi\frac{{m}_{F}}{F}\sin(2kz)].\\\
\end{split}$ (5)

This periodic potential only comes from the vector shift and the scalar term,
which gives a uniformed energy shift. For case 3, two laser beams with the
same polarized circular polarization counter-propagates along the z axis,
which can also produce the spatial intensity modulation to form a 1D optical
lattice,

$\begin{split}\Delta
U_{L3}\left(F,{{m}_{F}};\omega\right)=&-4A\cos^{2}kz[{\alpha}^{\left(0\right)}+{\alpha}^{\left(1\right)}\xi\frac{{m}_{F}}{F}].\\\
\end{split}$ (6)

The optical lattice potential for this case includes the scalar and vector
shift simultaneously. Therefore, we can study the tune-out wavelengths in
presence of contributions of both the scalar and vector shift in this case.
And for case 4, like case 2, the scalar shift is a constant only related to
the constant intensity of two orthogonal polarized counter-propagating laser
beams. But the different with former, is that the vector shift becomes zero
since it only produce the rotation of linear polarization in space. Hence it
can not produce any spatial modulation to form a 1D lattice.

$\begin{split}\Delta
U_{L4}\left(F,{{m}_{F}};\omega\right)=&-2A{\alpha}^{\left(0\right)}.\\\
\end{split}$ (7)

Here, for a 1D optical lattice along the external bias magnetic field
($|\cos\phi|=1$), it is convenient to define the left and right elliptical
polarizations in terms of the magnetic field quantization axis.

Figure 2: (Color online) Schematic of experimental setup and three cases for
different polarization configurations. (a)A 1D optical lattice is formed by
two counter-propagating laser fields. The external magnetic field is aligned
along with -z axis. (b) Case 1, the counter-propagating lasers have linear
parallel polarization. Case 2, the counter-propagating lasers have linear
orthogonal polarization. Case 3 and Case 4, the counter-propagating lasers
have the circular parallel or orthogonal polarizations.

##  III Experiment

The schematic of experimental setup is shown in Fig. 2(a). Ultracold 87Rb
atoms in $|F=2,{{m}_{F}}=2\rangle$ hyperfine state are loaded in a crossed
optical dipole trap Xiong  _et al._ (2010). Forced evaporation in the optical
trap is used to create the BEC with up to $5\times 10^{5}$ atoms. In order to
obtain the atoms in the different single spin states, BEC is transferred from
$|F=2,{{m}_{F}}=2\rangle$ into $|F=1,{{m}_{F}}=1\rangle$ via a rapid adiabatic
passage induced by a microwave-frequency field with duration of 10 ms at 3.9 G
of the bias magnetic field, where the frequency of center is 6.842935 GHz and
the width is 0.25 MHz. Then the atoms in $|F=1,{{m}_{F}}=1\rangle$ can further
be transferred into $|F=1,{{m}_{F}}=0\rangle$ state using a rapid adiabatic
passage induced by a radio frequency (rf) field at 28 G of the bias magnetic
field. The lattice beam is derived from a single frequency Ti:sapphire laser
with broad tuning range of the frequency. An acousto-optical modulator is used
to control the intensity of the lattice beam. The lattice beam passes through
the polarizing beam splitter to generate the perfect polarization. The
polarization extinction ratio of the polarizing beam splitter can reach 500:1.
Therefore the linear polarization purity of the lattice beam is about 0.2$\%$.
Furthermore the circular polarization purity of the lattice beam can reach
about 0.5$\%$. A lattice beam propagates with z axis and converges on BEC with
waist of 100 $\mu m$ by a lens (f = 300 mm). Then the beam is reflected by a
concave mirror ( curvature radius r=300 mm) and refocused on BEC with the
almost same waist size. The advantage of this configuration can reduce phase
jitter significantly. Here, we employ Kapitza-Dirac (or Raman-Nath) scattering
to measure the ac Stark shift. Kapitza-Dirac scattering is used to diffract
the BEC into a number of momentum states by a standing light wave, in which
the interaction is sufficiently short and strong Gould  _et al._ (1986). In
this process, BEC is kept in a crossed optical dipole trap and the lattice
potential imprints a phase modulation on matter wave in position space. Then
the phase modulation on matter wave is measured in momentum space via the
time-of-flight (TOF) absorption image. It is obvious that higher momentum
orders $\pm 2N\hbar k$ appear in the atomic density distribution of the TOF
absorption image, which depends on the potential depth and interaction time.
Here, we apply a 1D optical lattice short pulse for 4 $\mu s$ with the power
of 80 mW on BEC. Then immediately turn off the optical trap, let the atoms
ballistically expand in 12 ms and take the absorption images. We obtain the
lattice depth from the absorption images by applying the lattice at a fixed
laser power for different intervals of time and by observing the interval at
which the $n=0$ order atoms in the lattice vanish Ovchinnikov  _et al._
(1999); Cahn  _et al._ (1997); Denschlag  _et al._ (2002); Beswick  _et al._
(2019); Chen  _et al._ (2010). We define the recoil momentum $\hbar
k=2\pi\hbar/\lambda$ and recoil energy $E_{r}=(\hbar k)^{2}/2m=h\times 3.67$
kHz as the nature momentum and energy units, where $m$ is the mass of 87Rb
atom, $\lambda$ is the wavelength of the lattice laser.

Figure 3: (Color online) The 1D optical lattice with the linear parallel
polarization. (a) Measured data (squares, triangles and circles) and the
theoretical fit (the three types of lines overlapped together) of the lattice
potential depth as the function of the laser wavelength for the three
hyperfine states $|F=2,{{m}_{F}}=2\rangle$, $|F=1,{{m}_{F}}=1\rangle$ and
$|F=1,{{m}_{F}}=0\rangle$. It shows that this periodic potential is a spin-
independent lattice. Each point is the average of at least three measurements.
(b) Atomic density distribution in the TOF absorption images at different
wavelength of the lattice laser. Figure 4: (Color online) The 1D optical
lattice with the linear orthogonal polarization. (a) Measured data (squares,
triangles and circles) and the theoretical fit (dashed, solid and dotted
lines) of the lattice potential depth as the function of the laser wavelength
for the three hyperfine states $|F=2,{{m}_{F}}=2\rangle$,
$|F=1,{{m}_{F}}=1\rangle$ and $|F=1,{{m}_{F}}=0\rangle$. This periodic
potential is a spin-dependent lattice, which only depends on the vector shift.
(b) Atomic density distribution in the TOF absorption images at different
wavelength of the lattice laser. There is no lattice potential for the
$|F=1,{{m}_{F}}=0\rangle$ state ($\Delta{{m}_{F}}=0$) while the
$|{{m}_{F}}=\pm 2\rangle$ and $|{{m}_{F}}=\pm 1\rangle$ states
($\Delta{{m}_{F}}\neq 0$) always experience the lattice potential when
adjusting the lattice wavelength. Figure 5: (Color online) The 1D optical
lattice with the circular parallel polarization. (a) Measured data (squares,
triangles and circles) and the theoretical fit (dashed, solid and dotted
lines) of the lattice potential depth as the function of the laser wavelength
for the three hyperfine states $|F=2,{{m}_{F}}=2\rangle$,
$|F=1,{{m}_{F}}=1\rangle$ and $|F=1,{{m}_{F}}=0\rangle$. This periodic
potential is a spin-dependent lattice, which depends on the the scalar and
vector shift. The tune-out wavelengths are 792.462(22) nm, 790.020(25) nm and
785.146(12) nm for the $|F=1,{{m}_{F}}=1\rangle$, $|F=1,{{m}_{F}}=0\rangle$
and $|F=2,{{m}_{F}}=2\rangle$ respectively. (b) Atomic density distribution in
the TOF absorption images at different wavelength of the lattice laser.
(c1)-(c3) give the detailed measurements of the lattice potentials for the
$|F=1,{{m}_{F}}=1\rangle$, $|F=1,{{m}_{F}}=0\rangle$ and
$|F=2,{{m}_{F}}=2\rangle$ near the tune-out wavelengths respectively. Figure
6: (Color online) The 1D optical lattice with the circular orthogonal
polarization. (a) Measured data (squares, triangles and circles) and the
theoretical fit (the three types of lines overlapped together) of the lattice
potential depth as the function of the laser wavelength for the three
hyperfine states $|F=2,{{m}_{F}}=2\rangle$, $|F=1,{{m}_{F}}=1\rangle$ and
$|F=1,{{m}_{F}}=0\rangle$. There is no obvious periodic potential in this
spin-dependent lattice, because the scalar and vector don’t produce spatial
light intensity modulation. (b) Atomic density distribution of the time-of-
flight absorption images. There is no lattice potentials for all states for
all wavelength.

For case 1, a 1D optical lattice with the linear parallel polarization
produces the spatial intensity modulation, which only comes from the scalar
shift. Thus it is a spin-independent optical lattice and the potential depth
as the function of the lattice wavelength are plotted in Fig. 3(a). Here, the
positive and negative periodic potentials correspond to blue and red detuned
lattice laser. The higher momentum orders $\pm 2N\hbar k$ are observed in the
atomic density distribution of the TOF absorption images as shown in Fig.
3(b), which depends on the potential depth. We measure the tune-out
wavelengths by changing the lattice wavelength and find its location at
$\sim$790.020 nm for all spin states, which is in good agreement with the
previous works Leonard  _et al._ (2015); Schmidt  _et al._ (2016).

Figure 7: (Color online) The tune-out wavelengths as the function of the
amplitude of the external magnetic field for Case 3. (a), (b), (c) correspond
to $|F=2,{{m}_{F}}=2\rangle$, $|F=1,{{m}_{F}}=1\rangle$ and
$|F=1,{{m}_{F}}=0\rangle$. Blue (square sign) and red (circular sign) curves
correspond to the residual magnetic field in the (x, y, z) directions
suppressed from (0.4 G, 0.46 G, 0.38 G) to (0.25 G, 0.28 G, 0.2 G)
respectively.

For case 2, a laser beam with the linear polarization pass through a quarter-
wave plate and is reflected by a concave mirror, which produces the linear
orthogonal polarization configuration. We plot the potential depth vs
different wavelength as shown in Fig. 4(a). This periodic potential is a spin-
dependent lattice, which only comes from the contribution of the vector shift.
Therefore, there is no lattice potential for the $|F=1,{{m}_{F}}=0\rangle$
state ($\Delta{{m}_{F}}=0$) for all wavelength as shown in Fig. 4(b). The
$|{{m}_{F}}=\pm 2\rangle$ and $|{{m}_{F}}=\pm 1\rangle$ states
($\Delta{{m}_{F}}\neq 0$) always experience the lattice potential when
adjusting the lattice wavelength.

For case 3, a 1D optical lattice with the circular parallel polarization
produces the spatial intensity modulation, which includes the contribution of
the scalar and vector shift simultaneously. The potential depth of
$|F=1,{{m}_{F}}=1\rangle$, $|F=1,{{m}_{F}}=0\rangle$ and
$|F=2,{{m}_{F}}=2\rangle$ are given in Fig. 5(a) and (b). The tune-out
wavelengths are generated by balancing the scalar and vector shift, which are
792.462(22)nm, 790.020(25)nm and 785.146(12)nm for $|F=1,{{m}_{F}}=1\rangle$,
$|F=1,{{m}_{F}}=0\rangle$ and $|F=2,{{m}_{F}}=2\rangle$ respectively as shown
in Fig. 5(c). Note that the tune-out wavelengths in this case are sensitive to
the ellipticity of the lattice polarization, the alignment between the
direction of the 1D optical lattice and the external bias magnetic field. This
case provides us more controlled ways to generate the different kinds of spin-
dependent optical lattice. There is an interesting phenomenon that two
neighbouring spin states have the opposite lattice potential (blue and red
detuning respectively) by choosing the appropriate wavelength, for example,
$|F=1,{{m}_{F}}=1\rangle$ and $|F=1,{{m}_{F}}=0\rangle$ states have the
opposite lattice potential at the wavelength of 791.24 nm.

For case 4, two counter-propagating laser beams with the orthogonal circular
polarization can not generate any spatial modulation on the BEC. The potential
depth for $|F=1,{{m}_{F}}=1\rangle$, $|F=1,{{m}_{F}}=0\rangle$ and
$|F=2,{{m}_{F}}=2\rangle$ are given in Fig. 6(a) and (b), showing no effective
potential (no density modulation) for these states.

Furthermore, we study the dependence of the tune-out wavelengths on the
strength of the external bias magnetic field in more detail. The intersection
angle $\phi$ between $\hat{e}_{k}$ and $\hat{e}_{B}$ is expressed as

$\begin{split}cos(\phi)=\frac{B_{Bi}^{z}+B_{Re}^{z}}{\sqrt{(B_{Re}^{x})^{2}+(B_{Re}^{y})^{2}+(B_{Bi}^{z}+B_{Re}^{z})^{2}}},\\\
\end{split}$ (8)

where $B_{Bi}$ is the external bias magnetic field, $B_{Re}$ is the residual
magnetic field (such as the earth magnetic field). We measure the tune-out
wavelengths as the function of the strength of the external bias magnetic
field as shown in Fig. 7. By changing the bias magnetic field to small value,
the direction of the total magnetic field and the intersection angle $\phi$
can be changed. Therefore, the tune-out wavelengths change when the strength
of the bias magnetic field is near to the residual magnetic field value. The
strength of the external bias magnetic field in one direction is gradually
decreased to zero and then increased in the opposite direction. We find that
the tune-out wavelengths for $\xi=1$ jumps into $\xi=-1$ due to inversion the
external bias magnetic field direction for the spin $|F=1,{{m}_{F}}=1\rangle$
and $|F=2,{{m}_{F}}=2\rangle$ states as shown in Fig. 7(a) and (b). The slope
is sensitive to the strength of residual magnetic field in the perpendicular
direction of z axis. Here, three pairs of Helmholtz coils are employed to
compensate the background magnetic field. When the residual magnetic field in
the (x, y, z) directions are suppressed from (0.4 G, 0.46 G, 0.38 G) to (0.25
G, 0.28 G, 0.2 G) which are measured by a triaxial fluxgate magnetometer, the
slope is changed from -4.054 nm/G to -6.584 nm/G for $|F=1,{{m}_{F}}=1\rangle$
and 8.108 nm/G to 13.168 nm/G for $|F=2,{{m}_{F}}=2\rangle$ as shown in Fig.
7(a) and (b) respectively. Therefore, this method can be utilized to calibrate
and measure the residual magnetic field.

##  IV Conclusion

In conclusion, we present an experiment to measure the ac Stark shift around
the tune-out wavelengths of 87Rb BEC in the three different hyperfine ground
states $|F=1,{{m}_{F}}=1\rangle$, $|F=1,{{m}_{F}}=0\rangle$ and
$|F=2,{{m}_{F}}=2\rangle$ between D1 and D2 lines. Four different polarization
configurations of one-dimensional optical lattice, which are originated from
scalar shift, vector shift, both scalar and vector shift respectively by
manipulating the lattice polarizations, are investigated. Kapitza-Dirac
scattering technique is employed to probe the ac Stark shift of atoms in
optical lattice and the characteristics of spin-dependent optical lattice are
presented by scanning the lattice wavelength. We present the tune-out
wavelengths in more general cases of considering the contributions of both the
scalar and vector shift. We further study the dependence of the tune-out
wavelengths on the strength of the external bias magnetic field in more
detail. Our work provides a clear interpretation for spin-dependent optical
lattice and can be used for the realization of two species system, or the same
atoms (Rb) with different spin states, in which one of them move freely while
the others are trapped to different degrees of optical lattice potential. This
system can be a test-bed for observing or simulating phenomena such as entropy
cooling Catani  _et al._ (2009); Lamporesi  _et al._ (2010); Mckay and Demarco
(2010); Arora  _et al._ (2011); Bause  _et al._ (2020), Kondo effect Zhang
_et al._ (2016); Yao  _et al._ (2019) etc, and even have application in Sr
optical lattice clock Heinz  _et al._ (2020).

##  V Funding Information

This research was supported by the MOST (2016YFA0301602, 2018YFA0307601), NSFC
(Grant No. 11804203, 11974224, 11704234), the Fund for Shanxi ”1331 Project”
Key Subjects Construction.

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