# Measurement of the quadratic Zeeman shift of 85Rb hyperfine sublevels using
stimulated Raman transitions

Run-Bing Li  Lin Zhou  State Key Laboratory of Magnetic Resonance and Atomic
and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese
Academy of Sciences, Wuhan 430071, China  Center for Cold Atom Physics,
Chinese Academy of Sciences, Wuhan 430071, China  Graduate School, Chinese
Academy of Sciences, Beijing 100080, China  Jin Wang<EMAIL_ADDRESS>Ming-
Sheng Zhan  State Key Laboratory of Magnetic Resonance and Atomic and
Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy
of Sciences, Wuhan 430071, China  Center for Cold Atom Physics, Chinese
Academy of Sciences, Wuhan 430071, China

###### Abstract

Abstract

We demonstrate a technique for directly measuring the quadratic Zeeman shift
using stimulated Raman transitions. The quadratic Zeeman shift has been
measured yielding $\Delta\nu=1296.8$ $\pm 3.3$ Hz/G2 for magnetically
insensitive sublevels ($5S_{1/2},F=2,m_{F}=0\rightarrow$
$5S_{1/2},F=3,m_{F}=0$) of 85Rb by compensating the magnetic field and
cancelling the ac Stark shift. We also measured the cancellation ratio of the
differential ac Stark shift due to the imbalanced Raman beams by using two
pairs of Raman beams ($\sigma^{+}$, $\sigma^{+})$ and it is $1$:$3.67$ when
the one-photon detuning is $1.5$ GHz in the experiment.

###### pacs:

32.80.Qk,03.75.Dg, 37.25.+k

1\. Introduction

Since the atom interferometer was demonstrated in 1991Kasevich1991a , it has
been applied to rotation measurement, such as inertial navigation and even the
rotation rate of the earth Gustavson2000a ; Canuel2006a . Recently, an atom-
interferometer gyroscope of high sensitivity and long-term stability was
reported Durfee2006a . In order to improve the accuracy of the rotation rate
measurement by using an atom-interferometer gyroscope, the potential
systematic errors should be considered and controlled as well as possible. The
quadratic Zeeman shift is considered as a factor that influences the accuracy
of the rotation rate measurement in the atom-interferometer gyroscope.

The atom gyroscope generally uses two counter-propagating cold-atom clouds
launched in strongly curved parabolic trajectories Canuel2006a . The two cold
atom clouds should be overlapped completely in order to cancel common noise
and gravity acceleration, and cold collisions occur between atoms along
similar trajectories. For a dual atom-interferometer gyroscope, Rubidium is a
suitable candidate because it has a smaller collision frequency shift than
Cesium Santos2002a ; Kokkelmans1997a ; Gibble1995a ; Sortais2000a . In our
previous work Li2008a ; Wang2007a , we have experimentally investigated the
stimulated Raman transitions in the cold atom interferometer. Both the
accuracy and the fringe contrast of an atom-interferometer gyroscope can be
improved by studying the magnetic field dependence of the coherent population
transfer. A homogenous magnetic field must be applied along the Raman beams to
keep the quantization axis consistent and resolve degenerate magnetic
sublevels. This magnetic field will cause Zeeman shifts. The quadratic Zeeman
shift induces a relative frequency shift of the two coherent states, which
influences the accuracy of the rotation rate measurement. It is therefore
important to measure accurately and understand the quadratic Zeeman shift of
85Rb in the cold atom interferometer. Similarly, the quadratic Zeeman shift is
important in other applications such as microwave frequency standards
Ramsey1963a ; Thomas1982a ; Hemmer1986a , optical frequency standards
Kajita2005a ; Boyd2007a  and coherent population trapping clock Vanier2005a .
The quadratic Zeeman shift can be usually obtained from the Breit-Rabi formula
after the magnetic field is measured by the linear Zeeman effect Bize1999a .
We study this from the field-insensitive clock transitions whose linear Zeeman
shift is zero, thus the magnetic field is calibrated from other $u_{F}\neq 0$
states. We have also studied this quadratic Zeeman shift in the presence of
the ac Stark shift of the Raman pulses.

In this paper, we analyze the hyperfine sublevels of the ground states in the
magnetic field by using second-order perturbation theory, and demonstrate
experimentally the coherent population transfer of the different Zeeman
sublevels by stimulated Raman transitions. The quadratic Zeeman shift of the
ground state of 85Rb was measured by the two-photon resonance of the
stimulated Raman transition after the ac Stark shift was cancelled and the
residual magnetic field was compensated. The value of the magnetic field is
calibrated by the linear Zeeman shift. Our analysis shows that the quadratic
Zeeman shift can be measured to Hz level for magnetically insensitive states
($5S_{1/2},F=2,m_{F}=0\rightarrow$ $5S_{1/2},F=3,m_{F}=0$) in our experiment.
We also measured the cancellation ratio of the differential ac Stark shift due
to the imbalanced Raman beams by using two pairs of Raman beams. This study
provides useful data for higher precision measurement of the quadratic Zeeman
shift of 85Rb, even for improving the accuracy of the rotation rate
measurement of the atom-interferometer gyroscope.

2\. Quadratic Zeeman shift

Including the hyperfine interaction, the ground state energy levels will split
and shift in the magnetic field. The interaction Hamiltonian operator
Sobelman1996a ; Itano2000a  within the subspace of hyperfine sublevels
associated with the electronic levels is given by

$H^{{}^{\prime}}=hA_{S}I\cdot J+g_{J}\mu_{B}J\cdot B+g_{I}\mu_{B}I\cdot B$ (1)

where, $h$ is the Plank constant, $A_{S}$ is the hyperfine constant, $I$ and
$J$ are the nuclear spin operators and orbital angular momentum respectively,
$g_{J}$ and $g_{I}$ are the electronic $g$-factor and nuclear $g$-factor
respectively, $\mu_{B}$ is Bohr magneton. Second-order perturbation theory is
valid for low magnetic-field intensity, and the energies of the hyperfine
Zeeman sublevels for the ground states can be derived as following

For F=2

$E(\frac{1}{2},2,0,B)=E(\frac{1}{2})-\frac{7}{4}hA_{S}-\mathstrut\vskip 3.0pt
plus 1.0pt minus 1.0pt\frac{(g_{J}-g_{I})^{2}}{12hA_{S}}\mu_{B}^{2}B^{2}$ (2)

$\displaystyle E(\frac{1}{2},2,\pm 1,B)$
$\displaystyle=E(\frac{1}{2})-\frac{7}{4}hA_{S}\mp\frac{g_{J}-7g_{I}}{6hA_{S}}\mu_{B}B$
(3) $\displaystyle-\frac{2(g_{J}-g_{I})^{2}}{27hA_{S}}\mu_{B}^{2}B^{2}$

$\displaystyle E(\frac{1}{2},2,\pm 2,B)$
$\displaystyle=E(\frac{1}{2})-\frac{7}{4}hA_{S}\mp\frac{g_{J}-7g_{I}}{3hA_{S}}\mu_{B}B$
(4) $\displaystyle-\frac{5(g_{J}-g_{I})^{2}}{108hA_{S}}\mu_{B}^{2}B^{2}$

For F=3

$E(\frac{1}{2},3,0,B)=E(\frac{1}{2})+\frac{5}{4}hA_{S}+\frac{(g_{J}-g_{I})^{2}}{12hA_{S}}\mu_{B}^{2}B^{2}$
(5)

$\displaystyle E(\frac{1}{2},3,\pm 1,B)$
$\displaystyle=E(\frac{1}{2})+\frac{5}{4}hA_{S}\pm\frac{g_{J}+5g_{I}}{6hA_{S}}\mu_{B}B$
(6) $\displaystyle+\frac{2(g_{J}-g_{I})^{2}}{27hA_{S}}\mu_{B}^{2}B^{2}$

$\displaystyle E(\frac{1}{2},3,\pm 2,B)$
$\displaystyle=E(\frac{1}{2})+\frac{5}{4}hA_{S}\pm\frac{g_{J}+5g_{I}}{3hA_{S}}\mu_{B}B$
(7) $\displaystyle+\frac{5(g_{J}-g_{I})^{2}}{108hA_{S}}\mu_{B}^{2}B^{2}$

$E(\frac{1}{2},3,\pm
3,B)=E(\frac{1}{2})-\frac{7}{4}hA_{S}\pm\frac{g_{J}+5g_{I}}{2hA_{S}}\mu_{B}B$
(8)

Here, $E(J,F,m_{F},B)$ denotes the energy of the hyperfine sublevels,
including the effect of the hyperfine interaction and magnetic field
splitting. From eqs.(2) and (5), the quadratic Zeeman shift for the transition
$5S_{1/2},F=2,m_{F}=0\rightarrow 5S_{1/2},F=3,m_{F}=0$ is
$\Delta\nu=(g_{J}-g_{I})^{2}\mu_{B}^{2}B^{2}/6hA_{S}$, which is consistent
with the reference Steck  that is obtained from the Breit-Rabi formula when it
is extended to second order in the field strength.

3\. Experimental configuration

Figure 1: Experimental scheme: cold 85Rb atoms fly horizontally from the MOT
to the probe region. Three crossed pairs of Helmholtz coils are applied to
compensate the residual magnetic field in the stimulated Raman interaction
area. The combined Raman beams ($R_{1}$,$R_{2}$) and
($R_{1}^{{}^{\prime}}$,$R_{2}^{{}^{\prime}}$) are parallel to the magnetic
field $B$ and $B_{0}$ respectively. The laser-induced fluorescence signal is
detected by a PMT.

The experimental arrangement is shown in Fig.1, which is similar to our
previous work Li2008a ; Wang2007a . Briefly, the cold atoms are trapped in a
nonmagnetic stainless steel chamber with $14$ windows, where the trapping and
repumping beams are provided by a tapered amplifier diode laser (TOPTICA
TA100) and an external-cavity diode laser (TOPTIC DL100) respectively, whose
frequencies are stabilized using saturated absorption spectroscopy Wang2000a .
After the polarization gradient cooling (PGC) process, the atoms are guided by
a near-resonance laser pulse and fly transversely from the trapping region to
the probe region at a velocity of $24$ m/s Jiang2005a . Then, they are
completely pumped to the ground state $5S_{1/2},F=2$ as the initial state by a
perpendicular linearly polarized laser beam which is near resonance with the
transition $5S_{1/2},F=3\rightarrow 5P_{3/2},F=2$. Three crossed pairs of
Helmholtz coils are used to provide the magnetic field in the Raman
interaction area, where the current of the coils along the Raman beams (R1,
R2) is controlled by the DC power supply (MPS-901) and measured by the digital
multimeters (Flucke 8846A). The magnetic field intensity is calibrated by the
first-order Zeeman shift, whose uncertainty is less than one part in one
thousand. The combined Raman beams (R1, R2) and (R${}_{1}^{{}^{\prime}}$,
R${}_{2}^{{}^{\prime}}$) are applied along the magnetic fields $B$ and $B_{0}$
respectively in the stimulated Raman interaction region. The Raman beams (R1,
R2) are used to measure the frequency shift induced by the external fields
such as the Raman beams (R${}_{1}^{{}^{\prime}}$, R${}_{2}^{{}^{\prime}}$) and
the magnetic field $B$. The Raman beams (R1, R2) and (R${}_{1}^{{}^{\prime}}$,
R${}_{2}^{{}^{\prime}}$) are supplied from the same Raman laser. This
configuration has the benefit for the accurate measurement of the ac stark
shift because two pairs of Raman beams always have the same one-photon
detuning. The detailed description of the Raman laser arrangement is similar
to our previous work Wang2007a . The atoms are transferred to the state
$5S_{1/2},F=3$ from $5S_{1/2},F=2$ when they pass through a Raman $\pi$-pulse.
After coherent population transfer via a simulated Raman transition, the
population of the state is detected by a laser induced fluorescence (LIF)
signal, and we use a photo multiplier tube (PMT) to collect the LIF.

Figure 2: Population transfer dependence on two-photon detuning. The peaks
(-2,-2),(-1,-1),(0,0),(1,1),(2,2) are the resonance transition of the
hyperfine Zeeman sublevels when the Raman beams ($R_{1}$,$R_{2}$) are applied
along the magnetic field $B=220$ mG, where the Raman beams
($R_{1}^{{}^{\prime}}$,$R_{2}^{{}^{\prime}}$) and $B_{0}$ are not used.

4\. Results and analysis

Figure 3: The resonance frequency of the hyperfine Zeeman sublevels (-2,-2)
and (2,2) depends on the current of the Helmholtz coils. It is measured by
Raman beams (R1, R2) under R${}_{1}^{{}^{\prime}}$=$0$,
R${}_{2}^{{}^{\prime}}$=$0$, B0=$0$. The magnetic field can be scaled by the
first Zeeman shift after the linear fit, whose uncertainty is below one per
thousand.

The hyperfine level $F=2$ is split into five sublevels and $F=3$ into seven
sublevels, whose energies are expressed as in eqs.(2-8) when there exists a
magnetic field. After the magnetic field is compensated completely according
to our previous work Li2008a , all sublevels are degenerate. Coherent
population transfer can occur for the transition of the combined hyperfine
Zeeman sublevels $(-2,-2),(-1,-1),(0,0),(1,1),(2,2)$ when the Raman beams (R1,
R2) with ($\sigma^{+}$,$\sigma^{+}$) propagate along the magnetic field $B$ in
the stimulated Raman interaction region as shown in Fig.1 Li2008a ; Peters ;
Gustavson ; Petelsk , where the Raman beams (R${}_{1}^{{}^{\prime}}$,
R${}_{2}^{{}^{\prime}}$) and the magnetic field $B_{0}$ are not used. The
maximum population transfer is achieved when two-photon resonance is satisfied
with the transition selection rules shown in Fig.2. A perfect symmetric Raman
spectrum are achieved when the atoms are interacted with Raman beams
($\sigma^{+}$,$\sigma^{+}$). In Fig.2, the transition probability can be
explained using the oscillator strength of two-photon transition for the
different hyperfine Zeeman sublevels $(-2,-2),(-1,-1),(0,0),(1,1),(2,2)$
respectively. The energy separation of the different sublevels is well
explained by eqs.(2-8) when the bias magnetic field $B=220$ mG is applied. The
magnetic field is calibrated by the linear Zeeman shift of the hyperfine
Zeeman sublevels $(-2,-2),(2,2)$. For different magnetic field, we measured
the resonance frequency for the transitions $(-2,-2)$ and $(2,2)$, as shown in
Fig.3. After a linear fit, the slope is the magnetic field intensity
controlled by the current of the coils in the Raman interaction area. The
scaled method is similar to that of the quadratic Zeeman shift measurement
introduced in our paper. The scaled parameters come from earlier references
Steck ; Bender1958a ; Penselin1962a ; Arimondo1977a . The scale factor of the
magnetic field is $1576.9\pm 1.3$ mG/A after the averaged measurements.

Figure 4: The differential ac Stark shift caused by imbalanced Raman beams
versus the Raman light intensity. The dots are the frequency shift induced by
$R_{1}^{{}^{\prime}}$ while the squares are the frequency shift induced by
$R_{2}^{{}^{\prime}}$, where they are fitted linearly and the slopes are
$3.66$ kHz/(mW/cm2) and $-0.99$ kHz/(mW/cm2) respectively. The ac Stark shift
can be cancelled by adjusting the intensity ratio to $1:3.67$ for the one-
photon detuning $\Delta=1.5$GHz.

The differential ac Stark shift caused by the imbalanced Raman beams will
induce a measurement noise in the determination of the quadratic Zeeman shift.
The difference between the ac Stark shifts of two hyperfine sublevels,
$\delta^{AC}=\Omega_{F=3,m_{F}=0}^{AC}-\Omega_{F=2,m_{F}=0}^{AC}$, can be
cancelled by optimizing the ratio of two Raman beams weiss1994a . We measure
the frequency shift that is induced by one of the Raman beams separately. In
the experiment, we use two pairs of Raman beams (R1, R2) and
(R${}_{1}^{{}^{\prime}}$, R${}_{2}^{{}^{\prime}}$) along the magnetic field
$B$ and $B_{0}$, where $B$ and $B_{0}$ are $250$ mG and $100$ mG respectively.
The Raman beams (R1, R2) are used to measure the ac Stark shift induced by the
other Raman beams (R${}_{1}^{{}^{\prime}}$, R${}_{2}^{{}^{\prime}}$). We
carefully optimize the intensities of the Raman beams ($R_{1}$, $R_{2}$) along
the magnetic field $B$ to obtain a $\pi$-pulse. We scan the frequency
difference of the Raman beams (R1, R2), and the resonant frequency of the
hyperfine Zeeman sublevels ($0,0$) can be obtained by a Gaussian fit for the
different Raman light intensities (R${}_{1}^{{}^{\prime}}$, or
R${}_{2}^{{}^{\prime}}$). The detailed proceedure is similar to test of
quadratic Zeeman shift measurement in the paper. In the experiment, the Raman
beams (R1, R2) and (R${}_{1}^{{}^{\prime}}$, R${}_{2}^{{}^{\prime}}$) are
guided using single mode polarization maintained fiber. The intensity
instability is below one part in one thousand for each of the Raman beams. The
dots are the frequency shift that is induced by R${}_{1}^{{}^{\prime}}$, while
the squares are the frequency shift that is induced by R${}_{2}^{{}^{\prime}}$
in Fig.4, where they are fitted linearly. The slopes are $3.66$ kHz/(mW/cm2)
and $-0.99$ kHz/(mW/cm2) for R${}_{1}^{{}^{\prime}}$, and
R${}_{2}^{{}^{\prime}}$ respectively. The frequency shifts, induced by the
different Raman beams (R${}_{1}^{{}^{\prime}}$, or R${}_{2}^{{}^{\prime}}$),
are referenced to the separation of hyperfine sublevels ($3$ $035$ $732$
$436$) Penselin1962a . The non null values are mainly caused by the quadratic
Zeeman shift when the Raman beams (R${}_{1}^{{}^{\prime}}$,
R${}_{2}^{{}^{\prime}}$) are not applied in Fig.4. The ratio ($1:3.67$) of the
two slopes determines the cancellation of the ac Stark shift when the one-
photon detuning is $1.5$ GHz in our experiment. Therefore, we can cancel the
ac Stark shift by adjusting the ratio of two Raman beam intensities.

Figure 5: Frequency dependence of transition amplitude for a square Raman
$\pi$-pulse (R1, R2). The frequency is referenced to the hyperfine separation
of the two ground states. The dots are the experimental data with $B=600$ mG,
B0=$0$, R${}_{1}^{{}^{\prime}}$=$0$, R${}_{2}^{{}^{\prime}}$=$0$. while the
solid line is the Gaussian fit.

After the magnetic field compensation and the cancellation of the ac Stark
shift, their influence is considerably decreased in the measurement of the
quadratic Zeeman shift. The Raman beams are generated by an acousto-optical
modulator(Brimrose, $1.5$ GHz) driven by microwave generator (Agilent $8257$C)
which is locked by a H-maser. The arrangement of the Raman laser is similar to
our previous work Wang2007a . We carefully optimize the intensities of the
Raman beams ($R_{1}$, $R_{2}$) along the magnetic field $B$ to obtain a
$\pi$-pulse, where $B_{0}$, $R_{1}^{{}^{\prime}}$ and $R_{2}^{{}^{\prime}}$
are not used. The instability of the ratio of the Raman beams
($R_{1}:R_{2}=1:3.67$) is below $10^{-5}$ in the experiment. We scan the
frequency difference of the Raman beams (R1, R2), and observe a typical
stimulated Raman transition which shows the population versus frequency
difference between the two Raman beams in Fig.5 at a magnetic field $B=600$
mG, where the frequency is referenced to the separation between the two ground
states ($3$ $035$ $732$ $436$ Hz) Penselin1962a . In our experiment, the
intensity profile of the Raman beams is a Gaussian distribution and the line
width is mainly limited by the transition time because the spontaneous can be
ignored in large one-photon detuning. In such case, the population dependence
on the two-photon detuning is a Gaussian profile Demtroder2003a . The central
frequency is obtained from a Gaussian fit. We have made a series of such
curves for different magnetic fields, and the dependence of the frequency
shift on the magnetic field is shown in Fig.6. The frequency shift depends on
the magnetic field and it is fitted by a polynomial function (The maximum
power is $2$), while the quadratic dependence is for the quadratic Zeeman
shift. We measured a series of values as shown in table $1$, and the average
frequency shift induced by the quadratic Zeeman effect for the hyperfine
Zeeman sublevels ($5S_{1/2},F=2,m_{F}=0\rightarrow 5S_{1/2},F=3,m_{F}=0$) is
$1296.8$ Hz/G2. The measurement uncertainty comes mainly from the calibrated
magnetic field and the fitted error. As shown in table $1$, the averaged
uncertainty of the quadratic Zeeman shift is $2.1$ Hz/G2 and $2.5$ Hz/G2 for
the scaled magnetic field and the fitted error respectively. The final result
for the quadratic Zeeman shift is $1296.8\pm 3.3$ Hz/G2 by using an
independent error source model, which is in good agreement with the
calculation result Steck  within our measurement precision. The result shows
that the second perturbation theory is sufficient when the magnetic field is
less than $1$ mT Itano2000a . The ac Stark shifts induce a systematic shift of
the ground-state hyperfine splitting. This does not influence the value of the
quadratic Zeeman shift when a quadratic dependence term of the polynomial
function is chosen as shown in Fig.6. The fitted error, which is induced by
the instability of the Raman beams, is decreased when the cancellation ratio
of the Raman beams ($1:3.67$) is applied in the experiment.

Figure 6:  Dependence of resonance frequency on the magnetic field intensity.
The dots are the frequency shift in the different magnetic field that is
obtained from Fig. 5, while the line is the experimental fit by a polynomial
function.

In the atom interferometer, the bias magnetic field is applied through the
interference area. Although the atoms are always kept in magnetically
insensitive states with $m_{F}=0$, these states still show a quadratic Zeeman
shift that induces a relative frequency shift of two ground states. This
effect is big enough to require well controlled magnetic fields and extensive
magnetic field shielding to achieve the millihertz frequency stability
necessary for gravity measurements at the $1\mu$G level Peters . For the
rotation rate measurement, the quadratic Zeeman shift should be known
accurately when considering the accuracy necessary to determine the rotation
rate of the earth. The sensitivity of the rotation signal to the various bias
magnetic field was determined in detailly performed in the dual atomic
interferometer gyroscope, and the bias magnetic field caused a phase shift
$2\times 10^{-6}\Omega_{E}$/mG for the rotation measurement in the system
Gustavson , which is mainly induced by the quadratic Zeeman shift. In our
experiment, the precision of the quadratic Zeeman shift is mainly limited by
the measurement time, and it can be measured even more accurately by
decreasing the atomic flight velocity and increasing the Raman beam diameter,
and by using the separated oscillation field method in a weak magnetic field
Bize1999a . However, our result provides helpful data for higher precision
measurement of the quadratic Zeeman shift of 85Rb, even for the accuracy of
the rotation rate measurement of the atom-interferometer gyroscope.

Table $1$ Experimental data for the determination of the quadratic Zeeman
shift of hyperfine sublevels ($5S_{1/2},F=2,m_{F}=0\rightarrow$
$5S_{1/2},F=3,m_{F}=0$) of 85Rb.

$Run$ | $\begin{array}[c]{c}Frequency\\\ shift\\\ \text{(Hz/G}^{2}\text{)}\end{array}$ | $\begin{array}[c]{c}Scaled\\\ error\\\ \text{ (Hz/G}^{2}\text{)}\end{array}$ | $\begin{array}[c]{c}Fitted\\\ error\\\ \text{(Hz/G}^{2}\text{)}\end{array}$
---|---|---|---
$1$ | $1294.2$ | $2.1$ | $2.9$
$2$ | $1294.1$ | $2.1$ | $2.9$
$3$ | $1295.7$ | $2.1$ | $2.3$
$4$ | $1296.1$ | $2.1$ | $2.2$
$5$ | $1298.6$ | $2.1$ | $1.9$
$6$ | $1298.7$ | $2.1$ | $1.9$
$7$ | $1298.7$ | $2.1$ | $1.9$
$8$ | $1298.6$ | $2.1$ | $1.9$
$Average$ | $1296.8$ | $2.1$ | $2.5$
$Total$ |  $1296.8\pm 3.3$ (Hz/G2)

5\. Conclusion

In summary, we analyzed the energy of the hyperfine sublevels of two ground
states of 85Rb in the magnetic field. We demonstrated experimentally the
coherent population transfer of the hyperfine sublevels between two ground
states by the stimulated Raman transition. The ac Stark shift was
experimentally studied by measuring the ac Stark frequency shift dependence on
the Raman beam intensity, and it was cancelled by adjusting the ratio of two
Raman beam intensities. We measured the quadratic Zeeman shift of the ground
states using the coherent population transfer by a stimulated Raman
transition. The error analysis shows that the quadratic Zeeman shift was
measured to Hz level for magnetically insensitive states
$5S_{1/2},F=2,m_{F}=0\rightarrow$ $5S_{1/2},F=3,m_{F}=0$ in the experiment.
This result provides helpful data to improve the accuracy of the atom-
interferometer gyroscope in future.

Acknowledgments

We acknowledge the financial support from the National Basic Research Program
of China under Grant Nos. 2005CB724505, 2006CB921203, and from the National
Natural Science Foundation of China under Grant No.10774160. We thank
Professor J. P. Connerade and Professor L. You for useful comment and
discussion.

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