# Compact 459 nm Cs cell optical frequency standard with
$2.1\times{10}^{-13}/\sqrt{\tau}$ short-term stability

Jianxiang Miao  Tiantian Shi<EMAIL_ADDRESS>Jia Zhang  Jingbiao Chen  State
Key Laboratory of Advanced Optical Communication, System and Network, School
of Electronics, Peking University, Beijing 100871, China

###### Abstract

We achieve a compact optical frequency standard with an extended cavity diode
laser locked to the 459 nm 6S1/2 \- 7P1/2 transition of thermal 133Cs atoms in
a $\upphi~{}10~{}\mathrm{mm}\times 50~{}\mathrm{mm}$ glass cell, using
modulation transfer spectroscopy (MTS). The self-estimated frequency stability
of this laser is $1.4\times{10}^{-14}/\sqrt{\tau}$. With heterodyne
measurement, we verify the linewidth-narrowing effect of MTS locking and
measure the frequency stability of the locked laser. The linewidth of each
laser is reduced from the free-running 69.6 kHz to 10.3 kHz after MTS
stabilization, by a factor of 6.75. The Allan deviation measured via beat
detection is $2.1\times{10}^{-13}/\sqrt{\tau}$ for each MTS-stabilized laser.
In addition, we measure the hyperfine structure of the 7P1/2 energy level
based on the heterodyne measurements, and calculate the magnetic dipole
constant A of the Cs 7P1/2 level to be 94.38(6) MHz, which agrees well with
previous measurements. This compact optical frequency standard can also be
used in other applications that require high-stability lasers, such as laser
interferometry, laser cooling, geodesy, and so on.

##  I Introduction

In the past decade, optical clock technology has seen rapid advances, with
optical clocks based on lattice-trapped atoms and single-ion systems reaching
uncertainty levels below $10^{-18}$ [1, 2, 3, 4, 5, 6, 7]. With such an
unprecedented level of precision, optical clocks find applications in areas
such as geodesy [8, 4], testing of fundamental physics [9, 10, 7] and
gravitational wave detection [11]. However, state-of-the-art optical clocks
are usually confined to laboratory environments, due to their bulk and
complexity [12]. For applications with less stringent requirements on clock
performance, it is often advantageous to utilize optical clocks with thermal
atomic reference, whose compactness and portability surpass their laboratory-
bound counterparts. Taking advantage of their high operating frequency,
compact optical clocks achieve frequency stability surpassing or comparable to
microwave atomic clocks [13, 14].

Common experimental techniques for realizing a compact optical clock with
thermal atoms in a cell as frequency reference include saturation absorption
spectroscopy (SAS) [15, 16, 17], polarization spectroscopy (PS) [18, 19, 20],
two-photon spectroscopy [21, 22, 23, 24], dual-frequency sub-Doppler
spectroscopy (DFSDS) [25, 26, 27] and modulation transfer spectroscopy (MTS)
[28, 29, 30]. These methods all utilize the interaction between counter-
propagating lasers and thermal atoms to generate Doppler-free atomic spectra
as the basis of laser stabilization. With SAS, PS and MTS, the counter-
propagating geometry is used for velocity selection, so that only atoms with
zero velocity component along the laser propagation axis are interrogated. In
two-photon spectroscopy, since the Doppler shifts from the lasers in opposite
directions cancel each other, atoms in all velocity classes can contribute to
the signal. DFSDS uses the CPT effect in atoms with $\Uplambda$-type energy
levels to generate a sub-Doppler absorption peak with narrow linewidth. Here
we list the stability levels reached by these approaches in related works, as
shown in Table 1.

Among these stabilization techniques based on thermal atoms in a cell, MTS has
the advantages of insensitivity to background absorption and rejection of low-
frequency noise [31, 32, 33]. The 532 nm I2 optical frequency standard with
MTS locking has been employed as a wavelength standard due to its high
stability [34, 35]. Compared to iodine optical frequency standards, MTS
systems based on alkali metal atoms has several advantages: alkali metal
atomic cells can be made much smaller while retaining a similar level of
signal-to-noise ratio as iodine frequency standards, and a wide range of diode
lasers are available for interrogating different transitions of alkali metal
atoms [36, 37].

Table 1: Stability levels reached by different types of compact optical frequency standards.  System type | Frequency stability | Reference
---|---|---
SAS, Rb, 780 nm |  $1.7\times{10}^{-12}$ at 1 s | [17]
PS, Rb, 780 nm |  $1.5\times{10}^{-12}$ at 1 s | [20]
Two-photon, Rb, 778 nm | $4\times{10}^{-13}/\sqrt{\tau}$ | [23]
DFSDS, Cs, 895 nm | $1.1\times{10}^{-12}/\sqrt{\tau}$ | [27]
MTS, Cs, 852 nm | $2.6\times{10}^{-13}/\sqrt{\tau}$ | [37]
MTS, Cs, 459 nm | $2.1\times{10}^{-13}/\sqrt{\tau}$ | This work

In this paper, we experimentally study a compact Cs optical frequency standard
that is locked to the 459 nm 6S1/2 \- 7P1/2 transition of 133Cs. After
optimizing the operating parameters of the optical frequency standard, we
measure the frequency stability of the output laser via both self-estimation
and heterodyne measurement between two identical MTS systems. The self-
estimated Allan deviation is $1.4\times{10}^{-14}/\sqrt{\tau}$ and reaches a
minimum of $4\times{10}^{-15}$ at 30 s averaging time. By measuring the
frequency spectrum of the beat signal between two MTS-stabilized lasers and
the beat signal between one MTS-stabilized laser and free-running laser, we
verify the linewidth-narrowing effect of MTS locking, with the linewidth of an
individual laser reduced from 69.6 kHz to 10.3 kHz, by a factor of 6.75. The
Allan deviation of the beating signal is $3\times{10}^{-13}/\sqrt{\tau}$;
assuming the two MTS systems contribute equally to the fluctuations of the
beat signal, each MTS system has short-term frequency stability of
$2.1\times{10}^{-13}/\sqrt{\tau}$. We also measure the hyperfine level
splitting of the 7P1/2 level from the beat signal between two MTS systems
locked to the Cs 6S1/2 (F=4) - 7P1/2 (F’=3) transition and 6S1/2 (F=4) - 7P1/2
(F’=3&4) crossover transition, respectively, and calculate the magnetic dipole
constant A of the Cs 7P1/2 energy level to be 94.38(6) MHz, which agrees with
previous works [38, 39, 40, 41]. In addition to being used as a portable
optical frequency standard, this compact, narrow-linewidth, high-stability
laser can have various other applications, such as laser interferometry, laser
cooling, geodesy, and so on.

##  II Experiment

Figure 1:  (a) The experimental setup of the MTS system, with an identical MTS
system set up for heterodyne measurement. 459 nm IF-ECDL, 459 nm interference
filter configuration external cavity diode laser; HWP, half-wave plate; PBS,
polarizing beamsplitter; PD, photodetector; SIG, signal generator; PID,
proportion-integral-derivative locking system. Inset: hyperfine levels of the
Cs 6S1/2 and 7P1/2 energy levels. (b) Photo of the experimental setup of the
MTS system. All components are secured to an aluminum baseboard for better
mechanical stability, and the setup is enclosed during its operation to reduce
the effect of external disturbances. To the left of the photo are components
irrelevant to the MTS experiment, which are therefore omitted from the photo;
the MTS system can in principle be enclosed in the area prescribed by the
photo, which is $44\times 40~{}\mathrm{cm}^{2}$, and the box containing the
experimental setup is 16 cm high. The fiber-coupled heterodyne module and the
other MTS system are not shown in the photo.

The schematic of the 459 nm laser system frequency-stabilized by MTS is shown
in Fig. 1, and the photo of the system is shown in Fig. 1. We use a 459 nm
interference filter configuration external cavity diode laser (IF-ECDL) as
local oscillator for the MTS-stabilized optical clock. The frequency of the
IF-ECDL is tunable via adjusting the injection current and the voltage on a
piezoelectric ceramic (PZT) controlling the cavity length. The IF-ECDL has a
maximum output power up to 30 mW, and its linewidth is measured to be below 50
kHz. After passing a Faraday isolator and a beam expander, the output laser is
then split in two parts after passing through a half-wave plate (HWP) and a
polarizing beamsplitter (PBS), with about 2 mW of optical power for laser
frequency stabilization by MTS. The light used for stabilization is further
split into probe and pump beams, with optical power 0.33 mW and 1.5 mW,
respectively. The probe beam passes through a $\upphi~{}10~{}\mathrm{mm}\times
50~{}\mathrm{mm}$ cylindrical glass cell filled with 133Cs atoms, and is then
detected by the photodetector (PD1, Thorlabs PDA8A2). The cell is housed in a
casing fixed to the baseboard, with several layers of Teflon for heat-
insulation and two layers of cylindrical mu-metal magnetic shielding. The pump
beam is frequency modulated by an EOM, and it overlaps with the probe beam in
the cesium cell at a lin$\bot$lin counter-propagating configuration. The
temperature of the cesium cell is controlled by a commercial temperature
controller (Thorlabs TC200), which detects the temperature of the cesium cell
via a thermistor and adjusts the current sent to the heater accordingly. A HWP
is placed before the EOM to align the light polarization with the main axis of
the EOM crystal. The pump beam acquires sidebands from the EOM’s modulation,
with the $\pm 1$ sidebands dominant; and after four-wave mixing processes in
the cesium cell, the modulation is transferred to the probe beam. PD1 detects
the beatnote between the probe beam’s carrier and the two dominant sidebands.
After frequency mixing the signal from PD1 with a demodulation signal with the
same frequency as the modulation frequency, a dispersion-like signal is
obtained at the mixer’s output port. This signal is used as the error signal
for the PID locking system controlling the diode laser’s injection current and
PZT voltage, so the laser frequency is locked to atomic transition. The MTS
system occupies a space of $44\times 40\times 16~{}\mathrm{cm}^{3}$.

For the evaluation of the MTS-stabilized laser’s frequency stability, we use
the self-estimation method for a single MTS system and heterodyne measurement
between two identical MTS systems. The self-estimation method evaluates the
frequency stability by measuring the residual error signal after locking, and
is used for individually evaluating the frequency stability of an MTS-
stabilized laser. It is a commonly used method that characterizes how well the
laser frequency follows the atomic transition, and reflects on the in-loop
locking performance [33, 42, 43]. For heterodyne measurement, the output laser
of two identical MTS systems are coupled into two input ports of a 50:50
fiber-optical coupler, and the beat signal is detected by a high-bandwidth
photodetector (PD2, Hamamatsu c5658) at the output port. Two MTS systems with
identical setup are locked to the 6S1/2 (F=4) - 7P1/2 (F’=3) transition and
the 6S1/2 (F=4) - 7P1/2 (F’=3&4) crossover transition separately, and the beat
signal is sent into a frequency counter (Keysight 53230A), then the frequency
stability can be calculated from the recorded beat frequency. We also measure
the laser linewidth and the phase noise spectrum with a RF spectrum analyzer
(Keysight N9000B) and a phase noise analyzer (Keysight E5052B), respectively.

##  III Results

###  III.1 Optimization of the MTS setup

We first obtain SAS and MTS spectra of the 133Cs 6S1/2 \- 7P1/2 transition.
Figs. 2 and 2 show the MTS and SAS spectra with ground states 6S1/2 (F=3) and
6S1/2 (F=4), respectively. As can be seen from the figures, the 6S1/2 (F=4) -
7P1/2 (F=3) transition has the largest signal amplitude and slope under the
same working conditions (cell temperature, EOM modulation frequency, pump and
probe laser power, etc.), so we choose it as the reference transition for MTS
stabilization.

Figure 2: SAS and MTS spectrums of the Cs 6S1/2 \- 7P1/2 transition, starting
from the (a) F=3 and (b) F=4 hyperfine levels. Comparing (a) and (b), we find
the F=4 - F’=3 hyperfine transition has the largest MTS signal strength, and
choose it as the reference transition. The natural linewidth of the 133Cs
6S1/2 \- 7P1/2 transition is 6.053(7) MHz [44]. The linewidth of the MTS
signal corresponding to the F=4 - F’=3 hyperfine transition is measured to be
8.63 MHz from the data plot, using the hyperfine splitting (here we use the
approximate value of 377.4 MHz [38]) between F’=3 and F’=4 as a reference.

Since the laser is locked at the zero crossing of the MTS signal, its slope at
the zero crossing is a good indicator of the system parameters, as a higher
slope means a larger signal-to-noise ratio, as well as higher sensitivity to
frequency fluctuations. We measure the MTS signal slope under various cesium
cell temperatures and EOM modulation frequencies, as well as the short-term
stability measured via self-estimation, in order to find the optimal working
conditions. Results are shown in Figs. 3 and 4. In Fig. 3(a), the signal slope
first increases with the cell temperature due to the increase in atomic
density resulting in a larger signal, then decreases as the resonance
linewidth is affected by collision broadening; In Fig. 4(b), the signal slope
increases sharply around 1.68 MHz. The natural linewidth of the cesium 6S1/2
\- 7P1/2 transition is $\Gamma_{\mathrm{nat}}=6.053(7)~{}\mathrm{MHz}$ [44],
and the linewidth of the MTS signal corresponding to the 6S1/2 (F=4) - 7P1/2
(F’=3) hyperfine transition is measured to be
$\Gamma_{\mathrm{eff}}=8.63~{}\mathrm{MHz}$ from the data plot, using the
hyperfine splitting between 7P1/2 (F’=3) and 7P1/2 (F’=4) as a reference. This
increase in the measured linewidth from the natural linewidth can mostly be
attributed to saturation broadening [33]. Here we have the modulation
frequency $\omega_{m}=1.68~{}\mathrm{MHz}$ at approximately 0.2 times the
measured linewidth $\Gamma_{\mathrm{eff}}=8.63~{}\mathrm{MHz}$, which can be
regarded as the optimal modulation frequency for maximizing the slope of the
MTS signal [45]. The short-term stability measurements in Figs. 3(b) and 4(b)
also agree well with the signal slope measurements in Figs. 3(a) and 4(a). The
cell temperature is set at 109.4∘C, and the modulation frequency is set at
1.68 MHz, both corresponding to maximal signal slope and best self-estimated
short-term stability, thereby optimizing the performance of the MTS-stabilized
laser.

Figure 3:  (a) MTS signal slope at zero crossing and (b) self-estimated short-
term frequency stability, measured at different Cell temperatures, with the
modulation frequency set at 1.7 MHz. The temperature we choose for the
following experiments (except measurement of collision shift which changes the
cell temperature) is denoted with a star symbol in (a). Each data point in (a)
is the averaged result of 3 measurements, and the error bars represent the
standard deviation. Each data point in (b) is obtained by monitoring the error
signal for more than 100 seconds, then calculating the self-estimated Allan
deviation from the recorded error signal. The error bars in (b) represent the
1-sigma confidence interval of each Allan deviation value. Figure 4:  (a) MTS
signal slope at zero crossing and (b) self-estimated short-term frequency
stability, measured at different modulation frequencies, with the cell
temperature controlled at 109.4∘C. The modulation frequency we choose for the
following experiments is denoted with a star symbol in (a). Each data point in
(a) is the averaged result of 3 measurements, and the error bars represent the
standard deviation. Each data point in (b) is obtained by monitoring the error
signal for more than 100 seconds, then calculating the self-estimated Allan
deviation from the recorded error signal. The error bars in (b) represent the
1-sigma confidence interval of each Allan deviation value.

###  III.2 Characterization of MTS-stabilized laser

We first use the self-estimation method to evaluate the frequency stability of
the 459 nm laser after locking. The dispersion-like MTS signal is
approximately linear near resonance, and the frequency shift can be derived
from the residual error signal, as fluctuations of the error signal translate
to frequency fluctuations. As shown in Fig. 6, the short-term stability
measured with self-estimation at averaging time of 1 s is
$1.4\times{10}^{-14}/\sqrt{\tau}$, reaching $4\times{10}^{-15}$ at 30 s
averaging time, and maintains this level of stability until after 200 s, where
the Allan deviation then degrades slightly. The long-term self-estimated
frequency stability is mostly limited by the residual amplitude modulation
(RAM) of the EOM, and can be improved by methods such as controlling the
temperature of the EOM crystal [46, 47, 48]. Though self-estimation is only an
in-loop measurement and not truly representative of the laser frequency
stability, it reflects the how well the laser frequency “follows” the atomic
transition, which sets a lower limit to the attainable frequency stability
[33, 43, 42].

In addition to self-estimation evaluation, we measure the beat signal between
two MTS systems with identical setups. One system is locked to the 6S1/2 (F=4)
- 7P1/2 (F’=3) transition, with the other system locked to the 6S1/2 (F=4) -
7P1/2 (F’=3&4) crossover transition. After combining the laser output of the
two systems via a fiber-optical coupler, we use a high bandwidth photodetector
(Hamamatsu C5658, bandwidth 1 GHz) to measure the beat signal with frequency
equal to half of the frequency difference between the 7P1/2 (F’=3) and 7P1/2
(F’=4) hyperfine levels. The beat signal is sent into a spectrum analyzer
(Keysight N9002B) for measuring the laser’s linewidth. Results are shown in
Figs. 5 and 5. After repeated measurements, the mean beating linewidth between
two MTS-stabilized lasers is calculated to be 14.5 kHz, and the mean beating
linewidth between a MTS-stabilized laser and a free-running laser is 69.6 kHz.
Assuming equal contribution from each laser in the beating linewidth between
two MTS-stabilized lasers, the linewidth of each laser is 10.3 kHz, whereas
the beat linewidth between free-running and locked lasers mainly come from the
free-running one. Therefore, the laser linewidth is reduced by a factor of
6.75 after locking. This reduction is due to suppression of the laser
frequency noise by the MTS feedback [33].

The beat signal between two locked identical MTS systems is then sent into a
frequency counter (Keysight 53230A) for measurement of frequency stability.
The Allan deviation can be calculated from the measured beat frequency
sequence. The Allan deviation of the beatnote is
$3\times{10}^{-13}/\sqrt{{\tau}}$, and assuming the two identical MTS systems
contribute equally, the Allan deviation of each stabilized laser is
$1/\sqrt{2}$ of the beat sequence, i.e. $2.1\times{10}^{-13}/\sqrt{{\tau}}$,
as depicted by Fig. 6. The Allan deviation of the beatnote between a free-
running laser and a locked laser is $8.2\times{10}^{-10}$ for averaging time
below 10 s, and becomes $2\times{10}^{-10}\tau^{1/2}$ at higher averaging
times. The Allan deviation measured via heterodyne measurement is larger than
that of the self-estimation (in-loop measurement), mostly due to out-of-loop
elements such as temperature fluctuation of the cesium cells [37].

Figure 5: (a) Typical beating data of two identical MTS-stabilized 459 nm
lasers. The Lorentz-fitted FWHM of the beat linewidth is 12 kHz; Inset:
typical beating data with one laser free-running, the fitted FWHM is 72 kHz.
(b) Histogram of repeated measurements of the fitted beating linewidths with
both lasers locked, the mean linewidth after locking is 14.5 kHz, indicating a
linewidth of 10.3 kHz for each laser; Inset: histogram of repeated
measurements of the fitted linewidths with one laser free-running, mean
linewidth is 69.6 kHz. All linewidth measurements are made with the RBW of the
spectrum analyzer set at 10 kHz, in order to have a fair comparison between
the locked and free-running linewidths. Figure 6: Allan deviation of the 459
nm laser stabilized via MTS, measured by self-estimation (red solid circles)
and heterodyne measurement (black solid squares). Compared to the frequency
stability of the 852 nm laser locked to the Cs 6S1/2 \- 6P3/2 transition in
Ref. [37] (blue half-squares) and the 778 nm optical clock based on the Rb
5S1/2 \- 5D5/2 two-photon transition in Ref. [23] (green half-circles), our
MTS-stabilized laser has better short-term frequency stability at averaging
time below 100 s. The error bars in this figure represent the 1-sigma
confidence interval of each Allan deviation value.

We also measure the phase noise spectrum of the beat signal between two MTS-
stabilized lasers with a phase noise analyzer (Keysight E5052B), and the
result is shown in Fig. 7. However, since we only measured the phase noise
spectrum in the frequency range of 1 Hz to 1 MHz, it corresponds to timescales
less than 1 s, whereas our frequency stability measurement of the beatnote
only starts at 0.1 s, so we can’t make a direct comparison between the phase
noise spectrum and frequency stability measurements. Instead, the phase noise
spectrum supplements the frequency stability measurement and provides
information on the beat signal’s behavior at short timescales.

Figure 7: Measured phase noise spectrum of the beat signal between two lasers
locked to the 133Cs 6S1/2 (F=4) -7P1/2 (F’=3) transition and the F=4 - F’=4&3
crossover transition, respectively.

###  III.3 Hyperfine level measurements with MTS

Figure 8:  Experimental measurements of the beat frequencies between the Cs
6S1/2 (F=4) - 7P1/2 (F=3) transition (black squares) and the 6S1/2 (F=4) -
7P1/2 (F=3&4) crossover transition (red circles): (a) under different cell
temperatures; (b) under different total laser power. Each data point in this
figure is the averaged result from beat frequency data measured with a
frequency counter with more than 300 s measurement time, and the error bars
are the standard deviation calculated from each measurement result. The
frequency shift coefficients are obtained with least-square fitting, and the
linear fits are shown as dashed lines in the figure.

Since the beat frequency between the F=4 - F’=3 transition and the F=4 -
F’=3&4 crossover transition is equal to half of the hyperfine splitting of the
7P1/2 energy level, the heterodyne measurement can also be used to measure the
hyperfine structure of the 7P1/2 energy level. By varying the cell temperature
and the pump and probe laser power of one individual MTS system while keeping
the other constant, the effects of collision shift and power shift can be
accounted for and corrected. Figs. 8(a) and 8(b) show the relative frequency
shifts of the F=4 - F’=3 and crossover transitions under different cell
temperatures and total laser power, and the shift coefficients are obtained
via least-square fitting. The collision shift coefficients of the F=4 - F’=3
and crossover transitions are 2.64(17) kHz/K and 6.11(43) kHz/K, respectively.
The ac Stark shift coefficients of the F=4 - F’=3 and crossover transitions
are 77.89$\pm$9.81 kHz/mW and 115.65$\pm$4.54 kHz/mW, respectively. Note that
the frequency shift coefficients of the F=4 - F’=3 transition are of the
opposite sign of the fitted slopes in Figs. 8(a) and 8(b), since increasing
the F=4 - F’=3 transition frequency (therefore increasing the frequency of the
laser locked to the transition) will decrease the beat frequency, and vice
versa. The hyperfine splitting can be calculated by extrapolating to null
laser power and a cell temperature comparable with other experimental
measurements, such as Ref. [38] which uses optical double resonance (ODR) to
measure the hyperfine splitting at room temperature. The calculated hyperfine
splitting of the Cs 7P1/2 energy level is 377.52$\pm$0.24 MHz. The hyperfine
splitting from the transition’s center of gravity is given by

$W=\frac{1}{2}AK+B\frac{(3/2)K(K+1)-2I(I+1)J(J+1)}{2I(2I-1)2J(2J-1)},$ (1)

where $K=F(F+1)-I(I+1)-J(J+1)$; A and B are the magnetic dipole constant and
the electric quadrupole constant, respectively; I, J and F are the nuclear
spin, the total electronic angular momentum, and the total atomic angular
momentum, respectively. The hyperfine splitting between 7P1/2 F’=3 and 7P1/2
F’=4 equals to 4A, and the measurement result corresponds to a magnetic dipole
constant A of 94.38(6) MHz. As shown in Table 2, our measurement result agree
well with previous works, with a comparable level of measurement uncertainty
[38, 39, 40, 41]. By using a different method to measure the hyperfine $A$
constant, our results provide an independent verification for previous works.

Table 2: Measured magnetic dipole constant $A$ of the Cs 7P1/2 energy level, compared with earlier works.  A(MHz) | Method | Reference
---|---|---
94.35(4) | ODR | [38]
94.35(5) | SAS | [39]
94.40(5) | SAS | [40]
95.015 | Theory | [41]
94.38(6) | MTS | This work

###  III.4 Noise sources in the MTS system

Based on our measurements, we evaluate the major noise sources in our system
and give a brief analysis on their impact on the performance of our compact
optical clock. Since the total measured frequency instability of
$2.1\times{10}^{-13}/\sqrt{\tau}$, obtained by beating measurement, is an
order of magnitude larger than the in-loop self-estimated frequency
instability of $1.4\times{10}^{-14}/\sqrt{\tau}$, we mainly focus on the
effects of out-of-loop elements that limit the frequency stability of the
system.

In atomic frequency standards based on vapor cells, collision between the
thermal atoms introduce frequency shifts to the atomic energy levels. Since
the collision shift depends on the atomic density, fluctuations of the cell
temperature translates to fluctuations of the transition frequency. From the
linear fitting in Fig. 8(a), the collision shift coefficient is measured to be
2.64(17) kHz/K, or $\Delta f/f=(4.04\pm
0.26)\times{10}^{-12}~{}\mathrm{K}^{-1}$ in terms of relative frequency shift.
The cesium cell’s temperature instability is estimated to be 0.05 K at 1 s,
corresponding to a frequency instability of $2.02\times{10}^{-13}$.

External electromagnetic fields cause frequency shifts to the reference
transition, and their fluctuations also introduce instabilities. We measure
the B-field inside the two layers of mu-metal magnetic shielding to be
approximately 1 mG. In a static magnetic field, the energy shift of the ground
level can be calculated with the Breit-Rabi formula [49], which yields a
quadratic shift of 213.7 Hz/G2 for the 6S1/2 F=4 hyperfine level. As fort the
7P1/2 F=3 excited state, we make an estimate based on the fact that, the
Zeeman shift of the $m_{F}=0$ sublevel must be smaller than the splitting
between adjacent sublevels in the 7P1/2 F=3 level [37]. We therefore estimate
the Zeeman shift of the excited level to be below 466 kHz/G. With the B-field
below 1 mG, the total Zeeman shift of the reference transition is below 46 Hz.
Assuming the variation of the magnetic field to be less than 10%, the
frequency instability induced by the Zeeman effect is estimated to be below
$7\times{10}^{-15}$.

Meanwhile, due to ac Stark effect, any fluctuation in the laser power is
translated to fluctuation of the reference transition’s frequency shift.
Similar to the above, we calculate the relative frequency shift to be $\Delta
f/f=(1.19\pm 0.15)\times{10}^{-10}~{}\mathrm{mW}^{-1}$ from the measured ac
Stark shift under different laser power, as shown in Fig. 8(b). The measured
laser power instability is $1.2\times{10}^{-4}$ at 1 s, and the total laser
power is about 1.83 mW (pump laser power 1.5 mW and probe laser power 0.33
mW), so the laser power instability corresponds to $2.6\times{10}^{-14}$
frequency instability at 1 s.

The combined effect of cell temperature fluctuation, external magnetic field,
and laser power fluctuation produce a frequency instability of
$2.04\times{10}^{-13}$ at 1 s, which is in good agreement with the measured
short-term frequency of $2.1\times{10}^{-13}/\sqrt{\tau}$. Therefore, we
conclude that the main limitation to the short-term frequency stability is the
temperature fluctuation of the cesium cell. In future works, we plan to
utilize a double-layered atomic cell with vacuum heat insulation in
conjunction with a better temperature-control system, so as to reduce the
temperature fluctuation of the atomic ensemble and thereby improve the
frequency stability of the MTS-stabilized laser.

##  IV Conclusion

In this study, we investigate a compact optical frequency standard based on
modulation transfer spectroscopy of thermal cesium 133 atoms. By optimizing
the operating parameters of the MTS stabilization, we realize a self-estimated
short-term stability of $1.4\times{10}^{-14}/\sqrt{\tau}$, reaching
$4\times{10}^{-15}$ at 30 s. We measure the linewidth and Allan deviation via
heterodyne, verifying the linewidth-narrowing effect of MTS locking, with the
mean beating linewidth reduced from 69.6 kHz to 10.3 kHz, by a factor of 6.75.
The Allan deviation of the MTS-stabilized laser is calculated from the beat
frequency sequence to be $2.1\times{10}^{-13}/\sqrt{\tau}$. By measuring the
beat frequency between two MTS systems locked to the 6S1/2 (F=4) - 7P1/2
(F’=3) transition and the 6S1/2 (F=4) - 7P1/2 (F’=3&4) crossover transitions
respectively, we also measure the hyperfine structure of the 7P1/2 energy
level, and calculate the magnetic dipole constant A of the Cs 7P1/2 energy
level to be 94.38(6) MHz, which agrees with previous works [38, 39, 40, 41].
In future works, we expect to improve the long-term frequency stability of
this optical frequency standard by utilizing a double-layered atomic cell with
vacuum heat insulation in conjunction with a better temperature-control system
to reduce the temperature fluctuation of the atomic ensemble, as well as
actively controlling the temperature of the EOM to reduce the residual
amplitude modulation, both of which are limiting factors on the optical
frequency standard’s long-term stability. In addition to being used as an
optical frequency standard, this compact, narrow-linewidth, high-stability
laser may find various other applications, such as laser interferometry, laser
cooling, geodesy, and so on.

## References

  * Bloom  _et al._ [2014] B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, and J. Ye, An optical lattice clock with accuracy and stability at the ${10}^{-18}$ level, Nature 506, 71 (2014).
  * Ushijima  _et al._ [2015] I. Ushijima, M. Takamoto, M. Das, T. Ohkubo, and H. Katori, Cryogenic optical lattice clocks, Nat. Photonics 9, 185–189 (2015).
  * Campbell  _et al._ [2017] S. L. Campbell, R. B. Hutson, G. E. Marti, A. Goban, N. D. Oppong, R. L. McNally, L. Sonderhouse, J. M. Robinson, W. Zhang, B. J. Bloom, and J. Ye, A fermi-degenerate three-dimensional optical lattice clock, Science 358, 90 (2017).
  * McGrew  _et al._ [2018] W. F. McGrew, X. Zhang, R. J. Fasano, S. A. Schäffer, K. Beloy, D. Nicolodi, R. C. Brown, N. Hinkley, G. Milani, M. Schioppo, T. H. Yoon, and A. D. Ludlow, Atomic clock performance enabling geodesy below the centimetre level, Nature 564, 87 (2018).
  * Brewer  _et al._ [2019] S. M. Brewer, J. S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt, 27Al+ quantum-logic clock with a systematic uncertainty below ${10}^{-18}$, Phys. Rev. Lett. 123, 033201 (2019).
  * Norcia  _et al._ [2019] M. A. Norcia, A. W. Young, W. J. Eckner, E. Oelker, J. Ye, and A. M. Kaufman, Seconds-scale coherence on an optical clock transition in a tweezer array, Science 366, 93 (2019).
  * Takamoto  _et al._ [2020] M. Takamoto, I. Ushijima, N. Ohmae, T. Yahagi, K. Kokado, H. Shinkai, and H. Katori, Test of general relativity by a pair of transportable optical lattice clocks, Nat. Photonics 14, 411 (2020).
  * Grotti  _et al._ [2018] J. Grotti, S. Koller, S. Vogt, S. Häfner, U. Sterr, C. Lisdat, H. Denker, C. Voigt, L. Timmen, A. Rolland, F. N. Baynes, H. S. Margolis, M. Zampaolo, P. Thoumany, M. Pizzocaro, B. Rauf, F. Bregolin, A. Tampellini, P. Barbieri, M. Zucco, G. A. Costanzo, C. Clivati, F. Levi, and D. Calonico, Geodesy and metrology with a transportable optical clock, Nat. Phys. 14, 437 (2018).
  * Huntemann  _et al._ [2014] N. Huntemann, B. Lipphardt, C. Tamm, V. Gerginov, S. Weyers, and E. Peik, Improved limit on a temporal variation of ${m}_{p}/{m}_{e}$ from comparisons of Yb+ and Cs atomic clocks, Phys. Rev. Lett. 113, 210802 (2014).
  * Godun  _et al._ [2014] R. M. Godun, P. B. R. Nisbet-Jones, J. M. Jones, S. A. King, L. A. M. Johnson, H. S. Margolis, K. Szymaniec, S. N. Lea, K. Bongs, and P. Gill, Frequency ratio of two optical clock transitions in 171Yb+ and constraints on the time variation of fundamental constants, Phys. Rev. Lett. 113, 210801 (2014).
  * Kolkowitz  _et al._ [2016] S. Kolkowitz, I. Pikovski, N. Langellier, M. D. Lukin, R. L. Walsworth, and J. Ye, Gravitational wave detection with optical lattice atomic clocks, Phys. Rev. D 94, 124043 (2016).
  * Ludlow  _et al._ [2015] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, Optical atomic clocks, Rev. Mod. Phys. 87, 637 (2015).
  * Shang  _et al._ [2017] H. Shang, X. Zhang, S. Zhang, D. Pan, H. Chen, and J. Chen, Miniaturized calcium beam optical frequency standard using fully-sealed vacuum tube with $10^{-15}$ instability, Opt. Express 25, 30459 (2017).
  * Newman  _et al._ [2019] Z. L. Newman, V. Maurice, T. Drake, J. R. Stone, T. C. Briles, D. T. Spencer, C. Fredrick, Q. Li, D. Westly, B. R. Ilic, B. Shen, M.-G. Suh, K. Y. Yang, C. Johnson, D. M. S. Johnson, L. Hollberg, K. J. Vahala, K. Srinivasan, S. A. Diddams, J. Kitching, S. B. Papp, and M. T. Hummon, Architecture for the photonic integration of an optical atomic clock, Optica 6, 680 (2019).
  * Cuneo  _et al._ [1994] C. J. Cuneo, J. J. Maki, and D. H. McIntyre, Optically stabilized diode laser using high‐contrast saturated absorption, Appl. Phys. Lett 64, 2625 (1994).
  * Genov  _et al._ [2017] G. Genov, T. E. Lellinger, T. Halfmann, and T. Peters, Laser frequency stabilization by bichromatic saturation absorption spectroscopy, J. Opt. Soc. Am. B 34, 2018 (2017).
  * Strangfeld  _et al._ [2021] A. Strangfeld, S. Kanthak, M. Schiemangk, B. Wiegand, A. Wicht, A. Ling, and M. Krutzik, Prototype of a compact rubidium-based optical frequency reference for operation on nanosatellites, J. Opt. Soc. Am. B 38, 1885 (2021).
  * Wieman and Hänsch [1976] C. Wieman and T. W. Hänsch, Doppler-free laser polarization spectroscopy, Phys. Rev. Lett. 36, 1170 (1976).
  * Noh [2012] H.-R. Noh, Lineshapes in two-color polarization spectroscopy for cesium, Opt. Express 20, 21784 (2012).
  * Torrance  _et al._ [2016] J. S. Torrance, B. M. Sparkes, L. D. Turner, and R. E. Scholten, Sub-kilohertz laser linewidth narrowing using polarization spectroscopy, Opt. Express 24, 11396 (2016).
  * Touahri  _et al._ [1997] D. Touahri, O. Acef, A. Clairon, J.-J. Zondy, R. Felder, L. Hilico, B. de Beauvoir, F. Biraben, and F. Nez, Frequency measurement of the 5S1/2(F=3)-5D5/2(F=5) two-photon transition in rubidium, Opt. Commun. 133, 471 (1997).
  * Morzyński  _et al._ [2013] P. Morzyński, P. Wcisło, P. Ablewski, R. Gartman, W. Gawlik, P. Masłowski, B. Nagórny, F. Ozimek, C. Radzewicz, M. Witkowski, R. Ciuryło, and M. Zawada, Absolute frequency measurement of rubidium 5S - 7S two-photon transitions, Opt. Lett. 38, 4581 (2013).
  * Martin  _et al._ [2018] K. W. Martin, G. Phelps, N. D. Lemke, M. S. Bigelow, B. Stuhl, M. Wojcik, M. Holt, I. Coddington, M. W. Bishop, and J. H. Burke, Compact optical atomic clock based on a two-photon transition in rubidium, Phys. Rev. Appl. 9, 014019 (2018).
  * Newman  _et al._ [2021] Z. L. Newman, V. Maurice, C. Fredrick, T. Fortier, H. Leopardi, L. Hollberg, S. A. Diddams, J. Kitching, and M. T. Hummon, High-performance, compact optical standard, Opt. Lett. 46, 4702 (2021).
  * Hafiz  _et al._ [2016] M. A. Hafiz, G. Coget, E. D. Clercq, and R. Boudot, Doppler-free spectroscopy on the Cs D1 line with a dual-frequency laser, Opt. Lett. 41, 2982 (2016).
  * Brazhnikov  _et al._ [2019] D. Brazhnikov, M. Petersen, G. Coget, N. Passilly, V. Maurice, C. Gorecki, and R. Boudot, Dual-frequency sub-doppler spectroscopy: Extended theoretical model and microcell-based experiments, Phys. Rev. A 99, 062508 (2019).
  * Gusching  _et al._ [2021] A. Gusching, M. Petersen, N. Passilly, D. Brazhnikov, M. A. Hafiz, and R. Boudot, Short-term stability of Cs microcell-stabilized lasers using dual-frequency sub-doppler spectroscopy, J. Opt. Soc. Am. B 38, 3254 (2021).
  * Negnevitsky and Turner [2013] V. Negnevitsky and L. D. Turner, Wideband laser locking to an atomic reference with modulation transfer spectroscopy, Opt. Express 21, 3103 (2013).
  * Zi  _et al._ [2017] F. Zi, X. Wu, W. Zhong, R. H. Parker, C. Yu, S. Budker, X. Lu, and H. Müller, Laser frequency stabilization by combining modulation transfer and frequency modulation spectroscopy, Appl. Opt. 56, 2649 (2017).
  * Wu  _et al._ [2018] B. Wu, Y. Zhou, K. Weng, D. Zhu, Z. Fu, B. Cheng, X. Wang, and Q. Lin, Modulation transfer spectroscopy for D1 transition line of rubidium, J. Opt. Soc. Am. B 35, 2705 (2018).
  * Raj  _et al._ [1980] R. K. Raj, D. Bloch, J. J. Snyder, G. Camy, and M. Ducloy, High-frequency optically heterodyned saturation spectroscopy via resonant degenerate four-wave mixing, Phys. Rev. Lett. 44, 1251 (1980).
  * Shirley [1982] J. H. Shirley, Modulation transfer processes in optical heterodyne saturation spectroscopy, Opt. Lett. 7, 537 (1982).
  * Ito [2000] N. Ito, Doppler-free modulation transfer spectroscopy of rubidium $5^{2}\mathrm{S}_{1/2}$-$6^{2}\mathrm{P}_{1/2}$ transitions using a frequency-doubled diode laser blue-light source, Rev. Sci. Instrum. 71, 2655 (2000).
  * Zang  _et al._ [2007] E. J. Zang, J. P. Cao, Y. Li, C. Y. Li, Y. K. Deng, and C. Q. Gao, Realization of four-pass I2 absorption cell in 532-nm optical frequency standard, IEEE Trans. Instrum. Meas. 56, 673 (2007).
  * Schuldt  _et al._ [2017] T. Schuldt, K. Döringshoff, E. V. Kovalchuk, A. Keetman, J. Pahl, A. Peters, and C. Braxmaier, Development of a compact optical absolute frequency reference for space with $10^{-15}$ instability, Appl. Opt. 56, 1101 (2017).
  * Chang  _et al._ [2019] P. Chang, S. Zhang, H. Shang, and J. Chen, Stabilizing diode laser to 1 Hz$-$level Allan deviation with atomic spectroscopy for Rb four‑level active optical frequency standard, Appl. Phys. B: Lasers Opt. 125, 196 (2019).
  * Shang  _et al._ [2020] H. Shang, T. Zhang, J. Miao, T. Shi, D. Pan, X. Zhao, Q. Wei, L. Yang, and J. Chen, Laser with $10^{-13}$ short-term instability for compact optically pumped cesium beam atomic clock, Opt. Express 28, 6868 (2020).
  * Arimondo  _et al._ [1977] E. Arimondo, M. Inguscio, and P. Violino, Experimental determinations of the hyperfine structure in the alkali atoms, Rev. Mod. Phys. 49, 31 (1977).
  * Gerhardt  _et al._ [1972] H. Gerhardt, E. Matthias, F. Schneider, and A. Timmermann, Isotope shifts and hyperfine structure of the 6s-7p transitions in the cesium isotopes 133, 135, and 137, Z Physik A 288, 327 (1972).
  * Williams  _et al._ [2018] W. D. Williams, M. T. Herd, and W. B. Hawkins, Spectroscopic study of the 7P1/2 and 7P3/2 states in Cesium-133, Laser Phys. Lett. 15, 095702 (2018).
  * Grunefeld  _et al._ [2019] S. J. Grunefeld, B. M. Roberts, and J. S. M. Ginges, Correlation trends in the hyperfine structure for Rb, Cs, and Fr, and high-accuracy predictions for hyperfine constants, Phys. Rev. A 100, 042506 (2019).
  * Zhao  _et al._ [2004] Y. T. Zhao, J. M. Zhao, T. Huang, L. T. Xiao, and S. T. Jia, Frequency stabilization of an external-cavity diode laser with a thin Cs vapour cell, Journal of Physics D: Applied Physics 37, 1316 (2004).
  * Moon  _et al._ [2004] H. S. Moon, W. K. Lee, L. Lee, and J. B. Kim, Double resonance optical pumping spectrum and its application for frequency stabilization of a laser diode, Applied Physics Letters 85, 3965 (2004).
  * Toh  _et al._ [2019] G. Toh, N. Chalus, A. Burgess, A. Damitz, P. Imany, D. E. Leaird, A. M. Weiner, C. E. Tanner, and D. S. Elliott, Measurement of the lifetimes of the $7p{}^{2}\mathrm{P}_{3/2}$ and $7p{}^{2}\mathrm{P}_{1/2}$ states of atomic cesium, Phys. Rev. A 100, 052507 (2019).
  * Preuschoff  _et al._ [2018] T. Preuschoff, M. Schlosser, and G. Birkl, Optimization strategies for modulation transfer spectroscopy applied to laser stabilization, Opt. Express 26, 24010 (2018).
  * Li  _et al._ [2012] L. Li, F. Liu, C. Wang, and L. Chen, Measurement and control of residual amplitude modulation in optical phase modulation, Rev. Sci. Instrum. 83, 043111 (2012).
  * Zhang  _et al._ [2014] W. Zhang, M. J. Martin, C. Benko, J. L. Hall, J. Ye, C. Hagemann, T. Legero, U. Sterr, F. Riehle, G. D. Cole, and M. Aspelmeyer, Reduction of residual amplitude modulation to $1\times 10^{-6}$ for frequency modulation and laser stabilization, Opt. Lett. 39, 1980 (2014).
  * Tai  _et al._ [2016] Z. Tai, L. Yan, Y. Zhang, X. Zhang, W. Guo, S. Zhang, and H. Jiang, Electro-optic modulator with ultra-low residual amplitude modulation for frequency modulation and laser stabilization, Opt. Lett. 41, 5584 (2016).
  * Breit and Rabi [1931] G. Breit and I. I. Rabi, Measurement of nuclear spin, Phys. Rev. 38, 2082 (1931).