# Stellar $s$–process neutron capture cross section of Ce isotopes

R. N. Sahoo  The Hebrew University of Jerusalem, Jerusalem, Israel 91904  M.
Paul<EMAIL_ADDRESS>The Hebrew University of Jerusalem, Jerusalem,
Israel 91904  Y. Kashiv  University of Notre Dame, Notre Dame, IN 46556,
United States  M. Tessler  Soreq Nuclear Research Center, Yavne, Israel 81800
The Hebrew University of Jerusalem, Jerusalem, Israel 91904
M. Friedman  The Hebrew University of Jerusalem, Jerusalem, Israel 91904  S.
Halfon  Soreq Nuclear Research Center, Yavne, Israel 81800  A. Kreisel  Soreq
Nuclear Research Center, Yavne, Israel 81800  A. Shor  Soreq Nuclear Research
Center, Yavne, Israel 81800  L. Weissman  Soreq Nuclear Research Center,
Yavne, Israel 81800

###### Abstract

Stellar abundances of cerium are of high current interest based both on
observations and theoretical models, especially with regard to the
neutron–magic 140Ce isotope. A large discrepancy of $s-$process stellar models
relative to cerium abundance observed in globular clusters was highlighted,
pointing to possible uncertainties in experimental nuclear reaction rates. In
this work, the stellar neutron capture cross section of the stable cerium
isotopes 136Ce, 138Ce, 140Ce, and 142Ce, were re-measured. A natCe sample was
irradiated with quasi-Maxwellian neutrons at $kT=34.2$ keV using the
7Li($p,n$) reaction. The neutron field with an intensity of $3-5\times
10^{10}$ n/s was produced by irradiating the liquid-lithium target (LiLiT)
with a mA proton beam at an energy (1.92 MeV) just above the threshold at
Soreq Applied Research Accelerator Facility (SARAF). The activities of the
natCe neutron capture products were measured using a shielded High Purity
Germanium detector. Cross sections were extracted relative to that of the
197Au(n,$\gamma$) reaction and the Maxwellian-averaged cross section (MACS) of
the Ce isotopes were derived. The MACS values extracted from this experiment
are generally consistent with previous measurements and show for 140Ce a value
$\approx 15$% smaller than most recent experimental values.

##  I Introduction

The four stable isotopes of cerium (Z = 58, Fig. 1) display a showcase of
processes responsible for production of heavy elements. Since Cameron Cameron
(1957) and Burbidge et al. Burbidge  _et al._ (1957), neutron captures are
considered the main production gateways for these nuclides via the slow ($s$)
and the rapid ($r$) processes while very few proton-rich nuclides (two of
those being Ce isotopes) belong to the so-called $p$ process. About half of
the isotopes of the heavy elements ($A\gtrapprox 60$) are produced by the slow
($s$) process. There, the time between consecutive neutron captures is long
with respect to typical $\beta^{-}$ decay lifetimes along its path, resulting
in an evolution close to the valley of stability. The $s$ process is further
divided in a weak and main components. The weak $s$ process Käppeler  _et al._
(2011) operates in massive stars, $M_{\star}>8M_{\odot}$, before they explode
as type II supernovae and produces nuclides in the mass range $60\lessapprox
A\lessapprox 90$. The main component Käppeler  _et al._ (2011); Lugaro  _et
al._ (2003), which produces nuclides in the range $90\lessapprox A\leq 209$,
takes place in low-mass asymptotic giant branch (AGB) stars, $M_{\star}\leq
4M_{\odot}$. It operates on a timescale of a few $10^{5}$ yr. In a simplified
view, most of the neutron exposure experienced by seed nuclei is generated by
the ${}^{13}{\mathrm{C}}(\alpha,n)^{16}{\mathrm{O}}$ neutron source reaction,
operating in pulses of a few $10^{4}$ yr. The reaction is activated at $T\sim
0.9\times 10^{8}$ K ($\approx 8$ keV) and generates a low neutron number
density of $N_{n}=10^{6}-10^{8}$ cm-3.

Figure 1: Part of the nuclide chart showing the s-process nucleosynthesis path
in the Ce-Pr-Nd mass around the branching at A$=141-142$.

At the end of such a pulse when the temperature reaches $T\sim 2.5\times
10^{8}$ K ($\approx 22$ keV), a much shorter pulse is activated for a few
years during which the ${}^{22}\mathrm{Ne}(\alpha,n)^{25}\mathrm{Mg}$ is the
main neutron source. It generates a higher neutron number density of
$N_{n}\sim 10^{10}$ cm-3. Due to the short time it operates, the latter
neutron source accounts for only a small fraction of the total exposure.
However, the higher $N_{n}$ can affect branching points along the main
$s$-process path. The other about half of heavy nuclides are produced by the
rapid ($r$) process, in which the time between consecutive neutron captures is
short with respect to radioisotope $\beta^{-}$ decay lifetimes along its path.
The $r$ process operates on a timescale of 1 sec under explosive conditions
characterized by high temperature, $T>10^{9}$ K $(\approx 90$ keV), and high
neutron number density, $N_{n}>10^{20}$ cm-3 Freiburghaus  _et al._ (1999). A
third neutron capture process, termed the $i$ process, was proposed by Cowan
and Rose (1977). The neutron number density in this process,
$N_{n}=10^{12}-10^{16}$ cm-3, is intermediate between those the of $r$ and $s$
processes. It is considered to operate in some special cases Cowan and Rose
(1977); Denissenkov  _et al._ (2019); Choplin  _et al._ (2021, 2022).
Potential confirmation of the $i$ process comes from the chemical compositions
of carbon-enhanced metal-poor (CEMP)$-r/s$ stars, enriched in both $r-$ and
$s-$process elements. In addition to the neutron capture processes described
above, the $p$ process produces 35 neutron-deficient stable isotopes. It
operates in massive stars under explosive conditions by photonuclear reactions
e.g. $(\gamma,\alpha\setminus{n}\setminus{p})$, on pre-existing $r$ and $s$
isotopes Meyer (1994); Choplin, A.  _et al._ (2022).

Apart from the $p$-nuclides 136Ce and 138Ce, Ce is considered a primarily
(main) $s-$process element, with a small $r-$process fraction. Ce was indeed
observed in $s-$process enhanced stars Abia  _et al._ (2002); Aoki  _et al._
(2002), and in the CEMP$-r/s$ stars that may exhibit $i-$process composition
Denissenkov  _et al._ (2019); Choplin  _et al._ (2021); Karinkuzhi  _et al._
(2021). Cerium is however also observed in $r-$process enhanced stars Jonsell
_et al._ (2006); Siqueira Mello  _et al._ (2014); Holmbeck  _et al._ (2018)
and its presence was reported in the kilonova (AT 2017gfo) following the
gravitational wave event GW170817 Domoto  _et al._ (2022). A large body of
high-quality spectrometric data on the Galactic content of cerium has been
recently collected, see e.g. Contursi, G.  _et al._ (2023) and references
therein.

For a complete picture of the astronomical data of Ce, one needs to consider
presolar grains Zinner (2014) in addition to abundances in solar material.
Thermodynamic calculations show that Ce is refractory under both C-rich (C/O
$>$ 1) conditions Lodders and Fegley (1995) and under O-rich (C/O $<$ 1)
conditions Lodders (2003). The abundance of Ce was measured in single presolar
SiC grains (which condense under C-rich conditions) of different subtypes, the
source stars of which are believed to be mainly low-mass AGB stars of
different metallicities and supernovae Amari  _et al._ (1995). Recently, the
Ce abundance was measured for the first time also in a presolar silicate grain
(which condense under O-rich conditions), believed to originate in a low mass
AGB star Leitner and Hoppe (2022). Also recently, Lugaro et al. Lugaro  _et
al._ (2020) proposed correlating Ce/Y ratios in Ba stars with 88Sr/86Sr ratios
in large single presolar SiC grains to infer the type of their source stars.

In this paper we report new measurements of the radiative neutron capture
$(n,\gamma)$ Maxwellian-averaged cross sections (MACS) of the stable isotopes
at energies relevant to the $s$ and $i$ processes; see Käppeler  _et al._
(1996); Harnood  _et al._ (2000); Amaducci  _et al._ (2021) for previous
measurements and a compilation in kad  and references therein. Of special
interest is the $(n,\gamma)$ reaction on 140Ce which is a neutron-magic
($N=82$) nucleus with a typical high neutron binding energy and small neutron
capture cross section. 140Ce$(n,\gamma)$ feeds the potential $s-$ and
$i-$processes branching point 141Ce (terrestrial $t_{1/2}=32.504$ d,
calculated not to change at $s-$ and $i-$process temperatures Cosner K. and
Truran J. W. (1981); Takahashi and Yokoi (1987)). Koloczek et al.  Koloczek
_et al._ (2016) calculated that this cross section affects the production of
33 other isotopes in the $s$ process. A discrepancy between the Ce abundance
observed in the globular clusters M4 Young  _et al._ (2003) and M22 Roederer
_et al._ (2011) and that calculated by stellar models was recently pointed out
by Straniero et al. Straniero  _et al._ (2014) and was attributed possibly to
insufficient experimental knowledge of the stellar 140Ce$(n,\gamma)$ cross
section.

The article is organized as follows: Section II recounts the irradiation of
the natCe sample, Sec. III describes the detection of the radiation products
by $\gamma-$spectrometry, Sec. IV details the extraction of cross sections,
Sec. V is a short discussion and Sec. VI is a summary. Preliminary results of
this experiment were presented in Refs Sahoo, Rudra N.  _et al._ (2023);
Tessler  _et al._ (2016).

##  II  Ce sample irradiation

The neutron irradiation was performed using the Soreq Applied Research
Accelerator Facility (SARAF) and the Liquid Lithium Target (LiLiT); see Mardor
_et al._ (2018); Paul  _et al._ (2019) for review articles. Quasi-maxwellian
neutrons were produced by an intense proton beam (1-1.5 mA) from the SARAF
superconducting linear accelerator bombarding the LiLiT target at an energy of
1.925 MeV, just above the threshold (1881 keV) of the 7Li($p,n)^{7}$Be
reaction, following the method introduced in Ratynski & Käppeler Ratynski and
Käppeler (1988). The proton beam energy was measured by a Rutherford back
scattering detector located after the accelerator modules, with a typical
energy spread of 15 keV. The windowless LiLiT setup (Fig. 2), which consists
of liquid lithium (at $\approx 200^{\circ}$C) circulating in closed loop and
producing a target film of $\approx 1.5$ mm thickness and 18 mm wide, allows
for dissipation of the beam power (2-3 kW) by fast transport (2-3 m/s) to a
heat exchanger. Neutrons, emitted mostly in the forward direction due to the
reaction kinematics, bombarded secondary activation targets located in an
experimental chamber at rough vacuum separated by a 0.5 mm thick convex
stainless steel wall from the LiLiT chamber and accelerator vacuum. A high-
purity metallic natCe target (Ce-I, 25 mm diameter, 2.114(1) g) was sandwiched
between two Au foils labeled Au(1) and Au(2) (0.110(1) g and 0.119(1) g,
respectively), used as neutron fluence monitors (Table 1).

Figure 2:  Cross section diagram of the Liquid-Lithium Target (LiLiT) and activation target assembly. The proton beam (dashed thick arrow) impinges on the free-surface lithium film; inlet and outlet of the liquid lithium, circulating in a closed loop at $\approx 200^{\circ}$C, are indicated by thick downward arrows (see Paul  _et al._ (2019) for details). The activation target sandwich (Au-Ce-Au) is mounted on a circular target holder and positioned in the outgoing neutron field (dotted lines) at a distance of 6-8 mm from the lithium surface in a vacuum chamber separated from the LiLiT chamber by a 0.5 mm stainless steel wall convex to the beam. The retractable shaft (at left) is used to load and unload rapidly the target assembly. Table 1:  Areal density (atoms/cm2) of stable isotopes of cerium in targets Ce-I and Ce-II (see text), and Au(1), Au(2) targets for the above threshold irradiation. Stable Isotopes | Abundance | Ce-I | Ce-II
---|---|---|---
natCe, Au | ($\%$) | (atoms/cm2) | (atoms/cm2)
136Ce | 0.185(2) | 3.52(4)$\times$1018 | 3.46(4)$\times$1018
138Ce | 0.251(2) | 4.63(4)$\times$1018 | 4.55(4)$\times$1018
140Ce | 88.45(5) | 1.640(1)$\times$1021 | 1.612(1)$\times$1021
142Ce | 11.11(5) | 2.05(1)$\times$1020 | 2.02(1)$\times$1020
197Au(1) | 100 | 6.83$\times$1019 | 7.24$\times$1019
197Au(2) | 100 | 7.40$\times$1019 | 7.02$\times$1019

Figure 3 illustrates the time profile of the proton current during the
irradiation process monitored by a commercial fission detector located in the
neutron field at 0∘ relative to the incident proton beam at $\approx 80$ cm
downstream (in air) of the experimental chamber. The time profile was used to
calculate a correction for the decay of reaction products during the
irradiation, significant for the shortest-lived reaction product 137gCe
(t${}_{1/2}=9$ h).

Figure 3: Time profile of proton beam intensity during irradiation
($E_{p}=1.924$ MeV). The left vertical axis represents the count rate of 235U
fission events, produced in the fission detector (see text). The right
vertical axis displays the corresponding proton beam intensity, calibrated at
low intensity against an electron-suppressed Faraday cup located in front of
the LiLiT chamber. The low-intensity groups near time$=0$ and time$=6750s$
correspond to the calibration runs of the fission detector. Other gaps are
periods of SARAF accelerator instability. Figure 4: Simulated neutron spectrum
$\frac{dn_{sim}}{dE_{n}}$ impinging on the natCe- target (histogram) fitted to
a Maxwell-Boltzmann flux distribution at 34.2 keV (solid line).

The energy spectrum of the neutron field seen by the target sandwich could not
be experimentally determined and was calculated with the aid of the Monte
Carlo code SimLiT Friedman  _et al._ (2013) and the transport code GEANT4
Agostinelli  _et al._ (2003) using a detailed model of the LiLiT chamber and
target geometry. The SimLiT-GEANT4 codes were carefully benchmarked in
previous experiments using the same setup Tessler  _et al._ (2015); Paul  _et
al._ (2019). The simulated spectrum of the neutrons subtending the Ce target
is shown in Fig. 4 together with a Maxwell-Boltzmann flux distribution fitted
to the simulated spectrum for $T=34.2$ keV. The simulated experimental
spectrum (histogram in Fig. 4) includes 85% of the fitted Maxwell-Boltzmann
distribution at $kT$= 34.2 keV The validity of the simulated spectrum is
further demonstrated by the fact that the experimental and simulated number of
activated Au nuclei (denoted respectively by $N_{act}(Au)$ and
$N_{act}^{ENDF}(Au)$ (see below Eq. (8)) agree with each other within 1% after
normalization to proton charge.

It had been noted in previous experiments Tessler  _et al._ (2015); Paul  _et
al._ (2019) that in the LiLiT setup using a thick Li target, high-energy
$\gamma$ rays (17.6 and 14.6 MeV) due to the 7Li$(p,\gamma)^{8}$Be proton
capture reaction, produce $(\gamma,n)$ reactions in the irradiated target
which could interfere with the $(n,\gamma)$ reaction products. In the present
case for example, the 140Ce$(\gamma,n)^{139}$Ce reaction can interfere with
the 139Ce production via the 138Ce$(n,\gamma)^{139}$Ce investigated in this
work. In order to correct quantitatively for this effect, a separate Ce target
(Ce-II, Table 1) was irradiated with a pure $\gamma$ ray field from the
7Li$(p,\gamma)^{8}$Be reaction at a proton energy (1.810 MeV) below the
neutron threshold (no neutrons present). A Au foil monitored the $\gamma$
fluence via the 197Au$(\gamma,n)^{196}$Au and 196Au(6.2 d) activity, of well-
established cross section. Results of the $(n,\gamma)$ and $(\gamma,n)$
irradiations are presented in the next section.

##  III Detection of activated A+1Ce nuclei

Detailed information on the identified Ce isotopes, including their half-
lives, gamma-ray transitions, and intensities are given in Table 2. A shielded
HPGe detector was employed to identify and measure the induced activities of
reaction products. Prior to the activity measurement, the detector was
calibrated, and its efficiency was determined using a standard multi-gamma
radioactive source, including 22Na, 60Co, 88Y, 133Ba, 137Cs, 241Am, 152Eu, and
155Eu isotopes, positioned at a distance of 5 cm from the detector end-cap.
The efficiency curve obtained from this measurement is depicted in Fig. 5.

Figure 5: Full-energy peak efficiency of HPGe detector using standard multi-
gamma sources at 5cm apart from the end-cap.

The spectrum of irradiated natCe above neutron threshold, measured at 5 cm
distance, is presented in Fig. 6, revealing the identification of $\gamma$
lines corresponding to 137,139,141,143Ce isotopes and of a metastable state of
137Ce isotope (137mCe, $t_{1/2}=34.80h$, see Fig. 11). Figure 8 represents the
$\gamma-$ray spectrum Au(1) (upstream monitor) obtained in above threshold
irradiation.

Figure 6: $\gamma$-ray spectrum obtained after irradiating natCe with
Maxwellian neutron flux at SARAF-I above threshold energy. $\gamma$-transition
of all the isotopes of natCe(n,$\gamma$) reactions are identified along with
the isomeric 137mCe state. The figure is reproduced from Tessler  _et al._
(2016).

For either irradiation (above- or under-threshold), the residual number of
activated nuclei at time $t_{cool}$ after end of irradiation is obtained using
the following equation:

$n_{act}(t_{cool})=\frac{C_{\gamma}(t_{cool})}{\epsilon_{\gamma}I_{\gamma}K_{\gamma}(1-e^{-\lambda
t_{real}})}\frac{t_{real}}{t_{live}}\frac{1}{f_{b}};$ (1)

see decay curves of $n_{act}$ in Figs. 9, 10 for above-threshold and under-
threshold irradiation, respectively.

Figure 7: $\gamma$ -ray spectrum obtained after irradiating natCe with
$\gamma$-rays from 7Li(p,$\gamma$) reaction (under threshold energy). Only
139Ce and 141Ce $\gamma$ -transitions are observed and used for correction of
the $(n,\gamma)$ yields. The figure is reproduced from Tessler  _et al._
(2016). Figure 8: $\gamma$-ray spectrum obtained for the upstream Au(1)
monitor (above-threshold irradiation). The 198Au transitions (411.8 keV, and
675.9 keV) are labeled. Figure 9: Decay curves of 137,139,141,143Ce and 137mCe
(irradiation above neutron threshold). Data are fitted using the half-life
adopted in the literature. Figure 10: Decay curves of 139,141Ce (irradiation
under neutron threshold). Data are fitted using the half-life adopted in the
literature.

In Eq. (1), C${}_{\gamma}(t_{cool})$ is the number of counts in the full-
energy peak of a $\gamma$ line; $\epsilon_{\gamma}$, Iγ, and Kγ are the full-
energy peak efficiency (Fig. 5), $\gamma$ transition intensity and self-
absorption coefficient calculated from the target thickness data and the
different $\gamma$-ray energies, respectively (Table 2); $\lambda$, $t_{real}$
and $t_{live}$ are the decay constant of the transition, real and live
counting time, respectively. The notation $f_{b}$ refers to a correction for
nuclei decaying during the irradiation time, including a minor contribution
for 137gCe due to feeding from the isomeric state 137mCe, and is explained
below. We note here that the 447.2 keV $\gamma$-transition (Fig. 11), which
characterizes the decay of the 137Ce ground state, is practically degenerate
with the 447.45 keV (0.06%) transition in

Figure 11: Simplified decay scheme of 137Ce, showing the combined decay of
137(m+g)Ce.

143Ce decay within the resolution of the HPGe detector. In order to correct
for this contribution, the counts expected from the 447.45 keV transition were
estimated as follows, relative to those of the 293.26 (42.8 $\%$) keV $\gamma$
line in the decay of the same nucleus 143Ce:

$C_{447.45}=C_{293.26}\frac{I_{447.45}}{I_{293.26}}\frac{\epsilon_{447.45}}{\epsilon_{293.26}}\frac{K_{447.45}}{K_{293.26}}.$
(2)

$C_{\gamma}$, $I_{\gamma}$, $\epsilon_{\gamma}$ and $K_{\gamma}$ in Eq. (2)
represent the counts, intensity, efficiency of the detector and self-
absorption coefficient of the corresponding $\gamma$ transition, respectively.
The contribution of the 447.45 keV transition in 143Ce decay corresponds to
$\approx$15% of the full-energy peak observed at this energy and is subtracted
from the total peak counts.

The fraction of activated nuclei decaying during the irradiation time is
calculated via the expression

$f_{b}=\frac{\int_{0}^{t_{eoi}}\Phi_{n}(t)e^{-\lambda(t_{eoi}-t)}dt}{\int_{0}^{t_{b}}\Phi_{n}(t)dt},$
(3)

where $\Phi_{n}(t)$ represents the time profile of the neutron intensity on
target, taken as proportional to the (neutron-induced) fission chamber yield
(e.g Fig. 6 for above-threshold irradiation). Small additional corrections,
overall of the order of 1%, need to be applied in the case of 137gCe to take
into account feeding from the isomeric state 137mCe (Fig. 11) during
irradiation, cooling and counting; the expressions used for these corrections
are given in the Appendix. The final values of $f_{b}$ are given in Table 2.

Table 2:  Identified isotopes from natCe(n,$\gamma$) reactions, and their half-life, $\gamma$-ray transitions, and intensity NND . Calculated values of the self-absorption coefficient ($K_{\gamma}$) and correction factor $f_{b}$, including 137mCe feeding (see text), are listed. Identified | Half-life | Detected $\gamma$- | Intensity | Kγ | $f_{b}$
---|---|---|---|---|---
Isotopes |  | trans (keV) | ($\%$) |  |
137gCe | 9.11(3) h | 447.15(8) | 1.68(6) | 0.987 | 0.937
|  | 436.59(9) | 0.26(1) | 0.985 | 0.937
137mCe | 34.80(3) h | 254.29(5) | 11.1(4) | 0.966 | 0.982
139Ce | 137.64(2) d | 165.86(1) | 79.90(13) | 0.923 | 1.000
141Ce | 32.51(1) d | 145.443(1) | 48.30(7) | 0.912 | 0.999
143Ce | 33.039(6) h | 231.550(2) | 2.05(5) | 0.957 | 0.981
|  | 293.226(2) | 42.8(4) | 0.977 | 0.981
|  | 350.619(3) | 3.23(4) | 0.979 | 0.981
|  | 447.45(2) | 0.060(3) | 0.977 | 0.981
|  | 490.368(5) | 2.16(3) | 0.989 | 0.981
|  | 664.57(1) | 5.69(7) | 0.991 | 0.981
|  | 721.93(1) | 5.39(7) | 0.992 | 0.981
|  | 880.46(1) | 1.031(13) | 0.994 | 0.981
198Au | 2.6941(2) d | 411.80 | 95.62 | 0.999 | 0.990

The number $N_{act}^{above}$ ($N_{act}^{under}$) of activated ACe nuclei
produced during the irradiations above (under) neutron threshold (Table 3) was
extracted from an extrapolation of the respective decay curves to end of
irradiation (time $=0$, Figs. 9, 10). The net yield $N_{act}$ of the
ACe$(n,\gamma)$ reactions was obtained by subtracting the yield produced by
$(\gamma,n)$ reactions during the irradiation above threshold from the
measured yield $N_{act}^{above}$. The subtracted yield, significant only in
the case of the 139Ce product, was obtained from $N_{act}^{under}$ after
normalization of the respective target thicknesses and incident proton charges
(Table 3). We note here that the intensity of the high-energy $\gamma$ rays
produced by the 7Li$(p,\gamma)^{8}$Be reaction, responsible from the
$(\gamma,n)$ yield in our experiment, is dominated by a strong low-energy
resonance at E${}_{R}=441$ keV Zahnow  _et al._ (1995); Munch  _et al._
(2018). The difference between the thick-target average cross section for
production of high-energy $\gamma$ rays in the above-threshold ($E_{p}=1.912$
MeV) and under-threshold ($E_{p}=1.80$ MeV) irradiations is therefore
considered negligible.

Table 3:  Number of activated nuclei ($N_{act}$) obtained in the irradiation above neutron threshold ($N_{act}^{above}$) and under threshold ($N_{act}^{under}$) after implementing all corrections. The $(\gamma,n)$ yield during the above-threshold irradiation ($(\gamma,n)^{above}$) is obtained by normalization of $N_{act}^{under}$ for target thickness and proton charge and is subtracted from $N_{act}^{above}$ to obtain the net yield of $(n,\gamma)$ reactions ($N_{act}$). Errors listed are statistical errors only; see Table 7 for the overall uncertainty budget. Reaction | ${(n,\gamma)+(\gamma,n)}$ | $(\gamma,n)^{under}$ | $(\gamma,n)^{above}$ | $(n,\gamma)$
---|---|---|---|---
| $N_{act}^{above}$ | $N_{act}^{under}$ |  | $N_{act}$
Product | (108) | (108) | (108) | (108)
137gCe | 3.04(7) |  |  | 3.04(7)
137mCe | 0.310(6) |  |  | 0.310(6)
139Ce | 4.14(8) | 0.403(2) | 1.57(2) | 2.57(6)
141Ce | 47.0(7) | 0.020(1) | 0.078(2) | 46.96(7)
143Ce | 16.43(4) |  |  | 16.43(4)
198Au(1) | 138.8(2) |  |  | 138.8(2)
198Au(2) | 135.5(3) |  |  | 135.5(3)

##  IV ACe$(n,\gamma)$ experimental cross section and Maxwellian-averaged
cross sections

###  IV.1 Experimental cross sections

The net number of activated ACe nuclei $N_{act}$ can be written as

$N_{act}(x)=\sigma_{exp}(x)\phi_{total}n_{t}(x),$ (4)

where $\sigma_{exp}$(x) is our experimental $(n,\gamma)$ cross section,
sometimes termed the spectrum-averaged cross section, $\phi_{total}$ is the
time-integrated neutron rate and $n_{t}$(x) is the areal density of target
nuclei $x$. In our experiment the experimental cross section of cerium
isotopes is determined relative to that of 197Au used as monitor in the same
irradiation via the equation

$\sigma_{exp}(x)=\sigma_{exp}(Au)\frac{n_{t}(Au)N_{act}(x)}{n_{t}(x)N_{act}(Au)}.$
(5)

The spectrum-averaged cross section $\sigma_{exp}(Au)$ of the
197Au$(n,\gamma)^{198}$Au reaction is calculated via the expression

$\sigma_{exp}(Au)=\frac{\int\sigma_{ENDF}^{Au}(E_{n})\frac{dN_{exp}}{dE_{n}}dE_{n}}{\int\frac{dN_{exp}}{dE_{n}}dE_{n}},$
(6)

where the excitation function $\sigma_{ENDF}^{Au}(E_{n})$ is taken from the
ENDF/B-VIII.0 library Brown  _et al._ (2018). The ENDF/B-VIII.0 library was
found to reproduce closely experimental data Lederer  _et al._ (2011); Massimi
_et al._ (2014) of the 197Au$(n,\gamma)^{198}$Au reaction. The neutron
spectrum, $\frac{dN_{exp}}{dE_{n}}$, is obtained as described in Section II
from the SimLiT-GEANT4 simulation code Friedman  _et al._ (2013); Feinberg
_et al._ (2012) for each of the Au monitors. The average cross section
$\sigma_{exp}$(Au) for Au(1) is 571.9 mb and for Au(2) is 563.6 mb and the
value taken in Eq. (5) is 568(4) mb. The experimental cross sections of the
ACe isotopes using Eq. (5) and their uncertainty are given in Table 4.

Table 4:  Experimental neutron capture cross sections ($\sigma_{exp}$) of the stable isotopes of Ce. Reaction | Cross section
---|---
| ($\sigma_{exp}$) mb
136Ce(n,$\gamma$)137Ce | 262(10)
136Ce(n,$\gamma$)137mCe | 26.7(9)
138Ce(n,$\gamma$)139Ce | 163(6)
140Ce(n,$\gamma$)141Ce | 8.4(2)
142Ce(n,$\gamma$)143Ce | 23.5(9)

###  IV.2 MACS calculation

The Maxwellian$-$averaged cross section is defined Ratynski and Käppeler
(1988) as,

$MACS(kT)=\frac{2}{\pi}\frac{\int_{0}^{\infty}\sigma(E_{n})E_{n}e^{(-\frac{E_{n}}{kT})}dE_{n}}{\int_{0}^{\infty}E_{n}e^{(-\frac{E_{n}}{kT})}dE_{n}},$
(7)

where $\sigma(E_{n})$ is the true energy-dependent $(n,\gamma$) cross section.
Our measurements of the experimental cross sections of Section IV.1 allow us
to calibrate evaluated neutron cross section libraries, corrected to match the
measured $\sigma_{exp}$’s. Following the method described in Tessler  _et al._
(2015); Paul  _et al._ (2019), the SimLiT-GEANT4 code Friedman  _et al._
(2013) is used to calculate simultaneously the numbers of $(n,\gamma)$
activated nuclei for the two gold foils, Au(1) and Au(2), and the Ce target
using a neutron capture cross section library $\sigma_{lib}(E_{n})$ and the
detailed setup of our experiment. The correction factor $C_{lib}$ for
$\sigma_{lib}(E_{n})$ is then defined as follows:

$C_{lib}=\Big{[}\frac{N_{act}(Ce)}{N_{act}(Au)}\Big{]}\Big{/}\Big{[}\frac{N_{act}^{lib}(Ce)}{N_{act}^{ENDF}(Au)}\Big{]},$
(8)

where $N_{act}$(Ce) and $N_{act}$ (Au) are the number of $(n,\gamma)$
activated Ce-isotopes and Au nuclei determined experimentally. $N_{act}^{lib}$
(Ce) are the number of activated Ce-isotopes and Au nuclei determined from the
SimLiT-GEANT4 simulation using the cross section $\sigma_{lib}(E_{n})$ from a
given library. The ENDF/B-VIII.0 library was consistently used in the
simulation to calculate the number of 198Au activated nuclei
$N_{act}^{ENDF}$(Au). The Clib values derived for several evaluated libraries
are given in Table 5. Using this definition of $C_{lib}$, our experimental
MACS at temperature T is then determined as

$MACS_{lib}^{exp}(kT)=\frac{2}{\sqrt{\pi}}\frac{\int_{0}^{\infty}C_{lib}\sigma_{lib}(E_{n})E_{n}e^{(-\frac{E_{n}}{kT})}dE_{n}}{\int_{0}^{\infty}E_{n}e^{(-\frac{E_{n}}{kT})}dE_{n}},$
(9)

and the extracted values are given in Table 6 for $kT=30$ keV for the
respective libraries.

Table 5:  Calculated Clib values (see text) using Eq. (8). Target nucleus: |  136Ce |  138Ce |  140Ce |  142Ce
---|---|---|---|---
JENDL-5 | 1.04 | 1.23 | 1.06 | 1.29
JEFF-3.3 | 0.97 | 1.04 | 1.11 | 1.29
CENDL-3.2 | 0.85 | 3.93 | 1.06 | 1.29
ENDF/B-VIII.0 | 0.85 | 0.87 | 1.23 | 1.33
ROSFOND-10 | 0.85 | 0.87 | 1.23 | 1.29

Fluctuations in the $C_{lib}$ values are observed in Table 5, notably for the
outlying value of $C_{CENDL}$(138Ce), reflecting different evaluated
excitation functions (see Fig. 12 for 138Ce showing an energy dependence of
$\sigma_{CENDL}(E_{n})$ different from other libraries). Nevertheless, the
MACS(30 keV) values listed in Table 6 are remarkably stable; the standard
deviations are taken as representing the uncertainty in the extrapolation of
$\sigma_{exp}$ to a MACS at 30 keV. Table 7 summarizes all uncertainties
involved in our experimental MACS values.

Figure 12: Comparison of the 138Ce$(n,\gamma)$ excitation function evaluated in the different libraries. Table 6:  Experimental MACS (mb) at 30 keV using the Clib values and using $(n,\gamma)$ excitation function from the different libraries and their mean values. The standard deviation, resulting from the use of different values, is taken as representing the uncertainty in the extrapolation of $\sigma_{exp}$ to a MACS at 30 keV. Product nucleus: |  137gCe |  137mCe |  139Ce |  141Ce |  143Ce
---|---|---|---|---|---
JENDL-5 | 291.0 | 29.7 | 181.6 | 9.60 | 25.80
JEFF-3.3 | 290.4 | 29.6 | 181.5 | 10.5 | 25.82
CENDL-3.2 | 289.3 | 29.5 | 185.2 | 9.61 | 25.82
ENDF/B-VIII.0 | 289.3 | 29.5 | 181.6 | 9.59 | 24.90
ROSFOND | 289.3 | 29.5 | 181.6 | 9.59 | 25.90
Mean(std. dev.) | 290.0(8) | 29.6(1) | 182(2) | 9.7(4) | 25.8(4)
Table 7:  List of uncertainties ($\%$) of MACSexp of isotopes natCe.

Uncertainty are in $\%$ |
---|---
Source of uncertainty |  137Ce |  137mCe |  138Ce |  140Ce |  142Ce
Target thickness | 0.4 | 0.4 | 0.4 | 0.4 | 0.4
Activated nuclei(Ce) | 2.0 | 1.8 | 2.9 | 0.2 | 2.9
Activated nuclei(Au) | 0.2 | 0.2 | 0.2 | 0.2 | 0.2
HPGe efficiencya | 1.5 | 1.5 | 1.5 | 1.5 | 1.5
Intensity (Iγ) | 0.03 | 0.03 |  $\ll$ 0.01 |  $\ll$ 0.01 | 0.01
Simulationb | 1.8 | 1.8 | 1.8 | 1.8 | 1.8
$\sigma_{ENDF}$(Au) | 1.0 | 1.0 | 1.0 | 1.0 | 1.0
MACS${}^{exp}_{lib}$ c | 0.3 | 0.3 | 1.2 | 4.1 | 1.6
Total uncertainty | 3.3 | 3.2 | 4.1 | 4.9 | 4.2

  * •

a systematic error on $\gamma$ calibration sources.

  * •

b includes beam energy, energy spread and geometric positioning of the
activation targets.

  * •

c standard deviation of MACS values from the different libraries (Table 6).

Table 8 lists our final value and overall uncertainty of the the
${}^{A}Ce(n,\gamma$) cross section measured for all stable isotopes of cerium
along with the isomeric 137mCe state. The total 136Ce$(n,\gamma)^{137}$Ce
(isomeric $+$ ground state feeding) of astrophysical significance is also
listed. Table 9 lists the MACS values extrapolated to a larger range of
temperatures relevant to different astropysical sites, using the evaluated
library JENDL-5; no uncertainties are assigned for $kT\neq 30$.

Table 8:  Experimental MACS (mb) at 30 keV of stable isotopes of natural cerium from this work compared to values in the literature. Uncertainties include all contributions from Tables 3, 6 and 7 Reaction | This work | Käppeler  _et al._ (1996) | Harnood  _et al._ (2000) | kad
---|---|---|---|---
136Ce(n,$\gamma$)137gCe | 290(11) | 300(21) |  |
136Ce(n,$\gamma$)137mCe | 29.6(10) | 28.2(12) |  | 28.2(16)
136Ce(n,$\gamma$)137(g+m)Ce | 320(17) |  |  | 328(21)
138Ce(n,$\gamma$)139Ce | 182(8) | 179(5) |  | 179(5)
140Ce(n,$\gamma$)141Ce | 9.7(5) | 11.0(4) | 11.5(5) | 11.73(44)
142Ce(n,$\gamma$)143Ce | 25.8(11) | 28.3(10) |  | 29.9(10)
Table 9:  MACS of stable Ce isotopes in mb between $kT=10-120$ keV. JENDL-5
library is used to calculate the MACS at different $kT$ values. The MACS are
listed according to the product nucleus.

Temp(kT) |  136g+mCe |  138Ce |  140Ce |  142Ce
---|---|---|---|---
keV | MACS | MACS | MACS | MACS
10 | 542 | 299 | 14.8 | 53.9
20 | 381 | 215 | 11.4 | 32.5
30 | 320(17) | 182(8) | 9.7(5) | 25.8(11)
40 | 289 | 164 | 8.47 | 22.6
50 | 270 | 153 | 7.74 | 20.6
60 | 258 | 147 | 7.26 | 19.3
70 | 250 | 142 | 6.92 | 18.4
80 | 245 | 139 | 6.68 | 17.7
90 | 240 | 137 | 6.51 | 17.1
100 | 238 | 135 | 6.38 | 16.7
110 | 236 | 134 | 6.30 | 16.3
120 | 234 | 134 | 6.25 | 16.0

  * •

—

##  V  Discussion

Our results are in general agreement with the previous experimental studies of
Kaeppeler et al. Käppeler  _et al._ (1996) and Harnood et al. Harnood  _et
al._ (2000) with slightly lower uncertainties on the MACS of 136Ce. The MACS
value determined at 30 keV for the important case of 140Ce is lower by
$\approx$15% than reported in previous experimental studies Käppeler  _et al._
(1996); Harnood  _et al._ (2000) and in the Kadonis data base kad . This lower
value partially alleviates the discrepancy highlighted by Straniero et al.
Straniero  _et al._ (2014) for cerium in their study of the abundance of heavy
elements in globular clusters. We note however that the resonance shape
analysis done in Amaducci  _et al._ (2021) for an isolated $p-$wave resonance
at $E_{n}=$ 5.64 keV leads to a resonance strength larger than those extracted
from the evaluations JENDL-5 and ENDF/B-VIII.0 pointing to a possible conflict
with the trend of reduced MACS of 140Ce observed in our experiment.

##  VI Summary

We have determined the experimental cross section of neutron capture reactions
on the stable isotopes of cerium, averaged over a quasi-Maxwellian neutron
spectrum at 34.2 keV. The experiment used the mA proton beam of the SARAF
facility incident on the high-power Liquid-Lithium Target (LiLiT). In
conjunction with detailed simulations of the experimental system, the
experimental cross sections were extrapolated to MACS values at 30 keV and
other temperatures relevant to the stellar $s$ process. The MACS values
obtained for the important neutron-magic nucleus 140Ce is $\approx$15% smaller
than previous experimental determinations, partially resolving the discrepancy
in the Ce abundance observed in globular cluster stars.

##  VII  Acknowledgments

The SARAF and LiLiT (Soreq NRC) staff is gratefully acknowledged for their
dedicated help during the experiments. We thank S. Cristallo and D. Vescovi
for an enlightening discussion. This work is supported in part by the Pazy
Foundation (Israel) and the German Israeli Foundation (GIF) under Grant No.
I-1500-303.7/2019. MP acknowledges support by the European Union (ChETEC-
INFRA).

##  VIII  Appendix

The 137mCe isomer decays predominantly (99.2%) by internal transition to the
137gCe ground state as illustrated in Fig. 11. The additional feeding of the
ground state by decay of the isomeric state during irradiation, cooling, and
counting time leads to small corrections of the measured yield of 137gCe decay
to obtain its prompt feeding yield from the $(n,\gamma)$ reaction.

During irradiation, the correction factor $f_{b}^{137g}$ for 137gCe production
can is expressed by

$f_{b}^{{}^{137g}Ce}=f_{b}+\frac{b_{f}\sigma_{m}}{\sigma_{g}}\Big{[}\frac{(1-e^{-\lambda_{g}t_{b}})}{\lambda_{g}t_{b}}-\frac{(e^{-\lambda_{m}t_{b}}-e^{-\lambda_{g}t_{b}})}{(\lambda_{g}-\lambda_{m})t_{i}}\Big{]},$
(10)

where $\lambda_{g,m}$ are the respective decay constants of 137g,mCe, $t_{b}$
and $b_{f}$ the beam irradiation time and the $m-$to$-g$ branching ratio,
respectively. The correction to $f_{b}$ is of 0.17$\%$.

A fraction of the 137mCe nuclei produced during irradiation decays to the
ground state during the cooling time tcool between end of irradiation and
start of counting, given by the following expression:

$f^{{}^{\prime}}_{cool}=f_{cool}+\frac{\sigma_{m}f_{b,m}}{\sigma_{g}f_{b,g}}\Big{[}\frac{e^{-\lambda_{m}t_{cool}}-e^{-\lambda_{g}t_{cool}}}{\lambda_{g}-\lambda_{m}}\Big{]}\lambda_{m},$
(11)

where $f_{cool}$ = e${}^{-\lambda_{m}t_{cool}}$. The correction to the counted
137gCe decays is of 0.27%.

Similarly, a fraction of the 137mCe states decays to the ground state during
the counting time, expressed by

$\vspace{0.4cm}\begin{split}N_{act}^{{}^{137g}Ce}=N_{act}-\frac{b_{f}N_{m}}{\lambda_{g}-\lambda_{m}}\cdot\\\
\Big{[}\frac{\lambda_{m}(e^{-\lambda_{g}t_{m}}-1)-\lambda_{g}(e^{-\lambda_{m}t_{m}}-1)}{(1-e^{-\lambda_{m}t_{m}})}\Big{]},\end{split}$
(12)

where $N_{act}$ and $N_{m}$ are the respective number 137gCe and 137mCe
activated nuclei using Eq. (1) and $t_{m}$ represents counting time. In each
measurement, the contribution of the isomeric state varies from 0.4% to 0.7%
of the total 137gCe population.

The corrections above are included in the coefficients $f_{b}$ listed in Table
2.

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# Ramsey interferometry with arbitrary coherent-population-trapping pulse
sequence

Ruihuan Fang  Guangdong Provincial Key Laboratory of Quantum Metrology and
Sensing $\&$ School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai
Campus), Zhuhai 519082, China  State Key Laboratory of Optoelectronic
Materials and Technologies, Sun Yat-Sen University (Guangzhou Campus),
Guangzhou 510275, China  College of Physics and Optoelectronic Engineering,
Shenzhen University, Shenzhen 518060, China  Chengyin Han  College of Physics
and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China
Bo Lu  College of Physics and Optoelectronic Engineering, Shenzhen University,
Shenzhen 518060, China  Jiahao Huang<EMAIL_ADDRESS><EMAIL_ADDRESS>Guangdong Provincial Key Laboratory of Quantum Metrology
and Sensing $\&$ School of Physics and Astronomy, Sun Yat-Sen University
(Zhuhai Campus), Zhuhai 519082, China  College of Physics and Optoelectronic
Engineering, Shenzhen University, Shenzhen 518060, China  Chaohong Lee
<EMAIL_ADDRESS>College of Physics and Optoelectronic Engineering, Shenzhen
University, Shenzhen 518060, China

###### Abstract

Coherent population trapping (CPT) is a multi-level quantum coherence
phenomenon of promising applications in atomic clocks and magnetometers.
Particularly, multi-pulse CPT-Ramsey interferometry is a powerful tool for
improving the performance of CPT atomic clocks. Most studies on multi-pulse
CPT-Ramsey interferometry consider periodic pulse sequence and time-
independent detuning. However, to further improve the accuracy and precision,
one may modify the spectrum symmetry which involves pulse sequence with time-
dependent detuning or phase shift. Here, we theoretically analyze the multi-
pulse CPT-Ramsey interferometry under arbitrary pulse sequences of time-
dependent detuning and obtain a general analytical formula. Using our formula,
we analyze the popular CPT-Ramsey interferometry schemes such as two-pulse
symmetric and antisymmetric spectroscopy, and multi-pulse symmetric and
antisymmetric spectroscopy. Moreover, we quantitatively obtain the influences
of pulse width, pulse period, pulse number, and Rabi frequency under periodic
pulses. Our theoretical results can guide the experimental design to improve
the performance of atomic clocks via multi-pulse CPT-Ramsey interferometry.

##  I Introduction

Coherent population trapping (CPT) is a phenomenon of atoms trapped in a
coherent state that does not interact with external laser fields. Since the
first observation of CPT spectrum Alzetta  _et al._ (1976), it has been
extensively utilized in various applications of quantum engineering and
quantum sensing, such as all-optical manipulation Rogers  _et al._ (2014); Das
_et al._ (2018); Xia  _et al._ (2015); Santori  _et al._ (2006); Jamonneau
_et al._ (2016); Ni  _et al._ (2008), atomic cooling Aspect  _et al._ (1988),
atomic clocks Vanier (2005); Merimaa  _et al._ (2003); Yun  _et al._ (2017);
Liu  _et al._ (2017), and atomic magnetometers Scully and Fleischhauer (1992);
Nagel  _et al._ (1998); Schwindt  _et al._ (2004); Tripathi and Pati (2019).
Conventionally, CPT spectroscopy has the drawback of power broadening caused
by strong CPT light power. Using two CPT pulses to perform CPT-Ramsey
interferometry can narrow the spectral linewidth and improve the signal-to-
noise ratio (SNR) Zanon  _et al._ (2005); Merimaa  _et al._ (2003); Vanier
_et al._ (2003). In this case, the linewidth of CPT-Ramsey spectrum can be
narrower as the interval dark time between the two pulses increases Merimaa
_et al._ (2003); Vanier  _et al._ (2003). As the demands for higher
measurement precision and accuracy grow, various techniques are developed to
improve the spectrum SNR and resolution, as well as mitigate light shift.

The multi-pulse CPT-Ramsey interferometry has been developed in recent years
Guerandel  _et al._ (2007); Yun  _et al._ ; Warren  _et al._ (2018); Nicolas
_et al._ (2018). The spectrum linewidth can be narrowed and meanwhile, the
central peak can be identified due to the multi-pulse interference.
Understanding the mechanism of multi-pulse CPT-Ramsey interferometry is
beneficial to designing suitable CPT pulse sequences for frequency
measurement. Some typical multi-pulse CPT-Ramsey schemes can be analytically
analyzed. For example, under multiple pulses with identical periods and
duration, one can explain the multi-pulse CPT-Ramsey interference using a
simple model based on the Fourier analysis of the CPT pulse sequence Warren
_et al._ (2018); Jamonneau  _et al._ (2016). The Fourier analysis introduces a
characteristic number $N_{s}$ as the spectrum will reach steady-state for
large pulse number $N$ Jamonneau  _et al._ (2016). The other analytical
treatment is to compare the multi-pulse CPT-Ramsey interferometry to the
Fabry-Pérot resonator, which is valid from periodic CPT pulse sequence Nicolas
_et al._ (2018) to arbitrary time-independent CPT pulse sequence Fang  _et
al._ (2021).

However, to further improve the performances one may need to modulate the
pulse detuning or phase shift with time. For example, the auto-balance
technique uses a detuning change during the dark time to modify the symmetry
of the spectrum to achieve real-time clock servo. The frequency shift and
phase jump have been applied to change the vertical symmetry of the spectrum
and are used as the additional variables in auto-balanced CPT Abdel Hafiz  _et
al._ (2018a, b); Yudin  _et al._ (2018). Moreover, for quantum lock-in
amplifier Kotler  _et al._ (2011), one may even use the mixing between the
alternating magnetic field and CPT pulse sequence. The alternating magnetic
field may also induce alternating detuning. The modulation of frequency or
phase has been applied in experimental CPT schemes and become a potential
technique. Thus the analytical analysis of multi-pulse CPT-Ramsey
interferometry under time-dependent detuning and arbitrary pulse phase is of
great importance and broad applications.

In this article, we study the multi-pulse CPT-Ramsey interferometry with
arbitrary CPT pulse sequences. In particular, when the single-photon detuning
and Rabi frequencies are relatively small compared to the decay rate of the
excited state, we obtain an analytical formula to analyze various CPT-Ramsey
interferometry scenarios, such as the conventional CPT-Ramsey interferometry,
CPT-Ramsey interferometry with frequency shift or phase jump, multi-pulse CPT-
Ramsey interferometry, and multi-pulse CPT-Ramsey interferometry with
frequency shift. For the periodic CPT pulse sequence Yun  _et al._ , we can
analytically study the influences of pulse length, pulse strength, pulse
interval, and pulse number on the spectrum linewidth. The formula we derived
is a general solution that is valid for most situations with arbitrary CPT
pulse sequences.

##  II CPT in three-level $\Lambda$ system

###  II.1 Model

CPT can usually be achieved in a three-level $\Lambda$ structure as shown in
Fig. 1.

Figure 1: The three-level $\Lambda$ system and the timing sequence of CPT. (a)
Bi-chromatic light with the Rabi frequencies $\Omega_{1}$ and $\Omega_{2}$
couple the two ground states $|1\rangle$ and $|2\rangle$ to the excited state
$|3\rangle$. The $\delta_{1}$ and $\delta_{2}$ are the frequency detunings
from the $|1\rangle$ and $|2\rangle$ to $|3\rangle$. The decay rate of excited
state $|3\rangle$ is $\Gamma$. (b) The orange pulses are CPT pulses with the
Rabi frequency $\Omega$. The $\tau_{i}$ is the pulse length, $t_{i}$ is the
start point, and $\Phi_{i}$ is the phase of $i$-th CPT pulse. (c) The $\delta$
versus time $t$. The $\delta$ can be time-dependent randomly.

The bi-chromatic field with Rabi frequencies $\Omega_{1}$ and $\Omega_{2}$
couple the two ground states $|1\rangle$ and $|2\rangle$ to the excited state
$|3\rangle$. Here, $\delta_{1}$ and $\delta_{2}$ are the corresponding
detunings, and $\Gamma$ is the decay rate of the excited state $|3\rangle$.
Under the rotating wave approximation, the Hamiltonian in the interaction
picture reads Shahriar  _et al._ (2014)

$\displaystyle\hat{H}_{I}=$
$\displaystyle\hbar\left(\delta_{1}|1\rangle\langle
1|+\delta_{2}|2\rangle\langle 2|\right)$ (1)
$\displaystyle+\hbar\left(\frac{\Omega_{1}}{2}|1\rangle\langle
3|+\frac{\Omega_{2}}{2}|2\rangle\langle 3|+h.c.\right).$

According to the Lindblad equation, the time-evolution of the density matrix
$\rho$ obeys

$\frac{d\rho}{dt}=-\frac{\mathrm{i}}{\hbar}\left[\hat{H}_{I},\rho\right]+\frac{\Gamma}{2}\sum_{j=1}^{2}\left[\hat{L}_{j}\rho\hat{L}_{j}^{\dagger}-\frac{1}{2}\\{\hat{L}_{j}^{\dagger}\hat{L}_{j},\rho\\}\right].$
(2)

Here, $\hat{L}_{j}=|j\rangle\langle 3|$ and their Hermite conjugate
$\hat{L}_{j}^{\dagger}$ are the Lindblad operators. In most cases, the Rabi
frequencies of monochromatic light are equal, i.e.,
$|\Omega_{1}|=|\Omega_{2}|=\Omega$. Then the Rabi frequencies can be expressed
as $\Omega_{1}=\Omega\mathrm{e}^{\mathrm{i}\phi_{1}}$ and
$\Omega_{2}=\Omega\mathrm{e}^{\mathrm{i}\phi_{2}}$, with $\phi_{1}$ and
$\phi_{2}$ are the phases of monochromatic light. In this case, Eq. (2) can be
written as follows,

$\begin{cases}\frac{d}{dt}\rho_{11}=\frac{\Gamma}{2}\rho_{33}-\mathrm{i}\left(\frac{\Omega\mathrm{e}^{-\mathrm{i}\phi_{1}}}{2}\rho_{13}^{*}-\frac{\Omega\mathrm{e}^{\mathrm{i}\phi_{1}}}{2}\rho_{13}\right)\\\
\frac{d}{dt}\rho_{12}=-\mathrm{i}\frac{\Omega\mathrm{e}^{-\mathrm{i}\phi_{1}}}{2}\rho_{23}^{*}+\mathrm{i}\frac{\Omega\mathrm{e}^{\mathrm{i}\phi_{2}}}{2}\rho_{13}-\mathrm{i}\left(\delta_{1}-\delta_{2}\right)\rho_{12}\\\
\frac{d}{dt}\rho_{13}=-\frac{\Gamma}{2}\rho_{13}+\mathrm{i}\frac{\Omega\mathrm{e}^{-\mathrm{i}\phi_{1}}}{2}\rho_{11}-\mathrm{i}\frac{\Omega\mathrm{e}^{-\mathrm{i}\phi_{1}}}{2}\rho_{33}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\mathrm{i}\frac{\Omega\mathrm{e}^{-\mathrm{i}\phi_{2}}}{2}\rho_{12}-\mathrm{i}\delta_{1}\rho_{13}\\\
\frac{d}{dt}\rho_{12}^{*}=\mathrm{i}\frac{\Omega\mathrm{e}^{\mathrm{i}\phi_{1}}}{2}\rho_{23}-\mathrm{i}\frac{\Omega\mathrm{e}^{-\mathrm{i}\phi_{2}}}{2}\rho_{13}^{*}+\mathrm{i}\left(\delta_{1}-\delta_{2}\right)\rho_{12}^{*}\\\
\frac{d}{dt}\rho_{22}=\frac{\Gamma}{2}\rho_{33}-\mathrm{i}\left(\frac{\Omega\mathrm{e}^{-\mathrm{i}\phi_{2}}}{2}\rho_{23}^{*}-\frac{\Omega\mathrm{e}^{\mathrm{i}\phi_{2}}}{2}\rho_{23}\right)\\\
\frac{d}{dt}\rho_{23}=-\frac{\Gamma}{2}\rho_{23}+\frac{\mathrm{i}\Omega\mathrm{e}^{-\mathrm{i}\phi_{1}}}{2}\rho_{12}^{*}+\frac{\mathrm{i}\Omega\mathrm{e}^{-\mathrm{i}\phi_{2}}}{2}\rho_{22}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{\mathrm{i}\Omega\mathrm{e}^{-\mathrm{i}\phi_{2}}}{2}\rho_{33}-\mathrm{i}\delta_{2}\rho_{23}\\\
\frac{d}{dt}\rho_{13}^{*}=-\frac{\Gamma}{2}\rho_{13}^{*}-\mathrm{i}\frac{\Omega\mathrm{e}^{\mathrm{i}\phi_{1}}}{2}\rho_{11}+\mathrm{i}\frac{\Omega\mathrm{e}^{\mathrm{i}\phi_{1}}}{2}\rho_{33}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\mathrm{i}\frac{\Omega\mathrm{e}^{\mathrm{i}\phi_{2}}}{2}\rho_{12}^{*}+\mathrm{i}\delta_{1}\rho_{13}^{*}\\\
\frac{d}{dt}\rho_{23}^{*}=-\frac{\Gamma}{2}\rho_{23}^{*}-\frac{\mathrm{i}\Omega\mathrm{e}^{\mathrm{i}\phi_{1}}}{2}\rho_{12}-\frac{\mathrm{i}\Omega\mathrm{e}^{\mathrm{i}\phi_{2}}}{2}\rho_{22}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\mathrm{i}\Omega\mathrm{e}^{\mathrm{i}\phi_{2}}}{2}\rho_{33}+\mathrm{i}\delta_{2}\rho_{23}^{*}\\\
\frac{d}{dt}\rho_{33}=-\Gamma\rho_{33}+\frac{\mathrm{i}\Omega\mathrm{e}^{-\mathrm{i}\phi_{1}}}{2}\rho_{13}^{*}-\frac{\mathrm{i}\Omega\mathrm{e}^{\mathrm{i}\phi_{1}}}{2}\rho_{13}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\mathrm{i}\Omega\mathrm{e}^{-\mathrm{i}\phi_{2}}}{2}\rho_{23}^{*}-\frac{\mathrm{i}\Omega\mathrm{e}^{\mathrm{i}\phi_{2}}}{2}\rho_{23},\end{cases}$
(3)

where $\rho_{ij}=\langle i|\rho|j\rangle$. In the following, we solve these
equations analytically.

###  II.2 Analytical formula of ground-state coherence

Under the condition of the Rabi frequency is far smaller than the decay rate
$\Omega\ll\Gamma$, one can find $\frac{d}{dt}\rho_{33}\ll\Gamma\rho_{33}$, and
we can get

$\rho_{33}=\frac{1}{\Gamma}\left(\frac{\mathrm{i}\Omega\mathrm{e}^{-\mathrm{i}\phi_{1}}}{2}\rho_{13}^{*}-\frac{\mathrm{i}\Omega\mathrm{e}^{\mathrm{i}\phi_{1}}}{2}\rho_{13}+\frac{\mathrm{i}\Omega\mathrm{e}^{-\mathrm{i}\phi_{2}}}{2}\rho_{23}^{*}-\frac{\mathrm{i}\Omega\mathrm{e}^{\mathrm{i}\phi_{2}}}{2}\rho_{23}\right).$
(4)

Usually, the CPT works in the situation of near resonance
$\delta_{1,2}\ll\Gamma$. Due to the large decay rate of the excited state, the
population in the excited state can be ignored (compared with ground state
populations), i.e., $\rho_{33}\ll\rho_{11}$ and $\rho_{33}\ll\rho_{22}$. Thus
we can obtain

$\rho_{13}=\frac{1}{\Gamma}\left(\mathrm{i}\Omega\mathrm{e}^{-\mathrm{i}\phi_{1}}\rho_{11}+\mathrm{i}\Omega\mathrm{e}^{-\mathrm{i}\phi_{2}}\rho_{12}\right),$
(5)
$\rho_{23}=\frac{1}{\Gamma}\left(\mathrm{i}\Omega\mathrm{e}^{-\mathrm{i}\phi_{1}}\rho_{12}^{*}+\mathrm{i}\Omega\mathrm{e}^{-\mathrm{i}\phi_{2}}\rho_{22}\right),$
(6)
$\rho_{13}^{*}=\frac{1}{\Gamma}\left(-\mathrm{i}\Omega\mathrm{e}^{\mathrm{i}\phi_{1}}\rho_{11}-\mathrm{i}\Omega\mathrm{e}^{\mathrm{i}\phi_{2}}\rho_{12}^{*}\right),$
(7)
$\rho_{23}^{*}=\frac{1}{\Gamma}\left(-\mathrm{i}\Omega\mathrm{e}^{\mathrm{i}\phi_{1}}\rho_{12}-\mathrm{i}\Omega\mathrm{e}^{\mathrm{i}\phi_{2}}\rho_{22}\right).$
(8)

In a CPT process, the population is mostly in the ground states that
$\rho_{11}+\rho_{22}\approx 1$. Substituting Eqs. (5)- (8) into Eqs. (3) and
(4), we obtain

$\rho_{33}=\frac{\Omega^{2}}{\Gamma^{2}}\left(1+2\Re(\rho_{12}\mathrm{e}^{\mathrm{i}\phi(t)})\right),$
(9)

and

$\frac{d}{dt}\rho_{12}=-\frac{\Omega^{2}}{2\Gamma}\mathrm{e}^{-\mathrm{i}\phi(t)}-\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)\rho_{12}.$
(10)

Here, $\delta=\delta_{1}-\delta_{2}$ is the two-photon detuning, and
$\phi(t)=\phi_{1}-\phi_{2}$ is the phase of monochromatic light. Eq. (10) can
be analytically solved under a train of CPT pulses.

We can construct the response of Eq. (10) to an impulse taking place at
$t_{0}$, which is the Green’s function Berberan-Santos (2010)

$G(t,t_{0})=H(t-t_{0})\exp\left(-\int_{t_{0}}^{t}\left(\frac{\Omega^{2}(u)}{\Gamma}+\mathrm{i}\delta(u)\right)du\right),$
(11)

where $H(x)$ is Heaviside’s function. If the density matrix starts from a
mixture state $\rho=|1\rangle\langle 1|+|2\rangle\langle 2|$. Take the initial
value $\rho_{12}(0)=0$, we have

$\rho_{12}(t,\Omega)=-\int_{0}^{t}\frac{\Omega^{2}(t^{\prime})}{2\Gamma}\mathrm{e}^{-\mathrm{i}\phi(t^{\prime})}G(t,t^{\prime})dt^{\prime}$
(12)

being a function of $\delta(t)$ and $\Omega(t)$. We analyze Eq. (12) within
the context of a general CPT pulse sequence, as shown in Figs. 1(b,c). The
orange pulses represent the CPT pulses with Rabi frequency $\Omega$, variable
pulse duration $\tau_{i}$, and phase $\Phi_{i}$. The gray solid line
represents the time-dependent detuning $\delta(t)$. As a result, Eq. (12) can
be expressed as

$\displaystyle\rho_{12}(t,\Omega)=$
$\displaystyle-\frac{\Omega^{2}}{2\Gamma}\sum_{l=1}^{N}\left(\prod_{k=l+1}^{N}\exp\left(-\frac{\Omega^{2}}{\Gamma}\tau_{k}\right)\right)$
(13)
$\displaystyle\times\exp\left(-\int_{t_{l}+\tau_{l}}^{t}\mathrm{i}\delta(u)du\right)\mathrm{e}^{-\mathrm{i}\Phi_{l}}$
$\displaystyle\times\int_{t_{l}}^{t_{l}+\tau_{l}}\exp\left(-\int_{t^{\prime}}^{t_{l}+\tau_{l}}\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta(u)\right)du\right)dt^{\prime}.$

This is the general formula of ground-state coherence. Eq. (13) can be used to
describe most cases of pulse sequences including the two-pulse CPT-Ramsey
interferometry and multi-pulse CPT-Ramsey interferometry under fixed or time-
dependent frequency detuning and phase. However, the absence of single-photon
detunings $\delta_{1}$ and $\delta_{2}$ in the derivation of Eqs. (5)-(8)
means that the influences of light shift induced by the excited state are not
taken into account.

According to Eq. (9), if the phases of each CPT pulse are identical, the phase
value does not affect the observation $\rho_{33}$. This is because a global
phase of the Rabi frequencies can be gauged into the state without changing
the density matrix if we select the initial mixture state without non-diagonal
terms. That means the magnitude of Rabi frequencies can be real if we select
proper initial phases of $|1\rangle$ and $|2\rangle$. Generally, we gauge the
phase of the last CPT pulse $\Phi_{N}$ into zero such that the phase of Eq.
(9) can be eliminated, thus

$\rho_{33}=\frac{\Omega^{2}}{\Gamma^{2}}\left(1+2\Re\left(\rho_{12}\right)\right).$
(14)

And if the phases of CPT pulses are constant, all the phases can be gauged as
zero.

If the detuning $\delta(t)$ is fixed as $\delta$ during the CPT pulses and
varies with time $t$ during the dark period, Eq. (13) can be written in the
form of

$\rho_{12}=f\left(\delta\right)Q,$ (15)

where

$f(\delta)=-\frac{\Omega^{2}}{2\Gamma\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)},$
(16)

is the slow variant envelope and $Q$ is the multi-pulse CPT-Ramsey
interference term,

$\displaystyle Q=\sum_{l=1}^{N}$
$\displaystyle\left(\prod_{k=l+1}^{N}\mathrm{e}^{-\frac{\Omega^{2}}{\Gamma}\tau_{k}}\right)\exp\left(-\int_{t_{l}+\tau_{l}}^{t}\mathrm{i}\delta(u)du\right)\mathrm{e}^{-\mathrm{i}\Phi_{l}}$
(17)
$\displaystyle\times\left(1-\exp\left(-\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)\tau_{l}\right)\right).$

##  III Applications in CPT-Ramsey Spectroscopy

Our analytical formula Eq. (13) can be applied in various experimental CPT-
Ramsey scenarios. In the following, we show its applications in conventional
two-pulse CPT-Ramsey spectroscopy, anti-symmetric two-pulse CPT-Ramsey
spectroscopy with frequency shift, and multi-pulse CPT-Ramsey spectroscopy.

###  III.1 Two-pulse sequence

In most cases, the CPT-Ramsey interferometry is implemented with two pulses,
as shown in Fig. 2 (a). In this scheme, the CPT pulse sequence contains a
preparation pulse with duration $\tau_{1}$ and a detection pulse with duration
$\tau_{2}$, which are separated by a free evolution with dark time $T$ Zanon-
Willette  _et al._ (2005). Using Eq. (13) with pulse number $N=2$, we can
easily get the corresponding analytical results, which can be used as a
benchmark example. Our analytical results can also be used for optimizing the
CPT pulse sequence as we need.

####  III.1.1 Conventional Two-pulse CPT-Ramsey Spectroscopy

Considering the simple case, in which the detunings are time-independent and
the phases equal zero. Then, Eq. (13) can be simplified as

$\displaystyle\rho_{12}(t,\delta,\Omega)=$
$\displaystyle-\frac{\Omega^{2}}{2\Gamma\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)}\sum_{l=1}^{N}\left(\prod_{k=l+1}^{N}\exp\left(-\frac{\Omega^{2}}{\Gamma}\tau_{k}\right)\right)$
(18)
$\displaystyle\times\exp\left(-\mathrm{i}\delta\left(t-t_{l}-\tau_{l}\right)\right)$
$\displaystyle\times\left(1-\exp\left(-\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)\tau_{l}\right)\right).$

Figure 2: The two-pulse CPT-Ramsey spectrum. (a) The timing sequence of
conventional CPT-Ramsey includes the preparation pulse $\tau_{1}$, the
detection pulse $\tau_{2}$, and the dark time $T$. The Rabi frequency is
$\Omega=0.035\Gamma$. The CPT-Ramsey spectrum of both simulation(blue dots)
and analytical(black solid line) are obtained with dark time
$T=1000\times\frac{2\pi}{\Gamma}$ as well as the pulse sequences of (b)
$\tau_{1}=60\times\frac{2\pi}{\Gamma}$ and
$\tau_{2}=60\times\frac{2\pi}{\Gamma}$, (c)
$\tau_{1}=200\times\frac{2\pi}{\Gamma}$ and
$\tau_{2}=60\times\frac{2\pi}{\Gamma}$, (d)
$\tau_{1}=200\times\frac{2\pi}{\Gamma}$ and
$\tau_{2}=10\times\frac{2\pi}{\Gamma}$,
(e)$\tau_{1}=1000\times\frac{2\pi}{\Gamma}$ and
$\tau_{2}=10\times\frac{2\pi}{\Gamma}$.

For a two-pulse CPT-Ramsey interferometry, the CPT-Ramsey interference term
reads

$Q=Q^{T}_{1}+Q^{T}_{2},$ (19)

where

$\displaystyle Q_{1}^{T}=$
$\displaystyle\exp\left(-\frac{\Omega^{2}}{\Gamma}\tau_{2}\right)\left(1-\exp\left(-\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)\tau_{1}\right)\right)$
$\displaystyle\times\exp\left(-\mathrm{i}\delta\left(T+\tau_{2}\right)\right),$

and

$Q_{2}^{T}=\left(1-\exp\left(-\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)\tau_{2}\right)\right).$

The conventional CPT-Ramsey fringe is mainly dominated by $Q^{T}_{1}$ in Eq.
(19). It increases with the preparation pulse duration $\tau_{1}$ and
decreases with the detection pulse duration $\tau_{2}$. $Q^{T}_{2}$ in Eq.
(19) increases with the detection pulse $\tau_{2}$, which contributes a trend
of slow variance, resulting in the vertical asymmetry Zanon-Willette  _et al._
(2005).

As a benchmark example, we examine the conventional CPT-Ramsey pulse sequence
consisting of a preparation pulse of duration $\tau_{1}$, a detection pulse of
duration $\tau_{2}$, and the pulse interval $T=1000\times\frac{2\pi}{\Gamma}$,
as illustrated in Fig. 2 (a). The Rabi frequency of the CPT pulse is
$\Omega=0.035\Gamma$. For a detection pulse duration of
$\tau_{2}=60\times\frac{2\pi}{\Gamma}$, if the preparation pulse duration is
short, $\tau_{1}=60\times\frac{2\pi}{\Gamma}$, the CPT-Ramsey spectrum
contrast is low, as shown in Fig. 2 (b). The black solid line is the
analytical result of Eq. (19) and blue dots are the numerical result of
Lindblad equations. As the preparation pulse duration $\tau_{1}$ increases to
$\tau_{1}=200\times\frac{2\pi}{\Gamma}$, the $Q^{T}_{1}$ of Eq. (19) grows,
improving the contrast of the CPT-Ramsey fringe, see Fig. 2 (c). Meanwhile,
reducing the detection pulse duration to
$\tau_{2}=10\times\frac{2\pi}{\Gamma}$ further improves the contrast of the
CPT-Ramsey fringe and the spectrum becomes vertically symmetric, as shown in
Fig. 2 (d). When the preparation pulse duration
$\tau_{1}=1000\times\frac{2\pi}{\Gamma}$ is sufficient, the spectrum will
reach a saturation point, see Fig. 2 (e). As the period in frequency space of
the $Q^{T}_{1}$ is $\frac{2\pi}{T+\tau_{2}}$. The linewidth of the CPT-Ramsey
spectrum satisfies $\Delta\nu=\frac{1}{2(T+\tau_{2})}$. Thus, the CPT-Ramsey
with a long dark time $T$ can narrow the spectrum linewidth, which is
consistent with our common knowledge. These results indicate that our
analytical formula is valid for conventional two-pulse CPT-Ramsey scenarios.

####  III.1.2 Anti-symmetric Two-pulse CPT-Ramsey Spectroscopy with Frequency
shift

Usually, we need to perform two CPT-Ramsey interferometry with different
frequencies and compare their differences to obtain an antisymmetric error
signal for clock locking Yun  _et al._ .

Figure 3: Two-pulse CPT-Ramsey interferometry with frequency and phase shifts.
(a) The timing sequence of Rabi frequency $\Omega$ and detuning $\delta$ of
two-pulse CPT-Ramsey sequences with time $T=600\times\frac{2\pi}{\Gamma}$. The
amplitude of Rabi frequency is $0.035\Gamma$, the $\tau_{1}$ and $\tau_{2}$
are preparation and detection durations respectively, the $\Delta\delta$ and
$\Delta t=100\times\frac{2\pi}{\Gamma}$ are the magnitude and duration of
frequency shift. The $\Phi_{1}$ and $\Phi_{2}$ are the phases of preparation
and detection pulses, respectively. The CPT-Ramsey spectra of numerical
simulation (blue dots) and analytical formula (black solid line) with
$\tau_{1}=1000\times\frac{2\pi}{\Gamma}$,
$\tau_{2}=10\times\frac{2\pi}{\Gamma}$ as well as (b) zero frequency shift
$\Delta\delta=0$ and $\Phi_{1}=\Phi_{2}$, (c) frequency shift
$\Delta\delta=\frac{\pi/2}{\Delta t}$ and $\Phi_{1}=\Phi_{2}$, (d) zero
frequency shift $\Delta\delta=0$, $\Phi_{1}=0$ and $\Phi_{2}=-\frac{\pi}{2}$.
(e) The CPT-Ramsey spectra of numerical simulation (blue dots) and analytical
formula (black solid line) with $\tau_{1}=60\times\frac{2\pi}{\Gamma}$,
$\tau_{2}=60\times\frac{2\pi}{\Gamma}$ as well as zero frequency shift
$\Delta\delta=0$, $\Phi_{1}=0$ and $\Phi_{2}=-\frac{\pi}{2}$.

To achieve real-time antisymmetric spectra, we can directly apply a frequency
shift of $\Delta\delta$ during the dark time $T$ or implement a change in
phase $\Delta\Phi=\Phi_{2}-\Phi_{1}$ for the detection pulse based on
conventional CPT-Ramsey interferometry, as illustrated in Fig. 3 (a). Here,
$\Phi_{1}$ and $\Phi_{2}$ are the phases of the preparation pulse and
detection pulse, respectively. According to Eq. (13), the two-pulse CPT-Ramsey
interference term becomes

$Q=Q^{T\prime}_{1}+Q^{T}_{2},$ (20)

where

$Q^{T\prime}_{1}=Q^{T}_{1}\exp\left(-\mathrm{i}\Delta\delta\Delta
t+\mathrm{i}\Delta\Phi\right).$ (21)

As an example, we set $\tau_{1}=1000\times\frac{2\pi}{\Gamma}$,
$\tau_{2}=10\times\frac{2\pi}{\Gamma}$, and $T=1000\times\frac{2\pi}{\Gamma}$.
When the two phases are equal, i.e., $\Phi_{1}=\Phi_{2}$ and $\Delta\Phi=0$,
then it reduces to the conventional CPT-Ramsey interferometry as shown in Fig.
3 (b). While the spectra will become horizontal antisymmetric if the frequency
shift and phase shift satisfy $\Delta\delta\Delta t-\Delta\Phi=\frac{\pi}{2}$,
as shown in Fig. 3 (c) and (d). Then, the real-time processing of error
signals will be obtained. In auto-balance CPT-Ramsey interferometry Abdel
Hafiz  _et al._ (2018b), the $\Delta\Phi$ or $\Delta\delta$ can be used as the
additional parameters to compensate for the light shift. However, if
$\tau_{1}=60\times\frac{2\pi}{\Gamma}$ is short and
$\tau_{2}=60\times\frac{2\pi}{\Gamma}$ is substantial, the error signals will
not be horizontally antisymmetric as $Q^{T}_{2}$ is considerable compared to
$Q^{T\prime}_{1}$. It means that if the detection pulse duration is comparable
with the preparation pulse duration, the spectrum will become horizontally
asymmetric, as shown in Fig. 3 (e). The black solid lines are the analytical
result of Eq. (20) and the blue dots are the corresponding numerical results.
All the results match perfectly.

###  III.2 Multi-pulse sequence

In Sec. III.1, we analyzed conventional and frequency-shifted two-pulse CPT-
Ramsey spectra using our analytical formula. In this section, we will analyze
multi-pulse CPT-Ramsey interferometry. With a multi-pulse sequence, the
central peak becomes obvious due to constructive interference while the
neighboring peaks are suppressed through destructive interference. The multi-
pulse CPT-Ramsey interferometry makes the central peak easy to be identified
and makes the signal-to-noise ratio (SNR) better. Thus, the multi-pulse
sequence is useful for developing practical quantum sensors, such as atomic
clocks. However, for practical applications, the multi-pulse CPT-Ramsey
interferometry involves multiple pulses which need to be sophisticatedly
tuned. Our analytical formula provides a simple way to analyze and optimize
the pulse sequence as desired. We consider that the multi-pulse CPT Ramsey
interferometry starts with a preparation pulse $\tau_{1}$ to prepare the dark
state, followed by $N$ pulses of duration $\tau$ with pulse interval $T$, as
shown in Fig. 4 (a). By using our analytical formula, below we analyze the
roles of preparation pulse and periodic pulse sequence and provide an example
to achieve the anti-symmetric spectrum with frequency shift.

####  III.2.1 The influence of preparation pulse

Figure 4: Multi-pulse CPT-Ramsey interferometry. (a) The timing sequence of
Rabi frequency $\Omega$ of multi-pulse CPT-Ramsey sequences. Here, $\tau_{1}$
is the preparation pulse,$T=500\times\frac{2\pi}{\Gamma}$ is the dark time,
$\tau=10\times\frac{2\pi}{\Gamma}$ is the pulse duration and $N$ is the pulse
number. The multi-pulse CPT-Ramsey spectra with (b)
$\tau_{1}=60\times\frac{2\pi}{\Gamma}$ and $N=8$, (c)
$\tau_{1}=200\times\frac{2\pi}{\Gamma}$ and $N=8$, (d)
$\tau_{1}=60\times\frac{2\pi}{\Gamma}$ and $N=40$, (e)
$\tau_{1}=200\times\frac{2\pi}{\Gamma}$ and $N=40$.

We consider the phases of all CPT pulses are identical and so that we can set
the phases as zero. According to Eq. (10), the interference term

$Q=Q^{M}_{1}+Q^{M}_{2},$ (22)

includes two parts,

$\displaystyle Q^{M}_{1}=$
$\displaystyle\exp\left(-\frac{\Omega^{2}}{\Gamma}N\tau\right)\exp\left(-\mathrm{i}\delta
N\left(T+\tau\right)\right)$
$\displaystyle\times\left(1-\exp\left(-\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)\tau_{1}\right)\right),$

and

$\displaystyle Q^{M}_{2}=$
$\displaystyle\sum_{i=1}^{N}\exp\left(-\frac{\Omega^{2}}{\Gamma}(N-i)\tau\right)\exp\left(-\mathrm{i}\delta(N-i)\left(T+\tau\right)\right)$
$\displaystyle\times\left(1-\exp\left(-\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)\tau\right)\right).$

$Q^{M}_{1}$ is the fast oscillating term versus $\delta$. Obviously, the
preparation pulse duration $\tau_{1}$ only affects $Q^{M}_{1}$ and $Q^{M}_{1}$
increases with $\tau_{1}$. Therefore the amplitude of the spectrum increases
with the preparation pulse duration. As shown in Fig. 4 (b) and (c), the
spectra of multi-pulse CPT-Ramsey interferometry with longer preparation pulse
duration $\tau_{1}=200\times\frac{2\pi}{\Gamma}$ has a higher amplitude of the
peaks than that with shorter preparation pulse
$\tau_{1}=60\times\frac{2\pi}{\Gamma}$.

However, the subsequent CPT pulses also affect $Q^{M}_{1}$. The more or longer
subsequent CPT pulses, the smaller the $Q^{M}_{1}$ is. Hence, when the number
$N$ or the duration $\tau$ of the subsequent CPT pulses is large, the duration
of the preparation pulse $\tau_{1}$ has little impact on the spectra. As shown
in Fig. 4 (d) and (e), the preparation pulse has little influence on the
spectra of multi-pulse CPT-Ramsey interferometry when the pulse number $N$ is
large.

####  III.2.2 Periodic pulse sequence

With a large amount of pulse number $N$, the influence of the first CPT pulse
duration can be neglected. For simplicity, many experiments use periodic
multi-pulse sequences Yun  _et al._ . Under a periodic CPT pulse sequence with
interval $T$, pulse number $N$, pulse duration $\tau$, and Rabi
frequency$\Omega$, according to Eq. (13), we have

$\rho_{12}(\delta)=f(\delta)\sum_{l=0}^{N-1}\mathcal{R}^{l}\exp\left(-\mathrm{i}l\delta
T\right)\mathcal{T}.$ (23)

This is a temporally analog to light passing through the Fabry-Pérot resonator
Fang  _et al._ (2021), in which
$\mathcal{R}\equiv\exp\left(-\frac{\Omega^{2}}{\Gamma}\tau\right)$ takes the
role the reflection coefficient and $\mathcal{T}\equiv
1-\exp\left(-\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)\tau\right)$
corresponds to the transmission coefficient. Using the series summation, Eq.
(23) can be simplified as

$\rho_{12}(\delta)=f(\delta)\mathcal{T}S,$ (24)

with

$S=\frac{1-R^{N}\exp(-\mathrm{i}N\delta T)}{1-R\exp(-\mathrm{i}\delta T)}.$
(25)

For a multi-pulse CPT-Ramsey interferometry, $f(\delta)$ and $\mathcal{T}$ are
flatter than $S$ near the resonance point. The lineshape of $\rho_{12}$ is
mainly described by $S$, as shown in Fig. 5. If the pulse number $N\to\infty$,
the full width at half maximum (FWHM) $2\pi\Delta\nu_{\infty}$ of
$\Re{\rho_{12}}$ is determined by the reflection coefficient $\mathcal{R}$ and
the pulse period $T_{p}=T+\tau$, which is the Airy distribution Ismail  _et
al._ (2016),

$2\pi\Delta\nu_{\infty}=\frac{4}{T_{p}}\arcsin\left(\sqrt{\frac{\left(1-\mathcal{R}\right)^{2}}{2\left(\mathcal{R}^{2}+1\right)}}\right).$
(26)

Eq. (26) is valid in the saturation region of $\mathcal{R}^{N}\ll 1$.

For a finite pulse number, the FWHM of Eq. (24) cannot be exactly given
Pissadakis (2000). However, we can calculate the Lorenztian linewidth through
the Taylor expansion for the $\frac{1}{\mathrm{Re}(S)}$,

$\frac{1}{\mathrm{Re}\left(S\right)}=F_{0}+\frac{1}{2}F_{2}T_{p}^{2}\delta^{2}+\mathcal{O}\left(T^{4}\delta^{4}\right),$
(27)

with

$F_{0}=\frac{1-\mathcal{R}}{1-\mathcal{R}^{N}},$ (28)

and

$F_{2}=\frac{\mathcal{R}\left(1+\mathcal{R}\right)}{\left(1-\mathcal{R}\right)\left(1-\mathcal{R}^{N}\right)}-\frac{N^{2}\mathcal{\mathcal{R}}^{N}\left(1-\mathcal{\mathcal{R}}\right)+2N\mathcal{R}^{N+1}}{\left(1-\mathcal{R}^{N}\right)^{2}}.$
(29)

The real part of $S$ can be approximated as the Lorentzian form

$\mathrm{Re}(S)\approx\widetilde{S}=A\frac{\left(\frac{1}{2}W\right)^{2}}{\left(\frac{1}{2}W\right)^{2}+\delta^{2}}.$
(30)

Here, $A=\frac{1}{F_{0}}$ and
$W=\frac{2\sqrt{2}}{T}\sqrt{\frac{F_{0}}{F_{2}}}$ are the amplitude and FWHM
of $\widetilde{S}$. Substituting Eq. (28) and Eq. (29), we get that

$\displaystyle
W=\frac{2}{T_{p}}\left[\frac{2\left(\mathcal{R}^{2}+\mathcal{R}\right)}{\left(1-\mathcal{R}\right)^{2}}-\frac{2N^{2}\mathcal{R}^{N}}{1-\mathcal{R}^{N}}-\frac{4N\mathcal{R}^{N+1}}{\left(1-\mathcal{R}^{N}\right)\left(1-\mathcal{R}\right)}\right]^{-\frac{1}{2}}.$
(31)

Figure 5: The linewidth of multi-pulse CPT-Ramsey spectrum. (a) The linewidth
$W$ versus the pulse number $N$. (b) The linewidth $W$ versus the reflection
coefficient $\mathcal{R}$. (c) The numerical results of FWHM versus the pulse
number $N$ and the pulse duration $\tau$ with the Rabi frequency
$\Omega=0.04\Gamma$. (d) The numerical results of FWHM versus the pulse number
$N$ and the Rabi frequency $\Omega$ with the pulse duration
$\tau=10\times\frac{2\pi}{\Gamma}$.

In Fig. 5 (a) we show $W$ versus the pulse number $N$, where $W$ decreases
with the pulse number $N$. Intuitively, more pulses will lead to a narrower
linewidth. Since $\mathcal{R}$ decreases with both $\tau$ and $\Omega$, larger
$\Omega$ and $\tau$ will result in a larger linewidth. Larger values of
$\Omega$ and $\tau$ mean fewer pulses needed to reach saturation and fewer
pulses contributing to multi-pulse interference, therefore the linewidth
becomes broader. For illustration, Fig. 5 (c) shows the change of FWHM with
the pulse number $N$ and the pulse duration $\tau$ when $\Omega=0.04\Gamma$.
While Fig. 5 (d) displays the change of FWHM versus the pulse number $N$ and
the Rabi frequency $\Omega$ when $\tau=10\times\frac{2\pi}{\Gamma}$.

####  III.2.3 Anti-symmetric multi-pulse CPT-Ramsey spectroscopy with
frequency shift

Figure 6: Multi-pulse CPT-Ramsey interferometry with frequency shift. (a) The
timing sequence of Rabi frequency $\Omega$ with a frequency shift applied
right before the $k$-th subsequent pulse.
$\tau_{1}=200\times\frac{2\pi}{\Gamma}$ is the preparation time,
$T=500\times\frac{2\pi}{\Gamma}$ is the dark time,
$\tau=10\times\frac{2\pi}{\Gamma}$ and $N=40$ are respectively the duration
and number of the applied periodic pulses. $\Delta\nu=0.0025\Gamma$ and
$\Delta t=100\times\frac{2\pi}{\Gamma}$ are the magnitude and duration of
frequency shift. The multi-pulse CPT-Ramsey spectra with the frequency shift
happen (b) right after the preparation pulse $\tau_{1}$, (c) right before the
last $10$-th pulse, (d) right before the last third pulse, (e) right before
the last pulse.

As mentioned in Sec.III.1.2, one may prefer to use an anti-symmetric spectrum
for frequency locking. In a multi-pulse CPT-Ramsey interferometry, applying a
frequency shift $\Delta\delta$ for a duration of $\Delta t$ before the $k$-th
subsequent pulse [as illustrated in Fig. 6 (a)], or introducing a phase jump
$\Delta\Phi$, can alter the spectrum horizontal symmetry to be anti-symmetric.
However, how to determine pulse sequence with frequency shift or phase jump is
still challenging. Here, we use our analytical formula to address this issue.

Taking the frequency shift and the phase jump before the $k$-th subsequent
pulse, we can divide the interference term into three parts

$Q=Q^{M\prime}_{1}+Q^{M\prime}_{2}+Q^{M\prime}_{3},$ (32)

where

$\displaystyle Q^{M\prime}_{1}=$
$\displaystyle\exp\left(-\frac{\Omega^{2}}{\Gamma}N\tau\right)\left(1-\exp\left(-\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)\tau_{1}\right)\right)$
$\displaystyle\times\exp\left(-\mathrm{i}\delta
N\left(T+\tau\right)\right)\exp\left(-\mathrm{i}\Delta\delta\Delta
t+\mathrm{i}\Delta\Phi\right),$ $\displaystyle Q^{M\prime}_{2}=$
$\displaystyle\sum_{i=1}^{k-1}\exp\left(-\frac{\Omega^{2}}{\Gamma}(N-i)\tau\right)$
$\displaystyle\times\left(1-\exp\left(-\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)\tau\right)\right)$
$\displaystyle\times\exp\left(-\mathrm{i}\delta(N-i)\left(T+\tau\right)\right)\exp\left(-\mathrm{i}\Delta\delta\Delta
t+\mathrm{i}\Delta\Phi\right),$

and

$\displaystyle Q^{M\prime}_{3}=$
$\displaystyle\sum_{i=k}^{N}\exp\left(-\frac{\Omega^{2}}{\Gamma}(N-i)\tau\right)$
$\displaystyle\times\left(1-\exp\left(-\left(\frac{\Omega^{2}}{\Gamma}+\mathrm{i}\delta\right)\tau\right)\right)$
$\displaystyle\times\exp\left(-\mathrm{i}\delta(N-i)\left(T+\tau\right)\right).$

The frequency shift $\Delta\delta$ affects the horizontal symmetry of
$Q^{M\prime}_{1}$ and $Q^{M\prime}_{2}$, but $Q^{M\prime}_{3}$ remains
horizontally symmetric. If the frequency shift satisfies $\Delta\delta\Delta
t=\pi/2$, it will adjust $Q^{M\prime}_{1}$ and $Q^{M\prime}_{2}$ from
symmetric into antisymmetric in the horizontal direction. To obtain a
horizontally antisymmetric spectrum, the contribution of the horizontally
symmetric term $Q^{M\prime}_{3}$ should be small. Thus, it is a natural choice
to apply frequency shift right before the last pulse to suppress
$Q^{M\prime}_{3}$. In Fig. 6 (b)-(e), we applied the frequency shifts with
$\Delta\delta\Delta t=\pi/2$ after the preparation pulse $\tau_{1}$, before
the last $10$-th pulse, before the last third pulse, and before the last
pulse, respectively. The preparation pulse is
$\tau_{1}=200\times\frac{2\pi}{\Gamma}$ and the pulse interval is
$T=500\times\frac{2\pi}{\Gamma}$. The duration
$\tau=10\times\frac{2\pi}{\Gamma}$ and the pulse number $N=40$. Clearly, as
the delay of frequency shifts, $Q^{M\prime}_{3}$ decreases and the spectrum
tends to become antisymmetric. Thus, our analytic analysis can provide a
straightforward way to design the multi-pulse sequence for CPT-Ramsey
interferometry, which should be beneficial for developing high-accuracy
schemes such as auto-balanced Ramsey spectroscopy Abdel Hafiz  _et al._
(2018a, b); Yudin  _et al._ (2018).

##  IV Discussion

In conclusion, starting from the Lindblad equation, we derive an analytical
formula to describe the multi-pulse CPT-Ramsey interferometry with arbitrary
pulse sequence. The analytical formula can potentially optimize the pulse
sequence and analyze the influence of time-dependent detuning. We illustrate
the validity of the analytical result with the popular CPT-Ramsey scenarios
and obtain the sort of views.

For a two-pulse CPT-Ramsey interferometry, we study the influence of the
preparation and the detection pulse. We quantitatively show that the
preparation pulse should as long as possible to gain a larger spectrum
amplitude, and the detection pulse should be small to avoid destroying the CPT
coherence. The frequency shift or phase jump will change the spectrum
symmetry. The analytical results show that a long preparation pulse and a
small detection pulse are required to obtain the antisymmetric spectrum. For
multi-pulse cases, the role of preparation pulses becomes less significant as
the number of subsequent pulses increases. As the number of pulses increases,
the side peaks are continuously destroyed by interference, and we obtain a
high-contrast central peak. By adding a frequency shift or phase jump right
before the last pulse, we obtain an antisymmetric multi-pulse CPT-Ramsey
spectrum. Our theoretical results can be applied to design novel multi-pulse
and frequency modulated CPT-Ramsey schemes. Under the condition of the period
pulse sequence, we find out the approximate Lorenztian line shape of the
spectrum and get the relationship between FWHM and the parameters of the CPT
pulse sequence. It quantitatively grapes the role of multi-pulse interference.

For the multi-pulse and frequency regulation CPT-Ramsey interferometry, there
are many potential applications such as CPT clock and CPT magnetometers.
Effective optimization methods are conducive to efficiently improving the
measurement accuracy. In this work, we eliminate the one-photon detunings and
the effect of light shift is ignored. However, the real systems always contain
many states, deriving light shift from a three-level system is not meaningful.
As our analytical results can describe the situations of time-dependent
detuning, we can also analyze the impact of CPT pulses on the present light
shift by adding the light shift into detuning. These are deserved for further
investigations.

###### Acknowledgements.

This work is supported by the National Key Research and Development Program of
China (Grant No. 2022YFA1404104), the National Natural Science Foundation of
China (Grant No. 12025509), and the Key-Area Research and Development Program
of GuangDong Province (Grant No. 2019B030330001).

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