# An inhibited laser

Tiantian Shi1, Duo Pan1 & Jingbiao Chen1,∗***E-mail<EMAIL_ADDRESS>1State
Key Laboratory of Advanced Optical Communication Systems and Networks,
Institute of Quantum Electronics, School of Electronics, Peking University,
Beijing 100871, China

###### Abstract

Traditional lasers function using resonant cavities, in which the round-trip
optical path is exactly equal to an integer multiple of the intracavity
wavelengths to constructively enhance the spontaneous emission rate. By taking
advantage of the enhancement from the resonant cavity, the narrowest
sub-10-mHz-linewidth laser and a 10-16-fractional-frequency-stability
superradiant active optical clock (AOC) have been achieved. However, a laser
with atomic spontaneous radiation being destructively inhibited in an anti-
resonant cavity, where the atomic resonance is exactly between two adjacent
cavity resonances, has not been reported. Herein, we experimentally
demonstrate inhibited stimulated emission and termed it an inhibited laser.
Compared with traditional superradiant AOCs, which exhibit superiority in
terms of the high suppression of cavity noise, the suppression of the cavity-
pulling effect of an inhibited laser can be further improved by a factor of
${\left({{{{\text{2}}\mathcal{F}}\mathord{\left/{\vphantom{{{\text{2}}\mathcal{F}}\pi}}\right.\kern-1.2pt}\pi}}\right)^{2}}$,
i.e., 2.07 in this work, which was improved from 26 to 53 times. This study
will guide further development of AOCs with better stability, and thus, it is
significant for quantum metrology and may lead to new research in the laser
physics and cavity quantum electrodynamics fields.

## Introduction

The significantly enhanced spontaneous decay rate of the spin in a resonant
circuit, known as the Purcell effect Purcell1995Spontaneous , was first
reported by Purcell in 1946. It was practically observed in the 1980s using
atoms in resonant cavities both in the microwave Goy1983Observation  and
optical Heinzen1987Enhanced ; Martini1987Anomalous  domains. The enhanced
spontaneous radiation has important application potential in cavity quantum
electrodynamics (QED) Mabuchi2002Cavity ; Wallraff2004Strong ;
YoshieVacuum2004 , including for one-atom lasers McKeeverExperimental2003 ,
ion-trap lasers MatthiasExperimental2004 , and quantum logic gates
MonroeDemonstration1995  in quantum computers.

Essentially, the resonant cavity, whose cavity-mode frequency resonates with
the peak of the emission line for atomic transition, enhances the strength of
vacuum fluctuations, which promotes the atomic spontaneous radiation.
Conversely, the spontaneous decay rate is suppressed when the cavity is off
resonance, which was first proposed by Kleppner in 1981 and demonstrated
through inhibited blackbody absorption VaidyanathanInhibited1981  and
inhibited spontaneous emission Kleppner1981Inhibited . After, inhibited
spontaneous emission was experimentally demonstrated in microwave and optical
cavities in 1985 Hulet1985Inhibited  and 1987 Heinzen1987Enhanced ,
respectively. Heinzen Heinzen1987Enhanced  pointed out that in an anti-
resonant cavity, where the atomic frequency was exactly at the center of two
adjacent cavity resonances, the inhibition of the atomic spontaneous decay
rate was the greatest. More strikingly, through coupling with an anti-resonant
cavity, the atomic radiative level shift vanished and the spectral linewidth
narrowed Heinzen1987Vacuum , which is potentially useful for precision
measurements. Despite the experimental success of inhibited spontaneous
emission, it remains to be explored whether laser oscillations can be realized
when the spontaneous emission rate is suppressed to the greatest extent in an
anti-resonant cavity. The characteristics of an inhibited laser are
alternative to explore in depth for rich development in the fields of laser
physics.

Nevertheless, the demonstration of inhibited spontaneous emissions has
provided credible evidence for the observation of inhibited stimulated
emissions. The spontaneous emission can be viewed as a stimulated emission
originating from the vacuum fluctuations, and the spontaneous emission below
the threshold determines the spectrum of the laser above the threshold
Loudon2000The . It has significant potential to achieve inhibited lasing, with
the aid of a three- or four-level structure to increase the pumping efficiency
and the multi-atom system to reach the strong-coupling regime Fox2007Quantum .

Different from the traditional types of lasers working in the resonant-cavity
region, in this work, we propose a laser operating in the antiresonant-cavity
regime, where the atomic gain line is located exactly at the center of two
adjacent cavity resonances, which is termed an inhibited laser. The
characteristics of the inhibited laser, such as the intracavity photon
lifetime and primary laser power behavior, laser linewidth, and cavity-pulling
effect, under the conditions of an anti-resonant cavity, were proven
experimentally and theoretically. In particular, we proved that the frequency
shift of the laser oscillation vanished in an inhibited laser with reduced
sensitivity to the cavity-length fluctuations. Compared with resonant active-
optical-clock (AOC) lasing Chen2009Active ; Bohnet2012A ; PhysRevA.87.013821 ;
PhysRevLett.123.103601 ; Stefan2020PhysRevA ; PRL2020Holland , the influence
of the thermal cavity-length noise on the frequency of an inhibited laser is
further suppressed by a factor of
${\left({{{{\text{2}}\mathcal{F}}\mathord{\left/{\vphantom{{{\text{2}}\mathcal{F}}\pi}}\right.\kern-1.2pt}\pi}}\right)^{2}}$.

## Results

Experimental scheme. Here, we report an experimental demonstration of an
inhibited laser. The general setup is depicted schematically in Fig. 1,
sharing similarities with the proposed superradiant AOC based on thermal atoms
Shi2019OE . $N\approx 1.8\times{10^{11}}$ pure cesium (Cs) atoms are confined
to the TEM00 mode of a low-finesse optical cavity ($\mathcal{F}=3.07$), whose
dissipation rate is $\kappa_{0}=2{\rm{\pi}}\times 257$ MHz. Pumped by a 459 nm
laser (6S1/2-7P1/2), the atoms achieve stimulated emissions at a wavelength of
1470 nm (7S1/2-6P3/2). The relaxation rate of the atomic dipole
$\Gamma=2{\rm{\pi}}\times 10.04$ MHz is much smaller than $\kappa_{0}$.
Therefore, the laser works in a bad-cavity regime Chen2009Active ; Bohnet2012A
. Unlike traditional resonant lasers, the inhibited laser is realized with a
round-trip optical path equal to odd multiples of the half wavelength
$2L=\left({2q+1}\right){\lambda\mathord{\left/{\vphantom{\lambda
2}}\right.\kern-1.2pt}2}$. Information about the experimental details, as well
as a discussion about the working regime of the laser, is provided in the
Methods Sections I and II.

Figure 1: Working principle of the inhibited laser. a, Level scheme for 133Cs
atom showing the 459 nm transition used for pumping and the 1470 nm transition
as the inhibited lasing. Cs atoms are pumped by a 459 nm laser, which drives
the $\left|1\right\rangle$ ($6{{\rm S}_{1/2}}$) to $\left|2\right\rangle$
($7{{\rm P}_{1/2}}$) transition. Atoms are then transferred to the
$\left|3\right\rangle$ ($7{{\rm S}_{1/2}}$) state through spontaneous
radiation. Utilizing weak feedback of the cavity, population inversion is
built up between the $\left|3\right\rangle$ and $\left|4\right\rangle$
($6{{\rm P}_{3/2}}$) levels, and finally, 1470 nm lasing is realized. The
inhibited laser is realized in an anti-resonant cavity, where the atomic gain
line is exactly at the center of two adjacent cavity resonances. b, Sketch of
the inhibited laser. Cs atoms (blue spheres) along the direction of the cavity
mode are pumped by the 459 nm laser. Two cavity mirrors are coated with
reflectivities of ${{R}_{1}}={{R}_{2}}=34.5\%$ at 1470 nm. To change the
cavity-mode frequency, the location of the output mirror is tunable.
Generally, the round-trip optical path is exactly equal to an integer multiple
of the intracavity wavelengths, and the output mirror is located at
M${}_{2}^{{}^{\prime}}$. However, an inhibited laser (in red) is achieved when
the round-trip optical path is equal to an odd multiple of the half wavelength
$2L=(2q+1)\lambda/2$ with the output mirror being located at M2.

Enhanced and inhibited factors. Suppose that the atom emitting the first
photon by spontaneous radiation is located at the center of cavity, and the
reflectivities of cavity mirrors are ${R}_{1}={R}_{2}={R}$, the ratio of the
power of spontaneous radiation emitted into cavity, ${P_{\rm{c}}}$, to the
power into free space, ${P_{\rm{free}}}$, is given by Heinzen1987Vacuum

$\frac{{{P_{\rm{c}}}}}{{{P_{{\rm{free}}}}}}=\frac{{1-{R^{2}}}}{{1+{R^{2}}-2R\cos\left({{{\omega
2L}\mathord{\left/{\vphantom{{\omega
2L}{\rm{c}}}}\right.\kern-1.2pt}{\rm{c}}}}\right)}},$ (1)

where $\omega$ is the angular frequency of the radiation, and c the speed of
light. $\Delta\phi={{\omega 2L}\mathord{\left/{\vphantom{{\omega
2L}{\rm{c}}}}\right.\kern-1.2pt}{\rm{c}}}$ denotes the phase shift of the
intracavity reflected field, and it also indicates the detuning of the cavity
frequency ${\omega_{\rm{c}}}$ from the atomic resonance $\omega_{0}$. A phase
shift of ${\rm{2\pi}}$ between two consecutive round trips of the radiation
inside the cavity corresponds to the cavity-frequency detuning
${\omega_{\rm{c}}}-\omega_{0}$ of one free spectral range (FSR). According to
Eq. (1), the spontaneous decay rate from the atomic excited state is enhanced
and inhibited by a factor of $\frac{{1+R}}{{1-R}}$ compared with that in free
space when the cavity is resonant ($\Delta\phi={\rm{2}}\pi q$) and anti-
resonant ($\Delta\phi=\left({2q+1}\right){\rm{\pi}}$), respectively.
Accordingly, the suppression of the spontaneous emission rate induced by the
anti-resonant cavity is weak in the low-reflectivity case, which is conducive
to realize inhibited lasing.

Figure 2: Intracavity photon lifetime and primary laser power behavior. a,
Intracavity photon lifetime $\tau$ (black dots) is inversely proportional to
the loss coefficient $\eta$ described by Eq. (3). The experimental (light-blue
line) and simulated (dark-blue line) results of the output laser power
$P_{\rm{out}}$ match well under the condition of
$g=1.99\times{10^{5}}\,{\rm{s}^{-1}}$,
$\Omega=4.30\times{10^{7}}\,{\rm{s}^{-1}}$, and
$N_{\rm{eff}}=5.71\times{10^{9}}$. b, Output laser power $P_{\rm{out}}$ as a
function of the phase shift $\Delta\phi$ at different pumping light
intensities $I$ under vapor-cell temperature $T=100^{\circ}$C. The blue and
orange lines represent $P_{\rm{out}}$ vs. the pumping light intensity when the
cavity is on-resonant and anti-resonant, respectively. The blue circles and
the orange stars are the corresponding experimental results. The pumping light
intensities reaching the laser threshold were around 0.45 and 5.20 mW/mm2 when
the cavity was resonant and anti-resonant, respectively. c, $P_{\rm{out}}$ as
a function of $\Delta\phi$ at different vapor-cell temperatures $T$ under
$I=10$ mW/mm2. Theoretical and experimental $P_{\rm{out}}$ vs. $T$ when the
cavity is resonant (green) and anti-resonant (purple) are separately depicted
by the lines and dots, respectively. The corresponding temperatures for the
laser threshold were 72.5∘C and 94.5∘C, respectively.

Power characteristic of inhibited laser. The detuning,
$\Delta={\omega}-{\omega_{0}}$, of the radiation frequency $\omega$ from the
atomic transition frequency $\omega_{0}$, can also be given by
$\Delta=P\left({{\omega_{\rm{c}}}-{\omega_{\rm{0}}}}\right)$, where
$P\equiv{{d\omega}\mathord{\left/{\vphantom{{d\omega}{d{\omega_{\rm{c}}}}}}\right.\kern-1.2pt}{d{\omega_{\rm{c}}}}}$
represents the cavity-pulling coefficient Chen2009Active ; Bohnet2012A . In
the bad-cavity limit,
$P\approx{\Gamma\mathord{\left/{\vphantom{\Gamma{{\kappa_{0}}}}}\right.\kern-1.2pt}{{\kappa_{0}}}}\ll
1$ if the cavity is near resonant, and thus, ${\Delta^{2}}\ll
4{g^{2}}\left({n+1}\right)$ in this work. $n$ is the intracavity photon
number, and the atom-cavity coupling constant PRA1997Kyungwon
$g={\textstyle{\mu\over\hbar}}\sqrt{\frac{{\hbar{\omega_{0}}}}{{2{\varepsilon_{0}}{V_{\rm{c}}}}}}=1.99\times{10^{5}}{\mkern
1.0mu}$ s-1, where $\mu$ is the electric dipole moment, $\varepsilon_{0}$ is
the vacuum permittivity, and $V_{\rm{c}}$ is the equivalent mode volume.
Consequently, the detuning $\Delta$ in the laser rate equation
Christopher2002Multiple  is negligible (the exact calculations are given in
the Methods Section III). Here, we modify the loss term in the classical laser
rate equation to obtain a universal expression, which can be used to describe
any cavity-frequency detuning condition, as follows:

$\displaystyle\frac{{dn}}{{dt}}$
$\displaystyle={N_{\rm{eff}}}\frac{{{\rho_{33}}-{\rho_{44}}}}{{{\tau_{{\rm{cyc}}}}}}{\sin^{2}}\left({\sqrt{n+1}g{t_{{\rm{int}}}}}\right)-\frac{n}{\tau}$
$\displaystyle={N_{\rm{eff}}}\frac{{{\rho_{33}}-{\rho_{44}}}}{{{\tau_{{\rm{cyc}}}}}}{\sin^{2}}\left({\sqrt{n+1}g{t_{{\rm{int}}}}}\right)-\eta\kappa_{0}n.$
(2)

On the right-hand side, the first term represents the gain, and the second
term is the loss. For the gain term, the effective number of atoms that can be
pumped to the 7P1/2 state is $N_{\rm{eff}}=5.71\times{10^{9}}$ with a pumping
light intensity of $I=10$ mW/mm2 and vapor-cell temperature of
$T=100^{\circ}\rm{C}$. ${\rho_{ii}}$ denotes the population probability at
level $\left|i\right\rangle$ in Fig. 1a. ${\tau_{\rm{cyc}}}=28.0$ $\mu$s is
the cycle time for Cs atoms through a transition of $6{{\rm S}_{1/2}}\to
7{{\rm P}_{1/2}}\to 7{{\rm S}_{1/2}}\to 6{{\rm P}_{3/2}}$.
${t_{{\rm{int}}}}\approx 19.8$ ns is the interaction time between the atoms
and the cavity mode, and as a result, $g{t_{{\mathop{\rm
int}}}}\ll{\rm{\pi}}$. The loss term is inversely proportional to the
intracavity photon lifetime $\tau$. Typically, for the laser output from a
resonant cavity,
$\tau={1\mathord{\left/{\vphantom{1{{\kappa_{0}}}}}\right.\kern-1.2pt}{{\kappa_{0}}}}$.
However, if the cavity and the atomic-transition frequencies are not
identical,
$\tau<{1\mathord{\left/{\vphantom{1{{\kappa_{0}}}}}\right.\kern-1.2pt}{{\kappa_{0}}}}$.
$\tau$ is exactly expressed as $\tau=\frac{1}{{\eta\kappa_{0}}}$. The loss
coefficient $\eta$, reflecting the destructive interference of the intracavity
radiated fields, is defined as the ratio of the maximum power emitted into the
cavity at the resonant condition to the power at any cavity-frequency
detuning,

$\eta=\frac{{{P_{{\rm{cmax}}}}}}{{{P_{\rm{c}}}}}=1+{\left({\frac{{2\mathcal{F}}}{{\rm{\pi}}}}\right)^{2}}{\sin^{2}}\left({{{\omega
L}\mathord{\left/{\vphantom{{\omega
L}{\rm{c}}}}\right.\kern-1.2pt}{\rm{c}}}}\right).$ (3)

Here, the approximation of $\mathcal{F}=\frac{{{\rm{\pi}}\sqrt{R}}}{{1-R}}$ is
used. As for the resonant cavity, the loss coefficient exhibits a minimum of
${\eta_{\min}}=1$, and Eq. (Results) is reduced to the traditional expression
of the laser rate equation Christopher2002Multiple . Instead, $\eta$ reaches
the maximum
${\eta_{\max}}=1+{\left({\frac{{2\mathcal{F}}}{{\rm{\pi}}}}\right)^{2}}$,
which corresponds to the inhibited laser output from an anti-resonant cavity.
The black dots in Fig. 2a show the variation of $\tau$ with the change of
$\Delta\phi$.

Using Eqs. (Results) and (3), we obtain the steady-state solution of the
intracavity photon number $n$ as a function of $\Delta\phi$ and further obtain
the output laser power $P_{\rm{out}}$. $P_{\rm{out}}$ as a function of of
$\Delta\phi$ is represented by a dark-blue line in Fig. 2a, and the
experimental result is represented by the light-blue line. The deviation
between the experimental and theoretical values was caused by ambient
vibrations. This also confirmed that if the cavity was off-resonant, and the
intracavity radiated fields interfered destructively, resulting in a decreased
laser power. We measured $P_{\rm{out}}$ as a function of $\Delta\phi$ at
different pumping light intensities $I$ and different vapor-cell temperatures
$T$, as shown in Figs 2b and 2c, respectively. The power of the inhibited
laser could be further improved with higher pumping light intensities and
vapor-cell temperatures.

Table 1: Cavity-pulling coefficient in optical domain

Stimulated emission
---
Resonant |  Theoretical |  Experimental |  Simulated
$\frac{{{\Gamma}}}{{{\kappa_{0}}}}$ Chen2009Active ; Bohnet2012A  |  $0.038\pm 0.002$ |  $T=120^{\circ}$C
0.039 |  0.040
Anti-resonant |  $-{\left({\frac{\pi}{{2\mathcal{F}}}}\right)^{2}}\frac{{{\Gamma}}}{{{\kappa_{0}}}}$ |  $-0.019\pm 0.002$ |  $T=120^{\circ}$C
$-$0.012 |  $-$0.019

  * *

In this work, $\mathcal{F}=3.07$; ${\Gamma}=2\pi\times 10.04$ MHz;
$\kappa_{0}=2\pi\times 257$ MHz.

Laser Linewidth. Theoretically, analogous to the inhibited spontaneous
emission, the linewidth of the inhibited laser has the potential to be
narrower than that of the traditional resonant laser. Here, considering the
cavity-modification effect, as well as homogeneous and inhomogeneous
broadening, we provide the general expression of the laser linewidth as
follows:

$\displaystyle\Delta{\nu_{\rm{L}}}=$ $\displaystyle\frac{{{\Gamma^{2}}}}{{4\pi
n{\kappa_{0}}}}{N_{{\rm{sp}}}}\left[{\frac{1}{{{\xi^{2}}}}+\frac{{\left({1-\xi}\right){\Gamma_{\rm{g}}}+2\left({1+\xi}\right){\Gamma_{\rm{e}}}}}{{4{\xi^{2}}\left({{\Gamma_{\rm{e}}}+{\Gamma_{\rm{g}}}}\right)}}\frac{n}{{{n_{\rm{s}}}}}}\right]$
$\displaystyle\frac{1}{{1+{{\left({\frac{{2\mathcal{F}}}{{\rm{\pi}}}}\right)}^{2}}{{\sin}^{2}}\left({\frac{{\omega
L}}{{\rm{c}}}}\right)}},$ (4)

where ${N_{{\rm{sp}}}}=\frac{{{N_{\rm{e}}}}}{{{N_{\rm{e}}}-{N_{\rm{g}}}}}$ is
the spontaneous-emission factor; $N_{\rm{e}}$ and $N_{\rm{g}}$ represent the
populations of the excited and ground states, respectively;
${{\Gamma_{{\rm{e}}}}}$, ${{\Gamma_{{\rm{g}}}}}$, and ${{\Gamma_{{\rm{eg}}}}}$
are the decay rates of the atomic populations, and polarization; the
coefficient $\xi$ represents the inhomogeneous and homogeneous broadening; and
${n_{\rm{s}}}=\frac{{{\Gamma_{{\rm{eg}}}}}}{{2{g^{2}}}}\frac{{{\Gamma_{\rm{e}}}{\Gamma_{\rm{g}}}}}{{{\Gamma_{\rm{e}}}+{\Gamma_{\rm{g}}}}}$
is the homogeneous saturation intensity in units of number of photons.

Equation (Results) shows four extra features compared with the classical
Schawlow-Townes equation Schawlow1958Infrared : (i) the first term represents
the bad-cavity effect Kuppens1994Quantum  leading to linewidth narrowing, (ii)
${N_{{\rm{sp}}}}$ causes the linewidth broadening due to the incomplete
inversion, (iii) the factors in square brackets is induced by the Doppler
broadening and power broadening 1996PRAkHOURY , and (iv) the last term depicts
the cavity-induced modification Heinzen1987Enhanced  following the absorption
lineshape with the change of $\Delta\phi$.

It should be noted that the second term inside the square brackets is much
bigger than the first one under the conditions of this work. Therefore, Eq.
(Results) can be further reduced, and the result shows that the expression of
the laser linewidth is independent of the output power. Details about the
laser linewidth are given in the Methods Section V. According to the last term
of Eq. (Results), the laser linewidth is expected to be narrowed by a factor
of $\frac{1}{{1+{{\left({{{2{\cal
F}}\mathord{\left/{\vphantom{{2F}\pi}}\right.\kern-1.2pt}\pi}}\right)}^{2}}}}$
for the inhibited laser compared with the resonant one, and this can be
simplified to the classical bad-cavity expression 1996PRAkHOURY  under the
resonant condition. From Eq. (Results), the quantum-limited linewidths of the
resonant and inhibited lasers are 150 and 43 Hz, respectively (see Fig. 9).
Although the linewidth of the inhibited laser is affected by Doppler
broadening, this thermal-atom scheme has the advantages of a compact structure
and ease of operation, and it can solve the problem of pulse operations in the
cold-atom scheme. Moreover, if the Doppler broadening is suppressed by using
cold atoms as the gain medium or the atomic-beam scheme Chen2009Active ;
Kazakov2014EFTF ; PRL2020Holland ; PRA2021Holland , the quantum-limited
linewidth of the inhibited laser can be further narrowed to the Hz level
(details are given in Methods Section VI). In particular, a resonant
continuous-wave superradiant laser with a linewidth of 40 mHz based on a hot
atomic beam traversing an optical cavity has been proposed in Ref.
PRL2020Holland . If the gain in Eq. (Results) can be improved by increasing
the effective atomic number, so that meet the condition of laser oscillation,
we expect to realize the inhibited laser by using the atomic-beam scheme,
which is a feasible solution to achieve continuous-wave operation.

We measured the laser linewidth by beating the tested laser against the
reference laser, where the experimental setup is given in Fig. 5 in the
Methods section. The cavity frequency of the reference laser coincided with
the atomic transition frequency, while that of the tested laser was tunable by
changing the cavity length. Limited to the intensity sensitivity of the
photodetector, the cavity frequency of the reference laser should be
coincident with the atomic resonance to improve the light intensity for
optical heterodyning. It is difficult to measure the beat-note spectrum
between two inhibited lasers due to their weak laser powers. Figure 3a shows
the beat-note spectra of the reference laser and the tested laser, in which
the tested laser worked in the resonant and anti-resonant cavities,
respectively. The beating linewidths were both 1.2 kHz. Although the
experimental results cannot truly reflect the linewidth of the inhibited
laser, it can show that the linewidth of the inhibited laser will not be wider
than that of the resonant laser. In addition, limited by the technical noise,
such as the cavity-length change induced by the temperature fluctuations of
thermal atoms, the power fluctuations of the pumping laser, and the change of
external magnetic field, the measured laser linewidth was wider than its
corresponding quantum-limited linewidth. The influence of technical noise on
the linewidth broadening is analyzed in detail in Methods Section VII.

Figure 3: Linewidth and cavity-pulling characteristics. a, Beat-note spectra
between the reference laser and the tested laser. Their resolution bandwidths
were both 1 kHz with a sweep time of 362 ms. The beating spectrum (black
circles) between the reference laser and the resonant tested laser was fitted
by a Lorentzian function with a fitted linewidth of 1.2 kHz (red line). The
Lorentz fitting linewidth of the beat-note spectrum (black squares) between
the reference laser and the inhibited laser was also 1.2 kHz (grey line). b,
Frequency shift of the laser oscillation $\Delta$ as a function of cavity-
frequency detuning from the atomic transition frequency
${\omega_{\rm{c}}}-\omega_{0}$, whose adjustable range was one FSR. We modify
Eq. (5) with the laser rate equation to describe $\Delta$. The simulated
results of $\Delta$ at different temperatures for $\mathcal{F}=3.07$ are shown
as the solid lines, and the black triangles represent the experimental
results. Error bars denote standard deviations from three measurements. The
difference between the simulated data and experimental results is explained
further in the main text.

Cavity-pulling characteristic. As shown in Fig. 1, the inhibited laser works
in the flat anti-resonant regime, which is the center of two adjacent cavity
modes. Therefore, the inhibited laser has the advantage of an enhanced
suppression of the cavity-pulling effect. The relationship between the
frequency shift of the oscillation frequency, i.e., $\Delta$, and the cavity-
frequency detuning from the atomic transition ${\omega_{\rm{c}}}-\omega_{0}$,
is analyzed comprehensively for spontaneous emission Daniel1988Radiative ,
which is written as

$\Delta{\rm{=}}\frac{\Gamma}{{\rm{4}}}\frac{{{{\left({\frac{{2\mathcal{F}}}{{\rm{\pi}}}}\right)}^{2}}\sin\left({{{2\omega
L}\mathord{\left/{\vphantom{{2\omega
L}{\rm{c}}}}\right.\kern-1.2pt}{\rm{c}}}}\right)}}{{1+{{\left({\frac{{2\mathcal{F}}}{{\rm{\pi}}}}\right)}^{2}}{{\sin}^{2}}\left({{{\omega
L}\mathord{\left/{\vphantom{{\omega
L}{\rm{c}}}}\right.\kern-1.2pt}{\rm{c}}}}\right)}}.$ (5)

Therefore, for spontaneous radiation, the frequency shift caused by the
cavity-frequency detuning is eliminated, not only when the atomic resonance
coincides with one of the cavity resonances but also when the atomic resonance
is halfway between two adjacent cavity resonances.

From Eq. (5), the cavity-pulling coefficients are equal to
$\frac{{2\mathcal{F}}}{\pi}\frac{{{\Gamma}}}{{{\kappa_{0}}}}$ and ${\text{ -
}}\frac{{2\mathcal{F}}}{\pi}\frac{{{\Gamma}}}{{{\kappa_{0}}}}\frac{{\text{1}}}{{{\text{1
+
}}{{\left({{{{\text{2}}\mathcal{F}}\mathord{\left/{\vphantom{{{\text{2}}\mathcal{F}}\pi}}\right.\kern-1.2pt}\pi}}\right)}^{\text{2}}}}}$
utilizing $\mathcal{F}=\frac{{{\rm{\pi c}}}}{{L{\kappa_{0}}}}$, when the
cavity is resonant and anti-resonant, respectively. The ratio between the two
coefficients is approximately equal to
$-{\left({\frac{{2\mathcal{F}}}{\pi}}\right)^{2}}$. Analogous to the
spontaneous radiation, the ratio between the cavity-pulling coefficients when
the cavity is resonant and anti-resonant is
$-{\left({\frac{{2\mathcal{F}}}{\pi}}\right)^{2}}$ for the stimulated
emission. The difference is that the pulling coefficient is
$\frac{{{\Gamma}}}{{{\kappa_{0}}}}$ for the resonant bad-cavity laser
Chen2009Active ; Bohnet2012A . Accordingly, the cavity-pulling coefficient of
the inhibited laser is around
$-{\left({\frac{\pi}{{2\mathcal{F}}}}\right)^{2}}\frac{{{\Gamma}}}{{{\kappa_{0}}}}$.

More specifically, for the stimulated emission, we should consider the atom-
cavity interactions. Therefore, Eq. (5) is further modified by the laser rate
equation to obtain the frequency shift of the stimulated emission. The fitted
results are depicted by solid lines in Fig. 3b. In addition, we measured the
frequency shift as a function of the cavity-frequency detuning from the atomic
transition frequency. The experimental results (black triangles) were
consistent with the fitted results. For comparison, the cavity-pulling
coefficients discussed above are illustrated in Table 1. Compared with the
resonant condition, the suppression of the cavity-pulling effect of inhibited
lasers was enhanced from 26 to 53 times.

The deviation between the experimental and simulated results of the cavity-
pulling characteristics in Fig. 3b is analyzed. First, the low-reflectivity
mirrors result in a significant loss in the cavity. Therefore, the initial
measurement error may cause a deviation between the experimental and
theoretical values of the finesse, since the intracavity multiple round-trip
propagation of light will lead to error accumulation. Second, the
piezoelectric ceramic (PZT) was used to tune the cavity length, i.e., the
cavity-mode frequency. Ideally, the length change of the PZT should be linear
with the voltage applied to the PZT. However, due to unavoidable manufacturing
errors, this relationship is not perfectly linear, and leads to the deviation
of cavity-frequency detuning between the experimental and simulated results.
Third, the simulated result was obtained under the single-mode operation. In
our experiment, although the higher-order transverse mode was suppressed, the
environmental noise caused a the geometry change of the cavity, which would
destroy the perfect single-mode operation to some extent. In summary, these
factors will lead to the deviations between the experimental and simulated
results, as shown in Fig. 3b.

## Conclusions

In this work, we experimentally demonstrated an inhibited laser. It lased
exactly at the center of two adjacent cavity modes, which is different from
traditional resonant lasers. Compared with resonant superradiant lasers
Chen2009Active ; Bohnet2012A ; PhysRevA.87.013821 ; PhysRevLett.123.103601 ;
Stefan2020PhysRevA ; PRL2020Holland , the inhibited laser is inherently
insensitive to cavity-length fluctuations, characterized by a further enhanced
suppression on the cavity-pulling effect. The Doppler broadening can be
further inhibited by the cold-atom or atomic-beam scheme Kazakov2014EFTF ;
PRL2020Holland ; PRA2021Holland . In the future, using the cold-atom as gain
medium, we expect to achieve a sub-Hz-level linewidth inhibited laser with a
cavity-pulling coefficient on the order $10^{-5}$ and an output power at the
$\mu$W level by selecting an atomic transition with a lower decay rate, such
as the Cs 1359 nm transition, to realize lasing. This result is comparable to
the performances of resonant superradiant lasers using the natural linewidth
at only the mHz-level as the transition level but solves its problem of a low
output power. We expect to realize an anti-resonant superradiant AOC using the
principle of the inhibited laser, which will greatly facilitate precision
measurements for fundamental science, such as tests of the variations of
fundamental constants, the gravitational potential of Earth, and the search
for dark matter.

## Methods

I. Experimental details. To acquire sufficient gain, we take advantage of the
multilevel structure of the Cs atom and multiple atoms interacting with a
single mode of an optical cavity. As depicted in Fig. 1, a cloud of thermal Cs
atoms collected in the low-finesse F-P cavity were pumped by the 459 nm
continuous-wave laser. For typical lasers, the cavity length is exactly equal
to an integral multiple of the half-wavelength, i.e., the cavity-mode
frequency resonates with the peak of the emission line for an atomic
transition. However, the cavity length is equal to an odd multiple of the
quarter wavelength, namely, the atomic transition frequency is halfway between
two adjacent cavity modes, for the inhibited laser. In this work, the cavity
length was tunable through the PZT, of which the displacement range was more
than one-half wavelength of the laser oscillation. Therefore, the detuning
range of the cavity frequency was larger that one FSR$\sim$789 MHz.

To measure the frequency shift of the laser oscillation with the beat-note
method, we built another 1470 nm laser source as the frequency reference. The
working schematic is shown in Fig. 4, which consists of three modules: I. The
tested-laser generating module, whose cavity length is adjustable over a half-
wavelength transition at 1470 nm. II. The reference-laser generating module,
whose cavity-mode frequency is tuned to be exactly equal to the central
frequency of the gain medium. III. The heterodyne module, where the beat-note
signals of the tested laser and the reference laser are recorded. In modules I
and II, the 459 nm interference filter configuration external cavity diode
laser (IF-ECDL), which was frequency stabilized by modulation transfer
spectroscopy (MTS), pumped the Cs atoms inside the F-P cavity to realize the
1470 nm lasing. The displacement range of the PZT was adjustable in module I,
while it was tuned to a fixed value in module II. When tuning the length of
the F-P cavity in module I, we recorded the central frequency as well as the
full width at half maximum (FWHM) of each beat-note spectrum by a frequency
analyzer (FA).

II. Cavity-pulling coefficient. The integrated Invar F-P cavity consisted of a
plane mirror M1 and a plane-concave mirror M2 (radius of curvature $r=500$ mm)
separated by a distance $L=190$ mm. Therefore, the mode sustained by the
cavity had Gaussian transverse profiles, of which the spot radii on the cavity
mirrors M1 and M2 were ${w_{s1}}=0.429$ mm and ${w_{s2}}=0.337$ mm,
respectively. The equivalent mode volume was
$V_{\rm{c}}=\frac{1}{4}L\pi{\left({\frac{{{w_{s1}}+{w_{s2}}}}{2}}\right)^{2}}=21.89$
mm3. The cavity power decay rate was $\kappa_{0}=2\pi\times 257$ MHz under
resonant conditions, and the free spectral range was $\rm{FSR}=789$ MHz.
Therefore, the cavity finesse was ${\mathcal{F}}=3.07$ FS2006book .

The gain medium Cs atoms were pumped by the 459 nm laser through the velocity-
selective mechanism. It was assumed that the pumping light intensity $I$ = 10
mW/mm2, while the corresponding saturation light intensity ${I_{\rm{s}}}=\pi
h{\rm{c}}\Gamma/3{\lambda^{3}}=1.27$ mW/cm2. Therefore, the saturation
broadening of state $\left|2\right\rangle$ in Fig. 1 caused by the pumping
laser was

$\frac{{{\Gamma_{2}}}}{{2\pi}}=\frac{{{\Gamma_{21}}+{\Gamma_{23}}+{\Gamma_{24}}}}{{2\pi}}\sqrt{1+{s}}=26.38\,\rm{MHz},$
(6)

where ${\Gamma_{21}}=0.793\times{10^{6}}\,{{\rm{s}}^{-1}}$,
${\Gamma_{23}}=3.52\times{10^{6}}\,{{\rm{s}}^{-1}}$, and
${\Gamma_{24}}=1.59\times{10^{6}}\,{{\rm{s}}^{-1}}$ are the decay rates of the
$\left|2\right\rangle\to\left|1\right\rangle$,
$\left|2\right\rangle\to\left|3\right\rangle$, and
$\left|2\right\rangle\to\left|4\right\rangle$ transitions, respectively. $s$
is the saturation factor represented by
$s={I\mathord{\left/{\vphantom{I{{I_{\rm{s}}}}}}\right.\kern-1.2pt}{{I_{\rm{s}}}}}$.
According to the velocity-selective scheme, only atoms in the direction of the
cavity mode with a velocity less than
$\Delta\upsilon=\frac{{{\Gamma_{2}}}}{{2\pi}}\times{\lambda_{21}}$ can be
pumped to state $\left|2\right\rangle$ and then decay to state
$\left|3\right\rangle$. Consequently, the Doppler broadening of
$\left|3\right\rangle$ is
$\frac{{{\Gamma_{\rm{D}}}}}{{2\pi}}=\Delta\upsilon/{\lambda_{34}}$. Since the
spontaneous decay rate of the 1470 nm transition $\Gamma_{0}=2\pi\times 1.81$
MHz JOSA1961Heavens , the atomic decay rate was
${\Gamma}=\Gamma_{0}+\Gamma_{\rm{D}}=2\pi\times 10.04$ MHz, which was much
smaller than $\kappa$. Accordingly, the cavity-pulling coefficient in the
resonant cavity is
$P\approx{\Gamma\mathord{\left/{\vphantom{\Gamma\kappa}}\right.\kern-1.2pt}\kappa_{0}}=0.039$.

III. Intracavity photon number at steady state. For the atomic number density
$n^{\prime}=1.57\times{10^{13}}$/cm3 at a vapor-cell temperature of 100∘C
webCsD , the atomic number inside the cavity mode is
$N=\frac{1}{4}n^{\prime}\pi{L_{\rm{cell}}}{\left({\frac{{{w_{s1}}+{w_{s2}}}}{2}}\right)^{2}}=1.81\times{10^{11}}$,
where the vapor-cell length is $L_{\rm{cell}}=10$ cm. Only the atoms with
velocities between
${{-\Delta\upsilon}\mathord{\left/{\vphantom{{-\Delta\upsilon}{2\sim}}}\right.\kern-1.2pt}{2\sim}}{{\Delta\upsilon}\mathord{\left/{\vphantom{{\Delta\upsilon}2}}\right.\kern-1.2pt}2}$
can be pumped to the 7P1/2 state. According to the Maxwell speed distribution,
the effective atomic number $N_{\rm{eff}}$ is given by

$\displaystyle{N_{{\rm{eff}}}}$
$\displaystyle=N\int_{-\frac{1}{2}\Delta\upsilon}^{\frac{1}{2}\Delta\upsilon}{\sqrt{\frac{m}{{2\pi{k_{\rm{B}}}T}}}}\exp\left({\frac{{-m{\upsilon^{2}}}}{{2{k_{\rm{B}}}T}}}\right)d\upsilon$
$\displaystyle=5.71\times{10^{9}},$ (7)

where $m$ is the atomic mass, and ${{k_{\rm{B}}}}$ is the Boltzmann constant.

Utilizing the density matrix equations, the intracavity photon number at
steady state as a function of the phase shift $\Delta\phi$ (or cavity-
frequency detuning, ${\nu_{c}}-{\nu_{0}}$) is obtained. The atomic energy
level is shown in Fig. 1a, where the energy states are labelled as
$\left|i\right\rangle$. Under the conditions of this work, the homogeneous
intensity broadening was much larger than the inhomogeneous Doppler
broadening, which is analyzed in detail in Methods Section V. Using the
rotating wave approximation (RWA), the density matrix equations for Cs atoms
interacting with the 459 nm pumping laser are expressed as follows:

$\displaystyle\frac{{d{\rho_{11}}}}{{dt}}=-\Omega\rho_{{\text{12}}}^{I}+{\Gamma_{21}}{\rho_{22}}+{\Gamma_{41}}{\rho_{44}}+{\Gamma_{61}}{\rho_{66}},$
$\displaystyle\frac{{d{\rho_{22}}}}{{dt}}=\Omega\rho_{{\text{12}}}^{I}-\left({{\Gamma_{21}}+{\Gamma_{23}}+{\Gamma_{25}}}\right){\rho_{22}},$
$\displaystyle\frac{{d{\rho_{33}}}}{{dt}}={\Gamma_{23}}{\rho_{22}}-\left({{\Gamma_{34}}+{\Gamma_{36}}}\right){\rho_{33}}-\frac{{{\rho_{33}}-{\rho_{44}}}}{{{\tau_{\text{cyc}}}}}$
$\displaystyle{\left({\frac{{2g\sqrt{n+1}}}{{\sqrt{{\Delta^{2}}+4{g^{2}}\left({n+1}\right)}}}}\right)^{2}}{\sin^{2}}\left({\frac{{\sqrt{{\Delta^{2}}+4{g^{2}}\left({n+1}\right)}}}{2}{\tau_{{\text{cyc}}}}}\right),$
$\displaystyle\frac{{d{\rho_{44}}}}{{dt}}={\Gamma_{{\text{34}}}}{\rho_{{\text{33}}}}+{\Gamma_{{\text{54}}}}{\rho_{55}}-{\Gamma_{{\text{41}}}}{\rho_{{\text{44}}}}+\frac{{{\rho_{33}}-{\rho_{44}}}}{{{\tau_{\text{cyc}}}}}$
$\displaystyle{\left({\frac{{2g\sqrt{n+1}}}{{\sqrt{{\Delta^{2}}+4{g^{2}}\left({n+1}\right)}}}}\right)^{2}}{\sin^{2}}\left({\frac{{\sqrt{{\Delta^{2}}+4{g^{2}}\left({n+1}\right)}}}{2}{\tau_{{\text{cyc}}}}}\right),$
$\displaystyle\frac{{d{\rho_{{\text{55}}}}}}{{dt}}={\Gamma_{{\text{25}}}}{\rho_{{\text{22}}}}{\text{
-
}}\left({{\Gamma_{{\text{54}}}}+{\Gamma_{{\text{56}}}}}\right){\rho_{{\text{55}}}},$
$\displaystyle\frac{{d{\rho_{{\text{66}}}}}}{{dt}}={\Gamma_{{\text{36}}}}{\rho_{{\text{33}}}}{\text{
+
}}{\Gamma_{{\text{56}}}}{\rho_{{\text{55}}}}-{\Gamma_{{\text{61}}}}{\rho_{{\text{66}}}},$
$\displaystyle\frac{{d\rho_{{\text{12}}}^{I}}}{{dt}}=\frac{1}{2}\Omega\left({{\rho_{11}}-{\rho_{{\text{12}}}}}\right)+\rho_{{\text{12}}}^{R}\Delta^{\prime}-\frac{1}{2}{\Gamma_{21}}\rho_{{\text{12}}}^{I},$
$\displaystyle\frac{{d\rho_{{\text{12}}}^{R}}}{{dt}}=-\rho_{{\text{12}}}^{I}\Delta^{\prime}-\frac{1}{2}{\Gamma_{21}}\rho_{{\text{12}}}^{R},$
$\displaystyle\frac{{dn}}{{dt}}={N_{\rm{eff}}}\frac{{{\rho_{33}}-{\rho_{44}}}}{{{\tau_{{\text{cyc}}}}}}{\left({\frac{{2g\sqrt{n+1}}}{{\sqrt{{\Delta^{2}}+4{g^{2}}\left({n+1}\right)}}}}\right)^{2}}$
$\displaystyle{\sin^{2}}\left({\frac{{\sqrt{{\Delta^{2}}+4{g^{2}}\left({n+1}\right)}}}{2}{t_{{\text{int}}}}}\right)-\eta{\kappa_{0}}n.$
(8)

$\Omega$ is the Rabi frequency, and ${\Gamma_{ij}}$ represents the rate of
decay from $\left|i\right\rangle$ to $\left|j\right\rangle$.
${\tau_{{\rm{cyc}}}}=\frac{1}{\Omega}+\frac{1}{{{\Gamma_{23}}+{\Gamma_{25}}}}+\frac{1}{{{\Gamma_{34}}+{\Gamma_{36}}}}+\frac{1}{{{\Gamma_{41}}}}=28.0$
$\mu{\rm{s}}$ is the cycle time for Cs atoms through a complete transition of
$6{{\rm S}_{1/2}}\to 7{{\rm P}_{1/2}}\to 7{{\rm S}_{1/2}}\to 6{{\rm
P}_{3/2}}$. The interaction time between the atoms and the cavity mode is
given by
${t_{{\rm{int}}}}\approx\frac{1}{{{\Gamma_{34}}+{\Gamma_{36}}+{\Gamma_{41}}}}=19.8$
ns. Ideally, we would simplify the equations by setting the frequency detuning
between the pumping laser and the atomic transition of $\left|1\right\rangle$
to $\left|2\right\rangle$ to be zero. $\rho_{{\text{12}}}^{I}$ and
$\rho_{{\text{12}}}^{R}$ represent the energy shift and the power broadening,
respectively. ${\rho_{ii}}$ denotes the population probability of atoms in the
corresponding state, and the result is shown in Fig. 5.
$\Delta=\omega-{\omega_{0}}$ is the frequency detuning of the laser
oscillation from the atomic transition. Since ${\Delta^{2}}\ll
4{g^{2}}\left({n+1}\right)$, we assume that $\Delta=0$ in the main text. To
verify the correctness of this assumption, we give the most accurate
description of the detuning $\Delta$ in Eq. (Methods). The photon number at
steady state is calculated by inserting the fitted result of $\Delta$ in Fig.
3b into Eq. (Methods).

The results of the intracavity photon number at steady state with and without
considering the detuning $\Delta$ are shown as the green dotted line and the
red solid line in Fig. 6. This shows that there was little difference between
the photon number obtained by Eq. (Results) and the last equation of Eq.
(Methods). The difference between photon numbers obtained by the two equations
is shown in the inset of Fig. 6, which illustrates that the differences are
both zero when the cavity is resonant and anti-resonant. The difference is
eliminated when the mode frequency exactly coincides with the center frequency
of the gain profile. In addition, when the mode frequency is tuned to the
center of two adjacent cavity resonances, the effects of cavity-pulling of the
two adjacent cavity modes on the laser frequency are equal and opposite.
Hence, the difference is also zero for the inhibited laser. This result
demonstrates that the approximation used in the main text is reasonable.

To further characterize the output laser power as a function of the phase
shift at different pumping efficiencies and atomic densities, the pumping
light intensity and the vapor-cell temperature are adjustable, as depicted in
Fig. 2a. According to Eq. (Methods), the intracavity effective atomic number
is influenced by both the pumping light intensity and the vapor-cell
temperature, while the Rabi frequency of the pumping laser is relative to the
pumping light intensity, which is described as
$\Omega=\sqrt{\frac{{3{\lambda_{21}}^{3}{\Gamma_{21}}I}}{{2\pi hc}}}$.
$N_{\rm{eff}}$ and $\Omega$ as functions of $I$ and $N_{\rm{eff}}$ vs. $T$ are
shown in Fig. 7a and b, respectively.

IV. Photon number as function of cavity decay rate. According to Eq.
(Results), the function of the intracavity photon number $n$ with phase shift
$\Delta\phi$ varies with the cavity-mirror reflectivity $R$, namely, the
cavity power loss rate $\kappa$. When the cavity is anti-resonant
($\Delta\phi=\left({2q+1}\right)\pi$), the intracavity photon number decreased
with the increase in the reflectivity, which is shown in Fig. 8. The
intracavity photon number for the inhibited laser was smaller than 1 when the
reflectivity increased to 80$\%$ with $g=1.99\times{10^{5}}\,{\rm{s}^{-1}}$,
$\Omega=4.30\times{10^{7}}\,{\rm{s}^{-1}}$, and
$N_{\rm{eff}}=5.71\times{10^{9}}$. Nevertheless, $n$ could be further improved
with a higher pumping light intensity and a higher atomic number density.

V. General expression of laser linewidth. Considering the effects of
inhomogeneous and homogeneous broadening and the cavity-induced modifications,
this work provides a complete expression of the laser linewidth for the four-
level atomic structure. First, we show the definitions of the parameters
involved in the calculation as follows:

(1) ${{\Gamma_{{\rm{e}}}}}$ and ${{\Gamma_{{\rm{g}}}}}$: decay rates of the
atomic populations of the upper and lower levels, respectively.

(2) ${{\Gamma_{{\rm{eg}}}}}$: decay rate of the atomic polarization.

(3) $\Delta{\omega_{\rm{D}}}$: Doppler broadening.

(4) $\alpha=\frac{{2\Delta{\omega_{\rm{D}}}}}{{{\Gamma_{{\rm{eg}}}}}}$:
dimensionless inhomogeneous broadening width.

(5)
${n_{\rm{s}}}=\frac{{{\Gamma_{{\rm{eg}}}}}}{{4{g^{2}}}}\frac{{{\Gamma_{\rm{e}}}{\Gamma_{\rm{g}}}}}{{{\Gamma_{\rm{e}}}+{\Gamma_{\rm{g}}}}}$:
homogeneous saturation intensity in units of number of photons.

(6) ${n}$: intensity of the laser light in units of number of photons.

(7)
$\beta=\sqrt{1+{n\mathord{\left/{\vphantom{n{{n_{\rm{s}}}}}}\right.\kern-1.2pt}{{n_{\rm{s}}}}}}$:
dimensionless homogeneous intensity broadening width.

In this work, the thermal Cs atoms inside the optical cavity were pumped by
the 459 nm laser through q velocity-selective mechanism. For the pumping light
intensity of 10 mW/mm2, only atoms in the direction of the cavity mode with a
velocity of
$\left|\vartheta\right|\leq{{\Delta\vartheta}\mathord{\left/{\vphantom{{\Delta\vartheta}2}}\right.\kern-1.2pt}2}=6.05$
m/s could be pumped to the excited state. Therefore, the Doppler broadening
$\Delta{\omega_{\rm{D}}}=\frac{\vartheta}{{\rm{c}}}{\omega_{0}}=25.9\times{10^{6}}$
s-1, where $\omega_{0}$ represents the transition frequency. Here,
${\Gamma_{{\rm{eg}}}}=11.4\times{10^{6}}$ s-1,
${\Gamma_{{\rm{e}}}}=17.6\times{10^{6}}$ s-1,
${\Gamma_{{\rm{g}}}}=32.4\times{10^{6}}$ s-1. Thus,
$\alpha={{{\text{2}}\Delta{\omega_{\text{D}}}}\mathord{\left/{\vphantom{{{\text{2}}\Delta{\omega_{\text{D}}}}{{\Gamma_{{\text{eg}}}}}}}\right.\kern-1.2pt}{{\Gamma_{{\text{eg}}}}}}=4.54$.

The intracavity photon number at steady state is influenced by the detuning of
cavity-mode frequency from atomic transition. In this work, the intracavity
photon number $n$ was in the range of $5.51\times{10^{4}}$ to
$2.70\times{10^{5}}$ under the conditions of
$g=1.99\times{10^{5}}\,{\rm{s}^{-1}}$,
$\Omega=4.30\times{10^{7}}\,{\rm{s}^{-1}}$, and
$N_{\rm{eff}}=5.71\times{10^{9}}$. ${{n_{\rm{s}}}}=819$ with the atom-cavity
coupling constant $g=1.99\times{10^{5}}$ s-1. Therefore, $\beta$ was in the
range of $8.27$ to $18.18$. Consequently,
${\alpha\mathord{\left/{\vphantom{\alpha\beta}}\right.\kern-1.2pt}\beta}<1$,
and the intensity broadening was slightly larger than the Doppler broadening.

Moreover, the waist radius ${w_{0}}=\frac{{{w_{s1}}+{w_{s2}}}}{2}=0.383$ mm,
and the atomic velocity $\left|\vartheta\right|\leq 6.05$ m/s. Therefore, the
atomic transit time
$\tau\geq\frac{{2{w_{0}}}}{{\left|\vartheta\right|}}=0.127$ ms, which is much
longer than the atomic transition lifetime ${\Gamma_{{\rm{eg}}}}^{{\rm{-1}}}$
PRA2008Yu . Finally, we considered the influence of both inhomogeneous
broadening and the cavity-modified effect, giving a general expression of the
laser linewidth $\Delta{\nu_{\rm{L}}}$ under the assumption of
$\tau\gg{\Gamma_{{\rm{eg}}}}^{{\rm{-1}}}$, as follows:

$\displaystyle\Delta{\nu_{\rm{L}}}=$ $\displaystyle\frac{\kappa}{{4\pi
n}}{\left({\frac{{\Gamma^{\prime}}}{{\Gamma^{\prime}+\kappa}}}\right)^{2}}{N_{{\rm{sp}}}}\left[{1+\frac{{\left({1-\xi}\right){\Gamma_{\rm{g}}}+2\left({1+\xi}\right){\Gamma_{\rm{e}}}}}{{4\left({{\Gamma_{\rm{e}}}+{\Gamma_{\rm{g}}}}\right)}}\frac{n}{{{n_{\rm{s}}}}}}\right],$
(9)

where $\kappa=\eta{\kappa_{0}}$, $\kappa_{0}$ represents the cavity
dissipation rate under the resonant condition.
$\eta=1+{\left({\frac{{2\mathcal{F}}}{{\rm{\pi}}}}\right)^{2}}{\sin^{2}}\left({\frac{{\omega
L}}{{\rm{c}}}}\right)\geq 1$ is the loss coefficient, which is depicted in the
main text. Here, we have let
$\Gamma^{\prime}={\Gamma\mathord{\left/{\vphantom{\Gamma\xi}}\right.\kern-1.2pt}\xi}$
to make the expression of Eq. (Methods) more similar to the one depicting the
homogeneous broadening case PRA1993Kolobov . The coefficient $\xi$ describes
the inhomogeneous and homogeneous broadening, as

$\displaystyle\xi=\frac{2}{{\sqrt{\rm{\pi}}}}\frac{\beta}{\alpha}\frac{{\exp\left({-{{{\beta^{2}}}\mathord{\left/{\vphantom{{{\beta^{2}}}{{\alpha^{2}}}}}\right.\kern-1.2pt}{{\alpha^{2}}}}}\right)}}{{{\rm{erfc}}\left({{\beta\mathord{\left/{\vphantom{\beta\alpha}}\right.\kern-1.2pt}\alpha}}\right)}}-\frac{{2{\beta^{2}}}}{{{\alpha^{2}}}},$
$\displaystyle{\rm{erfc}}\left(z\right)=1-\frac{2}{{\sqrt{\rm{\pi}}}}\int_{0}^{z}{dt\exp\left({-{t^{2}}}\right)}.$
(10)

${N_{{\rm{sp}}}}=\frac{{{N_{\rm{e}}}}}{{{N_{\rm{e}}}-{N_{\rm{g}}}}}$ is the
spontaneous-emission factor, $N_{\rm{e}}$ and $N_{\rm{g}}$ represent the
populations of the excited and ground states, respectively.

Furthermore, in the bad-cavity limit, $\Gamma^{\prime}\ll\kappa$. Therefore,
we replace
${\left({\frac{{\Gamma^{\prime}}}{{\Gamma^{\prime}+\kappa}}}\right)^{2}}$ with
${\left({\frac{{\Gamma^{\prime}}}{\kappa}}\right)^{2}}$ and simplify Eq.
(Methods) as

$\displaystyle\Delta{\nu_{\rm{L}}}=$ $\displaystyle\frac{{{\Gamma^{2}}}}{{4\pi
n{\kappa_{0}}}}{N_{{\rm{sp}}}}\left[{\frac{1}{{{\xi^{2}}}}+\frac{{\left({1-\xi}\right){\Gamma_{\rm{g}}}+2\left({1+\xi}\right){\Gamma_{\rm{e}}}}}{{4{\xi^{2}}\left({{\Gamma_{\rm{e}}}+{\Gamma_{\rm{g}}}}\right)}}\frac{n}{{{n_{\rm{s}}}}}}\right]$
$\displaystyle\frac{1}{{1+{{\left({\frac{{2\mathcal{F}}}{{\rm{\pi}}}}\right)}^{2}}{{\sin}^{2}}\left({\frac{{\omega
L}}{{\rm{c}}}}\right)}}.$ (11)

The analysis of the features related to the laser linewidth is given in the
main text. It should be noted that the second term inside the square brackets
is much larger than 1 under the conditions of this work. Then, Eq. (Methods)
can be reduced as follows:

$\displaystyle\Delta{\nu_{\text{L}}}=$
$\displaystyle\frac{{{\Gamma^{2}}}}{{4\pi{\kappa_{0}}}}{N_{{\text{sp}}}}\left[{\frac{{\left({1-\xi}\right){\Gamma_{\text{g}}}+2\left({1+\xi}\right){\Gamma_{\text{e}}}}}{{4{\xi^{2}}\left({{\Gamma_{\text{e}}}+{\Gamma_{\text{g}}}}\right)}}\frac{{\text{1}}}{{{n_{\text{s}}}}}}\right]$
$\displaystyle\frac{1}{{1+{{\left({\frac{{2\mathcal{F}}}{\pi}}\right)}^{2}}{{\sin}^{2}}\left({\frac{{\omega
L}}{{\text{c}}}}\right)}}.$ (12)

Accordingly, the expression of the linewidth is independent of the output
power. Also, we calculated the quantum-limited linewidth with the change of
the phase shift of the cavity, as shown in the black-square dotted line of
Fig. 9. The results agree with both Eq. (Methods) and (Methods). This
indicates that the limit linewidths were 150 and 43 Hz when the cavity was
resonant and anti-resonant, respectively. This result is limited by the
Doppler broadening in the thermal atomic system. To explore the narrower
quantum-limited linewidth, next, we used the cold-atom and atomic-beam methods
to reduce the Doppler broadening and separately calculated the laser
linewidth.

VI. Laser linewidths using cold-atom and atomic-beam schemes. First, we
analyzed the cold-atom scheme. Assuming that the Cs atoms inside the optical
cavity were cooled and trapped by the magneto-optical trap (MOT) at 894 nm.
Meanwhile, the 459 nm laser pumped the cold Cs atoms to realize the 1470 nm
laser. This design was used to reduce the light shift induced by the cooling
and pumping light, because the energy levels of the 1470 nm transition and the
cooling and pumping light are separated.

Here, we assumed that the Cs atoms, which were confined in an optical cavity
with the finesse and dissipation rate being same as in the thermal-atom
conditions, could be cooled to a temperature of 200 $\mu$K using a two-
dimensional MOT PRL2019CJF . The effective atomic number could also reach
${N_{{\rm{eff}}}}=5.71\times{10^{9}}$. The only difference was that the atomic
decay rate $\Gamma\approx{\Gamma_{{\rm{eg}}}}$, because the Doppler broadening
was much smaller than the natural linewidth in the cold-atom system. Under
this condition, the Doppler broadening
$\Delta{\omega_{\rm{D}}}=\omega\sqrt{\frac{{2{k_{\rm{B}}}T}}{{m{{\rm{c}}^{2}}}}}=2{\rm{\pi}}\times
0.1$ MHz. Therefore, the ratio of the Doppler broadened inhomogeneous width to
the power broadened homogeneous width
$\frac{\alpha}{\beta}=\frac{{2\Delta{\omega_{\rm{D}}}}}{{{\Gamma_{{\rm{eg}}}}\left({1+{n\mathord{\left/{\vphantom{n{{n_{\rm{s}}}}}}\right.\kern-1.2pt}{{n_{\rm{s}}}}}}\right)}}\ll
1$. The main contribution of the linewidth broadening comes from power
broadening. The laser works in the homogeneous limit, and its linewidth can be
reduced to a simple expression as follows:

$\displaystyle\Delta{\nu_{\rm{L}}}$
$\displaystyle=\frac{\kappa}{{{\rm{4\pi}}n}}{N_{\rm{sp}}}{\left({\frac{\Gamma}{{\Gamma+\kappa}}}\right)^{2}}\left({1+\frac{{{\Gamma_{\rm{e}}}}}{{{\Gamma_{\rm{e}}}+{\Gamma_{\rm{g}}}}}\frac{n}{{{n_{\rm{s}}}}}}\right)$
$\displaystyle=\frac{{{\Gamma^{2}}}}{{4\pi
n{\kappa_{0}}}}{N_{{\rm{sp}}}}\left({1+\frac{{{\Gamma_{\rm{e}}}}}{{{\Gamma_{\rm{e}}}+{\Gamma_{\rm{g}}}}}\frac{n}{{{n_{\rm{s}}}}}}\right)\frac{1}{{1+{{\left({\frac{{2\mathcal{F}}}{{\rm{\pi}}}}\right)}^{2}}{{\sin}^{2}}\left({\frac{{\omega
L}}{{\rm{c}}}}\right)}}.$ (13)

Compared with Eq. (Methods), the difference of Eq. (Methods) comes from the
fourth term, which only reflects the power broadening rather than the
inhomogeneous broadening. The result is depicted by the blue dotted line in
Fig. 9. It shows that the quantum-limited linewidth was narrowed from 4.74 to
1.01 Hz when the cavity was tuned from resonant to anti-resonant. Moreover,
the limited linewidth could be further narrowed to the sub-Hz level if we
increased the cavity-dissipation rate or selected an atomic transition with a
lower decay rate, such as the 1359 nm transition, as the inhibited laser. In
addition, utilizing cold atoms as the gain medium, the cavity-pulling
coefficient of the inhibited laser could be significantly reduced to the order
of $10^{-5}$. Thus, the sensitivity of the laser frequency to the cavity
frequency fluctuations was strongly suppressed.

Another approach to suppress the Doppler broadening is to couple a beam of
moving atomic dipoles to a single mode of bad-cavity Chen2009Active ;
Kazakov2014EFTF ; PRL2020Holland ; PRA2021Holland . It has been proposed that
the 40-mHz-linewidth resonant suppradiant laser can be realized by using the
atomic-beam scheme PRL2020Holland . If the gain in laser rate equation can be
improved by increasing the effective atomic number, so that meet the condition
of laser oscillation, it is expected to realize the inhibited laser based on
the atomic-beam scheme. In this case, 40Ca atoms are evaporated to form an
atomic beam and effuse from the oven. After emerging from the oven, the atoms
are precooled and prepared in the excited state before entering the anti-
resonant cavity. The cavity is aligned transverse to the atomic beam to
suppress the motional and collisional effects. The residual cavity thermal
noise can be effectively restrained by placing the cavity inside a vacuum
chamber with an actively controlled vibration isolation table combined with an
additional thermally isolated passive heat shield Kessler2012NP .

Utilizing the atomic-beam system, the inhomogeneous Doppler broadening could
reach around 100 kHz, which is much smaller than the intensity broadening.
Therefore, the laser operates within the homogeneous limit, and the quantum-
limited linewidth is also expressed by Eq. (Methods). The linewidth of the
inhibited laser can be narrowed to the Hz level, and the problem of pulsed
operation in the cold-atom scheme can be solved.

VII. Linewidth broadening induced by technical noises. In this work, a cloud
of Cs atoms was pumped by the 459 nm laser through the velocity-selective
mechanism. To increase the effective atomic number, the atoms inside the vapor
cell were heated to around 100 ∘C. Due to the present experimental scheme,
there are mainly three kinds of technique noise that broaden the laser
linewidth. The analysis of each type of technical noise is given below.

a. Temperature fluctuations of atomic vapor cell.

First, assuming that the cavity-length is free running, we measured the
frequency shift of the 1470 nm laser vs. the vapor-cell temperature, which has
been reported in Ref. Pan2018IEEE . The results show that the frequency of the
laser changed almost linearly with the atomic temperature, with a slope of 692
kHz/∘C. The Allan deviation of the vapor-cell temperature was better than
$1\times{\rm{1}}{{\rm{0}}^{{\rm{-5}}}}$ at a temperature of 100∘C on a short
time scale. Therefore, the linewidth broadening caused by the temperature
fluctuations of the vapor cell was around 692 Hz.

Superficially, the frequency shift caused by temperature fluctuations is
serious. However, by introducing a reference light for the same measurement,
it was found that the frequency shift was actually introduced due to the
change of the equivalent cavity length caused by the cell ends’ glass wall
thermal expansion. The linewidth broadening did not indicate a collision
broadening of 692 Hz, because this broadening could be greatly suppressed by
eliminating the influence of cavity-length fluctuations, as shown in the
analysis below.

To analyze the impact of collision broadening on the laser linewidth caused by
temperature fluctuations individually, we stabilized the cavity length by the
optical phase loop locking (OPLL) technique in the system of the dual-
wavelength AOC, whose experimental details are shown in Ref. Shi2019OE . After
cavity-length stabilization, the slope of the frequency shift of the 1470 nm
laser with the change of the temperature fluctuations was below 45 $\pm$ 1.2
kHz/∘C. Also, the stability of the vapor-cell temperature was
$1\times{\rm{1}}{{\rm{0}}^{{\rm{-5}}}}$ at a temperature of 100∘C on the short
time scale. This indicated that the collision broadening caused by the atomic
temperature fluctuations was below 45 Hz. Therefore, the impact of collision
broadening on the short-term stability of the 1470 nm laser was small.
However, on the long timescales, the effect of the collision broadening on the
long-term relative frequency stability of the 1470 nm laser is indeed an issue
that needs to be solved.

In this work, the cavity-length was free running. Therefore, the linewidth
broadening caused by the temperature fluctuations of the atomic vapor cell was
around 700 Hz, where the collision broadening was about 45 Hz. To further
reduce the influence of temperature fluctuations on the laser linewidth, the
cavity will be placed into a vacuum chamber and a cold-atom system will be
built in our future work.

b. Power fluctuations of 459 nm pumping laser.

The linewidth of the 1470 nm laser was also influenced by the power stability
of the pumping laser. To further evaluate the influence of the power stability
of the pumping laser on the laser linewidth, we measured the laser frequency
variations with the power of the pumping laser, as shown in Ref. Shi2019OE .
An approximately linear relationship was obtained, with a slope of $-36.1\pm
0.94$ kHz/mW. The Allan deviation of the 459 nm laser power was
${\rm{1}}{\rm{.28}}\times{\rm{1}}{{\rm{0}}^{{\rm{-4}}}}$ at 1 s, and the
pumping power was about 10 mW, which means that the linewidth broadening
caused by the power fluctuations of the pumping laser was around 45 Hz. With
further power stabilization of the pumping laser, we expect a linewidth
broadening of the 1470 nm laser because the power fluctuations of the pumping
laser can be reduced to below the Hz level.

c. Change of external magnetic field.

To evaluate the impact of the magnetic-field changes on the 1470 nm laser, we
measured the frequency drift of the 1470 nm laser by applying an external
magnetic field. The laser frequency changed linearly with the magnetic field,
with a slope of about 600 kHz/G. Considering the magnetic shielding of the
system, we estimated that the linewidth broadening caused by the fluctuations
of the external magnetic field was a few Hz, and it would be reduced greatly
by designing a better magnetic shielding.

In summary, due to the influence of the above technical noise, the measured
linewidth of the resonant laser and the inhibited laser, which were both 1.2
kHz, were much larger than their quantum-limited linewidths of 150 and 43 Hz,
respectively. In addition to the above three types of technical noise, the
residual cavity-pulling effect will also cause linewidth broadening.
Therefore, the beat-note linewidths when the cavity was resonant and anti-
resonant looked the same, as shown in Fig. 3a.

The temperature fluctuations of the vapor cell were the main factor resulting
in the linewidth broadening, which will be further suppressed by cavity-length
stabilization and the optimization of the temperature stability. The influence
of the pumping-power change and the fluctuations of the residual magnetic
field were relatively small. With better power stabilization of the 459 nm
laser and the improvement of the temperature precision, we expect that the
linewidth of the inhibited laser can be narrowed to the quantum-limit level.

## Data availability

All the data that support the plots within this paper, including those in the
Methods section, are available on Zenodo with the digital object identifier
https://doi.org/10.5281/zenodo.6601106.

## Code availability

The codes implementing the calculations of this study are available from the
corresponding author upon request.

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## Acknowledgments

We acknowledge discussions with C. Peng and Z. Chen. This research was funded
by the National Natural Science Foundation of China (NSFC) (91436210), China
Postdoctoral Science Foundation (BX2021020), and Wenzhou Major Science &
Technology Innovation Key Project (ZG2020046).

## Author contributions

J. C. conceived the idea to use an anti-resonant cavity to realize the
inhibited laser as a stable active optical clock. T.S. performed the
experiments and carried out the theoretical calculations. T. S. wrote the
manuscript. D. P. and J. C. provided revisions. All authors contributed
equally to the discussions of the results.

## Competing interests

The authors declare no competing interests.

## Additional information

Correspondence and requests for materials should be addressed to T. Shi or J.
Chen.

Figure 4: Experimental setup of the heterodyne measuring system between the
reference laser and the inhibited laser. MTS, modulation transfer
spectroscopy; IF-ECDL, interference filter configuration external cavity diode
laser; PZT, piezoelectric ceramic; M1 and M2, cavity mirrors; DM1, dichroic
mirror coated with anti-reflection coating at 1470 nm and high-reflection
coating at 459 nm; HWP, halfwave plate; DM2, dichroic mirror coated with high-
reflection at 1470 nm and anti-reflection at 459 nm; HM, 1470 nm heterodyne
module; PD, photodetector; FA, frequency analyzer. Figure 5: Numerical results
of the population probability of each state ${\rho_{ii}}$ in Fig. 1a, under
$g=1.99\times{10^{5}}\,{\rm{s}^{-1}}$,
$\Omega=4.30\times{10^{7}}\,{\rm{s}^{-1}}$, and
$N_{\rm{eff}}=5.71\times{10^{9}}$. Figure 6: Photon number at steady state as
a function of the phase shift for $g=1.99\times{10^{5}}\,{\rm{s}^{-1}}$,
$\Omega=4.30\times{10^{7}}\,{\rm{s}^{-1}}$, and
$N_{\rm{eff}}=5.71\times{10^{9}}$. The red line and the green dotted line
represent the photon number when $\Delta=0$ and $\Delta\neq{\text{0}}$,
respectively. The inset shows the difference between the photon numbers when
$\Delta=0$ and $\Delta\neq{\text{0}}$. Figure 7: a, Effective atomic number
and Rabi frequency vs. the pumping light intensity at a vapor-cell temperature
of 100∘C. b, Effective atomic number as a function of the vapor-cell
temperature under a pumping light intensity of 10 mW/mm2. Figure 8:
Intracavity photon number $n$ as a function of the phase shift $\Delta\phi$ at
different cavity-mirror reflectivity $R$ values for
$g=1.99\times{10^{5}}\,{\rm{s}^{-1}}$,
$\Omega=4.30\times{10^{7}}\,{\rm{s}^{-1}}$, and
$N_{\rm{eff}}=5.71\times{10^{9}}$. The blue and red lines represent the photon
number when the cavity is resonant and anti-resonant, respectively. The orange
and the purple lines show the results at $R=34.5\%$ and $R=80\%$ ($n\leq 1$),
respectively. Figure 9: Quantum-limited linewidth $\Delta{\nu_{\rm{L}}}$ as a
function of the phase shift $\Delta\phi$. The black-square dotted line and the
blue dotted line represent the results of thermal- and cold-atom as gain
medium, respectively.