Abstract
Traditional lasers function using resonant cavities, in which the roundtrip optical path is equal to an integer multiple of the intracavity wavelengths to constructively enhance the spontaneous emission rate. Taking advantage of the cavity enhancement effect, the narrowest sub10mHzlinewidth laser and a 10^{−16}fractionalfrequencystability superradiant active optical clock (AOC) have been achieved. However, a laser with atomic spontaneous radiation being destructively inhibited in an antiresonant cavity, where the atomic resonance is exactly between two adjacent cavity resonances, has not been reported. Herein, we experimentally demonstrate the inhibited laser. Compared with traditional AOCs, which exhibit superiority in terms of the high suppression of cavity noise, the suppression of the cavitypulling effect of an inhibited laser can be further improved by a factor of \({\left(2{{{{{{{\mathcal{F}}}}}}}}/\pi \right)}^{2}\), which is 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^{1}, was first reported by Purcell in 1946. It was practically observed in the 1980s using atoms in resonant cavities both in the microwave^{2} and optical^{3,4} domains. The enhanced spontaneous radiation has important application potential in cavity quantum electrodynamics (QED)^{5,6,7}, including for oneatom lasers^{8}, iontrap lasers^{9}, and quantum logic gates^{10} in quantum computers.
Essentially, the resonant cavity, whose cavitymode 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 offresonance, which was first proposed by Kleppner in 1981 and demonstrated through inhibited blackbody absorption^{11} and inhibited spontaneous emission^{12}. After, inhibited spontaneous emission was experimentally demonstrated in microwave and optical cavities in 1985^{13} and 1987^{3}, respectively. Heinzen^{3} pointed out that in an antiresonant 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 antiresonant cavity, the atomic radiative level shift vanished, and the spectral linewidth narrowed^{14}, 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 antiresonant 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^{15}. It has significant potential to achieve inhibited lasing, with the aid of a three or fourlevel structure to increase the pumping efficiency and the multiatom system to reach the strongcoupling regime^{16}.
Different from the traditional types of lasers working in the resonantcavity region, in this work, we propose a laser operating in the antiresonantcavity 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 cavitypulling effect, under the conditions of an antiresonant 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 cavitylength fluctuations. Compared with resonant activeopticalclock (AOC) lasing^{17,18,19,20,21,22}, the influence of the thermal cavitylength noise on the frequency of an inhibited laser is further suppressed by a factor of \({\left(2{{{{{{{\mathcal{F}}}}}}}}/\pi \right)}^{2}\).
Results
Experimental scheme
Here, we report an experimental demonstration of an inhibited laser. The energy level scheme and general setup are depicted schematically in Fig. 1a, b, respectively, sharing similarities with the proposed superradiant AOC based on thermal atoms^{23}. N ≈ 1.8 × 10^{11} pure cesium (Cs) atoms are confined to the TEM_{00} mode of a lowfinesse optical cavity (\({{{{{{{\mathcal{F}}}}}}}}=3.07\)), whose dissipation rate is κ_{0} = 2π × 257 MHz. Pumped by a 459 nm laser (6S_{1/2}7P_{1/2}), the atoms achieve stimulated emissions at a wavelength of 1470 nm (7S_{1/2}6P_{3/2}). The relaxation rate of the atomic dipole Γ = 2π × 10.04 MHz is much smaller than κ_{0}. Therefore, the laser works in a badcavity regime^{17,18}. Unlike traditional resonant lasers, the inhibited laser is realized with a roundtrip optical path equal to odd multiples of the half wavelength \(2L=\left(2q+1\right)\lambda /2\), where q is natural number, and λ the laser wavelength. Information about the experimental details, as well as a discussion about the working regime of the laser, is provided in the “Experimental details” and “Cavitypulling coefficient” subsections of the Methods Section.
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_{c}, to the power into free space, P_{free}, is given by^{14}
where ω is the angular frequency of the radiation, and c the speed of light. \(\Delta \phi =\omega 2L/c\) denotes the phase shift of the intracavity reflected field, and it also indicates the detuning of the cavity frequency ω_{c} from the atomic resonance ω_{0}. A phase shift of 2π between two consecutive round trips of the radiation inside the cavity corresponds to the cavityfrequency detuning ω_{c}−ω_{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 \(\frac{1+R}{1R}\) compared with that in free space when the cavity is resonant (Δϕ = 2πq) and antiresonant (\(\Delta \phi =\left(2q+1\right)\pi\)), respectively. Accordingly, the suppression of the spontaneous emission rate induced by the antiresonant cavity is weak in the lowreflectivity case, which is conducive to realize inhibited lasing.
Power characteristic of inhibited laser
The detuning, Δ = ω−ω_{0}, of the radiation frequency ω from the atomic transition frequency ω_{0}, can also be given by \(\Delta =P\left({\omega }_{{{{{{{{\rm{c}}}}}}}}}{\omega }_{{{{{{{{\rm{0}}}}}}}}}\right)\), where \(P\equiv d\omega /d{\omega }_{{{{{{{{\rm{c}}}}}}}}}\) represents the cavitypulling coefficient^{17,18}. In the badcavity limit, \(P\approx \Gamma /{\kappa }_{0}\ll 1\) 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 atomcavity coupling constant^{24} \(g=\frac{\mu }{\hslash }\sqrt{\frac{\hslash {\omega }_{0}}{2{\varepsilon }_{0}{V}_{{{{{{{{\rm{c}}}}}}}}}}}=1.99\times 1{0}^{5}\) s^{−1}, where μ is the electric dipole moment, ε_{0} is the vacuum permittivity, and V_{c} is the equivalent mode volume. Consequently, the detuning Δ in the laser rate equation^{25} is negligible (the exact calculations are given in the “Intracavity photon number at steady state” subsection of the Methods Sections). Here, we modify the loss term in the classical laser rate equation to obtain a universal expression, which can be used to describe any cavityfrequency detuning condition, as follows:
On the righthand 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 7P_{1/2} state is N_{eff} = 5.71 × 10^{9} with a pumping light intensity of I = 10 mW/mm^{2} and vaporcell temperature of T = 100 ^{∘}C. ρ_{ii} denotes the population probability at level \(\lefti\right\rangle\) in Fig. 1a. τ_{cyc} = 28.0 μs is the cycle time for Cs atoms through a transition of 6S_{1/2}→7P_{1/2}→7S_{1/2}→6P_{3/2}. t_{int} ≈ 19.8 ns is the interaction time between the atoms and the cavity mode, and as a result, gt_{int} ≪ π. The loss term is inversely proportional to the intracavity photon lifetime τ. Typically, for the laser output from a resonant cavity, \(\tau =1/{\kappa }_{0}\). However, if the cavity and the atomictransition frequencies are not identical, \(\tau < 1/ {\kappa }_{0}\). τ is exactly expressed as \(\tau =\frac{1}{\eta {\kappa }_{0}}\). The loss coefficient η, 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 cavityfrequency detuning,
Here, the approximation of \({{{{{{{\mathcal{F}}}}}}}}=\frac{\pi \sqrt{R}}{1R}\) is used. As for the resonant cavity, the loss coefficient exhibits a minimum of \({\eta }_{\min }=1\), and Eq. (2) is reduced to the traditional expression of the laser rate equation^{25}. Instead, η reaches the maximum \({\eta }_{\max }=1+{\left(\frac{2{{{{{{{\mathcal{F}}}}}}}}}{\pi }\right)}^{2}\), which corresponds to the inhibited laser output from an antiresonant cavity. The black dots in Fig. 2a show the variation of τ with the change of Δϕ.
Using Eqs. (2) and (3), we obtain the steadystate solution of the intracavity photon number n as a function of Δϕ and further obtain the output laser power P_{out}. P_{out} as a function of of Δϕ is represented by a darkblue line in Fig. 2a, and the experimental result is represented by the lightblue line. The deviation between the experimental and theoretical values was caused by ambient vibrations. This also confirmed that if the cavity was offresonant, and the intracavity radiated fields interfered destructively, resulting in a decreased laser power. We measured P_{out} as a function of Δϕ at different pumping light intensities I and different vaporcell temperatures T, as shown in Fig. 2b, c, respectively. The power of the inhibited laser could be further improved with higher pumping light intensities and vaporcell temperatures.
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 cavitymodification effect, as well as homogeneous and inhomogeneous broadening, we provide the general expression of the laser linewidth as follows:
where \({N}_{{{{{{{{\rm{sp}}}}}}}}}=\frac{{N}_{{{{{{{{\rm{e}}}}}}}}}}{{N}_{{{{{{{{\rm{e}}}}}}}}}{N}_{{{{{{{{\rm{g}}}}}}}}}}\) is the spontaneousemission factor; N_{e} and N_{g} represent the populations of the excited and ground states, respectively; Γ_{e}, Γ_{g}, and Γ_{eg} are the decay rates of the atomic populations, and polarization; the coefficient ξ 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 (4) shows four extra features compared with the classical SchawlowTownes equation^{26}: (i) the first term represents the badcavity effect^{27} leading to linewidth narrowing, (ii) N_{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^{28}, and (iv) the last term depicts the cavityinduced modification^{3} following the absorption lineshape with the change of Δϕ.
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. (4) 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 “General expression of laser linewidth” subsection of the Methods Section. According to the last term of Eq. (4), the laser linewidth is expected to be narrowed by a factor of \(\frac{1}{1+{\left(2{{{{{{{\mathcal{F}}}}}}}}/ \pi \right)}^{2}}\) for the inhibited laser compared with the resonant one, and this can be simplified to the classical badcavity expression^{28} under the resonant condition. From Eq. (4), the quantumlimited linewidths of the resonant and inhibited lasers are 150 and 43 Hz, respectively, which is shown in the “General expression of laser linewidth” subsection of the Methods Sections. Although the linewidth of the inhibited laser is affected by Doppler broadening, this thermalatom scheme has the advantages of a compact structure and ease of operation, and it can solve the problem of pulse operations in the coldatom scheme. Moreover, if the Doppler broadening is suppressed by using cold atoms as the gain medium or the atomicbeam scheme^{17,22,29,30}, the quantumlimited linewidth of the inhibited laser can be further narrowed to the Hz level (details are given in the “Laser linewidths using coldatom and atomicbeam schemes” subsection of the Methods Section). In particular, a resonant continuouswave superradiant laser with a linewidth of 40 mHz based on a hot atomic beam traversing an optical cavity has been proposed in Ref. ^{22}. If the gain in Eq. (2) 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 atomicbeam scheme, which is a feasible solution to achieve continuouswave operation.
We measured the laser linewidth by beating the tested laser against the reference laser, where the experimental setup is given in the “Experimental details” subsection of 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 beatnote spectrum between two inhibited lasers due to their weak laser powers. Figure 3a shows the beatnote spectra of the reference laser and the tested laser, in which the tested laser worked in the resonant and antiresonant 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 cavitylength 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 quantumlimited linewidth. The influence of technical noise on the linewidth broadening is analyzed in detail in the “Linewidth broadening induced by technical noises” subsection of the Methods Section.
Cavitypulling characteristic
As shown in Fig. 1a, the inhibited laser works in the flat antiresonant regime, which is the center of two adjacent cavity modes. Therefore, the inhibited laser has the advantage of an enhanced suppression of the cavitypulling effect. The relationship between the frequency shift of the oscillation frequency, i.e., Δ, and the cavityfrequency detuning from the atomic transition ω_{c}−ω_{0}, is analyzed comprehensively for spontaneous emission^{31}, which is written as
Therefore, for spontaneous radiation, the frequency shift caused by the cavityfrequency 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 cavitypulling coefficients are equal to \(\frac{2{{{{{{{\mathcal{F}}}}}}}}}{\pi }\frac{\Gamma }{{\kappa }_{0}}\) and \(\,{{{}}}\,\frac{2{{{{{{{\mathcal{F}}}}}}}}}{\pi }\frac{\Gamma }{{\kappa }_{0}}\frac{\,{{{1}}}}{{{{1 +}}}\,{\left({{{2}}}{{{{{{{\mathcal{F}}}}}}}}/ \pi \right)}^{{{{2}}}}}\) utilizing \({{{{{{{\mathcal{F}}}}}}}}=\frac{\pi c}{L{\kappa }_{0}}\), when the cavity is resonant and antiresonant, 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 cavitypulling coefficients when the cavity is resonant and antiresonant 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 badcavity laser^{17,18}. Accordingly, the cavitypulling 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 atomcavity 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 cavityfrequency detuning from the atomic transition frequency. The experimental results (black triangles) were consistent with the fitted results. For comparison, the cavitypulling coefficients discussed above are illustrated in Table 1. Compared with the resonant condition, the suppression of the cavitypulling effect of inhibited lasers was enhanced from 26 to 53 times.
The deviation between the experimental and simulated results of the cavitypulling characteristics in Fig. 3b is analyzed. First, the lowreflectivity 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 roundtrip propagation of light will lead to error accumulation. Second, the piezoelectric ceramic (PZT) was used to tune the cavity length, i.e., the cavitymode 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 cavityfrequency detuning between the experimental and simulated results. Third, the simulated result was obtained under the singlemode operation. In our experiment, although the higherorder transverse mode was suppressed, the environmental noise caused a the geometry change of the cavity, which would destroy the perfect singlemode 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^{17,18,19,20,21,22}, the inhibited laser is inherently insensitive to cavitylength fluctuations, characterized by a further enhanced suppression on the cavitypulling effect. The Doppler broadening can be further inhibited by the coldatom or atomicbeam scheme^{22,29,30}. In the future, using the coldatom as gain medium, we expect to achieve a subHzlevel linewidth inhibited laser with a cavitypulling coefficient on the order 10^{−5} and an output power at the μW 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 mHzlevel as the transition level but solves its problem of a low output power. We expect to realize an antiresonant 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
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. 1b, a cloud of thermal Cs atoms collected in the lowfinesse FP cavity were pumped by the 459 nm continuouswave laser. For typical lasers, the cavity length is exactly equal to an integral multiple of the halfwavelength, i.e., the cavitymode 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 onehalf wavelength of the laser oscillation. Therefore, the detuning range of the cavity frequency was larger that one FSR∼789 MHz.
To measure the frequency shift of the laser oscillation with the beatnote 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 testedlaser generating module, whose cavity length is adjustable over a halfwavelength transition at 1470 nm. II. The referencelaser generating module, whose cavitymode frequency is tuned to be exactly equal to the central frequency of the gain medium. III. The heterodyne module, where the beatnote 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 (IFECDL), which was frequency stabilized by modulation transfer spectroscopy (MTS), pumped the Cs atoms inside the FP 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 FP cavity in module I, we recorded the central frequency as well as the full width at half maximum (FWHM) of each beatnote spectrum by a frequency analyzer (FA).
Cavitypulling coefficient
The integrated Invar FP cavity consisted of a plane mirror M_{1} and a planeconcave mirror M_{2} (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 M_{1} and M_{2} were w_{s1} = 0.429 mm and w_{s2} = 0.337 mm, respectively. The equivalent mode volume was \({V}_{{{{{{{{\rm{c}}}}}}}}}=\frac{1}{4}L\pi {\left(\frac{{w}_{{{{{{{{\rm{s1}}}}}}}}}+{w}_{{{{{{{{\rm{s2}}}}}}}}}}{2}\right)}^{2}=21.89\) mm^{3}. The cavity power decay rate was κ_{0} = 2π × 257 MHz under resonant conditions, and the free spectral range was FSR = 789 MHz. Therefore, the cavity finesse was \({{{{{{{\mathcal{F}}}}}}}}=3.07\)^{32}.
The gain medium Cs atoms were pumped by the 459 nm laser through the velocityselective mechanism. It was assumed that the pumping light intensity I = 10 mW mm^{−2}, while the corresponding saturation light intensity I_{s} = πhcΓ/3λ^{3} = 1.27 mW cm^{−2}. Therefore, the saturation broadening of state \(\left2\right\rangle\) in Fig. 1a caused by the pumping laser was
where Γ_{21} = 0.793 × 10^{6} s^{−1}, Γ_{23} = 3.52 × 10^{6} s^{−1}, and Γ_{25} = 1.59 × 10^{6} s^{−1} are the decay rates of the \(\left2\right\rangle \to \left1\right\rangle\), \(\left2\right\rangle \to \left3\right\rangle\), and \(\left2\right\rangle \to \left4\right\rangle\) transitions, respectively. s is the saturation factor represented by \(s=I/{I}_{{{{{{{{\rm{s}}}}}}}}}\). According to the velocityselective 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 \(\left2\right\rangle\) and then decay to state \(\left3\right\rangle\). Consequently, the Doppler broadening of \(\left3\right\rangle\) is \(\frac{{\Gamma }_{{{{{{{{\rm{D}}}}}}}}}}{2\pi }=\Delta \upsilon /{\lambda }_{34}\). Since the spontaneous decay rate of the 1470 nm transition Γ_{0} = 2π × 1.81 MHz^{33}, the atomic decay rate was Γ = Γ_{0} + Γ_{D} = 2π × 10.04 MHz, which was much smaller than κ. Accordingly, the cavitypulling coefficient in the resonant cavity is \(P\approx \Gamma /{\kappa }_{0}=0.039\).
Intracavity photon number at steady state
For the atomic number density \(n^{\prime} =1.57\times 1{0}^{13}\) cm^{−3} at a vaporcell temperature of 100 ^{∘}C^{34}, the atomic number inside the cavity mode is \(N=\frac{1}{4}n^{\prime} \pi {L}_{{{{{{{{\rm{cell}}}}}}}}}{\left(\frac{{w}_{{{{{{{{\rm{s1}}}}}}}}}+{w}_{{{{{{{{\rm{s2}}}}}}}}}}{2}\right)}^{2}=1.81\times 1{0}^{11}\), where the vaporcell length is L_{cell} = 10 cm. Only the atoms with velocities between \(\Delta \upsilon /2 \sim \Delta \upsilon /2\) can be pumped to the 7P_{1/2} state. According to the Maxwell speed distribution, the effective atomic number N_{eff} is given by
where m is the atomic mass, and k_{B} is the Boltzmann constant.
Utilizing the density matrix equations, the intracavity photon number at steady state as a function of the phase shift Δϕ (or cavityfrequency detuning, ν_{c}−ν_{0}) is obtained. The atomic energy level is shown in Fig. 1a, where the energy states are labelled as \(\lefti\right\rangle\). Under the conditions of this work, the homogeneous intensity broadening was much larger than the inhomogeneous Doppler broadening, which is analyzed in detail in the “General expression of laser linewidth” subsection of the Methods Sections. Using the rotating wave approximation (RWA), the density matrix equations for Cs atoms interacting with the 459 nm pumping laser are expressed as follows:
Ω is the Rabi frequency, and Γ_{ij} represents the rate of decay from \(\lefti\right\rangle\) to \(\leftj\right\rangle\). \({\tau }_{{{{{{{{\rm{cyc}}}}}}}}}=\frac{1}{{{\Omega }}}+\frac{1}{{\Gamma }_{23}+{\Gamma }_{25}}+\frac{1}{{\Gamma }_{34}+{\Gamma }_{36}}+\frac{1}{{\Gamma }_{41}}=28.0\) μs is the cycle time for Cs atoms through a complete transition of 6S_{1/2}→7P_{1/2}→7S_{1/2}→6P_{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 \(\left1\right\rangle\) to \(\left2\right\rangle\) to be zero. \({\rho }_{{{{12}}}}^{I}\) and \({\rho }_{{{{12}}}}^{R}\) represent the energy shift and the power broadening, respectively. ρ_{ii} denotes the population probability of atoms in the corresponding state, and the result is shown in Fig. 5. Δ = ω−ω_{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 Δ = 0 in the main text. To verify the correctness of this assumption, we give the most accurate description of the detuning Δ in Eq. (8). The photon number at steady state is calculated by inserting the fitted result of Δ in Fig. 3b into Eq. (8).
The results of the intracavity photon number at steady state with and without considering the detuning Δ 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. (2) and the last equation of Eq. (8). 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 antiresonant. 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 cavitypulling 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 vaporcell temperature are adjustable, as depicted in Fig. 2a. According to Eq. (7), the intracavity effective atomic number is influenced by both the pumping light intensity and the vaporcell 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_{eff} and Ω as functions of/and N_{eff} vs. T are shown in Fig. 7a, b, respectively.
Intracavity photon number as a function of cavity decay rate
According to Eq. (2), the function of the intracavity photon number n with phase shift Δϕ varies with the cavitymirror reflectivity R, namely, the cavity power loss rate κ. When the cavity is antiresonant (\(\Delta \phi =\left(2q+1\right)\pi\)), 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 × 10^{5} s^{−1}, Ω = 4.30 × 10^{7} s^{−1}, and N_{eff} = 5.71 × 10^{9}. Nevertheless, n could be further improved with a higher pumping light intensity and a higher atomic number density.
General expression of laser linewidth
Considering the effects of inhomogeneous and homogeneous broadening and the cavityinduced modifications, this work provides a complete expression of the laser linewidth for the fourlevel atomic structure. First, we show the definitions of the parameters involved in the calculation as follows:

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

(2)
Γ_{eg}: decay rate of the atomic polarization.

(3)
Δω_{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)
\(\beta =\sqrt{1+n/{n}_{{{{{{{{\rm{s}}}}}}}}}}\): dimensionless homogeneous intensity broadening width.
In this work, the thermal Cs atoms inside the optical cavity were pumped by the 459 nm laser through velocityselective mechanism. For the pumping light intensity of 10 mW/mm^{2}, only atoms in the direction of the cavity mode with a velocity of \(\left\vartheta \right\le \Delta \vartheta /2=6.05\) m s^{−1} could be pumped to the excited state. Therefore, the Doppler broadening \(\Delta {\omega }_{{{{{{{{\rm{D}}}}}}}}}=\frac{\vartheta }{c}{\omega }_{0}=25.9\times 1{0}^{6}\) s^{−1}, where ω_{0} represents the transition frequency. Here, Γ_{eg} = 11.4 × 10^{6} s^{−1}, Γ_{e} = 17.6 × 10^{6} s^{−1}, Γ_{g} = 32.4 × 10^{6} s^{−1}. Thus, \(\alpha =\,{{\mbox{2}}}\Delta {\omega }_{{{\mbox{D}}}}/{\Gamma }_{{{\mbox{eg}}}}=4.54\).
The intracavity photon number at steady state is influenced by the detuning of cavitymode frequency from atomic transition. In this work, the intracavity photon number n was in the range of 5.51 × 10^{4} to 2.70 × 10^{5} under the conditions of g = 1.99 × 10^{5} s^{−1}, Ω = 4.30 × 10^{7} s^{−1}, and N_{eff} = 5.71 × 10^{9}. n_{s} = 819 with the atomcavity coupling constant g = 1.99 × 10^{5} s^{−1}. Therefore, β was in the range of 8.27 to 18.18. Consequently, \(\alpha / \beta \, < \, 1\), and the intensity broadening was slightly larger than the Doppler broadening.
Moreover, the waist radius \({w}_{0}=\frac{{w}_{{{{{{{{\rm{s1}}}}}}}}}+{w}_{{{{{{{{\rm{s2}}}}}}}}}}{2}=0.383\) mm, and the atomic velocity \(\left\vartheta \right\le 6.05\) m s^{−1}. Therefore, the atomic transit time \(\tau \ge \frac{2{w}_{0}}{\left\vartheta \right}=0.127\) ms, which is much longer than the atomic transition lifetime \({{\Gamma }_{{{{{{{{\rm{eg}}}}}}}}}}^{{{{{{{{\rm{1}}}}}}}}}\)^{35}. Finally, we considered the influence of both inhomogeneous broadening and the cavitymodified effect, giving a general expression of the laser linewidth Δν_{L} under the assumption of \(\tau \gg {{\Gamma }_{{{{{{{{\rm{eg}}}}}}}}}}^{{{{{{{{\rm{1}}}}}}}}}\), as follows:
where κ = ηκ_{0}, κ_{0} represents the cavity dissipation rate under the resonant condition. \(\eta =1+{\left(\frac{2{{{{{{{\mathcal{F}}}}}}}}}{\pi }\right)}^{2}{\sin }^{2}\left(\frac{\omega L}{c}\right)\ge 1\) is the loss coefficient, which is depicted in the main text. Here, we have let \({\Gamma }^{\prime}=\Gamma /\xi\) to make the expression of Eq. (9) more similar to the one depicting the homogeneous broadening case^{36}. The coefficient ξ describes the inhomogeneous and homogeneous broadening, as
\({N}_{{{{{{{{\rm{sp}}}}}}}}}=\frac{{N}_{{{{{{{{\rm{e}}}}}}}}}}{{N}_{{{{{{{{\rm{e}}}}}}}}}{N}_{{{{{{{{\rm{g}}}}}}}}}}\) is the spontaneousemission factor, N_{e} and N_{g} represent the populations of the excited and ground states, respectively.
Furthermore, in the badcavity 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. (9) as
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. (11) can be reduced as follows:
Accordingly, the expression of the linewidth is independent of the output power. Also, we calculated the quantumlimited linewidth with the change of the phase shift of the cavity, as shown in the blacksquare dotted line of Fig. 9. The results agree with both Eqs. (11) and (12). This indicates that the limit linewidths were 150 and 43 Hz when the cavity was resonant and antiresonant, respectively. This result is limited by the Doppler broadening in the thermal atomic system. To explore the narrower quantumlimited linewidth, next, we used the coldatom and atomicbeam methods to reduce the Doppler broadening and separately calculated the laser linewidth.
Laser linewidths using coldatom and atomicbeam schemes
First, we analyzed the coldatom scheme. Assuming that the Cs atoms inside the optical cavity were cooled and trapped by the magnetooptical 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 thermalatom conditions, could be cooled to a temperature of 200 μK using a twodimensional MOT^{37}. The effective atomic number could also reach N_{eff} = 5.71 × 10^{9}. The only difference was that the atomic decay rate Γ ≈ Γ_{eg}, because the Doppler broadening was much smaller than the natural linewidth in the coldatom system. Under this condition, the Doppler broadening \(\Delta {\omega }_{{{{{{{{\rm{D}}}}}}}}}=\omega \sqrt{\frac{2{k}_{{{{{{{{\rm{B}}}}}}}}}T}{m{c}^{2}}}=2\pi \times 0.1\) MHz. Therefore, the ratio of the Doppler broadened inhomogeneous width to the power broadened homogeneous width \(\frac{\alpha }{\beta }=\frac{2\Delta {\omega }_{{{{{{{{\rm{D}}}}}}}}}}{{\Gamma }_{{{{{{{{\rm{eg}}}}}}}}}\left(1+n / {n}_{{{{{{{{\rm{s}}}}}}}}}\right)}\ll 1\). 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:
Compared with Eq. (11), the difference of Eq. (13) 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 quantumlimited linewidth was narrowed from 4.74 to 1.01 Hz when the cavity was tuned from resonant to antiresonant. Moreover, the limited linewidth could be further narrowed to the subHz level if we increased the cavitydissipation 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 cavitypulling 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 badcavity^{17,22,29,30}. It has been proposed that the 40mHzlinewidth resonant suppradiant laser can be realized by using the atomicbeam scheme^{22}. 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 atomicbeam scheme. In this case, ^{40}Ca 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 antiresonant 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^{38}.
Utilizing the atomicbeam 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 quantumlimited linewidth is also expressed by Eq. (13). The linewidth of the inhibited laser can be narrowed to the Hz level, and the problem of pulsed operation in the coldatom scheme can be solved.
Linewidth broadening induced by technical noises
In this work, a cloud of Cs atoms was pumped by the 459 nm laser through the velocityselective 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.
Temperature fluctuations of atomic vapor cell
First, assuming that the cavitylength is free running, we measured the frequency shift of the 1470 nm laser vs. the vaporcell temperature, which has been reported in Ref. ^{39}. The results show that the frequency of the laser changed almost linearly with the atomic temperature, with a slope of 692 kHz ^{∘}C^{−1}. The Allan deviation of the vaporcell temperature was better than 1 × 10^{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 cavitylength 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 dualwavelength AOC, whose experimental details are shown in Ref. ^{23}. After cavitylength stabilization, the slope of the frequency shift of the 1470 nm laser with the change of the temperature fluctuations was below 45 ± 1.2 kHz ^{∘}C^{−1}. Also, the stability of the vaporcell temperature was 1 × 10^{−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 shortterm stability of the 1470 nm laser was small. However, on the long timescales, the effect of the collision broadening on the longterm relative frequency stability of the 1470 nm laser is indeed an issue that needs to be solved.
In this work, the cavitylength 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 coldatom system will be built in our future work.
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. ^{23}. An approximately linear relationship was obtained, with a slope of −36.1 ± 0.94 kHz mW^{−1}. The Allan deviation of the 459 nm laser power was 1.28 × 10^{−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.
Change of external magnetic field
To evaluate the impact of the magneticfield 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^{−1}. 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 quantumlimited linewidths of 150 and 43 Hz, respectively. In addition to the above three types of technical noise, the residual cavitypulling effect will also cause linewidth broadening. Therefore, the beatnote linewidths when the cavity was resonant and antiresonant 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 cavitylength stabilization and the optimization of the temperature stability. The influence of the pumpingpower 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 quantumlimit level.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
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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Acknowledgements
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).
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J.C. conceived the idea to use an antiresonant 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.
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Shi, T., Pan, D. & Chen, J. An inhibited laser. Commun Phys 5, 208 (2022). https://doi.org/10.1038/s4200502200988y
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DOI: https://doi.org/10.1038/s4200502200988y
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