03 97 2 v 1 [ qu an tph ] 9 A ug 2 02 1 An inhibited laser

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 resonant cavity enhancement, the narrowest sub-10-mHz-linewidth laser [1] and a 10−16fractional-frequency-stability superradiant active optical clock (AOC) [2] have been achieved. However, never has a laser with atomic spontaneous radiation being destructively inhibited [3] in an anti-resonant cavity where the atomic resonance is exactly between two adjacent cavity resonances been proven. Herein, we present the first demonstration of the inhibited stimulated emission, which is termed an inhibited laser. Compared with traditional superradiant AOCs [4–9] exhibiting superiority for the high suppression of cavity noise in lasers, the effect of cavity pulling on the inhibited laser’s frequency can be further suppressed by a factor of −(2F/π). This study of the inhibited laser will guide further development of superradiant AOCs with better stability, thus significant for precision metrology, and may lead to new searches in the cavity quantum electrodynamics (QED) field.

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 resonant cavity enhancement, the narrowest sub-10-mHz-linewidth laser [1] and a 10 −16fractional-frequency-stability superradiant active optical clock (AOC) [2] have been achieved. However, never has a laser with atomic spontaneous radiation being destructively inhibited [3] in an anti-resonant cavity where the atomic resonance is exactly between two adjacent cavity resonances been proven. Herein, we present the first demonstration of the inhibited stimulated emission, which is termed an inhibited laser. Compared with traditional superradiant AOCs [4][5][6][7][8][9] exhibiting superiority for the high suppression of cavity noise in lasers, the effect of cavity pulling on the inhibited laser's frequency can be further suppressed by a factor of −(2F /π) 2 . This study of the inhibited laser will guide further development of superradiant AOCs with better stability, thus significant for precision metrology, and may lead to new searches in the cavity quantum electrodynamics (QED) field.
The significantly enhanced spontaneous decay rate of the spin in a resonant circuit, known as the Purcell effect [10], was first reported by Purcell in 1946. It was practically observed in the 1980s using atoms in resonant cavities both in the microwave [11] and optical [12,13] domains. The enhanced spontaneous radiation has important application potential in cavity quantum electrodynamics (QED) [14][15][16], including for one-atom lasers [17], ion-trap lasers [18], and quantum logic gates [19] 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 [20] and inhibited spontaneous emission [3]. After, inhibited spontaneous emission was experimentally demonstrated in microwave and optical cavities in 1985 [21] and 1987 [12], respectively. Heinzen [12] pointed out that in an anti-resonant cavity, where the atomic frequency was exactly between 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 decreased [22], which is potentially useful for precision measurements. Despite the experimental success of inhibited spontaneous emission, the working mechanism of inhibited laser is unknown.
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 [23]. 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 [24].
Here, we report the first 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 [25]. N ≈ 1.8 × 10 11 pure Cs atoms are confined to the TEM 00 mode of a low-finesse optical cavity (F = 3.07), whose dissipation rate is κ = 2π × 257 MHz. Pumped by a 459 nm laser (6S 1/2 -7P 1/2 ), the atoms achieve stimulated emissions at a 1470 nm transition (7S 1/2 -6P 3/2 ). The relaxation rate of the atomic dipole Γ = 2π × 10.04 MHz is much smaller than κ, which forms a bad-cavity regime [4,5]. Unlike traditional resonant lasers, the inhibited laser is realized with a round-trip optical path equal to odd multiples of the half wavelength 2L = (2q + 1) λ/2.
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 [22] where ω is the angular frequency of the radiation, and c the speed of light. ∆φ = ω2L/c denotes the phase shift of the intracavity reflected field, and it also reflects 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 cavity-frequency detuning ω c − ω 0 of one free spectral range (FSR). According to Eq. (1), the spontaneous ) to |2 (7P 1/2 ) transition. Atoms are then transferred to the |3 (7S 1/2 ) state through spontaneous radiation. Utilizing weak cavity feedback, population inversion is built up between the |3 and |4 (6P 3/2 ) levels, and finally, 1470 nm lasing is realized. An anti-resonant inhibited laser is realized when the cavity-mode frequency is tuned to the center of two adjacent cavity modes centered around the frequency of the atomic transition. 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 R1 = R2 = 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 . However, an inhibited laser (in red) is achieved when the optical path is equal to an odd multiple of the half wavelength 2L = (2q + 1)λ/2 with output mirror at M2. decay rate from the atomic excited state is enhanced and inhibited by a factor of 1+R 1−R compared with that in free space when the cavity is resonant (∆φ = q · 2π) and antiresonant (∆φ = (2q + 1) · π), respectively. Accordingly, the suppression of the spontaneous emission rate under the anti-resonant state is weak in the low-reflectivity cavity, which is conducive to lasing.
The detuning, ∆ = ω − ω 0 , of the radiation frequency ω from the atomic transition frequency ω 0 , can also be given by ∆ = P (ω c − ω 0 ), where P ≡ dω/dω c represents the cavity-pulling coefficient [4,26]. In the bad-cavity limit, P ≈ Γ/κ ≪ 1 when the cavity is near resonant, and thus, ∆ 2 ≪ 4g 2 (n + 1). n is the intracavity photon number, and g = µ ω0 2ε0Vc = 1.99 × 10 5 s −1 is the atom-cavity coupling constant, where µ is the electric dipole moment, ε 0 is the vacuum permittivity, and V c is the mode volume. Consequently, the detuning ∆ in the laser rate equation [27] is negligible (the exact calculations are given in the Methods section). 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: The first term on the right side 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 I = 10 mW/mm 2 and vapor-cell temperature T = 100 • C. ρ ii denotes the population probability at level |i in Fig. 1a. τ cyc is the cycle time for Cs atoms through a transition of 6S 1/2 → 7P 1/2 → 7S 1/2 → 6P 3/2 , and t int is the interaction time between the atoms and the cavity mode.
The loss term in Eq. (2) is inversely proportional to the intracavity photon lifetime τ . Typically, for the laser output from a resonant cavity, τ = 1/κ. However, if the cavity and the atomic-transition frequencies are not identical, τ < 1/κ. τ is expressed exactly as τ = 1 ηκ . 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 P c max to the power at any cavityfrequency detuning P c , As for the resonant cavity, the loss coefficient exhibits a minimum of η min = 1, and Eq.(2) is reduced to the traditional expression of the laser rate equation [27]. Instead, for the inhibited laser from an anti-resonant cavity. The black dots in Fig.  2a show the variation of τ with ∆φ.
Using Eqs. (2) and (3), we obtain the steady-state solution of the intracavity photon number n as a function of ∆φ. Utilizing P out = nhνκ, the output laser power P out with the change of ∆φ is represented by a dark-blue line in Fig. 2a, and the experimental result is indicated by the light-blue line. This reflects that if the cavity is tuned to off-resonance, the intracavity radiated fields interfere destructively, resulting in a decreased laser power. We measured P out as a function of ∆φ 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 can be further improved with higher pumping light intensities and temperatures.  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, we obtain the general expression of the laser linewidth as follows: where N sp = Ne Ne−Ng is the spontaneous-emission 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; and n s = Γeg 2g 2

ΓeΓg
Γe+Γg is the homogeneous saturation intensity in units of number of photons. Equation (4) shows four extra features compared with the classical Schawlow-Townes equation [28]: (i) broadening of the linewidth due to the incomplete inversion, (ii) a bad-cavity effect [26] leading to linewidth narrowing, (iii) power broadening [29], and (iv) cavity-induced modification [12] following the absorption lineshape with the change of ∆φ. From the last term, the laser linewidth is expected to be narrowed by a factor of 1 1+(2F /π) 2 for the inhibited laser compared with the resonant one, and simplify to the classical bad-cavity expression [29] under the resonant condition.
The laser linewidth was analyzed by measuring the beat-note spectrum between the tested laser and the reference laser, as shown in Fig. 3a. 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 beating. Although it is difficult to measure the beat-note spectrum between inhibited lasers due to the poor laser power, the measured linewidth of the inhibited laser was comparable to that of the resonant laser. a, Beat-note spectrum between the reference laser and the tested laser. The cavity frequency of the reference laser coincides with the atomic transition frequency, while that of the tested laser is tunable by the cavity length. 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). Moreover, 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 ∆ as a function of cavity-frequency detuning from the atomic transition frequency ωc − ω0, whose adjustable range was around one FSR. We modify Eq. (5) with the laser rate equation to describe ∆. The fitting results of ∆ at different temperatures for R = 28% are shown as the solid lines, and the black triangles represent the experimental results with R = 34.5% under T = 100 • C. The difference between the simulated data and experimental results is explained further in the main text.
Most importantly, 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., ∆, and the cavity-frequency detuning from the atomic transition ω c − ω 0 is analyzed comprehensively for spontaneous emission [30], which is written as 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. Replacing 2 √ R 1−R by 2F π , the cavity-pulling coefficients are equal to 2F π Γ κ and -2F π Γ κ 1 1 + (2F /π) 2 utilizing F = c 2L 2π κ , when the cavity is resonant and anti-resonant, respectively. The ratio between the two coefficients is approximately equal to − 2F π 2 .
Analogous to the spontaneous radiation, the ratio between the cavity-pulling coefficients when the cavity is resonant and anti-resonant is − 2F π 2 for the stimulated emission. The difference is that the pulling coefficient is Γ κ for the resonant bad-cavity laser [4,5]. Accordingly, the cavity-pulling coefficient is around − π 2F 2 Γ κ when the cavity is anti-resonant. 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 cavity-frequency detuning from the atomic transition frequency. The experimental results (black triangles) agreed with the dispersive lineshape and were consistent with the fitted results. For comparison, the cavity-pulling coefficients discussed above are illustrated in Table I. The measured pulling coefficient of the inhibited laser was 1.71-2.35 times smaller than that of the resonant condition. Such an inhibited laser is characterized by an enhanced suppression of the cavity-pulling effect compared with the traditional AOCs.
The deviation between the experimental and theoretical cavity-pulling coefficients is analyzed. First, the general calculation for the cavity finesse based on the highreflectivity approximation may not be preferably applicable to the low-reflectivity case, resulting in the deviation of the finesse calculation. Second, the non-linear hysteresis phenomenon of the piezoelectric ceramic used to adjust the cavity length could result in measurement errors of the cavity-frequency detuning. Third, the removal of the transverse mode degeneracy caused by the geometry change of cavity would disturb the single-mode requirement.
Differing from all existing types of lasers, the inhibited laser demonstrated here is inherently insensitive to cavity-length fluctuations, which leads to significantly smaller systematic perturbations compared to traditional AOCs [4][5][6][7][8][9]. In the future, with the cold atomic ensemble and the improved cavity finesse, we will expect a cavity-pulling coefficient of the order 10 −5 comparable with the results in reference [2], but solve its problem of pulsed operation. This novel superradiant AOC using the principle of inhibited laser might with better frequency sta-bility, having wide range applications in precision measurements, such as test of variation of fundamental constants, gravitational potential of Earth, and search for dark matter.

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 was 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, i.e., 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 adjustable range was more than one-half wavelength of the laser oscillation.
Cavity-pulling coefficient. The integrated Invar F-P 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 c = 1 4 Lπ ws1+ws2 2 2 = 21.89 mm 3 . The cavity power decay rate was κ = 2π × 257 MHz, and the free spectral range was FSR = 789 MHz. Therefore, the cavity finesse was F = 3.07 [31]. 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/mm 2 , while the corresponding saturation light intensity I s = πhcΓ/3λ 3 = 1.27 mW/cm 2 . Therefore, the saturation broadening of state |2 in Fig. 1 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 24 = 1.59 × 10 6 s −1 are the decay rates of the |2 → |1 , |2 → |3 , and |2 → |4 transitions, respectively. s is the saturation factor represented by s = I/I s . According to the velocity-selective scheme, only atoms in the direction of the cavity mode with a velocity less than ∆υ = Γ2 2π ×λ 21 can be pumped to state |2 and then decay to state |3 . Consequently, the Doppler broadening of |3 is ΓD 2π = ∆υ/λ 34 . Since the spontaneous decay rate of the 1470 nm transition Γ 0 = 2π × 1.81 MHz [32], the atomic decay rate was Γ = Γ 0 + Γ D = 2π × 10.04 MHz, which was much smaller than κ. Accordingly, the cavity-pulling coefficient in the resonant cavity is P ≈ Γ/κ = 0.039.
Intracavity photon number at steady state. For the atomic number density n ′ = 1.57 × 10 13 /cm 3 at a vapor-cell temperature of 100 • C [33], the atomic number inside the cavity mode is N = 1 4 n ′ πL cell Only the atoms with velocities between −∆υ/2 ∼∆υ/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 ∆φ (or cavity-frequency detuning, ν c − ν 0 ) is obtained. The atomic energy level is shown in Fig. 1a, where the energy states are labelled as |i . Using the rotating wave approximation (RWA) approximation, 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 |i to |j .
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 int ≈ 1 Γ34+Γ36+Γ41 . Ideally, we would simplify the equations by setting the frequency detuning between the pumping laser and the atomic transition of |1 to |2 to be zero. ρ I 12 and ρ R 12 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. 4. ∆ = ω −ω 0 is the frequency detuning of the laser oscillation from the atomic transition. Since ∆ 2 ≪ 4g 2 (n + 1), we assume that ∆ = 0 in the main text. To verify the correctness of this assumption, we give the most accurate description of the detuning ∆ in Eq. (8). The photon number at steady state is calculated by inserting the fitted result of ∆ in Fig. 3b into Eq. (8).
The results of the intracavity photon number at steady state with and without considering the detuning ∆ are shown as the green dotted line and the red solid line in Fig. 5. 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. 5, 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. (7), 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 Ω = 3λ21 3 Γ21I 2πhc . N eff and Ω as functions of I and N eff vs. T are shown in Fig. 6a and b, respectively.
Photon number as function of cavity decay rate. According to Eq. (2), the function of the intracavity photon number n with phase shift ∆φ varies with the cavitymirror reflectivity R, namely, the cavity power loss rate κ. When the cavity is anti-resonant (∆φ = (2q + 1) π), the intracavity photon number decreased with the increase in the reflectivity, which is shown in Fig. 7. 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.

Data availability
The data represented in Figs. 1-7 are available as Source Data. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.