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The development of active optical clock

The development of active optical clock The atomic clocks, whether operating at optical or microwave region, can be divided into two categories according to their working mode, namely the passive clocks and active clocks. The passive clocks, whose standard frequency is locked to an ultra-narrow atomic spectral line, such as laser cooled Cs beam or lattice trapped Sr atoms, depend on the spontaneous emission line. On the contrary, the active clocks, in which the atoms are used as the gain medium, are based on the stimulated emission radiation, their spectrum can be directly used as the frequency standard. Up to now, the active hydrogen maser has been the most stable microwave atomic clocks. Also, the Sr superradiant active atomic clock is prospects for a millihertz-linewidth laser. Moreover, the optical clocks are expected to surpass the performance of microwave clocks both in stability and uncertainty, since their higher working frequency. The active optical clock has the potential to improve the stability of the best clocks by 2 orders of magnitude. In this work, we introduce the development of active optical clocks, and their types is classified according to the energy-level struc- ture of atoms for stimulated radiation. Keywords Atomic clocks, Passive clocks, Active clocks, Stimulated emission radiation, Superradiant laser passive clocks, whose standard frequency is locked to an 1 Introduction ultra-narrow atomic spectral line, such as, laser cooled Cs As the most precise scientific device, atomic clocks are beam or lattice trapped Sr atoms, depend on spontane- widely used in numerous fields, such as military and ous emission line. On the contrary, the active clocks [10], national defense, cosmic exploration  [1–3], and scien- in which the atoms are used as gain medium, are based tific frontier research  [4–6]. According to operating fre - on the stimulated emission radiation, their spectrum can quency, atomic clocks can be divided into microwave be directly used as the frequency standard. clocks and optical clocks. The optical clocks, operating Currently, two types of optical clocks with the best per- at frequency domain by about five orders of magnitudes formance are both passive clocks, which are optical lat- higher than that of the microwave clocks, have been sur- tice clocks  [7, 11, 12] and ion optical clocks  [8, 13, 14]. pass the performance of microwave clocks both in sta- 1 3 Since the observation of the strongly forbidden S → P bility and uncertainty. The stability of state-of-the-art 0 0 −19 transition can be used as a clock transition by Lemonde optical clocks has reached the magnitude of 10  [7, 8], et  al.  [15] and Katori et al. [16] in 2003, Sr optical lattice which means it can verify general relativity within mil- clocks have developed rapidly. After that, the National limeter dimensions [9]. Atomic clocks also can be divided Institute of Standards and Technical (NIST), the Physi- into two categories according to their working mode, kalisch-Technische Bundesanstalt (PTB), the Obser- namely passive clocks and active clocks  (see Fig.  1). The vatoire de Paris, the National Research Council (NRC), the Joint Institute of Laboratory Astrophysics (JILA), the University of Tokyo, National Institute of Informa- *Correspondence: Tiantian Shi tion and Telecommunications Technology (NICT), and tts@pku.edu.cn others have conducted intensive research on Sr optical State Key Laboratory of Advanced Optical Communication Systems lattice clocks. Currently, the frequency stability of the and Networks, Institute of Quantum Electronics, School of Electronics, −17 Peking University, Yiheyuan Road 5, Beijing 100871, China Sr optical lattice clock can reach 4.8 × 10 / τ  [11], −18 and the system uncertainty can reach 2 × 10  [17]. In © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Zhang et al. AAPPS Bulletin (2023) 33:10 Page 2 of 17 Fig. 1 Classification of atomic clocks. According to the frequency band, atomic clocks can be divided into microwave clocks and optical clocks. Atomic clocks also can be divided into two categories according to their working mode, namely passive clocks and active clocks addition, the Yb optical lattice clock achieves a frequency Cao et  al. reported a robust, compact, and transport- −16 40 + stability of 1.5 × 10 / τ , a long-term frequency sta- able Ca single-ion optical clock with a system uncer- −19 −17 bility of 3.2 × 10  [7], and a system uncertainty of tainty of 1.1 × 10 , and the frequency stability reaches −18 −17 1.4 × 10  [18]. By cross-referencing with the Sr opti- 1.5 × 10 near 100,000 s [30]. cal lattice clock, the relative frequency uncertainties of In order to detect atomic transition spectra with mHz −16 the Hg optical lattice clocks achieved 1.8 × 10  [12] linewidth, the local oscillator linewidth must be nar- −17 and 8.4 × 10  [19], respectively. The National Institute row enough. At present, the ultra-narrow linewidth of Metrology (NIM) of China has realized the evaluation laser sources with low phase noise and high coherence and absolute frequency measurement of Sr optical lattice are mainly obtained by Pound-Drever-Hall (PDH) tech- −17 clock with a system uncertainty up to 2.8 × 10  [20, 21] nique  [31, 32]. It requires resonant cavity mirrors with −15 6 and a frequency stability of 1.8 × 10 / τ  [22]. The sys - high reflectivity coatings to achieve 10 or higher finesse. tem uncertainty of Yb optical lattice clock developed by To reduce the thermal noise of the resonant cavity  [33], −16 East China Normal University reaches 1.7 × 10  [23]. it is necessary to use single crystal silicon or microcrys- In recent years, breakthroughs have also been made in talline glass with ultra-low thermal expansion coeffi - ion-trapped optical clocks. In the 1980s, Dehmelt’s group cient as the cavity material [34]. In addition, the resonant proposed the idea of using the Paul trap to imprison cavity needs to be placed in an ultra-low temperature ions. Subsequently, the ion-trapped optical clocks with environment to reduce frequency drift  [35–37]. How- + + + + + Hg  [24], Sr  [25], Yb  [13], Al  [8, 26], and Ca  [27, 28] ever, even with the most stable resonant cavitiy, the sta- ions as quantum references developed rapidly. Among bilized laser is subject to cavity-length thermal noise, them, the PTB achieves a Yb optical clock with a sys- which causes phase drift that limits the laser linewidth −18 tem uncertainty of 3 × 10  [13] and tests the prin- to 0.125 − 0.3 Hz [38, 39]. Using super-stable resonant ciple of local position invariance  [29]. The NIST’s Al cavity results in expensive cost and complex systems, is −19 optical clock has reached the 10 order of magnitude environmentally sensitive, and does not solve the cavity- for system frequency accuracy assessment  [8]. In addi- length thermal noise problem essentially. tion, the system frequency uncertainty of the Ca opti- To overcome the problem that the optical cavity used cal clock realized by the Wuhan Institute of Physics for frequency stabilization is limited by thermal noise, and Mathematics (WIPM) of Chinese Academy of Sci- Chen proposed the concept of active optical clock (AOC) −17 ences is 5.1 × 10 , and the frequency stability reaches in 2005 [10, 40]. After the AOC was proposed, it received −17 7 × 10 at an average time of 20,000 s  [28]. Recently, wide attention from international peers. In the 2015 IEEE Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 3 of 17 International Frequency Control Symposium (IFCS), the 1470  nm active light field based on Cs thermal atoms AOC was listed as one of the three most emerging tech- with a linewidth of 53  Hz and demonstrate its superior nologies receiving the most attention in this field. Cur - cavity-pulling suppression property [62]. rently, there are research groups across the globe such Compared to passive optical clocks, AOCs output an as JILA  [41–43], NIST, University of Colorado, Vienna optical frequency standard directly based on the prin- University of Technology (TU Wien) [44, 45], University ciple of stimulated radiation. Since it works in the bad- of Copenhagen  [46], University of Amsterdam  [47, 48], cavity region, which means the cavity-mode linewidth is Aarhus University  [49, 50], Zhengzhou University  [51, larger than the atomic gain linewidth, it can effectively 52], Physical Research Laboratory (India)  [53], Univer- solve the cavity-length thermal noise problem of passive sity of Hamburg  [54], University of Innsbruck  [55, 56], optical clocks based on PDH technique. AOC has impor- Leibniz University Hannover  [57], Nicolaus Copernicus tant application prospects and research significance in University  [58], Academia Sinica  [59], Guru Nanak Dev a lot of fields, such as precision scientific measurement, University  [60], Université Sorbonne Paris Nord  [61], physical theory verification, quantum simulation, and and Peking University  [53, 62] conducting research on gravitational detection. bad-cavity superradiant laser based on various atomic In this paper, we start with the origin of AOC. In Sec- systems. Currently, the JILA research group achieves a tion  2, we will focus on the fundamental principles of superradiant pulsed lasing based on the ultra-narrow AOC. We then review the scheme of AOC in Section  3, transition linewidth of Sr atoms with a frequency stabil- which are classified into two-, three- and four-level −16 −15 ity of 6.7 × 10 at 1 s and an accuracy of 4 × 10  [43]. according to the quantum reference transition energy The University of Hamburg realizes a hyperbolic sinu - levels. In Section  4, we present some applications of soidal superradiant light field based on Ca atom AOC. Section 5 provides a summary and outlook. with intensity proportional to the particle number squared [54]. The Niels Bohr Research at the University of 2 Basic principle of active optical clock Copenhagen demonstrates a pulsed superradiant signal In AOC, multi-atom coherent stimulated radiation is based on Sr [46]. The Aarhus University realizes super - formed between atomic transition levels through the radiant pulsed laser with linewidth less than 2  Hz  [49]. weak feedback of optical resonator; the schematic dia- The Peking University achieves a continuous-wave (CW) gram of the AOC is shown in Fig.  2. AOCs are based on Fig. 2 Schematic diagram of an AOC. Due to the high phase coherence of collective atomic dipole, the stimulated radiation can be used as optical frequency standard directly. AOC works in bad-cavity region, and the gain linewidth is much smaller than the cavity-mode linewidth, so the center frequency of output laser depends on quantum transition frequency, which can effectively suppress the cavity-pulling effect Zhang et al. AAPPS Bulletin (2023) 33:10 Page 4 of 17 the principle of stimulated radiation, in which atoms are Therefore, passive optical clocks are inevitably affected pumped to the excited state under the action of pumping by the cavity-length thermal noise, which affects their light, creating a population inversion between two energy frequency stability. Conversely, the AOCs work in bad- levels. These atoms that have achieved population inver - cavity region, where Ŵ ≪ κ , P ≪ 1 . The effect of the cav - sion are placed in an optical resonant cavity as the gain ity-mode frequency variations on output laser frequency medium of the clock transition. Under the weak feed- is greatly suppressed, so the output laser frequency is back of the cavity, the coherent radiation output is real- immune to ambient noise. Here, the bad-cavity factor is ized as an active optical frequency standard. Due to the defined as the ratio of the atomic decay rate to the cav - phase coherence of collective dipole emission, the output ity dissipation rate, a = . Using the bad-cavity factor, laser has excellent phase coherence, which can exceed the cavity-pulling coefficient can also be expressed as the quantum-limited linewidth determined by spontane- P = . For AOC, the bad-cavity factor a ≫ 1 and the 1+a ous emission [42, 63]. AOC is an innovative way to obtain impact of cavity-length fluctuation on laser frequency is high coherence, ultra-narrow linewidth lasers. Using reduced drastically. quantum reference system as gain medium, its stimulated radiation can be directly used as the clock laser. Conven- 2.2 Linewidth characteristic tional passive optical clocks work in good-cavity region, The AOC output laser has excellent phase coherence its local oscillator laser generally use medium with broad and the laser linewidth can break the quantum-limited gain linewidth, and the cavity-mode linewidth is nar- hν linewidth �ν = , which is determined by sponta- 4π P out rower than the gain linewidth, so its output frequency is neous radiation. According to the modified Schawlow- mainly determined by the central frequency of the cav- Townes formula in bad-cavity regime, the linewidth of ity mode. When the external environment noise causes bad-cavity laser can be expressed as [64] the change of cavity length, the output laser’s frequency will change accordingly. Unlike conventional good-cavity hν κ Ŵ �ν = N (3) AOC sp laser, the atomic gain linewidth of AOC is narrower than 4π P Ŵ + κ out the cavity-mode linewidth. Therefore, the AOC works Here, it is assumed that the cavity-mode center frequency in the bad-cavity regime. The center frequency of the coincides with the atomic transition frequency. P is clock laser depends on quantum transition frequency, out the output laser power, h is Planck’s constant, ν is the which can effectively suppress the cavity-pulling effect atomic transition frequency, N = is the spon- and reduce the impact of the cavity-length thermal noise. sp N −N 1 2 taneous radiation factor, and N , N correspond to the Utilizing AOC, the laser linewidth is expected to reach 1 2 particle number in upper and lower levels, respectively. mHz level [41] and the frequency stability is expected to The first two terms of Eq. ( 3) represent the quantum- exceed existing optical clocks. There are several advan - limited linewidth determined by spontaneous radiation. tages of AOC, as described next. For good-cavity laser, the linewidth can be reduced to �ν = , with M being the average intracavity Good c 4πM 2.1 Cavity‑pulling characteristic photon number. For AOC operating in bad-cavity region, The relationship between the center frequency ν of AOC Ŵ ≪ κ , the linewidth can be simplified as �ν = . Bad πκM output laser and the frequency ν of atomic transition It is possible to break through the quantum-limited frequency can be expressed as [42] linewidth determined by spontaneous radiation and reach the order of mHz. ν − ν = (ν − ν ), (1) In summary, the AOC working in the bad-cavity limit 0 c 0 Ŵ + κ utilizes atoms as the gain medium, whose stimulated emission radiation can be directly used as the frequency where ν denotes the cavity-mode frequency, Ŵ is the standard. Therefore, compared with passive clocks, atomic decay rate, and κ is the cavity dissipation rate. The AOCs have two significant advantages of cavity-pulling output laser frequency changes with the cavity-mode fre- suppression effect and narrower laser linewidth. quency by an amount P called cavity-pulling coefficient. From Eq. (1), P can be expressed as 3 Research Schemes of AOC dν Ŵ P = = . (2) According to the energy-level structure of atoms for dν Ŵ + κ stimulated radiation, AOC can be divided into three In passive optical clock, Ŵ ≫ κ , the corresponding cav- categories: two-, three-, and four-level AOC. Among ity-pulling coefficient P ≈ 1 , that is, the output laser them, the two-level scheme includes atomic beam, Fara- frequency follows the cavity-mode frequency exactly. day atomic filter, and optical lattice type; the three-level Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 5 of 17 2 1 3 scheme includes optical lattice, bad-cavity Raman, and following energy levels are available: (4s ) S − (4s4p) P 0 1 2 3 3 ion trap type; and the four-level scheme includes thermal for Ca atoms, (3s ) P − (3s3p) P for Mg atoms, and 0 1 2 1 3 atom and optical-lattice-trapped cold atom type, which (5s ) S − (5s5p) P for Sr atoms. 0 1 are shown in Fig. 3. Take the Ca atom  [66] as an example to illustrate more specifically. A 657 nm laser pumps the collimated 2 1 3.1 Two‑level AOC Ca atoms from the ground state (4s ) S to metastable For two-level AOCs, international research is being car- state (4s4p) P , achieving a population inversion. Detec- ried out including JILA, Peking University, University tion is performed with a 423 nm blue laser, which is of Copenhagen, University of Hamburg, and so on. The located after the pump laser and divided into two parts. atoms used in the two-level AOCs are currently focused One is used to transfer unpumped ground state atoms on Ca and Sr. Some properties of the two-level AOCs, to (4s4p) P for electron-shelving detection. Once the including the clock transition levels, their wavelengths atoms are excited to metastable state, they fall back to and working types, are given in Table 1. the ground state after a few tens of centimeters of flight distance. Then, they are detected by another 423 nm 3.1.1 Atomic beam two‑level AOC laser beam. In this scheme, interaction time broaden- In terms of two-level AOCs, atomic beam type is appli- ing is the most dominant of all broadening mechanisms, cable to any two-level atomic system with metastable because it is much larger than the spontaneous radiation upper state. There are abundant quantum systems avail - rate at P . Therefore, the interaction time broadening able, atomic beams such as Mg [65], Ca [66], Sr [67], Ba, can be considered as the atom gain linewidth, which is and molecular beams such as CH  [68] and OsO  [69]. 2π × 150 kHz . The cavity-mode linewidth is taken as 4 4 The basic working principle of atomic beam type AOC 2π × 10 kHz . With proper design of bad-cavity struc- is as follows: gaseous atoms generated by a heating ture, the limiting linewidth can reach 0.1 Hz according to oven are collimated to form an atomic beam. The atoms the modified Schawlow-Townes formula. However, there in the ground state are pumped to the metastable state are two problems with the atomic beam scheme. Firstly, to achieve population inversion between clock transi- the remaining first-order Doppler effect due to the trans - tion energy levels. After that, metastable state atoms verse velocity distribution leads to the broadening of gain interact with an optical resonant cavity to realize the linewidth, which can be overcome through laser cooling stimulated radiation output when they reach threshold technology. Secondly, the second-order Doppler effect condition of laser oscillation. For different atoms, the causes asymmetry in the gain profile, which becomes the Fig. 3 Different types of AOC based on the transition energy level of quantum gain medium Table 1 Properties of selected two-level AOC, including the clock transitions, their wavelengths, and working types. The corresponding research groups and references are also shown Atom Clock transition /nm Type Research group References 40 1 3 657 atomic beam Peking University [66] Ca S - P 0 1 88 1 3 689 atomic beam University of Copenhagen [46] Sr S - P 0 1 40 1 3 657 atomic beam JILA [70] Ca S - P 0 1 88 1 3 Sr S - P 0 1 87 1 3 698 optical lattice JILA [43, 71] Sr S - P 0 0 40 1 3 657 optical lattice University of Hamburg [54] Ca S - P 0 1 133 2 2 Cs S P 852 Faraday Peking University [72] 1/2 3/2 Zhang et al. AAPPS Bulletin (2023) 33:10 Page 6 of 17 main limiting factor for clock accuracy. Thus, the perfor - course it can be extended to other kinds of alkaline earth mance of the two-level atomic beam AOC is ultimately metal atoms. The atoms can be continuously pumped to limited by the second-order Doppler shift, although it the excited state to obtain a constant population inver- adapts to a wide range of two-level atomic system with sion, thus overcoming the influence of Dick noise on metastable upper state. short stability and achieving CW superradiant laser out- Through laser cooling technique, the Doppler effect can put with a theoretical linewidth of mHz level. be largely suppressed. The Niels Bohr Institute at the Uni - versity of Copenhagen has chosen Sr atoms as a quantum 3.1.2 O ptical lattice two‑level AOC reference for laser frequency stabilization based on cavity- On the basis of laser cooling, atoms can be further loaded enhanced atomic interaction, using both passive and active into a magic wavelength optical lattice, which confines methods  [73]. In the passive scheme  [74, 75], a cavity- the atoms tightly within the Lamb-Dicke range along enhanced modulation transfer spectrum is employed and the cavity axis and eliminates the first-order Doppler the corresponding atomic phase shift is used as an error shift more effectively. In 2016, JILA achieved the first 3 1 signal. Since the atom-cavity coupling occurs in the bad- pulsed superradiant laser at 698 nm ( P → S ) using 0 0 cavity regime, the cavity-pulling effect can be signifi - Sr atoms trapped in the optical lattice [71]. Experimen- cantly suppressed compared to the conventional scheme tally, Sr atoms loaded into the optical lattice are pre- of locking the local oscillator to an ultra-stable cavity. In pared by a two-stage cooling process. Initial trapping and 7 88 the active scheme [46], 2 × 10 Sr atoms are confined in cooling are performed using the dipole-allowed 461 nm 1 1 a large waist cavity by laser cooling and magneto-optical S → P transition, and further cooling is performed 0 1 1 3 1 3 trapping (MOT). The 689 nm S → P dipole-forbidden using narrow linewidth 689 nm S → P transition. 0 1 0 1 transition with a natural linewidth of 7.5 kHz is used as Among them, the 689 nm laser is also used to pump Sr 1 3 clock transition, because it has a lifetime many orders of atoms from ground state S to P . Then the atoms trans - 0 1 magnitude longer than those of dipole-allowed transitions. fer to the upper level P of clock transition via an adiaba- By applying 689 nm π pulse, the Sr atoms cooled to mK tic passage by a 698 nm laser. Using an 813.4274 nm laser, are excited to upper level P . When the cavity-mode fre- which is close to the magic wavelength, as the lattice quency is resonant with the atomic transition, the atoms light, imparts near equal shifts to the ground and excited immediately establish coherence through the cavity field, states of the lasing transition so as to eliminate first-order achieving pulsed superradiance with high spectral purity. Doppler shift. Compared with independently radiating The laser operates in the bad-cavity region, where the pho - atoms, the atomic collective emissivity is enhanced more ton radiation is substantially enhanced due to the collec- than 10,000 times after coupling to the cavity. On this tive cooperativity. Its maximum output power is close to basis, JILA characterized the Sr atomic ultra-narrow 1µW . On the theoretical side, the atomic beam continu- linewidth superradiant laser in 2018 with a linewidth on −16 ous superradiant laser was studied by Université Sorbonne the order of 10 Hz , a frequency stability of 6.7 × 10 at −15 Paris Nord  [61]. They proposed a minimalistic model to 1  s, and an accuracy of 4 × 10  [43]. In this bad-cav- explain laser threshold, power, correlation properties, and ity regime, any fluctuation (thermal or mechanical) of linewidth. This model describes the dynamics of atoms the cavity length has much less influence on the output entering and leaving the cavity by a Hamiltonian process, laser spectrum, resulting in a cavity-pulling coefficient of −6 without stochastic approach. They demonstrated that the 2 × 10 obtained experimentally. For the Ca atom two- ultimate linewidth is set by the fundamental quantum fluc - level optical lattice AOC, the research group at the Uni- tuations of the collective atomic dipole and the continuous versity of Hamburg observed a hyperbolic secant shaped superradiant regime is tied to the growth of atom-atom superradiant pulse with intensity proportional to the correlations. square of the particle number in bad-cavity regime based 1 3 Cold-atom-based superradiant lasers have proven on S → P 657 nm transition [54]. The pulse duration 0 1 their superior performance, but parasitic heating from is much shorter than the natural lifetime of the P state, atomic repumping has so far limited these systems to and its decay time fluctuations are consistent with theo - pulsed operation [76]. This problem can be avoided using retical predictions. In this work, the population inversion a thermal atomic beam, because the pumping process is is achieved using incoherent pumping, which holds great performed outside the cavity. JILA proposed a new type promise for achieving continuous superradiant output. of superradiant laser using a hot atomic beam passing Theoretically, Zhang et  al. applied the Monte-Carlo through an optical cavity and show that the theoretical wave-function method (MCWF) method [50] to calculate minimum linewidth and maximum power are competi- the superradiant pulses with different initial atomic num - tive with the best ultracoherent clock laser  [70]. In this bers in the presence of atom loss, which is in agreement 40 88 article, Ca and Sr are analyzed as examples, but of with the experimental results in Ref. [71]. Since atoms are Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 7 of 17 lost rapidly in the experiment, steady-state superradiance between pulses  [44]. High-speed transport of ultra-cold cannot be formed. By introducing an incident flux of new atoms using a red-detuned one-dimensional optical lat- atoms, the laser linewidth can theoretically reach the tice has been experimentally achieved in 2006, with travel order of mHz despite rapid atom number fluctuations. distance up to 20 cm , transport speeds up to 6m/s , and 3 2 Zhang et al. also used a stochastic mean-field theory [52] accelerations up to 2.6 × 10 m/s  [77]. In the sequential to describe active frequency measurements of pulsed coupling and decoupling scheme, a blue-detuned optical superradiant emission generated by thousands of Sr lattice traps the atoms in the Lamb-Dicke region. When atoms trapped in an optical cavity. This theory combines an atomic ensemble is located inside the cavity and starts cavity quantum electrodynamics and quantum measure- to radiate, a second ensemble is excited to the upper state ment theory, and treats the whole atom ensemble as ten and moves toward the cavity. Once the first ensemble separate subensembles with different transition frequen - completes radiation, the second ensemble enters the cav- cies. The theoretically obtained superradiant beats sig - ity and goes on to radiating while still maintaining the nal, noisy power spectra, and frequency uncertainty are phase of the cavity. At the same time, the first ensem - in agreement with the experimental results reported in ble exits the cavity and a new inverted ensemble is pre- Ref. [43]. Meanwhile, this theory predicts that the short- pared. Repeating the above process, because the phase of −17 term frequency uncertainty can reach 7 × 10 / (τ/s) the cavity remains constant, the superradiant pulses are by using longer superradiance pulses of similar strength also highly phase-coherent with each other. It is worth and by reducing the time for single measurements. mentioning that the atoms can be prepared in the upper For the two-level optical lattice active superradiant lasing state outside the cavity, which can circumvent per- laser, Gogyan et al. [58] introduced a semiclassical theory turbations due to AC Stark shift. This sequential coupling of superradiant pulses generated by alkaline earth atoms method is a promising approach towards creating an and performed a feasibility analysis for an experimen- active optical frequency standard. Based on the sequen- tal implementation using the example of Ca atoms, tial coupling method proposed by Kazakov et  al.  [44], reported in Ref.  [54]. The results show that the inho - JILA has recently realized the transport of atoms using a mogeneous optical pumping procedure has a significant moving optical lattice. [78], which is expected to be used effect on the superradiant pulse characteristics. Uni - to realize a continuous superradiant laser. versity of Innsbruck has evaluated the effects of dipole- dipole interaction and collective spontaneous decay on 3.1.3 F araday type two‑level AOC the radiation properties of the superradiant laser  [55], Compared with the atomic beam and optical lattice such as linewidth, stability, and cavity-pulling, through scheme, the two-level Faraday type is simple in struc- direct numerical simulations of minority-atom systems ture and easy to implement. The Faraday laser uses an with different geometries and densities. Besides, they anti-reflective coated laser diode as the gain medium demonstrated that in the bad-cavity regime, by choos- and a Faraday atomic filter as the frequency-selective ing appropriate cavity detuning parameters, atoms can device  [79]. The laser frequency can be stabilized within be trapped and cooled by the cavity field generated by the transmission bandwidth of the atomic filter so that their own stimulated radiation [56]. Academia Sinica [59] the laser linewidth can be narrowed effectively through theoretically investigated the effect of long-range dipole- optical feedback  [80]. The AOC scheme is adopted to dipole interaction on the steady-state active superradi- optimize the Faraday laser’s frequency stability and ant laser. The cavity photon number and the coherence named Faraday AOC  [72]. Its core principle is that the between atoms have oscillation phenomenon with gain and quantum reference are independent of each interparticle distance of the atoms. The maximal cavity other, thus reducing the influence of noise in the gain part photon number and the minimal spectral linewidth are on the frequency stability. The gain can be provided by located under the condition of equidistant atomic arrays, materials such as semiconductors, solids, or dyes, while which can facilitate precision measurements and the narrow-band atomic filters provide a frequency refer - development of next-generation optical clocks. ence. By choosing suitable parameters to make the exter- For the schemes mentioned above using laser-cooled nal cavity-mode linewidth much larger than the atomic atomic beams and neutral atoms trapped in optical lat- filter bandwidth, the laser works in the bad-cavity region, tices, the superradiant laser output can only be oper- thus reducing the cavity-pulling effect and improving fre - ated at pulsed mode. There is no phase coherence quency stability. Meanwhile, the laser frequency is deter- between different individual pulses, and the pulse dura - mined by quantum transition frequency, which can be tion limits the stability of the output laser. To achieve directly used as a stable frequency standard. This scheme a continuous superradiant laser, sequential coupling can satisfy the laser oscillation threshold by increasing and decoupling can be used to maintain the coherence the pumping efficiency of the gain medium and obtain an Zhang et al. AAPPS Bulletin (2023) 33:10 Page 8 of 17 3 1 active optical frequency standard with narrow linewidth form the population inversion between P and S , thus 1 0 by compressing the atomic filter transmission bandwidth. realizing 657 nm optical frequency standard. This work Although this scheme is simple in structure, it is not presents the first neutral-atom-based optical lattice AOC, easy to narrow the transmission bandwidth to natu- which is expected to reach sub-Hz linewidth. In 2007, ral linewidth level, leading to a significant challenge in this group revealed that a 1 mHz linewidth optical clock further enhancement of bad-cavity factor. At present, could be realized by exploiting the phase-matching effect the linewidth of the Faraday active optical frequency of the three-level -type Sr atomic system confined in standard based on the thermal Cs atomic gas cell is magic wavelength optical lattice  [63]. When the nonadi- at 100 Hz order. It has not yet reached the theoretical abatic interaction of two quasimonochromatic fields with 1 1 3 88 value [72]. To solve this problem, cold atoms or ions can the states S , P , and P of Sr achieves phase coher- 0 1 0 be used as quantum frequency reference mode-selecting ence, a frequency difference field with 1 mHz linewidth devices  [81], which can effectively suppress the Doppler will be generated by the nonlinear crystal placed in a FP effect, reduce the transmission bandwidth, enhance the cavity. cavity-pulling suppression, and thus compress the laser A method to obtain a laser with mHz linewidth was linewidth. also proposed by the JILA research group in 2009  [41]: Sr atoms in an optical lattice collectively emit pho- 1 3 3.2 Three‑level AOC tons on the ultranarrow clock transition S → P , 0 0 The three-level AOC includes optical lattice, ion trap, into the mode of a high Q optical cavity. Since the cou- and bad-cavity Raman type. For optical lattice type, the pling between atoms and the optical field is completely atoms are trapped during the measurement period, so collective, i.e., the phase of different atomic dipoles are it can exploit the very narrow hyperfine-induced ns S 0 perfectly coherent, and the output laser linewidth is 3 87 88 -nsnp P , such as Sr and Sr . JILA research group 0 expected to be narrower than the natural linewidth. This study the bad-cavity Raman laser and implement an scheme assumes that the atoms are confined in an opti - active magnetometer based on it. For ion-trapped three- cal lattice within a fixed lattice point so that the intera - level AOC, there are only relevant theoretical studies, tomic coupling is maximized and these atoms are in and no real experimental realization has been made the same phase within a specific cavity mode. Repump - 1 3 yet. Properties of selected three-level AOC are shown ing lasers drive atoms from S to P , and then atoms 0 0 in Table  2, including the clock transitions, their wave- transfer to S . Due to spontaneous radiation, atoms in 3 3 3 lengths, and working types. the S state decay to the P and P , forming Raman 1 2 0 transition between these two states to implement side- 3.2.1 Optical lattice three‑level AOC band cooling to the vibrational ground state. Besides, the The Peking University firstly proposed an optical lattice repumping lasers also pump all atoms to the P meta- three-level AOC scheme in 2005  [40]. They use 423 nm stable level, thus satisfying inversion for laser transition. blue MOT and 657 nm red MOT to cool the Ca atoms Since the total relaxation rate Ŵ of the atomic dipole is 3 −1 and then confine them in a magic wavelength optical lat - at most on the order of 10 s and the cavity decay rate 5 −1 tice. The 423 nm and 1201 nm lasers are used to pump the κ is 9.4 × 10 s , the cavity-pulling suppression fac- 1 3 1 3 Ca atoms from ground state S to P through P , and tor is at least on the order of 10 . Theoretical calculation 0 2 1 the atoms are concentrated in P by repumping laser to shows laser linewidth can reach the mHz level. However, Table 2 Properties of selected three-level AOC, including the clock transitions, their wavelengths, and working types. The corresponding research groups and references are also shown Atom Clock transition /nm Type Research group Reference 40 1 3 657 Optical lattice Peking University [40] Ca S - P 0 1 88 1 3 698 Optical lattice Peking University [63] Sr S - P 0 0 87 1 3 698 Optical lattice JILA [41] Sr S - P 0 0 88 1 3 689 Optical lattice JILA [76] Sr S - P 0 1 88 1 3 689 “Hybrid” type TU Wien, JILA, EU [44, 47, 78, 82] Sr S - P 0 1 87 1 3 Sr S - P 0 0 87 2 2 780 Bad-cavity Raman JILA [42, 83–85] Rb S - P 1/2 3/2 171 2 2 435 Ion trap Peking University [86] Yb S - D 1/2 3/2 176 + 1 3 804 Ion trap TU Wien [45, 87] Lu S - D 0 2 Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 9 of 17 this result can only be achieved if the effective repump Astronomy at Aarhus University theoretically analyzed 3 −1 6 87 88 rate reaches 10 s and the atom number is at 10 level. Sr and Sr atoms also trapped within a one-dimen- Under this condition, the fluctuations in atomic tran - sional optical lattice in a bad-cavity region  [49]. Their 3 1 87 sition frequency introduced due to AC Stark shift are calculations confirm that using P → S of Sr atoms 0 1 3 1 88 negligible at mHz level. Moreover, the maximum laser and P → S of Sr atoms transitions, it is possible to 1 0 output power is proportional to the square of the atom realize the narrow linewidth of superradiant lasing. Spe- number. When the atom number is at 10 , the power can cially, under strong driving of the dipole-forbidden tran- 3 1 88 reach pW order, enough to be used for the phase locking sition P → S of Sr atoms, the superradiant laser 1 0 of a slave optical local oscillator. linewidth can be further narrowed due to the coherent Subsequently, the JILA group proposed the idea that excitation of the cavity field. using a high-finesse resonant cavity, based on alkaline- In 2014, Kazakov et al. discussed in detail two options earth metal atoms with an ultra-narrow-linewidth transi- for implementing active optical frequency standard: opti- tion to achieve a steady-state superradiance[88]. In order cal lattice AOC and atomic beam AOC. They analyzed to obtain the intensity fluctuations of the steady-state some parameters required to achieve the best frequency superradiant laser, JILA performed an analysis using stability as well as the implications and challenges in the Monte Carlo simulations and semiclassical approxima- implementation [90]. In addition, a “hybrid” method was tion methods  [89]. They found that the light exhibits proposed in which the atoms are prepared outside the bunching below threshold, is to a good approximation cavity, fed into the cavity by a “guide beam” or “optical coherent in the superradiant regime, and is chaotic above conveyors” to complete the stimulated radiation pro- the second threshold. Toward achieving mHz linewidth, cess. A blue-detuned optical lattice is used to prevent JILA studied superradiant lasing on the 7.5 kHz linewidth the atoms from moving along the cavity axis, suppress- 3 1 dipole-forbidden P → S transition at 689 nm , with an ing decoherence and first-order Doppler effects. JILA has 1 0 88 88 ensemble of Sr atoms tightly trapped in a 1D optical lat- recently realized continuous loading of ultra-cold Sr tice [76]. This laser is in a superradiant crossover regime, atoms into a high-finesse ring cavity and using a mov - which means it can be operated at the crossover between ing optical lattice to transport the atoms along the cav- good- and bad-cavity regimes. The cold-atom gain ity axis  [78]. Experimentally, the loading rate reaches medium can be repumped to achieve quasi-steady-state 2.1(3) × 10   atoms/s. This work lays the foundation for lasing, and the frequency of the emitted light is deter- the future implementation of continuous-type active mined by the atomic transition frequency when oper- superradiant lasers based on the mHz clock transition of ated in the bad-cavity regime. They also characterized Sr atom. the cavity-pulling suppression of the bad cavity, where Aiming to achieve a CW AOC, the European Union the laser frequency variation is reduced by an order of (EU) has set up the iqClock project, led by the Univer- magnitude. Experimentally, the cavity-pulling coeffi - sity of Amsterdam, in collaboration with six universities cient is 0.09(2). They also obtained heterodyne power (University of Amsterdam, University of Birmingham, spectral density (PSD) between output light and 689 nm Nicolaus Copernicus University, University of Copen- pump laser, with Lorentzian (Gaussian) full width at half hagen, Vienna University of Technology, University of maximum (FWHM) of 6.0(3) (4.7(3)) kHz. The measured Innsbruck) and six companies (Teledyne e2v, TOPTICA, linewidth is slightly narrower than the natural linewidth NKT Photonics, Acktar, Chronos Technology Ltd, British of the lasing transition (7.5  kHz) and far narrower than Telecom). A cold atomic beam scheme is used to achieve the linewidth imposed by repumping (100 kHz), exhibit- CW clock lasers by transporting Sr atoms through a ing the linewidth-narrowing characteristic of synchroni- moving optical lattice into a ring cavity. A clock laser zation in a laser. linewidth of 100 mHz is expected to be achieved within 5 Zhang et  al. theoretically explained this linewidth- years and linewidth on the order of mHz within 10 years. narrowing property  [51]. When the ultracold Sr In 2019, the University of Amsterdam realized a continu- atoms in the optical lattice are exposed to a magnetic ous guided atomic beam of Sr atoms with a phase-space −4 7 field, the ensemble of atoms with Zeeman-split excited density exceeding 10 and a flux of 3 × 10  atoms/s [47]. states exhibits lasing with very narrow linewidth, which With the optical guide, the atoms reach a velocity of is orders of magnitude smaller than both the cavity 8.4  cm/s and can be used to complement the gain linewidth and the incoherent atomic decay and exci- medium of the steady-state atom laser, which is an tation rates. The narrow-linewidth lasing is due to an important step towards the realization of a steady-state interplay of multiatom superradiant effects and the cou - superradiant AOC. In 2021, this group demonstrated a pling of bright and dark atom-light dressed states by the steady-state MOT of fermionic Sr atoms operating on the 1 3 magnetic field. In 2018, the Department of Physics and 7.5-kHz-wide S - P transition [48]. This MOT contains 0 1 Zhang et al. AAPPS Bulletin (2023) 33:10 Page 10 of 17 7 7 superradiant laser, the decay rate of a single particle can 8.4 × 10 atoms with a loading rate of 1.3 × 10   atoms/s be far balanced with the repumping rate by appropriate and an average temperature of 12 µK , which can be used design so as to increase the laser cooling and trapping to provide a high flux of ultracold atoms source for the time. realization of a continuous superradiant AOC. Based on Under this foundation, the JILA group investigated the steady-state MOT, this group has also achieved con- the oscillation relaxation, stability, and cavity feedback tinuous Bose-Einstein condensation  [91]. Through the characteristics of the bad-cavity Raman superradiant magic wavelength optical conveyor in the ring cavity, a laser  [84]. Moreover, they demonstrated a hybrid mode continuous source of ultracold Sr atoms in the excited in which the laser can switch between active sensing and state P can be realized, delivering several tens of mil- 88 87 passive phase measurements  [85]. The results culmi - lions of Sr atoms or millions of Sr atoms per sec- nate in a hybrid sensor that combines active sensing of ond [82]. The use of the ring cavity increases the transfer the collective atomic phase during superradiant emis- speed, reduces atom losses and decreases the density of sion with passive phase measurements using Ramsey-like the atoms, paving the way for CW superradiant AOC. evolution times, which are of guiding significance for the future development of ultra-narrow linewidth superradi- 3.2.2 Bad‑cavity Raman three‑level AOC ant laser. In 2012, JILA group proposed an bad-cavity Raman laser experimental scheme  [42, 83], which achieved a Raman 3.2.3 Ion tr ap three‑level AOC superradiant laser with an average photon number less In 2014, the Peking University proposed an active ion than 0.2 in the cavity. Experimentally, the laser operates optical clock scheme  [86] using cold ions trapped in in deep bad-cavity region, where the ratio of the trans- Paul trap as gain medium, which is expected to achieve verse decoherence rate to cavity decay rate of the laser −5 −3 6 active optical frequency standard with mHz linewidth. transition is in the range of 2 × 10 ∼ 10 . Ab out 10 The basic principle of this scheme is similar to that of the Rb atoms are trapped by 823 nm laser in one-dimen- optical lattice scheme, and theoretical studies have found sional optical lattice with a temperature of 40µK . The 171 191 + 137,138 + 43 + 87,88 + that Yb , Hg , Ba , Ca , and Sr ions cavity is coupled to an optically dressed state that mim- are suitable for active ion optical clock. Taking Yb as ics a long-lived optically excited state. A 795 nm linearly 2 2 an example, the 435.5 nm S (F = 0) − D (F = 1) 1/2 3/2 polarized dressing laser is applied to two magnetically transition with a natural linewidth of 3.1 Hz is chosen insensitive energy levels to induce Raman transition. If as clock laser. The cooling light and the repumping light the atoms are pumped continuously from ground state 2 2 correspond to 369.5 nm S (F = 0) − P (F = 1) 1/2 1/2 to metastable state while applying the dressing laser, a 2 2 and S (F = 1) − P (F = 0) transitions, respec- 1/2 1/2 quasi-continuous superradiance laser with a duration tively, and both can be used for pumping ions to upper of 20 − 140 ms can be obtained. Each atom can radiate 2 2 energy level D (F = 1, 2) . The ions at D (F = 2) 3/2 3/2 approximately 35 photons into the cavity mode. Under are pumped to D (F = 1) by 935 nm repumping 1/2 the action of light, collisions between atoms cause them laser. Eventually, most of the ions are transferred to to escape from the optical lattice, which eventually D (F = 1) to achieve population inversion and stim- 3/2 leads to a break in superradiance. The power spectrum ulated radiation output. Theoretically, the laser output (PSD) was obtained by heterodyning the superradiant power can be up to 37 pW when reaching steady state, laser and the dressing laser with a Gaussian FWHM of and it can be increased to 77pW by increasing the light 350 Hz and a Lorentzian FWHM of 4.5 Hz . Although intensity and the ions number. However, it is difficult to this result is much narrower than the spontaneous radia- significantly increase the ion number experimentally, and tion linewidth, it differs significantly from the theoreti - the ion AOC also suffers from the light shift caused by cally calculated 2(1) mHz linewidth due to the dispersion pumping laser, which ultimately affects the clock laser’s detuning of the cavity-mode frequency caused by atom performance. number changing. In this experiment, a surprisingly −5 −3 In 2017, Kazakov et  al. proposed that a bad-cav- tiny cavity-pulling coefficient P = 4 × 10 ∼ 2 × 10 ity laser may be realized using forbidden transitions is obtained, and it can be further enhanced by reducing in large ensembles of cold ions that form a spherical decay rate γ , which in turn suppresses the laser linewidth. Coulomb crystal in a linear Paul trap  [45], which can Overall, this scheme proves the feasibility and superiority guarantee longer trap lifetimes relative to neutral-atom- of AOC scheme, but there are problems of discontinuous optical-lattice type. Micromotion-induced shifts such output and weak power. The 795 nm dressing laser and as the second-order Doppler and DC Stark shifts can be 780 nm pump laser will introduce light shift, resulting in a suppressed by operating the ion trap at a magic fre- the bad-cavity Raman laser cannot be used as an opti- 176 + quency. Considering Lu ions imprisoned in a high cal frequency standard. To further achieve continuous Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 11 of 17 shows the simplified energy level of the three- and finesse ( F = 10 ) optical resonant cavity, an output 3 1 four-level AOC schemes. In the three-level scheme, power of 0.5  pW can be achieved based on the D - S 2 0 the light shift is inevitably introduced because the transition when the ion number reaches 10 and the con- clock laser shares the ground state |1� with the pump- finement frequency is 1  MHz  [87]. If proper continuous ing laser. Conversely, in the four-level scheme, atoms cooling and pumping are provided, a truly steady-state are pumped from ground state |1� to excited state |2� . AOC lasing in the bad-cavity regime is expected. This Due to spontaneous radiation, they are transferred to laser is promising to achieve an active optical frequency the clock transition upper energy level |3� , where popu- standard as a local oscillator for the next-generation opti- lation inversion is formed between |3� and |4� to achieve cal clock scheme. stimulated radiation. Since the pumping laser and the clock laser transition energy levels are independent of 3.3 Four‑level AOC each other, the light shift can be significantly reduced In the two-level atomic beam scheme, the Doppler by choosing a suitable quantum system with a signifi - effect of thermal atomic system is more influential. cant frequency difference. Using laser cooling technology can reduce the first- In a four-level AOC scheme, the quantum system can order Doppler effect, but this will complicate the sys - select alkali metal atoms such as K [92], Rb [93, 94], and tem, and the residual Doppler effect will still limit the Cs  [62, 95–98]. Taking Cs atom as an example, the Cs clock laser performance. The three-level optical lattice, atomic gas cell is placed in a low-finesse optical cavity to Paul trap scheme for imprisoning atoms or ions is lim- make cavity-mode linewidth larger than gain linewidth ited by light shift caused by pumping laser. Moreover, to satisfy bad-cavity condition. Using a 459 nm laser as most of the above two schemes implement pulse AOC pumping laser, the Cs atoms are pumped from ground signals. In contrast, the four-level AOC scheme has state 6S to the second excited state 7P , then 1/2 1/2 three advantages. Firstly, the clock transition energy dropped to 7S state by spontaneous radiation, cre- 1/2 level does not involve the ground state, which can ating a population inversion between 7S and 6P 1/2 3/2 reduce light shift introduced by pumping laser. Sec- when reaching steady state. Under the weak feedback of ondly, a magnetic dipole transition energy level can the optical cavity, the atomic dipoles are spontaneously be chosen, where the laser emission coefficient is not synchronized with high coherence, producing a 1470 nm limited by smaller atom-cavity coupling constant com- stimulated radiation clock laser. Different atoms and pared to narrower linewidth electric dipole transi- their corresponding energy level choices for four-level tion. Finally, the four-level AOC can be continuously AOC are shown in Table 3. pumped to output a stable CW AOC signal. Figure  4 Fig. 4 Simplified energy level diagram of (a) three-level and (b) four-level AOC Table 3 A few options for four-level active optical clocks Atom Pumping laser Pumping laser corresponding energy Clock laser Clock transition level Cs 455 nm 6S → 7P 6S → 7P 1470 nm 7S → 6P 7S → 6P 1/2 3/2 1/2 1/2 1/2 3/2 1/2 1/2 459 nm 1359 nm Rb 420 nm 5S → 6P 5S → 6P 1367 nm 6S → 5P 6S → 5P 1/2 3/2 1/2 1/2 1/2 3/2 1/2 1/2 421 nm 1323 nm K 405 nm 4S → 5P 1252 nm 5S → 4P 1/2 3/2 1/2 3/2 Zhang et al. AAPPS Bulletin (2023) 33:10 Page 12 of 17 3.3.1 Thermal atom four‑level AOC the natural linewidth of 1.81 MHz , has been achieved at The four-energy level AOC scheme was first proposed by room temperature. Yu et  al. in 2010  [99].Theoretically, a superradiant laser Based on Cs four-level AOC, Shi et  al. proposed an with intensity proportional to N and linewidth scales to anti-resonant laser  [103], which is very different from 1/N is studied. In addition, the stationary state solution the classical AOC operating in cavity resonance condi- of full atomic cooperativity is derived, and the stability tions. The lasing is realized when the atomic resonance of the superradiant laser is analyzed under the assump- is between two adjacent cavity resonances, that is, the tion of no spontaneous radiation. Subsequently, Wang cavity length equal to an odd multiple of a quarter wave- et  al. carried out a related experiment based on the Cs length. The linewidth of anti-resonant laser is not broad - atomic quantum system  [100]. By measuring the inten- ened compared to resonant laser, and its cavity-pulling sity of fluorescence signals at different wavelengths, such suppression characteristic is stronger. Using this anti- as 1470 nm and 1359 nm , they determined the formation resonant laser, the AOC can be extended from the cav- of population inversion between clock transition energy ity resonance region to the anti-resonance region, further levels. During the same period, a four-level scheme based improving the performance of the current active optical on alkali metal K  [92], Cs  [95], and Rb  [93] atoms was frequency standard. theoretically verified for feasibility based on the density matrix equation, and data such as the wavelength of the 3.3.2 O ptical lattice four‑level AOC pumping laser and the energy level satisfying the popu- In order to solve the Doppler broadening problem in lation inversion were obtained. Moreover, the magic the four-level AOC based on thermal atoms, the Peking wavelengths corresponding to Rb atoms 6S → 5P University, Guru Nanak Dev University, Beijing National 1/2 1/2 and 6S → 5P were theoretically calculated in order Laboratory for Condensed Matter Physics, and the Indian 1/2 3/2 to reduce the Doppler effect  [101]. Using an optical lat - Physical Research Laboratory calculated the magic wave- tice to imprison the laser-cooled atoms in the Lamb-Dick length corresponding to the 1470 nm Cs atom clock tran- region, it is estimated that the effect of AC Stark fre - sition in detail  [53]. The optical lattice four-level AOC quency shift on the clock laser frequency stability can be scheme predicts output laser power up to 24 µW with a −18 −15 reduced to less than 10 . mHz linewidth and a frequency uncertainty of 2 × 10 . The AOC lasing at a wavelength of 1470 nm using the Singh et  al. obtained the magic wavelengths between all Cs four-level system was firstly achieved in 2013  [102]. possible hyperfine levels of the transitions in Rb and Cs On this basis, the 1359 nm and 1470 nm clock laser out- atoms  [60], which will help build a more stable AOC. puts were achieved successfully by using 455 nm and They also gave the static dipole polarizabilities of Rb and 459 nm pumping laser. The bad-cavity factor reached Cs atoms to validate the results. Using cold atoms impris- more than 40, and the 1470 nm laser linewidth reached oned in an optical lattice as the gain medium can reduce 407.3 Hz . For the Rb atom, a 1367 nm continuous active the gain linewidth broadening. In principle, an active optical frequency standard was achieved using a 420 nm optical frequency standard with a mHz linewidth, much semiconductor laser with modulated transfer spectrum smaller than the clock transition natural linewidth, can stabilization  [94]. To further stabilize the resonant cav- be obtained. ity length and reduce the frequency drift caused by the To sum up, the above two-, three-, and four-level AOC residual cavity-pulling effect, a dual-wavelength good- schemes all have advantages and disadvantages. The two- bad-cavity AOC is proposed  [97]. In this scheme, two level structure is simple and easy to implement. However, lasers of different wavelengths are output simultane - its performance is ultimately limited by the second-order ously in a common cavity and separately operate in Doppler shift for the atomic beam and Faraday schemes, good- and bad-cavity regions, called the good- and bad- the pulsed mode operation and the pW laser power for cavity lasers. The good-cavity laser is locked to an ultra- the optical lattice scheme. Although the optical lat- stable cavity by the PDH technique to stabilize the cavity tice three-level scheme can achieve higher performance length. Therefore, as a clock laser, the bad-cavity laser using narrow linewidth quantum transition, it is not easy will be further optimized due to the cavity-length stabi- to realize experimentally due to the small atom number lization with good-cavity laser. To reduce the impact of and weak output power. Moreover, the clock laser is ulti- asynchronous cavity-length variation between two same mately limited by light shift. Because the pumping laser systems on the linewidth broadening of the clock laser, is separated from the clock laser, the four-level scheme the cavity-length stabilization was realized by utilizing avoids the light shift caused by the pumping laser. More- the phase locking technique of good-cavity laser  [62]. over, the output laser is a continuous laser with a higher Experimentally, a 1470 nm CW active optical field with a power which can reach approximately 100µW , but few tens of Hz linewidth, four orders of magnitude below this scheme requires a higher atom number. The use of Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 13 of 17 thermal atom gas cell can meet the corresponding atom measurement of the gravitational potential, the verifica - number requirement. However, it is affected by the Dop - tion of general relativity, the detection of dark matter, and pler effect, which leads to a broadening of the gain pro - other fields. file. To further improve long-term stability, a cold-atom scheme can be used in the four-level scheme. 4.2 Application in precision spectroscopy AOCs use the atomic stimulated radiation signal directly 4 Applications of AOC as the clock laser, whose output linewidth is narrower The applications of active optical frequency standards than the natural linewidth determined by spontane- and clocks are wide ranging. Thanks to the development ous radiation. In terms of spectroscopy, it can be called of atomic clocks, time and frequency are the most pre- stimulated radiation laser spectrum. Theoretically, the cise physical quantities that can be measured currently. linewidth of this “stimulated radiation laser spectrum” As a result, tremendous advances in atomic clocks have is 7–8 orders of magnitude narrower than the natu- made it possible to measure other physical and techni- ral linewidth of atomic transition. Therefore, narrow cal quantities that can be traced back to frequency with linewidth CW AOC can provide new principles and unprecedented precision. AOCs output an optical fre- methods for precision atomic spectroscopy. For exam- quency standard directly based on the principle of stimu- ple, the Peking University has been able to accurately lated radiation. Since it works in the bad-cavity region, measure the 1470  nm transition line (natural linewidth which means the atomic gain linewidth is narrower than 1.81  MHz  [104]) to the Hz level using the Cs four- the cavity-mode linewidth, it can effectively solve the level AOC scheme[62], which is 5–6 orders of magni- cavity-length thermal noise problem. Due to the phase tude more accurate than the previous 1470  nm spectral coherence of collective dipole emission, the output laser line measurement. This method can also be univer - has excellent phase coherence, which can exceed the sally extended to other alkali metal atoms, realizing the quantum-limited linewidth determined by spontaneous high-precision measurement of transition spectral lines emission. Thanks to the superior characteristics of the with MHz natural linewidths, which is expected to be AOC, it can be used in a lot of fields, such as precision upgraded to the measurement accuracy based on con- measurement, physical theory verification, gravitational ventional optical clocks. In this way, the new principle detection, testing of quantum electrodynamics and so on. and technology of the AOC can enrich the international high-precision spectral measurements and improve the 4.1 S erves as a local oscillator for passive optical clocks measurement precision of atomic spectroscopy with In order to detect the atomic transition spectrum with MHz natural linewidths. mHz linewidth, the linewidth of the local oscillator laser of a conventional passive optical clock must be narrow 4.3 Hyperfine‑structure measurement enough. At present, the ultra-narrow linewidth laser In the work of Shi et  al.[105], the Cs atomic four-level source is mainly obtained by the PDH technique. The use AOC can be used not only to realize narrow linewidth of ultra-stable cavities leads to expensive costs, complex active optical frequency standard but also to realize the systems, and environmental sensitivity. Moreover, it can- hyperfine level measurement of Cs 7P in combina- 1/2 not solve the cavity-length thermal noise problem essen- tion with the Doppler effect. Since the AOC uses quan - tially. The narrow linewidth optical field based on the tum reference system stimulated radiation directly as AOC scheme provides an excellent option to solve this the clock transition signal, the output linewidth is much problem. It can be used as the local oscillator laser for smaller than the natural linewidth determined by spon- passive optical clocks. At the same time, this optical field taneous radiation, which can improve the measurement has the absolute value characteristic of quantum transi- accuracy of atomic hyperfine level structure constants. tion frequency between atomic energy levels. Combined The experimental principle is similar to that of the with the optical frequency synthesizer, we can realize Cs four-level AOC, where the 459 nm pumping laser is the frequency comparison between the AOC and other locked to 6S (F = 4) → 7P (F = 3&4) by modula- 1/2 1/2 high-precision optical clocks, thus realizing the trans- tion transfer spectrum. The Cs atoms in the atomic gas mission, comparison, and application of high-precision cell are thermal atoms with Doppler velocity distribu- active quantum frequency standards. In the above cases, tion, and the Cs atoms in different velocity groups in the AOC scheme will break through the technical bot- the cavity sense different pumping laser frequencies. tleneck of the passive optical clock stability currently, According to the selection rule, the Cs atoms located in which is limited by the cavity-length thermal noise. It can the two velocity groups of the 6S (F = 4)state can be 1/2 ′ ′ be applied to the definition of the second, the quantiza - excited to 7P (F = 3) and 7P (F = 4) , resp e ctively . 1/2 1/2 tion of the International System of Units, the precision Cs atoms propagate with the pumping laser with velocity Zhang et al. AAPPS Bulletin (2023) 33:10 Page 14 of 17 6 87 v =  × �ν/2 are sensed at pumping laser fre- 1 459.3 nm Based on previous experiments, 10 Rb atoms quency corresponding to 6S (F = 4) − 7P (F = 3) . 1/2 1/2 were imprisoned in a one-dimensional optical lattice Conversely, when Cs atoms counter-propagate with the after laser cooling to 40µK . The volume of the atom −3 3 pumping laser with velocity v =− × �ν/2 2 459.3 nm cloud is roughly 2.1 × 10 mm using a low-finesse are pumped to 7P (F = 4) , where �ν is the hyper- 1/2 ( F = 710 ) resonant cavity. The atoms are continu - fine level spacing of the 7P state. Atoms pumped 1/2 ously pumped to 5 S (F = 2,m = 2) and decay to 1/2 F ′ ′ to both 7P (F = 3) and 7P (F = 4) can drop to 1/2 1/2 5 S (F = 1,m = 1) through Raman transition. Using 1/2 F ′′ 7S (F = 4) through spontaneous radiation, creat- 1/2 these two magnetic field-sensitive states, the pseudospin ing a population inversion between 7S and 6P . 1/2 3/2 1/2 regime is formed. The states sensitive to the mag - Eventually, the atoms of both velocity groups can out- netic field are chosen instead of the cavity resonant fre - put 1470 nm bad-cavity clock laser corresponding to quency, because the laser operates in bad-cavity region ′′ ′′′ 7S (F = 4) − 6P (F = 5) . The frequency differ - 1/2 3/2 with cavity-pulling suppression effect. Experimentally, ence f of the stimulated emission lights of atoms with the wideband sensitivity at continuous active oscilla- two velocity groups can be measured by optical het- tion was measured, and the narrowband sensitivity in a 459.3 m erodyne, where �f = × �ν . The hyperfine level passive Ramsay-like mode was obtained by narrowband 1469.9 nm spacing �ν of the 7P state can be deduced from the 1/2 detection using the spin-echo technique. Its sensitivity experimentally measured f , and then the magnetic reaches 190 pT/ Hz at 1 kHz , and the effective detection dipole hyperfine constant A of 7P state can be found 1/2 volume, which is the volume of the atomic cloud, is about −3 3 using �ν = A × 4. 2 × 10 mm . Future experimental work will focus on The scheme uses one system for measurement, which achieving truly continuous operation and realizing sensi- can eliminate common mode noise, but is limited by the tivity at the phase diffusion limit. cavity-pulling effect. Considering the errors introduced by the cavity-pulling effect, the measurement result is 5 Summary and outlook f = 118.0347 ± 0.1827 MHz . Hence, the hyperfine level The AOC, which is based on the principle of stimulated spacing of the 7P state is 377.628 ± 0.584 MHz , and 1/2 radiation, its output signal has excellent phase coher- the magnetic dipole hyperfine constant is 94.41(15) MHz . ence and can be used directly as an optical frequency Data comparison shows that this work’s result is in gen- standard. Working in the bad-cavity region, where the eral agreement with those obtained previously using atomic gain linewidth is narrower than the cavity-mode saturated absorption spectroscopy and two-photon reso- linewidth, the AOC can effectively solve the cavity-length nance spectroscopy. thermal noise problem in passive optical clocks. We describe in detail the basic principles and characteristics of AOC and classify AOC into two-, three-, and four- 4.4 M agnetometer based on bad‑cavity Raman AOC level schemes according to the energy-level structure of superradiant laser atoms for stimulated radiation. Currently, the two-level Most atomic magnetometers pass probe light through scheme includes atomic beam, Faraday atomic filter, and atomic vapor and sense the response of the atom to optical lattice type; the three-level scheme includes opti- magnetic field by measuring the polarization rotation or cal lattice, bad-cavity Raman, and ion trapping type; the phase shift of the probe light. In Ref. [106], a Raman laser four-level scheme includes thermal atom and optical- magnetometer based on cold atom superradiant was lattice-trapped cold atom type. It is worth noting that demonstrated, where the phase of stimulated radiation there are schemes that combine atomic beam and optical was directly detected to identify the phase response of lattice to achieve CW AOCs. For different energy levels, atomic dipole to external magnetic field. This experiment international research progress is introduced in detail, implemented a magnetometer using active mapping of including JILA, NIST research group, iqClock program atomic phase to optical field phase that can operate in in the EU, Peking University, Vienna University of Tech- both active and passive field sensing modes. Unlike typi - nology, University of Copenhagen, Aarhus University, cal good-cavity lasers, this laser can be operated in the and Physical Research Laboratory (India). bad-cavity region. Since the atomic gain medium is the Almost 20 years have passed since AOC was proposed primary reservoir of phase information, a passive oscilla- in 2005. During this period, the AOC has extensively tion mode is possible in this bad-cavity laser. The sensor developed. However, where should AOC goes in the can dynamically selectively switch between active oscilla- future? The basic direction is still to continue to improve tion and passive Ramsey-like phase evolution. This flex - the inaccuracy and stability of the AOC to make it closer ibility is achieved by controlling the optical radiation rate to the theoretical expectation. As the performance of the of atomic dipole via the intensity of Raman dressing laser. AOC improves, it is expected to achieve higher precision Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 15 of 17 3. A. Derevianko, M. Pospelov, Hunting for topological dark matter with measurement in gravitational wave detection, general atomic clocks. Nat. Phys. 10(12), 933–936 (2014) relativity verification, searching for time variation of fun - 4. C. Chou, D.B. Hume, T. Rosenband, D.J. 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Le Targat, J. Lodewyck, D. Nicolodi, Y. Le Coq, M. Abgrall, J. Guéna et al., Comparing a mercury Authors’ contributions optical lattice clock with microwave and optical frequency standards. J.C. proposed the concept of active optical clock. Under the guidance of T.S., New J. Phys. 18(11), 113002 (2016) J.Z. wrote the manuscript. J.M. and J.C. provided revisions. The authors read 13. N. Huntemann, C. Sanner, B. Lipphardt, C. Tamm, E. Peik, Single-ion and approved the final manuscript. −18 atomic clock with 3 × 10 systematic uncertainty. Phys. Rev. Lett. 116(6), 063001 (2016) Funding 14. Y. Huang, H. Guan, W. Bian, L. Ma, K. Liang, T. Li, K. Gao, A comparison of This research was funded by the National Natural Science Foundation of 40 −17 two Ca single-ion optical frequency standards at the 5 × 10 level China (NSFC) (91436210), Innovation Program for Quantum Science and and an evaluation of systematic shifts. Appl. Phys. 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The development of active optical clock

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The atomic clocks, whether operating at optical or microwave region, can be divided into two categories according to their working mode, namely the passive clocks and active clocks. The passive clocks, whose standard frequency is locked to an ultra-narrow atomic spectral line, such as laser cooled Cs beam or lattice trapped Sr atoms, depend on the spontaneous emission line. On the contrary, the active clocks, in which the atoms are used as the gain medium, are based on the stimulated emission radiation, their spectrum can be directly used as the frequency standard. Up to now, the active hydrogen maser has been the most stable microwave atomic clocks. Also, the Sr superradiant active atomic clock is prospects for a millihertz-linewidth laser. Moreover, the optical clocks are expected to surpass the performance of microwave clocks both in stability and uncertainty, since their higher working frequency. The active optical clock has the potential to improve the stability of the best clocks by 2 orders of magnitude. In this work, we introduce the development of active optical clocks, and their types is classified according to the energy-level struc- ture of atoms for stimulated radiation. Keywords Atomic clocks, Passive clocks, Active clocks, Stimulated emission radiation, Superradiant laser passive clocks, whose standard frequency is locked to an 1 Introduction ultra-narrow atomic spectral line, such as, laser cooled Cs As the most precise scientific device, atomic clocks are beam or lattice trapped Sr atoms, depend on spontane- widely used in numerous fields, such as military and ous emission line. On the contrary, the active clocks [10], national defense, cosmic exploration  [1–3], and scien- in which the atoms are used as gain medium, are based tific frontier research  [4–6]. According to operating fre - on the stimulated emission radiation, their spectrum can quency, atomic clocks can be divided into microwave be directly used as the frequency standard. clocks and optical clocks. The optical clocks, operating Currently, two types of optical clocks with the best per- at frequency domain by about five orders of magnitudes formance are both passive clocks, which are optical lat- higher than that of the microwave clocks, have been sur- tice clocks  [7, 11, 12] and ion optical clocks  [8, 13, 14]. pass the performance of microwave clocks both in sta- 1 3 Since the observation of the strongly forbidden S → P bility and uncertainty. The stability of state-of-the-art 0 0 −19 transition can be used as a clock transition by Lemonde optical clocks has reached the magnitude of 10  [7, 8], et  al.  [15] and Katori et al. [16] in 2003, Sr optical lattice which means it can verify general relativity within mil- clocks have developed rapidly. After that, the National limeter dimensions [9]. Atomic clocks also can be divided Institute of Standards and Technical (NIST), the Physi- into two categories according to their working mode, kalisch-Technische Bundesanstalt (PTB), the Obser- namely passive clocks and active clocks  (see Fig.  1). The vatoire de Paris, the National Research Council (NRC), the Joint Institute of Laboratory Astrophysics (JILA), the University of Tokyo, National Institute of Informa- *Correspondence: Tiantian Shi tion and Telecommunications Technology (NICT), and tts@pku.edu.cn others have conducted intensive research on Sr optical State Key Laboratory of Advanced Optical Communication Systems lattice clocks. Currently, the frequency stability of the and Networks, Institute of Quantum Electronics, School of Electronics, −17 Peking University, Yiheyuan Road 5, Beijing 100871, China Sr optical lattice clock can reach 4.8 × 10 / τ  [11], −18 and the system uncertainty can reach 2 × 10  [17]. In © The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. Zhang et al. AAPPS Bulletin (2023) 33:10 Page 2 of 17 Fig. 1 Classification of atomic clocks. According to the frequency band, atomic clocks can be divided into microwave clocks and optical clocks. Atomic clocks also can be divided into two categories according to their working mode, namely passive clocks and active clocks addition, the Yb optical lattice clock achieves a frequency Cao et  al. reported a robust, compact, and transport- −16 40 + stability of 1.5 × 10 / τ , a long-term frequency sta- able Ca single-ion optical clock with a system uncer- −19 −17 bility of 3.2 × 10  [7], and a system uncertainty of tainty of 1.1 × 10 , and the frequency stability reaches −18 −17 1.4 × 10  [18]. By cross-referencing with the Sr opti- 1.5 × 10 near 100,000 s [30]. cal lattice clock, the relative frequency uncertainties of In order to detect atomic transition spectra with mHz −16 the Hg optical lattice clocks achieved 1.8 × 10  [12] linewidth, the local oscillator linewidth must be nar- −17 and 8.4 × 10  [19], respectively. The National Institute row enough. At present, the ultra-narrow linewidth of Metrology (NIM) of China has realized the evaluation laser sources with low phase noise and high coherence and absolute frequency measurement of Sr optical lattice are mainly obtained by Pound-Drever-Hall (PDH) tech- −17 clock with a system uncertainty up to 2.8 × 10  [20, 21] nique  [31, 32]. It requires resonant cavity mirrors with −15 6 and a frequency stability of 1.8 × 10 / τ  [22]. The sys - high reflectivity coatings to achieve 10 or higher finesse. tem uncertainty of Yb optical lattice clock developed by To reduce the thermal noise of the resonant cavity  [33], −16 East China Normal University reaches 1.7 × 10  [23]. it is necessary to use single crystal silicon or microcrys- In recent years, breakthroughs have also been made in talline glass with ultra-low thermal expansion coeffi - ion-trapped optical clocks. In the 1980s, Dehmelt’s group cient as the cavity material [34]. In addition, the resonant proposed the idea of using the Paul trap to imprison cavity needs to be placed in an ultra-low temperature ions. Subsequently, the ion-trapped optical clocks with environment to reduce frequency drift  [35–37]. How- + + + + + Hg  [24], Sr  [25], Yb  [13], Al  [8, 26], and Ca  [27, 28] ever, even with the most stable resonant cavitiy, the sta- ions as quantum references developed rapidly. Among bilized laser is subject to cavity-length thermal noise, them, the PTB achieves a Yb optical clock with a sys- which causes phase drift that limits the laser linewidth −18 tem uncertainty of 3 × 10  [13] and tests the prin- to 0.125 − 0.3 Hz [38, 39]. Using super-stable resonant ciple of local position invariance  [29]. The NIST’s Al cavity results in expensive cost and complex systems, is −19 optical clock has reached the 10 order of magnitude environmentally sensitive, and does not solve the cavity- for system frequency accuracy assessment  [8]. In addi- length thermal noise problem essentially. tion, the system frequency uncertainty of the Ca opti- To overcome the problem that the optical cavity used cal clock realized by the Wuhan Institute of Physics for frequency stabilization is limited by thermal noise, and Mathematics (WIPM) of Chinese Academy of Sci- Chen proposed the concept of active optical clock (AOC) −17 ences is 5.1 × 10 , and the frequency stability reaches in 2005 [10, 40]. After the AOC was proposed, it received −17 7 × 10 at an average time of 20,000 s  [28]. Recently, wide attention from international peers. In the 2015 IEEE Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 3 of 17 International Frequency Control Symposium (IFCS), the 1470  nm active light field based on Cs thermal atoms AOC was listed as one of the three most emerging tech- with a linewidth of 53  Hz and demonstrate its superior nologies receiving the most attention in this field. Cur - cavity-pulling suppression property [62]. rently, there are research groups across the globe such Compared to passive optical clocks, AOCs output an as JILA  [41–43], NIST, University of Colorado, Vienna optical frequency standard directly based on the prin- University of Technology (TU Wien) [44, 45], University ciple of stimulated radiation. Since it works in the bad- of Copenhagen  [46], University of Amsterdam  [47, 48], cavity region, which means the cavity-mode linewidth is Aarhus University  [49, 50], Zhengzhou University  [51, larger than the atomic gain linewidth, it can effectively 52], Physical Research Laboratory (India)  [53], Univer- solve the cavity-length thermal noise problem of passive sity of Hamburg  [54], University of Innsbruck  [55, 56], optical clocks based on PDH technique. AOC has impor- Leibniz University Hannover  [57], Nicolaus Copernicus tant application prospects and research significance in University  [58], Academia Sinica  [59], Guru Nanak Dev a lot of fields, such as precision scientific measurement, University  [60], Université Sorbonne Paris Nord  [61], physical theory verification, quantum simulation, and and Peking University  [53, 62] conducting research on gravitational detection. bad-cavity superradiant laser based on various atomic In this paper, we start with the origin of AOC. In Sec- systems. Currently, the JILA research group achieves a tion  2, we will focus on the fundamental principles of superradiant pulsed lasing based on the ultra-narrow AOC. We then review the scheme of AOC in Section  3, transition linewidth of Sr atoms with a frequency stabil- which are classified into two-, three- and four-level −16 −15 ity of 6.7 × 10 at 1 s and an accuracy of 4 × 10  [43]. according to the quantum reference transition energy The University of Hamburg realizes a hyperbolic sinu - levels. In Section  4, we present some applications of soidal superradiant light field based on Ca atom AOC. Section 5 provides a summary and outlook. with intensity proportional to the particle number squared [54]. The Niels Bohr Research at the University of 2 Basic principle of active optical clock Copenhagen demonstrates a pulsed superradiant signal In AOC, multi-atom coherent stimulated radiation is based on Sr [46]. The Aarhus University realizes super - formed between atomic transition levels through the radiant pulsed laser with linewidth less than 2  Hz  [49]. weak feedback of optical resonator; the schematic dia- The Peking University achieves a continuous-wave (CW) gram of the AOC is shown in Fig.  2. AOCs are based on Fig. 2 Schematic diagram of an AOC. Due to the high phase coherence of collective atomic dipole, the stimulated radiation can be used as optical frequency standard directly. AOC works in bad-cavity region, and the gain linewidth is much smaller than the cavity-mode linewidth, so the center frequency of output laser depends on quantum transition frequency, which can effectively suppress the cavity-pulling effect Zhang et al. AAPPS Bulletin (2023) 33:10 Page 4 of 17 the principle of stimulated radiation, in which atoms are Therefore, passive optical clocks are inevitably affected pumped to the excited state under the action of pumping by the cavity-length thermal noise, which affects their light, creating a population inversion between two energy frequency stability. Conversely, the AOCs work in bad- levels. These atoms that have achieved population inver - cavity region, where Ŵ ≪ κ , P ≪ 1 . The effect of the cav - sion are placed in an optical resonant cavity as the gain ity-mode frequency variations on output laser frequency medium of the clock transition. Under the weak feed- is greatly suppressed, so the output laser frequency is back of the cavity, the coherent radiation output is real- immune to ambient noise. Here, the bad-cavity factor is ized as an active optical frequency standard. Due to the defined as the ratio of the atomic decay rate to the cav - phase coherence of collective dipole emission, the output ity dissipation rate, a = . Using the bad-cavity factor, laser has excellent phase coherence, which can exceed the cavity-pulling coefficient can also be expressed as the quantum-limited linewidth determined by spontane- P = . For AOC, the bad-cavity factor a ≫ 1 and the 1+a ous emission [42, 63]. AOC is an innovative way to obtain impact of cavity-length fluctuation on laser frequency is high coherence, ultra-narrow linewidth lasers. Using reduced drastically. quantum reference system as gain medium, its stimulated radiation can be directly used as the clock laser. Conven- 2.2 Linewidth characteristic tional passive optical clocks work in good-cavity region, The AOC output laser has excellent phase coherence its local oscillator laser generally use medium with broad and the laser linewidth can break the quantum-limited gain linewidth, and the cavity-mode linewidth is nar- hν linewidth �ν = , which is determined by sponta- 4π P out rower than the gain linewidth, so its output frequency is neous radiation. According to the modified Schawlow- mainly determined by the central frequency of the cav- Townes formula in bad-cavity regime, the linewidth of ity mode. When the external environment noise causes bad-cavity laser can be expressed as [64] the change of cavity length, the output laser’s frequency will change accordingly. Unlike conventional good-cavity hν κ Ŵ �ν = N (3) AOC sp laser, the atomic gain linewidth of AOC is narrower than 4π P Ŵ + κ out the cavity-mode linewidth. Therefore, the AOC works Here, it is assumed that the cavity-mode center frequency in the bad-cavity regime. The center frequency of the coincides with the atomic transition frequency. P is clock laser depends on quantum transition frequency, out the output laser power, h is Planck’s constant, ν is the which can effectively suppress the cavity-pulling effect atomic transition frequency, N = is the spon- and reduce the impact of the cavity-length thermal noise. sp N −N 1 2 taneous radiation factor, and N , N correspond to the Utilizing AOC, the laser linewidth is expected to reach 1 2 particle number in upper and lower levels, respectively. mHz level [41] and the frequency stability is expected to The first two terms of Eq. ( 3) represent the quantum- exceed existing optical clocks. There are several advan - limited linewidth determined by spontaneous radiation. tages of AOC, as described next. For good-cavity laser, the linewidth can be reduced to �ν = , with M being the average intracavity Good c 4πM 2.1 Cavity‑pulling characteristic photon number. For AOC operating in bad-cavity region, The relationship between the center frequency ν of AOC Ŵ ≪ κ , the linewidth can be simplified as �ν = . Bad πκM output laser and the frequency ν of atomic transition It is possible to break through the quantum-limited frequency can be expressed as [42] linewidth determined by spontaneous radiation and reach the order of mHz. ν − ν = (ν − ν ), (1) In summary, the AOC working in the bad-cavity limit 0 c 0 Ŵ + κ utilizes atoms as the gain medium, whose stimulated emission radiation can be directly used as the frequency where ν denotes the cavity-mode frequency, Ŵ is the standard. Therefore, compared with passive clocks, atomic decay rate, and κ is the cavity dissipation rate. The AOCs have two significant advantages of cavity-pulling output laser frequency changes with the cavity-mode fre- suppression effect and narrower laser linewidth. quency by an amount P called cavity-pulling coefficient. From Eq. (1), P can be expressed as 3 Research Schemes of AOC dν Ŵ P = = . (2) According to the energy-level structure of atoms for dν Ŵ + κ stimulated radiation, AOC can be divided into three In passive optical clock, Ŵ ≫ κ , the corresponding cav- categories: two-, three-, and four-level AOC. Among ity-pulling coefficient P ≈ 1 , that is, the output laser them, the two-level scheme includes atomic beam, Fara- frequency follows the cavity-mode frequency exactly. day atomic filter, and optical lattice type; the three-level Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 5 of 17 2 1 3 scheme includes optical lattice, bad-cavity Raman, and following energy levels are available: (4s ) S − (4s4p) P 0 1 2 3 3 ion trap type; and the four-level scheme includes thermal for Ca atoms, (3s ) P − (3s3p) P for Mg atoms, and 0 1 2 1 3 atom and optical-lattice-trapped cold atom type, which (5s ) S − (5s5p) P for Sr atoms. 0 1 are shown in Fig. 3. Take the Ca atom  [66] as an example to illustrate more specifically. A 657 nm laser pumps the collimated 2 1 3.1 Two‑level AOC Ca atoms from the ground state (4s ) S to metastable For two-level AOCs, international research is being car- state (4s4p) P , achieving a population inversion. Detec- ried out including JILA, Peking University, University tion is performed with a 423 nm blue laser, which is of Copenhagen, University of Hamburg, and so on. The located after the pump laser and divided into two parts. atoms used in the two-level AOCs are currently focused One is used to transfer unpumped ground state atoms on Ca and Sr. Some properties of the two-level AOCs, to (4s4p) P for electron-shelving detection. Once the including the clock transition levels, their wavelengths atoms are excited to metastable state, they fall back to and working types, are given in Table 1. the ground state after a few tens of centimeters of flight distance. Then, they are detected by another 423 nm 3.1.1 Atomic beam two‑level AOC laser beam. In this scheme, interaction time broaden- In terms of two-level AOCs, atomic beam type is appli- ing is the most dominant of all broadening mechanisms, cable to any two-level atomic system with metastable because it is much larger than the spontaneous radiation upper state. There are abundant quantum systems avail - rate at P . Therefore, the interaction time broadening able, atomic beams such as Mg [65], Ca [66], Sr [67], Ba, can be considered as the atom gain linewidth, which is and molecular beams such as CH  [68] and OsO  [69]. 2π × 150 kHz . The cavity-mode linewidth is taken as 4 4 The basic working principle of atomic beam type AOC 2π × 10 kHz . With proper design of bad-cavity struc- is as follows: gaseous atoms generated by a heating ture, the limiting linewidth can reach 0.1 Hz according to oven are collimated to form an atomic beam. The atoms the modified Schawlow-Townes formula. However, there in the ground state are pumped to the metastable state are two problems with the atomic beam scheme. Firstly, to achieve population inversion between clock transi- the remaining first-order Doppler effect due to the trans - tion energy levels. After that, metastable state atoms verse velocity distribution leads to the broadening of gain interact with an optical resonant cavity to realize the linewidth, which can be overcome through laser cooling stimulated radiation output when they reach threshold technology. Secondly, the second-order Doppler effect condition of laser oscillation. For different atoms, the causes asymmetry in the gain profile, which becomes the Fig. 3 Different types of AOC based on the transition energy level of quantum gain medium Table 1 Properties of selected two-level AOC, including the clock transitions, their wavelengths, and working types. The corresponding research groups and references are also shown Atom Clock transition /nm Type Research group References 40 1 3 657 atomic beam Peking University [66] Ca S - P 0 1 88 1 3 689 atomic beam University of Copenhagen [46] Sr S - P 0 1 40 1 3 657 atomic beam JILA [70] Ca S - P 0 1 88 1 3 Sr S - P 0 1 87 1 3 698 optical lattice JILA [43, 71] Sr S - P 0 0 40 1 3 657 optical lattice University of Hamburg [54] Ca S - P 0 1 133 2 2 Cs S P 852 Faraday Peking University [72] 1/2 3/2 Zhang et al. AAPPS Bulletin (2023) 33:10 Page 6 of 17 main limiting factor for clock accuracy. Thus, the perfor - course it can be extended to other kinds of alkaline earth mance of the two-level atomic beam AOC is ultimately metal atoms. The atoms can be continuously pumped to limited by the second-order Doppler shift, although it the excited state to obtain a constant population inver- adapts to a wide range of two-level atomic system with sion, thus overcoming the influence of Dick noise on metastable upper state. short stability and achieving CW superradiant laser out- Through laser cooling technique, the Doppler effect can put with a theoretical linewidth of mHz level. be largely suppressed. The Niels Bohr Institute at the Uni - versity of Copenhagen has chosen Sr atoms as a quantum 3.1.2 O ptical lattice two‑level AOC reference for laser frequency stabilization based on cavity- On the basis of laser cooling, atoms can be further loaded enhanced atomic interaction, using both passive and active into a magic wavelength optical lattice, which confines methods  [73]. In the passive scheme  [74, 75], a cavity- the atoms tightly within the Lamb-Dicke range along enhanced modulation transfer spectrum is employed and the cavity axis and eliminates the first-order Doppler the corresponding atomic phase shift is used as an error shift more effectively. In 2016, JILA achieved the first 3 1 signal. Since the atom-cavity coupling occurs in the bad- pulsed superradiant laser at 698 nm ( P → S ) using 0 0 cavity regime, the cavity-pulling effect can be signifi - Sr atoms trapped in the optical lattice [71]. Experimen- cantly suppressed compared to the conventional scheme tally, Sr atoms loaded into the optical lattice are pre- of locking the local oscillator to an ultra-stable cavity. In pared by a two-stage cooling process. Initial trapping and 7 88 the active scheme [46], 2 × 10 Sr atoms are confined in cooling are performed using the dipole-allowed 461 nm 1 1 a large waist cavity by laser cooling and magneto-optical S → P transition, and further cooling is performed 0 1 1 3 1 3 trapping (MOT). The 689 nm S → P dipole-forbidden using narrow linewidth 689 nm S → P transition. 0 1 0 1 transition with a natural linewidth of 7.5 kHz is used as Among them, the 689 nm laser is also used to pump Sr 1 3 clock transition, because it has a lifetime many orders of atoms from ground state S to P . Then the atoms trans - 0 1 magnitude longer than those of dipole-allowed transitions. fer to the upper level P of clock transition via an adiaba- By applying 689 nm π pulse, the Sr atoms cooled to mK tic passage by a 698 nm laser. Using an 813.4274 nm laser, are excited to upper level P . When the cavity-mode fre- which is close to the magic wavelength, as the lattice quency is resonant with the atomic transition, the atoms light, imparts near equal shifts to the ground and excited immediately establish coherence through the cavity field, states of the lasing transition so as to eliminate first-order achieving pulsed superradiance with high spectral purity. Doppler shift. Compared with independently radiating The laser operates in the bad-cavity region, where the pho - atoms, the atomic collective emissivity is enhanced more ton radiation is substantially enhanced due to the collec- than 10,000 times after coupling to the cavity. On this tive cooperativity. Its maximum output power is close to basis, JILA characterized the Sr atomic ultra-narrow 1µW . On the theoretical side, the atomic beam continu- linewidth superradiant laser in 2018 with a linewidth on −16 ous superradiant laser was studied by Université Sorbonne the order of 10 Hz , a frequency stability of 6.7 × 10 at −15 Paris Nord  [61]. They proposed a minimalistic model to 1  s, and an accuracy of 4 × 10  [43]. In this bad-cav- explain laser threshold, power, correlation properties, and ity regime, any fluctuation (thermal or mechanical) of linewidth. This model describes the dynamics of atoms the cavity length has much less influence on the output entering and leaving the cavity by a Hamiltonian process, laser spectrum, resulting in a cavity-pulling coefficient of −6 without stochastic approach. They demonstrated that the 2 × 10 obtained experimentally. For the Ca atom two- ultimate linewidth is set by the fundamental quantum fluc - level optical lattice AOC, the research group at the Uni- tuations of the collective atomic dipole and the continuous versity of Hamburg observed a hyperbolic secant shaped superradiant regime is tied to the growth of atom-atom superradiant pulse with intensity proportional to the correlations. square of the particle number in bad-cavity regime based 1 3 Cold-atom-based superradiant lasers have proven on S → P 657 nm transition [54]. The pulse duration 0 1 their superior performance, but parasitic heating from is much shorter than the natural lifetime of the P state, atomic repumping has so far limited these systems to and its decay time fluctuations are consistent with theo - pulsed operation [76]. This problem can be avoided using retical predictions. In this work, the population inversion a thermal atomic beam, because the pumping process is is achieved using incoherent pumping, which holds great performed outside the cavity. JILA proposed a new type promise for achieving continuous superradiant output. of superradiant laser using a hot atomic beam passing Theoretically, Zhang et  al. applied the Monte-Carlo through an optical cavity and show that the theoretical wave-function method (MCWF) method [50] to calculate minimum linewidth and maximum power are competi- the superradiant pulses with different initial atomic num - tive with the best ultracoherent clock laser  [70]. In this bers in the presence of atom loss, which is in agreement 40 88 article, Ca and Sr are analyzed as examples, but of with the experimental results in Ref. [71]. Since atoms are Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 7 of 17 lost rapidly in the experiment, steady-state superradiance between pulses  [44]. High-speed transport of ultra-cold cannot be formed. By introducing an incident flux of new atoms using a red-detuned one-dimensional optical lat- atoms, the laser linewidth can theoretically reach the tice has been experimentally achieved in 2006, with travel order of mHz despite rapid atom number fluctuations. distance up to 20 cm , transport speeds up to 6m/s , and 3 2 Zhang et al. also used a stochastic mean-field theory [52] accelerations up to 2.6 × 10 m/s  [77]. In the sequential to describe active frequency measurements of pulsed coupling and decoupling scheme, a blue-detuned optical superradiant emission generated by thousands of Sr lattice traps the atoms in the Lamb-Dicke region. When atoms trapped in an optical cavity. This theory combines an atomic ensemble is located inside the cavity and starts cavity quantum electrodynamics and quantum measure- to radiate, a second ensemble is excited to the upper state ment theory, and treats the whole atom ensemble as ten and moves toward the cavity. Once the first ensemble separate subensembles with different transition frequen - completes radiation, the second ensemble enters the cav- cies. The theoretically obtained superradiant beats sig - ity and goes on to radiating while still maintaining the nal, noisy power spectra, and frequency uncertainty are phase of the cavity. At the same time, the first ensem - in agreement with the experimental results reported in ble exits the cavity and a new inverted ensemble is pre- Ref. [43]. Meanwhile, this theory predicts that the short- pared. Repeating the above process, because the phase of −17 term frequency uncertainty can reach 7 × 10 / (τ/s) the cavity remains constant, the superradiant pulses are by using longer superradiance pulses of similar strength also highly phase-coherent with each other. It is worth and by reducing the time for single measurements. mentioning that the atoms can be prepared in the upper For the two-level optical lattice active superradiant lasing state outside the cavity, which can circumvent per- laser, Gogyan et al. [58] introduced a semiclassical theory turbations due to AC Stark shift. This sequential coupling of superradiant pulses generated by alkaline earth atoms method is a promising approach towards creating an and performed a feasibility analysis for an experimen- active optical frequency standard. Based on the sequen- tal implementation using the example of Ca atoms, tial coupling method proposed by Kazakov et  al.  [44], reported in Ref.  [54]. The results show that the inho - JILA has recently realized the transport of atoms using a mogeneous optical pumping procedure has a significant moving optical lattice. [78], which is expected to be used effect on the superradiant pulse characteristics. Uni - to realize a continuous superradiant laser. versity of Innsbruck has evaluated the effects of dipole- dipole interaction and collective spontaneous decay on 3.1.3 F araday type two‑level AOC the radiation properties of the superradiant laser  [55], Compared with the atomic beam and optical lattice such as linewidth, stability, and cavity-pulling, through scheme, the two-level Faraday type is simple in struc- direct numerical simulations of minority-atom systems ture and easy to implement. The Faraday laser uses an with different geometries and densities. Besides, they anti-reflective coated laser diode as the gain medium demonstrated that in the bad-cavity regime, by choos- and a Faraday atomic filter as the frequency-selective ing appropriate cavity detuning parameters, atoms can device  [79]. The laser frequency can be stabilized within be trapped and cooled by the cavity field generated by the transmission bandwidth of the atomic filter so that their own stimulated radiation [56]. Academia Sinica [59] the laser linewidth can be narrowed effectively through theoretically investigated the effect of long-range dipole- optical feedback  [80]. The AOC scheme is adopted to dipole interaction on the steady-state active superradi- optimize the Faraday laser’s frequency stability and ant laser. The cavity photon number and the coherence named Faraday AOC  [72]. Its core principle is that the between atoms have oscillation phenomenon with gain and quantum reference are independent of each interparticle distance of the atoms. The maximal cavity other, thus reducing the influence of noise in the gain part photon number and the minimal spectral linewidth are on the frequency stability. The gain can be provided by located under the condition of equidistant atomic arrays, materials such as semiconductors, solids, or dyes, while which can facilitate precision measurements and the narrow-band atomic filters provide a frequency refer - development of next-generation optical clocks. ence. By choosing suitable parameters to make the exter- For the schemes mentioned above using laser-cooled nal cavity-mode linewidth much larger than the atomic atomic beams and neutral atoms trapped in optical lat- filter bandwidth, the laser works in the bad-cavity region, tices, the superradiant laser output can only be oper- thus reducing the cavity-pulling effect and improving fre - ated at pulsed mode. There is no phase coherence quency stability. Meanwhile, the laser frequency is deter- between different individual pulses, and the pulse dura - mined by quantum transition frequency, which can be tion limits the stability of the output laser. To achieve directly used as a stable frequency standard. This scheme a continuous superradiant laser, sequential coupling can satisfy the laser oscillation threshold by increasing and decoupling can be used to maintain the coherence the pumping efficiency of the gain medium and obtain an Zhang et al. AAPPS Bulletin (2023) 33:10 Page 8 of 17 3 1 active optical frequency standard with narrow linewidth form the population inversion between P and S , thus 1 0 by compressing the atomic filter transmission bandwidth. realizing 657 nm optical frequency standard. This work Although this scheme is simple in structure, it is not presents the first neutral-atom-based optical lattice AOC, easy to narrow the transmission bandwidth to natu- which is expected to reach sub-Hz linewidth. In 2007, ral linewidth level, leading to a significant challenge in this group revealed that a 1 mHz linewidth optical clock further enhancement of bad-cavity factor. At present, could be realized by exploiting the phase-matching effect the linewidth of the Faraday active optical frequency of the three-level -type Sr atomic system confined in standard based on the thermal Cs atomic gas cell is magic wavelength optical lattice  [63]. When the nonadi- at 100 Hz order. It has not yet reached the theoretical abatic interaction of two quasimonochromatic fields with 1 1 3 88 value [72]. To solve this problem, cold atoms or ions can the states S , P , and P of Sr achieves phase coher- 0 1 0 be used as quantum frequency reference mode-selecting ence, a frequency difference field with 1 mHz linewidth devices  [81], which can effectively suppress the Doppler will be generated by the nonlinear crystal placed in a FP effect, reduce the transmission bandwidth, enhance the cavity. cavity-pulling suppression, and thus compress the laser A method to obtain a laser with mHz linewidth was linewidth. also proposed by the JILA research group in 2009  [41]: Sr atoms in an optical lattice collectively emit pho- 1 3 3.2 Three‑level AOC tons on the ultranarrow clock transition S → P , 0 0 The three-level AOC includes optical lattice, ion trap, into the mode of a high Q optical cavity. Since the cou- and bad-cavity Raman type. For optical lattice type, the pling between atoms and the optical field is completely atoms are trapped during the measurement period, so collective, i.e., the phase of different atomic dipoles are it can exploit the very narrow hyperfine-induced ns S 0 perfectly coherent, and the output laser linewidth is 3 87 88 -nsnp P , such as Sr and Sr . JILA research group 0 expected to be narrower than the natural linewidth. This study the bad-cavity Raman laser and implement an scheme assumes that the atoms are confined in an opti - active magnetometer based on it. For ion-trapped three- cal lattice within a fixed lattice point so that the intera - level AOC, there are only relevant theoretical studies, tomic coupling is maximized and these atoms are in and no real experimental realization has been made the same phase within a specific cavity mode. Repump - 1 3 yet. Properties of selected three-level AOC are shown ing lasers drive atoms from S to P , and then atoms 0 0 in Table  2, including the clock transitions, their wave- transfer to S . Due to spontaneous radiation, atoms in 3 3 3 lengths, and working types. the S state decay to the P and P , forming Raman 1 2 0 transition between these two states to implement side- 3.2.1 Optical lattice three‑level AOC band cooling to the vibrational ground state. Besides, the The Peking University firstly proposed an optical lattice repumping lasers also pump all atoms to the P meta- three-level AOC scheme in 2005  [40]. They use 423 nm stable level, thus satisfying inversion for laser transition. blue MOT and 657 nm red MOT to cool the Ca atoms Since the total relaxation rate Ŵ of the atomic dipole is 3 −1 and then confine them in a magic wavelength optical lat - at most on the order of 10 s and the cavity decay rate 5 −1 tice. The 423 nm and 1201 nm lasers are used to pump the κ is 9.4 × 10 s , the cavity-pulling suppression fac- 1 3 1 3 Ca atoms from ground state S to P through P , and tor is at least on the order of 10 . Theoretical calculation 0 2 1 the atoms are concentrated in P by repumping laser to shows laser linewidth can reach the mHz level. However, Table 2 Properties of selected three-level AOC, including the clock transitions, their wavelengths, and working types. The corresponding research groups and references are also shown Atom Clock transition /nm Type Research group Reference 40 1 3 657 Optical lattice Peking University [40] Ca S - P 0 1 88 1 3 698 Optical lattice Peking University [63] Sr S - P 0 0 87 1 3 698 Optical lattice JILA [41] Sr S - P 0 0 88 1 3 689 Optical lattice JILA [76] Sr S - P 0 1 88 1 3 689 “Hybrid” type TU Wien, JILA, EU [44, 47, 78, 82] Sr S - P 0 1 87 1 3 Sr S - P 0 0 87 2 2 780 Bad-cavity Raman JILA [42, 83–85] Rb S - P 1/2 3/2 171 2 2 435 Ion trap Peking University [86] Yb S - D 1/2 3/2 176 + 1 3 804 Ion trap TU Wien [45, 87] Lu S - D 0 2 Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 9 of 17 this result can only be achieved if the effective repump Astronomy at Aarhus University theoretically analyzed 3 −1 6 87 88 rate reaches 10 s and the atom number is at 10 level. Sr and Sr atoms also trapped within a one-dimen- Under this condition, the fluctuations in atomic tran - sional optical lattice in a bad-cavity region  [49]. Their 3 1 87 sition frequency introduced due to AC Stark shift are calculations confirm that using P → S of Sr atoms 0 1 3 1 88 negligible at mHz level. Moreover, the maximum laser and P → S of Sr atoms transitions, it is possible to 1 0 output power is proportional to the square of the atom realize the narrow linewidth of superradiant lasing. Spe- number. When the atom number is at 10 , the power can cially, under strong driving of the dipole-forbidden tran- 3 1 88 reach pW order, enough to be used for the phase locking sition P → S of Sr atoms, the superradiant laser 1 0 of a slave optical local oscillator. linewidth can be further narrowed due to the coherent Subsequently, the JILA group proposed the idea that excitation of the cavity field. using a high-finesse resonant cavity, based on alkaline- In 2014, Kazakov et al. discussed in detail two options earth metal atoms with an ultra-narrow-linewidth transi- for implementing active optical frequency standard: opti- tion to achieve a steady-state superradiance[88]. In order cal lattice AOC and atomic beam AOC. They analyzed to obtain the intensity fluctuations of the steady-state some parameters required to achieve the best frequency superradiant laser, JILA performed an analysis using stability as well as the implications and challenges in the Monte Carlo simulations and semiclassical approxima- implementation [90]. In addition, a “hybrid” method was tion methods  [89]. They found that the light exhibits proposed in which the atoms are prepared outside the bunching below threshold, is to a good approximation cavity, fed into the cavity by a “guide beam” or “optical coherent in the superradiant regime, and is chaotic above conveyors” to complete the stimulated radiation pro- the second threshold. Toward achieving mHz linewidth, cess. A blue-detuned optical lattice is used to prevent JILA studied superradiant lasing on the 7.5 kHz linewidth the atoms from moving along the cavity axis, suppress- 3 1 dipole-forbidden P → S transition at 689 nm , with an ing decoherence and first-order Doppler effects. JILA has 1 0 88 88 ensemble of Sr atoms tightly trapped in a 1D optical lat- recently realized continuous loading of ultra-cold Sr tice [76]. This laser is in a superradiant crossover regime, atoms into a high-finesse ring cavity and using a mov - which means it can be operated at the crossover between ing optical lattice to transport the atoms along the cav- good- and bad-cavity regimes. The cold-atom gain ity axis  [78]. Experimentally, the loading rate reaches medium can be repumped to achieve quasi-steady-state 2.1(3) × 10   atoms/s. This work lays the foundation for lasing, and the frequency of the emitted light is deter- the future implementation of continuous-type active mined by the atomic transition frequency when oper- superradiant lasers based on the mHz clock transition of ated in the bad-cavity regime. They also characterized Sr atom. the cavity-pulling suppression of the bad cavity, where Aiming to achieve a CW AOC, the European Union the laser frequency variation is reduced by an order of (EU) has set up the iqClock project, led by the Univer- magnitude. Experimentally, the cavity-pulling coeffi - sity of Amsterdam, in collaboration with six universities cient is 0.09(2). They also obtained heterodyne power (University of Amsterdam, University of Birmingham, spectral density (PSD) between output light and 689 nm Nicolaus Copernicus University, University of Copen- pump laser, with Lorentzian (Gaussian) full width at half hagen, Vienna University of Technology, University of maximum (FWHM) of 6.0(3) (4.7(3)) kHz. The measured Innsbruck) and six companies (Teledyne e2v, TOPTICA, linewidth is slightly narrower than the natural linewidth NKT Photonics, Acktar, Chronos Technology Ltd, British of the lasing transition (7.5  kHz) and far narrower than Telecom). A cold atomic beam scheme is used to achieve the linewidth imposed by repumping (100 kHz), exhibit- CW clock lasers by transporting Sr atoms through a ing the linewidth-narrowing characteristic of synchroni- moving optical lattice into a ring cavity. A clock laser zation in a laser. linewidth of 100 mHz is expected to be achieved within 5 Zhang et  al. theoretically explained this linewidth- years and linewidth on the order of mHz within 10 years. narrowing property  [51]. When the ultracold Sr In 2019, the University of Amsterdam realized a continu- atoms in the optical lattice are exposed to a magnetic ous guided atomic beam of Sr atoms with a phase-space −4 7 field, the ensemble of atoms with Zeeman-split excited density exceeding 10 and a flux of 3 × 10  atoms/s [47]. states exhibits lasing with very narrow linewidth, which With the optical guide, the atoms reach a velocity of is orders of magnitude smaller than both the cavity 8.4  cm/s and can be used to complement the gain linewidth and the incoherent atomic decay and exci- medium of the steady-state atom laser, which is an tation rates. The narrow-linewidth lasing is due to an important step towards the realization of a steady-state interplay of multiatom superradiant effects and the cou - superradiant AOC. In 2021, this group demonstrated a pling of bright and dark atom-light dressed states by the steady-state MOT of fermionic Sr atoms operating on the 1 3 magnetic field. In 2018, the Department of Physics and 7.5-kHz-wide S - P transition [48]. This MOT contains 0 1 Zhang et al. AAPPS Bulletin (2023) 33:10 Page 10 of 17 7 7 superradiant laser, the decay rate of a single particle can 8.4 × 10 atoms with a loading rate of 1.3 × 10   atoms/s be far balanced with the repumping rate by appropriate and an average temperature of 12 µK , which can be used design so as to increase the laser cooling and trapping to provide a high flux of ultracold atoms source for the time. realization of a continuous superradiant AOC. Based on Under this foundation, the JILA group investigated the steady-state MOT, this group has also achieved con- the oscillation relaxation, stability, and cavity feedback tinuous Bose-Einstein condensation  [91]. Through the characteristics of the bad-cavity Raman superradiant magic wavelength optical conveyor in the ring cavity, a laser  [84]. Moreover, they demonstrated a hybrid mode continuous source of ultracold Sr atoms in the excited in which the laser can switch between active sensing and state P can be realized, delivering several tens of mil- 88 87 passive phase measurements  [85]. The results culmi - lions of Sr atoms or millions of Sr atoms per sec- nate in a hybrid sensor that combines active sensing of ond [82]. The use of the ring cavity increases the transfer the collective atomic phase during superradiant emis- speed, reduces atom losses and decreases the density of sion with passive phase measurements using Ramsey-like the atoms, paving the way for CW superradiant AOC. evolution times, which are of guiding significance for the future development of ultra-narrow linewidth superradi- 3.2.2 Bad‑cavity Raman three‑level AOC ant laser. In 2012, JILA group proposed an bad-cavity Raman laser experimental scheme  [42, 83], which achieved a Raman 3.2.3 Ion tr ap three‑level AOC superradiant laser with an average photon number less In 2014, the Peking University proposed an active ion than 0.2 in the cavity. Experimentally, the laser operates optical clock scheme  [86] using cold ions trapped in in deep bad-cavity region, where the ratio of the trans- Paul trap as gain medium, which is expected to achieve verse decoherence rate to cavity decay rate of the laser −5 −3 6 active optical frequency standard with mHz linewidth. transition is in the range of 2 × 10 ∼ 10 . Ab out 10 The basic principle of this scheme is similar to that of the Rb atoms are trapped by 823 nm laser in one-dimen- optical lattice scheme, and theoretical studies have found sional optical lattice with a temperature of 40µK . The 171 191 + 137,138 + 43 + 87,88 + that Yb , Hg , Ba , Ca , and Sr ions cavity is coupled to an optically dressed state that mim- are suitable for active ion optical clock. Taking Yb as ics a long-lived optically excited state. A 795 nm linearly 2 2 an example, the 435.5 nm S (F = 0) − D (F = 1) 1/2 3/2 polarized dressing laser is applied to two magnetically transition with a natural linewidth of 3.1 Hz is chosen insensitive energy levels to induce Raman transition. If as clock laser. The cooling light and the repumping light the atoms are pumped continuously from ground state 2 2 correspond to 369.5 nm S (F = 0) − P (F = 1) 1/2 1/2 to metastable state while applying the dressing laser, a 2 2 and S (F = 1) − P (F = 0) transitions, respec- 1/2 1/2 quasi-continuous superradiance laser with a duration tively, and both can be used for pumping ions to upper of 20 − 140 ms can be obtained. Each atom can radiate 2 2 energy level D (F = 1, 2) . The ions at D (F = 2) 3/2 3/2 approximately 35 photons into the cavity mode. Under are pumped to D (F = 1) by 935 nm repumping 1/2 the action of light, collisions between atoms cause them laser. Eventually, most of the ions are transferred to to escape from the optical lattice, which eventually D (F = 1) to achieve population inversion and stim- 3/2 leads to a break in superradiance. The power spectrum ulated radiation output. Theoretically, the laser output (PSD) was obtained by heterodyning the superradiant power can be up to 37 pW when reaching steady state, laser and the dressing laser with a Gaussian FWHM of and it can be increased to 77pW by increasing the light 350 Hz and a Lorentzian FWHM of 4.5 Hz . Although intensity and the ions number. However, it is difficult to this result is much narrower than the spontaneous radia- significantly increase the ion number experimentally, and tion linewidth, it differs significantly from the theoreti - the ion AOC also suffers from the light shift caused by cally calculated 2(1) mHz linewidth due to the dispersion pumping laser, which ultimately affects the clock laser’s detuning of the cavity-mode frequency caused by atom performance. number changing. In this experiment, a surprisingly −5 −3 In 2017, Kazakov et  al. proposed that a bad-cav- tiny cavity-pulling coefficient P = 4 × 10 ∼ 2 × 10 ity laser may be realized using forbidden transitions is obtained, and it can be further enhanced by reducing in large ensembles of cold ions that form a spherical decay rate γ , which in turn suppresses the laser linewidth. Coulomb crystal in a linear Paul trap  [45], which can Overall, this scheme proves the feasibility and superiority guarantee longer trap lifetimes relative to neutral-atom- of AOC scheme, but there are problems of discontinuous optical-lattice type. Micromotion-induced shifts such output and weak power. The 795 nm dressing laser and as the second-order Doppler and DC Stark shifts can be 780 nm pump laser will introduce light shift, resulting in a suppressed by operating the ion trap at a magic fre- the bad-cavity Raman laser cannot be used as an opti- 176 + quency. Considering Lu ions imprisoned in a high cal frequency standard. To further achieve continuous Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 11 of 17 shows the simplified energy level of the three- and finesse ( F = 10 ) optical resonant cavity, an output 3 1 four-level AOC schemes. In the three-level scheme, power of 0.5  pW can be achieved based on the D - S 2 0 the light shift is inevitably introduced because the transition when the ion number reaches 10 and the con- clock laser shares the ground state |1� with the pump- finement frequency is 1  MHz  [87]. If proper continuous ing laser. Conversely, in the four-level scheme, atoms cooling and pumping are provided, a truly steady-state are pumped from ground state |1� to excited state |2� . AOC lasing in the bad-cavity regime is expected. This Due to spontaneous radiation, they are transferred to laser is promising to achieve an active optical frequency the clock transition upper energy level |3� , where popu- standard as a local oscillator for the next-generation opti- lation inversion is formed between |3� and |4� to achieve cal clock scheme. stimulated radiation. Since the pumping laser and the clock laser transition energy levels are independent of 3.3 Four‑level AOC each other, the light shift can be significantly reduced In the two-level atomic beam scheme, the Doppler by choosing a suitable quantum system with a signifi - effect of thermal atomic system is more influential. cant frequency difference. Using laser cooling technology can reduce the first- In a four-level AOC scheme, the quantum system can order Doppler effect, but this will complicate the sys - select alkali metal atoms such as K [92], Rb [93, 94], and tem, and the residual Doppler effect will still limit the Cs  [62, 95–98]. Taking Cs atom as an example, the Cs clock laser performance. The three-level optical lattice, atomic gas cell is placed in a low-finesse optical cavity to Paul trap scheme for imprisoning atoms or ions is lim- make cavity-mode linewidth larger than gain linewidth ited by light shift caused by pumping laser. Moreover, to satisfy bad-cavity condition. Using a 459 nm laser as most of the above two schemes implement pulse AOC pumping laser, the Cs atoms are pumped from ground signals. In contrast, the four-level AOC scheme has state 6S to the second excited state 7P , then 1/2 1/2 three advantages. Firstly, the clock transition energy dropped to 7S state by spontaneous radiation, cre- 1/2 level does not involve the ground state, which can ating a population inversion between 7S and 6P 1/2 3/2 reduce light shift introduced by pumping laser. Sec- when reaching steady state. Under the weak feedback of ondly, a magnetic dipole transition energy level can the optical cavity, the atomic dipoles are spontaneously be chosen, where the laser emission coefficient is not synchronized with high coherence, producing a 1470 nm limited by smaller atom-cavity coupling constant com- stimulated radiation clock laser. Different atoms and pared to narrower linewidth electric dipole transi- their corresponding energy level choices for four-level tion. Finally, the four-level AOC can be continuously AOC are shown in Table 3. pumped to output a stable CW AOC signal. Figure  4 Fig. 4 Simplified energy level diagram of (a) three-level and (b) four-level AOC Table 3 A few options for four-level active optical clocks Atom Pumping laser Pumping laser corresponding energy Clock laser Clock transition level Cs 455 nm 6S → 7P 6S → 7P 1470 nm 7S → 6P 7S → 6P 1/2 3/2 1/2 1/2 1/2 3/2 1/2 1/2 459 nm 1359 nm Rb 420 nm 5S → 6P 5S → 6P 1367 nm 6S → 5P 6S → 5P 1/2 3/2 1/2 1/2 1/2 3/2 1/2 1/2 421 nm 1323 nm K 405 nm 4S → 5P 1252 nm 5S → 4P 1/2 3/2 1/2 3/2 Zhang et al. AAPPS Bulletin (2023) 33:10 Page 12 of 17 3.3.1 Thermal atom four‑level AOC the natural linewidth of 1.81 MHz , has been achieved at The four-energy level AOC scheme was first proposed by room temperature. Yu et  al. in 2010  [99].Theoretically, a superradiant laser Based on Cs four-level AOC, Shi et  al. proposed an with intensity proportional to N and linewidth scales to anti-resonant laser  [103], which is very different from 1/N is studied. In addition, the stationary state solution the classical AOC operating in cavity resonance condi- of full atomic cooperativity is derived, and the stability tions. The lasing is realized when the atomic resonance of the superradiant laser is analyzed under the assump- is between two adjacent cavity resonances, that is, the tion of no spontaneous radiation. Subsequently, Wang cavity length equal to an odd multiple of a quarter wave- et  al. carried out a related experiment based on the Cs length. The linewidth of anti-resonant laser is not broad - atomic quantum system  [100]. By measuring the inten- ened compared to resonant laser, and its cavity-pulling sity of fluorescence signals at different wavelengths, such suppression characteristic is stronger. Using this anti- as 1470 nm and 1359 nm , they determined the formation resonant laser, the AOC can be extended from the cav- of population inversion between clock transition energy ity resonance region to the anti-resonance region, further levels. During the same period, a four-level scheme based improving the performance of the current active optical on alkali metal K  [92], Cs  [95], and Rb  [93] atoms was frequency standard. theoretically verified for feasibility based on the density matrix equation, and data such as the wavelength of the 3.3.2 O ptical lattice four‑level AOC pumping laser and the energy level satisfying the popu- In order to solve the Doppler broadening problem in lation inversion were obtained. Moreover, the magic the four-level AOC based on thermal atoms, the Peking wavelengths corresponding to Rb atoms 6S → 5P University, Guru Nanak Dev University, Beijing National 1/2 1/2 and 6S → 5P were theoretically calculated in order Laboratory for Condensed Matter Physics, and the Indian 1/2 3/2 to reduce the Doppler effect  [101]. Using an optical lat - Physical Research Laboratory calculated the magic wave- tice to imprison the laser-cooled atoms in the Lamb-Dick length corresponding to the 1470 nm Cs atom clock tran- region, it is estimated that the effect of AC Stark fre - sition in detail  [53]. The optical lattice four-level AOC quency shift on the clock laser frequency stability can be scheme predicts output laser power up to 24 µW with a −18 −15 reduced to less than 10 . mHz linewidth and a frequency uncertainty of 2 × 10 . The AOC lasing at a wavelength of 1470 nm using the Singh et  al. obtained the magic wavelengths between all Cs four-level system was firstly achieved in 2013  [102]. possible hyperfine levels of the transitions in Rb and Cs On this basis, the 1359 nm and 1470 nm clock laser out- atoms  [60], which will help build a more stable AOC. puts were achieved successfully by using 455 nm and They also gave the static dipole polarizabilities of Rb and 459 nm pumping laser. The bad-cavity factor reached Cs atoms to validate the results. Using cold atoms impris- more than 40, and the 1470 nm laser linewidth reached oned in an optical lattice as the gain medium can reduce 407.3 Hz . For the Rb atom, a 1367 nm continuous active the gain linewidth broadening. In principle, an active optical frequency standard was achieved using a 420 nm optical frequency standard with a mHz linewidth, much semiconductor laser with modulated transfer spectrum smaller than the clock transition natural linewidth, can stabilization  [94]. To further stabilize the resonant cav- be obtained. ity length and reduce the frequency drift caused by the To sum up, the above two-, three-, and four-level AOC residual cavity-pulling effect, a dual-wavelength good- schemes all have advantages and disadvantages. The two- bad-cavity AOC is proposed  [97]. In this scheme, two level structure is simple and easy to implement. However, lasers of different wavelengths are output simultane - its performance is ultimately limited by the second-order ously in a common cavity and separately operate in Doppler shift for the atomic beam and Faraday schemes, good- and bad-cavity regions, called the good- and bad- the pulsed mode operation and the pW laser power for cavity lasers. The good-cavity laser is locked to an ultra- the optical lattice scheme. Although the optical lat- stable cavity by the PDH technique to stabilize the cavity tice three-level scheme can achieve higher performance length. Therefore, as a clock laser, the bad-cavity laser using narrow linewidth quantum transition, it is not easy will be further optimized due to the cavity-length stabi- to realize experimentally due to the small atom number lization with good-cavity laser. To reduce the impact of and weak output power. Moreover, the clock laser is ulti- asynchronous cavity-length variation between two same mately limited by light shift. Because the pumping laser systems on the linewidth broadening of the clock laser, is separated from the clock laser, the four-level scheme the cavity-length stabilization was realized by utilizing avoids the light shift caused by the pumping laser. More- the phase locking technique of good-cavity laser  [62]. over, the output laser is a continuous laser with a higher Experimentally, a 1470 nm CW active optical field with a power which can reach approximately 100µW , but few tens of Hz linewidth, four orders of magnitude below this scheme requires a higher atom number. The use of Zhang  et al. AAPPS Bulletin (2023) 33:10 Page 13 of 17 thermal atom gas cell can meet the corresponding atom measurement of the gravitational potential, the verifica - number requirement. However, it is affected by the Dop - tion of general relativity, the detection of dark matter, and pler effect, which leads to a broadening of the gain pro - other fields. file. To further improve long-term stability, a cold-atom scheme can be used in the four-level scheme. 4.2 Application in precision spectroscopy AOCs use the atomic stimulated radiation signal directly 4 Applications of AOC as the clock laser, whose output linewidth is narrower The applications of active optical frequency standards than the natural linewidth determined by spontane- and clocks are wide ranging. Thanks to the development ous radiation. In terms of spectroscopy, it can be called of atomic clocks, time and frequency are the most pre- stimulated radiation laser spectrum. Theoretically, the cise physical quantities that can be measured currently. linewidth of this “stimulated radiation laser spectrum” As a result, tremendous advances in atomic clocks have is 7–8 orders of magnitude narrower than the natu- made it possible to measure other physical and techni- ral linewidth of atomic transition. Therefore, narrow cal quantities that can be traced back to frequency with linewidth CW AOC can provide new principles and unprecedented precision. AOCs output an optical fre- methods for precision atomic spectroscopy. For exam- quency standard directly based on the principle of stimu- ple, the Peking University has been able to accurately lated radiation. Since it works in the bad-cavity region, measure the 1470  nm transition line (natural linewidth which means the atomic gain linewidth is narrower than 1.81  MHz  [104]) to the Hz level using the Cs four- the cavity-mode linewidth, it can effectively solve the level AOC scheme[62], which is 5–6 orders of magni- cavity-length thermal noise problem. Due to the phase tude more accurate than the previous 1470  nm spectral coherence of collective dipole emission, the output laser line measurement. This method can also be univer - has excellent phase coherence, which can exceed the sally extended to other alkali metal atoms, realizing the quantum-limited linewidth determined by spontaneous high-precision measurement of transition spectral lines emission. Thanks to the superior characteristics of the with MHz natural linewidths, which is expected to be AOC, it can be used in a lot of fields, such as precision upgraded to the measurement accuracy based on con- measurement, physical theory verification, gravitational ventional optical clocks. In this way, the new principle detection, testing of quantum electrodynamics and so on. and technology of the AOC can enrich the international high-precision spectral measurements and improve the 4.1 S erves as a local oscillator for passive optical clocks measurement precision of atomic spectroscopy with In order to detect the atomic transition spectrum with MHz natural linewidths. mHz linewidth, the linewidth of the local oscillator laser of a conventional passive optical clock must be narrow 4.3 Hyperfine‑structure measurement enough. At present, the ultra-narrow linewidth laser In the work of Shi et  al.[105], the Cs atomic four-level source is mainly obtained by the PDH technique. The use AOC can be used not only to realize narrow linewidth of ultra-stable cavities leads to expensive costs, complex active optical frequency standard but also to realize the systems, and environmental sensitivity. Moreover, it can- hyperfine level measurement of Cs 7P in combina- 1/2 not solve the cavity-length thermal noise problem essen- tion with the Doppler effect. Since the AOC uses quan - tially. The narrow linewidth optical field based on the tum reference system stimulated radiation directly as AOC scheme provides an excellent option to solve this the clock transition signal, the output linewidth is much problem. It can be used as the local oscillator laser for smaller than the natural linewidth determined by spon- passive optical clocks. At the same time, this optical field taneous radiation, which can improve the measurement has the absolute value characteristic of quantum transi- accuracy of atomic hyperfine level structure constants. tion frequency between atomic energy levels. Combined The experimental principle is similar to that of the with the optical frequency synthesizer, we can realize Cs four-level AOC, where the 459 nm pumping laser is the frequency comparison between the AOC and other locked to 6S (F = 4) → 7P (F = 3&4) by modula- 1/2 1/2 high-precision optical clocks, thus realizing the trans- tion transfer spectrum. The Cs atoms in the atomic gas mission, comparison, and application of high-precision cell are thermal atoms with Doppler velocity distribu- active quantum frequency standards. In the above cases, tion, and the Cs atoms in different velocity groups in the AOC scheme will break through the technical bot- the cavity sense different pumping laser frequencies. tleneck of the passive optical clock stability currently, According to the selection rule, the Cs atoms located in which is limited by the cavity-length thermal noise. It can the two velocity groups of the 6S (F = 4)state can be 1/2 ′ ′ be applied to the definition of the second, the quantiza - excited to 7P (F = 3) and 7P (F = 4) , resp e ctively . 1/2 1/2 tion of the International System of Units, the precision Cs atoms propagate with the pumping laser with velocity Zhang et al. AAPPS Bulletin (2023) 33:10 Page 14 of 17 6 87 v =  × �ν/2 are sensed at pumping laser fre- 1 459.3 nm Based on previous experiments, 10 Rb atoms quency corresponding to 6S (F = 4) − 7P (F = 3) . 1/2 1/2 were imprisoned in a one-dimensional optical lattice Conversely, when Cs atoms counter-propagate with the after laser cooling to 40µK . The volume of the atom −3 3 pumping laser with velocity v =− × �ν/2 2 459.3 nm cloud is roughly 2.1 × 10 mm using a low-finesse are pumped to 7P (F = 4) , where �ν is the hyper- 1/2 ( F = 710 ) resonant cavity. The atoms are continu - fine level spacing of the 7P state. Atoms pumped 1/2 ously pumped to 5 S (F = 2,m = 2) and decay to 1/2 F ′ ′ to both 7P (F = 3) and 7P (F = 4) can drop to 1/2 1/2 5 S (F = 1,m = 1) through Raman transition. Using 1/2 F ′′ 7S (F = 4) through spontaneous radiation, creat- 1/2 these two magnetic field-sensitive states, the pseudospin ing a population inversion between 7S and 6P . 1/2 3/2 1/2 regime is formed. The states sensitive to the mag - Eventually, the atoms of both velocity groups can out- netic field are chosen instead of the cavity resonant fre - put 1470 nm bad-cavity clock laser corresponding to quency, because the laser operates in bad-cavity region ′′ ′′′ 7S (F = 4) − 6P (F = 5) . The frequency differ - 1/2 3/2 with cavity-pulling suppression effect. Experimentally, ence f of the stimulated emission lights of atoms with the wideband sensitivity at continuous active oscilla- two velocity groups can be measured by optical het- tion was measured, and the narrowband sensitivity in a 459.3 m erodyne, where �f = × �ν . The hyperfine level passive Ramsay-like mode was obtained by narrowband 1469.9 nm spacing �ν of the 7P state can be deduced from the 1/2 detection using the spin-echo technique. Its sensitivity experimentally measured f , and then the magnetic reaches 190 pT/ Hz at 1 kHz , and the effective detection dipole hyperfine constant A of 7P state can be found 1/2 volume, which is the volume of the atomic cloud, is about −3 3 using �ν = A × 4. 2 × 10 mm . Future experimental work will focus on The scheme uses one system for measurement, which achieving truly continuous operation and realizing sensi- can eliminate common mode noise, but is limited by the tivity at the phase diffusion limit. cavity-pulling effect. Considering the errors introduced by the cavity-pulling effect, the measurement result is 5 Summary and outlook f = 118.0347 ± 0.1827 MHz . Hence, the hyperfine level The AOC, which is based on the principle of stimulated spacing of the 7P state is 377.628 ± 0.584 MHz , and 1/2 radiation, its output signal has excellent phase coher- the magnetic dipole hyperfine constant is 94.41(15) MHz . ence and can be used directly as an optical frequency Data comparison shows that this work’s result is in gen- standard. Working in the bad-cavity region, where the eral agreement with those obtained previously using atomic gain linewidth is narrower than the cavity-mode saturated absorption spectroscopy and two-photon reso- linewidth, the AOC can effectively solve the cavity-length nance spectroscopy. thermal noise problem in passive optical clocks. We describe in detail the basic principles and characteristics of AOC and classify AOC into two-, three-, and four- 4.4 M agnetometer based on bad‑cavity Raman AOC level schemes according to the energy-level structure of superradiant laser atoms for stimulated radiation. Currently, the two-level Most atomic magnetometers pass probe light through scheme includes atomic beam, Faraday atomic filter, and atomic vapor and sense the response of the atom to optical lattice type; the three-level scheme includes opti- magnetic field by measuring the polarization rotation or cal lattice, bad-cavity Raman, and ion trapping type; the phase shift of the probe light. In Ref. [106], a Raman laser four-level scheme includes thermal atom and optical- magnetometer based on cold atom superradiant was lattice-trapped cold atom type. It is worth noting that demonstrated, where the phase of stimulated radiation there are schemes that combine atomic beam and optical was directly detected to identify the phase response of lattice to achieve CW AOCs. For different energy levels, atomic dipole to external magnetic field. This experiment international research progress is introduced in detail, implemented a magnetometer using active mapping of including JILA, NIST research group, iqClock program atomic phase to optical field phase that can operate in in the EU, Peking University, Vienna University of Tech- both active and passive field sensing modes. Unlike typi - nology, University of Copenhagen, Aarhus University, cal good-cavity lasers, this laser can be operated in the and Physical Research Laboratory (India). bad-cavity region. Since the atomic gain medium is the Almost 20 years have passed since AOC was proposed primary reservoir of phase information, a passive oscilla- in 2005. 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Journal

AAPPS BulletinSpringer Journals

Published: Apr 24, 2023

Keywords: Atomic clocks; Passive clocks; Active clocks; Stimulated emission radiation; Superradiant laser

References