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We demonstrate a dual wavelength optical frequency standard based on the dual optical transition modulation transfer spectroscopy (DOTMTS) between different quantum transitions of the rubidium D1 (795 nm) and D2 (780 nm) lines. In a single rubidium atomic ensemble, modulation frequency sidebands from the 780 nm pump beam are simultaneously transferred to both the 780 nm and 795 nm probe lasers. The DOTMTS enables the simultaneous stabilization of 780 nm and 795 nm lasers on a single vapor cell. Both lasers exhibit a frequency instability in the low 10 ^(-14) range at 1 s of averaging, as estimated from the residual error signal. A theoretical model is developed based on the V type atomic level structure to illustrate the dual-wavelength spectroscopy. This approach can be extended to develop a multiwavelength optical frequency standard within a single atomic ensemble, broadening its applicability in fields such as precision metrology, wavelength standards, optical networks, and beyond.

We realize a high-performance compact optically pumped cesium beam clock using Faraday laser simultaneously as pumping and detection lasers. The Faraday laser, which is frequency stabilized by modulation transfer spectroscopy (MTS) technique, has narrow linewidth and superior frequency stability. Measured by optical heterodyne method between two identical systems, the linewidth of the Faraday laser is 2.5 kHz after MTS locking, and the fractional frequency stability of the Faraday laser is optimized to $1.8\times{10}^{-12}/\sqrt{\tau}$. Based on this high-performance Faraday laser, the cesium beam clock realizes a signal-to-noise ratio (SNR) in 1 Hz bandwidth of $39600$ when the cesium oven temperature is 130{\deg}C. Frequency-compared with Hydrogen maser, the fractional frequency stability of the Faraday laser pumped cesium beam clock can reach $1.3\times{10}^{-12}/\sqrt{\tau}$ and drops to $1.4\times{10}^{-14}$ at 10000 s when the cesium oven temperature is 110{\deg}C. %, which is the best reported result compared with other cesium beam clocks. This Faraday laser pumped cesium beam clock demonstrates its excellent performance, and its great potential in the fields of timekeeping, navigation, and communication. Meanwhile, the Faraday laser, as a high-performance optical frequency standard, can also contribute to the development of other applications in quantum metrology, precision measurement and atomic physics.

$^{133}$Cs, which is the only stable cesium (Cs) isotope, is one of the most investigated elements in atomic spectroscopy and was used to realize the atomic clock in 1955. Among all atomic clocks, the cesium atomic clock has a special place, since the current unit of time is based on a microwave transition in the Cs atom. In addition, the long lifetime of the $6{\text{P}}_{3/2}$ state and simple preparation technique of Cs vapor cells have great relevance to quantum and atom optics experiments, which suggests the use of the $6{\text{S}} - 6{\text{P}}$ D2 transition as an optical frequency standard. In this work, using one laser as the local oscillator and Cs atoms as the quantum reference, we realized two atomic clocks in the optical and microwave frequencies, respectively. Both clocks could be freely switched or simultaneously output. The optical clock based on the vapor cell continuously operated with a frequency stability of $3.89 \times {10^{ - 13}}$ at 1 s, decreasing to $2.17 \times {10^{ - 13}}$ at 32 s, which was frequency stabilized by modulation transfer spectroscopy and estimated by an optical comb. Then, applying this stabilized laser for an optically pumped Cs beam atomic clock to reduce the laser frequency noise, we obtained a microwave clock with a frequency stability of $1.84 \times {10^{ - 12}}/\sqrt \tau $, reaching $5.99 \times {10^{ - 15}}$ at $10^5$ s. This study demonstrates an attractive feature for the commercialization and deployment of optical and microwave clocks and will guide further development of integrated atomic clocks with better stability. Thus, this study lays the groundwork for future quantum metrology and laser physics., Comment: 8 pages, 4 figures

Lasing in the bad-cavity regime has promising applications in precision measurement and frequency metrology due to the reduced sensitivity of the laser frequency to cavity length fluctuations. Thus far, relevant studies have been mainly focused on conventional cavities whose finesse is high enough that the resonance linewidth is sufficiently narrow compared to the cavity's free spectral range, though still in the bad-cavity regime. However, lasing output from the cavity whose finesse is close to the limit of 2 has never been experimentally accessed. Here, we demonstrate an extremely bad-cavity laser, analyze the physical mechanisms limiting cavity finesse, and report on the worst ever laser cavity with finesse reaching 2.01. The optical cavity has a reflectance close to zero and only provides a weak optical feedback. The laser power can be as high as tens of $\mu$W and the spectral linewidth reaches a few kHz, over one thousand times narrower than the gain bandwidth. In addition, the measurement of cavity pulling reveals a pulling coefficient of 0.0148, the lowest value ever achieved for a continuous wave laser. Our findings open up an unprecedentedly innovative perspective for future new ultra-stable lasers, which could possibly trigger the future discoveries in optical clocks, cavity QED, continuous wave superradiant laser, and explorations of quantum manybody physics.

We demonstrate transition between the fine structure splitting of the ground state of triply ionized zirconium (Zr IV) is suitable for a terahertz (THz) atomic clock. Its transition frequency is about 37.52 THz and is mainly guided by the magnetic dipole (M1) transition and can be accessible by a readily available laser. We suggest to consider stable even isotopes of Zr and $M_J= \pm 1/2$ sublevels (i.e. $|4D_{3/2},M_J=\pm 1/2\rangle \rightarrow |4D_{5/2},M_J=\pm 1/2\rangle$ clock transition) for the experimental advantage. By performing necessary calculations, we have estimated possible systematics due to blackbody radiation, ac Stark, electric quadrupole and second-order Zeeman shifts along with shifts due to the second-order Doppler effects. The proposed THz atomic clock can be very useful in quantum thermometry and frequency metrology., Comment: 1 figure, 11 pages, accepted for publication in Physical Review A

A single-mode Cs atom 852 nm Faraday laser based on the corner cube feedback is demonstrated, and termed as corner-cube-feedback Faraday laser. Using the corner-cube retroreflector as external cavity feedback element in Faraday laser, mechanical robustness can be greatly improved due to the precise reflection of the incident light beam back to its original direction. This Faraday laser can achieve laser oscillation at a large angle, which between the incident light and the optical axis of corner cube, ranging from +3{\deg} to -3{\deg}. The most probable linewidth is 8 kHz measured by heterodyne beating with two identical lasers. Moreover, its output frequency remains close to the Cs atomic Doppler-broadened transition line, even though the diode current changes from 55 mA to 155 mA and the diode working temperature varies from 11.8 to 37.2 degrees Celsius. The corner-cube-feedback Faraday laser with high mechanical robustness as well as narrow linewidth can be widely used in quantum precision measurement, such as atomic clocks, atomic gravimeters, and atomic magnetometers, etc.

We report the development of laser systems -- a "Voigt laser" -- using a Voigt anomalous dispersion optical filter as the frequency-selective element, working at the wavelength of 780 nm of $^{87}$Rb-D2 resonance line. Compared with Faraday anomalous dispersion optical filter, the Voigt anomalous dispersion optical filter can generate a stronger and more uniform magnetic field with a compact size of magnet, and obtains a transmission spectrum with narrower linewidth and more stable lineprofile. In this case, the frequency stability of the Voigt laser reaches 5$\times$10$^{-9}$ at the averaging time of 200 s, and the wavelength fluctuation of 8-hours free operation is $\pm$0.1 pm. Besides, the Voigt laser has greater immunity to diode current than the Faraday laser, with a wavelength fluctuation of $\pm$0.5 pm in the current range from 73 mA to 150 mA. Finally, the Voigt laser frequency can be controlled by the cell temperature of the Voigt optical filter, which is expected to achieve a frequency detuning of 20 GHz. Consequently, the Voigt laser, whose frequency could correspond to the atomic transition frequency by tuning the cell temperature, obtains good robustness to the current and temperature fluctuation of laser diode, and could realize a compact optical standard for precise measurement once stabilized by modulation transfer spectroscopy., Comment: 4 pages, 5 figures. The following article has been submitted to [Appled Physics Letters]

The quantum entanglement enables the precision measurement and frequency metrology beyond the standard quantum limit that is imposed by the quantum projection noise and photon shot noise. Here we propose employing the entangled atoms in the synchronous differential measurement to enhance the sensitivity of the spatial-shift detection. Two ways of engineering the entangled atoms are studied. The synchronous comparison between two pixels within an entangled atomic cloud leads to a sensitivity enhancement factor of 1.4 over the standard quantum limit. Increasing the atom number hardly further improves the sensitivity. In contrast, the synchronous comparison between two independent pixels that are individually composed of entangled atoms allows for a strong sensitivity enhancement by a factor of, for example, 9.7 with $10^{3}$ entangled atoms in each pixel, corresponding to a reduction of the averaging time by a factor of about $10^{2}$. A large atom number may further elevate the sensitivity. Our work paves the way towards the entanglement-enhanced detection of the gravitational redshift by means of the \emph{in situ} imaging spectroscopy., Comment: 9 pages, 9 figures

We achieve a compact optical frequency standard with an extended cavity diode laser locked to the 459 nm 6S$_{1/2}$ - 7P$_{1/2}$ transition of thermal $^{133}$Cs atoms in a $\phi$ 10 mm $\times$ 50 mm glass cell, using modulation transfer spectroscopy (MTS). The self-estimated frequency stability of this laser is $1.4\times{10}^{-14}/\sqrt{\tau}$. With heterodyne measurement, we verify the linewidth-narrowing effect of MTS locking and measure the frequency stability of the locked laser. The linewidth of each laser is reduced from the free-running 69.6 kHz to 10.3 kHz after MTS stabilization, by a factor of 6.75. The Allan deviation measured via beat detection is $2.1\times{10}^{-13}/\sqrt{\tau}$ for each MTS-stabilized laser. In addition, we measure the hyperfine structure of the 7P$_{1/2}$ energy level based on the heterodyne measurements, and calculate the magnetic dipole constant $A$ of the Cs 7P$_{1/2}$ level to be 94.38(6) MHz, which agrees well with previous measurements. This compact optical frequency standard can also be used in other applications that require high-stability lasers, such as laser interferometry, laser cooling, geodesy, and so on., Comment: 9 pages, 8 figures, submitted to Physical Review Applied, under review

Traditional lasers function using resonant cavities, in which the round-trip optical path is exactly equal to an integer multiple of the intracavity wavelengths to constructively enhance the spontaneous emission rate. By taking advantage of the enhancement from the resonant cavity, the narrowest sub-10-mHz-linewidth laser and a $10^{-16}$-fractional-frequency-stability superradiant active optical clock (AOC) have been achieved. However, a laser with atomic spontaneous radiation being destructively inhibited in an anti-resonant cavity, where the atomic resonance is exactly between two adjacent cavity resonances, has not been reported. Herein, we experimentally demonstrate inhibited stimulated emission and termed it an inhibited laser. Compared with traditional superradiant AOCs, which exhibit superiority in terms of the high suppression of cavity noise, the suppression of the cavity-pulling effect of an inhibited laser can be further improved by a factor of $(2F/pi)^2$, i.e., 2.07 in this work, which was improved from 26 to 53 times. This study will guide further development of AOCs with better stability, and thus, it is significant for quantum metrology and may lead to new research in the laser physics and cavity quantum electrodynamics fields., Comment: 14 pages, 9 figures, 1 table

An active optical clock based on highly charged Nd$^{9+}$ ion is proposed for the first time. The clock can offer ultra-narrow linewidth at the $\upmu$Hz-level which is more than two-order of magnitude below the currently recorded laser linewidth. Operating at 605(90) nm superradiation lasing transition between the $5p^2~4f$ ground state and one of the long-lived $5p~4f^2$ excited state, the proposed active clock is inherently immune against the cavity noise which provides high stability. The clock serves as a sensitive probe with high sensitivity to variation of the fine-structure constant with accuracies below 10$^{-19}$ level. Sophisticated relativistic many-body methods are employed to predict related atomic properties that have corroborated the above findings., Comment: 5 papges, 2 figures, 1 SM

A thermal atomic ensemble-based laser source with superior frequency stability is proposed that relies on the accumulated contributions from an abundance of nonzero-transverse-velocity atomic ensembles. Compared with the traditional case in which only atoms with near-zero transverse velocities are utilized, the amplitude of the optical Ramsey fringes for a thermal calcium beam can be dramatically enhanced by three orders of magnitude or more, thus, the signal-to-noise ratio can be improved 33-fold. Based on the recent results of atomic interferometry-based laser stabilization, a quantum projection noise-limited frequency instability less than 2E-17/tau^0.5 is feasible. Such an ultrastable laser has promising applications in diverse areas, including metrology and astronomy.

We demonstrate state-of-the-art technique of an active clock to provide a continuous superradiant lasing signal using an ensemble of trapped Cs atoms in the optical lattice. A magic wavelength of the proposed |7S1/2; F = 4, MF = 0> - |6P3/2; F = 3, MF = 0> clock transition in Cs atom is identified at 1181 nm for constructing the optical lattice. Pertinent optical lines are also found for pumping and repumping atoms from their ground states. A fractional uncertainty about 10-15 level to the clock frequency has been predicted by carrying out rigorous calculations of several atomic properties. The bad-cavity operational mode of the active clock is anticipated to improve its short-term stability remarkably by suppressing intrinsic thermal fluctuations. Thus, a composite clock system with better in both short-term and long-term stabilities can be built by combining the above proposed active clock with another high-accuracy passive optical clock.

Atomic comagnetometers, which measure the spin precession frequencies of overlapped species simultaneously, are widely applied to search for exotic spin-dependent interactions. Here we propose and implement an all-optical single-species Cs atomic comagnetometer based on the optical free induction decay (FID) signal of Cs atoms in hyperfine levels $F_g=3~\&~4$ within the same atomic ensemble. We experimentally show that systematic errors induced by magnetic field gradients and laser fields are highly suppressed in the comagnetometer, but those induced by asynchronous optical pumping and drift of residual magnetic field in the shield dominate the uncertainty of the comagnetometer. With this comagnetometer system, we set the constraint on the strength of spin-gravity coupling of the proton at a level of $10^{-18}$ eV, comparable to the most stringent one. With further optimization in magnetic field stabilization and spin polarization, the systematic errors can be effectively suppressed, and signal-to-noise ratio (SNR) can be improved, promising to set more stringent constraints on spin-gravity interactions.

The residual cavity-pulling effect limits further narrowing of linewidth in dual-wavelength (DW) good-bad-cavity active optical clocks (AOCs). In this paper, we for the first time experimentally realize the cavity-length stabilization of the 1064/1470 nm DW-AOCs by utilizing the phase locking technique of two independent 1064 nm good-cavity lasers. The frequency tracking accuracy between the two main-cavities of DW-AOCs is better than $3 \times {10^{ - 16}}$ at 1 s, and can reach $1 \times {10^{ - 17}}$ at 1000 s. Each 1470 nm bad-cavity laser achieves a most probable linewidth of 53 Hz, which is about a quarter of that without phase locking. The influence of the asynchronous cavity-lengths variation between two DW laser systems is suppressed., Comment: 15 pages, 9 figures

We experimentally observed the Mollow quintuplet (MQ) in F=3/2 hyperfine structure state of 3He atoms. The metastability-exchange collisions (MECs) transfer the Mollow Triplet (MT) from the ground states of 3He atoms to the metastable states, and the MQ is demonstrated by four Zeeman levels of F=3/2 hyperfine states with linearly polarized light. The similar effect also achieves in the mixture cell of 3He and 4He., Comment: 4 pages, 6 figures. arXiv admin note: text overlap with arXiv:1810.11779

We study the dressed states of 3He atoms and experimentally observe the Mollow triplet (MT) induced with an ultra-low-frequency (ULF) oscillating magnetic field as low as 4 Hz. The ULF MT signatures from the ground states of 3He atoms are transferred to the metastable states by metastability-exchange collisions (MECs) and measured optically, which demonstrates 2 s coherence time in the dressed ground states. The result shows the possibility of ULF magnetic field amplitude measurement and a new scheme for optical frequency modulation.

In this work, sidebands suppression brought by buffer gas argon (Ar) in cesium Faraday anomalous dispersion optical filter (FADOF) at 852 nm is investigated. FADOF performances at different Ar pressures (0 torr, 1 torr, 5 torr and 10 torr) are compared, and a single-peak transmittance spectrum with peak transmittance up to 80% is achieved at the high Ar pressure. A detailed analysis shows that, this sidebands suppression comes from the depopulation enhancement by the buffer gas. This result can be generalized to other FADOFs with similar level structures such as the D2 lines of other alkali metal atoms.

Taking advantages of ultra-narrow bandwidth and high noise rejection performance of the Faraday anomalous dispersion optical filter (FADOF), simultaneously with the coherent amplification of atomic stimulated emission, a stimulated amplified Faraday anomalous dispersion optical filter (SAFADOF) at cesium 1470 nm is realized. The SAFADOF is able to significantly amplify very weak laser signals and reject noise in order to obtain clean signals in strong background. Experiment results show that, for a weak signal of 50 pW, the gain factor can be larger than 25000 (44 dB) within a bandwidth as narrow as 13 MHz. Having this ability to amplify weak signals with low background contribution, the SAFADOF finds outstanding potential applications in weak signal detections.

In this letter, we introduce the concept and experimentally demonstrate an ultralong Faraday laser as an optical frequency standard in principle. The ultralong Faraday laser is based on the Faraday anomalous dispersion optical filter (FADOF) with ultra-narrow bandwidth and the ultralong fiber extended cavity of $800$ m. The ultra-narrow FADOF is based on atomic transition line of isotope $^{87}$Rb, which has an ultra-narrow bandwidth of $26.0$ MHz and a transmission of $23.6\%$ at $780$ nm. Fibers of length $150$ m and $800$ m are used as ultralong fiber extended cavities, which provide optical feedback and give extremely small FSR of $0.667$ MHz and $0.125$ MHz, respectively. The mechanism of the proposed ultralong Faraday laser is to combine FADOF's ultra-narrow bandwidth and ultralong cavity's small free spectral range to limit the lasing frequency within FADOF bandwidth covered by the semiconductor gain. The active lasing frequency of the ultralong Faraday laser is determined by the center frequency of FADOF transmission, which is corresponding to atomic transition $5^{2}S_{1/2},\ F = 2 \ \rightarrow\ 5^{2}P_{3/2},\ F^{\prime} = 2,\ 3$ of isotope $^{87}$Rb. The Allan deviation of the fractional frequency of the ultralong Faraday laser output signal in 0.1 s -- 1 s measuring time is around $1\times10^{-11}$., Comment: Corrected the calculation method of the result of Allan deviation of fractional frequency

We propose the mechanism of active Faraday optical clock, and experimentally demonstrate active Faraday optical frequency standards based on 852 nm narrow bandwidth Faraday atomic filter by the method of velocity-selective optical pumping of cesium vapor. The center frequency of the active Faraday optical frequency standards is determined by the cesium 6 $^{2}S_{1/2}$ $F$ = 4 to 6 $^{2}P_{3/2}$ $F'$ = 4 and 5 crossover transition line. The optical heterodyne beat between two similar independent setups shows that the frequency linewidth reaches 996(26) Hz, which is 5.3 $\times$ 10$^{3}$ times smaller than the natural linewidth of the cesium 852 nm transition line. The maximum emitted light power reaches 75 $\upmu$W. The active Faraday optical frequency standards reported here have advantages of narrow linewidth and reduced cavity pulling, which can readily be extended to other atomic transition lines of alkali and alkaline-earth metal atoms trapped in optical lattices at magic wavelengths, making it useful for new generation of optical atomic clocks., Comment: 4 pages, 4 figures

We experimentally realize the dual-wavelength active optical clock for the first time. As the Cs cell temperature is kept between 118 $^{\circ }C$ and 144 $^{\circ }C$, both the 1359 nm and the 1470 nm stimulated emission output of Cs four-level active optical clock are detected. The 1470 nm output linewidth of each experimental setup of Cs four-level active optical clock is measured to be 590 Hz with the main cavity length unstabilized. To stabilize the cavity length of active optical clock, the experimental scheme of 633 nm and 1359 nm good-bad cavity dual-wavelength active optical clock is proposed, where 633 nm and 1359 nm stimulated emission is working at good-cavity and bad-cavity regime respectively. The cavity length is stabilized by locking the 633 nm output frequency to a super-cavity with the Pound-Drever-Hall (PDH) technique. The frequency stability of 1359 nm bad-cavity stimulated emission output is then expected to be further improved by at least 1 order of magnitude than the 633 nm PDH system due to the suppressed cavity pulling effect of active optical clock, and the quantum limited linewidth of 1359 nm output is estimated to be 77.6 mHz., Comment: 12 pages, 5 figures

We experimentally demonstrate the collective emission behavior and suppressed cavity-pulling effect of four-level active optical clock with Cesium atoms. Thermal Cesium atoms in a glass cell velocity selective pumped with a 455.5 nm laser operating at 6S$_{1/2}$ to 7P$_{3/2}$ transition are used as lasing medium. Population inverted Cesium atoms between 7S$_{1/2}$ and 6P$_{3/2}$ levels are optical weakly coupled by a pair cavity mirrors working at deep bad-cavity regime with a finesse of 4.3, and the ratio between cavity bandwidth and gain bandwidth is approximately 45. With increased 455.5 nm pumping laser intensity, the output power of cesium active optical clock at 1469.9 nm from 7S$_{1/2}$ level to 6P$_{3/2}$ level shows a threshold and reach a power of 13 $\mu$W. Active optical clock would dramatically improve the optical clock stability since the lasing frequency does not follow the cavity length variation exactly, but in a form of suppressed cavity pulling effect. In this letter the cavity pulling effect is measured using a Fabry-Perot interferometer (FPI) to be reduced by a factor of 38.2 and 41.4 as the detuning between the 1469.9 nm cavity length of the Cs active optical clock and the Cs 1469.9 nm transition is set to be 140.8 MHz and 281.6 MHz respectively. The mechanism demonstrated here is of great significance for new generation optical clocks and can be applied to improve the stability of best optical clocks by at least two orders of magnitude., Comment: 12 pages, 8 figures

Considering the strong inhomogeneous spatial polarization and intensity distribution of spontaneous decay fluorescence due to the Hanle effect, we propose and demonstrate a universe Hanle detection configuration of electron-shelving method for optical clocks. Experimental results from Ca atomic beam optical frequency standard with 423 nm electron-shelving method show that a designed Hanle detection geometry with optimized magnetic field direction, detection laser beam propagation and polarization direction, and detector position can improve the fluorescence collection rate by more than one order of magnitude comparing with that of inefficient geometry. With the fixed 423 nm fluorescence, the improved 657 nm optical frequency standard signal intensity is presented. And the potential application of the Hanle detection geometry designed for facilitating the fluorescence collection for optical lattice clock with a limited solid angle of the fluorescence collection has been discussed. This Hanle detection configuration is also effective for ion detection in ion optical clock and quantum information experiments. Besides, a cylinder fluorescence collection structure is designed to increase the solid angle of the fluorescence collection in Ca atomic beam optical frequency standard., Comment: 5 pages, 6 figures

Active optical clock, a new conception of atomic clock, has been proposed recently. In this report, we propose a scheme of active optical clock based on four-level quantum system. The final accuracy and stability of two-level quantum system are limited by second-order Doppler shift of thermal atomic beam. To three-level quantum system, they are mainly limited by light shift of pumping laser field. These limitations can be avoided effectively by applying the scheme proposed here. Rubidium atom four-level quantum system, as a typical example, is discussed in this paper. The population inversion between $6S_{1/2}$ and $5P_{3/2}$ states can be built up at a time scale of $10^{-6}$s. With the mechanism of active optical clock, in which the cavity mode linewidth is much wider than that of the laser gain profile, it can output a laser with quantum-limited linewidth narrower than 1 Hz in theory. An experimental configuration is designed to realize this active optical clock., Comment: 5 pages

After pumped from $5s_{1/2}$ ground state to $6p_{1/2}$ state, the population inversion between $6s_{1/2}$ and $5p_{1/2,3/2}$ will be established for Rubidium four-level active optical clock. In this paper, we calculate AC Stark shift due to lattice trapping laser which dominates the frequency shift of clock transition in lattice trapped Rubidium four-level active optical clock. Several magic wavelengths are found that can form desired optical lattice trapping potential. By choosing a proper intensity and linewidth of trapping laser, the fractional frequency uncertainty of clock transition due to AC Stark shift of trapping laser, is estimated to be below 10^-18, Comment: 5 pages

In this paper, we propose a scheme of active ion optical clock with detailed pumping method, lasing states, output power, linewidth and light shift. Using 171Yb+ ions in a Paul Trap we propose to utilize a Fabry-Perot resonator to realize lasing of active optical frequency standards. The quantum-limited linewidth of active 171Yb+ ion optical clock is narrower than 1 mHz. Based on the mechanism and advantages of active optical clock at ion optical clock transition frequency, this new laser light source as a stable local oscillator, will be beneficial to the single-ion optical clock, which currently is the most accurate clock., Comment: 4 pages, 4 figures

We present a theory of the multi-threshold second-order phase transition, and experimentally demonstrate the multi-threshold second-order phase transition phenomenon. With carefully selected parameters, in an external cavity diode laser system, we observe second-order phase transition with multiple (three or four) thresholds in the measured power-current-temperature three dimensional phase diagram. Such controlled death and revival of second-order phase transition sheds new insight into the nature of ubiquitous second-order phase transition. Our theory and experiment show that the single threshold second-order phase transition is only a special case of the more general multi-threshold second-order phase transition, which is an even richer phenomenon., Comment: 5 pages, 3 figures