The theory of thermodynamic relativity

We introduce the theory of thermodynamic relativity, a unified theoretical framework for describing both entropies and velocities, and their respective physical disciplines of thermodynamics and kinematics, which share a surprisingly identical description with relativity. This is the first study to generalize relativity in a thermodynamic context, leading naturally to anisotropic and nonlinear adaptations of relativity; thermodynamic relativity constitutes a new path of generalization, as compared to the "traditional" passage from special to general theory based on curved spacetime. We show that entropy and velocity are characterized by three identical postulates, which provide the basis of a broader framework of relativity: (1) no privileged reference frame with zero value; (2) existence of an invariant and fixed value for all reference frames; and (3) existence of stationarity. The postulates lead to a unique way of addition for entropies and for velocities, called kappa-addition. We develop a systematic method of constructing a generalized framework of the theory of relativity, based on the kappa-addition formulation, which is fully consistent with both thermodynamics and kinematics. We call this novel and unified theoretical framework for simultaneously describing entropy and velocity "thermodynamic relativity". From the generality of the kappa-addition formulation, we focus on the cases corresponding to linear adaptations of special relativity. Then, we show how the developed thermodynamic relativity leads to the addition of entropies in nonextensive thermodynamics and the addition of velocities in Einstein's isotropic special relativity, as in two extreme cases, while intermediate cases correspond to a possible anisotropic adaptation of relativity. Using thermodynamic relativity for velocities, we start from the kappa-addition of velocities and construct the basic formulations of the linear anisotropic special relativity; e.g., the asymmetric Lorentz transformation, the nondiagonal metric, and the energy-momentum-velocity relationships. Then, we discuss the physical consequences of the possible anisotropy in known relativistic effects, such as, (i) matter-antimatter asymmetry, (ii) time dilation, and (iii) Doppler effect, and show how these might be used to detect and quantify a potential anisotropy.

Evolving scientific discovery by unifying data and background knowledge with AI Hilbert

The discovery of scientific formulae that parsimoniously explain natural phenomena and align with existing background theory is a key goal in science. Historically, scientists have derived natural laws by manipulating equations based on existing knowledge, forming new equations, and verifying them experimentally. However, this does not include experimental data within the discovery process, which may be inefficient. We propose a solution to this problem when all axioms and scientific laws are expressible as polynomials and argue our approach is widely applicable. We model notions of minimal complexity using binary variables and logical constraints, solve polynomial optimization problems via mixed-integer linear or semidefinite optimization, and prove the validity of our scientific discoveries in a principled manner using Positivstellensatz certificates. We demonstrate that some famous scientific laws, including Kepler's Law of Planetary Motion and the Radiated Gravitational Wave Power equation, can be derived in a principled manner from axioms and experimental data.

Linewidth Measurement of a Narrow-Linewidth Laser: Principles, Methods, and Systems

Narrow-linewidth lasers mainly depend on the development of advanced laser linewidth measurement methods for related technological progress as key devices in satellite laser communications, precision measurements, ultra-high-speed optical communications, and other fields. This manuscript provides a theoretical analysis of linewidth characterization methods based on the beat frequency power spectrum and laser phase noise calculations, and elaborates on existing research of measurement technologies. In addition, to address the technical challenges of complex measurement systems that commonly rely on long optical fibers and significant phase noise jitter in the existing research, a short-delay self-heterodyne method based on coherent envelope spectrum demodulation was discussed in depth to reduce the phase jitter caused by 1/f noise. We assessed the performance parameters and testing conditions of different lasers, as well as the corresponding linewidth characterization methods, and analyzed the measurement accuracy and error sources of various methods.

High-Precision Fiber Noise Detection and Comparison over a 260 km Field Fiber Link

In this paper, we present a high-precision optical frequency noise detection and comparison technique using a two-way transfer method over a 260 km field fiber link. This method allows for the comparison of optical frequencies between remote optical references without the need for data transfer through communication. We extend a previously established two-way comparison technique to obtain all data at the local site. Two optical carrier signals are injected into the bidirectional fiber from both ends, and one carrier is reflected back from the remote end. This enables the phase comparison of the two carrier signals at a single site without the need to transmit experimental data. The common-mode frequency noise induced by the bidirectional fiber link is detected and effectively suppressed without the need for sophisticated active fiber noise control. Our demonstration system, which uses a 260 km field fiber link and a common laser source, achieves a fractional instability of 2.5×10-17 at 1 s averaging time and scales down to 3.5×10-21 at 8000 s. This scheme offers the distinct advantage of completing the comparison at a single site, eliminating the need for remote data transfer via communication. This method is expected to enhance reliability for high-precision frequency comparisons between remote optical clocks and advanced atomic clocks.

Noise characterization of an ultra-stable laser for optical clocks

We report on the development and performance evaluation of an ultra-stable laser for an 27Al+ optical clock. After a series of noise suppressions, especially the vibrational and temperature fluctuation noise, the 30 cm long cavity stabilized laser obtains a frequency instability of 1.3 × 10-16 @1 s. This result is predicted by noise summation and confirmed by the three-cornered hat method. The 27Al+ optical clock transition is also used to characterize the laser frequency noise, and consistent results are yielded. This is the first reported instance of using single ion optical clocks to measure the frequency noise of ultra-stable lasers, as far as we know. With the implementation of the ultra-stable clock laser, an ultra-narrow linewidth clock transition of 2.8 Hz is obtained.

Excited-Band Coherent Delocalization for Improved Optical Lattice Clock Performance

We implement coherent delocalization as a tool for improving the two primary metrics of atomic clock performance: systematic uncertainty and instability. By decreasing atomic density with coherent delocalization, we suppress cold-collision shifts and two-body losses. Atom loss attributed to Landau-Zener tunneling in the ground lattice band would compromise coherent delocalization at low trap depths for our ^{171}Yb atoms; hence, we implement for the first time delocalization in excited lattice bands. Doing so increases the spatial distribution of atoms trapped in the vertically oriented optical lattice by ∼7 times. At the same time, we observe a reduction of the cold-collision shift by 6.5(8) times, while also making inelastic two-body loss negligible. With these advantages, we measure the trap-light-induced quenching rate and natural lifetime of the ^{3}P_{0} excited state as 5.7(7)×10^{-4}  E_{r}^{-1} s^{-1} and 19(2) s, respectively.

Narrow-linewidth and low RIN Tm/Ho co-doped fiber laser based on self-injection locking

A narrow-linewidth and low relative intensity noise (RIN) Tm/Ho co-doped fiber laser based on a saturable absorber and self-injection locking was demonstrated for the first time. Utilizing self-injection locking technology, the frequency noise power spectral density is remarkably reduced by more than 17.1 dB from 1.21 × 106 Hz2/Hz to 7.30 × 103 Hz2/Hz when the frequency is approximately 1 kHz. Furthermore, a laser with a linewidth compressed to a quarter of the original linewidth from 44.386 kHz to 2.850 kHz, a RIN of less than -127.74 dB/Hz, and an optical signal-to-noise ratio of more than 71.6 dB can be obtained. Using a delay fiber, the relaxation oscillation peak frequencies move to lower frequencies, from 27.9 kHz to 15.8 kHz. The proposed laser is highly competitive in advanced coherent light detection fields, including coherent Doppler wind lidar, high-speed coherent optical communication, and precise absolute distance coherent measurement.

Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity

We consider two interacting systems when one is treated classically while the other system remains quantum. Consistent dynamics of this coupling has been shown to exist, and explored in the context of treating space-time classically. Here, we prove that any such hybrid dynamics necessarily results in decoherence of the quantum system, and a breakdown in predictability in the classical phase space. We further prove that a trade-off between the rate of this decoherence and the degree of diffusion induced in the classical system is a general feature of all classical quantum dynamics; long coherence times require strong diffusion in phase-space relative to the strength of the coupling. Applying the trade-off relation to gravity, we find a relationship between the strength of gravitationally-induced decoherence versus diffusion of the metric and its conjugate momenta. This provides an experimental signature of theories in which gravity is fundamentally classical. Bounds on decoherence rates arising from current interferometry experiments, combined with precision measurements of mass, place significant restrictions on theories where Einstein's classical theory of gravity interacts with quantum matter. We find that part of the parameter space of such theories are already squeezed out, and provide figures of merit which can be used in future mass measurements and interference experiments.

Node-downloadable frequency transfer system based on a mode-locked laser with over 100 km of fiber

To meet the requirements of time-frequency networks and enable frequency downloadability for nodes along the link, we demonstrated the extraction of stable frequency signals at nodes using a mode-locked laser under the condition of 100 km laboratory fiber. The node consists of a simple structure that utilizes widely used optoelectronic devices and enables plug-and-play applications. In addition, the node can recover frequency signals with multiple frequencies, which are useful for scenarios that require different frequencies. Here, we experimentally demonstrated a short-term frequency instability of 2.83 × 10-13@1 s and a long-term frequency instability of 1.18 × 10-15@10,000 s at the node, which is similar to that at the remote site of the frequency transfer system. At the same time, frequency signals with different frequencies also achieved stable extraction with the same performance at the node. Our results can support the distributed application under large-scale time-frequency networks.

Optimal Generators for Quantum Sensing

We propose a computationally efficient method to derive the unitary evolution that a quantum state is most sensitive to. This allows one to determine the optimal use of an entangled state for quantum sensing, even in complex systems where intuition from canonical squeezing examples breaks down. In this paper we show that the maximal obtainable sensitivity using a given quantum state is determined by the largest eigenvalue of the quantum Fisher information matrix (QFIM) and the corresponding evolution is uniquely determined by the coinciding eigenvector. Since we optimize the process of parameter encoding rather than focusing on state preparation protocols, our scheme is relevant for any quantum sensor. This procedure naturally optimizes multiparameter estimation by determining, through the eigenvectors of the QFIM, the maximal set of commuting observables with optimal sensitivity.

An integrated high-flux cold atomic beam source for strontium

We present the design, construction, and characterization of an integrated cold atomic beam source for strontium (Sr), which is based on a compact Zeeman slower for slowing the thermal atomic beam and an atomic deflector for selecting the cold flux. By adopting arrays of permanent magnets to produce the magnetic fields of the slower and the deflector, we effectively reduce the system size and power compared to traditional systems with magnetic coils. After the slower cooling, one can employ additional transverse cooling in the radial direction and improve the atom collimation. The atomic deflectors employ two stages of two-dimensional magnetic-optical trapping (MOT) to deflect the cold flux, whose atomic speed is lower than 50 m/s, by 20° from the thermal atomic beam. We characterize the cold atomic beam flux of the source by measuring the loading rate of a three-dimensional MOT. The loading rates reach up to 109 atoms/s. The setup is compact, highly tunable, lightweight, and requires low electrical power, which addresses the challenge of reducing the complexity of building optical atomic clocks and quantum simulation devices based on Sr.

A lab-based test of the gravitational redshift with a miniature clock network

Einstein's theory of general relativity predicts that a clock at a higher gravitational potential will tick faster than an otherwise identical clock at a lower potential, an effect known as the gravitational redshift. Here we perform a laboratory-based, blinded test of the gravitational redshift using differential clock comparisons within an evenly spaced array of 5 atomic ensembles spanning a height difference of 1 cm. We measure a fractional frequency gradient of [ - 12.4 ± 0. 7(stat) ± 2. 5(sys)] × 10-19/cm, consistent with the expected redshift gradient of - 10.9 × 10-19/cm. Our results can also be viewed as relativistic gravitational potential difference measurements with sensitivity to mm scale changes in height on the surface of the Earth. These results highlight the potential of local-oscillator-independent differential clock comparisons for emerging applications of optical atomic clocks including geodesy, searches for new physics, gravitational wave detection, and explorations of the interplay between quantum mechanics and gravity.

Combining data and theory for derivable scientific discovery with AI-Descartes

Scientists aim to discover meaningful formulae that accurately describe experimental data. Mathematical models of natural phenomena can be manually created from domain knowledge and fitted to data, or, in contrast, created automatically from large datasets with machine-learning algorithms. The problem of incorporating prior knowledge expressed as constraints on the functional form of a learned model has been studied before, while finding models that are consistent with prior knowledge expressed via general logical axioms is an open problem. We develop a method to enable principled derivations of models of natural phenomena from axiomatic knowledge and experimental data by combining logical reasoning with symbolic regression. We demonstrate these concepts for Kepler's third law of planetary motion, Einstein's relativistic time-dilation law, and Langmuir's theory of adsorption. We show we can discover governing laws from few data points when logical reasoning is used to distinguish between candidate formulae having similar error on the data.

Bosonic Pair Production and Squeezing for Optical Phase Measurements in Long-Lived Dipoles Coupled to a Cavity

We propose to simulate bosonic pair creation using large arrays of long-lived dipoles with multilevel internal structure coupled to an undriven optical cavity. Entanglement between the atoms, generated by the exchange of virtual photons through a common cavity mode, grows exponentially fast and is described by two-mode squeezing of effective bosonic quadratures. The mapping between an effective bosonic model and the natural spin description of the dipoles allows us to realize the analog of optical homodyne measurements via straightforward global rotations and population measurements of the electronic states, and we propose to exploit this for quantum-enhanced sensing of an optical phase (common and differential between two ensembles). We discuss a specific implementation based on Sr atoms and show that our sensing protocol is robust to sources of decoherence intrinsic to cavity platforms. Our proposal can open unique opportunities for next-generation optical atomic clocks.

Fully digital platform for local ultra-stable optical frequency distribution

This article reports on the use of a Field Programmable Gate Array (FPGA) platform for local ultra-stable optical frequency distribution through a 90 m-long fiber network. This platform is used to implement a fully digital treatment of the Doppler-cancellation scheme required by fiber links to be able to distribute ultra-stable frequencies. We present a novel protocol that uses aliased images of a digital synthesizer output to directly generate signals above the Nyquist frequency. This approach significantly simplifies the setup, making it easy to duplicate within a local fiber network. We demonstrate performances enabling the distribution of an optical signal with an instability below 10^-17 at 1 s at the receiver end. We also use the board to implement an original characterization method. It leads to an efficient characterization of the disturbance rejection of the system that can be realized without accessing the remote output of the fiber link.

Adaptive optics LEO uplink pre-compensation with finite spatial modes

Adaptive optics pre-compensation of free-space optical communications uplink from ground to space is complicated by the "point ahead angle" due to spacecraft velocity and the finite speed of light, as well as anisoplanatism of the uplink beam and the wavefront beacon. This Letter explores how pre-compensation varies with the number of spatial modes applied and how it varies with a beacon at the point-ahead angle versus a downlink beacon. Using a w₀ = 16 cm Gaussian beam propagating through a modified Hufnagel-Valley model as an example, we find pre-compensation performance plateaus beyond ∼100 applied modes regardless of integrated turbulence strength, and that a point ahead beacon provides a 1-4 dB gain in median received power and an order-of-magnitude reduction in scintillation over a downlink beacon at wavelengths typical of optical communications. Modeling tailored to specific scenarios should be conducted to determine whether implementing a resource-intensive point ahead beacon is the optimum path to meeting link requirements.

Effective linewidth compression of a single-longitudinal-mode fiber laser with randomly distributed high scattering centers in the fiber induced by femtosecond laser pulses

Femtosecond lasers can be used to create many functional devices in silica optical fibers with high designability. In this work, a femtosecond laser-induced high scattering fiber (HSF) with randomly distributed high scattering centers is used to effectively compress the linewidth of a fiber laser for the first time. A dual-wavelength, single-longitudinal-mode (SLM) erbium-doped fiber laser (EDFL) is constructed for the demonstration, which is capable of switching among two single-wavelength operations and one dual-wavelength operation. We find that the delayed self-heterodyne beating linewidth of the laser can be reduced from >1 kHz to <150 Hz when the length of the HSF in the laser cavity increases from 0 m to 20 m. We also find that the intrinsic Lorentzian linewidth of the laser can be compressed to several Hz using the HSF. The efficiency and effectiveness of linewidth reduction are also validated for the case that the laser operates in simultaneous dual-wavelength lasing mode. In addition to the linewidth compression, the EDFL shows outstanding overall performance after the HSF is incorporated. In particular, the optical spectrum and SLM lasing state are stable over long periods of time. The relative intensity noise is as low as <-150 dB/Hz@>3 MHz, which is very close to the shot noise limit. The optical signal-to-noise ratios of >85 dB for single-wavelength operation and >83 dB for dual-wavelength operation are unprecedented over numerous SLM fiber lasers reported previously. This novel method for laser linewidth reduction is applicable across gain-medium-type fiber lasers, which enables low-cost, high-performance, ultra-narrow linewidth fiber laser sources for many applications.

An elementary quantum network of entangled optical atomic clocks

Optical atomic clocks are our most precise tools to measure time and frequency^1-3. Precision frequency comparisons between clocks in separate locations enable one to probe the space-time variation of fundamental constants^4,5 and the properties of dark matter^6,7, to perform geodesy^8-10 and to evaluate systematic clock shifts. Measurements on independent systems are limited by the standard quantum limit; measurements on entangled systems can surpass the standard quantum limit to reach the ultimate precision allowed by quantum theory-the Heisenberg limit. Although local entangling operations have demonstrated this enhancement at microscopic distances^11-16, comparisons between remote atomic clocks require the rapid generation of high-fidelity entanglement between systems that have no intrinsic interactions. Here we report the use of a photonic link^17,18 to entangle two ⁸⁸Sr⁺ ions separated by a macroscopic distance¹⁹ (approximately 2 m) to demonstrate an elementary quantum network of entangled optical clocks. For frequency comparisons between the ions, we find that entanglement reduces the measurement uncertainty by nearly [Formula: see text], the value predicted for the Heisenberg limit. Today's optical clocks are typically limited by dephasing of the probe laser²⁰; in this regime, we find that entanglement yields a factor of 2 reduction in the measurement uncertainty compared with conventional correlation spectroscopy techniques^20-22. We demonstrate this enhancement for the measurement of a frequency shift applied to one of the clocks. This two-node network could be extended to additional nodes²³, to other species of trapped particles or-through local operations-to larger entangled systems.

Dual optical frequency combs with ultra-low relative phase jitters from 550 nm to 1020 nm for precision spectroscopy

Here, ultra-low relative phase jitters over a wide optical spectrum were achieved for dual Ti:Sapphire optical frequency combs. The two optical frequency combs were independently phase-locked to a Sr optical lattice clock laser delivered through a commercial optical fiber network. We confirmed that the relative phase jitters between the two combs integrated from 8.3 mHz to 200 kHz were below 1 rad, corresponding to a relative linewidth of below 8.3 mHz, over the entire wavelength of the optical frequency combs ranging from 550 nm to 1020 nm. Our work paves the way for ultrahigh-precision dual-comb spectroscopy covering a wide optical spectral range with a simple setup, and provides an absolute optical frequency reference with great stability over a wide range of wavelengths.

Differential clock comparisons with a multiplexed optical lattice clock

Rapid progress in optical atomic clock performance has advanced the frontiers of timekeeping, metrology and quantum science^1-3. Despite considerable efforts, the instabilities of most optical clocks remain limited by the local oscillator rather than the atoms themselves^4,5. Here we implement a 'multiplexed' one-dimensional optical lattice clock, in which spatially resolved strontium atom ensembles are trapped in the same optical lattice, interrogated simultaneously by a shared clock laser and read-out in parallel. In synchronous Ramsey interrogations of ensemble pairs we observe atom-atom coherence times of 26 s, a 270-fold improvement over the measured atom-laser coherence time, demonstrate a relative instability of [Formula: see text] (where τ is the averaging time) and reach a relative statistical uncertainty of 8.9 × 10^-20 after 3.3 h of averaging. These results demonstrate that applications involving optical clock comparisons need not be limited by the instability of the local oscillator. We further realize a miniaturized clock network consisting of 6 atomic ensembles and 15 simultaneous pairwise comparisons with relative instabilities below [Formula: see text], and prepare spatially resolved, heterogeneous ensemble pairs of all four stable strontium isotopes. These results pave the way for multiplexed precision isotope shift measurements, spatially resolved characterization of limiting clock systematics, the development of clock-based gravitational wave and dark matter detectors^6-12 and new tests of relativity in the lab^13-16.

Resolving the gravitational redshift across a millimetre-scale atomic sample

Einstein's theory of general relativity states that clocks at different gravitational potentials tick at different rates relative to lab coordinates-an effect known as the gravitational redshift¹. As fundamental probes of space and time, atomic clocks have long served to test this prediction at distance scales from 30 centimetres to thousands of kilometres^2-4. Ultimately, clocks will enable the study of the union of general relativity and quantum mechanics once they become sensitive to the finite wavefunction of quantum objects oscillating in curved space-time. Towards this regime, we measure a linear frequency gradient consistent with the gravitational redshift within a single millimetre-scale sample of ultracold strontium. Our result is enabled by improving the fractional frequency measurement uncertainty by more than a factor of 10, now reaching 7.6 × 10^-21. This heralds a new regime of clock operation necessitating intra-sample corrections for gravitational perturbations.

Floquet Engineering Hz-Level Rabi Spectra in Shallow Optical Lattice Clock

Quantum metrology with ultrahigh precision usually requires atoms prepared in an ultrastable environment with well-defined quantum states. Thus, in optical lattice clock systems deep lattice potentials are used to trap ultracold atoms. However, decoherence, induced by Raman scattering and higher order light shifts, can significantly be reduced if atomic clocks are realized in shallow optical lattices. On the other hand, in such lattices, tunneling among different sites can cause additional dephasing and strongly broadening of the Rabi spectrum. Here, in our experiment, we periodically drive a shallow ^{87}Sr optical lattice clock. Counterintuitively, shaking the system can deform the wide broad spectral line into a sharp peak with 5.4 Hz linewidth. With careful comparison between the theory and experiment, we demonstrate that the Rabi frequency and the Bloch bands can be tuned, simultaneously and independently. Our work not only provides a different idea for quantum metrology, such as building shallow optical lattice clock in outer space, but also paves the way for quantum simulation of new phases of matter by engineering exotic spin orbit couplings.

Quantum probe of space-time curvature

An atom interferometer measures the quantum phase due to gravitational time dilation.

Wavelength-switchable ultra-narrow linewidth fiber laser enabled by a figure-8 compound-ring-cavity filter and a polarization-managed four-channel filter

We propose and demonstrate a high-performance wavelength-switchable erbium-doped fiber laser (EDFL), enabled by a figure-8 compound-ring-cavity (F8-CRC) filter for single-longitudinal-mode (SLM) selection and a polarization-managed four-channel filter (PM-FCF) for defining four lasing wavelengths. We introduce a novel methodology utilizing signal-flow graph combined with Mason's rule to analyze a CRC filter in general and apply it to obtain the important design parameters for the F8-CRC used in this paper. By combining the functions of the F8-CRC filter and the PM-FCF assisted by the enhanced polarization hole-burning and polarization dependent loss, we achieve the EDFL with fifteen lasing states, including four single-, six dual-, four tri- and one quad-wavelength lasing operations. In particular, all the four single-wavelength operations are in stable SLM oscillation, typically with a linewidth of <600 Hz, a RIN of ≤-154.58 dB/Hz@≥3 MHz and an output power fluctuation of ≤±3.45%. In addition, all the six dual-wavelength operations have very similar performances, with the performance parameters close to those of the four single-wavelength operations, superior to our previous work and others' similar work significantly. Finally, we achieve the wavelength-spacing tuning of dual-wavelength operations for photonic generation of tunable microwave signals, and successfully obtain a signal at 23.10 GHz as a demonstration.

Narrow-linewidth single-polarization fiber laser using non-polarization optics

Single longitudinal mode and single polarization are basic requirements of high performance fiber lasers, while their realizations are nontrivial, owing to the long laser cavity and lack of polarization selection of ordinary optical fibers. Here, we demonstrate an all-fiber narrow-linewidth laser realized on an external high-Q fiber ring, with combined functions of single-longitude-mode selection and linewidth reduction. A single-longitude-mode laser with a high polarization extinction ratio of ∼40dB and low white frequency noise at 0.3Hz²/Hz is achieved, corresponding to a fundamental linewidth of ∼0.92Hz. Using all non-polarization fiber components and ordinary gain fiber, our scheme shows the realization of narrow-linewidth single-polarization fiber lasers in a simple and cost-effective way, promising for broadband applications.

Vibration modes of a transportable optical cavity

Many factors still need to be evaluated to fully understand the physical mechanisms determining optical cavity vibration, which are crucial for designing and constructing a transportable ultra-stable laser. Herein, a detailed dynamic analysis is used to characterize the vibration modes of a transportable optical cavity. The first five resonance modes are identified in the presence of a cavity support, and the guidelines to achieve a high first-order resonance frequency are proposed; thereby, high robustness is described using a 50 mm length optical cavity. The results demonstrate that the first-order resonance of up to 681 Hz with a gross weight of 2.51 kg can be achieved using an optimization strategy for given cavity support. The theoretical results are consistent with simulation and experiment. With the optimal group, a 1.34 Hz linewidth transportable ultra-stable laser at 1550 nm is established, and a linewidth of 1.5 Hz and frequency instability of 9.5×10^-15@1s are obtained after the test including 100 km actual road transportation and 34 min continuous vibration. The test condition is equivalent to ∼ 1000 km road transportation. Our method can be readily extended to other transportable optical cavities, providing a powerful tool for improving the robustness of vibration, particularly important for transportable environments.

Dipole-Dipole Frequency Shifts in Multilevel Atoms

Dipole-dipole interactions lead to frequency shifts that are expected to limit the performance of next-generation atomic clocks. In this work, we compute dipolar frequency shifts accounting for the intrinsic atomic multilevel structure in standard Ramsey spectroscopy. When interrogating the transitions featuring the smallest Clebsch-Gordan coefficients, we find that a simplified two-level treatment becomes inappropriate, even in the presence of large Zeeman shifts. For these cases, we show a net suppression of dipolar frequency shifts and the emergence of dominant nonclassical effects for experimentally relevant parameters. Our findings are pertinent to current generations of optical lattice and optical tweezer clocks, opening a way to further increase their current accuracy, and thus their potential to probe fundamental and many-body physics.

Narrow-linewidth self-injection locked diode laser with a high-Q fiber Fabry-Perot resonator

Narrow-linewidth lasers are essential for various applications, but are limited by their size, weight, power, and cost requirements. Here we demonstrate a self-injection locked diode laser fabricated with a high quality factor fiber Fabry-Perot resonator, with a 145 Hz free-running linewidth. The locking scheme is all-fiber for plug-and-play operation. White frequency noise of 50Hz²/Hz is measured with over 42 dB reduction from the low-cost TO-can distributed feedback laser diode, and shows its wide applications in a compact and cost-effective way.

Measurement of gravitational coupling between millimetre-sized masses

Gravity is the weakest of all known fundamental forces and poses some of the most important open questions to modern physics: it remains resistant to unification within the standard model of physics and its underlying concepts appear to be fundamentally disconnected from quantum theory^1-4. Testing gravity at all scales is therefore an important experimental endeavour^5-7. So far, these tests have mainly involved macroscopic masses at the kilogram scale and beyond⁸. Here we show gravitational coupling between two gold spheres of 1 millimetre radius, thereby entering the regime of sub-100-milligram sources of gravity. Periodic modulation of the position of the source mass allows us to perform a spatial mapping of the gravitational force. Both linear and quadratic coupling are observed as a consequence of the nonlinearity of the gravitational potential. Our results extend the parameter space of gravity measurements to small, single source masses and low gravitational field strengths. Further improvements to our methodology will enable the isolation of gravity as a coupling force for objects below the Planck mass. This work opens the way to the unexplored frontier of microscopic source masses, which will enable studies of fundamental interactions^9-11 and provide a path towards exploring the quantum nature of gravity^12-15.

Demonstration of the Systematic Evaluation of an Optical Lattice Clock Using the Drift-Insensitive Self-Comparison Method

The self-comparison method is a powerful tool in the uncertainty evaluation of optical lattice clocks, but any drifts will cause a frequency offset between the two compared clock loops and thus lead to incorrect measurement result. We propose a drift-insensitive self-comparison method to remove this frequency offset by adjusting the clock detection sequence. We also experimentally demonstrate the validity of this method in a one-dimensional 87Sr optical lattice clock. As the clock laser frequency drift exists, the measured frequency difference between two identical clock loops is (240 ± 34) mHz using the traditional self-comparison method, while it is (−15 ± 16) mHz using the drift-insensitive self-comparison method, indicating that this frequency offset is cancelled within current measurement precision. We further use the drift-insensitive self-comparison technique to measure the collisional shift and the second-order Zeeman shift of our clock and the results show that the fractional collisional shift and the second-order Zeeman shift are 4.54(28) × 10−16 and 5.06(3) × 10−17, respectively.

Point-to-point stabilized optical frequency transfer with active optics

Timescale comparison between optical atomic clocks over ground-to-space and terrestrial free-space laser links will have enormous benefits for fundamental and applied sciences. However, atmospheric turbulence creates phase noise and beam wander that degrade the measurement precision. Here we report on phase-stabilized optical frequency transfer over a 265 m horizontal point-to-point free-space link between optical terminals with active tip-tilt mirrors to suppress beam wander, in a compact, human-portable set-up. A phase-stabilized 715 m underground optical fiber link between the two terminals is used to measure the performance of the free-space link. The active optical terminals enable continuous, cycle-slip free, coherent transmission over periods longer than an hour. In this work, we achieve residual instabilities of 2.7 × 10-6 rad2 Hz-1 at 1 Hz in phase, and 1.6 × 10-19 at 40 s of integration in fractional frequency; this performance surpasses the best optical atomic clocks, ensuring clock-limited frequency comparison over turbulent free-space links.

The Uniformly Accelerated Frame as a Test Bed for Analysing the Gravitational Redshift

Ever since Eddington’s analysis of the gravitational redshift a century ago, and the arguments in the relativity community that it produced, fine details of the roles of proper time and coordinate time in the redshift remain somewhat obscure. We shed light on these roles by appealing to the physics of the uniformly accelerated frame, in which coordinate time and proper time are well defined and easy to understand; and because that frame exists in flat spacetime, special relativity is sufficient to analyse it. We conclude that Eddington’s analysis was indeed correct—as was the 1980 analysis of his detractors, Earman and Glymour, who (it turns out) were following a different route. We also use the uniformly accelerated frame to pronounce invalid Schild’s old argument for spacetime curvature, which has been reproduced by many authors as a pedagogical introduction to curved spacetime. More generally, because the uniformly accelerated frame simulates a gravitational field, it can play a strong role in discussions of proper and coordinate times in advanced relativity.

Sub-ps resolution clock-offset measurement over a 114 km fiber link using linear optical sampling

We demonstrate a sub-ps resolution clock-offset measurement based on linear optical sampling technique via a 114 km fiber link by transferring a dual optical frequency comb. The time deviation between two distance clocks is 110 fs at 1 s and 22 fs at 100 s averaging, and the standard deviation of the measured clock offset is 237 fs. This sub-ps level of clock offset measurement should benefit many time synchronization applications via long fiber links.

Quantum clocks observe classical and quantum time dilation

At the intersection of quantum theory and relativity lies the possibility of a clock experiencing a superposition of proper times. We consider quantum clocks constructed from the internal degrees of relativistic particles that move through curved spacetime. The probability that one clock reads a given proper time conditioned on another clock reading a different proper time is derived. From this conditional probability distribution, it is shown that when the center-of-mass of these clocks move in localized momentum wave packets they observe classical time dilation. We then illustrate a quantum correction to the time dilation observed by a clock moving in a superposition of localized momentum wave packets that has the potential to be observed in experiment. The Helstrom-Holevo lower bound is used to derive a proper time-energy/mass uncertainty relation.

Laboratory Courses on Laser Spectroscopy and Atom Trapping

We present an overview of experiments covered in two semester-length laboratory courses dedicated to laser spectroscopy and atom trapping. These courses constitute a powerful approach for teaching experimental physics in a manner that is both contemporary and capable of providing the background and skills relevant to a variety of research laboratories. The courses are designed to be accessible for all undergraduate streams in physics and applied physics as well as incoming graduate students. In the introductory course, students carry out several experiments in atomic and laser physics. In a follow up course, students trap atoms in a magneto-optical trap and carry out preliminary investigations of the properties of laser cooled atoms based on the expertise acquired in the first course. We discuss details of experiments, impact, possible course formats, budgetary requirements, and challenges related to long-term maintenance.

An Evaluation of the Zeeman Shift of the 87Sr Optical Lattice Clock at the National Time Service Center

The Zeeman shift plays an important role in the evaluation of optical lattice clocks since a strong bias magnetic field is applied for departing Zeeman sublevels and defining a quantization axis. We demonstrated the frequency correction and uncertainty evaluation due to Zeeman shift in the 87Sr optical lattice clock at the National Time Service Center. The first-order Zeeman shift was almost completely removed by stabilizing the clock laser to the average frequency of the two Zeeman components of mF = ±9/2. The residual first-order Zeeman shift arose from the magnetic field drift between measurements of the two stretched-state center frequencies; the upper bound was inferred as 4(5) × 10−18. The quadratic Zeeman shift coefficient was experimentally determined as –23.0(4) MHz/T2 and the final Zeeman shift was evaluated as 9.20(7) × 10−17. The evaluation of the Zeeman shift is a foundation for overall evaluation of the uncertainty of an optical lattice clock. This measurement can provide more references for the determination of the quadratic coefficient of 87Sr.

Seeded optical parametric oscillator light source for precision spectroscopy

Precision spectroscopy of fundamental bands of molecules in the mid-infrared (MIR) region is of great interest in applications of trace detection and testing fundamental physics, where high-power and narrow-linewidth MIR lasers are needed. By using a frequency-stabilized near-infrared laser as a seed of the signal light of a continuous-wave optical parametric oscillator, we established a broadly tunable MIR light source that has an output power of several hundred milliwatts and a linewidth of a few tens of kilohertz. The MIR laser frequency drift was reduced to below 1 kHz by using an optical frequency comb to stabilize the frequency of the 1064 nm pumping laser. The performance of the light source was investigated and tested by measuring the saturated absorption spectroscopy of a few molecular transitions at 3.3 µm.

An experimental test of the geodesic rule proposition for the noncyclic geometric phase

The geometric phase due to the evolution of the Hamiltonian is a central concept in quantum physics and may become advantageous for quantum technology. In noncyclic evolutions, a proposition relates the geometric phase to the area bounded by the phase-space trajectory and the shortest geodesic connecting its end points. The experimental demonstration of this geodesic rule proposition in different systems is of great interest, especially due to the potential use in quantum technology. Here, we report a previously unshown experimental confirmation of the geodesic rule for a noncyclic geometric phase by means of a spatial SU(2) matter-wave interferometer, demonstrating, with high precision, the predicted phase sign change and π jumps. We show the connection between our results and the Pancharatnam phase. Last, we point out that the geodesic rule may be applied to obtain the red shift in general relativity, enabling a new quantum tool to measure gravity.

Detection of the 5p - 4f orbital crossing and its optical clock transition in Pr9

Recent theoretical works have proposed atomic clocks based on narrow optical transitions in highly charged ions. The most interesting candidates for searches of physics beyond the Standard Model are those which occur at rare orbital crossings where the shell structure of the periodic table is reordered. There are only three such crossings expected to be accessible in highly charged ions, and hitherto none have been observed as both experiment and theory have proven difficult. In this work we observe an orbital crossing in a system chosen to be tractable from both sides: Pr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{9+}$$\end{document}9+. We present electron beam ion trap measurements of its spectra, including the inter-configuration lines that reveal the sought-after crossing. With state-of-the-art calculations we show that the proposed nHz-wide clock line has a very high sensitivity to variation of the fine-structure constant, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α, and violation of local Lorentz invariance; and has extremely low sensitivity to external perturbations.

Atomic clocks are based on the frequency of optical transitions and offer high precision. Here the authors demonstrate a configuration crossing in the highly charged ion praseodymium (Pr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{9+}$$\end{document}9+) and determine the frequency of a potential reference transition for a highly charged ion clock.

Four-wavelength-switchable SLM fiber laser with sub-kHz linewidth using superimposed high-birefringence FBG and dual-coupler ring based compound-cavity filter

We propose and demonstrate a four-wavelength-switchable erbium-doped fiber laser (4WS-EDFL) with a four-channel superimposed high-birefringence fiber Bragg grating (SI-HBFBG) and a dual-coupler ring based compound-cavity (DCR-CC) filter. Both for the first time, a SI-HBFBG as a four-channel reflective filter is used in a multi-wavelength switchable fiber laser to define wavelength channels and a DCR-CC filter is used to select a single mode from dense longitudinal-modes in a fiber laser. We present in detail how to design, fabricate, and characterize the DCR-CC filter with both theoretical analysis and experimental results, which we believe is the first systematic approach for making a compound-cavity based filter used for selecting single-longitudinal mode (SLM) in a fiber laser. The enhanced polarization hole burning effect in a 2.9 m long erbium-doped fiber, coiled inside a three-loop polarization controller, and the polarization-mismatch-induced losses are introduced into the laser cavity to achieve wavelength-switching operations. We show that the 4WS-EDFL can be switched among fifteen lasing states, including four single-wavelength operations, six dual-wavelength operations, four three-wavelength operations and one four-wavelength operation, all with high stability. For demonstration, in switchable single-wavelength operations, the four SLM lasing outputs measured are all with an optical signal to noise ratio of >80 dB, a linewidth of <700 Hz, a relative intensity noise of ≤-156.7 dB/Hz at frequencies over 3 MHz, an output power fluctuation of ≤0.555 dB and excellent polarization characteristics.

Reciprocity of propagation in optical fiber links demonstrated to 10-21

We present a study of the fundamental limit of fiber links using dedicated link architecture. We use an experimental arrangement that enables us to detect the forward and backward propagation noise independently and simultaneously in optical fiber and where the optical phase evolution is expected to be driven by the only contribution of the reference arms of the Michelson interferometer ensemble. In this article, we demonstrate indeed the high correlation between the optical phase evolution and the temperature variation of the interferometer ensemble, leading to a frequency offset of (4.4±2.3)×10^-21. Using a simple temperature model and a Bayesian analysis to evaluate the model parameters, we show that the temperature effect can be compensated with post-processing, removing the frequency offset down to (0.5±2.0)×10^-21. The residual slope of the optical phase evolution over 33 days is 350 yoctosecond/s. Using a global temperature parameter, we divide these 33 days dataset in four subsets and analyse their uncertainties. We show that they are self-consistent when the temperature is taken into account. This provides an alternative method to evaluate the accuracy of a fiber link, especially when the dataset includes large dead times. The result is finally interpreted as a test of the reciprocity of the propagation delay in an optical fiber. This unprecedented transfer capability could enable the comparisons of future optical clocks with expected performance at 10^-20 level and open new possibilities for stringent tests of special and general relativity.

Interference of clocks: A quantum twin paradox

The phase of matter waves depends on proper time and is therefore susceptible to special-relativistic (kinematic) and gravitational (redshift) time dilation. Hence, it is conceivable that atom interferometers measure general-relativistic time-dilation effects. In contrast to this intuition, we show that (i) closed light-pulse interferometers without clock transitions during the pulse sequence are not sensitive to gravitational time dilation in a linear potential. (ii) They can constitute a quantum version of the special-relativistic twin paradox. (iii) Our proposed experimental geometry for a quantum-clock interferometer isolates this effect.

Free-space transfer of comb-rooted optical frequencies over an 18 km open-air link

Phase-coherent transfer of optical frequencies over a long distance is required for diverse photonic applications, including optical clock dissemination and physical constants measurement. Several demonstrations were made successfully over fiber networks, but not much work has been done yet through the open air where atmospheric turbulence prevails. Here, we use an 18 km outdoor link to transmit multiple optical carriers extracted directly from a frequency comb of a 4.2 THz spectral width. In stabilization to a high-finesse cavity with a 1.5 Hz linewidth, the comb-rooted optical carriers are simultaneously transferred with collective suppression of atmospheric phase noise to -80 dBc Hz^-1. Microwaves are also delivered by pairing two separate optical carriers bound with inter-comb-mode coherence, for example a 10 GHz signal with phase noise of -105 dBc Hz^-1 at 1 Hz offset. Lastly, an add-on demonstration is given for multi-channel coherent optical communications with the potential of multi-Tbps data transmission in free space.

Seconds-scale coherence on an optical clock transition in a tweezer array

Coherent control of high-quality factor optical transitions in atoms has revolutionized precision frequency metrology. Leading optical atomic clocks rely on the interrogation of such transitions in either single ions or ensembles of neutral atoms to stabilize a laser frequency at high precision and accuracy. We demonstrate a platform that combines the key strengths of these two approaches, based on arrays of individual strontium atoms held within optical tweezers. We report coherence times of 3.4 seconds, single-ensemble duty cycles up to 96% through repeated interrogation, and frequency stability of 4.7 × 10^-16 (τ/s)^-1/2 These results establish optical tweezer arrays as a powerful tool for coherent control of optical transitions for metrology and quantum information science.

Bell's theorem for temporal order

Time has a fundamentally different character in quantum mechanics and in general relativity. In quantum theory events unfold in a fixed order while in general relativity temporal order is influenced by the distribution of matter. When matter requires a quantum description, temporal order is expected to become non-classical-a scenario beyond the scope of current theories. Here we provide a direct description of such a scenario. We consider a thought experiment with a massive body in a spatial superposition and show how it leads to entanglement of temporal orders between time-like events. This entanglement enables accomplishing a task, violation of a Bell inequality, that is impossible under local classical temporal order; it means that temporal order cannot be described by any pre-defined local variables. A classical notion of a causal structure is therefore untenable in any framework compatible with the basic principles of quantum mechanics and classical general relativity.

Ramsey-Bordé Matter-Wave Interferometry for Laser Frequency Stabilization at 10^{-16} Frequency Instability and Below

We demonstrate Ramsey-Bordé (RB) atom interferometry for high performance laser stabilization with fractional frequency instability <2×10^{-16} for timescales between 10 and 1000s. The RB spectroscopy laser interrogates two counterpropagating ^{40}Ca beams on the ^{1}S_{0}-^{3}P_{1} transition at 657 nm, yielding 1.6 kHz linewidth interference fringes. Fluorescence detection of the excited state population is performed on the (4s4p) ^{3}P_{1}-(4p^{2}) ^{3}P_{0} transition at 431 nm. Minimal thermal shielding and no vibration isolation are used. These stability results surpass performance from other thermal atomic or molecular systems by 1 to 2 orders of magnitude, and further improvements look feasible.

Quadrupole Shift Cancellation Using Dynamic Decoupling

We present a method that uses radio-frequency pulses to cancel the quadrupole shift in optical clock transitions. Quadrupole shifts are an inherent inhomogeneous broadening mechanism in trapped ion crystals and impose one of the limitations forcing current optical ion clocks to work with a single probe ion. Canceling this shift, at each interrogation cycle of the ion frequency, reduces the complexity in using N>1 ions in clocks, thus allowing for a reduction of the instability in the clock frequency by sqrt[N] according to the standard quantum limit. Our sequence relies on the tensorial nature of the quadrupole shift, and thus also cancels other tensorial shifts, such as the tensor ac stark shift. We experimentally demonstrate our sequence on three and seven ^{88}Sr^{+} ions trapped in a linear Paul trap, using correlation spectroscopy. We show a reduction of the quadrupole shift difference between ions to the ≈10  mHz level where other shifts, such as the relativistic second-order Doppler shift, are expected to limit our spectral resolution. In addition, we show that using radio-frequency dynamic decoupling we can also cancel the effect of first-order Zeeman shifts.

Direct phase-locking of a Ti:Sapphire optical frequency comb to a remote optical frequency standard

We report on an ultralow noise optical frequency transfer from a remotely located Sr optical lattice clock laser to a Ti:Sapphire optical frequency comb through telecom-wavelength optical fiber networks. The inherent narrow linewidth of the Ti:Sapphire optical frequency comb eliminates the need for a local reference high-finesse cavity. The relative fractional frequency instability of the optical frequency comb with respect to the remote optical reference was 6.7(1) × 10^-18 at 1 s and 1.05(3) × 10^-19 at 1,000 s including a 2.9 km-long fiber network. This ensured the optical frequency comb had the same precision as the optical standard. Our result paves the way for ultrahigh-precision spectroscopy and conversion of the highly precise optical frequency to radio frequencies in a simpler setup.