Atomic-device perturbation compensator and associated method

Abstract

A perturbation-compensation method includes splitting an optical beam from a laser into a first input beam and a second input beam. The first input beam propagates along a first beam-path that traverses a first vapor cell of a first spectroscopic channel and terminates at a first photodetector of the first spectroscopic channel. The second input beam that propagates along a second beam-path that terminates at a second photodetector of a second spectroscopic channel. The method also includes generating a synthetic error signal (SES) by applying a filter to a weighted sum of a first error signal and a second error signal obtained from the first photodetector and the second photodetector, respectively. The method also includes, adjusting, based on the SES, at least one of (i) a control signal of the laser and (ii) an operating parameter of the first spectroscopic channel.

Description

PATENT Attorney Docket No. VAPC.P2001WO / 00631646 ATOMIC-DEVICE PERTURBATION COMPENSATOR AND ASSOCIATED METHOD CROSS-REFERENCE TO RELATED APPLICATION

[0001] This application claims the benefit of U.S. Provisional Application No. 63 / 553,584 filed on 14 February 2024 the disclosure of which is incorporated herein by reference in its entirety. BACKGROUND

[0002] Atomic devices (such as atomic clocks and atomic sensors) can provide high-accuracy, high-precision, and high-sensitivity measurements of electromagnetic fields, radiation, and inertial forces, among other physical quantities. However, the performance of atomic devices is frequently limited by perturbations that depend on the parameters of the measurement configuration. One such example is light shifts: the intensity of light used to spectroscopically interrogate the atomic device shifts the center absorption frequency for atomic spectra and broadens the linewidth. Fluctuations in the intensity produce fluctuations in the measured center frequency, which degrades the stability and accuracy of atomic clocks, optical frequency standards, and optical frequency references. Compensating for these perturbations can greatly increase the performance of the devices.

[0003] To perform compensation, atomic technologies typically use active optoelectronics to periodically apply known perturbations and perform a fit function on that sequence of measurements to extract a zero-perturbation-estimate. This increases system complexity and cost as active optoelectronic devices are often expensive, bulky, and delicate. Further, the interrogation sequence in this modality is elongated and often reduces the device bandwidth. Examples of such optoelectronic devices include acousto-optic modulators (AOMs), electro-optic modulators (EOMs), liquid crystal modulators, and mechanical or micro-electromechanical systems (MEMS) shutters. These components are often sensitive to temperature, vibration, electronics noise, electronics drift, wear and tear, and aging. 1 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 SUMMARY OF THE EMBODIMENTS

[0004] As an alternative to these active optoelectronic devices, the present embodiments use an ensemble of spectrometers with different configurations (e.g., electronic, positional, or otherwise) to extract a perturbation estimate (FIG.5, FIG.4) periodically and / or in real time. This ensemble of spectrometers creates an ensemble error signal that is synthesized to create a robust, precise, accurate, and stable synthetic error signal by combining high signal-to-noise ratio (SNR) measurements with high- stability measurements and high-accuracy measurements in a way that preserves all desired features.

[0005] These features may include at least one of: long term stability, immunity to vibration, immunity to injected power instabilities via fiber optics, short term stability, laser frequency lock bandwidth, average and instantaneous power consumption, alkali vapor contaminants (such as contaminant gasses at low partial pressures), long-term positional and dimensional drift in mechanical components (e.g. angular alignment), accuracy, magnetic field fluctuations, temperature gradients, temperature fluctuations, alkali condensation, and multi-photon detection stability. The synthetic error signal is able to reject disturbances, utilize high SNR signals for short term stabilization and low-SNR, high-accuracy signals for long-term stabilization.

[0006] In a first aspect, a perturbation compensator is disclosed. The perturbation compensator includes a first spectroscopic channel, a second spectroscopic channel, a beam-array generator, and processing circuitry. The first spectroscopic channel includes a first photodetector and a first vapor cell. The second spectroscopic channel includes a second photodetector. The beam-array generator splits an optical beam from a laser into (a) a first input beam that propagates along a first beam-path that traverses the first vapor cell and terminates at the first photodetector and (b) a second input beam that propagates along a second beam-path that terminates at the second photodetector. The processing circuitry is electrically coupled to the first spectroscopic channel, the second spectroscopic channel, and control electronics of the laser. The processing circuitry operable to generate a synthetic error signal (SES) by applying a filter to a weighted sum of a first error signal and a second error signal obtained from the first photodetector and the second photodetector, respectively. In response to receiving the SES, at least one or more of (i) 2 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 the control electronics adjusts a control signal of the laser and (ii) hardware of the first spectroscopic channel adjusts an operating parameter of the first spectroscopic channel.

[0007] In embodiments, a perturbation-compensation method includes splitting an optical beam from a laser into a first input beam and a second input beam. The first input beam propagates along a first beam-path that traverses a first vapor cell of a first spectroscopic channel and terminates at a first photodetector of the first spectroscopic channel. The second input beam that propagates along a second beam-path that terminates at a second photodetector of a second spectroscopic channel. The method also includes generating a synthetic error signal (SES) by applying a filter to a weighted sum of a first error signal and a second error signal obtained from the first photodetector and the second photodetector, respectively. The method also includes, adjusting, based on the SES, at least one of (i) a control signal of the laser and (ii) an operating parameter of the first spectroscopic channel. BRIEF DESCRIPTION OF THE FIGURES

[0008] FIG.1 is a functional block diagram of a perturbation compensator, in an embodiment.

[0009] FIG.2 is a plot showing spectra, error signals, and an SES, which are examples of output and error signals, and synthetic error signal of FIG.1, respectively.

[0010] FIG.3 is a flowchart illustrating an embodiment of perturbation- compensation method that may be executed by the perturbation compensator of FIG.1.

[0011] FIGs.4–6 are schematics of respective perturbation compensators, each of which is an example of the perturbation compensator of FIG.1.

[0012] FIG.7 is a schematic of a pre-processor, which may be included in embodiments of compensator of FIG.1 as part of its processing circuitry. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0013] In embodiments, an ensemble of spectrometers includes passive components, active components, or a combination thereof. One or more optical beams from one or more interrogation light sources, such as a laser, may be split into an array of beams with different or identical parameters (e.g., amplitude, polarization, Poynting vector, spatial intensity distribution, spectral bandwidth, optical absorption / transmission tailoring e.g., dichroics, etc.). This array of beams may then be 3 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 directed through (or onto) the ensemble of spectrometers. Each spectrometer may be designed to generate an error signal optimized for the measurement of spectra (atomic and / or molecular) under different perturbations and / or optimized for a combination of stability, accuracy, precision, redundancy, robustness to physical changes, size, weight, and power (SWaP), cost, algorithmic complexity, systematic limitations, and software control. The physical changes may range from shocks to drifts, and may be one or more of chemical, mechanical, thermal, and electrical. Examples of systematic limitations include mitigating the effect of transient vapor condensation in optical paths, gain- bandwidth limits in photoreceivers, and optical collection efficiency.

[0014] The ensemble of beams additionally gives redundancy in light transmission through a volume in the case when a region of the transmission is blocked by a droplet or crystal of material deposited from the vapor phase, which would otherwise block a beam. Hence, multiple beams decrease the probability that light transmission will be blocked by deposits.

[0015] The ensemble of spectrometers generates a set of error signals (the number of error signals may be greater than the number of spectrometers e.g., measuring a single error signal using different filters and / or demodulations). The set of error signals may be combined with an algorithm that optimizes adaptively to compensate for all measured and estimated perturbations. The set of error signals is used periodically to measure the sensitivity of the device to perturbations by applying or witnessing perturbations in time; the extraction of the coefficients of disturbance is critical to applying stable, accurate, and precise corrections. The computed estimate of the perturbations is then corrected by delivering an error signal from a feedback mechanism to the laser, to the spectrometers, or to both.

[0016] Common feedback mechanisms include modifying the injected current into a laser, modifying the operating temperature setpoint(s) of a laser, modifying an offset frequency or voltage that controls the frequency of an optical beam (e.g., AOMs, voltage-controlled oscillators (VCOs) for phase locked lasers, piezoelectric devices in an optical cavity, single sideband generation in EOMs). A correction signal may be directly applied to some other measurement system such as an interferometer, where the calculated displacement from the interferometric measurement may be digitally corrected in software. 4 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646

[0017] In some embodiments, a method of correction applies offsets to output parameters (e.g., adjusting the laser wavelength (frequency) by tuning the electrical current). Another method of correction is by outputting a computed value (e.g., outputting a magnetic-field estimate digitally). Another method of correction includes applying a compensation. Examples of a compensation include a magnetic field, a thermal gradient (e.g., to remove condensation in the optical path), reactivating a getter of a vapor cell, and a chemical reaction to a vapor cell of a spectrometer channel. The result of this effort is a method of making an atomic device less sensitive to fundamental, systematic, and environmental perturbations by actively and continuously correcting for them via measurement and estimation in a low-cost, reconfigurable, software-updatable, mass-manufacturable, low-SWaP package.

[0018] In some embodiments, the atomic device provides a state estimate of the atomic payload (e.g., the central absorption frequency for an optical signal or the magnetic field experienced by atoms or molecules). The state estimate is effectively equivalent to a measurement. The state estimate contains noise based on limited information. The perturbation estimate is an update to the state estimate which enhances the total state estimate synthetically using feedback (digital, analog, computational). This is similar to a Kalman filter, which combines a model (of the atomic physics of an atomic device) with measurements that include the expected measurement and correction measurements that ultimately reduce the uncertainty and improve the accuracy of the devices. In addition, perturbations may be applied to spectrometers or the synthetic error signal is tailored to achieve precision offsets from the un-perturbed atomic state to be estimated using measurements in tandem with models.

[0019] In embodiments, an operation sequence includes: 1. Estimate the sensitivity to perturbations to be corrected. This step may include modulating control parameters and / or using lookup tables or mathematical models for these sensitivity coefficients. The lookup tables may be locally or remotely, e.g., distributed over a network. 2. measure some or all spectrometers; and 3. generate a set of error signals, combine the set of error signals adaptively to optimize for performance (some combination of robustness, sensitivity, precision, accuracy, and stability) under varying perturbations. 5 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646

[0020] The operation sequence may include an adaptive synthesis process, which may use analog or digital logic. One example is when a spectrometer of the spectrometer array fails, in which case hardware that generates the synthetic error signal will respond to this failure using logic rules based on the measurements, and update the weights of a network to remove the contribution of this faulty measurement on the synthetic error signal. The sequence may include repeating the prior step until a self-check sequence is called. Self-check sequences may be called periodically or when a perturbation or state estimate raises a flag. Flags may be raised as a Boolean (True / False) or analog (continuum) value and may be connected to monitor outputs. 1. Individual Error Signals

[0021] This section describes processing of individual error signals (^^^^^^^) to yield Corrected Individual Error Signals ^^^^^^,^^^^^^^^^^^^^^.

[0022] Using ^ distinct spectrometers (indexed by ^ =1, 2,…,^) with different optical intensities ^^, polarization states ^, beam diameters ^^, beam angles ^^, and demodulation signals ^^, we obtain ^ multiple measurements of the spectrum.(M≥N; M>N means at least one signal from a spectrometer is analyzed at least two different ways creating two distinct outputs). Each spectrometer yields a measurement that may be modeled as the true atomic response plus a spectrometer-specific shift. For example, if ^^is the true atomic transition frequency absent any perturbations, the apparent line center ^^^,^^^^^^^^ observed by spectrometer ^ may be written as: ^^^,^^^^^^^^ ^ ^^ ^ Δ^^^^^, ^^, … ^ ^ "^^^^ ^ #^^$"%^&'"($"^, ^^ . (1)Herein,stated otherwise.

[0023] In equation (1), Δ^^^^^ represents the systematic frequency shift for spectrometer ^ (due to its intensity, polarization, etc.), "^^^^ is measurement noise, and #^^^^ are environmental perturbations. Embodiments disclosed herein determine the unperturbed atomic transition (e.g., ^^) by combining these readings in a way that cancels out the measurement, systematic, and environmental perturbations). Each spectrometer has a specific sensitivity (and insensitivity) to various perturbations and an ensemble of spectrometers with different sensitivities to perturbations may be used to subtract the correlated estimates of the total perturbations. In general, there is no 6 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 measurement that measures ^^directly, every measurement always produces an offset value to this unperturbed value. 2. Synthetic Error Signal (SES) Construction from Error Components

[0024] This section describes a synthetic error signal (SES) is constructed as a weighted, filtered sum of ^^^^^,^^^^^^^^^}^^^. An SES functions as a single feedback signal that isolates the real frequency error and provides a zero crossing when the laser is on- resonance with the inferred atomic resonance frequency.

[0025] Each ^^+spectrometer produces an individual error signal ^^^^which typically measures the difference between the laser frequency ^,^^^^^^^ and that local spectrometer’s best estimate ^^^,^^^^}^^^ of the atomic line center. However, ^^^^is typically contaminated by systematic offsets which, in embodiments, is estimated and compensated to reduce the dependence on a fluctuating parameter. Examples of fluctuating parameters include intensity fluctuations, magnetic field fluctuations, and temperature fluctuations.

[0026] To remove measurement, systematic, and environmental offsets (collectively known as ‘offsets’), the ^^^^may be processed by estimating the contribution to perturbations and subtracting that value. For example, residual amplitude modulation (RAM) on the laser at a given demodulation frequency induces a finite offset when demodulating the signal which may be corrected.

[0027] Equation (2) is an example mathematical expression for ^^^^. Near the measured line center ^^^,^^^^}^^^, the absolute value the individual error signal is anincreasing functiondifference (^,^^^^^^^ − ^^^,^^^^}^^^^, such that the errorsignal may be minimized by changing the laser frequency, which in turn minimizes the frequency difference. 3(2)

[0028] sensitivity of the ^^+spectrometer to a given offset factor, Δ2^^^ are the various offset factors, and "^^^^ are noise terms. The summation is from 8 = 1 to 9, where 9 is the number of measured offset factors. 7 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646

[0029] Herein, ^,^^^^^^^ is the frequency of the laser at time ^, and also depends on operating parameters such as injection current(s), temperature(s), and piezo voltage(s). Because changing these operating parameters changes the emitted frequency, the error signal may be used to steer the frequency of the laser to an atomic transition. When the laser frequency is below the target frequency, the slope of the error signal gives direction to the steering to increase the frequency (and vice versa).

[0030] Function .^may be related to the combined specific line-shape function and spectroscopy transduction realized in the spectrometer producing units of [V] and / or [V] / [Hz] and crossing zero at ^^^,^^^^}^^^. In embodiments, function .^is a transduction function that converts the raw spectroscopy curve into a voltage that has a slope that crosses zero at ^^^,^^^^}. Function .^may be a proportional function, that is, one that multiplies the by a scalar. The scalar may equal one, in which casefunction .^ is an identity .^^:^ = :.

[0031] Equation (2) is applicable when ^,^^^^^^^ − ^^ + ;∑2 ^^2^^^Δ2^^^ +"^^^^= ~ Γ where Γ is the approximate linewidth of an atomic transition; away from thisthe ^^^^is not necessarily well-behaved. We can estimate the 9 offsets from the ensemble of ^^^^because each ^^^^responds to Δ2^^^ differently.

[0032] In embodiments, sensitivity-coefficient matrix ^^2^^^ is initialized based on the design of the individual spectrometers. It may be estimated by monitoring changes in the field and estimating the sensitivity of ^^^^^^^ passively. The values can also be calibrated periodically. The values can also be actively measured by applying offset factors (e.g., applying a change in intensity). Any combinations of these may be performed periodically.

[0033] The total offset may be estimated according to eqn. (3). K3JIn Thesquares fitting which may be solved numerically or using invertible matrices. There are9 elements to the vector of estimates of the perturbing fields and there may be at least9 + 1 linearly independent (i.e., not redundant) individual error functions ^^^^^^^.8 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 ^^^^^^^ may be a measured error signal, e.g., obtained by applying a peak-finding algorithm to spectroscopic signal that includes a resonance.

[0034] A single spectrometer channel can provide multiple ^^^^^^^ particularly by sweeping the laser frequency and measuring the total spectrum and extracting parameters of interest (e.g. the atomic transition linewidths and lineshapes, the separation between atomic transitions, and the comparative amplitudes of magnetically sensitive and magnetically insensitive transitions to estimate laser linewidth, angular misalignment, buffer gas broadening, and magnetic field, etc.) with a single spectrometer but processed multiple different ways generating a sequence of ^^^^^^^.

[0035] We can now correct each ^^^^^^^. Eqn. (4) is an example expression for a corrected IES. In eqn. (4), ΔL2^^^ is obtained from the right side of eqn. (3). K= ^^^^^^^ − 1 ^^2^^^ΔL2^^^ (4)A synthetican eqn. .K^^^^^^ = 1 N^O^ (5)In eqn. (5), O^[… ]pass,normalization, convolutional filters, convolutional neural networks, etc.) and N^are weighting coefficients for each ^^^^^,^^^^^^^^^}^^^. For example, if one channel fails (e.g., due to isolated degradation), the weighting coefficient should drive to zero to remove the contribution of the failed spectrometer channel.

[0036] In embodiments, perturbation compensators disclosed herein include a servo loop. This servo loop drives the ^^^^^^ to zero to lock the laser frequency to the best estimate of the unperturbed atomic frequency, intrinsically compensating for a variety of offsets. 3. Optimization of the SES

[0037] A cost function R that quantifies total frequency inaccuracy, with penalty terms for environmental perturbations and different time-scale instabilities, guiding thechoice of SES weights / filters via optimization. The weights, N^, and filter functions,O^[… ] of eqn. (5) may be optimized according to a user cost function R, an example ofwhich is shown in eqn. (6). 9 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 R= S5〈[^,^^^^^^^ − ^ J^] 〉VWXYZ[ + SJ〈[^,^^^^^^^ − ^ J^] 〉V\Y]^ + 1 Λ2〈Δ2^^^J〉 (6)2Forthat S5 ≫ SJ and S5 ≫ Λ2. In other contexts, particularly in heavy industrial machineintegrations, the suppression of EMI is critical over the long-term, such that Λ2 ≫ S5.This cost function enables optimization of the SES to achieve a desiredperformance and suppress offsets by minimizing R. This cost function enables re-optimization when the SES or ^^^^^,^^^^^^^^^}^^^ may indicate instability, drift, inaccuracy, or imprecision over time due to external effects. This may occur near large EMI sources such as motors, spindles, and rotary tools which may be periodic (i.e., may turn on and off).

[0039] Cost function R may be evaluated via frequency-domain metrics as well. For instance, short-term and long-term stability correspond to high-frequency and low- frequency components of the error spectrum. One can equivalently express J as an integral of the error’s power spectral density ^^ab}^c^ weighted by a filter d^c^ that emphasizes frequencies associated withfluctuations, etc. The chosen J formulation ensures that the SI meter realization (via the laser frequency) remains accurate (little bias) and stable (little noise or drift) in both the short and long term. 4. Adaptive Re-Optimization via Cost Function Optimization as a Result of Frequency Spectrum Analysis

[0040] This section describes an adaptive criterion that monitors the SES spectrum and triggers re-optimization of the weights and filters if new errors appear, thereby maintaining optimal performance.

[0041] Embodiments of perturbation compensators described herein continually monitors the performance of the SES and adapts if new noise or shift contributions emerge. The SES’s frequency spectrum ^^efe}^c^ may be analyzed in real-time or in regular intervals to check for anomalies. When the spectral content of the error signal exceeds predefined boundaries (indicating that an unmodeled disturbance or a change in noise profile is degrading the lock) the SES may be automatically re-optimized. In practical terms, if there exists some frequency c for which the error spectrum surpasses 10 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646an allowed threshold (i.e., ^^efe}^c^ > ^^hij}(c)) or if any component of the cost functionJ increases beyond acceptable limits, then an adaptation is triggered.

[0042] In other words, whenever the error spectrum violates its design envelope, the control system finds a new set of SES parameters. The update seeks new weights (and possibly filter bandwidths or other settings) that minimize the cost function J under the current conditions. The adaptation may use gradient descent or other search algorithms to adjust the weights in small increments, or it may use predetermined calibration data to quickly switch to new optimal values when a known type of disturbance is detected.

[0043] The SES is not static, rather, it evolves to remain optimal. For instance, ifincreased EMI at some frequency is detected (perhaps a new electromagnetic noise at100 Hz appears), the system might increase the weight on a filter that specificallysenses and cancels that 100-Hz noise, or adjust the bandwidth of the servo to reject it. If a slow thermal drift starts to bias the lock, the system might put more weight on a temperature compensation signal or integrate the error over a longer period (addressing drift). This closed-loop adaptation ensures that the laser frequency error remains minimized over time, despite changing conditions. 5. Atomic-device perturbation compensator

[0044] FIG.1 is a functional block diagram of a compensator 100, which is an example of an atomic-device perturbation compensator disclosed herein. Compensator 100 includes a beam-array generator 120, processing circuitry 180, and channels 103, which include channels 103(1) and 103(2). Compensator 100 may also include at least one of a laser 101 and control electronics 170 communicatively coupled thereto. Processing circuitry 180 may be electrically coupled to at least one of channels 103 and to control electronics 170. Control electronics 170 may include feedback circuitry 172, such as a PID controller, and may be part of processing circuitry 180.

[0045] Control electronics 170 may output a control signal 178 to laser 101. Control signal 178 may controls one or more of: an injected current to laser 101, an operating temperature of laser 101, a state of a tuning element of laser 101. The tuning element may be one of an acousto-optic modulator, an electrooptic modulator, a voltage-controlled oscillator, a piezoelectric device, and a thermo-optic device. 11 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646

[0046] Control electronics 170 may control the power of optical beam 102 may be modified for stability or to increase or decrease the power. Control electronics 170 may adjust the modulation signal of laser 101 to ‘move’ the demodulation away from interferants (e.g., EMI, power supply pickup). The modulation signal may include sweeps for capturing spectral information like linewidths of spectra for monitoring for contaminant gases. The modulation signal may include sweeps for measuring the magnetic field using an optically pumped magnetometer configuration.

[0047] Control electronics 170 may change or update, via control signal 179, the input temperature setpoint of the vapor cell 140, e.g., if the spectra is noticed to have an unexpected OD (e.g., if the thermistor drifts). Control signal 179 may update an input temperature gradient setpoint e.g., if a photodetector stops receiving light to remove condensed Rb from specified zones.

[0048] The reverse bias voltage on a photodetector 160 may be changed (particularly reduced) by control electronics 170 if the temperature of the photodiode causes large dark currents when reverse biased at normal voltages. The reverse bias voltage may be changed by control electronics 170 to eliminate gain peaking in a transimpedance amp. A bias magnetic field may be updated and applied by control electronics 170 if a large magnetic field is expected or estimated.

[0049] A perturbation to a channel 103 caused by a given parameter (such as intensity) follows an equation that may be Taylor expanded for small values and / or linearized to create a low-order model. Examples of such a perturbation include offset factors Δ2^^^ introduced in eqn. (2). One of these models includes power broadening which, when extracted, gives estimates of the purity and leak rate of a vapor cell 140. In embodiments, the number of measurements provides enough information to unambiguously fit to the perturbation curve. For example, the number of beams (e.g., ^) may be proportional to the number of measurements, which should be greater than or equal to the number of unknowns. Unknowns may be the theoretical parameters associated with perturbation sensitivity curves, a linearized zone within the theoretical sensitivity curves, an empirical sensitivity curve, or a sufficiently estimated sensitivity curve.

[0050] A channel 103 may be a spectroscopic channel that outputs a spectroscopic signal. The number of optical channels equals ^, where ^ ≥ 2. Each optical channel 103 includes at least one of a vapor cell 140 and a photodetector 160. A 12 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 photodetector 160 may be, or include, at least one of a PN photodetector, a PIN photodetector, a thermal detector, a THz radiation detector, a split detector (e.g., a quadrant photodetector), an avalanche photodiode, and a photomultiplier tube.

[0051] Herein, a number in parenthesis following a reference number is an instance of the referent of the reference number. For example, each spectroscopic channel 103(^) is an instance of spectroscopic channel 103, where 8 is a non-negative integer herein. Multiple channels 103 may have a common vapor cell, such that vapor cells 140(^5) and 140(^J) are the same vapor cell, and ^5and ^Jare non-negative integers.

[0052] Any two of vapor cells 140, e.g., vapor cells 140(1) and 140(2), may differ in at least one of the following aspects: temperature, spatial dimensions, mechanical configuration(s), materials, optical coatings thereon, and gas content (atomic or molecular species, mass and molar mass ratios, pressures). A vapor cell 140 maycontain a species that emit light at an unperturbed transition frequency, introduced as^^ before eqn. (1).A vapor cell 140 may be traversed by a single optical beam or by multiple optical beams. 5.1. Spectrometer types and intra-spectrometer differences

[0053] Each channel 103 may include, or be, one of several different types of spectrometers. Each spectrometer type may operate in one of several different modes or configurations, hereinafter intra-spectrometer differences. Channels 103 may be orthogonal (linearly independent) as a result of each channel 103 differing from each other channel 103 in terms of either or both of its spectrometer type and intra- spectrometer differences.

[0054] Example spectrometer types include: multiphotonic, photoacoustic, Raman, optical pumping, velocity selective, time of flight, pump-probe, cavity-enhanced, cavity-ringdown, terahertz, Fourier-transform, ultrafast / direct frequency comb spectroscopy, electron spin resonance, nuclear spin resonance, Zeeman modulation, and Rydberg spectroscopy, sub-Doppler dichroic atomic vapor laser lock spectroscopy (DAVLL), and polarization-enhanced absorption spectroscopy (POLEAS).

[0055] Some intra-spectrometer differences pertain the spectrometer apparatus. These include total dimensions, physical orientation, power consumption, vibration damping, optical coupling into spectrometers (free-space, fiber, photonic circuit), 13 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 distinct local oscillators for modulation / demodulation, and separate data acquisition hardware. Example properties of such hardware include ADC resolution, sample frequency, synchronization, and readout schemes. Examples of physical orientation include (i) mechanical mounting angle, which influences measurements / samples of vector values such as orientation to external fields, and (ii) placement within a system, which influences measurements / samples of gradients / proximity to offset sources like power supplies generating heat / EMI.

[0056] Intra-spectrometer differences may pertain to properties of light and its propagation within spectrometers. These include angular beam variation within spectrometer, laser beam power, wavefront shaping / spatial distribution of the optical intensity, differential pump vs probe power, polarization modification of beams, polarization cleanup of beams, focus / defocus of beams, power sampling of beams, beam wavelengths, different beams can carry different signal encodings (modulations), optical power modulation as functions of time, optical phase modulation as functions of time, sequence of the applications of different beams in time, beam waist, beam focus, beam waist location, and wavelength multiplexing / demultiplexing (e.g. dichroic optics, fiber combiners, etc.).

[0057] Properties of electric fields, magnetic fields, and / or electromagnetic fields within the spectrometer are also examples of intra-spectrometer differences. The field may be applied across vapor cell 140 of channel 103. These fields include time- dependent electric (E), magnetic (B), radio frequency (RF) / microwave (MW) / terahertz (THz) fields, shielding of E, B, RF / MW / THz, and ancillary measurement of E, B, RF / MW / THz fields.

[0058] Two channels 103 of the same spectrometer type may have different vapor cells. Two instances of vapor cell 140 may different in at least one of the following: species of atoms / molecules, additional gases (buffer gases), physical dimensions (internal and external length, width, height), temperature, atomic / molecular source generating materials (such as dispensers), getters for contaminant gases, thin film optical filters (transmissive, reflective, polarization, as functions of wavelength), thin film electrical traces (application of fields, application of heat, measurement of temperature), thin film protective barriers (reduce diffusion, adhesion, and selective precipitation), thermal isolation, vacuum packaging, materials for substrates (ceramic, silicon, glass, metals, polymers), diffractive optics, microfluidic 14 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 wicking, electrical connections (e.g. two vapor cells may be connected to differently), dopants and wall coatings, partial pressures of contained gases, fluorescence emission through thin film filters normal to the input beam / orthogonal to the input beam / at a determined angle or solid angle to the input beam, backscatter readout.

[0059] Signal processing of two channels 103 may differ. Differences in linear signal processing include properties of averaging, filtering (bandpass, band-stop, low- pass, high-pass, combinations, gain, attenuation), demodulation. Differences in nonlinear signal processing may pertain to properties of neural networks, curve fitting and parameter estimation, convolutional filters, noise reduction, Kalman filters, sensor fusion, correlation.

[0060] Two channels 103 may differ in terms of servo control algorithms applied thereto, or parameters of a same type of control algorithm Example algorithms include PID, state space methods, model-predictive controllers. 5.2. Advantage of multiple vapor cells

[0061] The presence of two or more vapor cells 140 in perturbation compensator 100 has additional benefits over the presence a just one vapor cell 140 in compensator 100. One simple benefit is if radiation trapping and light shifts are isolated to one atomic ensemble at least by line of sight. This may be achieved with a micromachined vapor cell with various chambers without line-of-sight access to the neighboring chambers between spectroscopically-probed ensembles. In embodiments, vapor cells 140(1) and 140(2) are made of different compositions of buffer gas, then one of vapor cells 140 is much more sensitive to magnetic fields and serves as an in-situ witness of the neighboring magnetic field (particularly environmental fields). The other cell may be highly pure using nonevaporable getters. The pure cell would be more suitable for optical spectroscopy.

[0062] A combination of cells 140 with different features may be selected to achieve an ensemble set of measurements to produce a synthetic error signal with high robustness to perturbations and achieve a target accuracy, short-term stability, and long-term stability to meet objectives in the resulting device. An optical clock based on the two-photon transition in rubidium, for example, would emphasize extreme reduction in perturbations to achieve long-term and short-term stability and immunity to magnetic fields, intensity fluctuations (especially at the demodulation 15 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 frequency / frequencies), and temperature stability. A different application includes the generation of entangled photons using a frequency reference where the stability needs may be relaxed in favor of compactness.

[0063] As the injected optical intensity may vary over orders of magnitude, the optical path (or other dimensions) of a vapor cell 140 may be made longer to achieve a higher optical depth (OD) for a given temperature. A vapor cell 140 may be made smaller to reduce the power requirement to heat to a given temperature. This has a secondary benefit that the coefficient of perturbation of collisional broadening may be measured using temperature modulation. A vapor cell 140 may be made larger in thermal mass (thicker sidewalls, windows, and / or internal thermal mass) to allow for a longer thermal time constant, which improves the short-term instability.

[0064] In an example mode of operation, laser 101 emits an optical beam 102. Beam-array generator 120 splits optical beam 102 into a ^ input beams 104, denotes as beams 104(1–^). In spectroscopic channel 103(1), input beams 104(1) propagates along a beam-path that traverses vapor cell 140(1) and terminates at photodetector 160. For any spectroscopic channel 103(8), input beam 104(8) propagates along a beam-path that terminates at photodetector 160(8) and may traverse vapor cell 140(8). Each photodetector 160 outputs a respective photodetector output 169.

[0065] Processing circuitry 180 may include one of an application-specific integrated circuit and a field-programmable gate array that implement the functionality of compensator 100 described herein. In embodiments, processing circuitry 180 includes a processor 186 and a memory 181 storing non-transitory machine-readable instructions that, when executed by processor 186, cause the processor to implement the functionality of compensator 100 described herein.

[0066] FIG.1 depicts error signals 190, weights 192, a filter 193, filter parameters 194, and an optimizer 184. Error signals 190 include a synthetic error signal (SES) 199 and a respective individual error signal 191 corresponding to each channel 103. Error signals 190, weights 183, and filter parameters 194 may be stored in memory 181. Processing circuitry 180 includes at least one of a coefficient calculator 110 and an optimizer 184, each of which may be stored in memory 181 as machine- readable instructions. Optimizer 184 may implement a cost function 185. Each error signal 191 is an example of ^^^^^^^ of eqns. (2), (3), and (4). 16 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646

[0067] In embodiments, coefficient calculator 110 computes sensitivity coefficients 187 from at least one of system channel parameters 105 of channels 103 and photodetector outputs 169. Memory 181 may store sensitivity coefficients 187 and may store coefficient calculator 110 as machine-readable instructions. Optimizer 184 may employ one or more sensitivity coefficients 187 when minimizing cost function 185. Parameters 105 are examples of offset factor Δ2^^^ introduced in eqn. (2).Sensitivity coefficients 187 are examples of elements of sensitivity-coefficient matrix^^2^^^ introduced in eqn. (2). Control electronics 170 may be communicatively coupledto processing circuitry 180, e.g., to receive sensitivity coefficients 187 for use in adjusting channels 103. Control electronics 170 may adjust parameters of a channel 103 via a control signal 179.

[0068] Examples of system channel parameters 105 include laser injection current, temperatures of different hardware components or as measured at a specific location, laser control points (applied voltages or currents for diode current injection), DPSS optical pumping, temperature, thermo-optic / electro-optic tuning of photonically integrated resonators / lasers, piezo tuning of cavities, power modulation, frequency modulation), and vapor cell ensemble control points (applied magnetic fields / orientations / frequencies, vapor cell temperature and temperature gradient).

[0069] Sensitivity coefficients 187 may include a coefficient for one or more of the following physical parameters: light intensity, vapor cell integrity / leak rate / hermeticity (especially to helium), vapor cell aging (e.g. consumption of alkali metal by surfaces or bulk, especially glass), vapor cell temperature, vapor cell temperature gradient, vapor cell time constant, vapor cell impurity (gas contaminant), laser modulation frequency, laser pointing stability, polarization fluctuations, speckle stability, magnetic field amplitude / orientation / frequency spectrum, and photodiode temperature sensitivity (dark current, responsivity, capacitance, bias stability).

[0070] The individual error signal of eqn. (2) and the corrected error signal of eqn. (4) are examples of error signals 191. An error signal 191(8) may be determined from multiple measurements obtained by channel 190(8), where each of the multiple measurements may be a spectroscopic measurement. The SES of eqn. (5) is an exampleof SES 199, where weights Nl and parameters of filter O^[… ] are examples of weights183 and filter parameters 194, respectively. Each weight 183 may be a weight 17 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 coefficient, as in eqn. (5). The cost function of eqn. (6) is an example of cost function 185.

[0071] Examples of filter parameters 194 include a time constant of the fractional frequency instability of an error signal 191, a low-pass cutoff frequency, a high-pass cutoff frequency, a filter bandwidth, a frequency offset, and a scaling factor. Changing filter parameter 194 may result in modifying filter 193 by applying an operator thereto, where examples of the operator include a discrete derivative, a thresholding operator, and a normalization operator

[0072] In embodiments, processing circuitry 180 generates SES 199 from error signals 191 in part based on weights 192, which may be adjusted over time or as physical parameters change out of set bounds. Weights 192 may therefore be dynamic.

[0073] Weights 192 may include parameters associated with individual error signals 191 when combined into SES 199. In some embodiments, weights 192 include the relative contribution of different signals (i.e., 10% of one signal and 90% of another). In other embodiments, additional parameters are included such as DC offsets and frequency-dependent filters to be associated with each individual error signal 191. Some individual error signals 191 may be low-amplitude and need amplification at low- frequencies with high-frequency noise removed. Other error signals 191 may need DC offsets such that a zero crossing is adjusted to a more accurate value (e.g., offsets associated with residual amplitude modulation demodulated into a DC offset).

[0074] In an example mode of operation, when the injected optical power is suddenly decreased, e.g., by 90%, optimizer 184 decreases weight 192 of the lowest intensity signal (because it will have a consequently lower SNR) and increase weight 192 of channel 103 with the highest SNR without amplifier saturation. The specific choice of the set of weights 192 depends on the cost function to be optimized and the nominally-uncontrolled inputs which can vary in time. Cost function 185 may also vary in time to go into high-accuracy-and-slow mode or low-accuracy-and-high- agility / bandwidth mode, for example. In embodiments, weights 192 are dynamic by virtue of this time dependence and adaptivity to the specifics of the parameters.

[0075] In embodiments, processing circuitry 180 compares coefficients of disturbance to live fluctuations of the perturbative values which are measured in real time by perturbation compensator 100. Using one or more of a principal component analysis (PCA), maximum likelihood estimation (ML), and a rank-ordered heuristic, the 18 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 selection of the weights of the combined error signal may be used to minimize a cost function. In embodiments, cost function 185 at least partially minimizes the instability which would favor minimizing the coefficients of the highest-fluctuating parameters.

[0076] Cost function 185 may penalize inaccuracy-driven perturbations more than total SNR (related to the short term instability). This may look to favor terms that might modify the SNR but which do not comparably reduce the accuracy. An example is the ‘residual doppler shift’ which reduces the linewidth (and therefore the Q factor and SNR) but barely modifies the line center (the broadening is symmetric with no shift when the pump and probe have a nonzero intersection angle).

[0077] Error signals 190 may be combined with a synthesis algorithm that optimizes adaptively to compensate for all measured and estimated perturbations to yield SES 199. In embodiments, the synthesis algorithm depends on cost function 185, which changes for different applications. Some combination of weights, e.g., weights 192, in the cost function may reward a combination of one or more of the following: SNR, bandwidth, signal recovery, accuracy, robustness, dynamic input range, input wavelength, etc. For space applications, the sensitivity to shocks and thermal cycling (inside / outside line of sight of the sun) is relevant alongside He impermeability (when they flush the space station) and the long-term instability may be rewarded to enhance the averaging power of atomic sensors like atom interferometers.

[0078] For industrial applications, the sensitivity to vibrations, potential humidity, and harsh temperatures is critical and the short-term stability and accuracy are far less important than robustness. For handheld measurement instruments, low size, weight, and power (SWaP) are useful which may emphasize low-temperature, low- bandwidth, and compact designs. In particular, the modulation and demodulation frequencies of the individual error signals may be adaptively chosen to avoid frequencies associated with periodic or high-amplitude noise.

[0079] Control electronics 170 may include a perturbator 175, which applies respective perturbations to one or more vapor cells 140. The perturbations may include at least one of a radio-frequency field, a magnetic field, and temperature. Perturbator 175 may include one or more of a microheater, an RF transmitter (e.g., an on-chip RF transmitter module), and a radiating and / or static field generator. Field generators include permanent magnets, inductive coils / loops that may be energized by an applied current, microstrips, radio antenna, coplanar waveguides, and horn antenna. In 19 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 particular, electromagnetic coils may be used to compensate for applied fields. Also, some embodiments will use magnetic shielding to minimize the effect of a perturbing magnetic field. For certain types of channels 103, a magnetic field may be applied to shift spectroscopic features for controlled-offset frequency references (e.g., laser cooling). The spectroscopy type for such channels 103 may include sub-Doppler dichroic atomic vapor laser lock spectroscopy (DAVLL) and / or polarization-enhanced absorption spectroscopy (POLEAS). Such channels may be configured to exhibit electromagnetically induced transparency and / or coherent population trapping, Also, an applied magnetic field may modulate the atomic response such that an error signal may be recovered (e.g., Zeeman locking).

[0080] In embodiments, channels 103 include one or more tailored spectrometers providing a set of measurements optimized to deduce the set of perturbations. The set of perturbations are estimated. The set of perturbations may be used as inputs to a cost function (e.g., cost function 185) that penalizes some combination of robustness, stability, precision, and accuracy at different timescales. In embodiments, optimization to minimize this cost function produces a set of weights and parameters (e.g., weights 192 and parameters 194) that combines the set of measurements into a synthetic error signal 199.

[0081] In embodiments, processing circuitry 180 is communicatively coupled to at least one of laser 101 and perturbator 175. Laser 101 may receive SES 199, the function of which may include locking laser 101 to an atomic spectral line and maintain said locking in the presence of perturbations.

[0082] One example of such a perturbation is magnetic noise from an AC source, such as an electrical outlet or a switching mode power supply. A component of the synthetic error signal may be delivered to perturbator 175 to apply a live correction. Another example is changing the temperature of the vapor cell based on the error signal to increase or decrease SNR. When the signal is weak, the cell temperature may be decreased to avoid the exponential absorption at the minimum of an absorption resonance which distorts the hyperfine spectrum. A third example is if one vapor cell is subjected to an AC magnetic field, the spectrum will shift (especially under a polarimeter) at the AC magnetic field frequency. This may be used to create an error signal even when the laser light is unmodulated and is called a Zeeman lock or Zeeman 20 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 spectroscopy. An additional example is when the magnetic field is measured and contained in error signal 199, in which case it may be used to generate a correction field applied to perturbator 175 such that the effective magnetic field seen by the atoms is nulled.

[0083] FIG.2 is a plot 200 showing spectra 269, error signals 291, and an SES 299, which are examples of output 169 and error signals 191, and synthetic error signal 199, respectively. Spectra 269(1–3) correspond to input laser power having low, medium, and high relative power, which results in spectra 269 having relative SNR, quality factor, and frequency shift as denoted in FIG.2. FIG.2 includes an inset that describes relative values of laser power, SNR, quality factor Q, and frequency shift that, in embodiments, are applicable to spectral 269.

[0084] Error signals 291 may be obtained from spectra 269 via a peak-finding method, which includes processing spectra 269 to find peaks and generate zero- crossing at those peaks. A single spectroscopic signal may be processed multiple times / ways to generate multiple zero-crossing error signals (e.g., first derivative and third derivative). Error signals 291(1,2,3a) may represent respective first-harmonic demodulation of spectra 269(1,2,3), and are similar to a first-derivative of the origin spectra. Error signal 291(3b) may represent a third-harmonic demodulation of spectrum 269(3), which resembles third derivative, which is insensitive to linear slopes. SES 299 crosses zero at estimated unperturbed atomic transition, and relative to error signals 291, has a high effective SNR, tight frequency discrimination, and a wide capture range. 6. Perturbation compensation method

[0085] FIG.3 is a flowchart illustrating a perturbation-compensation method 300. In embodiments, method 300 is implemented within one or more aspects of compensator 100. One or more steps of method 300 may be implemented by processing circuitry 180, e.g., by processor 186 executing non-transitory computer-readable instructions of software stored in memory 181. Method 300 includes at least one of steps 310, 320, and 330. Step 330 may include at least one of steps 333, 334, 335, 336, and 337.

[0086] Method 300 may include one or more of steps 333–337 as part of step 330 and / or independently of step 330. For example, embodiments of method 300 that 21 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 do not include step 330 may include at least one of steps 333–337. Also, method 300 may include step 330 and hence include at least one of steps 333–337 as part of step 330, while also including one or more of steps 333–337 that are not part of step 330.

[0087] The following description of method 300 includes parenthetical numbers following terms recited by the method. The parenthetical number indicates that the element associated with the number in parenthesis is an example of the term. For example, the description of step 310 below recites “an optical beam (102),” which means that optical beam 102 introduced in FIG.1 is an example of the optical beam introduced in step 310.

[0088] Step 310 includes splitting an optical beam (101) from a laser (100) into (a) a first input beam (104(1)) that propagates along a first beam-path that traverses a first vapor cell (140(1)) of a first spectroscopic channel (103(1)) and terminates at a first photodetector (160(1)) of the first spectroscopic channel and (b) a second input beam (104(2)) that propagates along a second beam-path that terminates at a second photodetector (160(2)) of a second spectroscopic channel (103(2)).

[0089] Step 320 includes generating a synthetic error signal (SES, 199) by applying a filter (193) to a weighted sum of a first error signal (191(1)) and a second error signal (191(2)) obtained from the first photodetector and the second photodetector, respectively.

[0090] Step 330 includes stabilizing the laser with the SES (199). Control electronics 170 may execute at least part of step 330. Step 333 includes adjusting, based on the SES, at least one of a control signal (179) of the laser and an operating parameter of the first spectroscopic channel and / or the second spectroscopic channel. Control electronics 170 may execute step 333.

[0091] In embodiments, the first vapor cell contains atoms that emit light at an unperturbed transition frequency, e.g., ^^. In such embodiments, method 300 may include at least one of steps 334, 335, 336, and 337, each of which may be executed when the SES has an amplitude that exceeds a threshold value.

[0092] Step 334 includes determining a cost-function value of cost function (185) that is an increasing function (i) a deviation of the laser frequency from the unperturbed transition frequency and / or (ii) one or more offset factors, e.g., Δ2^^^ introduced in eqn. (2). Step 335 includes adjusting at least one of weight coefficients (192) of the weighted sum and parameters (194) of the filter based on the cost-function value. Step 22 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 336 includes regenerating the SES by applying the adjusted filter to a weighted sum of either (i) the first error signal and the second error signal or (ii) a subsequent first error signal and subsequent second error signal. Step 337 includes repeating step 333 with the regenerated SES replacing the SES. 7. Embodiments of perturbation compensators

[0093] FIG.4 is a schematic of a perturbation compensator 400. Perturbation compensator 400 includes beam-array generator 120, channels 403, and processing circuitry 180. Perturbation compensator 400 may also include laser 101. Beam-array generator 120 that generates ^ optical beams 404(1–^), each of which propagate through a respective channel 403(1–^). Channels 403 and optical beams 404 are respective examples of channels 103 and input beams 104 of FIG.1.

[0094] Channel count ^ denotes the number of channels of perturbation compensator 400. Perturbation compensator 400 also includes ^ pre-cell optical elements 430(1–^), m vapor cells 440(n = 1, 2, …, m), ^ post-cell optical elements 450(1–^), and ^ photodetectors 460(1–^). Each vapor cell 440 is an example of vapor cell 140.

[0095] Both cell count m and channel count ^ are positive integers. Cell count m may be less than channel count ^, for example, when multiple optical beams 404 traverse the same vapor cell 440. Herein, ^ and n are integers that denote a channel index and a vapor-cell index, respectively.

[0096] Pre-cell optical elements 430 may function as a magnifying system that can modify the beam waist, modify the beam focus, rotate the polarization state, etc. Perturbation compensator 400 may include optical elements that result in an optical beam 404(^) traversing a vapor cell 440 twice, after retroreflection for example, such that the same optical element functions as both pre-cell optical element 430(^) and post-cell optical element 450(^). Channel index 8 is an integer between one and ^, inclusive. In such embodiments, this optical element may recapture optical beam 404(^) and focuses it onto photodetector 160(^).

[0097] An optical beam 404(^) may be “unconditioned,” that is, it propagates from beam-array generator 420 through a vapor cell 440 and to a photodetector 160 without traversing one or both of a pre-cell optical element 430 and a post-cell optical element 450. That is, one or more values of channel index ^, perturbation compensator 23 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 400 may lack one or both of pre-cell optical element 430(^) and post-cell optical element 450(^). Such unconditioned optical beams 404 have spectroscopic applications e.g., for identifying the unmodified optical state using spectroscopy to identify high- variance frequencies of interest and / or sources of large-amplitude, short-term, and / or long-term perturbations. In such embodiments, perturbation compensator 400 may include fewer than ^ pre-cell optical elements 430 and / or fewer than ^ post-cell optical elements 450.

[0098] An error signal 191 generated by an unconditioned optical beam 404 injected into perturbation compensator 400 compared to a synthetic error signal will have different frequency spectra, which may be measured to adaptively modify the bandwidth and weights of the synthetic algorithm to reject disturbances actively. For example, the unconditioned optical beam 404 may see significant noise at 10 kHz so a digital band-stop filter may be applied to the synthetic error signal 199 to minimize this noise source. Alternatively the loop gain may be equalized or tailored to combat high- amplitude sources of perturbation. Also, the residual amplitude modulation may be measured directly with no vapor cell in the optical path to create a feedback signal to offsets in FM spectroscopy due to intrinsic amplitude and phase noise of the laser.

[0099] Functions of post-cell optical elements 450 include polarization state measurement (i.e., polarizers / polarizing beam splitters) and focusing an optical beam 404 to a photodetector 160 to enhance the bandwidth by achieving a smaller shunt capacitance in a small-area photodiode. An optical beam 404 may also be expanded to decrease the intensity when using a high-intensity light source. An optical beam 404 may be analyzed for Poynting vector angle using a device such as a pinhole or other spatial mode filter to monitor beam pointing stability.

[0100] Post-cell optical elements 450 may also include a diffraction grating used to send various intensities to different photodetectors 460 that have different gains. This becomes most relevant when the input optical power needs to have a high dynamic range / rejection to high intensities / sensitivity to low intensities. Post-cell optical elements 450 may also include a quarter wave plate followed by a partial reflector to achieve saturated absorption spectroscopy using a polarizing beam splitter dumping to a photodiode. Nonpolarizing elements may be used to reject polarization-fluctuation- related perturbations. Post-cell optical elements 450 may also be a kinematic element that allows for retroreflection or angled reflection tuning. 24 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646

[0101] Table 1 describes several channels that may be included in embodiments of perturbation compensator 400. Each channel is numbered by channel index 8 in the first column of Table 1. A channel includes a vapor cell 440 and may also include at least one of a pre-cell optical element 430, a post-cell optical element 450, and a photodetector 160. The output of each photodetector may be pre-processed by circuitry of processing circuitry 180, for example, as described in the right-most column of Table 1. Embodiments of perturbation compensator 400 may include, in any combination, one or more channels described in Table 1. Each vapor cell 440 is indexed by a vapor-cell index n. Candidate channels of perturbation compensator 400 include any combination of a pre-cell optical element 430(^5), a vapor cell 440^n^, a post-cell optical element 450(^J), and a photodetector 160(^o), where each of channel indices ^5, ^J, and ^ois a positive integer less than or equal to ten, per the range of ^ in Table 1. Table 1: Example channels of a perturbation compensator Channel pre-cell optical vaporpost-cell optical element 450(^) Photodetectorpost-^element 430(^) cell 160(^) processing440(n) operation applied to electrical signal by circuitry 710(^) 403(1) ND filter 440(1) neutral density reflective filter or 460(1) partially-transmissive mirror 403(2) polarizer 440(1) neutral density reflective filter or 460(2) partially-transmissive mirror 403(3) focusing lens 440(1) neutral density (or selectively 460(3), e.g., a Slow sweep at and / or reflective bandpass filter) reflective PMT or APD ~10–600 kHz magnification filter, may be in vapor cell 440(1). for blue light depending on system detection laser frequency noise for intermodulation noise contributions. 403(4) Diffractive optical 440(1) Diffractive optical element and / or a 460(4) element and / or a metalens. metalens 403(5) quarter 440(2) Polarizer, e.g., a polarizing beam 460(5), e.g., Balanced pair waveplate splitter cube Large-area of photodiode pair of followed by a detectors (low subtractor with bandwidth; bipolar swings one for each polarization channel) 403(6) dual-beam 440(2) Pair of polarization state measuring 460(6) generator optics (may measure the difference followed by two in atomic spectra due to a magnetic quarter splitting which may be induced with 25 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 Channel pre-cell optical vaporpost-cell optical element 450(^) Photodetectorpost-^element 430(^) cell 160(^) processing440(n) operation applied to electrical signal by circuitry 710(^) waveplates a Helmholtz generating coil / electromagnet / permanent opposing helicity magnet 403(7) 440(3) 460(7) 403(8) 440(4) 460(8) 403(9) 440(5) 460(9) 403(10) 440(6) 460(10)

[0102] Pre-cell optical element 430(4) and / or post-cell optical element 450(4) may be a diffractive optical element and / or a metalens. It may have controllable optical characteristics such as polarization, amplitude, k vector, wavelength selection, beam profile. Post-cell optical element 430(4) may generate additional channels,

[0103] Each of vapor cell 440 (2) and 440(3) may include a buffer gas, the pressure of which may be between 1 and 604 Torr. Vapor cell 440(3) may be an ultra- clean cell with getters, and may have isotopic enrichment which may reduce the SNR in favor of a simplified spectrum. Vapor cell 440(4) may function as a backup cell and include materials that cause selective precipitation. Vapor cell 440(5) may be an all- dielectric cell composed of glass. For example, vapor cell 440(4) may be made of aluminosilicate glass, which has higher helium impermeability.

[0104] Examples of vapor cell 440(6) include, but are not limited to: (1) ceramic thin films such as AlJOoto mitigate He permeability and minimize diffusion of gases through the surface, (2) conductive thin film ceramics that may be transparent to minimize the potential for charging of highly excited species or other ionic content generators (the film thickness should be thin enough that the relaxation time constant of an applied electric field is sufficiently long but short enough to mitigate electric field drifts from ions), (3) conductive thin films for controlling the temperature gradient and applying ohmic, inductive, or radiative heating, (4) optical thin films for anti-reflection, high-reflection, wavelength-selection, and / or diffractive properties, 26 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 (5) witness layers that may change optical behavior (wavelength dependent transmission and / or reflection, electrical conductivity, or colorimetric changes) upon exposure to one or more gases, (6) thin films that may getter excess contaminant gases, (7) thin films that may pro-duce / evolve spectroscopically desirable gases (including, but not limited to, iodine, acetylene, HCN, Rb, Cs, Na, K, H, O2, Sr, Yb, Hg), (8) cell with external thin film coatings that may be used to electrically heat the cell, produce optical coatings, and / or diffusion barrier coatings.

[0105] FIG.5 is a schematic of a perturbation compensator 500 that includes beam-array generator 520, pre-cell optical elements 530, vapor cells 140(1,2), and photodetectors 160(1–4). Pre-cell optical elements 530 include a quarter waveplate 531, a polarizer 532, and ND filter 533, and an ND filter 534.

[0106] Beam-array generator 520 that splits optical beam 102 into optical beams 504 and may include a diffractive optical element 522 and a lens 524. Perturbation compensator 500, beam-array generator 520, pre-cell optical elements 530, and optical beams 504 are respective examples of compensator 100, beam-array generator 120, pre-cell optical elements 430, and input beams 104. A beam 504(^) may be retroreflected within a vapor cell 140 and generate two-photon signals that may be emitted in one or more directions to be captured by a photodetector 160.

[0107] FIG.6 is a schematic of a perturbation compensator 600, which is an example of compensator 100. Perturbation compensator 600 includes laser 101, an optical element 620, a vapor cell 140, photodetectors 660, control electronics 170, and processing circuitry 180. Perturbation compensator 600 may also include at least one of pre-cell optical elements 430 and post-cell optical elements 450.

[0108] Optical element 620 may be or include a planar substrate and may have a thickness that is sufficient to spatially separate optical beams reaching photodetectors 660(1) and 660(2). This thickness may be, for example, between one and six millimeters. To ensure beam separation, the thickness may exceed the product of the incident beam diameter and the sine of angle of incidence of the beam incident on optical element 620. Optical element 620 may create a plurality of parallel beams with different attenuations and other physical properties.

[0109] In embodiments, optical element 620 may include at least one of a diffraction grating, a metalens (and / or additional metamaterial optical elements), and a 27 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 collimating lens to produce sets of optical states that include Poynting vector direction / distribution, polarization rotation, beam geometry modification for transit time broadening and saturation intensity variation, and optical filters for two photon and multiphoton spectroscopy (and / or additional metamaterial optical coatings). Optical element 620 may have one or more of spatially-varying transmittances and spatially-varying reflectances at one or more wavelengths.

[0110] Here, the injected power into the device is monitored on photodetector 660(1) using a partial sampling of the injected beam. Photodetector 660(4) monitors the transmitted ‘pump’ beam at high relative intensity through a vapor cell in a saturated absorption spectroscopy (SAS) configuration. Photodetector 660(5) samples the probe of this first SAS spectroscopy signal, which will be weaker. The optical signal that reaches photodetector 660(5) has traversed vapor cell 140 twice. Photodetector 660(3) monitors a lower-relative-intensity SAS signal. Photodetector 660(2) measures the standard 1-photon absorption spectrum (not pump-probe) for removal of the Doppler-broadened spectrum (colloquially, the Doppler spectrum).

[0111] As the injected power fluctuates, the various spectra are perturbed as functions of their absolute intensity. However, as the various spectrometers (one for each photodetector in this case) have different but proportionally related intensities, the intensity-dependent shift (also known as the ‘light shift’) may be extracted. In typical cases, the light from the laser is delivered via fiber optic; intensity fluctuations from environmental perturbations on the optical fiber are common. If the intensity becomes so weak that photodetector 660(2) and photodetector 660(3) do not generate signals with signal to noise ratios above a threshold value, then the synthetic error signal will depend largely on photodetector 660(4) and photodetector 660(5) and far less so on photodetector 660(2) and photodetector 660(3).

[0112] Conversely, when the injected power is large such that the estimated light shift on photodetector 660(5) and photodetector 660(4) is large or the photodetector signals are saturated (or clipped in some way), the synthetic error signal depends largely on photodetector 660(3) and photodetector 660(2). Photodetector 660(1) may be used to correlate fluctuations in the error signal with intensity fluctuations and extract other types of perturbations (such as transient magnetic fields). The combined information from this set of spectrometers enables a robust, multimodal rejection of varying systematic, environmental, and fundamental perturbation mechanisms. 28 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646

[0113] FIG.7 is a schematic of a pre-processor 700, which may be included in embodiments of compensator 100 as part of processing circuitry 180. Pre-processor 700 includes circuitry 710(1–^), which processes signals output by respective photodetectors 160(1–^) of perturbation compensator 100, e.g., when determining error signals 191. Circuitry 710(1–^) that receives outputs of respective photodetectors 160(1–^) and generates a respective error signal 191from each. Circuitry 710 may include at least one of an amplifier, a digitizer, a switching mixer, a comparator, an amplifier, an RF mixer, and / or an electrical filter. For example, circuitry 710 may include a circuit 712 shown in FIG.7. * * *

[0114] Features described above, as well as those claimed below, may be combined in various ways without departing from the scope hereof. The following enumerated examples illustrate some possible, non-limiting combinations.

[0115] Changes may be made in the above methods and systems without departing from the scope of the present embodiments. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. Herein, and unless otherwise indicated the phrase “in embodiments” is equivalent to the phrase “in certain embodiments,” and does not refer to all embodiments.

[0116] As used in this specification, any appendices thereto, and the appended claims, the singular forms “a,” “an,” and “the” include plural referents unless the content clearly dictates otherwise. It should also be noted that the term “or” is generally employed in its sense including “and / or” unless the content clearly dictates otherwise. Regarding instances of the terms “and / or” and “at least one of,” for example, in the cases of “A and / or B,” “at least one of A and B,” and “at least one of A or B,” such phrasing encompasses the selection of (i) A only, or (ii) B only, or (iii) both A and B. In the cases of “A, B, and / or C,” “at least one of A, B, and C,” and “at least one of A, B, or C,” such phrasing encompasses the selection of (i) A only, or (ii) B only, or (iii) C only, or (iv) A and B only, or (v) A and C only, or (vi) B and C only, or (vii) each of A and B and C. This may be extended for as many items as are listed.

[0117] The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween. 29 LEGAL\75364491\10

Claims

Attorney Docket No. VAPC.P2001WO / 00631646 CLAIMS We claim:

1. A perturbation compensator, comprising: a first spectroscopic channel including a first photodetector and a first vapor cell; a second spectroscopic channel including a second photodetector; a beam-array generator that splits an optical beam from a laser into (a) a first input beam that propagates along a first beam-path that traverses the first vapor cell and terminates at the first photodetector and (b) a second input beam that propagates along a second beam-path that terminates at the second photodetector; and processing circuitry that is electrically coupled to the first spectroscopic channel, the second spectroscopic channel, and control electronics of the laser, and operable to generate a synthetic error signal (SES) by applying a filter to a weighted sum of a first error signal and a second error signal obtained from the first photodetector and the second photodetector, respectively; wherein, in response to receiving the SES, at least one or more of (i) the control electronics adjusts a control signal of the laser and (ii) hardware of the first spectroscopic channel adjusts an operating parameter of the first spectroscopic channel.

2. The perturbation compensator of claim 1, the filter having filter parameters and the weighted sum having weight coefficients, the first vapor cell containing species that emit light at an unperturbed transition frequency; the processing circuitry being further operable to, when SES has an amplitude that exceeds a threshold value: determine a cost-function value of cost function that is an increasing function (i) a deviation of the laser frequency from the unperturbed transition frequency and / or (ii) one or more offset factors; 30 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 adjust at least one of the weight coefficients and the filter parameters based on the cost-function value; regenerate the SES by applying the adjusted filter to a weighted sum to either the first error signal and the second error signal or a subsequent first error signal and subsequent second error signal; wherein, in response to receiving the regenerated SES, at least one or more of (i) the control electronics adjusts a control signal of the laser and (ii) hardware of the first spectroscopic channel adjusts an operating parameter of the first spectroscopic channel and / or the second spectroscopic channel.

3. The perturbation compensator of claim 2, the processing circuitry minimizing the SES when decreasing the value of the SES.

4. The perturbation compensator of claim 2, the adjusted filter parameter including at least one of: a time constant of a fractional frequency instability of the first error signal and / or the second error signal, a low-pass cutoff frequency, a high-pass cutoff frequency, a filter bandwidth, a frequency offset, and a scaling factor.

5. The perturbation compensator of claim 2, the filter parameter including at least one of a discrete derivative, a thresholding operator, and a normalization operator.

6. The perturbation compensator of claim 1, wherein the control signal controls one or more of: an injected current to the laser, an operating temperature of the laser, a state of a tuning element of the laser.

7. The perturbation compensator of claim 6, the tuning element being one of an acousto-optic modulator, an electrooptic modulator, and a voltage-controlled oscillator, and a piezoelectric device.

8. The perturbation compensator of claim 1, the first spectroscopic channel including a first type of spectrometer, the second spectroscopic channel including a second type of spectrometer that differs from the first type of spectrometer. 31 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 9. The perturbation compensator of claim 8, each of the first type and the second type being one of: multiphotonic, photoacoustic, Raman, optical pumping, velocity selective, time of flight, pump-probe, cavity-enhanced, cavity-ringdown, terahertz, Fourier-transform, ultrafast / direct frequency comb spectroscopy, electron spin resonance, nuclear spin resonance, Zeeman modulation, and Rydberg.

10. The perturbation compensator of claim 1, the hardware of the first spectroscopic channel including a perturbator that applies a perturbation to the first vapor cell, wherein the perturbator includes one or more of a microheater, an RF transmitter, an electromagnetic field generator, a static electric field generator, a static magnetic field generator, and a time-varying magnetic field generator.

11. The perturbation compensator of claim 1, the second spectroscopic channel comprising: a second-channel vapor cell, the second beam-path traversing the second-channel vapor cell before terminating at the second photodetector, the second-channel vapor cell being either the first vapor cell or a second vapor cell that differs from the first vapor cell.

12. The perturbation compensator of claim 1, the first spectroscopic channel further comprising: a first pre-cell optical element on the first beam-path between the beam-array generator and the first vapor cell, the first pre-cell optical element including at least one of a neutral density filter, a wavelength-selective filter, a variable reflector, a polarization optic, a refractive optical element, and a diffractive optical element.

13. The perturbation compensator of claim 1, the first spectroscopic channel further comprising: a first post-cell optical element on the first beam-path between the first vapor cell and the first photodetector; and 32 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 the first post-cell optical element including at least one of a neutral density filter, a polarization optic, a refractive optical element, and a diffractive optical element.

14. The perturbation compensator of claim 1, further comprising: a third-channel spectroscopic channel including a third-channel vapor cell and a third photodetector; the beam-array generator further generating (c) a third input beam that propagates along a third beam-path that traverses the third-channel vapor cell and terminates at the third photodetector; the third-channel vapor cell being either the first vapor cell or a third vapor cell that differs from the first vapor cell.

15. The perturbation compensator of claim 1, further comprising the laser and / or the control electronics.

16. The perturbation compensator of claim 1, the processing circuitry being further operable to: generate the first error signal from a first output signal of the first photodetector; and generate the second error signal from a second output signal of the first photodetector.

17. The perturbation compensator of claim 1, the processing circuitry comprising: a processor; and a memory storing non-transitory machine-readable instructions that, when executed by the processor, cause the processor to [input steps of claim 1]. 33 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 18. The perturbation compensator of claim 1, the processing circuitry comprising one of an application-specific integrated circuit and a field-programmable gate array.

19. A perturbation-compensation method, comprising: splitting an optical beam from a laser into (a) a first input beam that propagates along a first beam-path that traverses a first vapor cell of a first spectroscopic channel and terminates at a first photodetector of the first spectroscopic channel and (b) a second input beam that propagates along a second beam- path that terminates at a second photodetector of a second spectroscopic channel; generating a synthetic error signal (SES) by applying a filter to a weighted sum of a first error signal and a second error signal obtained from the first photodetector and the second photodetector, respectively.

20. The method of claim 19, further comprising stabilizing the laser with the SES 21. The method of claim 20, said stabilizing comprising: adjusting, based on the SES, at least one of (i) a control signal of the laser and (ii) an operating parameter of the first spectroscopic channel and / or the second spectroscopic channel.

22. The method of claim 20: the first vapor cell containing atoms that emit light at an unperturbed transition frequency; and further comprising, when SES has an amplitude that exceeds a threshold value, and said stabilizing comprising: determining a cost-function value of cost function that is an increasing function (i) a deviation of the laser frequency from the unperturbed transition frequency and / or (ii) one or more offset factors; adjusting at least one of weight coefficients of the weighted sum and parameters of the filter based on the cost-function value; regenerating the SES by applying the adjusted filter to a weighted sum of either (i) the first error signal and the second error signal or a (ii) subsequent first error signal and subsequent second error signal; and 34 LEGAL\75364491\10Attorney Docket No. VAPC.P2001WO / 00631646 adjusting, based on the regenerated SES, at least one of (i) a control signal of the laser and (ii) adjusting an operating parameter of the first spectroscopic channel and / or the second spectroscopic channel.

23. A perturbation compensator, comprising: a first spectroscopic channel including a first photodetector and a first vapor cell; a second spectroscopic channel including a second photodetector; a beam-array generator that splits an optical beam from a laser into (a) a first input beam that propagates along a first beam-path that traverses the first vapor cell and terminates at the first photodetector and (b) a second input beam that propagates along a second beam-path that terminates at the second photodetector; and processing circuitry that is electrically coupled to the first spectroscopic channel, the second spectroscopic channel, and the laser, and operable to: generate a synthetic error signal (SES) by applying a filter to a weighted sum of a first error signal and a second error signal obtained from the first photodetector and the second photodetector, respectively.

24. The perturbation compensator of claim 23, the circuitry being further operable to stabilize the laser with the SES. 35 LEGAL\75364491\10

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