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Abstract
Ellipse fitting is widely used in the extraction of the differential phase between atom interferometers amid substantial common phase noise. This study meticulously examines the dependency of extraction noise on the differential phase between atom interferometers during ellipse fitting. It reveals that the minimum extraction noise can manifest at distinct differential phases, contingent upon the dominance of different noise types. Moreover, the outcomes are influenced by whether the interferometers undergo simultaneous detection or not. Our theoretical simulations find empirical validation in a compact horizontal atom gravity gradiometer. The adjustment of the differential phase significantly enhances measurement sensitivity, culminating in a differential gravity resolution of 1.6 × 10−10 g @ 4800 s.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The atom interferometer (AI) [1] stands as a novel and robust tool for precision measurements, finding wide applications in fundamental physics research [2–8] and various engineering fields [9–18]. An atom gravity gradiometer (AGG) [19–26], comprised of multiple AIs spaced apart, gauges gravity differences by extracting differential phases among the AIs. Over more than two decades of development, the AGG has proven instrumental in measuring the gravitational constant [5,6] and precisely detecting subterranean targets [27]. Ellipse fitting, a favored technique for extracting differential phases between AIs [28,29], notably suppresses common-mode noises associated with interference phases, including laser source phase noise, vibration noise, and light shift noise, surpassing traditional fringe fitting methods [19,20].
Differential phase noise denotes non-common-mode phase variations, potentially stemming from Raman laser frequency noise or magnetic field disturbances, significantly contributing to measurement noise [22]. Detection noise, typically arising from frequency and intensity fluctuations of the detection laser, presents another significant factor. As detection occurs post-interference, it seems that ellipse fitting cannot effectively suppress detection noise consistently, which becomes a limiting factor for measurement sensitivity in many AGGs [25,26,30–32].
The value of the differential phase may affect the extraction noise. In fringe fitting, a higher signal-to-noise ratio prevails at smaller differential phases [20,33]. However, in ellipse fitting, the differential phase noise is most effectively constrained when the differential phase approximates π/2 [34,35]. This paper delves into the intricate relationship between extraction noise and the differential phase, considering both the differential phase noise and detection noise.
Through theoretical analysis and simulations, we uncover a nuanced link between extraction noise and differential phase. In scenarios where only detection noise is considered, and all AI detections occur simultaneously, the lowest ellipse fitting noise emerges near a differential phase close to zero. Conversely, when differential phase noise dominates or AI detections happen sequentially, the lowest ellipse fitting noise emerges at a differential phase near π/2. Additionally, when differential phase noise and detection noise are of comparable magnitude, and AI detections occur simultaneously, the lowest ellipse fitting noise arises at a differential phase ranging between 0 and π/2. Experimentally, our compact horizontal AGG validates the correlation between ellipse fitting noise and the differential phase, showcasing significant improvement of measurement sensitivity through optimization of the operational differential phases.
2. Theory and simulation
A uniaxial Atom Gravity Gradiometer (AGG) comprises two AIs separated by a distance (L). Cold atom clouds undergo manipulation via a shared sequence of π/2-π-π/2 Raman laser pulses, experiencing splitting, reflection, and recombination. The gravitational acceleration information at their respective positions is encoded into interference phases via a scaling relationship
where Δφ represents the phase shift of the interference signal, keff denotes the effective wave vector of the Raman laser, and T stands for the free evolution time of the atoms between Raman pulses.Under ideal circumstances, the final population signals of the two AIs can be expressed as
Here, Ai and Bi (i = 1, 2) symbolize the amplitude and offset of the two interference signals, while φc and φd reflect the common-mode phase and differential phase, housing information regarding the common-mode and differential components of gravitational accelerations at the two positions. When two sinusoidal signals maintain a fixed phase difference φd, their Lissajous figure assumes an elliptical form. Employing the least squares method for fitting the ellipse equation
and resolving the parameters (a, b, c, d, e, f), facilitates the extraction of the differential phase asBased on the scaling relationship
we can ascertain the measurement of the gravity gradient Γ.For realistic operational conditions, considering various noises present in the signals, Eqs. (2a) and (2b) can be reformulated as
In this context, δφc pertains to the common-mode phase noise, typically induced by vibration noise, common-mode light shift noise, Raman laser source phase noise, and similar factors. Meanwhile, δφd corresponds to differential phase noise, often stemming from Raman laser frequency noise, magnetic field noise, non-common-mode light shift noise, Coriolis noise, and related sources. δAi represents interference amplitude noise, typically arising from Raman laser intensity fluctuations and shifts in the positions of the atom clouds; δQPNi refers to quantum projection noise, constrained by the total number of atoms; δdeti denotes detection noise, usually linked to fluctuations in the detection laser's parameters, resulting in intensity fluctuation of emitted fluorescence from background and interference atoms. Instruments like single AI-based atom gravimeters often encounter significant phase noise due to vibrations. However, in AGGs incorporating two or more AIs, this noise manifests in δφc and can be effectively suppressed in normal operations through ellipse fitting, no longer dominating as a primary noise source. Consequently, differential phase noise δφd (resulting from Raman laser frequency noise) [22] and detection noise δdet [25,26,30–32] have been noted as significant limiting factors affecting measurement sensitivity, due to the lack of a standard method for suppressing them in typical operations.
In AGGs conducting sequential detection on dual AIs, such as those vertical gravity gradiometers where both AIs share the same detection region and unit [21,30], δdeti in the two interference signals are entirely random and uncorrelated. Conversely, in AGGs employing synchronous detection on dual AIs, including horizontal AGGs [22,24] and vertical AGGs with separate detection systems for the two AIs [20,23,25–27], due to a common detection laser source, the two detection beams undergo synchronous frequency and intensity fluctuations. When detection noise primarily arises from these fluctuations, δdeti in the two interference signals are identical. Consequently, assuming detection noise dominates and discounting other noise effects, Eqs. (6a) and (6b) can be simplified as
In this scenario, Fig. 1 showcases two interference signals with varying φd and their corresponding elliptical signals. As φd ranges between (0, π/2), the two signals are represented by Fig. 1(a) and (b). As φd decreases, the ellipse contour approaches a line with a positive slope. When Ai and Bi of the two interference signals are equal, the line takes the form y = x. Here, synchronous fluctuations of the two signals, with a coefficient of 1 + δdet, depicted as positive fluctuation of δdet > 0 at time t1 and negative fluctuation of δdet < 0 at time t2, manifest as data points moving along the ellipse, causing minimal disturbance to its shape. When φd falls within the interval (π/2, π), represented by the two interference signals in Fig. 1(a) and (c), the ellipse contour approximates a line with a negative slope as φd increases (as in Fig. 1(e)). When Ai and Bi of the two interference signals match, this line can be expressed as y = 2B-x. In this instance, synchronized fluctuations cause the data points to move perpendicular to the ellipse contour, markedly reducing the signal-to-noise ratio of the ellipse. This results in heightened fitted phase noise during ellipse fitting, ultimately diminishing the gravity gradient measurement sensitivity.
Fig. 1. The impact of synchronous detection noise on ellipses with varying phases. (a) The interference signal of AI1; (b) The interference signal of AI2 with a minor phase difference (0<φd<π/2) compared to AI1; (c) The interference signal of AI2 with a substantial phase difference (π/2<φd<π) compared to AI1; (d) Lissajous ellipse with a minor differential phase formed by signals (a) and (b); (e) Lissajous ellipse with a substantial differential phase formed by signals (a) and (c). Solid lines and shaded areas depict the ideal interference signals and those incorporating synchronous detection noise. At times t1 and t2, detection noise-induced synchronous jitters with δdet > 0 and det < 0 in AI1 and AI2, respectively. The disturbance caused by this noise on the ellipse shape in large φd conditions significantly surpasses that in small φd conditions.
To quantitatively analyze the impact of the differential phase on elliptical fitting phase noise, we conducted a series of simulations based on the theoretical principles involved. Considering typical instrument parameters, we set both interference signals’ contrast to 15%, detection noise standard deviations in both AIs to 2‰, and the differential phase noise to 20 mrad/shot. Both detection noise and differential phase noise were modeled as Gaussian white noise. Under conditions of synchronous and uncorrelated detection noise, we generated sets of interference data with different φd using a computer. Each group was comprised of 10,000 data pairs (P1, P2). A least-squares fit was performed every 40 pairs, and the standard deviations of the resulting 250 fitted phases were calculated. This process was iterated 10 times, and the average values and deviations are graphed in Fig. 2.
Fig. 2. Simulation outcomes illustrating the relationship between ellipse fitting noise and the differential phase. (a) Detection of two AIs performed sequentially, resulting in uncorrelated detection noise in the AI signals. (b) Detection of two AIs performed simultaneously, resulting in synchronous detection noise. Each data point reflects an average of 10 simulations, and in each simulation 10,000 interference data pairs were employed.
The figures illustrate that considering the differential phase noise δφd alone, the ellipse fitting phase noise demonstrates a minimum value at a differential phase of π/2. When solely accounting for detection noise, in conditions where the two AIs’ detection noise is uncorrelated (typical in AGGs performing sequential detection on interferometers), the ellipse fitting noise remains independent of the differential phase. Conversely, in scenarios where the two AIs experience synchronous detection noise (as in AGGs operating in synchronous detection mode), the ellipse fitting noise increases with the differential phase. For differential phases less than π/2, the synchronously detected AGGs exhibit lower fitting noise compared to the sequentially detected AGGs, highlighting the common-mode restraining effect on the detection noise. However, for differential phases larger than π/2, the fitting noise in synchronously detected AGGs surpasses that of sequentially detected AGGs, indicating an amplification effect on the detection noise. At a differential phase of 0.1 π in synchronous detection mode, the fitted phase noise of the ellipses is merely 0.16 times that at 0.9 π, signifying a 4.5 times common-mode suppression compared to sequential detection mode.
When considering both differential phase noise and detection noise, they interact in a compositional manner. As depicted in Fig. 2(a), sequential AI detection leads to uncorrelated detection noise, consistently resulting in the minimum ellipse fitting noise at a differential phase of π/2. However, in simultaneous AI detection scenarios, the lowest ellipse fitting noise can manifest at varying differential phases based on the dominant noise types. Our simulations maintain a fixed detection noise of 2‰ while varying the differential phase noise from 10 mrad/shot to 50 mrad/shot. This range approximately corresponds to Raman laser frequency noise from 160 kHz to 800 kHz within an AGG with L = 45 cm and T = 100.5 ms. The outcomes displayed in Fig. 3 unveil distinct patterns: at a lower differential phase noise (e.g., 10 mrad/shot in Fig. 3), the ellipse fitting noise diminishes with the differential phase. Conversely, with relatively higher differential phase noise (e.g., 50 mrad/shot in Fig. 3), the ellipse fitting noise minimizes when the differential phase nears π/2. However, when the differential phase noise aligns with the detection noise (e.g., 30 mrad/shot in Fig. 3), the lowest ellipse fitting noise occurs within a differential phase range between 0 and π/2 (0.35 π for 30 mrad/shot in Fig. 3).
Fig. 3. Simulation results showcasing the relationship between ellipse fitting noise and the differential phase at varying differential phase noise levels. The detection noise remains fixed at 2‰. The lowest extraction noise occurs at different differential phases as the differential phase noise varies. Each data point represents an average of 10 simulations, employing 10,000 interference data pairs in each simulation.
3. Horizontal atom gravity gradiometer
We conducted a verification experiment using our compact horizontal AGG, extensively detailed in [36]. The sensor unit, illustrated in Fig. 4, comprises vacuum chambers, optical components, magnetic field units, microwave components, mechanical structures, and peripheral magnetic shielding. Structurally, it can be perceived as two separate fountain-type AIs symmetrically arranged by 45 cm horizontally. Each AI includes a two-dimensional magneto-optical trap (2D-MOT), a three-dimensional magneto-optical trap (3D-MOT), and regions for state preparation, detection, and interference. These AI vacuum chambers, made of all-quartz materials, offer smaller volumes and larger optical apertures, connected by a titanium alloy pipe at their interference regions. Two parallel Raman laser beams enter through the left window, traverse both interference regions, exit from the right, and return along the original path after reflection via a mirror, enabling synchronized excitation of the atom clouds. As the Raman laser beams propagate entirely in a vacuum between the AIs, they remain undisturbed by any medium, ensuring optimal common-mode noise suppression. The entire sensor unit measures 70 cm (L) × 42 cm (W) × 35 cm (H), possessing a volume of 103 L, demonstrating high compactness.
Fig. 4. Schematic diagram of the sensor head of the compact horizontal AGG. Symmetrical elements in this figure are labeled only on one side, as the two atom interferometers exhibit near mirror-symmetric spatial structures.
The operational sequences of both AIs are completely synchronized. After transverse precooling in 2D-MOTs, Rb-85 atoms move to 3D-MOTs through horizontal pushing beams. These trapped atoms are launched by moving molasses, reaching 12 cm apexes after 156 ms before falling back. To avoid co-propagating Raman excitations, the atoms are launched at a 3° angle in the vertical direction, acquiring a horizontal velocity of approximately 7.8 cm/s, resulting in a Doppler shift of about 100 kHz. 26 ms post-launch, the atoms reach state preparation zones, where microwave and blow away laser pulses prepare them to the magnetically insensitive state $|{F = 2,\textrm{ }{m_F} = 0} \rangle$. Another 29 ms later, the atoms arrive at the interference regions and undergo manipulation by a set of π/2-π-π/2 Raman laser pulses spaced by 100.5 ms. During the splitting, reflection, and recombination processes of atom wave packets, local gravity information gets imprinted in the interferometer phases. Upon falling back to the detection region, atoms in states $|{F = 2,\textrm{ }{m_F} = 0} \rangle$ and $|{F = 3,\textrm{ }{m_F} = 0} \rangle$ are sequentially excited by detection lasers, and by collecting fluorescence signals, population ratios of $|{F = 3} \rangle$ state in the two AIs are obtained via the following equation:
where SiF = 3 and SiF = 2 denote fluorescence signals of $|{F = 3} \rangle$ and $|{F = 2} \rangle$ states, respectively, and ηi represents relative detection efficiencies between the states.We recorded the detection signals of the original falling atoms in both AIs and those of state-prepared atoms further excited by a single π/2 Raman pulse. Results are displayed in Fig. 5. Both signals, uninvolved in the interference process and free from phase noise. Additionally, the π/2 pulse-excited signals are normalized using Eq. (8a) and (8b), thereby eliminating atom number fluctuation noise, while the signals of original falling atoms don’t include the microwave related noise. Figure 5 shows significant synchronicity in the noises of both AIs, aligning with the premise of Eq. (7).
Fig. 5. Detection signals of the original falling atoms (a) and the π/2 pulse-excited atoms (b) in the two AIs. The signals from both AIs exhibit significant synchronicity, with calculated Pearson correlation coefficients of 0.87 and 0.91, respectively.
4. Verification experiment based on magnetic field modulation
Throughout the interference process, although the atoms consistently reside in the $|{{m_F} = 0} \rangle$ magnetic sublevels, any non-uniformity in the quantization bias magnetic field within the interference region may still induce energy level shifts due to the quadratic Zeeman effect, consequently altering the output phase of the AI. The relationship between the Zeeman phase shift ΔφZeeman and the bias magnetic field B is described by
Fig. 6. Ellipses at different magnetic coil driving currents. Each ellipse contains 1000 data points, with the phase increasing incrementally with current.
Utilizing the correlation between the magnetic coil current and the ellipse phase φd, as depicted in Fig. 7, we conducted measurements to gauge the fitted elliptical phase noise at various φd values by adjusting the coil current. In our AGG setup, a single measurement cycle lasts 470 ms. Over a span of 30 minutes at each phase, we accumulated around 3800 interference data points. Employing Eqs. (3) and (4), we fitted the ellipse every 40 data points, yielding approximately 95 phase values, and calculated the standard deviation σφ. Successively, we performed measurements at 9 phase points within the (0, π) range, cycling 4 times to eliminate the impact of time-varying factors. The data from the 4 cycles at each phase point were averaged, revealing the relationship between the fitted phase noise and the ellipse phase, φd, illustrated in Fig. 8. The results are very close to the simulation outcomes in Fig. 2(b), and shows that when φd > 0.1 π, the fitted phase noise increases in relation to φd, with the noise at larger phases reaching up to 1.7 times that of smaller phases. Nonetheless, reducing the elliptical phase indefinitely doesn't uniformly decrease the fitting phase noise. Notably, at extremely small phase values like 0.02 π, the fitted phase noise increases abnormally due to the loss of the elliptical shape.
Fig. 8. Dependency of the fitted phase noise on the ellipse phase. Data points compile statistics from 4 scan cycles. Each scan entails sequential measurements of ellipse phase noises at 9 phase points. Within each measurement, 3800 raw data points yield 95 fitted phases for standard deviation calculation.
Taking the experimental findings into account, we chose to operate the AGG at a bias magnetic coil current of 67.5 mA, aligning with an ellipse phase of roughly 260 mrad. The comparison between Allan deviations in this small phase mode and when the AGG functions at 2.65 rad is displayed in Fig. 9. The phase measurement sensitivity in the small phase mode is 23.5 mrad/Hz1/2, significantly outperforming the large phase mode. Following an integration time of 4800 s, we achieved a measurement resolution of 240 µrad, corresponding to a differential gravity resolution of 1.6 × 10−10 g and a gravity gradient resolution of 3.3 E (1 E = 1 × 10−9 /s2).
Fig. 9. Comparison of the Allan deviations in the small phase mode (260 mrad) and large phase mode (2.6 rad).
5. Discussion and conclusion
The detection noise background in our horizontal AGG has been previously measured to be 2‰∼ 3‰ and the Raman laser frequency noise was slightly over 100 kHz. Given that the simulation parameters align with empirical data, the results in Fig. 2(b) closely mirror the experiments depicted in Fig. 7. These findings suggest that detection noise is a predominant factor in our horizontal AGG, while the differential phase noise (stemming from Raman laser frequency noise) is also significantly influential.
Ellipse fitting serves as a common method for extracting the differential phase between atom interferometers, especially in scenarios with substantial common phase noise. This study rigorously examines how extraction noise varies concerning the differential phase between atom interferometers during ellipse fitting. Our initial analysis dissects the transfer process of detection noise into ellipse fitting phase noise. We discovered that larger ellipse phases within the (π/2, π) range intensify the transition of synchronous detection noise into ellipse fitting phase noise, consequently diminishing measurement sensitivity. Conversely, smaller ellipse phases within (0, π/2) weaken this transfer, thereby enhancing measurement sensitivity. When considering both differential phase noise and detection noise, the lowest extraction noise can emerge at various points within (0, π/2) depending on the dominant noise type. Optimizing the operational ellipse phase in our compact horizontal AGG substantially bolstered measurement sensitivity. Setting an integration time of 4800 s, we achieved a gravity gradient resolution of 3.3 E.
Funding
Innovation Program for Quantum Science and Technology (2021ZD0300603); National Key Research and Development Program of China (2017YFC0601602).
Acknowledgements
We acknowledge the support from the China National Precise Gravity Measurement Facility (PGMF).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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