Digital Closed-Loop Thermal Atomic-Beam Interferometer

• The digital closed-loop thermal atomic-beam interferometer is a matter-wave sensor architecture that uses thermal rubidium beams and Raman pulses for inertial measurements.
• It employs dynamic, real-time phase feedback to cancel velocity-dependent Sagnac shifts, ensuring scale-factor independence and contrast restoration.
• Simultaneous acceleration and rotation sensing is achieved through dual-channel phase extraction and digitally encoded detuning updates within a pseudo-inertial frame.

A digital closed-loop thermal atomic-beam interferometer is an atom-interferometric inertial sensor architecture in which thermal rubidium atomic beams traverse spatially separated Raman interaction regions while a digital controller updates two-photon detunings or synthetic phases in real time to null interferometric phase excursions. In the 2024 closed-loop gyroscope, the central problem is the velocity-dependent Sagnac phase shift combined with the longitudinal velocity distribution of the atoms, which restricts measurements of large angular velocities; the reported remedy is a pseudo-rotation effect generated by appropriate Raman two-photon detunings, restoring contrast and making the gyroscope scale factor independent of the longitudinal velocity distribution (Sato et al., 2024). Subsequent work generalized the approach to simultaneous absolute acceleration-rotation sensing through synchronized phase biasing, momentum-kick reversal, and detuning feedback in simulation (Sato et al., 7 Sep 2025), and then demonstrated dual-channel closed-loop atomic beam interferometry beyond the half-fringe limit with decoupled feedback control of acceleration- and rotation-induced phases (Jia et al., 16 Mar 2026).

1. Physical architecture and interferometer geometry

The experimentally demonstrated gyroscope architecture uses dual spatial-domain Mach–Zehnder interferometers on counter-propagating $^{87}$Rb beams. Two ovens emit collimated thermal Rb-87 atomic beams with mean longitudinal speed $v_0\approx330\,\mathrm{m/s}$ and a Gaussian-like spread $\Delta v/v_0\approx10\%$. Each beam is optically pumped into $|F=1,m_F=0\rangle$ and directed through three spatially separated Raman zones with $L=70\,\mathrm{mm}$ spacing. In each zone, two counter-propagating beams of wavelength $\lambda=780\,\mathrm{nm}$, detuned by $\Delta\approx1.5\,\mathrm{GHz}$ from the $5P_{3/2}$ state, drive Doppler-sensitive $\pi/2$–$\pi$–$v_0\approx330\,\mathrm{m/s}$0 Raman pulses. The first $v_0\approx330\,\mathrm{m/s}$1 zone splits, the $v_0\approx330\,\mathrm{m/s}$2 zone redirects, and the final $v_0\approx330\,\mathrm{m/s}$3 recombines; detection is performed via state-selective fluorescence after the last Raman zone. A compact digital closed-loop three-axis thermal-beam atom-interferometer gyroscope can be built by arranging three independent pairs of counter-propagating thermal beams along orthogonal axes $v_0\approx330\,\mathrm{m/s}$4, with a common set of three Raman interaction regions overlaid for all three axes by rotating the Raman beams in turn or by multiplexing three orthogonal Raman-beam pairs (Sato et al., 2024).

The 2025 theoretical proposal retains the three-zone Mach–Zehnder structure but specifies three spatially separated, retro-reflected Raman beam pairs labeled A, B, and C, with electro-optic modulators on the Raman beams and a capillary-source $v_0\approx330\,\mathrm{m/s}$5Rb beam inclined by $v_0\approx330\,\mathrm{m/s}$6. Its operating sequence combines phase biasing and momentum-kick reversal over a four-step cycle, yielding four fluorescence outputs from right- and left-going beams, before and after k-reversal: $v_0\approx330\,\mathrm{m/s}$7, $v_0\approx330\,\mathrm{m/s}$8, $v_0\approx330\,\mathrm{m/s}$9, and $\Delta v/v_0\approx10\%$0 (Sato et al., 7 Sep 2025).

The 2026 dual-channel demonstration uses continuous, transversely cooled $\Delta v/v_0\approx10\%$1Rb atomic beams and two counter-propagating Raman Mach–Zehnder interferometers. In that formulation, the two interferometer outputs $\Delta v/v_0\approx10\%$2 and $\Delta v/v_0\approx10\%$3 are explicitly written so that acceleration and rotation enter with different signs, enabling later separation by half-sum and half-difference operations (Jia et al., 16 Mar 2026).

These implementations share a common design logic: thermal atomic beams provide continuous interrogation in a spatial-domain interferometer, while counter-propagating geometries supply differential channels that are suitable for closed-loop inertial readout. A plausible implication is that the “digital closed-loop thermal atomic-beam interferometer” is best understood not as a single apparatus, but as a design class defined by thermal-beam Raman Mach–Zehnder interferometry plus active digital phase compensation.

2. Velocity-dependent phase dispersion and pseudo-rotation compensation

The defining difficulty for thermal-beam gyroscopes is that the Sagnac phase is velocity dependent. For an atom of velocity $\Delta v/v_0\approx10\%$4 in an area-enclosed interferometer, the reported rotation phase is

$\Delta v/v_0\approx10\%$5

Because the atomic beam has a longitudinal velocity distribution, different velocity classes accumulate different phases, causing dephasing and contrast deterioration. The closed-loop gyroscope introduces a compensating velocity-dependent phase

$\Delta v/v_0\approx10\%$6

generated by linearly chirping the Raman phases, where $\Delta v/v_0\approx10\%$7 are the two-photon detunings of the three pulses and $\Delta v/v_0\approx10\%$8. The dispersion-cancellation condition is

$\Delta v/v_0\approx10\%$9

Substituting this into the differential phase between counter-propagating beams gives

$|F=1,m_F=0\rangle$0

which nullifies the $|F=1,m_F=0\rangle$1 Sagnac dispersion for all $|F=1,m_F=0\rangle$2 (Sato et al., 2024).

The same work writes the detected population as

$|F=1,m_F=0\rangle$3

thereby making the role of the velocity distribution explicit. In open loop,

$|F=1,m_F=0\rangle$4

so the inferred angular velocity depends on $|F=1,m_F=0\rangle$5. In closed loop,

$|F=1,m_F=0\rangle$6

which is independent of both $|F=1,m_F=0\rangle$7 and $|F=1,m_F=0\rangle$8 (Sato et al., 2024).

This compensation is described as a pseudo-rotation effect. In the gyroscope realization, the angular velocity of the system can be estimated through the detuning point where the phase difference between the two interferometers is zero. The significance is not merely contrast restoration: the measurement variable is transferred from a velocity-weighted phase observable to a directly readable detuning frequency. This suggests a shift from open-loop fringe metrology to digitally encoded inertial metrology.

3. Dual-channel inertial decomposition and pseudo-inertial-frame control

The 2025 proposal extends closed-loop compensation from rotation alone to simultaneous acceleration and rotation sensing. It defines, for atoms of velocity $|F=1,m_F=0\rangle$9,

$L=70\,\mathrm{mm}$0

and writes the four phase channels as

$L=70\,\mathrm{mm}$1

$L=70\,\mathrm{mm}$2

$L=70\,\mathrm{mm}$3

$L=70\,\mathrm{mm}$4

From these, $L=70\,\mathrm{mm}$5 is inferred as

$L=70\,\mathrm{mm}$6

and, after nulling $L=70\,\mathrm{mm}$7 by slow optical-path feedback, the pure inertial phases follow as

$L=70\,\mathrm{mm}$8

The two-photon detuning control variables are a static offset $L=70\,\mathrm{mm}$9 on beam B and a linear ramp $\lambda=780\,\mathrm{nm}$0 across A$\lambda=780\,\mathrm{nm}$1B$\lambda=780\,\mathrm{nm}$2C. By choosing

$\lambda=780\,\mathrm{nm}$3

all $\lambda=780\,\mathrm{nm}$4- and $\lambda=780\,\mathrm{nm}$5-terms, and $\lambda=780\,\mathrm{nm}$6-terms, cancel so that $\lambda=780\,\mathrm{nm}$7 for every velocity class (Sato et al., 7 Sep 2025).

The 2026 experiment presents a closely related but experimentally realized diagonalization. The two signal phases are combined as

$\lambda=780\,\mathrm{nm}$8

with

$\lambda=780\,\mathrm{nm}$9

$\Delta\approx1.5\,\mathrm{GHz}$0

This diagonalizes the inertial coupling and assigns acceleration to $\Delta\approx1.5\,\mathrm{GHz}$1 and rotation to $\Delta\approx1.5\,\mathrm{GHz}$2 (Jia et al., 16 Mar 2026).

Across these formulations, the common objective is decoupled feedback control. The 2025 proposal states that the closed-loop realization eliminates cross-coupling, with no acceleration leak into the $\Delta\approx1.5\,\mathrm{GHz}$3 channel and vice versa. The 2026 demonstration states that acceleration- and rotation-induced phases are independently extracted, tracked across multiple fringes, and actively compensated through Raman frequency modulation. In both cases, the interferometer is digitally maintained in a pseudo-inertial frame (Sato et al., 7 Sep 2025, Jia et al., 16 Mar 2026).

4. Digital signal extraction, phase tracking, and feedback laws

The 2024 gyroscope describes a digital PI controller running at $\Delta\approx1.5\,\mathrm{GHz}$4. The two interferometer outputs are demodulated, for example via lock-in at $\Delta\approx1.5\,\mathrm{GHz}$5, and differenced to form an error signal $\Delta\approx1.5\,\mathrm{GHz}$6. The control law is

$\Delta\approx1.5\,\mathrm{GHz}$7

with $\Delta\approx1.5\,\mathrm{GHz}$8 updated so as to maintain $\Delta\approx1.5\,\mathrm{GHz}$9. The locked detuning gives the inertial readout through

$5P_{3/2}$0

The same source also gives implementation notes for inertial navigation: real-time phase-demodulation, error computation, PI control, and DDS detuning updates at $5P_{3/2}$1 loop rate on FPGA/DSP hardware, with fully digital readout of $5P_{3/2}$2 and inferred $5P_{3/2}$3 enabling seamless interfacing with inertial-navigation Kalman filters (Sato et al., 2024).

The 2025 proposal uses a four-step cycle lasting $5P_{3/2}$4. An electro-optic modulator on the middle beam pair imparts an alternating phase bias of $5P_{3/2}$5 every transit time $5P_{3/2}$6, so that each bias produces a net interferometer phase shift $5P_{3/2}$7. After two such phase-bias steps, all three EOM drive frequencies are shifted by $5P_{3/2}$8 to reverse the effective momentum kick. From the four measured intensities $5P_{3/2}$9 and $\pi/2$0 at $\pi/2$1, the phase is extracted by

$\pi/2$2

A digital processor then forms the phase combinations, nulls optical-path drifts, and implements closed-loop cancellation of acceleration and rotation by adjusting $\pi/2$3 and $\pi/2$4 (Sato et al., 7 Sep 2025).

The 2026 dual-channel experiment digitizes $\pi/2$5 at $\pi/2$6 and computes analytic signals via a digital Hilbert transform. Quadrature demodulation at $\pi/2$7 yields

$\pi/2$8

from which the instantaneous phase $\pi/2$9 is unwrapped cycle-to-cycle to obtain a continuous, multi-$\pi$0 track of $\pi$1 and $\pi$2. Two independent digital PID loops then operate at $\pi$3: one updates $\pi$4 for rotation and one updates $\pi$5 for acceleration. The closed-loop transfer function for rotation is chosen as a single-pole integrator plus proportional path with a unity-gain bandwidth of order $\pi$6, limited by atomic-beam transit and detection latency; the acceleration loop bandwidth is again $\pi$7, set high enough to follow turbulent inertial inputs but below the $\pi$8 data-acquisition limit (Jia et al., 16 Mar 2026).

Taken together, these reports establish a specifically digital operating mode: inertial observables are represented as synthesizer detunings and tracked by discrete-time feedback rather than inferred solely from static fringe amplitudes. The 2026 paper explicitly states that this phase-encoded readout replaces the traditional cosine-amplitude fringe measurement and already yields a full $\pi$9 open-loop dynamic range, with phase unwrapping extending this to several $v_0\approx330\,\mathrm{m/s}$00 before contrast decay sets in (Jia et al., 16 Mar 2026).

5. Demonstrated and simulated performance

In the 2024 $v_0\approx330\,\mathrm{m/s}$01Rb gyroscope, the atomic beam parameters are $v_0\approx330\,\mathrm{m/s}$02, $v_0\approx330\,\mathrm{m/s}$03 for a $v_0\approx330\,\mathrm{m/s}$04 source, $v_0\approx330\,\mathrm{m/s}$05, and $v_0\approx330\,\mathrm{m/s}$06. The achieved enclosed area is $v_0\approx330\,\mathrm{m/s}$07, and the detuning range is $v_0\approx330\,\mathrm{m/s}$08 up to $v_0\approx330\,\mathrm{m/s}$09 for $v_0\approx330\,\mathrm{m/s}$10 compensation. Open-loop contrast falls to $v_0\approx330\,\mathrm{m/s}$11 at $v_0\approx330\,\mathrm{m/s}$12, whereas closed-loop contrast remains $v_0\approx330\,\mathrm{m/s}$13 up to $v_0\approx330\,\mathrm{m/s}$14, which was table limited. Numerical modeling predicts $v_0\approx330\,\mathrm{m/s}$15 contrast up to $v_0\approx330\,\mathrm{m/s}$16 before higher-order $v_0\approx330\,\mathrm{m/s}$17 terms matter. In comparison with a commercial FOG of stability $v_0\approx330\,\mathrm{m/s}$18, open loop shows nonlinearity $v_0\approx330\,\mathrm{m/s}$19, while closed loop remains linear $v_0\approx330\,\mathrm{m/s}$20 up to $v_0\approx330\,\mathrm{m/s}$21. Under applied acceleration of $v_0\approx330\,\mathrm{m/s}$22 via $v_0\approx330\,\mathrm{m/s}$23 tilt, the closed-loop scale factor shift is $v_0\approx330\,\mathrm{m/s}$24, whereas the open-loop shift is $v_0\approx330\,\mathrm{m/s}$25. Closed-loop operation maintains full contrast and achieves sensitivity $v_0\approx330\,\mathrm{m/s}$26 in the $v_0\approx330\,\mathrm{m/s}$27–$v_0\approx330\,\mathrm{m/s}$28 band. The reported demonstration achieved a measurement of angular velocity of $v_0\approx330\,\mathrm{m/s}$29 even with an acceleration of $v_0\approx330\,\mathrm{m/s}$30 on a three-axis rotation table (Sato et al., 2024).

The 2025 proposal reports simulations for a $v_0\approx330\,\mathrm{m/s}$31Rb beam from a capillary source at $v_0\approx330\,\mathrm{m/s}$32, with most-probable speed $v_0\approx330\,\mathrm{m/s}$33, $v_0\approx330\,\mathrm{m/s}$34, flux $v_0\approx330\,\mathrm{m/s}$35, beam separation $v_0\approx330\,\mathrm{m/s}$36, inclination $v_0\approx330\,\mathrm{m/s}$37, Doppler width $v_0\approx330\,\mathrm{m/s}$38, and simulated steady-state fringe contrast $v_0\approx330\,\mathrm{m/s}$39. Using the stated shot-noise-limited expressions,

$v_0\approx330\,\mathrm{m/s}$40

the proposal gives $v_0\approx330\,\mathrm{m/s}$41 and $v_0\approx330\,\mathrm{m/s}$42. Each closed-loop measurement cycle takes $v_0\approx330\,\mathrm{m/s}$43; the closed-loop transfer function has a $v_0\approx330\,\mathrm{m/s}$44 bandwidth on order $v_0\approx330\,\mathrm{m/s}$45, and simulations of step responses show full settling in $v_0\approx330\,\mathrm{m/s}$46, corresponding to bandwidth $v_0\approx330\,\mathrm{m/s}$47. Because the loop holds $v_0\approx330\,\mathrm{m/s}$48, the proposal states that fringe contrast is maintained even for large applied $v_0\approx330\,\mathrm{m/s}$49 up to $v_0\approx330\,\mathrm{m/s}$50 and $v_0\approx330\,\mathrm{m/s}$51 of tens of $v_0\approx330\,\mathrm{m/s}$52 (Sato et al., 7 Sep 2025).

The 2026 experiment reports unambiguous ranges of $v_0\approx330\,\mathrm{m/s}$53 for rotation and $v_0\approx330\,\mathrm{m/s}$54 for acceleration while maintaining high fringe contrast, corresponding to nearly two orders-of-magnitude extension beyond the conventional half-fringe limit. The long-term stability at $v_0\approx330\,\mathrm{m/s}$55 averaging time is $v_0\approx330\,\mathrm{m/s}$56 for rotation and $v_0\approx330\,\mathrm{m/s}$57 for acceleration. Short-term sensitivity is reported as $v_0\approx330\,\mathrm{m/s}$58 for rotation and a few $v_0\approx330\,\mathrm{m/s}$59 for acceleration (Jia et al., 16 Mar 2026).

These results span three levels of maturity: an experimentally demonstrated closed-loop gyroscope with dispersion compensation, a theoretical proposal for simultaneous absolute acceleration-rotation sensing with high bandwidth, and an experimental dual-channel closed-loop realization beyond the half-fringe limit. The shared pattern is that dynamic range increases when inertial phase is driven to zero rather than allowed to accumulate.

6. Advantages, limitations, and relation to inertial navigation

The reported advantages are explicit. For the 2024 gyroscope, the closed-loop method extends dynamic range by $v_0\approx330\,\mathrm{m/s}$60 without laser-cooling or mechanical tip-tilt, provides scale-factor self-calibration because $v_0\approx330\,\mathrm{m/s}$61 depends directly on detuning frequency rather than atom-beam speed, rejects drifts in velocity distribution and Raman beam alignment, and preserves compactness because no extra optics or mechanical rotation of beams are required (Sato et al., 2024). The 2025 proposal adds simultaneous, absolute sensing of $v_0\approx330\,\mathrm{m/s}$62 and $v_0\approx330\,\mathrm{m/s}$63 in a single device, minimal alignment errors, a truly self-contained INS, and an intrinsic optical reference given by the atomic transition, implying no long-term scale-factor drift; it frames the technology as suitable for GPS-denied environments (Sato et al., 7 Sep 2025). The 2026 experiment states that by converting the intrinsically periodic interferometric response into stabilized phase-encoded inertial channels, the scheme advances matter-wave sensors toward practical quantum inertial navigation under dynamic conditions (Jia et al., 16 Mar 2026).

The principal limitations are also stated. In the 2024 system, noise sources include Raman-laser phase noise, atom-beam shot noise, and detection electronics; closed loop suppresses Doppler-dispersion noise, but the numerical modeling predicts a limit when higher-order $v_0\approx330\,\mathrm{m/s}$64 terms matter (Sato et al., 2024). In the 2026 dual-channel experiment, the dominant noise sources are detection photon shot noise, laser phase and microwave synthesizer phase noise, and residual velocity-distribution dephasing from higher-order terms beyond first order for rotation; loop bandwidth is limited by atomic-beam transit and detection latency (Jia et al., 16 Mar 2026). The 2025 proposal likewise depends on a four-step synchronized sequence and slow optical-path feedback to null $v_0\approx330\,\mathrm{m/s}$65, indicating that optical-path mismatches remain a distinct systems problem even when inertial cross-coupling is canceled (Sato et al., 7 Sep 2025).

A common misconception is that closed-loop operation simply enlarges the linear range of the conventional fringe. The cited work describes a stronger claim: the interferometer output is reformulated as a digitally stabilized phase channel, with inertial information appearing as detuning updates $v_0\approx330\,\mathrm{m/s}$66 rather than solely as an amplitude displacement on a cosine fringe. Another misconception is that contrast preservation implies complete immunity to thermal-beam velocity spread. The reported analyses are more specific: first-order velocity-dependent terms can be canceled for all velocity classes, but higher-order terms and residual dephasing remain relevant at sufficiently large rotation or acceleration (Sato et al., 2024, Jia et al., 16 Mar 2026).

In this sense, the digital closed-loop thermal atomic-beam interferometer marks a transition from open-loop matter-wave sensing, constrained by dephasing and half-fringe ambiguity, to a control-defined operating regime in which pseudo-rotation, pseudo-inertial-frame tracking, and dual synthetic-phase feedback produce wide-dynamic-range inertial observables directly in the digital domain.