A microwave control system for continuous spatial-domain atom interferometers

Light-pulse atomic interferometers (LPAIs) have emerged in recent years as crucial tools for precision measurements, enabling highly accurate assessments of various physical quantities such as rotation rate,^1–9 acceleration,^5–11 gravity,^12–14 and gravity gradients.^15,16 In typical Raman LPAIs, atomic wave packets are coherently manipulated using two-photon stimulated Raman transitions.^12,17 The Raman lasers’ relative phases encode the position information of the atomic wave packets, allowing for the measurement of inertial quantities through sequences of π/2 and π Raman pulses that split and recombine the wave packets. These phases directly contribute to the atom interferometer phase shifts as a non-inertial term.

Raman lasers, with two frequency components offset by the clock frequency of alkali metal atoms (6.835 GHz for ⁸⁷Rb), are usually generated using acousto-optic modulators (AOMs),¹⁸ electro-optic modulators (EOMs),¹⁹ or optical phase-locked loops (OPLLs)²⁰ driven by low-phase-noise microwave sources. Phase noise from these microwave sources transfers to the lasers, becoming a dominant noise source and representing a limiting factor for atom interferometer performance.²¹ In addition, microwave sources must control the phase, frequency, and power of Raman light pulses, allowing flexible manipulation of atomic interference phase shifts.

Various microwave sources or frequency synthesizers with low phase noises have been proposed for quantum experiment applications. They commonly utilize frequency sources with ultra-high frequency stability as references, including ultra-stable lasers²² and Crystal Sapphire Oscillators (CSOs),^23,24 as well as Oven-Controlled Crystal Oscillators (OCXOs) and atomic clocks.^25–34 The latter are particularly popular because of their compact size and low power consumption.²⁵ The frequency conversion process is performed, based on these references, to achieve the required microwave frequencies. Typical approaches to frequency conversion include phase-locked loops (PLLs),²⁶ sampling mixing,²⁷ frequency multipliers,²⁸ digital synthesis,^35,36 and harmonic generation techniques that employ step recovery diodes (SRDs)^29,30 or nonlinear transmission lines (NLTLs).^25,31–34

For instance, Boudot et al. developed a single-channel 9.392 GHz microwave synthesizer for the CPT cesium atomic clock based on an NLTL, which achieves a fractional frequency stability better than 4 × 10⁻¹⁷ over one day.³¹ In 2015, they established cost-effective single-channel frequency synthesizers with 4.596 and 6.834 GHz outputs using frequency multipliers, contributing a Dick effect at the 10⁻¹⁴ level to high-performance vapor cell atomic clocks.²⁸ Lautier et al. reported on a mobile 6.834 GHz synthesizer with a Raman frequency and one selective frequency channel for transportable cold atom interferometric gravimeters utilizing PLLs.²⁶ This system achieved a residual noise level of −65 dB rad²/Hz with a compact structure and a low power consumption of ∼30 W. These microwave systems have demonstrated significant potential in terms of precise atomic manipulation.

Typically, these microwave sources have only one main output channel and rely on time-sequence control to drive Raman light pulses in atom interferometers. This operation mode is suitable for atomic clocks^{22,23,25,27–32,34} and time-domain atom interferometers and for some spatial-domain atom interferometers with pulsed atomic clouds as matter wave sources.^26,36 In these cases, Raman pulse sequences can coherently manipulate cold atomic clouds in time-division multiplexing mode.

However, for spatial-domain atom interferometers with continuous atomic beam sources,^1,8,37,38 spatially separated Raman light pulses continuously interact with atomic wave packets and ideally require independent control of the phase, frequency, or even waveform of each Raman light pulse. While a single-channel microwave system can drive multiple Raman lasers for interference manipulation,^8,37 its flexibility in controlling the interference phase is limited. This limitation necessitates the use of additional phase devices such as piezoelectric transducers (PZTs) and liquid crystal phase plates. Although a combination of multiple commercial signal sources can generate multiple microwave outputs,^1,38 this approach lacks overall integration and is neither cost-effective nor conducive to long-term use in field applications. Therefore, it is essential to develop a dedicated and integrated microwave control system with multiple low-phase-noise signal channels that is capable of independently adjusting waveform parameters for continuous spatial-domain atom interferometers.

In this paper, we propose an integrated low-phase-noise microwave control system designed for a spatial-domain ⁸⁷Rb atomic interferometer. The system features three adjustable microwave outputs at ∼6835 MHz. Starting with an atomic clock and an ultra-stable quartz crystal oscillator, it generates a 6800 MHz low-phase-noise signal through an NLTL-based phase-locked loop. This signal is then mixed with frequency outputs from a multi-channel digital signal source to achieve outputs of the three channels that are independently adjustable. In addition, the system integrates a closed-loop feedback architecture for servo control of the microwave power, ensuring a stable power ratio between Raman frequency components. The absolute phase noise of the synthesizer and relative stability between channels are measured before the system is applied to an atom interferometric inertial sensor,⁹ demonstrating its high applicability as a practical component in spatial-domain LPAIs.

In our atomic interferometer,⁹ a pair of counter-propagating ⁸⁷Rb atomic beams is generated at opposite sides of the vacuum chamber. Three spatially separated Raman pulses, configured in a π/2–π–π/2 sequence, are used to split, reflect, and recombine the atomic wave packets to form dual Mach–Zehnder (M–Z) atom interferometers. Each Raman pulse utilizes a retro-reflection structure and contains two frequency components (ω₁ and ω₂) with a difference of ∼6835 MHz to drive two-photon stimulated Raman transitions of atoms. Figure 1(a) illustrates the Raman laser generation methods. Three coherent Raman lasers are derived from one locked seed laser and undergo independent modulation, amplification, and frequency doubling processes. A three-channel microwave control system is used to drive the EOMs to generate the carrier and sidebands needed for the Raman lasers.

Figure 1(b) depicts the detailed architecture of the proposed microwave system. A 10 MHz OCXO (HSO14, Rakon) is phase-locked to a 10 MHz Rubidium atomic clock (FS725c, SRS) to achieve both short- and long-term stability. In addition, a 100 MHz oscillator (STD-APM1-10-100, Synchronization Technology) is referenced to the OCXO using an internal phase-locked loop to generate a low-phase-noise 100 MHz signal, which is then amplified to ∼+19 dBm to drive an NLTL (MLPNC-7100, MACOM). This induces a frequency comb with harmonics up to 20 GHz. The input power is carefully optimized to suppress the NLTL’s residual noise.^25,34 The addition of attenuators to the synthesizer ensures impedance matching between the NLTL and subsequent components.³⁴ 

Next, the 68th harmonic (6800 MHz) is selected using a bandpass filter (VBFZ-6260-S+, Mini Circuits) and amplified. This signal is used as a reference to phase-lock a Dielectric Resonator Oscillator (DRO, RDRO-5.0-7.9, RADITEK) tuned to 6800 MHz. The phase error between the signals is discriminated by a double-balanced frequency mixer (ZX05-73L+, Mini Circuits) followed by a filter. A customized servo controller is subsequently employed to convert the error into a control voltage using a home-built potential-induced degradation (PID) module. The loop bandwidth is optimized to be approximately 100 kHz, leveraging the phase noise advantages of both the DRO and NLTL output across different frequency offsets to achieve the best overall phase-locking performance. The locked DRO’s output is split into three paths, with each path being single-band mixed with an ∼35 MHz signal from a digital signal source to generate the final outputs.

We also implemented a servo loop to stabilize the power ratio between the two frequency components of the Raman laser by controlling the microwave output power. As shown in Fig. 1, the ratio is monitored by a Fabry–Perot cavity (FPC) and is compared with the set-point to generate a voltage feedback signal for the voltage-controlled attenuator (VCA, ZX73-2500-S+, Mini Circuits) in the microwave system. This approach mitigates fluctuations in light shifts caused by unstable microwave power and temperature variations, helping to improve the long-term stability of the atom interferometer.³⁹ 

Three microwave outputs with the same frequency satisfying the transition resonance condition are applied to drive the atom interferometer.

The phase of one of the output paths is swept in this mode. Consequently, the initial phase shift ϕ₀ and the overall interference phase are also swept, achieving an effect equivalent to that in the phase modulation mode.

The microwave drives of both π/2 pulses are operated with opposite frequency shifts, simulating the Doppler frequency shift induced by the rotation of the atom interferometer. This enables the electronic simulation or compensation of rotation for developing a closed-loop gyroscope.^40,41

The microwave control system is measured and characterized in terms of its absolute phase noise, relative frequency and phase stability between channels, and power ratio stability. It is finally employed to drive a spatial-domain atom interferometer to demonstrate its feasibility for precise atomic manipulation.

The absolute phase noise of the system was measured using a signal source analyzer (E5052B, Keysight). To reduce the analyzer’s measurement floor, the internal cross-correlation function was employed. This method utilizes two independent frequency references to separately measure the system’s phase noise, and through correlation analysis, it suppresses the analyzer’s internal noise, providing a more accurate measurement of the system’s phase noise.^25,28 The phase noise data across the entire measured frequency range are presented in Fig. 2. Only the phase noise of one channel’s output is shown, as the spectra of all three channels are nearly identical. The absolute phase noise of the 6835 MHz output is measured at −55, −100, and −124 dBc/Hz at frequency offsets of 1 Hz, 1 kHz, and 1 MHz, respectively. For frequency offsets below 10 Hz, the measured noise is dominated by the analyzer’s noise floor, which limits the precision of the measurement in these offset bands and represents the upper bound of the actual noise.

The phase noise spectrum of the 6800 MHz locked DRO is essentially identical to that of the final 6835 MHz output and is significantly higher than the noise from the 35 MHz generator. This makes it the system’s primary phase noise contributor. According to the phase-locked loop characteristics, the noise is primarily limited by the reference signal within the loop bandwidth, specifically the 6800 MHz harmonic generated by the NLTL. It is further determined by the phase noise of PLL-VCXO and the residual phase noise introduced in the NLTL chain. The degradation of phase noise between the 100 MHz PLL-VCXO and 6800 MHz signals is almost consistent with theoretical expectations (20lg68 ≈ 37 dB) at 1–100 Hz frequency offsets. However, the actual degradation is more severe at higher offsets, possibly due to additional residual noise from the NLTL and other amplifiers in the synthesizer.

To evaluate this, we adjust the frequency outputs of two channels to create a 2 MHz beat note. We measured this beat note using a phase noise analyzer (53100A, Microchip) with the same reference signal from the 10 MHz locked OCXO. This setup suppresses noise from the common reference while extracting relative noise originating from different microwave paths. The frequency deviation of the beat note (Δf) is shown in Fig. 3(a), with an RMS value of 9.6 μHz. The Allan deviation Δf/f₀ (where f₀ = 6836.9 MHz is the carrier frequency) is also presented in Fig. 3(b), characterizing the relative frequency stability between channels. The deviation decreases to ∼4 × 10⁻¹⁸ at an averaging time of 4000 s, corresponding to a remarkably small relative frequency deviation of 27 nHz.

The beat notes’ phase variation during the same measurement cycle is also illustrated in Fig. 3(c). It exhibits a periodic fluctuation with a period of about 10 minutes. The peak-to-peak value of the variation is ∼1.12 mrad, with an RMS value of ∼251 μrad over a timescale of 40 000 s. Correspondingly, the Allan deviation shown in Fig. 3(d) exhibits a bump around the averaging time of 10~1000 s, reflecting the phase instability between the channels. This variation is primarily caused by environmental temperature fluctuations in the laboratory, differences in devices between channels (e.g., power splitters, amplifiers, and mixers), and relative phase instability between the channels of the digital signal source. The RMS value of the latter fluctuation is measured as ∼83 μrad, indicating that it is a notable factor of the total variation. Further suppression of the relative phase instability can be achieved through closed-loop control of the phase of microwave outputs. This can be implemented using an external phase shifter or the external phase modulation capability of the digital signal source in the system.

As described in Sec. II, a servo loop is established to control the microwave system’s output power, aiming to eliminate the atom interferometer’s light shift. An FPC driven by a scanning voltage with a period of 1 s is used to obtain multiple sidebands of the Raman laser. The power ratio is calculated from the average of these sidebands to reduce measurement noise from the FPC. The feedback rate of a single microwave channel is 0.33 Hz, as the power ratios for the three channels are acquired and controlled sequentially. The Allan deviation, as illustrated in Fig. 4, demonstrates that the power ratio stabilization technique reduces the drift by ∼10-fold over an average time of around 800 s, facilitating the long-term elimination of the light shift in the atom interferometer.

The microwave control system drives our spatial-domain atom interferometer⁹ with dual continuous atomic beams in phase-sweeping mode, as described in Sec. II. In this mode, the phase of the first π/2 pulse is swept from −π to π (rad) at a rate of 10 Hz by modulating the phase of the first channel’s microwave output. Consequently, the term ϕ₀ in Eq. (1) is synchronously scanned in the interference phases of both AI-1 and AI-2. We employ the lock-in technique to extract these phases from the interference signals. A frequency shift of f_r = 200 Hz is applied to the π Raman pulse relative to both π/2 Raman pulses, generating a linear phase modulation ϕ_m = 2π · 2f_rt. A 2f_r = 400 Hz signal from the common reference is used to demodulate the signal with a lock-in amplifier. As shown in Fig. 5, the extracted interference phases (bottom) are swept along with the microwave scanning at the same rate of 10 Hz, inducing sinusoidal fringes in the dual interference signals (top). The amplitudes and phases correspond to each other, as they are simultaneously acquired from the same interference signal through quadrature demodulation. The phase difference between the dual interferometers primarily arises from the differing components in the interference phases, such as the inertial phase shift caused by acceleration and the initial phase shift introduced by the relative positioning of the Raman lasers’ reflecting mirrors. The resolution of the interference phase sweeping can reach ∼0.1°, limited by the scanning capability of the digital signal source. During this process, the interference phases are flexibly manipulated and directly read out by adjusting microwave parameters, without the need for additional phase devices.

In this paper, we propose an integrated microwave control system specifically designed for continuous spatial-domain atomic interferometers. Utilizing a low-phase-noise phase-locked loop based on a nonlinear transmission line and single-band mixing with a multi-channel digital signal source, the synthesizer produces editable 6835 MHz microwave outputs across three channels. The absolute phase noise of a single channel is measured as −55 and −124 dBc/Hz at 1 Hz and 1 MHz frequency offsets, respectively. We also evaluate the relative stability between channels, demonstrating an RMS value of 251 μrad for the relative phase fluctuations over 40 000 s. A servo loop was established to stabilize the power ratio between the frequency components of the Raman lasers. This microwave control system, characterized by low phase noise and multiple channels with independently adjustable parameters, provides a foundation for developing high-precision atom-interferometric inertial sensors in a continuous spatial-domain configuration.

In the future, we aim to further optimize the absolute phase noise of the microwave control system by analyzing and mitigating residual phase noise within the synthesizer. We will also work on controlling the microwave phase to stabilize the relative phase between multi-channels. Furthermore, we will design additional functions to enhance the adoption of the system as a standard component in various spatial-domain atom interferometers.

This work was supported by the National Natural Science Foundation of China (Grant No. 61473166).

The authors have no conflicts to disclose.

Weichen Jia: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Peiqiang Yan: Conceptualization (supporting); Data curation (supporting); Formal analysis (supporting); Methodology (supporting); Software (supporting); Validation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Yanying Feng: Conceptualization (lead); Data curation (supporting); Formal analysis (supporting); Funding acquisition (lead); Project administration (lead); Resources (lead); Supervision (lead); Writing – original draft (supporting); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.