Dynamical low-noise microwave source for cold-atom experiments Open Access

Atom interferometers belong to today’s most precise sensors with broad applications for navigation, geodesy, timekeeping, as well as fundamental research. An atom interferometric measurement relies on the determination of the phase difference between two atomic states. In many cases, these atomic states are hyperfine levels in alkali atoms. For magnetic field sensing or frequency measurements, the hyperfine transitions can be directly driven with microwave radiation. For inertial sensors, such as gravimeters, accelerometers, or gyroscopes, the hyperfine transitions can be driven by a two-photon optical transition with a microwave frequency detuning. In analogy to optical interferometers, microwave pulses driving these transitions act as beam splitters or mirrors between the atomic states. The phase noise of the microwave is directly converted into fluctuations of the measured interferometer phase. The rapidly increasing sensitivity in the field of atom interferometry poses increasing demands on the noise characteristics of the employed microwave sources. This is specifically the case for interferometric experiments with squeezed atomic samples,¹ which are not restricted by the Standard Quantum Limit (SQL) and are thus even more sensitive to the microwave’s phase and intensity noise.

In our experiments, we create entangled many-particle states in Bose–Einstein condensates by spin-changing collisions.^2–5 The microwave source serves four purposes: the initial preparation of all atoms in a specific hyperfine/Zeeman level, the dressing of Zeeman levels to activate spin-changing collisions, the manipulation of the states in an interferometric protocol, and the final rotation of the state for detection and state analysis. These tasks pose the following requirements on the design of the microwave source.

Variable frequency. For the preparation of the atoms, a sequence of microwave pulses with variable frequencies is desired.

Low phase noise. The high-fidelity manipulation of the atomic states demands a microwave with small phase fluctuations as the phase serves as the reference for all atomic measurements.⁶ In our atom interferometric applications, the fluctuation of the microwave’s phase during the interferometric cycle is directly converted into fluctuations of the quantity of interest. For the future, we would like to operate the interferometers with microwave pulses of 10 µs length and free evolution times of several tens of milliseconds such that a fluctuating phase is relevant from 10 Hz up to 100 kHz. We aim at an interferometric phase resolution close to the Heisenberg limit ΔΦ ≳ 1/N, with the number of employed particles N. The generation of entangled states is currently performed with N = 10⁴ atoms. A Heisenberg-limited resolution for such atom numbers, of course, requires the suppression of more technical noise sources. For example, we were able to improve the so far limiting detection noise to single-atom resolution for several hundred atoms.⁷ To ensure that the microwave phase noise is not limiting, it must be suppressed to a value below ΔΦ < 2π/N ≈ 600 µrad.

Low intensity noise. For tomographic measurements of the entangled states, the microwave should perform stable couplings between the employed spin levels. To resolve the expected patterns with single-particle resolution, the microwave must offer spin rotations with a relative intensity stabilization ΔI/I < 2/N ≈ 0.02% that is stable for a typical measurement time on the order of 1 h.

Fast dynamical adjustments. The spin preparation benefits from short microwave pulses that are executed closely after each other. Short pulses bear the advantage of a low sensitivity to small microwave detunings due to an increasingly broad peak in Fourier space. This is, however, not sufficient for an advantageous spectral distribution. Short pulses can lead to many side peaks in the Fourier spectrum if they are simple box pulses. This may lead to an unwanted population of neighboring levels. The entire width of the spectrum can be drastically reduced by applying tailored pulse shapes instead of rectangular pulses. Pulse shaping requires a microwave amplitude modulation below the microsecond scale, and the short pulse durations can only be realized by a source with sufficient microwave power. Furthermore, some interferometry and state tomography schemes are based on a fast and reproducible adjustment of the microwave’s phase. The combined modulation of frequency, phase, and intensity enables the application of composite pulses,⁸ which can be designed to suppress the sensitivity to various technical noise sources.

Intensity-stable dressing field. Finally, we wish to apply an independent microwave dressing field during the experimental sequence to achieve precise control of individual energy levels. This application is particularly sensitive to fluctuations of the microwave field’s intensity because the desired energy shift depends linearly on the power of the microwave signal. Hence, a stable microwave power is desired for quasi-adiabatic state preparation by ramped microwave dressing as we use it for the generation of highly entangled twin-Fock states.^9,10 Typically, for a 1 G magnetic field, we shift the |F = 1, m_F = 0⟩ level via microwave dressing by 72 Hz. For high-accuracy adiabatic generation, we require maximal frequency fluctuations of 10 mHz over typical ramping times of a few seconds, corresponding to microwave intensity fluctuations of 10 mHz/72 Hz = 0.014%. In summary, our experiments require a microwave source with amplitude, frequency, and phase modulation in the microsecond range, low phase fluctuations, and an independent, stable microwave dressing field.

There exist commercial frequency generators in the microwave regime that allow for a dynamical adjustment of frequency and amplitude. However, these systems typically lack an independent adjustment of the phase or suffer from slow update speeds in the millisecond range. There are important scientific developments toward microwave sources with minimal phase fluctuations, which serve as local oscillators for microwave clocks.^11–15 However, for these applications, a fast dynamical adjustment is not intended.

Publications reporting on low-phase-noise microwave sources that are designed for a fast dynamical adjustment of frequency, phase, and amplitude are rare. Chen et al.¹⁶ presented a microwave source for spin-squeezing experiments in ⁸⁷Rb based on a self-built nonlinear transmission line that is phase-locked to a 10 MHz atomic clock. Controllability of parameters in the range of 24–32 µs is achieved by a single-sideband modulator and a direct digital synthesizer (DDS). The obtained integrated phase noise amounts to 3 mrad in the 10 Hz to 100 kHz bandwidth. Morgenstern et al.¹⁷ demonstrated a microwave source at 1.8 GHz, controlled by a field programmable gate array (FPGA), to realize a microwave dressing in sodium Bose–Einstein condensates. The source features parameter changes within 4 µs. The phase stability is not specified as it is not critical for their application.

Here, we present a microwave source with a low integrated phase noise of 480 µrad in the 10 Hz to 100 kHz bandwidth. The microwave frequency is derived from a commercial 7 GHz source and a DDS, which controls frequency, amplitude, and phase with high accuracy. An interplay of components from the open-source hardware family Sinara¹⁸ and the control system ARTIQ¹⁹ enables single-parameter updates within 712 ns. The setup does not comprise self-built components and fits into a 19 in. rack. The source consists of two independent DDS-controlled channels, which can be applied for two-photon transitions or for a background dressing field. The described microwave source thus offers the ability to control atomic spins with highest fidelity on short time scales and the simultaneous application of a dressing field.

This article is organized as follows: Section II covers the system design and the installed components. In Sec. III, we characterize the system regarding phase noise and dynamic features. Finally, we summarize our findings and outline the future application of the system in our spin-squeezing experiments.

Figure 1 shows a schematic of the final design of our microwave source, and Fig. 2 shows the corresponding picture. The basic idea is to combine the low phase noise of a multiplied crystal oscillator with the dynamic properties of an FPGA-controlled DDS. Our design presents an easy-to-assemble setup with commercial components²⁰ and very low phase noise, and our implementation opens up highly dynamical features.

An ultra-low-phase-noise oscillator Wenzel MXO-PLD is used as a local oscillator. It provides a stable microwave frequency at 7 GHz through integrated multiplication of a fundamental 100 MHz oscillation, which is accessible through an additional output. The oscillator also has the option of an external reference, which is currently not used in our setup.

For the dynamic parameter change of the radio frequency component, we employ the commercially available DDS module Urukul.²¹ This module contains four Analog Devices AD9910²² DDS chips and associated radio frequency components for adequate filtering, variable attenuation, and switching. Control over the DDS module is achieved by the FPGA module Kasli. A digital I/O module DIO_BNC offers the possibility for triggered execution and controlling additional components, such as microwave switches. All three modules are part of the Sinara ecosystem and can be bought commercially as a preconfigured device. The DDS module features four independently controllable radio frequency channels with an adjustable frequency of up to 400 MHz. All channels share a common reference of 1 GHz, given by the local oscillator’s 100 MHz multiplied by 10. The 1 GHz frequency reference allows for direct clocking of the DDS channels. Compared to the case of indirect clocking, where the 1 GHz clock is generated by internally multiplying the local oscillator’s 100 MHz in the DDS module, we measured an improved phase noise performance of 100 µrad. For the microwave source, two of these DDS outputs operate the two individual microwave output stages. In addition to frequency setting, the amplitude and also the phase offset for absolute phase control can be defined (Table I).

For our ⁸⁷Rb atoms, we have to apply frequencies at f_Rb = 6.835 GHz with a tuning range of ±5 MHz to account for the employed magnetic fields. In both output channels, the frequency is reached by mixing a shared local oscillator (LO) signal at f_LO = 7 GHz with an individual DDS frequency of f_DDS = 165 ± 5 MHz. The lower sideband after the mixing presents the desired resonant frequency at f_Rb = f_LO − f_DDS. The two output stages for pulses and dressing include final amplifiers to obtain powers of 40 and 10 W, respectively.

For the mixing process, a Marki Microwave SSB-0618 single sideband mixer and a Mini Circuits ZMX-8GLH mixer are used for the pulse path and the dressing path, respectively. The single sideband mixer in the more critical channel is used to lower the unwanted peaks at the local oscillator frequency f_LO and the upper sideband at f_LO + f_DDS. A narrow Wainwright WBCQV3 bandpass filter with a passband of 20 MHz and 50 dB attenuation for frequencies of ±250 MHz from the center frequency enables the reduction of the LO and the upper sideband level in both channels.

Directly after mixing, the pulse and dressing paths measure maximally −7.5 and −6.5 dBm, respectively. Low-noise pre-amplifiers Mini Circuits ZX60-83LN-S+ raise the signal to the desired 0 dBm input for the high-power amplifiers.

Because the internal power switch of the DDS channel only offers a maximal attenuation of 31.5 dB, the source is equipped with additional power switches Mini Circuits ZFSWA2-63DR+ with 25 ns rise/fall time and an attenuation of 40 dB. The employed switches—designed for frequencies up to 6 GHz—are a cost-effective solution, coming with disadvantages of larger insertion loss and reflections. The insertion loss is compensated by the pre-amplifiers. Isolators prevent unwanted back reflections, in general, and are installed in front of the switches and the antenna, as well as behind the local oscillator.

For the amplification of the pulse path, a water-cooled 40 W amplifier Microwave Amps AM43 is employed. The amplifier in the dressing path is a Kuhne KU PA 640720-10 A with an output power of 10 W. The pulse path contains a −30 dB dual directional coupler RF-Lambda RFDDC2G8G30 for monitoring the output power and the power that is reflected from the antenna. A −20 dB directional coupler MCLI C39-20 after the dressing amplifier allows for monitoring the dressing power. For addressing the atoms, the microwave frequency is coupled to free space by an impedance-matched, open-ended waveguide. In Sec. III, the performance of the described microwave source is evaluated.

First of all, we examine the spectrum of the microwave source to quantify the suppression of disturbing frequencies that could drive unwanted transitions, resulting in decreased efficiency of the state preparation. Figure 3 shows the spectrum of our microwave source for both the dressing path and the pulse path with a chosen output frequency of 6.835 GHz. The spectrum was recorded with a Rohde&Schwarz FSWP8 phase noise and spectrum analyzer.

The unwanted frequency contributions close to the main peak are caused by the DDS, while the two peaks at 7 and 7.165 GHz are residuals of the local oscillator frequency and the upper sideband of the mixing process. Both of these peaks are suppressed by the narrow bandpass filter: For the pulse path, the local oscillator and the upper sideband peak are suppressed by 69.8 and 87.9 dB, respectively. For the dressing path, the suppression is 79.5 and 70.3 dB due to the different choice for the frequency mixer.

Because of the bandpass filters, the peak power slightly varies with the frequency of the microwave source by about ± 0.2 dB in the relevant frequency range of 6.835 GHz ±  5 MHz and ± 1 dB for 6.835 GHz ±  25 MHz. The small effect in the relevant range can be calibrated, and the power can be adjusted by the ARTIQ software.

Furthermore, we evaluate the phase noise $L(f)=12Sφ(f)$⁠, where S_φ(f) is the one-sided spectral density of phase fluctuations and f is the offset frequency from the carrier.²⁴ A low phase noise is desired for a high-fidelity state preparation and manipulation (Sec. I). Figure 4 shows the phase noise contributions for the microwave source, measured for an output frequency of 6.835 GHz with the Rohde&Schwarz FSWP8. The colors of the lines correspond to different measurement points in the system, as visualized in Fig. 1.

The phase noise of the two microwave output signals is very similar, although they follow different amplification paths. The overall phase noise is limited by the DDS for frequencies from 2 kHz up to 20 kHz and by the local oscillator for all other frequencies. It is below −120 dBc/Hz, above 350 Hz, and below −130 dBc/Hz for frequencies greater than 2.5 kHz and approaches a floor of −140 dBc/Hz at high frequencies. The small peak at 16 kHz stems from residual DDS noise²⁵ and may originate from power supply noise. For both paths, we obtain a favorably small integrated phase noise of ΔΦ = 480 µrad in the important range of 10 Hz to 100 kHz (Sec. I), which provides the central figure of merit of the constructed microwave source.

With a reference connected to the local oscillator, the long-term stability could be improved, and therefore, the phase noise for small frequency offsets could be decreased.

The amplitude (AM) noise S_a(f) of the presented microwave source is also measured and shown in Fig. 5. Because the amplitude noise is important also for very low offset frequencies and measurements in that range have a very long duration, high sensitivity limits had to be accepted as a consequence. Hence, the amplitude noise of the microwave source can only be given as an upper bound. For the bandwidth of 278 µHz (measurement duration of one hour) to 100 kHz, it evaluates to an integrated AM noise of 0.015% and, therefore, an intensity noise of ΔI/I = 0.03%. It is thus in the desired order of magnitude to achieve Heisenberg-limited resolution.

The dynamics of the microwave source is controlled by the ARTIQ system, which is connected to a computer via an optical fiber and a LAN switch. The commands are implemented on the computer in the Python programming language but compiled and executed on the FPGA hardware, thereby allowing for a minimal timing resolution of 4 ns. The code generating our experiments is available online.²⁶ 

The AD9910 DDS chip in the Urukul module also provides a 1024 × 32 bit random-access memory (RAM) to retrieve and output different waveforms that are designed by the user. It is possible to program frequency, amplitude, phase, or polar (amplitude and phase) nonlinear ramps with a step width of 4 ns into one out of eight profiles. For our purposes, time-dependent amplitude functions are used to reduce the effective Fourier width (Fig. 6) and the coupling to unwanted transitions. For the dressing scenario, adiabatic power ramps ensure that the system remains in the desired dressed state.

The phase of the microwave signal is deterministically referenced to the local oscillator phase in a coherent way. This allows for composite pulse techniques, where one single pulse is replaced by a sequence of pulses with different phases, amplitudes, and durations to reduce the impact of various noise sources.^8,27 Figure 7 shows a so-called Knill sequence²⁸ that reduces sensitivity to both intensity and frequency fluctuations. Besides the phase setting, it is advantageous for this technique to execute the pulses in quick succession to avoid unwanted fluctuations during dead times. Updates of parameters such as frequency or phase require 712 ns in our implementation. This corresponds to a minimal pulse duration of a few μs when two totally different pulses should be executed in direct succession. Start and end times of pulses on different DDSs have to be equal or at least differ by 424 ns if they are on the same Urukul module. Two separate modules could, however, be operated completely independent without this delay. Figure 8 finally shows an amplitude-shaped pulse where the phase of the radio frequency is altered halfway using one common profile instead of two distinguished pulses. This example demonstrates the highly dynamical phase control.

In summary, we have presented a microwave source with sub-µs update speed and adjustable phase. Composite pulses and nonlinear ramps can be delivered while maintaining a low integrated phase noise of 480 µrad. The experimental sequences are implemented on a computer with Python and the ARTIQ software, allowing for a nanosecond timing resolution. The microwave source is based on commercially available components and adaptable to other atomic species than ⁸⁷Rb. The design with two independently controllable paths opens up various possibilities, such as the addressing of two-photon processes and the Rabi coupling to a dressed state.

We thank É. Wodey for his support regarding the ARTIQ implementation. We acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project-ID 274200144-SFB 1227 DQ-mat within the project A02, under Germany’s Excellence Strategy-EXC-2123 QuantumFrontiers (Grant No. 390837967) and from the European Union through the QuantERA Grant No. 18-QUAN-0012-01 (CEBBEC). F.A. and B.M.-H. acknowledge support from the Hannover School for Nanotechnology (HSN). B.M.-H. acknowledges the use of the QuTiP²⁹ Python toolbox for Bloch sphere illustrations.

The authors have no conflicts to disclose.

Bernd Meyer-Hoppe: Conceptualization (lead); Investigation (lead); Software (supporting); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Maximilian Baron: Investigation (supporting); Software (lead); Visualization (supporting); Writing – review & editing (equal). Christophe Cassens: Investigation (supporting); Writing – review & editing (equal). Fabian Anders: Conceptualization (supporting); Supervision (supporting); Writing – review & editing (equal). Alexander Idel: Conceptualization (supporting); Writing – review & editing (equal). Jan Peise: Conceptualization (supporting); Supervision (supporting); Writing – review & editing (equal). Carsten Klempt: Conceptualization (lead); Funding acquisition (lead); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

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