A prototype differential atom interferometer for fundamental physics

Cooling sequence

The cold-atom apparatus used in this experiment has previously been described in refs. ^42,56. To prepare samples of cold ⁸⁷Sr, the atoms are first collected over 1.5 s in a blue three-dimensional MOT that uses the ¹S₀ → ¹P₁ transition at 461 nm and a field gradient of 3.5 mT cm⁻¹. Atoms that leak into the metastable ³P₂ manifold are recycled into the MOT using repump lasers at 679 nm and 707 nm. For efficient repumping of ⁸⁷Sr, frequency sidebands at 585 MHz and 487 MHz are applied to the 707-nm light using an electro-optic modulator to create frequency components near-resonant with transitions from all five hyperfine manifolds of ³P₂ (ref. ⁵⁷).

When the blue MOT is switched off, the atoms are captured in a red MOT operating on the ¹S₀ F = 9/2 to \({}^{3}P_{{1}}\,{F}^{{\prime} }=11/2\) transition at 689 nm, using a field gradient of 390 μT cm⁻¹. Sidebands at 1,463.265 MHz are applied to the 689-nm light using a resonant electro-optic modulator, such that the F = 9/2 to \({F}^{{\prime} }=9/2\) transition stirs the atoms between Zeeman sublevels of the ground state, thus mitigating losses into sublevels where atoms are weakly confined²⁸. During the first 220 ms in the red MOT, an intensity of 1,800I_sat is used for each of the six MOT beams, where I_sat = 3 μW cm⁻² is the saturation intensity of the 689-nm transition. To capture the wide range of Doppler-shifted atoms released from the blue MOT, a sawtooth-wave modulation is applied to the 689-nm light at a sweep frequency of 20 nm and a peak-to-peak sweep range of 6 MHz (ref. ⁵⁸). For the following 100 nm, while in the ‘narrowband’ red MOT, the sawtooth frequency modulation is switched off and the intensities of the six MOT beams are ramped linearly from 490I_sat to 40I_sat. To help support the atoms against the force of gravity, a seventh, unbalanced MOT beam—the ‘up’ beam—is introduced in the vertical direction during the narrowband MOT. The up beam is necessary for creating narrowband red MOTs below 100I_sat without causing significant atom loss. Upon completion of the narrowband red MOT, the atoms have a temperature of 2 μK and are compressed into a region comparable in size with the optical dipole trap.

Dipole trap and state preparation

Two crossed optical dipole traps, separated vertically by 1 mm, are formed by separate 2.5-W horizontal beams at 1,064 nm with horizontal and vertical 1/e² radii of 220 μm and 23 μm, respectively, crossed with a shared 840-mW vertical beam at 813 nm with 1/e² radii of 60 μm in both transverse axes. Overlapping with the top crossed dipole trap, a 4-mW transparency beam at 488 nm, detuned by 25 GHz from the 5s5p ³P₁ → 5s5d ³D₂ transition, is applied with a 1/e² radius of 40 μm to protect the atoms from scattered 689-nm light after they are loaded into the top crossed dipole trap region.

Immediately after the free-space red MOT stages described above, the dipole trapping beams, the transparency beam and repumpers at 679 nm and 707 nm are switched on; the red MOT is then held for 100 ms in a ‘top-trap loading’ stage, during which the bias magnetic fields, beam intensities and detunings of the red MOT are optimized to load the atoms into the upper of the two dipole traps. During the top-trap loading stage, the red MOT intensity is linearly ramped from 20I_sat to 4I_sat to steadily reduce the atom temperature. Next, to load the bottom optical dipole trap, the red MOT is released for 3 ms by switching off the 689-nm beams. During this time, the cold atoms already in the top trap are held in place, while the hotter atoms fall towards the bottom trap. While the atoms are falling, the vertical bias magnetic field is stepped such that the zero of the quadrupole magnetic field is close to the bottom dipole trap. After 3 ms of free fall, the red MOT beams are switched back on for 100 ms in a ‘bottom-trap loading’ stage using the same parameters as the top-trap loading stage, except for the different bias magnetic field. All but the hottest atoms in the top trap remain in the top trap during the bottom-trap loading stage, as they are protected by the 488-nm transparency beam against scattered 689-nm light.

After both dipole traps are loaded, the MOT beams are switched off, a horizontal bias field is applied and the trapped atoms are optically pumped into the stretched state M_F = 9/2 by applying a horizontal bias field of 38 μT and delivering a 20-ms pulse of circularly polarized light at 689 nm, resonant with the ¹S₀ F = 9/2 to ³P₁ \({F}^{{\prime} }\) = 9/2 transition. During the optical pumping, sawtooth-wave frequency modulation is applied to the 689-nm light at a rate of 30 kHz over a range of 6 MHz. Finally, all beams except the dipole trap are switched off, and the bias magnetic field is adiabatically ramped to the final field used for atom interferometry: 31 μT aligned with the linear polarization of the vertical 698-nm clock beam.

Velocity selection on the clock transition

The clock beam at 698 nm propagates vertically upwards through both dipole trap regions with a waist of 600 μm. The clock laser linewidth is verified against an independent cavity-stabilized laser to ensure that it is below 2 Hz before delivery of the light to atoms through an uncompensated 10-m fibre⁴². Clock spectroscopy sequences are carried out immediately after atoms are released from both dipole traps. The excitation fraction is detected using a 200-μs fluorescence pulse at 461 nm to detect the number of atoms in the ground state ¹S₀, which is followed by 3.5-ms repumping pulses at 679 nm and 707 nm and another 200-μs fluorescence pulse at 461 nm to detect atoms that are in the ³P₀ state after the interferometer sequence. Scattered light from each 461-nm spectroscopy pulse is gathered in separate exposures of an electron-multiplying charge-coupled device (EMCCD) camera (Andor iXon Ultra 897), and a separate EMCCD image without atoms present is used to subtract background counts.

At the maximum available clock power of 640 mW, a Rabi π-pulse time of 44 μs is measured. However, the clock transition was observed to have a peak excitation fraction of 0.3 and a Doppler-broadened linewidth of 60 kHz, which is considerably larger than the 20-kHz Fourier limit. To improve the fidelity of the Rabi pulses in the atom-interferometer sequence, a velocity selection procedure is used. The clock beam is pulsed on for 200 μs at 20 mW, which implements a π pulse that excites the slowest atoms to the upper clock state ³P₀. The atoms in the ground state are then pushed away using a 500-μs pulse at 461 nm, leaving only the slow atoms in the ³P₀ state to enter the interferometer sequence. After this velocity selection sequence, a resonant, 44-μs Rabi π pulse yielded a peak de-excitation fraction of 90%.

Clock atom interferometry

The clock atom interferometry consists of a sequence of three resonant pulses on the 698-nm clock transition, with pulse areas π/2 − π − π/2, a π-pulse time t_π = 44 μs and a dark time T = 200 μs between each consecutive pulse. For the data in Fig. 4, the phase of the clock light is always stepped deterministically during the dark times such that the phases of the first, second and third pulses are 0, ϕ and 4ϕ, respectively, with ϕ ranging from 0 to 2π in 100 steps in a randomized order. Each data point in the right-hand side of Fig. 4 is the result of 2 × 100 samples, interleaved between HLN and LLN samples. For the HLN samples, extra phase steps were applied during the interferometer dark times (Fig. 3). The HLN samples were drawn independently from a Gaussian distribution with a standard deviation of 4π rad and mean of 0 rad.

It is important to distinguish between the two types of randomization employed in this work. For both the LLN and HLN datasets, the clock laser phase is scanned deterministically through 100 values in randomized order; this scan-order randomization ensures that any spurious time-oscillatory signals, such as 50 Hz from room lights, are not aliased to look like apparent fringes. For the HLN dataset, we additionally applied large, uncorrelated phase jumps between shots, which fully randomize the absolute phase of each individual interferometer on a shot-by-shot basis. This per-shot phase randomization mimics the regime expected in long-baseline atom interferometers, where integrated laser frequency noise over multi-second interrogation times will produce phase excursions of many radians (see ‘Laser phase noise estimate for a kilometre-scale detector’ section). Under these conditions, a single atom interferometer retains no recoverable phase information, so this provides a stringent test of the noise rejection capability of differential measurements. The phase randomization fully masks the fringes in each individual interferometer but does not affect the measurement of the relative phase of the two interferometers.

Laser phase noise estimate for a kilometre-scale detector

The phase noise imparted onto the atoms by the laser can generally be calculated from the spectral density of the frequency fluctuations in the laser beam⁵⁹. In our prototype, the laser phase imprinted on each atom interferometer in one repetition of the interferometer sequence beginning at time t is approximately ϕ_laser = φ(t) − 2φ(t + T) + φ(t + 2T), where φ(t) is the time-dependent phase of the laser field oscillating as \(\cos (kz-{\omega }_{0}t+\varphi
(3)