High-visibility large-area atom interferometry
We demonstrate phase-stable high-visibility atom interferometry for up to 112 photon recoil momenta () path separation. The signal visibility of at the largest separation greatly exceeds the performance of earlier free-space interferometers. In addition to the symmetric interferometer geometry and Bose-Einstein condensate source, the robust scalability of our approach relies crucially on the suppression of undesired diffraction phases through a careful choice of atom optics parameters. The interferometer phase evolution is quadratic with number of recoils, reaching a rate as large as radians/s. We discuss the applicability of our method towards a new measurement of the fine-structure constant and a test of QED.
The precision of atom interferometry methods Cronin et al. (2009) can be fruitfully directed towards applications such as inertial sensing Peters et al. (2001); McGuirk et al. (2002); Durfee et al. (2006); Geiger et al. (2011), and fundamental physics such as tests of the equivalence principle Fray et al. (2004); Schlippert et al. (2014) and of quantum electrodynamics (QED) Bouchendira et al. (2011). Light-pulse interferometers, central to these endeavors, operate by using standing-wave optical pulses as beamsplitters and mirrors, which impart momenta in units of photon momentum to the atoms. Such interferometers gain in measurement sensitivity by the increase of the enclosed space-time area with momentum-boosting acceleration pulses Muller et al. (2009); Chiow et al. (2011). Phase-stable interferometers with large momentum separation are thus an overarching goal in atom interferometry.
Path separations with up to have been demonstrated Chiow et al. (2011), however interferometer phase-stability foo (2017a) was not observed due to technical noise from mirror vibrations. Vibration-immunity and resultant phase-stability can be recovered by operating two simultaneous interferometers in a conjugate or dual geometry Chiow et al. (2009, 2011). However the operation of such interferometers has been limited to Muller et al. (2008); Chiow et al. (2011); Asenbaum et al. (2017). Visibility of has been reported for in a guided atom interferometer McDonald et al. (2013), but such methods have to contend with the additional systematic effects from a confining potential.
In this paper we demonstrate phase-stable and scalable atom interferometry with very large momentum separation ( photon recoils), achieving 30% visibility at the largest . The resulting interferometer phase grows quadratically with momentum, reaching a rate as high as radians/s. Undesirable diffraction phases are theoretically and experimentally analyzed and kept under control by our choice of atom-optics parameters. When applied to a precision measurement for the fine-structure constant and test of QED, our interferometer demonstrates a favorable scaling. Our techniques for high signal visibility and suppressed diffraction phases for large interferometers should also positively impact other applications of atom interferometry including inertial sensing Peters et al. (2001); McGuirk et al. (2002); Durfee et al. (2006); Geiger et al. (2011), measuring gravity Chung et al. (2009); Altin et al. (2013); Sugarbaker et al. (2013) and the Newtonian gravitational constant Fixler et al. (2007); Rosi et al. (2014).
Our contrast interferometer (CI), (Fig. 1) operates on a Bose-Einstein condensate (BEC) atom source and consists of four atom-optics elements: splitting pulse, mirror pulse, acceleration pulses, and readout pulse. The splitting pulse places each atom into an equal superposition of 3 momentum states: , , and , referred to as paths 1, 2, and 3 respectively. The mirror pulse reverses the momenta of paths 1 and 3 from to . The acceleration pulses increase the momentum separation of paths 1 and 3 to a variable amount during two sets of free evolution times . After the final deceleration sequence brings the outer paths back to , all three paths overlap in space and form an atomic density grating whose amplitude varies in time. By pulsing on a traveling “readout” laser beam and collecting the Bragg-reflection off this matter-wave grating we obtain the characteristic CI signal:
Here is the signal envelope related to the coherence of the source and are the phases accumulated by the different paths. Relative to path 2, paths 1 and 3 accumulate phase from their kinetic energies and thus oscillates at a frequency of , where is the recoil frequency and is the mass of the atom. Importantly, effects from mirror vibrations on the optical standing wave phases cancel in this expression. Distinct from earlier realizations Gupta et al. (2002); Jamison et al. (2014), a dramatic enhancement of phase accumulation of is achieved in this work due to the scalable application of multiple acceleration pulses.
Our atom source consists of ytterbium (Yb) BECs of atoms which we prepare in a crossed-beam optical dipole trap operating at nm. After condensate formation, we decompress the trap to a mean frequency of Hz. To reduce the density and atomic interactions further, we allow ms time-of-flight after trap turn-off before beginning the interferometry sequence.
Our atom optics consist of diffraction beams near the (nm, kHz) intercombination transition and a readout beam near the (nm, MHz) transition, both of which are derived from our Yb laser cooling sources. The two diffraction beams are detuned from the atomic resonance by and counter-propagate horizontally to form a standing wave. For precise relative frequency control, each beam is derived from the first diffraction order of a 200 MHz acousto-optic modulator (AOM) driven by an Analog Devices AD9910 direct digital synthesizer. Each AOM output is passed through a polarization-maintaining single mode fiber to ensure a clean Gaussian mode and a stable linear polarization. The diffraction beams have a waist of mm, allowing lattice depths of up to 50 at the atoms for the available power. We stabilized the diffraction beam intensities by feeding back to the AOMs, keeping fluctuations in the diffraction pulse peak lattice depth to .
The splitting pulse has a width of 7s, and operates within the Kapitza-Dirac regime Gupta et al. (2001). The mirror pulse is a second-order Bragg pulse with Gaussian 1/e full-width 54s and peak lattice depth 14. Each acceleration pulse is a third-order Bragg pulse delivering of momentum, with Gaussian 1/e full-width 54s and peak lattice depth 26.6. We accelerate the outer paths sequentially as shown in Fig. 1(a,b) with successive third-order pulses separated by 130s. Although this sequential acceleration scheme breaks the symmetric form of the interferometer, the suppression of systematic effects from the symmetry of the three-path geometry Gupta et al. (2002); Jamison et al. (2014) is still largely retained if the time between acceleration pulses for paths 1 and 3 is short, as in our case. We use light at (399nm) for the readout beam which Bragg reflects at 44 degrees from the nm period matter wave grating to form the CI signal, and eliminates many signal-to-noise issues associated with stray reflections when using Bragg back-scattering at 556nm as in earlier work Jamison et al. (2014).
Figure 1(c-f) shows contrast readout signals for various values of , each of which is an average of multiple experimental iterations (shots). The phase stability of the interferometer is apparent in the high visibility of these fringes even for the largest () momentum splitting used in this work. To our knowledge, this is the highest momentum splitting in any atom interferometer that produces stable visible fringes. We attribute this capability to the inherent vibration insensitivity of the CI and the suppression of diffraction phases as discussed below. Note that these results are obtained without any active vibration isolation. To extract fringe visibility, we fit these averaged signals with the expression using the currently-accepted value of and a Gaussian envelope foo (2017b). Here is the time from the start of the readout pulse and is a vertical offset. The visibility of our signal (Fig. 2(a)) is defined as [(Max-Min)/(Max+Min)], where Max and Min are determined by our fitted values for the vertical offset and peak envelope amplitude. The signal vertical offset is due to the 7% spontaneous scattering probability from the readout pulse which is detuned by .
We analyze single experimental shots by extracting the amplitude and phase of the Fourier component at . Figure 2(a) shows how the amplitude, relative to the smallest interferometer, varies with . We find that our data is well described by a simple model based on finite acceleration pulse efficiency. The fraction of atoms remaning in path 1 or 3 that contribute to the final CI signal is , where is the efficiency per of our acceleration pulses, and is the total number of photon recoils from the acceleration pulses only, for path 1 or 3. A fit to the amplitude data using this model (Fig. 2(a)) returns , or 91% per third-order Bragg pulse, consistent with a direct measurement of our -pulse efficiency from absorption imaging of the atoms. This amplitude model together with the signal vertical offset from spontaneous scattering foo (2017c), yields a visibility model which captures all the main features of our data (dashed blue line in Fig. 2(a)). The phase-stability of the interferometer is characterized by the standard deviation of extracted single-shot phases (Fig. 2(b)), which is around rad for low , growing to rad for the largest . The calculated reduction in visibility due exclusively to these phase fluctuations (dotted green line in Fig. 2(a)) is negligible compared to the effect from the decreasing amplitude. We conclude that the current limitation in reaching even higher is the acceleration pulse efficiency, which can be improved in future setups.
For a free evolution time of , we define the signal phase at the start of the readout pulse to be the CI phase . Here contains a number of phase shifts that are common to interferometers of different , as well as contributions from systematic effects. The evolution of the CI phase with time is shown in Fig. 3 for various . The fitted slopes are in good agreement with the expected and may be used to precisely measure .
The quadratic scaling of the CI phase with is a distinct benefit for precision measurements, however it comes at the cost of a systematic effect from diffraction phases. This effect stems from momentum-dependent phase shifts during Bragg diffraction and can be significant for different interferometer geometries Buchner et al. (2003); Jamison et al. (2014); Estey et al. (2015); Giese et al. (2016). A critical gauge of the viability of the CI scheme is therefore the scaling of the diffraction phase with large momentum separation. An important element for our favorable scaling was the selection of acceleration pulse parameters that suppressed the diffraction phase contributions to the CI phase fluctuations, in addition to providing good atom optics efficiency.
The presence of an optical lattice modifies the atomic dispersion relations leading to momentum-dependent (and therefore path-dependent) phase shifts which affect the CI signal according to Eqn.1. We experimentally characterized the diffraction phase effect for our acceleration and mirror pulse parameters by varying the peak pulse intensity around the pulse condition (point) and observing the variation of the CI phase. As shown in Fig. 4, our observations agree well with a numerical model of the diffraction processes which is equivalent to those described in Jamison et al. (2014); Estey et al. (2015). The diffraction phase from the splitting pulse is negligible and the diffraction phases at the points for the Bragg mirror and acceleration pulses constitute a -independent offset to the CI phase. Thus, in standard interferometer operation we need only consider the variations of the diffraction phase around the point from the intensity fluctuations of our lattice. The black curve in Fig. 4 implies that 2% intensity variations (our upper bound) in the mirror pulse contribute 70 mrad to the CI phase standard deviation. The first acceleration pulse contributes about mrad (blue curve), while for larger the behavior converges to the dotted orange curve. For the largest interferometer there is less than mrad diffraction phase fluctuations per shot, including the effects from all the pulses. The operation of the interferometer with the chosen parameters for the acceleration -pulses is crucial for its scalability. For instance, pulse widths four times longer lead to order of magnitude greater diffraction phases, which we have verified both experimentally and theoretically.
The CI phase fluctuations can be improved by applying a shot-by-shot diffraction phase correction based on the correlation between the CI phase and the recorded diffraction pulse amplitudes. The correction is significant, reducing the phase noise to mrad for up to 88 (Fig. 2(b)). We also observe a small improvement in the visibilities of the corrected averaged data (Fig. 2(a)).
We have considered the effects from atomic interactions on our results. Our condensate source is in the Thomas-Fermi regime and its time-of-flight expansion can be analyzed accordingly Castin and Dum (1996); Jamison et al. (2011). The estimated phase fluctuations arising from the fluctuations in initial splitting asymmetry are far less than those observed in Fig. 2(b).
The remaining phase fluctuations in Fig. 2(b) are likely due to momentum transfer timing variations caused by shot-to-shot fluctuations in the time-varying Rabi frequencies of the acceleration pulses. This effect, which grows with , can be reduced in future setups with more robust stabilization methods for the diffraction pulses.
We now turn to the application of the large CI technique towards a photon recoil and measurement. The precision in can be written as:
where is the range of free evolution times over which the slope of is measured, is the uncertainty in the measured CI phase, and is the number of experimental shots. In our current CI setup, the free evolution time is constrained by the atoms falling out of the horizontally oriented diffraction beams, and is optimized to radians for , and ms. This represents an improvement of two orders of magnitude in total interferometer phase compared to our earlier CI realization Jamison et al. (2014). For these parameters, the maximum separation of interfering states is 1.5mm. The observed =250 mrad at (Fig. 2(b)) then gives a precision of in in 200 shots.
The interferometer cycle time is dominated by the BEC production time. While this is s for this work, we have demonstrated Yb BEC cycle times as low as s in our group Roy et al. (2016). Using s as a reasonable benchmark for longterm measurements, the above numbers scale to in in 10 hrs of integration time. We are initiating a new CI configuration with vertically oriented diffraction beams where the limitation on free evolution time is lifted and ms (keeping ) is possible in a cm vertical region. The above scaling then indicates a precision of in in 10 hrs foo (2017d). The corresponding precision in , which can be determined by combining and measurements of other fundamental constants Weiss et al. (1993) is a factor of two better. Together with potential improvements in and from better diffraction pulse control, this approach holds promise for a level measurement of and QED test Hanneke et al. (2008); Bouchendira et al. (2011); Aoyama et al. (2012).
In summary, we have developed a BEC-based high-visibility phase-stable atom interferometer with momentum splitting up to , which is more than three times larger in momentum separation than earlier realizations with phase-stable free-space interferometers. The robust scalability is acquired from the inherent vibration insensitivity of the interferometer geometry as well as diffraction phase control via our atom optics parameter choices. We demonstrated a quadratic growth of interferometer phase with momentum splitting and a favorable scaling of the performance towards a precision measurement of . The large momentum splitting interferometers developed in this work can also be adapted towards other applications such as gravity gradiometry McGuirk et al. (2002) and atom interferometer based gravitational wave sensing Dimopoulos et al. (2008). Finally, our results also represent an important advance in the use of alkaline-earth-like atoms for precision atom interferometry, where their ground-state magnetic field insensitivity and the presence of narrow intercombination transitions can be exploited Riehle et al. (1991); Jamison et al. (2014); Tarallo et al. (2014); Mazzoni et al. (2015); Graham et al. (2013); Hartwig et al. (2015); Norcia et al. (2017).
Acknowledgements.We thank Eric Cooper, Brendan Saxberg, and Ryan Weh for important technical contributions, and Alan Jamison for valuable discussions and a critical reading of the manuscript. This work was supported by the National Science Foundation.
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