Coriolis compensation via gravity in a matter-wave interferometer

Matter-wave interferometry offers insights into fundamental physics and provides a precise tool for sensing. Improving the sensitivity of such experiments requires increasing the time particles spend in the interferometer, which can lead to dephasing in the presence of velocity-dependent phase shifts such as those produced by the Earth’s rotation. Here we present a technique to passively compensate for the Coriolis effect using gravity, without the need for any moving components. We demonstrate the technique with fullerenes in a long-baseline molecule interferometer by measuring the gravitational and Coriolis phase shifts and obtaining the maximum visibility one would expect in the absence of the Coriolis effect.

A major challenge facing ground-based interferometers is the Coriolis effect, which reduces the interference fringe visibility for slow non-monochromatic beams. Various approaches for compensating the Coriolis force or employing a Sagnac phase as compensation for another dispersive force have been proposed [15,16] and employed [17][18][19], for example via mechanical actuation of the gratings. The Coriolis compensation scheme presented here requires no moving components which may induce vibrations nor velocity-resolved measurements which require increased integration times, making it particularly well-suited for long-baseline molecule interferometry.
Gravitational and Sagnac phases were first observed in a series of neutron interferometry experiments [20][21][22] and later in atom and electron interferometers [23,24]. Here we use the gravitational phase induced by a tilt of our interferometer to compensate the phase due to the Earth's rotation. We demonstrate this compensation technique with the long-baseline universal matter-wave interferometer (LUMI), a two-meter long Talbot-Lau interferometer which is compatible with both supersonic atomic beams and slow molecular beams. Coriolis compensation is critical for reaching high interference visibilities in the LUMI experiment [3].

Theory
Talbot-Lau interferometers [3,[25][26][27][28][29][30][31] require at least two gratings. The first grating, G 1 , prepares transverse coherence in the beam, while G 2 acts as the diffraction element. The grating structure of G 2 is imprinted into the density of the molecular beam in the near field behind the second grating. It is common to employ a third grating, G 3 , as a transmission mask which is moved transversely to detect these fringes as a sinusoidal variation of the transmitted flux. We use a symmetric scheme, in which the gratings are spaced equidistantly by L and have equal periods d. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
The phase of the interference fringes after traversing such an interferometer subject to a constant transverse acceleration a is [32,33] where v is the longitudinal beam velocity. We define θ as the roll angle of the three gratings around the molecular axis, measured with respect to gravity (see figure 1). We consider q 1,  giving a transverse acceleration , where the first term is due to the Coriolis effect and the second due to gravity. We can therefore express the total phase as is the rotational frequency of Earth and  48 is the geographical latitude of our experiment. We neglect higher order effects such as the contribution of the centrifugal force or the vertical Coriolis shift due to the East-West velocity component.
Velocity-dependent phase shifts reduce the visibility of the interference pattern when averaged over the velocity distribution of the beam, which is typically broad in molecule interferometry. For a Gaussian velocity distribution r v ( ) with center velocity v 0 and spread σ, the velocity-averaged fringe visibility ¢ A becomes We neglect the negative tail of the Gaussian distribution in the normalization, which is a good approximation for the velocities and spreads we consider. The velocity dependence of the visibility amplitudes A, assuming fixed grating open fractions, is given in [34].
We define the reduced visibility as the ratio of the visibility in the presence of gravity and rotation to the visibility without any velocity-dependent phase shifts. If we set q = 0 by aligning the gratings to gravity, only the Coriolis shift contributes in equation (2). In figure 2 the numerically integrated R demonstrates the strong visibility reduction caused by the Coriolis effect alone.
This can be compensated by choosing a grating roll θ such that the gravitational phase term makes f nearly constant over the velocity range of interest. The roll angle which optimizes R is numerically determined for each velocity, with the improved visibility shown in figure 2.
Several approximations can be made to obtain analytic forms of the reduced visibility R and the optimal roll angle. First, we assume A to be constant over the velocity range of interest, such that = ¢

Experimental setup
In the LUMI experiment, the first and third gratings are silicon nitride nano-structures with period = d 266 nm, while the center grating is a phase grating formed by a back-reflected 532 nm laser. Such a mixed mechanicaloptical grating scheme is advantageous for observing interference of slow beams of highly polarizable molecules [28]. C 60 fullerenes were used for these measurements since they form a stable thermal beam and their optical polarizability at 532 nm is known [35]. Detection was via electron impact ionization followed by quadrupole mass selection and ion counting.
We studied the Coriolis compensation mechanism by measuring the relative contributions of gravity and the Coriolis force to the phase of the interference fringes as a function of velocity. This was done by modulating the beam with a periodic pseudo-random sequence [36] and cross-correlating the beam signal with the sequence to retrieve the time-of-flight distribution. The third grating was moved transversely and a time-of-flight measurement taken at each position step.
This procedure yields an intensity map of the flux as a function of both transverse grating position and time, such as those shown in figures 4(a)-(c). Each line-cut of the time axis contains a small spread of times determined by the resolution of the time-of-flight measurement. For typical parameters this spread is 3% fullwidth-at-half-maximum, small enough that velocity averaging over a given line-cut can be safely neglected in the data analysis.

Results
Coriolis compensation allows high interference visibility to be retained when interference data is integrated over all velocities in the beam. This is illustrated in figure 3, which shows the improvement in the integrated interference visibility when the gratings are rolled to q q » 0 as compared to the uncompensated case of q » 0. A Figure 2. The reduced visibility R in LUMI for uncompensated (blue, lower curves) and compensated (red, upper curves) schemes. Solid lines are calculated numerically for a beam of C 60 using equations (3) and (4), where R is maximized over θ at each velocity for the compensated case. Dashed lines are the approximate analytic expressions (5) and (7). The two pairs of lines for each scheme correspond to σ/v=0.1 and 0.2, with the latter corresponding to the lower R values in each case.
visibility of 21% for an optical grating power of 7 W would be expected in the absence of the Coriolis effect, while 15% was achieved with compensation, despite a large velocity spread of 0.44 v . 0 This reduced visibility R of 0.65 is in reasonable agreement with the predicted value of 0.73 for the given parameters. The lesser degree of experimental compensation compared with theory can likely be explained by a slight relative roll misalignment  to which the visibility is very sensitive when working with vertically extended beams [35], as were required for the large velocity spread.
To analyze the compensation systematically we performed a series of time-resolved interference scans as a function of θ. The roll was adjusted for the three gratings equally in order to maintain their relative alignment while introducing a gravitational phase shift.
A subset of these measurements is shown in figures 4(a)-(c), in which the gravitational phase shift, which dominates at large roll angles, is visible as a shearing of the contours. The power of the optical grating was held fixed at 7 W for these measurements.
To extract the phase shift as a function of velocity we take horizontal line-cuts of the time-resolved interference signal and fit a sine curve to each of these cuts. This is done first for the optimal-roll setting q 0 shown in figure 4(b), with the extracted phases shown in figure 4(d). The value of q 0 used here was determined by an optimization of the visibility as a function of roll, rather than calculating it via equation (6), since there was some uncertainty regarding the initial grating rolls with respect to gravity.
The turning point observed at about 170 m s −1 indicates optimal compensation near this velocity. It is also clear evidence that the observed phase shifts are not merely gravitational, as these would be monotonically increasing or decreasing with velocity. We fit equation (2) to the observed phase shifts with an additional constant offset and the roll q as free parameters. The best-fit value of q 0 is -0.96 mrad, which provides optimal Coriolis compensation at a velocity of 173 m s −1 , as estimated from equation (6).
The other roll settings of q q =  0. Similar experiments can also be used to measure local gravity or perform equivalence principle tests if g in equation (2) is left as a free parameter.

Conclusion
A passive scheme to compensate for the velocity-dependent Coriolis force is demonstrated in a two-meter long Talbot-Lau molecule interferometer. The scheme uses a grating roll offset to give a gravitational phase shift which compensates the Coriolis shift in the velocity band of interest. The technique provides a simple and robust means to compensate for the Coriolis effect in matter-wave interferometers with non-monochromatic beams.