Coriolis compensation via gravity in a matter-wave interferometer

Increasing the interrogation time in matter-wave interferometers is desirable for a range of applications, including the demonstration of macroscopic superpositions [1–3], inertial sensing [4–7], weak equivalence principle tests [8], and precision measurements of the fine-structure constant [9] and the gravitational constant [10]. Long interaction times can be achieved via particle slowing or trapping [11], moving to a free-fall environment [12–14], or increasing the interferometer baseline [3].

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 [17–19], 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 [20–22] 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].

1.1. Theory

Talbot–Lau interferometers [3, 25–31] require at least two gratings. The first grating, G₁, prepares transverse coherence in the beam, while G₂ acts as the diffraction element. The grating structure of G₂ 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, G₃, 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 $d.$

The phase of the interference fringes after traversing such an interferometer subject to a constant transverse acceleration $a$ is [32, 33]

$$\begin{eqnarray}&&\phi =\displaystyle \frac{2\pi }{d}a\displaystyle \frac{{L}^{2}}{{{v}}^{2}},\end{eqnarray} \tag{ 1 }$$

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 $\theta \ll 1,$ giving a transverse acceleration $a\approx 2{\rm{\Omega }}v+\,g\theta ,$ where the first term is due to the Coriolis effect and the second due to gravity. We can therefore express the total phase as

$$\begin{eqnarray}&&\phi =\displaystyle \frac{2\pi }{d}\left[\displaystyle \frac{2{\rm{\Omega }}{L}^{2}}{{v}}+\displaystyle \frac{g\theta {L}^{2}}{{{v}}^{2}}\right].\end{eqnarray} \tag{ 2 }$$

Here, ${\rm{\Omega }}={{\rm{\Omega }}}_{{\rm{E}}}\,\sin \,{48}^{\circ },$ where ${{\rm{\Omega }}}_{{\rm{E}}}$>0 is the rotational frequency of Earth and $48^\circ$ 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.

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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 $\rho ({v})$ with center velocity ${{v}}_{0}$ and spread σ, the velocity-averaged fringe visibility $A^{\prime}$ becomes

$$\begin{eqnarray}&&{A}^{{\prime} }=\left|{\int }_{0}^{{\rm{\infty }}}\rho \left(v\right)A\left(v\right){e}^{i\phi \left(v\right)}dv\right|.\end{eqnarray} \tag{ 3 }$$

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

$$\begin{eqnarray}&&R=\frac{{A}^{{\prime} }}{\left|{\int }_{0}^{{\rm{\infty }}}\rho \left(v\right)A\left(v\right)dv\right|}\end{eqnarray} \tag{ 4 }$$

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 $\theta =0$ 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.

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This can be compensated by choosing a grating roll θ such that the gravitational phase term makes $\phi$ 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 $R=A^{\prime} /A.$

For the $\theta =0$ case we expand $\phi$ to first order around ${{v}}_{0},$ giving

$$\begin{eqnarray}&&{R}_{{\rm{u}}{\rm{n}}{\rm{c}}{\rm{o}}{\rm{m}}{\rm{p}}}\approx \exp \left[-8{\left(\displaystyle \frac{\pi {\rm{\Omega }}\sigma {L}^{2}}{{\rm{d}}{{{v}}_{0}}^{2}}\right)}^{2}\right].\end{eqnarray} \tag{ 5 }$$

For the compensated case we choose a roll angle ${\theta }_{0}$ such that $\phi$ is constant to first order for ${v}={{v}}_{p},$ giving

$$\begin{eqnarray}&&{\theta }_{0}=-\displaystyle \frac{{\rm{\Omega }}\,{{v}}_{p}}{g}.\end{eqnarray} \tag{ 6 }$$

Expanding $\phi$ to second order around ${{v}}_{0}$ and setting ${{v}}_{0}={{v}}_{p}$ to achieve maximal compensation at each velocity gives

$$\begin{eqnarray}&&{R}_{comp}\approx {\left[1+{\left(\frac{4\pi {\sigma }^{2}{\rm{\Omega }}{L}^{2}}{d{v}_{0}^{3}}\right)}^{2}\right]}^{-\frac{1}{4}}.\end{eqnarray} \tag{ 7 }$$

Equations (5) and (7) are plotted as the dashed lines in figure 2 together with the numerically integrated values, showing reasonable agreement especially for small velocity spreads.

1.2. Experimental setup

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 $\theta \approx {\theta }_{0}$ as compared to the uncompensated case of $\theta \approx 0.$ A 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.

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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.

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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 ${\theta }_{0}$ shown in figure 4(b), with the extracted phases shown in figure 4(d). The value of ${\theta }_{0}$ 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 $\theta$ as free parameters. The best-fit value of ${\theta }_{0}$ 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 $\theta ={\theta }_{0}\pm 0.5$ mrad and $\theta ={\theta }_{0}\pm 1.0$ mrad can be similarly analyzed. The results are shown in figure 4(e). With ${\theta }_{0}$ fixed, the only free parameter is the arbitrary constant phase offset. The fits show excellent agreement with the data.

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.

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.

This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant Nr. 320694), the Austrian Science Fund (FWF) within program P-30176 and W1210-N25.