# Shaping the distribution of vertical velocities of antihydrogen in GBAR

## Abstract

GBAR is a project aiming at measuring the free-fall acceleration of gravity for antimatter, namely antihydrogen atoms (\(\overline{\mathrm {H}}\)). The precision of this timing experiment depends crucially on the dispersion of initial vertical velocities of the atoms as well as on the reliable control of their distribution. We propose to use a new method for shaping the distribution of the vertical velocities of \(\overline{\mathrm {H}}\), which improves these factors simultaneously. The method is based on quantum reflection of elastically and specularly bouncing \(\overline{\mathrm {H}}\) with small initial vertical velocity on a bottom mirror disk, and absorption of atoms with large initial vertical velocities on a top rough disk. We estimate statistical and systematic uncertainties, and we show that the accuracy for measuring the free fall acceleration \(\overline{g}\) of \(\overline{\mathrm {H}}\) could be pushed below \(10^{-3}\) under realistic experimental conditions.

## 1 Introduction

Gravitational properties of antimatter have never been measured directly. A promising experimental method to do so consists in producing sufficiently cold antihydrogen atoms (\(\overline{\mathrm {H}}\)) and timing their free fall in the Earth’s gravity field. This approach is being pursued by AEGIS [1], ATHENA-ALPHA [2], ATRAP [3] and GBAR [4] collaborations.

In order to get the highest accuracy for measuring the free-fall acceleration \(\overline{g}\) of \(\overline{\mathrm {H}}\), one has to cool atoms down to low temperatures and to measure, or at least to deduce from design and calculations, the initial velocity distribution. We discuss here the method proposed by Walz and Hänsch [5], which is used in the GBAR project to reach very low temperatures: \({\overline{\mathrm {H}}}^+\) ions are trapped and cooled down to the lowest quantum state in a Paul trap and \(\overline{\mathrm {H}}\) is then produced by photo-detaching the excess positron. The photo-detachment pulse is the START signal for the free-fall timing measurement, while the STOP signal is provided by the annihilation of \(\overline{\mathrm {H}}\) atoms on a detection plate placed at a height \(H\) below the center of the ion trap.

The precision of this measurement depends crucially on the dispersion of vertical velocities before the free fall, which corresponds to the residual kinetic energy of the atoms after the cooling process. The aim of the present paper is to propose a new filtering method to further reduce the initial distribution of the vertical velocities and thus improve the accuracy in the measurement of \(\overline{g}\).

In Sect. 2 we justify our choice of characteristic values for the spatial localization of the initial atomic cloud by considering the spreading of the freely falling wavepacket of \(\overline{\mathrm {H}}\) in the gravitational field. We describe in Sect. 3 the new method for shaping vertical velocities of \(\overline{\mathrm {H}}\) in the quasi-classical approximation, and we show in Sect. 4 that the improvement of accuracy due to the velocity selection overcomes the degradation associated with the decrease of the statistics. We then present in Sect. 5 a quantum-mechanical description of the experiment in order to validate the quasi-classical estimations of the preceding sections. In Sect. 6 we list possible systematic effects and show that they scale down compared to those in the case of unrestricted free fall of \(\overline{\mathrm {H}}\). We then deduce the accuracy which could be reached on the measurement of \(\overline{g}\) under realistic experimental conditions.

We neglect throughout this paper systematic effects related to the energy-dependent probability of quantum reflection of \(\overline{\mathrm {H}}\) from the detection plate [6]. The atomic recoil in the photo-detachment process induces an additional velocity dispersion which is discussed in the last section on systematic effects.

## 2 Spreading of a freely falling wavepacket

After their release from the trap at time \(t=0\), atoms start falling freely in the Earth’s gravity field until they reach the detection plate placed at a height \(H\) below the center of the trap. The time of fall is measured as the delay \(t\) from their release to their annihilation on the detection plate. The acceleration of gravity \(\overline{g}\) for antihydrogen is then deduced from the distribution of these fall times. This acceleration \(\overline{g}\) for antihydrogen is related to the analog quantity \(g\) defined for hydrogen by \(\overline{g}=Mg/m\), where \(M\) is the gravitational mass of \(\overline{\mathrm {H}}\).

We now discuss the distribution of free-fall times, assuming for simplicity that this distribution is determined by initial dispersions of the vertical velocity and position (other sources of uncertainty negligible). If the initial quantum state is poorly localized (large values of \(\zeta \)), then the spread of the fall times is too large, because of the initial dispersion of height. In the opposite case where the wavepacket is too localized (small values of \(\zeta \)), the spread of the fall times is too large, because of the initial dispersion of the vertical velocity. An optimum localization of the initial quantum state should be found as a compromise between these two limiting cases.

## 3 Shaping the distribution of vertical velocities of \(\overline{\mathrm {H}}\) in GBAR

The current design for GBAR is a classical free-fall experiment which aims at an accuracy of the order of 1 % [4]. With a quantum detection technique, one could get a significantly higher precision, in analogy to spectroscopy [7] or interferometry [8] of near-surface quantum states [9] of ultracold neutrons (UCNs) [10, 11]. However, these techniques require high energy resolution and sufficient statistics [12, 13]. The method that we propose in this paper is an intermediate step in this direction which is less precise than the full quantum detection technique but allows for better statistics and simpler design.

In the zone between the two disks, atoms with sufficiently small vertical velocities bounce on the bottom mirror disk due to the high efficiency of quantum reflection in the Casimir–Polder potential [6]. If the top surface of the mirror disk is flat, smooth and horizontal, the horizontal velocity component as well as the total energy of the vertical motion do not change and atoms thus pass through the shaping device with high probability. This last statement would be precisely valid for ideal quantum reflection from the mirror surface; otherwise corresponding corrections have to be taken into account (more discussion below). On the other hand, atoms with large vertical velocities rise in the Earth’s gravity field to the height of the rough surface of the top disk and scatter non-specularly on this surface. As this scattering mixes horizontal and vertical velocity components, it leads to rapid loss of scattered \(\overline{\mathrm {H}}\) through annihilation on the top or bottom disk.

A few remarks are useful at this point: (1) the shaping device has to be coupled with the Paul trap (not shown on the figure); this point is not discussed in this paper except for the role of the openings left in the center of the disks for operating the Paul trap; note that the disks may consist of several sectors not covering the complete \(2\pi \) horizontal angle in order to include the Paul trap in the overall design; (2) annihilation events are assumed to be detected with position-sensitive and time-resolving detectors; this will allow one to account for the time spent in the shaping device (see below); (3) due to the cylindrical symmetry of the device, all atoms with small enough vertical velocity components and any value and direction of the horizontal velocity component can pass through it with high probability.

We want also to stress that the time \(T\) appears as a systematical delay in the free-fall timing experiment so that its knowledge is crucial for accuracy. Here the fact that annihilation event detectors are position sensitive is important. Measuring the horizontal distance \(L\) between the initial spot and the detection point indeed gives the actual horizontal velocity of the atom \(L/T_{\mathrm {tot}}\) with \(T_{\mathrm {tot}}\) the time between escape from the trap and annihilation on the detector and allows one to correct the timing measurement for the time spent in the shaping device \(T=R T_{\mathrm {tot}}/L\).

At the exit of the shaping device, the height lies in the interval \(\left[ H,H+h\right] \), while the vertical velocity lies in the interval \(\left[ -\Delta v,+\Delta v\right] \). As discussed in the next section, this affects the resolution of the timing measurement in the same manner as the dispersion of velocities did affect the free-fall measurement discussed in Sect. 2. In order to optimize the various parameters, in particular the value of the radius \(R\), we have to simulate the whole experiment, that is, the photo-detachment, the passage through the shaping device, the free fall from its output slit to the detection plate, the timing of annihilation events, and the correction from the time spent in the device. In the present paper, we use simpler arguments to estimate the resulting accuracy of the measurement.

## 4 Estimation of statistical uncertainty

- 1.
Equation (4.4) shows that \(h\) should be smaller than \(\approx h_{\max }/2\pi \) for the shaping device to improve the resolution of the experiment. We choose as an example \(h=1\) mm, so that the statistics is \(N\approx 3.3\times 10^3\). The opening radius has to be smaller than \(r_{\max }\simeq 3.2 \sqrt{\varepsilon }\) mm and the disk radius should be larger than \(R_{\min }\simeq 13 \sqrt{\varepsilon }\) mm. The statistical accuracy is then \(\Delta \overline{g}/\overline{g}\sqrt{N} \approx 2.0\times 10^{-3}\). Note that, for a conducting mirror and a maximal vertical velocity \(\sqrt{2 g h}\approx 0.14\) m/s, the reflection probability for an atom is \(78\,\%\) [6]. To simultaneously improve the resolution and reduce losses from annihilation on the bottom mirror, we move to smaller values of \(h\).

- 2.
For \(h<50\)\(\upmu \)m, the atom flux through the slit can no longer be evaluated from classical arguments and the quantum behavior of \(\overline{\mathrm {H}}\) in the slit between the disks has to be taken into account [7, 14, 15]. At the boundary \(h=50\)\(\upmu \)m, the statistics is \(N \approx 7.3\times 10^2\) and the statistical accuracy is \(\Delta \overline{g}/\overline{g}\sqrt{N} \approx 1.0{\times }{10}^{-3}\). The opening radius has to be smaller than \(r_{\max }\simeq 0.7 \sqrt{\varepsilon }\) mm. Note that the reflection probability for an atom with the maximal velocity \(\sqrt{2 g h}=3.1\times 10^{-2}\) m/s is 94 % for a conducting mirror.

- 3.
For \(h< 20\,\upmu \)m, only atoms in the lowest quantum state can pass through the slit. The reflection probability approaches unity in this case which also corresponds to the highest accuracy for the free-fall timing measurement. This quantum limit is analyzed in Sects. 5.2 and 5.3.

## 5 Quantum-mechanical description

We now perform a quantum-mechanical description of the experiment, which will turn out to reproduce the main features and estimations of the quasi-classical treatment given above.

### 5.1 Free fall of a wavepacket

### 5.2 Gravitational quantum states in the shaping device

As a quantitative illustration, Fig. 3 shows the probability of transmission for atoms in the two lowest gravitational states, \(\Psi _1\) and \(\Psi _2\), when the length of the shaping device is \(R-r=5\) cm and the roughness amplitude of the top absorber is 1 \(\upmu \)m. A slit size \(h=24\)\(\upmu \)m provides 72 % transmission probability for the first state but only 0.3 % for the second state. This implies that a nearly pure ground state or a superposition of a few lowest gravitational states can be prepared by a suitable choice of the parameters of the shaping device.

### 5.3 Free-fall experiment after the velocity shaping

The output of the velocity shaping device is a superposition of gravitational quantum states \(\Psi _n\), determined by the propagator (5.12) calculated for a time \(t=(R-r)/v_\mathrm {hor}\) for an atomic horizontal velocity \(v_\mathrm {hor}\). This shaped superposition then falls freely to the detection plate so that the time distribution of annihilation events depends on the properties of the shaped state. We stress again at this point that this presupposes that the time \(R/v_\mathrm {hor}\) spent in the shaping device, and before its entrance, is corrected in the data analysis, \(v_\mathrm {hor}\) being deduced from the position of the annihilation event.

The preceding argument disregards the coherence between the components \(\Psi _n\) in the superposition prepared by the shaping device. This approximation can be justified qualitatively by considering that the effects of coherence are washed out in the averaging associated with free-fall propagation as well as the horizontal velocity dispersion. However, it cannot be considered as exact, and it will have to be confirmed by more precise simulations, to be published in forthcoming papers.

## 6 Estimation of systematic effects

For our proposal to be useful as an improved option of the GBAR measurement, one must ensure that there are no large systematic uncertainties which could contribute at a level comparable to the estimated statistical uncertainty of \({\ 10}^{-3}\).

We first examine the additional velocity dispersion caused by the photo-detachment recoil. As discussed in [23], the vertical velocity dispersion due to the absorption of the photon and the positron emission can be kept small (\(\sim 0.5\) m/s) by using a horizontal polarized laser beam with an energy tuned at around \(\Delta E\approx 10\)\(\upmu \)eV \(\approx 0.1\) cm\(^{-1}\) above threshold. The photo-detachment cross section near threshold follows the Wigner law and can be estimated by using the available information in the literature to be \(\sigma =6.8\times 10^{-26} (\Delta E/1 \text { cm}^{-1})^{3/2}\approx 2\times 10^{-27}\) m\(^2\) [24, 25, 26, 27]. With a \(P=1\) W laser beam tuned close the threshold energy \(E_T=6,\!083\) cm\(^{-1}=0.76\) eV focused on an area \(A=10\) \(\upmu \)m \(\times 10\) \(\upmu \)m covering the Paul trap center, the photo-detachment rate is \(R=\sigma P/A E_T=130\) s\(^{-1}\).

In GBAR, antihydrogen ions can be produced only every 110 s, the ejection period of the antiproton decelerator at CERN. This time is sufficient to photo-detach the excess positron with high efficiency. The method is to illuminate the ion during a short enough time so as to define the start time with high precision, at a low enough repetition rate so that in case of successful photo-detachment, the free fall is completed before the next laser shot. For example, since the free-fall time on 30 cm is only 250 ms, laser shots of 100 \(\upmu \)s duration at a repetition rate of 2 Hz during 100 s allows the start time to be known with enough precision (\(4\times 10^{-4}\)), it also avoids ambiguity on identifying the successful shot, and it leads to a photo-detachment efficiency larger than of 90 %.

Since the velocity dispersion induced by the atomic recoil is of the same order as that from the confinement in the Paul trap, one would not gain by trying to get closer to the optimal cloud size. Finally, this effect is equivalent to a slightly warmer antihydrogen cloud, which changes the effective value of the frequency \(\omega \) to be used in the calculations, without affecting the principle of the method.

- 1.
Uncertainty of shaping/measuring the distribution of the vertical velocity components of \(\overline{\mathrm {H}}\) within the range of acceptance of the two-disk system.

- 2.
Finite positioning and timing resolution for the detection of annihilation events.

- 3.
Accuracy and reliability for the correction for the time spent in the shaping device.

- 4.
Diffraction of atoms on the mirror edges.

- 5.
Residual electromagnetic effects, and in particular the patch effect on mirror surfaces.

- 6.
Defects of mechanical alignments, such as inclinations of the disks and detection plate.

- 7.
Finite precision of production and adjustment of optical elements.

- 8.
Vibrations able to cause parasitic transitions between gravitational quantum states.

## 7 Conclusion

In this paper, we have proposed a new method for shaping vertical velocities of antihydrogen atoms in the timing experiment to be performed by the GBAR collaboration [4]. We have given first estimations of the corresponding statistic uncertainties and listed possible systematic effects. The conclusion of these preliminary estimations, to be confirmed by further analysis, is that the accuracy in the measurement of the free-fall acceleration \(\overline{g}\) of \(\overline{\mathrm {H}}\) atoms could be pushed below 10\(^{-3}\) in realistic experimental conditions.

Statistical uncertainties in the experiment are improved for smaller slit heights, which lead to better defined vertical velocities of \(\overline{\mathrm {H}}\). This means that a better selection of the range of the vertical velocities overweighs the loss in statistics. Systematical uncertainties are expected to decrease even more dramatically for smaller heights of the slit between the two disks in the proposed experimental design. In the optimum experiment where the atomic wavepacket is shaped to the lowest quantum state, the effective temperature corresponding to the vertical motion of \(\overline{\mathrm {H}}\) is as low as 10 nK.

These preliminary estimations have to be confirmed by more complete simulations. We are currently working to develop a fully quantum treatment of the shaping device as well as a complete Monte-Carlo simulations.

Let us also mention that an even better accuracy could in principle be obtained by studying interference effects in the time-of-arrival distribution of a coherent superposition of a few lowest-lying gravitational quantum states [12, 13].

## Notes

### Acknowledgments

The authors thank the ESF Research Networking Programme CASIMIR (casimir-network.org), the GRANIT collaboration and the GBAR collaboration (gbar.in2p3.fr) for providing excellent possibilities for discussions and exchange.

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