Advances in High Energy Physics

Volume 2015 (2015), Article ID 173572, 7 pages

http://dx.doi.org/10.1155/2015/173572

## Can We Observe the Gravitational Quantum States of Positronium?

^{1}ETH Zurich, Institute for Particle Physics, Otto-Stern-Weg 5, 8093 Zurich, Switzerland^{2}Institut Laue-Langevin, 6 rue Jules Horowitz, 38046 Grenoble, France^{3}Lebedev Physical Institute, 53 Leninsky Prospect, Moscow 119991, Russia

Received 6 June 2014; Revised 17 August 2014; Accepted 29 August 2014

Academic Editor: Stefan Baessler

Copyright © 2015 P. Crivelli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We consider the feasibility of observing the gravitational quantum states of positronium. The proposed scheme employs the flow-throw technique used for the first observation of this effect with neutrons. Collimation and Stark deceleration of Rydberg positronium atoms allow selecting the required velocity class. If this experiment could be realized with positronium, it would lead to a determination of for this matter-antimatter system at the few % level. As discussed in this contribution, most of the required techniques are currently available but important milestones have to be demonstrated experimentally before such an experiment could become reality. Those are the efficient focusing of a bunched positron beam, Stark deceleration of Rydberg positronium, and its subsequent excitation into states with large angular momentum. We provide an estimate of the efficiencies we expect for these steps and assuming those could be confirmed we calculate the signal rate.

#### 1. Introduction

Quantum gravitational states were observed for the first time with neutrons by measuring their transmission through a slit made of a mirror and an absorber [1]. If the distance between the mirror and the absorber (which is a rough surface used as a scatterer to mix the velocity components) is much higher than the turning point for the corresponding gravitational quantum state, the neutrons pass through the slit without significant losses. As the slit size decreases the absorber starts approaching the size of the neutron wave function and the probability of neutron loss increases. If the slit size is smaller than the characteristic size of the neutron wave function in the lowest quantum state, the slit is not transparent for neutrons as this was demonstrated experimentally. The height and energies of the gravitational quantum states can be determined analytically and the solution of the Schrödinger equation contains airy functions. A more transparent and simple equation can be derived using a semiclassical approach [2]. This solution reproduces the energy of the gravitational states within 1% and is given bywhere is the particle mass, the gravitational acceleration, the principal quantum number, and the reduced Planck constant. The characteristic scale for the gravitational quantum states is equal to

The corresponding classically allowed heights are given bywhere are the zeros of the airy function. For neutrons the height of the lowest gravitational level is 13.7 m. For positronium, the electron-positron bound state, that is, 1000 times lighter than a neutron, one gets a 100 larger size corresponding to mm while the energy is 10 times smaller, peV. The observation time to resolve a quantum gravitational state can be estimated using the Heisenberg uncertainty principle to be of the order of ms. This value is much larger than the long lived triplet positronium lifetime in the ground state which is 142 ns (the Ps singlet state lives only 125 ps and thus in the following we will only consider the triplet state and refer to it as Ps). Luckily, the Ps lifetime can be increased by exciting it to a higher level. In a Rydberg state the Ps lifetime against annihilation is increased by a factor of , where is the principal quantum number, because of the decrease of the overlap of the positron and the electron wave functions. As for the case of a measurement of the gravitational free fall of Ps proposed by Mills and Leventhal [3], the usable lifetime to observe a quantum gravitation state of a Rydberg Ps atom (hereafter ) is the one before it emits the first photon. In fact, after that the recoil will modify its trajectory and vertical energy; thus the Ps atom will be lost inside the slit. For this reason, the excited Ps has to be spun up to high values with circularly polarized microwave radiation.

#### 2. Experimental Technique

A scheme of the proposed experimental setup is shown in Figure 1. Positronium is formed by implanting keV positrons from a remoderated pulsed slow positron beam in a positron-positronium converter (see Section 3.3). To observe the quantum mechanical behavior of Ps in the gravitational field its vertical velocity should be of the same order as the gravitational energy levels and thus m/s. Furthermore to resolve the quantum state the Ps atom has to interact long enough with the slit and therefore it has to be laser excited to a Rydberg state with and maximum quantum number. To keep a reasonable size of the experimental setup (i.e., a slit size of the order of 0.5 m) and minimize the number of detectors the velocities in the horizontal plane should be smaller than m/s. Similar to neutrons a collimator will be used to select the velocity components , of the positronium distribution. However since no reliable thermal cold source of positronium exists, the velocity component perpendicular to the surface has to be lowered by some other means. Relying to the fact that atoms in Rydberg states have a large dipole moment, Stark deceleration can be used for this purpose. After deceleration the Ps^{*} are driven by circularly polarized microwaves to a state with high . If the slit width is smaller than the first expected gravitational state (i.e., <1 mm), this will not be transparent and therefore no signal will be detected above the expected background in the detectors. If the width is increased to a value lying between the first and the second gravitational state (i.e., <2 mm) the Ps wavefunction can propagate and a signal is expected to be detected. This quantum jump provides the unambiguous indication of the observation of a quantum gravitational state of positronium.