# Prospects for studies of the free fall and gravitational quantum states of antimatter

###### Abstract

Different experiments are ongoing to measure the effect of gravity on cold neutral antimatter atoms such as positronium, muonium and antihydrogen. Among those, the project GBAR in CERN aims to measure precisely the gravitational fall of ultracold antihydrogen atoms. In the ultracold regime, the interaction of antihydrogen atoms with a surface is governed by the phenomenon of quantum reflection which results in bouncing of antihydrogen atoms on matter surfaces. This allows the application of a filtering scheme to increase the precision of the free fall measurement. In the ultimate limit of smallest vertical velocities, antihydrogen atoms are settled in gravitational quantum states in close analogy to ultracold neutrons (UCNs). Positronium is another neutral system involving antimatter for which free fall under gravity is currently being investigated at UCL. Building on the experimental techniques under development for the free fall measurement, gravitational quantum states could also be observed in positronium. In this contribution, we review the status of the ongoing experiments and discuss the prospects of observing gravitational quantum states of antimatter and their implications.

This work reviews contributions made at the GRANIT 2014 workshop on prospects for the observation of the free fall and gravitational quantum states of antimatter.

## Introduction

At present, together with the dark matter problem, one of the most tantalizing open questions in physics is the baryon-antibaryon asymmetry, i.e. why are we living in a matter-dominated Universe? Where did all the antimatter go? Different theoretical and experimental efforts trying to address this question are ongoing, including activities focusing on the gravitational behavior of antimatter [1, 2, 3, 4, 5]. No compelling theoretical argument seems to support that a difference between the gravitational behavior of matter and antimatter should be expected [6], although some attempts have been made to show the contrary [7, 8, 9, 10, 11]. Moreover observations and experiments have been interpreted as evidence against the existence of “antigravity” type forces [12, 13, 14, 15]. However those could be argued to be model dependent and therefore a simple free fall measurement is preferable. This justifies ongoing experimental efforts in that direction. A first attempt in this direction was made recently by the ALPHA collaboration [1] that bounds the ratio of gravitational mass to inertial mass of antihydrogen between 65 and 110.

The idea of directly measuring the gravitational force acting on antiparticles in the Earth’s field goes back many decades, from the work of Witteborn and Fairbank, attempting to measure the acceleration of gravity for electrons and, eventually, positrons [16] to the PS200 experiment at CERN in the 1980’s that included measurements on antiprotons [17]. Such measurements are extremely difficult because measuring the force of gravity on a charged particle requires a physically unrealistic (it would seem) elimination of stray electromagnetic fields. The obvious solution to this problem is to use neutral antimatter particles. However, at the present time it is not technically feasible to do so; antineutrons cannot be produced in a controllable manner and antineutrinos are similarly elusive to experimenters. One may instead consider using composite systems that are electrically neutral, in which case it is only necessary to contend with dipole moments. Only a few systems that are composed of, or contain some fraction of, antimatter are available for scientific study. These are antihydrogen, muonium and positronium, which have all been suggested as possible candidates for gravity measurements \bibnoteSee for example articles 4-6 in Antimatter and Gravity, International Journal of Modern Physics: Conference Series, 30, January 2014.

Selecting between different experimental methods, one should aim at precision experiments as they are much more strongly motivated theoretically. Among these, the method of quantum gravitational spectroscopy stands out by its remarkable statistical sensitivity and its cleanness from a systematic point of view.

Gravitational quantum states are solutions of the Schrödinger equation in a gravity field above a surface. They are characterized by the following energy () and spatial scale () :

(0.1) | |||

(0.2) | |||

(0.3) |

Here is the gravitational mass of the particle, is its inertial mass (we distinguish between and in view of discussing EP tests), is the gravitational field intensity near the Earth surface, is the acceleration of the particle in that field and is the Airy function [19, 20]. For neutrons and antihydrogen atoms, the height of the lowest gravitational level is 13.7 m. For positronium, whose mass is approximately 1000 times smaller, it extends over 1.3 mm. The frequency of transitions between first and second quantum states equals 254 Hz for neutrons and antihydrogen, and 26 Hz for positronium. The corresponding characteristic times needed to form quantum states are 0.5 ms and 5 ms respectively.

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 in the GRANIT experiment [21]. 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 was demonstrated experimentally.

Here we analyze in detail several experiments which will study the free fall of anti-atoms and argue that the observation of gravitational quantum states of antimatter is feasible. In section 1 we describe the forthcoming experiment GBAR. We explain in section 2 the quantum reflection mechanism which allows the formation of gravitational quantum states of above material surfaces. In section 3 we show how the filtering scheme of the GRANIT experiment could be implemented in GBAR and in section 4 we describe a possible spectroscopy of gravitational quantum states of . Section 5 reviews the status of positronium free fall experiment at UCL and section 6 explores the possibility of observing gravitational quantum states of positronium.

## 1 GBAR status report

GBAR is an experiment in preparation at CERN. Its goal is to measure the gravitational acceleration () imparted to freely falling antihydrogen atoms, in order to perform a direct experimental test of the Weak Equivalence Principle with antimatter. The objective is to reach a relative precision on of in a first stage, with the perspective to reach a much higher precision using quantum gravitational states in a second stage, as is described in section 4.

The principle of the experiment is described in detail in [22] and is briefly recalled here. It is based on an idea proposed in [23]. Antihydrogen ions are produced, trapped and sympathetically cooled to around 10 K. The excess positron is detached by a laser pulse, which gives the start signal for the free fall of the ultracold antihydrogen atom . The subsequent annihilation on a plate is detected and provides the information to measure . The choice of producing ions to get ultracold antihydrogen atom is the specificity of the GBAR experiment. It is very costly in statistics, but makes the cooling to K temperatures a realistic aim.

We report in this section on three recent progresses in the preparation of the experiment: estimations of the production cross sections, accumulation of positrons, and cooling of the ions.

### 1.1 Production cross sections of ions

The production proceeds in two steps: (1) followed by (2). The Ps symbol stands for positronium. The cross-sections of these reactions are not well known and are very low. The matter counterpart of the first one has been measured. It is around cm (10 barn) for tens of keV protons [24]. The second one is estimated to be around cm (10 barn) [25].

New calculations of these reactions have been performed in which the first excitations levels for the Ps (up to ) and the (up to ) have been considered. The results suggest that the production of can be efficiently enhanced by using either a fraction of Ps(2p) and a 2 keV antiproton beam or a fraction Ps(3d) and antiprotons with kinetic energy below 1 keV [26]. The product of the cross sections of reactions (1) and (2) reaches values around cm (10 barn) for an optimized fraction of excited Ps. Simulations are underway to estimate the effective gain with a realistic experimental setup.

This shows that very low energy antiprotons are needed. The Extremely Low Energy Antiproton (ELENA) ring which is in construction at CERN and which will complement the Antiproton Decelerator (AD), will provide 75 ns rms bunches of 100 keV antiprotons every 100 s. Those have to be further decelerated and cooled to match the GBAR requirements. The decelerator is under construction at CSNSM (“Centre de Sciences Nucléaires et de Sciences de la Matière”) in Orsay, France.

### 1.2 Positron accumulation

In addition to a high flux of low energy antiprotons, the production of via reactions (1) and (2) require to form a dense cloud of positronium. It has been shown that Ps can be efficiently produced by dumping few keV positrons on mesoporous silica films. Yields of 30 to 40 depending on the incident positron energy (few keV) have been measured [27, 28]. The accumulation of a very large number of positrons, around , between two ejections of antiprotons from ELENA is thus necessary to produce a dense enough positronium cloud.

A demonstration facility for the production and accumulation of positrons is currently running at CEA/Saclay. It consists of a low energy electron linear accelerator (LINAC), a high field Penning-Malmberg trap from the Atomic Physics Laboratory in RIKEN, Japan, and a dedicated beam line for further studies of positron-positronium conversion and for applications in material science. In addition, a laser system is now being built at LKB (“Laboratoire Kastler-Brossel”) in Paris to test the excitation of the positronium which will be formed downstream of the trap.

The LINAC produces a 4.3 MeV electron bunched beam. The bunch length is 200 s, and the LINAC runs at 200 Hz, producing a mean current of 120 A. Electrons are sent onto a tungsten mesh moderator. A flux of typically slow (few 100 eV) positrons per second is driven towards the Penning trap through a vacuum tube equipped with solenoid coils producing a 80 mT field. They are accelerated to around 1 keV to enter the high magnetic field (5 T) region. They reach the Penning trap which is made of 23 cylindrical electrodes, surrounded by 4 additional long electrodes to control the admission and the trapping of incident particles. Positrons make a round trip in less than 100 ns. In order to trap them, it is necessary to compress the 200 s bunch. This is done by applying a varying voltage (20 to 150 V) when extracting the slow positron from the moderator. In this way, it is possible to close the entrance of the trap before the bunch escapes. This method allows to trap one single bunch.

In order to accumulate a large number of bunches, positrons have to be slowed down and stored in a dedicated potential well formed by a subset of electrodes of the trap (see Fig. 1) before the next bunch arrives. Positrons are cooled by passing through a preloaded electron plasma in another dedicated potential well. This method has been set up and demonstrated in [29] with a continuous positron beam issued from a Na source. With such a beam, it is not possible to close the entrance gate, and positrons must be slowed down in one step. This was done with a remoderator downstream of the trap. An efficiency of 1 was obtained. With a bunched positron beam, the remoderator is not necessary, and a much higher efficiency is expected.

The cooling time fixes the maximal LINAC frequency, and depends on the density of the electron plasma. With e/m, simulations show that the cooling time is around 3 ms.

The principle of this accumulation scheme has now been successfully demonstrated at Saclay. The details of the experimental setup used during accumulation are given in [30]. The result of a successful set of accumulation trials is shown in Fig. 2.

Given the characteristics of the demonstrator facility at Saclay, a realistic objective is now to accumulate around 10 positrons in the trap within 2 minutes.

### 1.3 Cooling of the ions

Recent progresses have been made for the design of the cooling of ions. The cooling proceeds in two steps: Doppler cooling at the mK level, Raman side band cooling to reach 10 K.

In the first step, ions are captured in a linear Paul trap inside which Be ions are preloaded and laser cooled. In the original scheme of GBAR, it was assumed that the ions would be cooled by Coulomb interactions with the Be ions (“sympathetic cooling”). Simulations show that this process is very slow. This is due to the large mass ratio between the ions. As a consequence, one cannot reach the mK level in a short enough time to avoid the destruction of the : the laser cooling of Be induces the photodetachment of the excess positron in a fraction of a second. However, the simulations show also that the addition of a third species of ions of intermediate mass, namely HD ions, make the process efficient enough [31]. Starting with 1800 Be and 200 HD ions, cooling times of ms are achievable.

Second step: to reach the 10 K level necessary for the free fall experiment, a Be/ ion pair must be transferred to a precision trap to undergo a ground state Raman side band cooling. Calculations show that one may achieve the desired cooling in less than a second. This is shown in [31] and references therein. This method will be tested with matter ions (Ca /Be , H /Be) before being implemented for the GBAR experiment. Traps are being mounted at LKB in Paris and at Mainz University.

Since the uncertainty on the measurement of is fully dominated by the initial velocity dispersion due to both the vertical velocity after cooling and the recoil due to the positron photodetachment, the implementation of a vertical velocity selector will allow a drastic gain in the statistics needed to reach the 1 precision on as is described in section 3.

## 2 Quantum reflection of antihydrogen on material surfaces

In the ultracold regime, the interaction of antihydrogen () atoms with a surface is governed by the phenomenon of quantum reflection. Although the atoms are strongly attracted to the surface, the atomic wave function can be partly reflected on the steep atom-surface potential, leading to a non-zero probability of classically forbidden reflection. This effect is relevant to experiments such as GBAR where ultracold atoms are detected by annihilation on a plate (see section 1).

A single atom placed in vacuum near a material surface experiences an attractive Casimir-Polder (CP) force [32, 33]. This force is a manifestation of the electromagnetic quantum fluctuations which are coupled to the atomic dipole. Quantum reflection occurs if an atom impinges with low velocity on such a rapidly varying potential [34]. We will give a more explicit condition later on.

In this section we first describe how the CP potential is calculated for realistic experimental conditions. We then go on to compute the scattering amplitudes of an atom on this potential. We show that quantum reflection can be understood as a deviation from the semiclassical approximation. Finally we describe materials from which quantum reflection is enhanced and above which gravitationally bound states of could be observed.

### 2.1 Calculation of the Casimir-Polder potential

We use the scattering approach to Casimir forces [35] to give a realistic estimation of the atom-surface interaction energy. In this approach the interacting objects are described by reflection matrices for the electromagnetic field. Reflection on a plane is described by Fresnel coefficients, while reflection on the atom is treated in the dipolar approximation and depends on the dynamic polarizability [36]. This allows an evaluation of the CP potential for any material when its optical properties are known. Those used here are detailed in [37]. Note that since the typical length scale for quantum reflection (100 nm) is below the thermal wavelength at 300 K (1 m), we carried out all calculations at null temperature.

The CP potential for at a distance of a perfectly conducting plane and thick silicon and silica slabs are presented in Fig. 3. For a perfectly conducting mirror in the long-distance regime we recover the historic result of Casimir and Polder [32, 33]:

(2.1) |

where is the static polarizability of the atom.

For real mirrors, the potential is reduced but it shows the same power law dependence in the van der Waals (short distance) and retarded (long distance) regimes:

(2.2) |

where is a typical wavelength associated with the optical response of atom and plane.

### 2.2 Scattering on the Casimir-Polder potential

We now solve the Schrödinger equation for an atom of energy scattering on the CP potential :

(2.3) |

with the classical momentum. We write the exact wave function as a sum of counter-propagating WKB waves whose coefficients are allowed to vary:

(2.4) |

Upon insertion in the Schrödinger equation we obtain coupled first order equations for the coefficients [38]. The annihilation of on the material surface translates as a fully absorbing boundary condition on the surface: . This is in contrast with matter atoms, for which more complicated surface physics is involved in the boundary condition. Close to the surface, the energy becomes negligible compared with the potential, which takes the van der Waals form and can be solved for analytically [37]. The equations are then integrated numerically until become constants.

The reflection probability is plotted against the energy in Fig. 4 for various semi-infinite media. Note that the quantum reflection probability is larger for materials with a weaker CP interaction, such as silica.

### 2.3 The badlands function

To understand this surprising result we look more closely at what distinguishes the exact solution of the Schrödinger equation from the reflectionless WKB approximation. If are no longer allowed to vary, one can show that the wave function (2.4) obeys a modified Schrödinger equation where is replaced by [38]. is known as the badlands function since the WKB approximation is not valid in regions where it is non-negligible:

(2.5) |

For the CP potential, the badlands function exhibits a peak in the region where but goes to zero both as (where the potential cancels) and as (where the classical momentum diverges).

As the energy is decreased, the semiclassical approximation breaks down and the badlands function’s peak becomes larger. But for a given energy, the peak is larger and closer to the surface when the potential is weak, as shown in Fig. 5. The difference between exact and WKB solutions is larger in weaker CP potentials, leading to enhanced quantum reflection.

### 2.4 Enhancing quantum reflection

Quantum reflection first appears as a bias in the context of the GBAR experiment, since it tends to exclude low energy atoms from the statistics. However this phenomenon opens perspectives for the storage and guiding of antimatter with material walls. With this in mind we consider materials which couple weakly to the electromagnetic field and are therefore good mirrors for atoms, as we have seen in the previous paragraph.

A simple strategy is to remove matter from the reflective medium, by using thin slabs or porous materials for example. Our versatile approach allowed us to compute the CP interaction near thin slabs, an undoped graphene sheet [37] and nanoporous materials [39]. The latter consist in a solid matrix which forms an array of nanometric pores. Aerogels, which are obtained by supercritically drying a silica gel, are a well known example. We also consider porous silicon and powders of diamond nanoparticles formed by explosive shock.

From a distance larger than the typical pore size, such materials can be modeled as homogeneous effective media with properties averaged between that of vacuum and of the solid matrix. In consequence their effective dielectric constant is extremely low, as a result of which quantum reflection is exceptionally efficient. In table 1 we show the lifetime of an antihydrogen atom in the first gravitationally bound state above a surface (see section 4 for more details).

Surface (porosity) | Lifetime (s) |
---|---|

perfect conductor | 0.11 |

bulk silicon | 0.14 |

bulk silica | 0.22 |

nano-diamond powder (95%) | 0.89 |

porous silicon (95%) | 0.94 |

silica aerogel (98%) | 4.6 |

Note that this approach does not take into account the possible presence of stray charges on the surface, a question that would have to be addressed to observe the predicted reflection probabilities. Moreover, the effective medium approximation is applicable only for low atom velocities, such that the atom is reflected far enough from the surface. With these caveats, nanoporous materials are an outstanding candidate for the manipulation and study of antihydrogen and its gravitationally bound states, with lifetimes above the second.

## 3 Shaping of vertical velocity components of antihydrogen atoms for GBAR

The main source of uncertainty on the determination of in the GBAR experiment is the width of the vertical velocity distribution of the atom at the beginning of the free fall. This spread in velocities is due to the quantum uncertainty on the momentum of in the ground state of the harmonic Paul trap and to the additional recoil associated with the photo-detachment of the extra positron (see section 1).

In this section we give an estimation of the uncertainty on the arrival time associated with the initial vertical velocity spread and show how it can be reduced by filtering out the fastest atoms. Since slow antihydrogen atoms bounce on material surfaces thanks to quantum reflection (see previous section), the filtering scheme used in GRANIT with ultracold neutrons [21] can also be applied in GBAR.

### 3.1 Width of the arrival time distribution

We consider a wave-packet falling in a linear gravitational potential and want to determine the arrival time distribution on a fixed horizontal plane, supposing there is no reflection from that (ideal) detector. In this case classical and quantum calculations give identical results, as can be seen by noticing that the Wigner quasi-distribution function obeys the classical equations of motion if the potential is at most quadratic. Therefore a given initial phase-space distribution simply propagates along the classical trajectories.

For a wave-packet initially centered at a height above the detector, with zero mean velocity and uncorrelated vertical position and velocity distributions of width and respectively, the spread of the arrival time distribution is

(3.1) |

with the classical free fall time. This translates as a statistical uncertainty on the determination of after independent measurements.

If the particle is initially in the ground state of a harmonic trap, the distribution is Gaussian and saturates the Heisenberg inequality: . Then the time uncertainty is minimal for

(3.2) |

For cm and this evaluates to m/s, and the relative uncertainty on the arrival time is . However the current expected value for GBAR is three orders of magnitude larger m/s, which leads to a relative uncertainty of .

The uncertainty in GBAR is largely dominated by the vertical velocity dispersion. If the initial velocity dispersion can be reduced from to by filtering out the hottest atoms, the single-shot precision and the number of atoms are both reduced by a factor . Despite the loss in statistics, this results in a net reduction of the statistical uncertainty on .

### 3.2 Shaping of the vertical velocity distribution

Our proposal [40] to realize this filtering is to let the atoms pass through a horizontal slit between two disks. The bottom disk has a smooth top surface on which atoms reflect with high probability whereas the top disk has a rough bottom surface which effectively acts as an absorber for the atoms (see Fig. 6). Antihydrogen is initially trapped in the center of the two disks (openings are made in the center of the disks to allow operation of the trap). If its vertical velocity is high enough to reach the rough surface, it is reflected non-specularly and remains inside the device until it annihilates with high probability. On the contrary, if it cannot reach the top disk the atom will exit the device with high probability after bouncing on the bottom mirror a few times. It then falls freely to a detector a height below. Since the horizontal velocity is conserved, the knowledge of the total time between photo-detachment and annihilation and of the total horizontal distance traveled, allows one to correct for the time spent inside the device before the free fall.

If is the height of the slit, the velocity spread at the output is and the proportion of atoms that exit the device is . Using the shaping device therefore reduces the statistical uncertainty on by a factor which scales as .

Classically going to ever smaller slit heights leads to arbitrarily good precision. For example for mm, m/s, and the accuracy is improved by a factor 2, whereas for m, m/s, and the accuracy improved by a factor 4.

There are however two limits: the number of repetitions of the experiment must be large enough that at least some atoms make it through the filter, but more fundamentally, the wave function of the atom must fit inside the slit. Indeed for slit sizes below 50 m the discrete spectrum of states bound by gravity must be taken into account. For a slit size of 20 m only the ground state can travel through the guide, below that the transmission drops to zero.

This fact has been used to demonstrate the existence of gravitationally bound states for neutrons [21]. The next section explores the possibilities of similar experiments with antihydrogen to further increase the precision of equivalence principle tests on antimatter.

## 4 Resonance spectroscopy of gravitational states of antihydrogen near material surface

In this section we will study a motion of an atom, localized in a gravitational state near a horizontal plane mirror. The existence of such states though counterintuitive is explained by the phenomenon of quantum reflection of ultracold (anti-)atoms from a steep attractive Casimir-Polder atom-surface potential. Such states have similar properties with those discovered for neutrons [21, 41, 42, 43, 44].

To account for the interaction of with a material wall, the gravitational quantum states (0.1) receive a complex energy shift , with for a perfectly conducting wall [45]. All states therefore acquire equal width, which is a function of a material surface substance . This width corresponds to the lifetime of s in case of a perfectly conducting surface and is twice longer for silica [46, 37, 39] for instance.

The interest to study gravitational quasi-stationary states of is due to their comparatively long life-time on one hand and easy identification of certain state because of it’s mesoscopic spatial scale. This opens an interesting perspective to apply potentially very precise resonance spectroscopy method to establish the gravitational properties of anti-atoms. These methods are based on inducing an observation of resonance transitions between gravitational states. One of possible approach is to use an alternating inhomogeneous magnetic field for such a purpose.

The interaction of a magnetic field with a ground state atom moving through the field [47, 48, 49] is dominated by the interaction of an average magnetic moment of the atom [20] in a given hyperfine state with the magnetic field. We are going to focus on an alternating magnetic field with a gradient in the vertical direction. This condition is needed for coupling the field and the center of mass (c.m) motion in the gravitational field of the Earth. It allows one to induce resonant transitions between quantum gravitational states of [45].

We will consider the magnetic field in the following form:

(4.1) |

Here is the amplitude of a constant, vertically aligned, component of magnetic field, is the value of magnetic field gradient, is a distance measured in the vertical direction, is a distance measured in the horizontal direction, parallel to the surface of a mirror. A time-varying magnetic field (4.1) is accompanied with an electric field (). However, for the velocities of ultracold atoms, corresponding interaction terms are small and thus will be omitted.

An inhomogeneous magnetic field couples the spin and the spatial degrees of freedom. A wave function is described in this case using a four-component column (in a non-relativistic treatise) in the spin space, each component being a function of the c.m. coordinate , relative coordinate and time . The corresponding Schrödinger equation is:

(4.2) |

A subscript in this equation indicates one of four spin states of the system. The meaning of the interaction terms is the following. is an atom-mirror interaction potential, which turns into the Casimir-Polder potential at an asymptotic atom-mirror distance (see [46, 50] and references therein). is the Hamiltonian of the internal motion, which includes the hyperfine interaction:

(4.3) |

Here , is the antiproton mass, is the positron mass, , is the hyperfine constant, is the operator of the total spin of the antiproton and the positron. We will treat only atoms in a -state (below we will show that the excitation of other states in the studied process is improbable). The term is a model operator, which effectively accounts for the hyperfine interaction and reproduces the hyperfine energy splitting correctly. The term describes the field-magnetic moment interaction:

(4.4) |

Here and are magnetic moments of the positron and the antiproton respectively, , is a spin operator, acting on spin variables of positron (antiproton), , is a corresponding identity operator. As far as the field changes in space and in time, this term couples the spin and the c.m. motion.

We will assume that in typical conditions of a spectroscopy experiment the velocity component parallel to the mirror surface (directed along -axis) is of the order of a few and is much larger than a typical vertical velocity in lowest gravitational states (which is of the order of ). We will treat the motion in a frame moving with the velocity of the atom along the mirror surface. Thus we are going to consider the x-component motion as a classical motion with a given velocity , and we will substitute a -dependence by a -dependence. We will also assume that , where cm is a typical size of an experimental installation of interest. This condition is needed for ”freezing” the magnetic moment of an atom along the vertical direction; it provides the maximum transition probability.

We will be interested in the weak field case, such that the Zeeman splitting is much smaller than the hyperfine level spacing . The hierarchy of all mentioned above interaction terms could be formulated as follows:

(4.5) |

and thus it justifies the use of the adiabatic expansion for solving Eq. (4.2); it is based on the fact that an internal state of an atom follows adiabatically the spatial and temporal variations of an external magnetic field. Neglecting non-adiabatic couplings, an equation system for the amplitude of a gravitational state has the form:

(4.6) |

The transition frequency is determined by the gravitational energy level spacing. This fact is used in the proposed approach to access the gravitational level spacing by means of scanning the applied field frequency, as will be explained in the following.

Within this formalism the role of the coupling potential is played by the energy of an atom in a fixed hyperfine state thought of as a function of (slowly varying) distance and time .

(4.7) |

Here is the gravitational state wave function, which is known in terms of the Airy function [45].

The energy is the eigenvalue of the internal and magnetic interactions , where the c.m. coordinate and time are treated as slow-changing parameters. Corresponding expressions for the eigen-energies of a manifold are:

(4.8) | |||||

(4.9) |

Subscripts are standard notations for hyperfine states of a manifold in a magnetic field. The presence of a constant field produces the Zeeman splitting between states and . As far as the energy of states depends on magnetic field linearly, while for states it depends quadratically, only transition between states take place in case of a weak field. In the following we will consider only transitions between gravitational states in a manifold.

A qualitative behavior of the transition probability is given in the Rabi formula, which can be deduced by means of neglecting the high frequency terms compared to the resonance couplings of only two states, initial and final , in case the field frequency is close to the transition frequency :

(4.10) |

The factor appears in front of the right-hand side of the above expression due to the fact that only two of four hyperfine states participate in the magnetically induced transitions.

It is important that the transition frequencies do not depend on the anti-atom-surface interaction up to the second order in the splitting . This is a consequence of the already mentioned fact that all energies of gravitational states acquire equal shift due to the interaction with a material surface.

A resonant spectroscopy of gravitational states could consist of observing atoms localized in the gravitational field above a material surface at a certain height as a function of the applied magnetic field frequency. A “flow-through type” experiment, analogous to the one discussed for the spectroscopy of neutron gravitational states [51], includes three main steps. A sketch of a principle scheme of an experiment proposed in [40] is shown in Fig. 7 (see also section 3).

First, an atom of is shaped in a ground gravitational state. This is achieved by means of passing through a slit, formed by a mirror and an absorber, which is placed above the mirror at a given height . The mirror and the absorber form a waveguide with a state-dependent transmission [44]. The choice of m implies that only atoms in the ground gravitational state pass through the slit. Second, atoms are affected by an alternating magnetic field (4.1) while they are moving parallel to the mirror. An excited gravitational state is resonantly populated. Third, the number of atoms in an excited state is measured by means of counting the annihilation events in a detector, which is placed at a height above the mirror. The value of is chosen to be larger than the spatial size of the gravitational ground state and smaller than the spatial size of the final state (0.2), , so that the ground state atoms pass through, while atoms in the excited state are detected.

We present a simulation of the number of detected annihilation events as a function of the field frequency in Fig. 8 for the transition from the ground to the -th excited state, based on a numerical solution of the equation system Eq. (4.6). The corresponding resonance transition frequency is Hz. The value of the field gradient, optimized to obtain the maximum probability of transition during the time of flight s, turned to be equal Gs/m, the corresponding guiding field value, which guarantees the adiabaticity of the magnetic moment motion, is Gs.

It follows from (0.1) that the gravitational mass could be deduced from the measured transition frequency as follows:

(4.11) |

Let us mention that in the above formula means the gravitational field intensity near the Earth surface, a value which characterizes properties of the field and is assumed to be known with a high precision. At the same time all the information about gravitational properties of is included in the gravitational mass . Equality of the gravitational mass and the inertial mass , imposed by the Equivalence principle, results in the following expression:

(4.12) |

Estimation of the accuracy of the above expression requires account of different effects, including dynamical Stark shift of the resonance line, non-adiabatic corrections to the transition probability, interaction of alternating magnetic field with a mirror, etc. The detailed study of different systematic effects is under way. Assuming that the spectral line width is determined by the lifetime s of gravitational states, we estimate that the gravitational mass can be deduced with the relative accuracy for annihilation events for the transition to the -th state.

## 5 Gravitational free fall of cold positronium

Antihydrogen, muonium and positronium are the possible candidates
for gravity measurements on antimatter, with various pros and cons.
Antihydrogen and muonium [4] are extremely difficult to
produce, requiring large facilities (i.e., PSI, CERN), whereas
positronium is relatively easy to produce in smaller university
laboratories. However, Ps has an inconvenient propensity to
self-annihilate; the triplet ground state vacuum lifetime of only
142 ns, would seem to preclude using this system for a free fall
measurement. As has been pointed out by various authors, in
particular A. P. Mills, Jr., [52], this is not the
case, since one need only excite Ps atoms into long-lived Rydberg
states to prevent self-annihilation. Indeed, for any Ps state with
the radiative lifetime is always less than the annihilation
lifetime^{1}^{1}1The only excited state for which this is not true
is the metastable 2s state.. That is to say, for excited states the
overlap of the positron and electron wave functions is sufficiently
low that annihilation can be considered to be negligible (see Fig.
9).

The radiative lifetimes of excited Ps states, shown in Fig. 9 , are almost twice those of the corresponding states in hydrogen. For practical reasons the smallest Ps beam deflections one can expect to observe will be 10’s of micrometers or more. Therefore, if Ps falls with the usual gravitational acceleration, it would be necessary to produce states with lifetimes of the order of a few ms to observe such deflections. As is evident from Fig. 9, achieving such long radiative lifetimes requires either exciting low Rydberg levels (i.e., s or d) to extremely high principal quantum numbers, or going to lower states (perhaps around = 30 or so) and then transferring the atoms to circular states \bibnoteFor a discussion of the properties of circular states, and methods for producing them, see [59] (or, if not true circular states, at least states with higher angular momentum).

Aside from the creation of sufficiently long-lived Rydberg levels, conducting a Ps free fall experiment will require solving many other problems. In order to accomplish an experiment of the type first outlined by Mills and Leventhal [55] it will be necessary to produce a small (10-50 micron) “point” source of slow positronium in a cryogenic environment. The resulting long-lived Rydberg atoms will then have to be formed into a beam, perhaps by electrostatic manipulation (focusing, and deceleration) via their electric dipole moments [56], and finally detected with good spatial resolution (as a function of flight time) in order to observe a deflection due to gravity. Possible methods to accomplish some of these tasks are considered elsewhere [5, 55].

The production of Ps Rydberg states with principle quantum numbers around 30 can be accomplished using a two-step process (1s2pd), and has already been experimentally demonstrated using broad-band ( 100 GHz) lasers to accommodate the large Doppler-broadened width of the transitions [57]. However, this methodology is not well suited to the requirement that these atoms are subsequently transferred to higher angular momentum states, and in order to achieve the required state selectivity it may be necessary to use a different excitation mechanism; i.e., a Doppler-free two-photon transition from the ground state directly to a well-defined Rydberg Stark state \bibnoteThis has not yet been demonstrated for positronium; the feasibility of doing so is considered in T. E. Wall, D. B. Cassidy and S. D. Hogan, to be published..

As is well known, Rydberg atoms exhibit exaggerated properties [59] (see table 2). In the present case this is critical, since we seek to produce Ps atoms with very long lifetimes, and also to take advantage of the large electric dipole moments of Rydberg atoms to create and control an atomic beam. However, insofar as we are compelled to make use of electrically neutral systems to measure the weak gravitational force acting on antimatter particles without extraneous electromagnetic fields dominating their motion, excitation to states with very large dipole moments brings us back to the original problem of extraneous field effects. The situation is considerably less dire when dealing with electric dipoles (and, to a much lesser extent, magnetic dipoles) since in this case only field gradients give rise to forces. Nevertheless, in an experiment designed to probe the weak force of gravity with Rydberg atoms, forces due to stray fields must be taken into account.

-scaling | Ps | H | He | |

Binding energy (meV) | -7.56 | -15.11 | -15.12 | |

State separation (meV) | 0.48 | 0.96 | 0.96 | |

Orbital radius () | 2694 | 1347 | 1347 | |

Radiative lifetime (s) | 28.4 | 14.2 | 14.2 | |

Dipole moment/mass (/amu) | 2.2 | 1206 | 304 |

When an atom is placed in an external electric field of strength and direction , the field mixes the atom’s angular momentum states. To first order the state is mixed with states of adjacent but the same and , [60]. The resulting Stark states repel each other, causing them to spread out as the electric field strength is increased, as shown in Fig. 11. Following the example of hydrogen, where the first order Stark shift is analytically calculable, the Schrödinger equation for an atom in an electric field can be written in cylindrical coordinates, where the relevant quantum numbers are , and the parabolic quantum numbers and , which together satisfy the condition .

The Stark states in a given manifold are described by the index , where has values in the range from up to (with ). The first order Stark shift in Ps is given by , where the electric dipole moment has magnitude (where the Ps Bohr radius is (almost) twice that of hydrogen, i.e., ). For a high- Rydberg state with low angular momentum, for example the 30d state with , the value of the electric dipole moment can be very large. The Stark state with has an electric dipole moment of . This large electric dipole moment arises because within this -state there are many degenerate angular momentum states with the same value of that are coupled by the electric field. While this is extremely useful for atomic control [56, 61, 62] it presents a significant problem for gravity measurements, since the electric field gradient experienced by a Ps atom in this state that would result in a force equal to that of “normal” gravity ( N ) is only V/m. Although this is by no means insignificant, it does compare favorably with the V/m electric field that would apply a -like force to a bare positron (or electron).

States with the maximum absolute values for the orbital and magnetic quantum numbers for a given , the so-called circular states, experience no first-order Stark shift. For these states , meaning that there is only one Stark state associated with this value of , which has , and thus, to first order, no electric dipole moment (see Fig. 11). The explanation for this is that, within a given -manifold, the circular states have unique values of , and thus are not coupled to any other degenerate angular momentum states. In the classical limit these states correspond to circular orbits, in which the average -position of the electron is zero, resulting in no electric dipole moment, unlike the lower angular momentum Rydberg states where, in the classical limit, the electronic orbit is highly anisotropic, with the electron having a large average displacement from the atomic core. Although there is no atomic core or nucleus in the case of Ps, the wave function is nevertheless hydrogenic, and the same arguments apply. The circular states do experience a second-order Stark shift, from coupling of adjacent -states, however, this shift is extremely weak. It should, therefore, be possible to produce Ps states with high angular momentum and minimize the effects of stray fields while simultaneously extending the lifetimes to useful levels. However, any manipulation techniques that rely on large dipole moments will obviously have to be performed after the optical excitation to the relevant states, but before transferring the atoms to states with high angular momentum.

Performing a gravity measurement on any system containing antimatter is clearly very challenging, and many of the potential obstacles are currently being investigated. The ability to produce controllable beams of Ps atoms may also open the door to other types of experiments, such as interferometry [63, 64], which could provide an alternative route to an antimatter gravity measurement.

## 6 Can we observe gravitational quantum states of positronium ?

Positronium is about 1000 times lighter than a neutron or antihydrogen. Therefore the expected height of the gravitational quantum state is 100 larger corresponding to a macroscopic size of mm while the energy is 10 times smaller, peV (see Eqs. 0.1-0.2). 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 only lives 125 ps and thus in the following we will only consider the triplet state and refer to it as Ps). Hence, as for the case of a measurement of the gravitational free fall of Ps described in the previous section, the Ps lifetime can be increased by excitation to a higher level. A possible scheme to observe the Ps gravitational quantum states could employ the flow-through technique used for the first observation of this effect with neutrons (see Fig. 12). Greater detail of the proposed experimental set-up and technique are described in a dedicated contribution to this workshop \bibnoteP. Crivelli, V.V. Nesvizhevsky, and A.Yu. Voronin. Can we observe the gravitational quantum states of Positronium? Advances in High Energy Physics, this issue, 2014.. Here we describe the main idea.

Positronium is formed by implanting keV positrons from a re-moderated pulsed slow positron beam in a positron-positronium converter. To observe the quantum mechanical behavior of Ps in the gravitational field its vertical velocity should be of the same order of 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 (see previous section). 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 could 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 on the fact that atoms in Rydberg states have a large dipole moment Stark deceleration can be used for this purpose. This method has been demonstrated for different atomic species (including hydrogen) [62] and molecules [66]. Atoms in Rydberg states have large dipole moments thus electric field gradients can be used to manipulate them. The acceleration/deceleration imparted to the Rydberg atoms is given by:

(6.1) |

where is the gradient of the electric field in V cm, the mass of the decelerated particles in atomic units, and the Stark state quantum numbers. H atoms in and an initial velocity of 700 m/s can be brought at rest in 3 mm [62].

For Ps being 1000 times lighter decelerations exceeding m/s could be realized and therefore the vertical velocity of Ps emitted from thin silica films with initial velocities of the order of m/s [67, 68] could be reduced to below 100 m/s. Since one is interested only in decelerating the distribution that is almost perpendicular to the surface of the Ps target, one can expect for those atoms an efficiency close to 100 %. This is confirmed by preliminary simulations \bibnotePrivate communication with Dr. C. Seiler.. The collimator will be placed after the deceleration stage and the microwave region where circularly polarized radiation will spin up the Ps to the maximum so that kicks to the momentum imparted to the atoms in the vertical direction during these processes will be accounted for.

The fraction of atoms with m/s is estimated to be of the order of . After the collimator the Ps will fly through the slit made of a mirror and the absorber. If the distance between them is smaller than the first expected gravitational state (i.e mm) this will not be transparent and therefore no signal will be detected above the expected background in the detectors. If the width of the slit is increased to a value lying between the first and the second gravitational state (i.e. mm) the Ps wave function can propagate and a signal is expected to be detected via field-ionization and subsequent detection with MCPs. This quantum jump would provide the unambiguous indication of the observation of a quantum gravitational state of positronium.

As a mirror for Ps we propose to exploit a gradient of magnetic field created using wires arranged parallel to each other with a constant current to create a uniform gradient of the magnetic field. Only the Ps triplet atoms with have a non-zero net magnetic moment. For the the electron and the positron magnetic moments cancel and therefore those are insensitive to the magnetic field. Therefore, only one third of the initial population will be reflected. To equate the peV a field of few mG at the wire surface will be sufficient.

Because of the large spacial size of gravitational quantum states and the very large characteristic length of the mirror needed to form the gravitational states that is much larger than a characteristic inter-wire distance, we expect that the very weak magnetic gradient will not perturb the gravitational states. The strict theoretical analysis of this clearly mathematically defined problem is ongoing. A matter mirror could also be considered. Due to the large spacial size of the gravitational quantum state, the surface potential is expected to be very sharp and therefore result in efficient quantum reflection (see section 2).In both cases (magnetic or material mirror) we expect to have effectively (quasi-classically) only a few collisions with the surface. Nevertheless, the transitions rates due to quenching and ionization caused by the electric or magnetic fields have to be calculated. The absorber as for the neutrons is a rough surface on which the impinging Ps will mix its velocity components and therefore be lost.

With such a scheme assuming a mono-energetic slow positron beam flux of e/s (this the highest intensity reported so far reached at the FMR II NEMOPUC source in Munich [70]) an event rate of 0.8 events/day with a background 0.05 events/day might be achievable with a realistic extrapolation of current technologies. Possible losses due to spurious effects like stray electric or magnetic fields or black body radiation seems to be negligible but as for the case of a free gravity fall further calculations and preliminary experiments should be done to confirm this assumption and that all the required efficiencies (e.g. Ps excitation in the , state) can be attained.

To note that the expected height of the gravitational state is related to the gravitational mass by Eq. (0.2). This means that for an uncertainty in the determination of of one can get an accuracy in the determination of at the level of where is the number of detected signals. Assuming an uncertainty of mm which is mainly determined by the finite source size the value of can be determined to 3% in three months. This is comparable to the accuracy that is aimed for antihydrogen experiments at CERN [71, 72, 73]. Therefore, observation of Ps gravitational quantum states offers a complementary approach to test the effect of gravity on a pure leptonic system. Most of the techniques required for such an experiment are under development for the ongoing free gravity fall experiment of Ps (see section 5) and Rydberg Ps deceleration experiments are being considered at ETH Zurich where Prof. B. Brown’s (Marquette University) buffer gas trap is being commissioned. The advantage of using gravitational quantum states is that unpredicted perturbations of the Ps atoms will not result in a systematic effect for the experiment but will only affect the signal rate. Therefore as for the case of antihydrogen this approach seems promising to provide a much higher accuracy than a free fall experiment.

## Conclusion

In this review, we have reported the progress of ongoing experiments to measure gravitational free fall of antimatter. The GBAR experiment will produce antihydrogen atoms in the ultracold regime where quantum reflection from surfaces takes place. Quantum reflection will allow the observation of gravitational quantum states of antimatter that promise to lead to a very sensitive probe of the effect of gravity on anti-atoms (2 orders of magnitude improvement compared to the free fall experiments).

The techniques developed in experiments designed to produce a cold beam of Ps for a free fall measurement will also eventually find application in creating ultracold Ps atoms, as required for observing gravitational quantum states. They will also enable a wide variety of other experimental areas, such as precision spectroscopy.

Antimatter atoms in gravitational quantum states also provide a unique opportunity to constrain experimentally extra short-range forces between the mirror and the anti-atom with about the same sensitivity as we do for normal matter [74].

## Acknowledgements

The authors wish to thank the GRANIT collaboration and the GBAR collaboration (gbar.in2p3.fr) for providing excellent possibilities for discussions and exchange. In addition to the involved institutes and laboratories, preparatory work for GBAR has been funded by the “Conseil Général de l’Essonne”, by the “Agence Nationale de la Recherche” in France, and by the LABEX P2IO.

P.C. acknowledges the support by the Swiss National Science Foundation (grant PZ00P2_132059) and ETH Zurich (grant ETH-47-12-1). DBC and TEW gratefully acknowledge funding from the EPSRC (EP/K028774/1), the Leverhulme Trust (RPG-2013-055) and the ERC (CIG 630119).

## References

- [1] The ALPHA Collaboration and A. E. Charman. Description and first application of a new technique to measure the gravitational mass of antihydrogen. Nature Communications, 4:1785, April 2013.
- [2] AEGIS Collaboration and Marco G. Giammarchi. AEGIS at CERN: Measuring Antihydrogen Fall. Few-Body Systems, 54(5-6):779–782, May 2012.
- [3] P. Perez and Y. Sacquin. The GBAR experiment: gravitational behaviour of antihydrogen at rest. Classical and Quantum Gravity, 29(18), September 2012.
- [4] Klaus Kirch and Kim Siang Khaw. Testing antimatter gravity with muonium. International Journal of Modern Physics: Conference Series, 30:1460258, January 2014.
- [5] D. B. Cassidy and S. D. Hogan. Atom control and gravity measurements using Rydberg positronium. International Journal of Modern Physics: Conference Series, 30:1460259, January 2014.
- [6] M. M. Nieto and T. Goldman. The arguments against “antigravity” and the gravitational acceleration of antimatter. Physics Reports, 205(5):221–281, 1991.
- [7] J. Scherk. Antigravity: A crazy idea? Physics Letters B, 88(3–4):265–267, December 1979.
- [8] G. Chardin and J. M. Rax. CP violation. A matter of (anti)gravity? Physics Letters B, 282(1–2):256–262, May 1992.
- [9] G. Chardin. CP violation and antigravity (revisited). Nuclear Physics A, 558:477–495, June 1993.
- [10] G. Chardin. Motivations for antigravity in General Relativity. Hyperfine Interactions, 109(1-4):83–94, August 1997.
- [11] V. Alan Kostelecký and Jay D. Tasson. Matter-gravity couplings and Lorentz violation. Physical Review D, 83(1):016013, January 2011.
- [12] Myron L. Good. K$_2^0$ and the Equivalence Principle. Physical Review, 121(1):311–313, January 1961.
- [13] Sandip Pakvasa, Walter A. Simmons, and Thomas J. Weiler. Test of equivalence principle for neutrinos and antineutrinos. Physical Review D, 39(6):1761–1763, March 1989.
- [14] A Apostolakis, E Aslanides, G Backenstoss, P Bargassa, O Behnke, A Benelli, V Bertin, F Blanc, P Bloch, P Carlson, M Carroll, E Cawley, G Chardin, M. B Chertok, M Danielsson, M Dejardin, J Derre, A Ealet, C Eleftheriadis, L Faravel, W Fetscher, M Fidecaro, A Filipčič, D Francis, J Fry, E Gabathuler, R Gamet, H. J Gerber, A Go, A Haselden, P. J Hayman, F Henry-Couannier, R. W Hollander, K Jon-And, P. R Kettle, P Kokkas, R Kreuger, R Le Gac, F Leimgruber, I Mandić, N Manthos, G Marel, M Mikuž, J Miller, F Montanet, A Muller, T Nakada, B Pagels, I Papadopoulos, P Pavlopoulos, G Polivka, R Rickenbach, B. L Roberts, T Ruf, L Sakeliou, M Schäfer, L. A Schaller, T Schietinger, A Schopper, L Tauscher, C Thibault, F Touchard, C Touramanis, C. W. E Van Eijk, S Vlachos, P Weber, O Wigger, M Wolter, D Zavrtanik, D Zimmerman, John Ellis, N. E Mavromatos, and D. V Nanopoulos. Tests of the Equivalence Principle with neutral kaons. Physics Letters B, 452(3–4):425–433, April 1999.
- [15] G. Gabrielse, A. Khabbaz, D. S. Hall, C. Heimann, H. Kalinowsky, and W. Jhe. Precision Mass Spectroscopy of the Antiproton and Proton Using Simultaneously Trapped Particles. Physical Review Letters, 82(16):3198–3201, April 1999.
- [16] F. C. Witteborn and W. M. Fairbank. Experimental Comparison of the Gravitational Force on Freely Falling Electrons and Metallic Electrons. Physical Review Letters, 19(18):1049–1052, October 1967.
- [17] Michael H. Holzscheiter. Constraints on gravitational properties of antimatter from cyclotron-frequency measurements. International Journal of Modern Physics: Conference Series, 30:1460260, January 2014.
- [18] See for example articles 4-6 in Antimatter and Gravity, International Journal of Modern Physics: Conference Series, 30, January 2014.
- [19] Milton Abramowitz and Irene A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Wiley, New York, Etats-Unis, 1972.
- [20] Lev Davidovitch Landau, Evgenii Mikhailovich Lifshitz, John Bradbury Sykes, and John Stewart Bell. Quantum mechanics: non-relativistic theory. Oxford, Royaume-Uni, 1965.
- [21] Valery V. Nesvizhevsky, Hans G. Börner, Alexander K. Petukhov, Hartmut Abele, Stefan Baeßler, Frank J. Rueß, Thilo Stöferle, Alexander Westphal, Alexei M. Gagarski, Guennady A. Petrov, and Alexander V. Strelkov. Quantum states of neutrons in the Earth’s gravitational field. Nature, 415(6869):297–299, January 2002.
- [22] G. Chardin, P. Grandemange, D. Lunney, V. Manea, A. Badertscher, P. Crivelli, A. Curioni, A. Marchionni, B. Rossi, A. Rubbia, V. Nesvizhevsky, P.-A. Hervieux, G. Manfredi, P. Comini, P. Debu, P. Dupré, L. Liszkay, B. Mansoulié, P. Pérez, J.-M. Rey, N. Ruiz, Y. Sacquin, A. Voronin, F. Biraben, P. Cladé, A. Douillet, A. Gérardin, S. Guellati, L. Hilico, P. Indelicato, A. Lambrecht, R. Guérout, J.-P. Karr, F. Nez, S. Reynaud, V.-Q. Tran, A. Mohri, Y. Yamazaki, M. Charlton, S. Eriksson, N. Madsen, D.-P. van der Werf, N. Kuroda, H. Torii, and Y. Nagashima. Proposal to measure the Gravitational Behaviour of Antihydrogen at Rest. Technical Report CERN-SPSC-2011-029. SPSC-P-342, CERN, Geneva, September 2011.
- [23] Jochen Walz and Theodor W. Hänsch. A Proposal to Measure Antimatter Gravity Using Ultracold Antihydrogen Atoms. General Relativity and Gravitation, 36(3):561–570, March 2004.
- [24] J. P. Merrison, H. Bluhme, J. Chevallier, B. I. Deutch, P. Hvelplund, L. V. Jørgensen, H. Knudsen, M. R. Poulsen, and M. Charlton. Hydrogen Formation by Proton Impact on Positronium. Physical Review Letters, 78(14):2728–2731, April 1997.
- [25] H. R. J. Walters and C. Starrett. Positron and positronium scattering. physica status solidi (c), 4(10):3429–3436, September 2007.
- [26] P. Comini and P.-A. Hervieux. $\bar{\textrm{H}}^{+}$ ion production from collisions between antiprotons and excited positronium: cross sections calculations in the framework of the GBAR experiment. New Journal of Physics, 15(9):095022, September 2013.
- [27] L. Liszkay, C. Corbel, P. Perez, P. Desgardin, M.-F. Barthe, T. Ohdaira, R. Suzuki, P. Crivelli, U. Gendotti, A. Rubbia, M. Etienne, and A. Walcarius. Positronium reemission yield from mesostructured silica films. Applied Physics Letters, 92(6):063114, February 2008.
- [28] D. B. Cassidy, T. H. Hisakado, H. W. K. Tom, and A. P. Mills. Photoemission of Positronium from Si. Physical Review Letters, 107(3):033401, July 2011.
- [29] N. Oshima, T. M. Kojima, M. Niigaki, A. Mohri, K. Komaki, and Y. Yamazaki. New Scheme for Positron Accumulation in Ultrahigh Vacuum. Physical Review Letters, 93(19):195001, November 2004.
- [30] Pierre Grandemange. Piégeage et accumulation de positons issus d’un faisceau pulsé produit par un accélérateur pour l’étude de l’interaction gravitationnelle de l’antimatière. PhD thesis, Université Paris Sud - Paris XI, December 2013.
- [31] Laurent Hilico, Jean-Philippe Karr, Albane Douillet, Paul Indelicato, Sebastian Wolf, and Ferdinand Schmidt Kaler. Preparing single ultra-cold antihydrogen atoms for free-fall in GBAR. International Journal of Modern Physics: Conference Series, 30:1460269, January 2014.
- [32] H. B. G. Casimir and D. Polder. Influence of retardation on the London-van der Waals forces. Nature, 158:787–788, 1946.
- [33] H. B. G. Casimir. On the attraction between two perfectly conducting plates. Proc. K. Ned. Akad. Wet, 51:793, 1948.
- [34] H. Friedrich, G. Jacoby, and C. G. Meister. Quantum reflection by Casimir-van der Waals potential tails. Physical Review A, 65(3):032902, 2002.
- [35] A Lambrecht, P A Maia Neto, and S Reynaud. The Casimir effect within scattering theory. New Journal of Physics, 8:243–243, 2006.
- [36] Riccardo Messina, Diego A. R. Dalvit, Paulo A. Maia Neto, Astrid Lambrecht, and Serge Reynaud. Dispersive interactions between atoms and nonplanar surfaces. Phys. Rev. A, 80(2):022119, August 2009.
- [37] G. Dufour, A. Gérardin, R. Guérout, A. Lambrecht, V. V. Nesvizhevsky, S. Reynaud, and A. Yu. Voronin. Quantum reflection of antihydrogen from the Casimir potential above matter slabs. Physical Review A, 87(1):012901, January 2013.
- [38] M V Berry and K E Mount. Semiclassical approximations in wave mechanics. Reports on Progress in Physics, 35(1):315–397, January 1972.
- [39] G. Dufour, R. Guérout, A. Lambrecht, V. V. Nesvizhevsky, S. Reynaud, and A. Yu. Voronin. Quantum reflection of antihydrogen from nanoporous media. Physical Review A, 87(2):022506, February 2013.
- [40] G. Dufour, P. Debu, A. Lambrecht, V. V. Nesvizhevsky, S. Reynaud, and A. Yu Voronin. Shaping the distribution of vertical velocities of antihydrogen in GBAR. European Physical Journal C, 74(1):2731, January 2014.
- [41] V.V Nesvizhevsky, H Börner, A.M Gagarski, G.A Petrov, A.K Petukhov, H Abele, S Bäßler, T Stöferle, and S.M Soloviev. Search for quantum states of the neutron in a gravitational field: gravitational levels. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 440(3):754–759, February 2000.
- [42] V. V. Nesvizhevsky, H. G. Börner, A. M. Gagarski, A. K. Petoukhov, G. A. Petrov, H. Abele, S. Baeßler, G. Divkovic, F. J. Rueß, Th. Stöferle, A. Westphal, A. V. Strelkov, K. V. Protasov, and A. Yu. Voronin. Measurement of quantum states of neutrons in the Earth’s gravitational field. Physical Review D, 67(10):102002, May 2003.
- [43] V. V. Nesvizhevsky, A. K. Petukhov, H. G. Borner, T. A. Baranova, A. M. Gagarski, G. A. Petrov, K. V. Protasov, A. Y. Voronin, S. Baessler, H. Abele, A. Westphal, and L. Lucovac. Study of the neutron quantum states in the gravity field. European Physical Journal C, 40(4):479–491, April 2005. WOS:000231122900004.
- [44] A. Yu. Voronin, H. Abele, S. Baeßler, V. V. Nesvizhevsky, A. K. Petukhov, K. V. Protasov, and A. Westphal. Quantum motion of a neutron in a waveguide in the gravitational field. Physical Review D, 73(4):044029, February 2006.
- [45] A. Yu Voronin, P. Froelich, and V. V. Nesvizhevsky. Gravitational quantum states of Antihydrogen. Physical Review A, 83(3):032903, 2011.
- [46] A. Y. Voronin, P. Froelich, and B. Zygelman. Interaction of ultracold antihydrogen with a conducting wall. Physical Review A, 72(6):062903, 2005.
- [47] Willis E. Lamb. Fine Structure of the Hydrogen Atom. III. Physical Review, 85(2):259–276, January 1952.
- [48] L. P. Gor’kov and I. E. Dzyaloshinskii. Contribution to the theory of the Mott exciton in a strong magnetic field. Sov Phys JETP, 26:449–453, 1968.
- [49] Yu. E. Lozovik and S. Yu. Volkov. Hydrogen atom moving across a magnetic field. Physical Review A, 70(2):023410, August 2004.
- [50] A Y Voronin and P Froelich. Quantum reflection of ultracold antihydrogen from a solid surface. Journal of Physics B: Atomic, Molecular and Optical Physics, 38:L301–L301, 2005.
- [51] M. Kreuz, V. V. Nesvizhevsky, P. Schmidt-Wellenburg, T. Soldner, M. Thomas, H. G. Börner, F. Naraghi, G. Pignol, K. V. Protasov, D. Rebreyend, F. Vezzu, R. Flaminio, C. Michel, N. Morgado, L. Pinard, S. Baeßler, A. M. Gagarski, L. A. Grigorieva, T. M. Kuzmina, A. E. Meyerovich, L. P. Mezhov-Deglin, G. A. Petrov, A. V. Strelkov, and A. Yu. Voronin. A method to measure the resonance transitions between the gravitationally bound quantum states of neutrons in the GRANIT spectrometer. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 611(2–3):326–330, December 2009.
- [52] A.P. Mills Jr. Positron moderation and remoderation techniques for producing cold positron and positronium sources. Hyperfine Interactions, 44(1-4):105–123, March 1989.
- [53] Hans Albrecht Bethe and Edwin E. Salpeter. Quantum Mechanics Of One- And Two-Electron Atoms. Dover Publications, Incorporated, 1957.
- [54] For a discussion of the properties of circular states, and methods for producing them, see [59].
- [55] A. P. Mills Jr. and M. Leventhal. Can we measure the gravitational free fall of cold Rydberg state positronium? Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 192(1–2):102–106, May 2002.
- [56] Ch. Seiler, S. D. Hogan, H. Schmutz, J. A. Agner, and F. Merkt. Collisional and Radiative Processes in Adiabatic Deceleration, Deflection, and Off-Axis Trapping of a Rydberg Atom Beam. Physical Review Letters, 106(7):073003, February 2011.
- [57] A. C. L. Jones, T. H. Hisakado, H. J. Goldman, H. W. K. Tom, Jr. Mills, A. P., and D. B. Cassidy. Doppler-corrected Balmer spectroscopy of Rydberg positronium. Physical Review A, 90(1):012503, July 2014.
- [58] This has not yet been demonstrated for positronium; the feasibility of doing so is considered in T. E. Wall, D. B. Cassidy and S. D. Hogan, to be published.
- [59] Thomas F. Gallagher. Rydberg Atoms. Cambridge University Press, 1994.
- [60] T. F. Gallagher. Rydberg atoms. Reports on Progress in Physics, 51(2):143, February 1988.
- [61] E. Vliegen and F. Merkt. Normal-Incidence Electrostatic Rydberg Atom Mirror. Physical Review Letters, 97(3):033002, July 2006.
- [62] S. D. Hogan and F. Merkt. Demonstration of Three-Dimensional Electrostatic Trapping of State-Selected Rydberg Atoms. Physical Review Letters, 100(4):043001, January 2008.
- [63] Thomas J. Phillips. Antimatter gravity studies with interferometry. Hyperfine Interactions, 109(1-4):357–365, August 1997.
- [64] M. K. Oberthaler. Anti-matter wave interferometry with positronium. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 192(1–2):129–134, May 2002.
- [65] P. Crivelli, V.V. Nesvizhevsky, and A.Yu. Voronin. Can we observe the gravitational quantum states of Positronium? Advances in High Energy Physics, this issue, 2014. arXiv:1408.7051.
- [66] S. D. Hogan, Ch. Seiler, and F. Merkt. Rydberg-State-Enabled Deceleration and Trapping of Cold Molecules. Physical Review Letters, 103(12):123001, September 2009.
- [67] D. B. Cassidy, P. Crivelli, T. H. Hisakado, L. Liszkay, V. E. Meligne, P. Perez, H. W. K. Tom, and A. P. Mills. Positronium cooling in porous silica measured via Doppler spectroscopy. Physical Review A, 81(1):012715, January 2010.
- [68] Paolo Crivelli, Ulisse Gendotti, André Rubbia, Laszlo Liszkay, Patrice Perez, and Catherine Corbel. Measurement of the orthopositronium confinement energy in mesoporous thin films. Physical Review A, 81(5):052703, May 2010.
- [69] Private communication with Dr. C. Seiler.
- [70] C. Hugenschmidt, B. Löwe, J. Mayer, C. Piochacz, P. Pikart, R. Repper, M. Stadlbauer, and K. Schreckenbach. Unprecedented intensity of a low-energy positron beam. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 593(3):616–618, August 2008.
- [71] A. Kellerbauer, M. Amoretti, A.S. Belov, G. Bonomi, I. Boscolo, R.S. Brusa, M. Büchner, V.M. Byakov, L. Cabaret, C. Canali, C. Carraro, F. Castelli, S. Cialdi, M. de Combarieu, D. Comparat, G. Consolati, N. Djourelov, M. Doser, G. Drobychev, A. Dupasquier, G. Ferrari, P. Forget, L. Formaro, A. Gervasini, M.G. Giammarchi, S.N. Gninenko, G. Gribakin, S.D. Hogan, M. Jacquey, V. Lagomarsino, G. Manuzio, S. Mariazzi, V.A. Matveev, J.O. Meier, F. Merkt, P. Nedelec, M.K. Oberthaler, P. Pari, M. Prevedelli, F. Quasso, A. Rotondi, D. Sillou, S.V. Stepanov, H.H. Stroke, G. Testera, G.M. Tino, G. Trénec, A. Vairo, J. Vigué, H. Walters, U. Warring, S. Zavatarelli, and D.S. Zvezhinskij. Proposed antimatter gravity measurement with an antihydrogen beam. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 266(3):351–356, February 2008.
- [72] D. P. van der Werf. The GBAR experiment. International Journal of Modern Physics: Conference Series, 30:1460263, January 2014.
- [73] A. I. Zhmoginov, A. E. Charman, R. Shalloo, J. Fajans, and J. S. Wurtele. Nonlinear dynamics of anti-hydrogen in magnetostatic traps: implications for gravitational measurements. Classical and Quantum Gravity, 30(20):205014, October 2013.
- [74] I. Antoniadis, S. Baessler, M. Buchner, V. V. Fedorov, S. Hoedl, A. Lambrecht, V. V. Nesvizhevsky, G. Pignol, K. V. Protasov, S. Reynaud, and Y. Sobolev. Short-range fundamental forces. Comptes Rendus Physique, 12(8):755–778, October 2011.