Positronium density measurements using polaritonic eects Erika Cortese1David B. Cassidy2and Simone De Liberato1 1School of Physics and Astronomy University of Southampton Southampton SO17 1BJ United Kingdom

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Positronium density measurements using polaritonic eﬀects

Erika Cortese,1David B. Cassidy,2and Simone De Liberato1

1School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom

2Department of Physics and Astronomy, University College London,

Gower Street, London WC1E 6BT, United Kingdom

Recent experimental advances in Positronium (Ps) physics have made it possible to produce dense

Ps ensembles in which Ps-Ps interactions may occur, leading to the production of Ps2molecules and

paving the way to the realization of a Ps Bose-Einstein Condensate (BEC). In order to achieve this

latter goal it would be advantageous to develop new methods to measure Ps densities in real-time.

Here we describe a possible approach to do this using polaritonic methods: using realistic experi-

mental parameters we demonstrate that a dense Ps gas can be strongly coupled to the photonic ﬁeld

of a distributed Bragg reﬂector microcavity. In this strongly coupled regime, the optical spectrum

of the system is composed of two hybrid positronium-polariton resonances separated by the vacuum

Rabi splitting, which is proportional to the square root of the Ps density. Given that polaritons

can be created on a sub-cycle timescale, a spectroscopic measurement of the vacuum Rabi splitting

could be used as an ultra-fast Ps density measurement in regimes relevant to Ps BEC formation.

Moreover, we show how positronium-polaritons could potentially enter the ultrastrong light-matter

coupling regime, introducing a novel platform to explore its non-perturbative phenomenology.

I. INTRODUCTION

Positronium (Ps), the electron-positron bound state, is

a meta-stable two-body atomic system that has a lifetime

against self-annihilation of 142 (0.125) ns in the triplet

(singlet) ground state [1]. Since Ps is composed only of

leptons it is almost fully described by bound-state quan-

tum electrodynamics (QED) [2], and can therefore be

used to test QED theory via precision measurements of

Ps energy levels or decay rates [3].

The existence of Ps atoms was ﬁrst suggested by

Mohoroviˇci´c in 1934 [4], with subsequent, independent,

predictions by Pirenne [5], Ruark [6] and Wheeler [7].

Wheeler also considered what he called polyelectrons,

which are systems containing more than one electron

and/or positron, the simplest case being the Ps atom.

He showed that three-body Ps ions, comprising two elec-

trons and one positron (or two positrons and one elec-

tron), would also form meta-stable bound states. Al-

though Wheeler was unable to determine if four-body Ps2

molecules would be stable, this was subsequently shown

to be the case by Hylleraas and Ore [8].

Ps atoms were ﬁrst produced experimentally in 1951

by Deutsch using a gas cell apparatus [9]. The develop-

ment of slow positron beams [10] in the 1970’s allowed for

more controlled Ps production using solid surfaces [11],

and later also the production of the negative Ps ion [12]

and Ps2molecules [13]. In addition to creating polyelec-

trons, a long term goal of Ps physics has been the for-

mation of an ensemble of Ps atoms that are cold/dense

enough to create a Bose-Einstein Condensate (BEC) [14].

The primary motivation for producing a Ps BEC is that

such a system may exhibit the phenomenon of stimu-

lated annihilation [15,16], allowing for the creation of a

gamma-ray laser, but the properties of a Ps BEC are also

of interest from a theoretical perspective (e.g., [17–20]).

The low mass of Ps means that the transition tempera-

ture (that is, the temperature at which a dense ensemble

will undergo a phase transition to form a condensate)

is considerably higher than it is for all other atoms; for

example room temperature Ps condensates could form

at densities on the order of 1020 cm−3[21]. Since Ps

can be cooled via collisions to ambient cryogenic tem-

peratures in microcavities [22] this density requirement

may be reduced to the 1019 cm−3level for experimentally

accessible temperatures. The speciﬁcs of various Ps pro-

duction methods may also allow for signiﬁcant density

enhancements (e.g., [23,24]).

It is evident that any practical scheme designed to pro-

duce a Ps BEC requires a high density positron beam,

and an eﬃcient means to generate a correspondingly high

Ps density. Recent advances in positron trapping and

control methods [25] have made it possible to produce

Ps in porous silica ﬁlms at densities that allow for Ps-Ps

interactions to occur [26], resulting in the formation of a

spin polarized Ps gas with an average density on the or-

der of 1016 cm−3[27], with higher Ps densities expected

in the future [28].

The optimization of any experimental schemes to gen-

erate a Ps BEC would beneﬁt from direct measurements

of the Ps temperature and density. However, these are

not trivial measurements: Ps temperatures can be di-

rectly measured using the angular correlation of annihi-

lation radiation [29], but this requires the atoms to be

magnetically quenched and decay via a two-photon anni-

hilation process. The rate of annihilation events follow-

ing Ps-Ps scattering can be measured using single-shot

lifetime methods [30], from which one may infer the Ps

density. However, once the Ps gas has become fully spin-

polarized this method no longer provides any signal [31].

Another way to determine Ps densities would be to mea-

sure the density-dependant collisional frequency shift [32]

of Ps atomic transitions. This technique has not been

demonstrated for conﬁned Ps, and may be aﬀected by

the detailed interactions between the Ps gas and its en-

vironment [33].

arXiv:2210.09875v1 [physics.atom-ph] 18 Oct 2022

n=1

n=2

n=12

n=13

n=3

n=4

Energy (meV)

-6803

-1700

-775

-425

-68

-47

-40

Lyman-alpha

transition

Rydberg-Rydberg

transition

2 4 6 8 10 12 14 16 18

Quantum number n

Dipole moment (D)

Figure 1. (a) Sketch of Ps CQED system, representing Ps

atoms interacting with the electromagnetic ﬁeld trapped in

a cavity formed by two DBRs. (b) Energy levels of Ps with

highlighted the Lyman-alpha and Rydberg-Rydberg transi-

tions explicitly considered in this paper (not to scale). (c)

Dipole moment dn→n+1 in Debyes as a function of the prin-

cipal quantum number nof the initial energy state.

In this paper we describe an alternative approach to

this problem that exploits concepts and techniques of

cavity quantum electrodynamics (CQED), the ﬁeld that

investigates the interaction between dipolar active tran-

sitions in atoms, molecules or other materials, and single

photons inside an optical cavity [34]. When the light-

matter coupling strength, referred to as Vacuum Rabi

Frequency (VRF), becomes larger than the loss rates of

the light and matter excitations, the system enters the so-

called strong coupling (SC) regime [35,36]. In this regime

the physics of the system can be correctly described only

in terms of the light–matter hybrid eigenmodes of the

coupled system, often named polaritons [37,38], which

manifest themselves as spectral resonances split by twice

the VRF. The collective nature of light-matter coupling

inside the cavity leads to a VRF splitting that scales as

the square root of the number of optically active dipoles

within the cavity mode volume [39]. The possibility of

achieving SC between light and collective matter excita-

tions, and thus to observe polaritons in a variety of solid-

state systems, has led to the achievement of landmarks as

room-temperature BEC [40] and room-temperature su-

perﬂuidity [41].

Here we theoretically investigate the light-matter inter-

action between the electronic transitions of a collection of

Ps atoms and the conﬁned electromagnetic ﬁeld of an op-

tical cavity, and explore the conditions under which such

a system can access the SC regime where it would be

possible to observe positronium-polariton modes. Since

the cavity-induced splitting in the SC regime is propor-

tional to the square root of the density of the atoms,

producing a Ps ensemble inside an optical cavity would

lead us to a direct measurement of the Ps density via

spectroscopic measurements. Moreover, since the forma-

tion of polaritons can occur on sub-period time scale [42],

this mechanism can provide an ultrafast measurement of

the Ps density, allowing for real-time measurements even

under conditions where the Ps density is varying.

Distributed Bragg Reﬂectors (DBRs), cavities com-

posed of a pair of mirrors composed of alternating layers

of semiconductor materials, are a ﬂexible and successful

resonator technology widely used in polaritonics [37,43].

Although usually covering the near-infrared and visible

ranges, DBRs can also be eﬀective at shorter and longer

wavelengths, and thus can be compatible with a wide

variety of Ps transitions. Deep-UV DBRs have been re-

cently fabricated via deposition of AlGaN/AlN alternat-

ing layers with very promising Q-factors [44]. For mid- to

far-infrared ranges, DBRs consisting of silicon slabs in air

have been used [45], with high quality factors obtained

owing to a large refractive index mismatch.

Open cavity systems have been largely employed to

overcome the issue of growing the active material on top

of the mirrors, allowing both non-invasive investigation

and good control over the microcavity resonances and

mode volumes [46,47]. These systems are particularly

suitable to the present investigation as diﬀerent kinds of

microscopic Ps producing materials, such as porous silica

layers [48] or MgO powders [49], can be placed upon a

prefabricated DBR with diﬀerent thickness and surface

extension [see Fig. 1(a)].

II. LIGHT-MATTER COUPLING IN

POSITRONIUM

A. Theoretical framework

At the Bohr level, the energy levels of Ps atoms are

formally equivalent to those of the hydrogen atom, with

reduced mass µPs =me/2 and Bohr radius aPs = 2a0,

where meis the electron mass and a0the Bohr radius.

Fine structure corrections are considerably diﬀerent in

Ps [3] but these diﬀerence are unimportant in the present

case. The Bohr energy spectrum, sketched in Fig. 1(b),

similar to that for the hydrogen atom, becomes

En=−Ry

2n2eV,(1)

where nrefers to the principal quantum number, and

the Rydberg constant Ry ≈13.605 eV. We will consider

transitions between states of the form |nSiand |(n+1)Pi,

corresponding to energies ~ωn→n+1 =En+1 −Enand

dipoles dn→n+1 =−eh(n+ 1)P|z|nSi. We choose the

phases to make the dipoles real and positive [see Fig.

1(c)].

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PositroniumdensitymeasurementsusingpolaritoniceectsErikaCortese,1DavidB.Cassidy,2andSimoneDeLiberato11SchoolofPhysicsandAstronomy,UniversityofSouthampton,Southampton,SO171BJ,UnitedKingdom2DepartmentofPhysicsandAstronomy,UniversityCollegeLondon,GowerStreet,LondonWC1E6BT,UnitedKingdomRecentexperiment...

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Positronium density measurements using polaritonic eects Erika Cortese1David B. Cassidy2and Simone De Liberato1 1School of Physics and Astronomy University of Southampton Southampton SO17 1BJ United Kingdom

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