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arXiv:1402.5159 · Physical Review A

Spinor Bose-Einstein condensates of positronium

Yi-Hsieh Wang, Brandon Anderson, Charles W. Clark

Published Apr 28, 2014• 20 citations
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Abstract

Bose-Einstein condensates (BECs) of positronium (Ps) have been of experimental and theoretical interest due to their potential application as the gain medium of a $\ensuremath{\gamma}$-ray laser. Ps BECs are intrinsically spinor due to the presence of ortho-positronium ($o$-Ps) and para-positronium ($p$-Ps), whose annihilation lifetimes differ by three orders of magnitude. In this paper, we study the spinor dynamics and annihilation processes in the $p$-Ps/$o$-Ps system using both solutions of the time-dependent Gross-Pitaevskii equations and a semiclassical rate-equation approach. The spinor interactions have an O(4) symmetry which is broken to SO(3) by an internal energy difference between $o$-Ps and $p$-Ps. For an initially unpolarized condensate, there is a threshold density of $\ensuremath{\approx}{10}^{19}$ cm${}^{\ensuremath{-}3}$ at which spin mixing between $o$-Ps and $p$-Ps occurs. Beyond this threshold, there are unstable spatial modes accompanied by spin mixing. To ensure a high production yield above the critical density, a careful choice of external field must be made to avoid the spin mixing instability.

DOI
10.1103/physreva.89.043624

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Bose-Einstein condensates (BECs) of positronium (Ps) have been of experimental and theoretical interest due to their potential application as the gain medium of a \ensuremathγ\ensuremath{\gamma}\ensuremathγ-ray laser. Ps BECs are intrinsically spinor due to the presence of ortho-positronium (ooo-Ps) and para-positronium (ppp-Ps), whose annihilation lifetimes differ by three orders of magnitude. In this paper, we study the spinor dynamics and annihilation processes in the ppp-Ps/ooo-Ps system using both solutions of the time-dependent Gross-Pitaevskii equations and a semiclassical rate-equation approach. The spinor interactions have an O(4) symmetry which is broken to SO(3) by an internal energy difference between ooo-Ps and ppp-Ps. For an initially unpolarized condensate, there is a threshold density of \ensuremath≈1019\ensuremath{\approx}{10}^{19}\ensuremath≈1019 cm\ensuremath−3{}^{\ensuremath{-}3}\ensuremath−3 at which spin mixing between ooo-Ps and ppp-Ps occurs. Beyond this threshold, there are unstable spatial modes accompanied by spin mixing. To ensure a high production yield above the critical density, a careful choice of external field must be made to avoid the spin mixing instability.
Realization of Bose-Einstein Condensate (BEC) of positronium (Ps), a purely leptonic particle-antiparticle bound system, is a challenging subject [1, 2]. Recently, Ivanov et al. [3] theoretically investigated the Ps-Ps scattering and showed theoretically that the scattering length for the o-Ps atoms is positive, indicating that a stable o-Ps BEC is actually possible. The condition for the BEC of the ideal Bose gas is λth > (2.612/n)1/3, where λth = h/ √ 2πmkT is the thermal de Broglie wavelength of the atom and n is the number density. This gives the critical temperature for the Ps BEC as Tc = 1.46×10−15n2/3, where Tc is in K and n is in m−3. This condition is much less severe than for the usual alkali atom BEC because λth is longer by five orders of magnitude. However, Ps has a finite lifetime: the lifetime of singlet state, para-positronium (p-Ps), is 125 ps and that of triplet state, ortho-positronium (o-Ps), is 142 ns. Thus a many-Ps system must be cooled within the lifetime of o-Ps. The short lifetime of Ps demands quick cooling and high density. The laser cooling which is employed for the accomplishment of alkali atom BEC is likely to be impractical because it takes longer time. It is also subject to relatively large recoil of the Ps atom when hit by a photon. Moreover, unless an effective method for trapping Ps in free space is developed, the system of Ps atoms diffuses and lowers its density. Platzmann and Mills [1] suggested to confine Ps in a small cavity of insulating material and let the Ps atoms give away their kinetic energy to the walls of the cavity. If one injects an intense polarized positron beam onto a material which has a micro cavity near surface, some of the positrons form Ps and are emitted into the cavity and stay there. This is expected because the work function of most of the material for Ps is negative, i.e., Ps atoms emitted into the cavity after thermalization in the material will never penetrate back into the material. They just hit the cavity walls and approach thermal equilibrium by exciting phonons. The probability of the annihilation of o-Ps with a surface electron is small if there are no paramagnetic centers on the cavity walls. It is necessary to use a polarized positron beam to make Ps. Otherwise all the o-Ps atoms are lost through mutual spin conversion into p-Ps [1, 4]. In this report we would like to discuss prospect of a Ps BEC with special attention to the following aspects [4]: (1) use of 18F for a source of an intense spin-polarized positron beams [5], (2) development of high efficiency Wmesh positron moderator, (3) emission of Ps from solid surface (4) the cooling of Ps in a cavity, and (5) the detection of a Ps BEC.
The s-wave scattering parameters for the Positronium–Positronium (Ps–Ps) interaction are estimated semi-phenomenologically with the long-range van der Waals force and the short-range repulsive potential. The obtained values of the scattering length and the effective range are a ∼0.44 nm, and r 0 ∼0.2 nm for the symmetric channel. Stability of them is also checked with two kinds of potential with different parametrizations. Spin coupling schemes for the Ps–Ps system are discussed and the formulae of scattering length for each spin channel are derived. Using this value of the scattering length, the Gross-Pitaevskii equation can be fixed for the Positronium Bose-Einstein condensates (Ps BEC). The static properties of Ps BEC are discussed using the solutions of that equation. The critical temperature shift of the phase-transition due to the Ps–Ps interaction is also evaluated in the mean-field approximation.
Although it can probably only exist within a confining potential, a positronium (Ps) Bose-Einstein condensate (BEC) would be interesting because it could be viewed as the ultimate member of Wheeler’s polyelectron series, e+, Ps, Ps−, Ps+, Ps2, …, Ps-BEC [1], not counting high density neutral e+-e− plasmas. A Ps-BEC could be a source of very cold Ps atoms for precision spectroscopy, for gravitation experiments, and as the gain medium for an annihilation gamma ray laser [2]. It would also be interesting to look for other effects that are non-linear in the positron density, such as the production of Ps+ ions and highly polarized positrons at a metal surface. Specifically, a process in which one thermalized bulk positron passes into the vacuum accompanied by an electron and a surface positron (surface pseudo-Ps) to form a Ps+ ion in vacuum, e+solid + Pssurf → Ps+vac + □E, is energetically allowed for a Ni(100) remoderator surface, for which + □E = 1.00 eV. The positive ions could be used in an experiment in which one positron undergoes a process in a material and the other positron reports on the process via its subsequent Ps emission, etc. Since the two positrons involved in the formation of Ps+ and Ps2 must have opposite spins, at sufficiently high implanted positron densities the minority positron spins will be significantly depleted by these processes, resulting in the enhancement of the spin polarization of the reemitted positrons which might be useful for spin polarized angle-resolved positronium emission spectroscopy of the electronic structure of solids and surfaces.
We present an analytical model for the theoretical analysis of spin dynamics and spontaneous symmetry breaking in a spinor Bose-Einstein condensate (BEC). This allows for an excellent intuitive understanding of the processes and provides good quantitative agreement with the experimental results of Scherer et al. [Phys. Rev. Lett. 105, 135302 (2010)]. It is shown that the dynamics of a spinor BEC initially prepared in an unstable Zeeman state mF=0{m}_{F}=0mF​=0 ($\left|0\right\ensuremath{\rangle}$) can be understood by approximating the effective trapping potential for the state $\left|\ifmmode\pm\else\textpm\fi{}1\right\ensuremath{\rangle}$ with a cylindrical box potential. The resonances in the creation efficiency of these atom pairs can be traced back to excitation modes of this confinement. The understanding of these excitation modes allows for a detailed characterization of the symmetry-breaking mechanism, showing how a twofold spontaneous breaking of spatial and spin symmetry can occur. In addition, a detailed account of the experimental methods for the preparation and analysis of spinor quantum gases is given.
This thesis covers three applications of Bose-Einstein condensates and related phenomena, in the theme of pair creation and pair annihilation. First, Bose-Einstein condensates (BEC) are viewed as one of the candidates to implement a sonic black hole. This can lead to the observation of analog Hawking radiation, resulting from a phonon pair creation at a black-hole horizon (BH). Such implementation has been achieved in a resent experiment by J. Steinhauer, in which a black-hole/white-hole pair has been produced. He also reported the observations of self-amplifying Hawking radiation, via a lasing mechanism operating between the black and white-hole horizons. Through our simulations, we find that the observations should be attributed not to the black hole laser effect, but rather to a growing zero-frequency bow wave, generated at the white-hole horizon. The relative motion of the two horizons produces a Doppler shift of the bow wave at the BH, where it stimulates a monochromatic Hawking radiation. We also find that shot-to-shot atom number variations and quantum fluctuations give density-density correlations consistent with those reported in the experiments. In particular, atom number variations can produce a spurious correlation signal. Secondly, a sonic black hole/white hole pair and phonon pair creation can also be realized using a ring-shaped condensate. Here we focus on the phonon spectroscopy of a ring condensate. We probe the phonon excitation spectrum by applying a harmonically driven barrier to a 23Na Bose-Einstein condensate in a ring-shaped trap. When excited resonantly, these wavepackets display a regular periodic structure. The resonant frequencies depend upon the particular configuration of the barrier, but are commensurate with the orbital frequency of a sound wave traveling around the ring. Energy transfer to the condensate over many cycles of the periodic wavepacket motion causes enhanced atom loss from the trap at resonant frequencies. Solutions of the Gross-Pitaevskii (GP) equation exhibit quantitative agreement with the experimental data. Thirdly, positronium (Ps) BECs have been of experimental and theoretical interest due to their potential application as the gain medium of a gamma-ray laser. Ps BECs are intrinsically spinor due to the presence of ortho- (o-Ps) and para-positronium (p-Ps), whose annihilation lifetimes differ by three orders of magnitude. We study the spinor dynamics and annihilation processes in the p-Ps/o-Ps system using both solutions of the GP equations and a rate-equation approach. For an initially unpolarized condensate, there is a threshold density at which spin mixing between o-Ps and p-Ps occurs. Beyond this threshold, there are unstable spatial modes accompanied by spin mixing. To ensure a high production yield above the critical density, a careful choice of external field must be made to avoid the spin mixing instability.
The s-wave scattering length for the Positronium-Positronium interaction is estimated semi-phenomenologically with the long-range van der Waals force and the short-range repulsive potential that represents the hard core between two positronium. The obtained value of the scattering length is a∼0.44nma \sim 0.44{\rm nm}a∼0.44nm, and its stability is also checked for different parametrizations. Using this value, the Gross-Pitaevskii equation can be fixed for the Positronium Bose-Einstein condensates (Ps BEC). The static properties of Ps BEC are studied from the solutions of that equation. The phase-transition temperature shift of the Ps BEC due to the Ps-Ps interaction is also evaluated with the mean-field approximation.