Possibility of forming a stable Bose-Einstein condensate of 2 3 S 1 positronium atoms
PPossibility of forming a stable Bose-Einstein condensate of S positronium atoms Y. Zhang , M.-S. Wu , J.-Y. Zhang , , ∗ Y. Qian , X. Gao , and K. Varga State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China Beijing Computational Science Research Center, Beijing 100193, China Department of Computer Science and Technology,East China Normal University, Shanghai 200062, China and Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA (Dated: June 19, 2019)The confined variational method in conjunction with the orthogonalizing pseudo-potential methodand the stabilization method is used to study the low energy elastic scattering between two spin-polarized metastable positronium Ps(2 S ) atoms. Explicitly correlated Gaussian basis functionsare adopted to properly describe the complicated Coulomb interaction among the four chargedparticles. The calculated s -wave scattering length ( ≈ . a ) is positive, indicating the possibilityof forming a stable Bose-Einstein condensate of fully spin-polarized Ps(2 S ) atoms. Our resultswill open a new way of experimental realization of Ps condensate and development of γ -ray andPs(2 S ) atom lasers. PACS numbers: 34.80.Bm, 34.80.Uv, 03.65.Nk
The positronium (Ps) atom, a hydrogen-like boundsystem of an electron and a positron, has two groundstates, the singlet 1 S state and the triplet 1 S state,known respectively as para -Ps ( p -Ps) and ortho -Ps ( o -Ps). P -Ps has a lifetime of 0.125 ns [1] and decaysinto two gamma photons, and o -Ps has a lifetime of142 ns [2] and decays into three gamma photons. Thelonger lived o -Ps atoms were amongst the first candidates[3] for achieving Bose-Einstein condensation (BEC), aphase transition of a bose gas where a macroscopic num-ber of bosons occupy a same quantum state below a criti-cal temperature. BEC is one of the most interesting phe-nomena in quantum systems of bosons that has impor-tant applications such as in testing the weak equivalenceprinciple and in studying gravitational effects on quan-tum systems [4]. Historically, the first BEC was realizedin an ensemble of Rb atoms in 1995 [5], which opened anew era of ultracold physics. For o -Ps, the smallness ofits mass allows for much higher BEC temperature of 20-30 K than ordinary atoms around 200 nK [6, 7]. However,a realization of BEC of o -Ps atoms has been hindered byits very short lifetime.Since Platzman and Mills suggestion of a possible wayof creating an o -Ps BEC in 1994 [3], some significantprogress has been made both theoretically and experi-mentally. Low energy scattering between two ground-state Ps atoms has been extensively studied for calcu-lating scattering cross sections and the s -wave scatteringlengths [8–12]. These quantities are critical for determin-ing possibility of forming a stable ground-state Ps BECand for designing experimental configurations. In orderto probe Ps densities, some low-energy scattering prop-erties of the ground- and 2 s -state of Ps have been com-puted using hyperspherical coordinates [13]. For model-ing a BEC process of o -Ps atoms confined in a poroussilica material, Morandi et al. [14] showed that the con- densation process is compatible with the o -Ps lifetime,which strongly depends on the external electromagneticfield [15]. There are also some theoretical works on γ -ray laser [16, 17] and spinor dynamics [18, 19] based onBEC of Ps atoms. From experimental side, significantprogress has been made in the area of Ps-laser physicsdue to the breakthrough development of the Surko typebuffer gas positron trap [20–22]. Recently, implantinghigh density bursts of polarized positrons into a poroussilica film in a high magnetic field, Cassidy et al. pro-duced a highly spin-polarized (96%) o -Ps gas [23]. More-over, the suppression of the Zeeman mixing of the 2 P and 2 P states of Ps observed in high magnetic fields [24]makes laser cooling of Ps feasible [25–28]. Typically, Psatoms produced in most porous materials can approachroom temperature at the currently highest achievabledensity 10 cm − [23]. To form a Ps BEC, one needs notonly increase the Ps density but also significantly reducethe temperature of Ps gas. However, the short lifetimeof o -Ps seriously limits application of advanced coolingtechniques, such as laser cooling, developed for ordinaryatoms [29]. So far, o -Ps has been the only focus of allPs-BEC related studies, although some experimental andtheoretical researches have been conducted on the longer-lived metastable and Rydberg states of Ps [30–38].In this Letter, we will explore an alternative possibil-ity of forming a BEC using metastable Ps ∗ (2 S ) atoms.In the following, the notation Ps ∗ (2 S ) is abbreviatedas Ps ∗ . The Ps ∗ has a lifetime of 1136 ns that is eighttimes as long as the lifetime of o -Ps [2]. By calculatingthe s -wave scattering length for the spin-aligned Ps ∗ -Ps ∗ elastic scattering that governs the interaction betweenPs ∗ atoms at low temperatures, we will see whether it isa positive value, which is a necessary condition for form-ing a stable BEC. Ps ∗ -atom scattering problem is oneof the most difficult problems in atomic collision the- a r X i v : . [ phy s i c s . a t o m - ph ] J un ory because both projectile and target are composite ob-jects with their internal structures. One has to deal withmulti-center integrals of interaction matrix elements. Afurther complication to these calculations lies in the factthat both colliding Ps ∗ atoms are in the excited 2 S state so that one should have basis functions to be ableto accurately describe both short- and long-range (vander Waals) interactions, in particular for low energy scat-tering.Based on the existing computational techniques [39,40], a novel method i.e. the confiend variation method(CVM) has been proposed recently [41, 42] for studyinglow-energy elastic scattering between a simple or compos-ite projectile with an atom. The CVM combined with theorthogonalizing pseudo-potential (OPP) method [43–46]will be applied to study the s -wave elastic scattering be-tween two spin-aligned Ps ∗ atoms. The principal resultof this work is that, for the first time, we have estab-lished a definitive value of the s -wave scattering lengththat has positive sign, indicating that a stable BEC ofspin-aligned Ps ∗ atoms can be formed. Theory for the spin-aligned Ps ∗ -Ps ∗ scattering .—Thenonrelativistic Hamiltonian for the four-body system of( e + e + e − e − ) can be written in the form (in atomic units) H = − (cid:88) i =0 ∇ r i | r − r | − | r − r | − | r − r |− | r − r | − | r − r | + 1 | r − r | , (1)where r and r are the two-positron position vectors,and r and r the two-electron position vectors. Forcalculating the s -wave elastic scattering of Ps ∗ -Ps ∗ , weuse the OPP method [43–46] to prevent any electron-positron pair from forming the ground state Ps(1 S ).The OPP operator is constructed by summing over thePs(1 S ) projection operators λ ˆ P = λ (cid:88) i =0 3 (cid:88) j =2 ˆ P ij (2)= λ (cid:88) i =0 3 (cid:88) j =2 | φ S ( r i − r j ) (cid:105)(cid:104) φ S ( r i − r j ) | , where λ is a large positive number and φ S ( r i − r j )is the wave function of Ps(1 S ). Since a wave functionwith a nonzero overlap with the Ps(1 S ) orbital tends toincrease the energy, an eigenfunction of H + λ ˆ P for a lowenergy level will have a very small overlap with φ S .The OPP method was first introduced by Krasnopolskyand Kukulin [43] in 1974. Mitroy and Ryzhikh performeda comprehensive numerical investigation on the effectsdue to different λ and different sizes of basis sets [46]and found that the energies calculated with OPP willconverge to those of the ˆ Q ˆ H ˆ Q Hamiltonian in the pro-jection operator method [47], a method that has been widely used in studying atomic and molecular resonantand excited states. Compared to the ˆ Q ˆ H ˆ Q method, theOPP method is easier to apply for scattering problems.The ( e + e + e − e − ) system possesses rich symmetries in-cluding the electron interchange symmetry, the positroninterchange symmetry, the inversion parity, and thecharge parity. These symmetries can be described bya permutation group isomorphous to the molecular pointgroup D h . A detailed analysis of the symmetries ispresented in Ref. [48]. For the fully spin-aligned Ps ∗ -Ps ∗ scattering, the total spin operators ˆ S and ˆ S z havegood quantum numbers S = S z = 2. In the calculationof the s -wave elastic scattering, even parities are usedfor both inversion and charge conjugation. After takingthese symmetries into account, the total symmetry pro-jector applied to the spacial part of the basis function is(1 + P P )(1 − P )(1 − P ), where P ij is the permu-tation of the spatial coordinates of particles i and j .Of crucial importance to this work is the use of ex-plicitly correlated Gaussians (ECGs) [39, 49–51] to de-scribe the Coulomb interaction between the charged par-ticles. An ECG basis can not only describe the correla-tions among the charged particles, but also allows us toevaluate the Hamiltonian matrix elements analytically.After separating out the center-of-mass motion from the( e + e + e − e − ) system, an ECG can be written in the formΦ n = exp − (cid:88) i ≥ j ≥ A nij x i · x j , (3)where x i = r i − r . The independent parameters A nij con-tained in symmetric matrices A n are optimized throughthe energy minimization using the confined variationalmethod (CVM). The CVM is simple and powerful in thesense that it converts a problem of continuum states toa problem of bound states by adding a confining poten-tial χ cp ( ρ ) to the Hamiltonian. This method providesa framework for optimizing wave functions in the inter-action region using bound-state techniques. The advan-tages of using the CVM have been demonstrated by solv-ing some long-term intractable problems, including the e + -H scattering and Ps-H scattering, where the calcu-lated annihilation parameters are, for the first time, inagreement with precise experimental values [52, 53]. Inthis work, the confining potential is chosen to be χ cp ( ρ ) = 0 , ρ < R , (4) χ cp ( ρ ) = G ( ρ − R ) , ρ ≥ R , (5)where ρ is the distance between the two centers of massof two electron-positron pairs, and G is a small positivenumber. We set R = 50 a because the long-range in-teraction V L = − C /ρ , where C = 27320 a.u. [54], isrequired by the CVM to be weak at the boundary R .Due to the exchange symmetries between the identicalparticles and their indistinguishability, the Schr¨odingerequation for the confined Ps ∗ -Ps ∗ system can be writtenin the form (cid:104) H + λ ˆ P + χ cp ( ρ ) + χ cp ( ρ ) (cid:105) Ψ i ( x ) = E i Ψ i ( x ) , (6)where ρ = | x + x − x | / ρ = | x − x + x | / x stands for { x , x , x } collectively. Besides theECG basis for the short-range interaction region, as asupplement a set of exterior basis functions are designedto describe the long-range interaction between the twoPs ∗ , as listed belowΦ ext = exp (cid:0) − α i ρ (cid:1) φ Ps ∗ ( x ) φ Ps ∗ ( x − x ) , (7)where φ Ps ∗ is the Ps ∗ wave function written as a lin-ear combination of 20 ECGs that give rise to an energyeigenvalue of − .
062 499 999 999 a.u. very close to theexact value of -0.0625 a.u. for 2 S . A total of 10 even-tempered exponents α i are generated using α i = α /T i − with α = 0 . T = 1 . C ( ρ )to the asymptotical density distribution in the range ρ ∈ [46 a , a ], where C ( ρ ) is defined as C ( ρ ) = (cid:82) d x d x d x δ (( x + x − x ) / − ρ ) × | Ψ( x , x , x ) | . (8)To take into account the effect of the long range potential,the radial Schr¨odinger equation of the potential V L ( ρ )scattering was numerically integrated inwards with theasymptotical wave function B sin( kρ + δ k ) /ρ . Then afterfitting the phase shift δ k for small wave number k to theeffective-range expansion [55] k cot( δ k ) = − A + 12 r k , (9)one obtained the scattering length A . Results and discussion .— Table I presents the threelowest wave numbers, their corresponding phase shifts,and the determined s -wave scattering length at variousstages of optimization for the inner basis functions. Dur-ing the optimization, the OPP parameter λ = 1 and theCVM parameter G = 1 . × − were used in order toavoid linear dependence as we enlarged the size of ba-sis set as large as possible. However, in order to obtainmore accurate scattering parameters listed in Table I, thelargest λ that we could choose without loss of precisionwas 5000, which was used for the stabilization calcula-tions instead of λ = 1. As the size of basis becomeslarger, the scattering length A becomes smaller. Fig-ure 1 plots the phase shift δ k for small wave numberscalculated with three different values of λ . These lines FIG. 1. S -wave phase shift δ k for the fully spin-aligned Ps ∗ -Ps ∗ scattering as a function of wave number k computed withthree different values of the OPP parameter λ . The linesrepresent effective-range fits to the phase shifts using Eq. (9). represent the effective-range fits to these data points ob-tained with the largest basis set composed of 5000 inte-rior basis functions and 10 exterior basis functions. Inaddition, the s -wave scattering lengths A obtained withdifferent λ are summarized in Table II. One can see fromFigure 1 and Table II that, as λ increases from 100 to5000, the overall accuracy of the phase shift and the scat-tering length improve. From the trends of A in Tables Iand II, we estimated the exact value of A around 26 . a .The validity of the above scattering calculation canbe independently evaluated using the relation betweenthe scattering parameters and the bound state param-eters according to the quantum defect theory [56–58].Zhang et al. recently predicted the existence of a quasi-bound resonance state of Ps ∗ ( S ) [59]. Using the OPPmethod and ECG basis, the calculated resonance positionis at − .
127 11 a.u. [59], which is about 8 . × − a.u.lower than the resonance position − .
126 24 a.u. [59] esti-mated with the complex rotation (CR) method [60]. Ac-cording to the angular-momentum-insensitive quantumdefect theory (AQDT) [56] for a diatomic system withthe − C /ρ long-rang interaction, the relation between s -wave scattering length and binding energy can be ex-pressed as (cid:15) s (cid:39) − A s − ¯ A s ) − .
438 755 2( A s − ¯ A s ) + 0 .
216 313 9( A s − ¯ A s ) . (10)In the above, (cid:15) s is the scaled energy defined by (cid:15) s = (cid:15) ( (cid:126) / µ )(1 /β ) (11) TABLE I. Wave numbers k i , corresponding phase shifts δ k i , and determined s -wave scattering length A for the fully spin-aligned Ps ∗ -Ps ∗ elastic scattering, with the OPP parameter λ = 5000. In the first column, the first entry is the dimension ofthe interior region basis and the second entry is for the exterior basis. N k ( a − ) k ( a − ) k ( a − ) δ k (rad) δ k (rad) δ k (rad) A ( a )4000+10 0.004531 0.008925 0.01415 –0.1286 –0.2547 –0.3904 28.94500+10 0.004515 0.008887 0.01408 –0.1236 –0.2377 –0.3676 27.35000+10 0.004502 0.008853 0.01400 –0.1195 –0.2272 –0.3524 26.3TABLE II. S -wave scattering length for the fully spin-alignedPs ∗ -Ps ∗ elastic scattering obtained with different values of theOPP parameter λ . λ
100 1000 5000 A ( a ) 23.8 25.2 26.3 and ¯ A s = 2 π/ (Γ(1 / is the mean s -wave scatteringlength [61] scaled by the length factor β = (2 µC / (cid:126) ) / .In addition, A s = A /β is the scaled scattering lengthand µ is the reduced mass. Using Eq. (10), we obtained A ≈ . a and A ≈ . a , respectively, for the OPPand CR binding energies against the threshold − .
125 a.uof two Ps ∗ atoms. The consistence between AQDT andstabilization calculations illustrates that A ≈ . a isa reliable estimate for the scattering length.The positive value of A means that in princi-ple it is possible to produce a stable BEC of fullyspin-aligned Ps ∗ atoms. For a low-density Bose gas,the interaction is dominated by the low-energy s -wave elastic scattering. Therefore, the s -wave scat-tering length plays an important role in the accuratedescription of static and dynamic properties of thelow-density Bose gas. For a gas of bosonic atomstrapped in a harmonic oscillator potential with an an-gular frequency ω HO and an oscillator length a HO = (cid:112) (cid:126) / ( mω HO ), for example, the interaction is scaled bythe ratio A /a HO . For a Ps ∗ -BEC, this ratio becomes A (Ps ∗ ) /a HO = 7 . − / √ (cid:126) ω HO , which is largerthan A ( o -Ps) /a HO = 0 .
80 (eV) − / √ (cid:126) ω HO for o -Ps [8].Compared with A (H) /a HO = 9 . − / √ (cid:126) ω HO for apolarized H [62], however, the interaction effect in Ps ∗ -BEC is obviously weaker and thus it is effectively closerto the ideal BEC than the H-BEC. Using the value of A (Ps ∗ ) /a HO , not only the Gross-Pitaevskii equationcan be solved to study the properties of the ground-statePs ∗ -BEC but also the dynamics of phase transition canbe investigated using the mean-field theory [63].The necessary conditions for a realization of Ps ∗ -BECare as follows. Firstly, one should be able to producea sufficiently high number of polarized Ps ∗ atoms in aconfined void. Secondly, the Ps ∗ gas has to be cooleddown to sufficiently low temperature. For cooling itcan be achieved by thermalization through collisions ofPs ∗ with the walls of the void, Ps ∗ -Ps ∗ scattering, lasercooling, and other cooling methods. There has been a long interest in producing Ps ∗ [64–66] due to their po-tential applications [37] in testing quantum electrody-namics (QED), in atom interferometry, and in gravi-tational interaction of antimatter. The available tech-niques of Ps ∗ production include radio frequency tran-sition from laser-excited Ps(2 P) in a weak magneticfield [64], two-photon Doppler-free Ps(1 S)-Ps(2 S) laserexcitation [65, 66], single-photon excitation of Ps(1 S)to Ps(2 P) in an electric field [34], and radiative de-cay of Ps(3 P) generated by single-photon excitation ofPs(1 S) [35, 36, 38]. In particular, the efficiency of Ps ∗ production has recently been increased to 30% by stim-ulating the Ps(3 P)-Ps ∗ transition using a laser pulse,and further improvement in efficiency is still possible [38].In addition, the highly polarized o -Ps gas and its corre-sponding production techniques [23] will benifet to pro-ducing of highly polarized Ps ∗ gas. The produced Ps ∗ canapproach the room temperature 300 K after thermaliza-tion through collisions of Ps ∗ with the walls of a void andthrough Ps ∗ -Ps ∗ scattering. However, the density is fiveor more orders of magnitude lower than the required oneat the corresponding temperature. Therefore, it is nec-essary to further cool down the Ps ∗ ensemble using lasercooling or other advanced cooling methods. So far o -Pslaser cooling has not been experimentally realized dueto the serious limitation of its short lifetime [24–28]. Amajor advantage of using Ps ∗ over o -Ps to realize a BECis that Ps ∗ has much longer lifetime than o -Ps. How-ever, laser cooling Ps ∗ is very challenging and new coolingmeathodologies and technologies are required [25].The Ps ∗ -BEC can be applied to study fundamentalphysics and create new technologies once it is formed.Ideally, it is possible to realize the transformation fromPs ∗ -BEC to o -Ps-BEC through the stimulated transitionfrom Ps ∗ to o -Ps. It is also possible to produce gamma-ray laser through the stimulated transition from Ps ∗ to p -Ps followed by the corresponding two-photon annihila-tion. Moreover, a coherent beam of Ps ∗ atoms, the so-called Ps ∗ atom laser, can be generated from a Ps ∗ BEC.Employing a Ps ∗ atom laser as the Ps ∗ source will signifi-cantly improve the accuracy of measurements on matter-antimatter gravitational interaction and on Ps precisionspectroscopy [67, 68] which have been proposed to testQED and physics beyond the Standard Model [69, 70]such as the dark matter, and it will be of great bene-fit to producing cold antihydrogen atoms [71, 72]. Fur-thermore, a coherent beam of Ps ∗ atoms as a tool willenrich Ps chemistry to study various interactions withother atoms and molecules. Summary .— In this letter, the near-zero-energy s -waveelastic scattering between two fully-spin-aligned Ps ∗ hasbeen studied using a combined approach of OPP method,CVM, and stabilization method. The calculated s -wavescattering length represents the first determination of thisquantity. The positive value of the scattering length( ≈ . a ) is particularly significant since it demon-strates the feasibility of forming a stable BEC of fully-spin-aligned Ps ∗ atoms and hence it becomes possible fordeveloping γ -ray and Ps ∗ lasers based on Ps ∗ -BEC. Acknowledgments — J. Y. Z acknowledges S. Yi forvaluable discussion and hospitality during his visit at theInstitute of Theoretical Physics, Chinese Academy of Sci-ences. We would also like to thank Z.-C. Yan and W.-M.Liu for their helpful discussion. J. Y. Z. was supported bythe Hundred Talents Program of the Chinese Academyof Sciences. X. G. is supported by the National NaturalScience Foundation of China (Grant Nos. 11774023 andU1530401), the National Key Research and DevelopmentProgram of China (Grant No. 2016YFA0302104). ∗ Email address: [email protected][1] P. A. M. Dirac, Proc. Camb. Philos. Soc. , 361 (1930).[2] A. Ore and J. L. Powell, Phys. Rev. , 1696 (1949).[3] P. M. Platzman and A. P. Mills, Phys. Rev. B , 454(1994).[4] D. E. Bruschi, C. Sab´ın, A. White, V. Baccetti, D. K. L.Oi, and I. Fuentes, New J. Phys. , 053041 (2014).[5] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E.Wieman, and E. A. Cornell, Science , 198 (1995).[6] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E.Wieman, and E. A. Cornell, , 198 (1995).[7] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G.Hulet, Phys. Rev. Lett. , 1687 (1995).[8] I. A. Ivanov, J. Mitroy, and K. Varga, Phys. Rev. Lett. , 063201 (2001).[9] J. Shumway and D. M. Ceperley, Phys. Rev. B , 165209(2001).[10] K. Oda, T. Miyakawa, H. Yabu, and T. Suzuki, J. Phys.Soc. Jpn , 1549 (2001).[11] I. A. Ivanov, J. Mitroy, and K. Varga, Phys. Rev. A ,022704 (2002).[12] K. M. Daily, J. von Stecher, and C. H. Greene, Phys.Rev. A , 012512 (2015).[13] M. D. Higgins, K. M. Daily, and C. H. Greene,arXiv:1904.04295 (2019).[14] O. Morandi, P.-A. Hervieux, and G. Manfredi, Phys.Rev. A , 033609 (2014).[15] N. Cui, M. Macovei, K. Z. Hatsagortsyan, and C. H.Keitel, Phys. Rev. Lett. , 243401 (2012).[16] H. K. Avetissian, A. K. Avetissian, and G. F. Mkrtchian,Phys. Rev. Lett. , 023904 (2014).[17] H. K. Avetissian, A. K. Avetissian, and G. F. Mkrtchian,Phys. Rev. A , 023820 (2015). [18] Y.-H. Wang, B. M. Anderson, and C. W. Clark, Phys.Rev. A , 043624 (2014).[19] C. Zheng, P. Zhang, and S. Yi, Commun. Theor. Phys. , 236 (2017).[20] C. M. Surko, M. Leventhal, and A. Passner, Phys. Rev.Lett. , 901 (1989).[21] T. J. Murphy and C. M. Surko, Phys. Rev. A , 5696(1992).[22] J. R. Danielson, D. H. E. Dubin, R. G. Greaves, andC. M. Surko, Rev. Mod. Phys. , 247 (2015).[23] D. B. Cassidy, V. E. Meligne, and A. P. Mills, Phys.Rev. Lett. , 173401 (2010).[24] D. B. Cassidy, T. H. Hisakado, H. W. K. Tom, and A. P.Mills, Phys. Rev. Lett. , 173401 (2011).[25] E. P. Liang and C. D. Dermer, Opt. Commun. , 419(1988).[26] H. Iijima, T. Hirose, M. Irako, M. Kajita, T. Kumita,H. Yabu, and K. Wada, J. Phys. Soc. Jpn , 3255(2001).[27] T. Hirose, T. Asonuma, H. Iijima, M. Irako, K. Kadoya,T. Kumita, B. Matsumoto, N. N. Mondal, K. Wada, andH. Yabu, Laser Cooling of Ortho-Positronium: TowardRealization of Bose-Einstein Condensation (2014).[28] K. Shu, X. Fan, T. Yamazaki, T. Namba, S. Asai,K. Yoshioka, and M. Kuwata-Gonokami, J. Phys. B ,104001 (2016).[29] D. B. Cassidy, Eur. Phys. J. D , 53 (2018).[30] K. P. Ziock, R. H. Howell, F. Magnotta, R. A. Failor,and K. M. Jones, Phys. Rev. Lett. , 2366 (1990).[31] J. Estrada, T. Roach, J. N. Tan, P. Yesley, andG. Gabrielse, Phys. Rev. Lett. , 859 (2000).[32] F. Castelli, I. Boscolo, S. Cialdi, M. G. Giammarchi, andD. Comparat, Phys. Rev. A , 052512 (2008).[33] D. B. Cassidy, T. H. Hisakado, H. W. K. Tom, and A. P.Mills, Phys. Rev. Lett. , 043401 (2012).[34] A. M. Alonso, S. D. Hogan, and D. B. Cassidy, Phys.Rev. A , 033408 (2017).[35] S. Aghion et al. (AEgIS Collaboration), Phys. Rev. A ,013402 (2018).[36] C. Amsler et al. (AEgIS Collaboration), Phys. Rev. A , 033405 (2019).[37] D. B. Cassidy, H. W. K. Tom, and A. P. Mills, AIP Conf.Proc. , 66 (2008).[38] M. Antonello et al. (AEgIS Collaboration),arXiv:1904.09004 (2019).[39] Y. Suzuki and K. Varga, Stochastic Variational Approachto Quantum-Mechanical Few-Body Problems (Springer,New York, 1998).[40] J. Y. Zhang and J. Mitroy, Phys. Rev. A , 012703(2008).[41] J. Mitroy, J. Y. Zhang, and K. Varga, Phys. Rev. Lett. , 123201 (2008).[42] J. Y. Zhang, J. Mitroy, and K. Varga, Phys. Rev. A ,042705 (2008).[43] V. M. Krasnopolskij and V. I. Kukulin, Yad. Fiz. , 883(1974).[44] A. K. Bhatia, A. Temkin, and J. F. Perkins, Phys. Rev. , 177 (1967).[45] A. K. Bhatia and A. Temkin, Phys. Rev. A , 2018(1975).[46] J. Mitroy and G. Ryzhikh, Comput. Phys. Comm. ,103 (1999).[47] G. G. Ryzhikh, J. Mitroy, and K. Varga, J. Phys. B ,3965 (1998). [48] D. M. Schrader, Phys. Rev. Lett. , 043401 (2004).[49] S. F. Boys, Proc. R. Soc. London A , 402 (1960).[50] K. Singer, Proc. R. Soc. London A , 412 (1960).[51] W. Cencek and J. Rychlewski, J. Chem. Phys. , 1252(1993).[52] J.-Y. Zhang, J. Mitroy, and K. Varga, Phys. Rev. Lett. , 223202 (2009).[53] J.-Y. Zhang, M.-S. Wu, Y. Qian, X. Gao, Y. Y.-J., K. Varga, Z.-C. Yan, and U. Schwingenschl¨ogl,arXiv:1803.03026 (2018).[54] Y. Zhang, M.-S. Wu, and J.-Y. Zhang, (unpublished).[55] G. W. F. Drake, Springer Handbook of Atomic, Molecularand Optical Physics (Springer, New York, 2006) p. 668.[56] B. Gao, J. Phys. B , 4273 (2004).[57] X. Gao and J.-M. Li, Phys. Rev. A , 022710 (2014).[58] X. Gao, X.-Y. Han, and J.-M. Li, J. Phys. B , 214005(2016).[59] Y. Zhang, M.-S. Wu, J.-Y. Zhang, and K. Varga, (un-published).[60] Y. K. Ho, Phys. Rep. , 1 (1983).[61] G. F. Gribakin and V. V. Flambaum, Phys. Rev. A ,546 (1993).[62] M. J. Jamieson, A. Dalgarno, and M. Kimura, Phys. Rev. A , 2626 (1995).[63] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari,Rev. Mod. Phys. , 463 (1999).[64] A. P. Mills, S. Berko, and K. F. Canter, Phys. Rev. Lett. , 1541 (1975).[65] S. Chu, A. P. Mills, and J. L. Hall, Phys. Rev. Lett. ,1689 (1984).[66] M. S. Fee, A. P. Mills, S. Chu, E. D. Shaw, K. Danzmann,R. J. Chichester, and D. M. Zuckerman, Phys. Rev. Lett. , 1397 (1993).[67] A. P. Mills and M. Leventhal, Nucl. Instrum. MethodsPhys. Res. B , 102 (2002).[68] M. Oberthaler, Nucl. Instrum. Methods Phys. Res. B , 129 (2002).[69] S. G. Karshenboim, Phys. Rev. Lett. , 220406 (2010).[70] S. Kotler, R. Ozeri, and D. F. J. Kimball, Phys. Rev.Lett. , 081801 (2015).[71] A. S. Kadyrov, C. M. Rawlins, A. T. Stelbovics, I. Bray,and M. Charlton, Phys. Rev. Lett. , 183201 (2015).[72] B. Mansouli´e and on behalf of the GBAR Collaboration,Hyperfine Interact.240