Candidate theories to explain the anomalous spectroscopic signatures of atomic H in molecular H 2 crystals
aa r X i v : . [ c ond - m a t . o t h e r] A p r Candidate theories to explain the anomalous spectroscopic signatures of atomic H inmolecular H crystals Kaden R.A. Hazzard ∗ and Erich J. Mueller Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853
We analyze a number of proposed explanations for spectroscopic anomalies observed in atomichydrogen defects embedded in a solid molecular hydrogen matrix. In particular, we critically evaluatethe possibility that these anomalies are related to Bose-Einstein condensation (both global and local).For each proposed mechanism we discuss which aspects of the experiment can be explained, andmake predictions for future experiments.
I. INTRODUCTION AND MOTIVATION
Quantum solids, where the zero point motion of theatoms is greater than roughly 10% of their separation,form a fascinating class of materials. A principal questionwith these materials is to what extent they are quantumcoherent, and under what conditions they can be super-solid – supporting dissipationless mass flow. Examples ofquantum solids include He , solid hydrogen , Wignercrystals, and atomic hydrogen defects in solid molecu-lar hydrogen . Here we theoretically study the lastsystem, giving a critical evaluation of scenarios of Bose-Einstein condensation of atomic hydrogen defects. Wemake testable predictions for these scenarios.Solid hydrogen, with a Lindemann ratio of 0.18, is theonly observed molecular quantum crystal. It is a rich sys-tem with many rotational order/disorder transitions .The phenomenology of H solids is even more interestingwhen atomic H defects are introduced . Here we focuson the spectroscopic properties of this system, and howthey may be related to quantum coherence.Our work is motivated by recent experiments in low-temperature ( T ∼ n ∼ cm − ) of atomichydrogen defects. These experiments observe unex-plained internal state populations . Ahokas et al. provocatively conjectured that the anomalies may be re-lated to Bose-Einstein Condensation (BEC) of the atomicdefects. Here our goal is to explore and constrain thisand alternative scenarios. We conclude that global Bose-Einstein condensation could not realistically explain theexperiments, and that local condensation would lead todistinct signatures in future experiments. II. EXPERIMENTS
We review Ahokas et al. ’s experimental apparatus,results, and observed anomalies. A. General introduction: physics of atomichydrogen embedded in solid hydrogen
Hyperfine structure of atomic hydrogen.
Fig. 1schematically shows the level structure for a H atom in a B = 4 .
6T magnetic field, similar to that used in theexperiments of interest . At these large fields, the levelsbreak into two nearly degenerate pairs. Levels within apair are separated by radio frequencies and the pairs areseparated by microwave frequencies. The electronic spinin states a and b is aligned with the magnetic field andis anti-aligned in the other states. | a i = cos θ |↓ −↑i − sin θ |↑ −↓i | b i = |↓ −↓i| d i = |↑ −↑i | c i = cos θ |↑ −↓i + sin θ |↓ −↑i FIG. 1: Hyperfine level diagram for hydrogen atom in a B =4 .
6T magnetic field, where the mixing angle is θ = 3 × − .Vertical axis schematically denotes energy, while horizontalaxis has no physical meaning. Arrows denote electron (no-slash arrow) and nuclear (slashed arrow) spin projections. Ahokas et al. observe that within the solid hydrogenmatrix the atomic hydrogen’s spectra is modified. The a-d energy splitting decreases while the b-c energy splittingincreases by the same amount. Spectroscopy of atomic hydrogen.
We considerthree different spectroscopic probes, distinguished by thestates they connect: nuclear magnetic resonance (NMR),electron spin resonance (ESR), and electron-nuclear dou-ble resonance (ENDOR). NMR uses radio waves to cou-ple the a and b states, while ESR uses microwaves tocouple the a and d or b and c states. ENDOR uses a twophoton transition to couple a and c . All of these probescan be used in the linear regime, where they give in-formation about the splittings and occupation numbers,or in the non-linear regime. As an example of the latter,Ahokas et al. study what happens when they apply highRF power, saturating the NMR line. Molecular hydrogen.
At these temperatures, twostates of molecular hydrogen are relevant: the “para”and “ortho” states. In the “para” or “p-” configurationthe relative nuclear wavefunction ψ is symmetric underexchanging nuclei and the nuclear spins form a singlet. Inthe “ortho” or “o-” configuration ψ is antisymmetric andthe nuclear spins form a triplet. In all cases, the electronsare in a symmetric bonding orbital and the electronicspins are consequently anti-aligned.The true ground state of solid hydrogen is formed ofp-H . In practice, however, it is very rare to have a purep-H sample. The ortho state is long lived , requiringhours for the concentration to change by 1%. This ortho-para conversion can be a source of heating in experimentswith an energy ∆ /k B = 170 K released per molecule. Inthe experiments of Ahokas et al. the exact quantity ofo-H is unknown, but given the growth technique it islikely to be at least ten percent.At standard pressure the p-H in the solid is highlyspherical: interaction with neighboring H negligibly dis-torts the p-H . Modeling the hydrogen-hydrogen inter-actions by their vacuum values quite accurately describesquantities such as the speed of sound . If sufficientortho-hydrogen is present, orientational ordering transi-tions may occur around 1K. The models we consider donot rely upon any orientational ordering. Depending onsample-preparation conditions, either hcp or fcc crystalsmay be produced . Atomic hydrogen in the solid lattice.
The mo-tion of atomic hydrogen in molecular hydrogen has beenwidely studied . Both thermally activated and quan-tum tunneling contribute to defect motion, but quan-tum tunneling dominates at these low temperatures (thetwo dominant tunneling pathways have energy barriersof 4600K and 100K). One motivation for these studies isto explore the possibility of Bose-Einstein condensationof these defects. This is conceptually related to superso-lidity driven by condensation of vacancies.Kumada argues on the basis of experimental data thatthe exchange reaction H + H → H + H is the dominantdiffusion mechanism at low temperatures. Other tunnel-ing pathways are possible, including correlated, collectiverelaxation and “physical” diffusion.Ahokas et al. achieve populations of 50ppm H defectsin their solid, and argue that H sits at substitutionalsites. The observed lifetime of these defects was weeks.The dominant decay mechanism should be the recombi-nation of two hydrogen defects. One therefore expectsthat this rate is determined by the diffusion rate of thedefects. The long lifetime is therefore inconsistent withthe diffusion rates predicted by phonon assisted tunnel-ing. One possible explanation is the suppression of tun-neling by the strain-induced mismatch of energy levelsbetween neighboring sites . Crystal growth.
Ahokas et al. grow solid hydro-gen from a gas of electron-spin polarized metastable hy-drogen atoms, which undergo two-hydrogen recombina-tion to form molecules – these events are only allowed inthe presence of walls. The H solid grows layer-by-layerfrom at a rate of 0.5-1 molecular layer per hour. After ∼ ± . B. Anomalies and experimental results
Ahokas et al. observed four anomalies: (1) severalorders-of-magnitude too fast “Overhauser” relaxation,(2) a non-Boltzmann a-b population ratio, (3) satura-tion of the a-b spectroscopic line fails to give a 1:1 pop-ulation ratio, and (4) recombination rates are extremelylow. Items 2 and 3 will be our main focus. Overhauser relaxation.
The c to a relaxation isexpected to be extremely small at the 4 .
6T fields of theexperiments. This can be seen from the small mixingangle θ ≈ × − , indicated in Fig. 1. The mixing angleappears in the states as | a i = cos θ |↓ −↑i − sin θ |↑ −↓i , and | b i = cos θ |↑ −↓i − sin θ |↓ −↑i . Any c-a decay mechanismby photon emission is suppressed by θ α with α ∼ > et al. observe no such suppression: the c-a linedecays with a time constant of ∼ < Equilibrium populations.
The polarization p ≡ n a − n b n a + n b (1)characterizes the a and b state population. Assuminga Boltzmann distribution n a /n b = exp (∆ ab /T ) where∆ ab ≈ p = 0 .
14 at 150 mK. On thecontrary, Ahokas et al. measure p = 0 . ∼ Saturation of a-b line.
An extremely strong rf fieldshould saturate the a:b line, driving the population ratioto 1:1, or p = 0. Ahokas et al. obtained a minimum of p = 0 . Low recombination rates.
As previously described,at 150mK Ahokas et al. ’s recombination rate is muchsmaller than expected, negligible on a time scale of weeks.In contrast, at T = 1K their recombination rates areconsistent with previous studies . Hole burning.
Ahokas et al. applied a magneticfield gradient and a rf field to saturate the a-b line in amillimeter sized region of the sample. The spectral holerecovered in a time similar to that in the homogeneouscase, indicating that the nuclear spin-relaxation is some-what faster than spin migration. This would seem toindicate that the atomic hydrogen defects are immobileon regions much larger than a millimeter.
III. SCENARIOS
In this section we evaluate some previously proposedscenarios for these phenomena and suggest a new one .For each scenario, we present the idea, examine its con-sistency with Ahokas et al. ’s experiments, consider pos-sible microscopic mechanisms, and give testable predic-tions. A. Bose statistics and Bose-Einstein condensation
1. Idea
Ahokas et al. suggested Bose-Einstein condensation(BEC) as a possible mechanism to explain the departurefrom the Boltzmann distribution. In a BEC the lowestenergy mode becomes “macroscopically occupied,” lead-ing to an excess of a -H. Neglecting interactions, the BECtransition temperature T c for a homogeneous system ofspinless particles with density ρ and effective mass m ∗ is T c = (cid:18) ρζ (3 / (cid:19) / π ~ k B m ∗ . (2)Here ζ is the Riemann zeta function; ζ (3 / ≈ . .Including the b states is straightforward and makes onlysmall changes: for example if ∆ ab = 0 one would dividethe density by a factor of 2. Note that as m ∗ becomeslarger the transition temperature becomes smaller.
2. Phenomena explainable
This scenario can in principle explain the non-Boltzmann equilibrium ratio n b /n a . It does not pro-vide an explanation of the inability to saturate the a - b line, the slow recombination, or the fast Overhausercross-relaxation.
3. Consistency with experiment
Transition temperature and densities.
As re-ported in Ahokas et al. , for density ρ ∼ cm − andeffective mass m ∗ similar to bare mass m , the ideal Bosegas transition temperature in Eq. (2) is T c ∼ et al. suggest that phase separa-tion may concentrate the defects to locally higher densi-ties. For example, if the defects phase separated so thattheir density was ρ ∼ × cm − , then with m ∗ = m the transition temperature would be T c = 170 mK,and one would reproduce the observed ratio of n a /n b at T = 150 mK. Such a powerful concentrating mechanismwould have additional consequences, such as increasedrecombination rates.A key question is how large the effective mass is. Inthe following subsection we use the experimental hole-burning data to constrain the effective mass, finding thatit is sufficiently large to completely rule out simple Bose-Einstein condensation of defects at the experimental tem-peratures. Estimate of effective mass.
We expect that theeffective mass m ∗ of the defects is much higher than thatof the free atoms. Here we bound the effective mass byconsidering Ahokas et al. ’s measurement of the lifetimeof a localized spectral hole. They found that a w ∼ . τ pers > a -state, spin exchangecollisions can lead to spin diffusion, or a -state atoms candiffuse back into that region of space. Neglecting all butthe last process gives us an upper bound on the atomicdiffusion constant D , D ∼ < w τ pers ≈ − cm /s. (3)Throughout this argument we will aim to produce anorder-of-magnitude estimate, and use the symbol “ ∼ ” toindicate that we neglect constants of order unity. Thisdiffusion constant can be related to the microscopic col-lision time τ coll and the mean velocity v by D ∼ vℓ. (4)with ℓ the mean free path. In the effective mass approxi-mation, one would expect thermal effects to yield a meanvelocity v ∼ p k B T /m ∗ . An lower bound for the meanfree path is given by the lattice constant ℓ ∼ > d ∼ .Thus we arrive at the following lower bound for the ef-fective mass m ∗ ∼ > k B T (cid:18) d − cm / s (cid:19) ≈ m H , (5)where m H is the bare hydrogen mass and the last ap-proximate equality is for the experimental temperature T = 150mK. This effective mass is several orders of mag-nitude too large to allow BEC at experimentally relevanttemperature scales, regardless of the defect concentra-tion. This argument has neglected interactions and in-homogeneities: phase separation or some “local” BEC’sthat are uncoupled, thus disallowing global transport,would invalidate the arguments leading to our bound.
4. Microscopic mechanism
Microscopic estimates of tunneling matrix elements arebeyond the scope of this work. As previously discussed,the exchange reaction mechanism is expected to be thedominant pathway .
5. Experimental predictions
Polarization temperature dependence.
Perhapsthe most easily testable prediction of this model isthe temperature dependence of the polarization, p =( n a /n b − / ( n a /n b + 1) with n a n b = R d k n [ ǫ a ( k ) / ( k B T )] R d k n [ ǫ b ( k ) / ( k B T )] (6)with n ( x ) ≡ / ( e x − ǫ a ( k ) ≈ ~ k / m ∗ , and ǫ b ( k ) ≈ ~ k / m ∗ +∆ ab , where∆ ab is the b-a energy difference. Above the BEC transi-tion temperature T c , the integrals yield n a n b = g / (cid:0) e µ/T (cid:1) g / (cid:0) e ( µ − ∆ ab ) /T (cid:1) (7)where g α ( x ) = P j x j /j α is the polylog function and µ isself-consistently determined to set N , for a homogeneous,three-dimensional gas. The same expression holds in di-mension d with 3 / d/
2. Below T = T c , oneinstead finds n a n b = (cid:18) T c T (cid:19) / ζ (3 /
2) + g / (cid:0) e − ∆ ab /T c (cid:1) g / (cid:0) e − ∆ ab /T (cid:1) − . (8)While the effective mass sets the density at T c , it does notappear in this expression. The polarization depends onlyon ∆ ab /T and T /T c . To produce the observed n a /n b = 3at T = 150mK, one needs T c = 170mK.Fig. 2 shows p ( T ) for Boltzmann and Bose condensed(assuming T c = 200mK) gases. Accurately measur-ing p ( T ) would clearly distinguish Bose and Boltzmannstatistics.Equation (7) shows that Bose statistics can affect theratio n a /n b even if the system is non-condensed. How-ever, Eq. (7) bounds n a /n b < ζ (3 / /g / ( e − ∆ ab /T ) =2 .
3, which is insufficient to explain the experimentallyobserved n a /n b = 3. H mK L p = n a - n b n a + n b FIG. 2: The polarization versus temperature for the Boltz-mann case (solid line) and the Bose-condensed case (dashedline), from Eq. 8.
Transport.
A second signature of BEC is superflow.For example, the sample could be incorporated into a tor-sional oscillator, providing a measurement of a possiblenonclassical moment of inertia I : below T c , the superflowdecouples from the cell, and I decreases. Mounting the sophisticated hydrogen growth and measurement equip-ment in an oscillator would be challenging, as would thedifficulty of working with such small samples. Bimodal Cold Collision Shifts.
Bose Einstein con-densation also has implications for the cold collision shiftsin the atomic spectra. Insofar as the interaction may bedescribed by the s-wave scattering length the a - d line willbe shifted by δω = − π ~ m g (0) ( a ↑↓ − a ↓↓ ) h n i (9)where g ( r ) ≡ (cid:10) ψ † ( r ) ψ † (0) ψ (0) ψ ( r ) (cid:11) / h n i , h n i is theaverage density, and a ↑↓ , a ↓↓ are the a - a and b - b scat-tering lengths, respectively. For a noninteracting BEC g (0) = 1 while for a normal gas g (0) = 2. For apartially condensed gas one in fact sees a bimodal spec-trum with two peaks: one from the condensed atomsand one from the noncondensed atoms. This techniquerevealed BEC in magnetically trapped spin-polarized hy-drogen gas . Thermodynamics and collective excitations.
The BEC phase transition can in principle be directlyobserved by monitoring thermodynamic quantities suchas specific heat. Due to the small number of H atoms,the signal should be quite small. Similarly, the presenceof a superfluid component would lead one to expect asecond-sound mode, which could be excited (for exam-ple) via localized heating of the sample. B. Local
Bose-Einstein condensation
1. Idea
Next we pursue the idea of “local BEC”, where thedefects aggregate in small disconnected regions, each ofwhich contains a condensate, but which have no relativephase coherence.
2. Phenomena explainable
This model can explain the non-Boltzmann ratio n b /n a , and the slow transport observed in the hole burn-ing experiments. It does not provide an explanation ofthe slow recombination, failure to saturate, or fast Over-hauser relaxation.
3. Consistency with experiment
The arguments from Section III A about the polariza-tion go through without change. The slow recovery inthe hole burning experiment is readily explained if thedisconnected condensates are smaller than 0.1mm. Fur-thermore, local clusters are naturally expected if there isthe dramatic sort of concentrating mechanism describedin Section III A.
4. Microscopic mechanism
An attractive interaction between defects can lead toclustering. A long distance phonon mediated attractionis expected for this system . Inhomogeneities in themolecular sample, or its environment could also lead toclustering. For example, it has been observed that theortho and para molecules phase separate. Furthermore,the surface that the sample sits on creates inhomogeneousstrains, which couple to defects. This can possibly causethem to accumulate.
5. Experimental predictions
The temperature dependence calculated for globalBEC is unchanged for local BEC, and one again expectsa double-peaked ESR spectrum. Jointly observing thesewould provide a “smoking gun” for local BEC. We notethat at sufficiently cold temperatures the puddles phaselock giving a global BEC. However, the transition tem-perature would be effectively zero since atoms would haveto tunnel macroscopic distances between concentrated re-gions. The local concentration of H ↑ , regardless of BEC,may be diagnosed by examining the dipolar shift of spec-tral lines due to H-H interactions. C. Nuclear spin dependent Density-of-states
Here we outline a non-BEC scenario for the anomalousobservations.
1. Idea
If the degeneracy of the a and b states were g a and g b ,one would expect that n a /n b = ( g a /g b ) e β ∆ ab . Thus if amechanism could be found to enhance ( g a /g b ), then onecould explain the observed ratio of n a /n b . Assuming sucha relative enhancement of the density of states, a strongRF field would lead to a saturated ratio ( n a /n b ) sat = g a /g b .
2. Phenomena explainable
Both the equilibrium n a /n b and the RF saturated( n a /n b ) sat can be explained by this model. We also pro-vide a scenario whereby the Overhauser relaxation is en-hanced. Within this model, the low recombination ratescould be a consequence of the defects being immobile.
3. Consistency with experiment
At strong excitation powers the polarization p =( n tot ,a − n b ) / ( n tot ,a + n b ) saturates to p sat = g − g + 1 (10)where g ≡ g a /g b . Meanwhile, the thermal polarization is p therm = g exp (∆ ab / ( k B T )) − g exp (∆ ab / ( k B T )) + 1 ≈ g − . g + 0 .
75 (11)where the last equation holds for Ahokas et al. ’s experi-ments, where ∆ ab = 43mK and T = 150mK. If one takes g = 2 one finds p sat = 0 .
33 and p therm = 0 .
45. Ahokas et al. experimentally find p sat = 0 . p therm = 0 . p sat = 0 . p therm = 0 .
4. Microscopic mechanism
There are very few mechanisms whereby the molecu-lar hydrogen matrix can change the degeneracies of theatomic hydrogen hyperfine states. The most plausiblesource would be to consider nuclear spin dependent in-teractions with o-H . Such interactions can be producedthrough spin-orbit coupling in the presence of a bias mag-netic field. Hybridization of the molecular and atomiclevels could in principle lead to sufficiently drastic re-arrangements of the hyperfine states to affect their de-generacy. Such a strong interaction would presumablyhave other spectroscopic implications, such as a severerenormalization of the a - b splitting. Unfortunately, theexperiments observe that the a - b splitting is changed byonly 0.1% compared to its vacuum value.If there is significant hybridization of the atomic andmolecular states, then the symmetry which forbids the a - c transition would generically be broken. This wouldbe a source of the fast Overhauser relaxation.
5. Experimental predictions
One would expect that at low temperatures, g shouldbe roughly independent of temperature. Thus the tem-perature dependence of the polarization in Eq. (11) canbe compared with the experiment. For any given mech-anism additional predictions are possible. As one ex-ample, in a spin-orbit mechanism hinted at above, thephysics should also manifest in the populations of theo-H states, which can be probed spectroscopically. Fi-nally, under further assumptions, one may be able to pre-dict the power dependence of the polarization saturationexperiments. IV. OTHER OBSERVATIONS
We would like to point out three other possibly rel-evant observations. First, Ceperley et al. have shownthat the surface of small p-H clusters in a vacuum aresuperfluid . Similarly, Cazorla et al. have shown that onsmall length scales 2D p-H has superfluid correlations .Analogous effects are predicted for defects in solid He,including grain boundaries, dislocations, and amorphousregions . Ahokas et al. ’s crystals are of very highquality, and it is doubtful that grain boundaries and dis-locations are playing an important role. On the otherhand, one could imagine that the the interface of o- andp-H clusters could play a similar role.Second, it seems possible that the spectroscopicanomalies are related in some way to torsional oscilla-tor mass decoupling observed in solid hydrogen at sim-ilar temperatures. It is important to note that Clark etal. ruled out global supersolidity in their experimentsby comparing the response of the system in an open andblocked annulus. Local supersolidity, however, can alsogive rise to a small period drop, which would be presentin both the open and blocked annuli.A third observation is that magnetic ordering transi-tions — for example ferromagnetism — would alter theratio of a -state to b -state population. One can eliminateferromagnetic ordering, as it would give rise to an observ-able energy shift. On the other hand, perhaps a differentform of magnetic order is playing a role. V. SUMMARY, CONCLUSIONS, ANDCONSEQUENCES
We reviewed the unexplained phenomena seen in ex-periments of Ahokas et al. . We enumerated a number ofpossible mechanisms which could be involved in produc- ing the observed phenomena. In particular, we gave de-tailed consideration to the idea, first introduced in Ref. that the non-Boltzmann ratio n a /n b may be due to Bose-Einstein condensation of atomic hydrogen. We concludethat global Bose-Einstein condensation is not consistentwith other experimental observations.Although we present several other scenarios, we findthat none of them are wholly satisfactory. Although someof the phenomena can be explained by local BEC, it failsto provide a mechanism for the unexpected saturationpopulation ( n a /n b ) sat when a strong RF field is applied.We do find that all of the phenomena would be consistentwith a nuclear spin dependent density of states. However,we are unable to provide a microscopic mechanism forthis density of states.Ultimately, substantial experimental work will be nec-essary to clarify the situation. Our arguments make astrong case that measuring the polarization’s tempera-ture dependence is a promising first step, and suggestsother experimental signatures — especially in transportand spectral features — that would clarify the phenom-ena.During the preparation of this paper, new results cameout from Ahokas et al. , which introduced new myster-ies. In particular they observe substantial density andsubstrate dependence of the population ratio n a /n b . Allof our considerations remain valid, with the additionalclue that whatever the underlying mechanism is, it mustinvolve the surface of the sample, and be sensitive to den-sity. For example, the formation of superfluid domainscould be influenced by the substrate, or magnetic impuri-ties in the substrate could interact with atomic hydrogen,leading in some way to the unexpected density of states. Acknowledgments.
We would like to acknowl-edge discussions with David Lee, J¨arno Jarvinen, SergeiVasiliev, and Cyrus Umrigar. This work was supportedby NSF Grant PHY-0758104. ∗ Electronic address: [email protected] N. Prof’ev, Advances in Physics , 381 (2007). E. Kim and M. Chan, Nature , 225 (2004), 10.1038/na-ture02220. E. Kim and M. H. W. Chan, Science , 1941 (2004). A. S. C. Rittner and J. D. Reppy, Physical Review Letters , 175302 (2007). A. S. C. Rittner and J. D. Reppy, Physical Review Letters , 165301 (2006). T. Kumada, Phys. Rev. B , 052301 (2003). J. Ahokas, J. J¨arvinen, V. V. Khmelenko, D. M. Lee, andS. Vasiliev, Physical Review Letters , 095301 (2006). S. Vasilyev, J. J¨arvinen, V. V. Khmelenko, and D. M. Lee,AIP Conference Proceedings , 81 (2006). Nat Phys , 651 (2006), 10.1038/nphys435. I. F. Silvera, Rev. Mod. Phys. , 393 (1980). Quantum Tunnelling in Condensed Media, edited by Y.Kagan and A. J. Leggett (Elsevier Science, Amsterdam,North-Holland, 1992). A. V. Ivliev, A. S. Iskovskikh, A. Y. Katunin, I. I. Luka-shevich, V. V. Sklyarevkii, V. V. Suraev, V. V. Filippov,N. I. Filippov, and V. A. Shevtsov, JETP Lett. , 379(1983). N. V. Prokof’ev and G. V. Shlyapnikov, Sov. Phys. JETP , 1204 (1987). A. V. Ivliev, A. Y. Katunin, I. I. Lukashevich, V. V. Skl-yarevskii, V. V. Suraev, V. V. Filippov, N. I. Filippov, andV. A. Shevtsov, JETP Letters , 472 (1982). C. J. Pethick and H. Smith, Bose-Einstein condensation indilute gases (Cambridge University Press, The EdinburghBuilding, Cambridge CB2 2RU, UK, 2001). M. O. Oktel and L. S. Levitov, PRL , 6 (1999). M. O. Oktel, T. C. Killian, D. Kleppner, and L. S. Levitov,Phys. Rev. A , 033617 (2002). D. G. Fried, T. C. Killian, L. Willmann, D. Landhuis, S. C.Moss, D. Kleppner, and T. J. Greytak, Phys. Rev. Lett. , 3811 (1998). G. D. Mahan, Many-Particle Physics (Kluwer Aca- demic/Plenum Publishers, 233 Spring Street, New York,New York 10013, 2000). S. A. Khairallah, M. B. Sevryuk, D. M. Ceperley, and J. P.Toennies, arxiv 0612161 (2006). C. Cazorla and J. Boronat, arXiv:0807.0307v1 (2008). L. Pollet, M. Boninsegni, A. B. Kuklov, N. V. Prokof’ev,B. V. Svistunov, and M. Troyer, Physical Review Letters , 135301 (2007). M. Boninsegni, A. B. Kuklov, L. Pollet, N. V. Prokof’ev,B. V. Svistunov, and M. Troyer, Physical Review Letters , 035301 (2007). S¸. G. S¨oyler, B. Capogrosso-Sansone, N. V. Prokof’ev, andB. V. Svistunov, Physical Review A (Atomic, Molecular,and Optical Physics) , 043628 (2007). A. C. Clark, X. Lin, and M. H. W. Chan, Physical ReviewLetters , 245301 (2006). J. Ahokas, O. Vainio, S. Novotny, J. J¨arvinen, V. V. Khme-lenko, D. M. Lee, and S. Vasiliev, Phys. Rev. B , 104516(2010).27