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Supershell structure in trapped dilute Fermi gases Y. Yu, M. O¨gren, S. ˚Aberg, and S. M. Reimann Division of Mathematical Physics, LTH, Lund University. P.O. Box 118, S-221 00 Lund, Sweden M. Brack Institut fu¨r Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany 6 (Dated: November11, 2005) 0 0 We show that a dilute harmonically trapped two-component gas of fermionic atoms with a weak 2 repulsive interaction has a pronounced super-shell structure: the shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the “magic numbers” n occurringbetweenthebeatnodesbyhalfaperiod. Thelengthandamplitudeofthisbeatingmode a depend on the strength of the interaction. We give a simple interpretation of the beat structure in J terms of a semiclassical trace formula for the symmetry breaking U(3) → SO(3). 8 1 PACSnumbers: 03.75.Ss05.30.Fk ] r e In finite systems of fermions, quantum effects lead to ofatoms or by applying a magnetic field: Feshbachreso- h bunching of energy levels resulting in shell structure. nances[9]allowtotunethescatteringlengthfromalarge ot Well-known examples are the shell structures in atoms positivetoalargenegativevalue. (Here,wefocus onthe . or nuclei [1], determining their chemical properties and repulsive case.) t a stability. In fact, shell effects and the so-called “magic Due to the Pauli principle the δ interaction only ap- m numbers”,correspondingtosphericalshellclosings,have plies to fermions of pairwise opposite spin. We consider - been discovered in a variety of other finite fermion sys- a fully unpolarizedtwo-componentsystemwith twospin d tems. Metal clusters in which the delocalized valence states, so that the total particle density is composed n electrons are bound in the field of the metallic ions [2], of two different densities of equal magnitude, n(r) = o or quantum dots in semiconductor heterostructures [3] n↑(r)+n↓(r) = 2n↑(r). In the weak-interaction regime, c [ arefamousexamples. Morerecentexperimentalprogress the interaction energy density is given by gn↑(r)n↓(r)= makes it possible to study yet another species of finite gn2(r)/4, where the coupling strength parameter g is in- 3 quantal systems: atomic gases,often weakly interacting, troduced by v confined e.g. by an optical dipole trap [4]. 6 9 In this paper, we show that a harmonically trapped g =4π~2a/m. (1) 0 gas of fermionic atoms interacting by a weak repulsive This leads to the single-particle Hartree-Fock equation 2 two-body force may exhibit super-shell structure: the 0 shell oscillations of the spherical harmonic oscillator are ~2 5 modulated by a beat structure, whereby the positions of − ∆+gn↑(r)+V (r) ψ↓(r)=ǫ ψ↓(r), (2) 0 (cid:20) 2m ho (cid:21) i i i the magic numbers are shifted by half a period between / t successive beats. We can explain this surprising result a whereV istheharmonicoscillator(HO)trappotential. m semiclassically by the interference of diameter and cir- Thedhiloutenessconditionnecessarytotreattheinterac- cle orbits surviving the breaking of the U(3) symmetry - tion as a two-body process is that the interparticle spac- d of the harmonic oscillator by the leading anharmonicity ingn¯−1/3 ismuchlargerthantherangeoftheinteraction n term in the mean field. and that n¯a3 ≪ 1. This dimensionless parameter also o Similarsuper-shellstructurewaspredictedformetallic limits the life time ofthe two-componentatomicfermion c clusters [5], inspiredby a semiclassicalanalysisofBalian : gas due to dimer formation, which is a three-body pro- v and Bloch in terms of the periodic orbits in a spherical cess[11]. To guaranteethatthis conditionis fulfilled, we i cavity [6], and observed experimentally [7]. Analogous X calculate the central density n(0) in the Thomas-Fermi ideascouldbeappliedtothedescriptionofshellstructure r approximation and plot level curves of log(a3n(0)) in a a in transport properties of quantum wires [8]. g-N landscape seen in Fig. 1. The diluteness condition Let us consider a dilute gas of fermionic atoms, con- is seen to be fulfilled for all considered combinations of fined by a spherical harmonic potential modeling an ex- particle numbers and interaction strengths. This also ternaltrap[4],interactingthrougharepulsivezero-range outrules the possibility of phase separation discussed in two-body potential. The many-body Hamiltonian is [10]. Assuming spherical symmetry, Eq.(2) reduces to its N p2 m 4π~2a H = i + ω2r2 + δ3(r −r ), radial part (cid:18)2m 2 i(cid:19) m i j Xi=1 Xi<j ~2 1 ∂ ∂ ~2 l(l+1) − r2 + +U(r) ψ↓ =ǫ ψ↓, where a is the s-wave scattering length. The value and (cid:20) 2mr2∂r(cid:18) ∂r(cid:19) 2m r2 (cid:21) i i i the sign of a can be varied either by changing the type (3) 2 with U(r)=gn↑(r)+ 1mω2r2 being the effective mean- given by 2 field potential. Each state has a (2l + 1)-fold angular momentumdegeneracy. WesolveEq.(3)self-consistently Eho ≃(3N)23 ~ω ∞ (−1)k cos 2πk(3N)31 . (5) on a grid. The interaction term is updated (with some osc 2π2 k2 weight factors) in each iteration according to gn↑(r) = kX=1 (cid:16) (cid:17) g i|ψi↑(r)|2. Hereby k is the repetition number of the primitive clas- P sical periodic orbit of the system with action S (E) = 0 100 2πE/ω. The argument of the cosine function in Eq. (5) is simply k times S (E)/~, taken at the Thomas-Fermi −0.5 0 80 value of the Fermi energy E (N) = (3N)1/3~ω. The F gross-shell structure is governed by the lowest harmonic with k =1. 60 1/3 −1 Switching on the interaction, this scenario changes. A N beating modulation of the rapid oscillations is found. In 40 −1.5 Fig. 2 we show the shell energy versus N1/3 for three −2 values of the interaction strength, g=0.2, 0.4 and 2. A 20 −3 beating modulation of the amplitude of the shell energy, −5 i.e., a super-shell structure, is clearly seen to appear for all cases. At small particle numbers and particularly 0.5 1 1.5 2 2.5 g for small g values, the shell energy is that of the non- interacting system, given by Eq. (5). For larger inter- FIG. 1: (Color online)Level curves of 10log a3n(0) calcu- actionstrengths the super-shellstructure is more clearly latedwithintheThomas-Fermiapproximation(cid:0). g=4π(cid:1)~2a/m seen, and several beating nodes appear for g=2. With (~=ω=m=1). increasinginteractionstrengththeamplitudeoftheshell energy oscillations becomes smaller. For example, for Afterconvergenceisobtained,theground-stateenergy particle numbers around 803 ≈500000,the amplitude of oftheN-particlesystemisgivenby(E =Fermienergy) theshellenergyisabout40~ω,whichisonlyabout10−6 F of the total ground-state energy. Through Fourier analysis of the calculated shell en- E (g,N)= ǫ −g n2(r)d3r. (4) ergy, two frequencies are seen to smoothly appear with tot i ↑ Z increasing g value around the HO frequency (9.06), see ǫi≤EXF,σ=↑,↓ Fig. 3. The exact values of the two frequencies depend Ingeneral,theground-stateenergyasafunctionofN can on the range of particle numbers included in the analy- be written as the sum of a smooth average part and an sis. The super-shell features appear when the contribu- oscillating part, E =E +E . The oscillating part, tion to the effective potential from the interaction, gn↑, tot av osc referred to as the shell-correction energy, or shell energy is sufficiently large, i.e., at large values of g and N. We inshort,reflectsthequantizedlevelspectrum{ǫ }. Fora alsoobservethat(almost)until the firstsuper-node,i.e., i non-interactingFermi gasin a sphericallysymmetric 3D N1/3 ≈ 28 in Fig. 4, the magic numbers agree with the harmonic trap, the leading-order term for the average HOones(g =0). Betweenthefirsttwosuper-nodes,i.e., energy is found in the Thomas-Fermi approximation to 28 ≤ N1/3 ≤ 49 in Fig. 4, the magic numbers for the be[12]Eho =(3N)4/3~ω/4. Fortherepulsiveinteracting interacting system are situated in the middle of two HO av case, we find E (g >0) ∝ Nα with a larger exponent magic numbers, i.e., they appear at the maxima of the av α > 4/3. However, Eq. (2) with an interaction term fast shell oscillations. Then, after the second super-node linear in the density is only valid for moderate g values theyroughlyagreewiththeunperturbedHOonesagain. andinpracticeweareclosetoα=4/3(e.g.,α≈1.35for In the following we outline a semiclassical interpreta- g =2). Contrarytothe non-interactingcase,andalsoto tion of these features [13]. The U(3) symmetry of the self-saturatingfermionsystems(suchasnucleiandmetal unperturbedHO systemisbrokenbythe termδU =gn↑ clusters)with anearlyconstantparticledensity, itisnot in (2), resulting in the SO(3) symmetry of the interact- possible here to obtainthe smoothpart of the energy by ing system. The shortest periodic orbits in this system a simple expansion in volume, surface and higher-order are the pendulating diameter orbits and the circular or- terms. We therefore perform a numerical averaging of bits with a radius corresponding to the minimum of the the energy (4) over the particle number N in order to effective potential in (3) including the centrifugal term. extract its oscillating part. Thesetwoorbitsleadtotheobservedsupershellbeating. Inthenon-interactingcase(g=0)theshellenergyE The above symmetry breaking has so far not been dis- osc oscillates with a frequency 2π31/3 ≈ 9.06 as a function cussed in the semiclassical literature. In a perturbative of N1/3 and has a smoothly growing amplitude ∝N2/3. approach[14],itcanbeaccountedforbyagroupaverage This followsfromthe exacttraceformula[12]forE of of the lowest-order action shift ∆S(o) brought about by osc the 3D harmonic oscillator, whose leading-order term is theperturbationofthesystem: he~i ∆S(o)io∈U(3). Hereby 3 o is an element of the group U(3) characterizing a mem- 4-dimensional manifold CP2 [15], which for a perturba- ber of the unperturbed HO orbit family (ellipses or cir- tion δU(r)=εr4 can be done analytically [13]. cles). Forthe averageitissufficientto integrateoverthe 300 200 100 0 −100 c os −200 E −300 −400 −500 −600 −700 10 20 30 40 50 60 70 80 90 100 N1/3 FIG.2: (Coloronline)Theoscillatingpartofthegroundstateenergyinunitsof~ωasafunctionofN1/3 forg=0.2(blue),0.4 (red)and2(green). Thetwolowercurvesaredisplacedby400~ω and600~ω,respectively. Theverticaldottedlinescorrespond to theHO magic numbersN =M(M+1)(M +2)/3 for M =1,2,... mag In the perturbative regime (ε≪1) we find the following perturbed trace formula: ] s t ni m2ω4 ∞ (−1)k kS kS u Epert(N)= sin c −sin d ,(6) ry osc 2επ3Xk=1 k3 (cid:20) (cid:18) ~ (cid:19) (cid:18) ~ (cid:19)(cid:21) a 8 10 12 14 16 18 r bit where kSd and kSc are the classical actions of the diam- r eter and circle orbits, respectively. In the limit ε → 0, a [ their difference goes as k(S −S ) → kεπE2(N)/m2ω5, a c d F r so that (6) tends to the pure HO limit (5). Extracting t c ε from a polynomial fit to the numerical potential U(r) e 8 10 12 14 16 18 p in (3), one can qualitatively describe the beating of the s r shell energy E (N). With ε of order ∼ 5×10−4g, Eq. e osc ri (6) approximately reproduces the curves seen in Fig. 2 u o uptothebeginningofthesecondsupershell. Itexplains, F inparticular,alsothephasechangeinthepositionofthe 8 10 12 14 16 18 magic numbers Nmag shown in Fig. 4. ω (N1/3) Tocoverlargervaluesofε(andN),wehavedeveloped [13]ananalyticaluniformtraceformulaforthepotential U(r)=mω2r2/2+εr4, which contains the contributions FIG. 3: (Color online) Fourier spectra of theshell energy for g = 0.2 (blue), 0.4 (red) and 2 (green). A peak splitting aroundω=9.06isresolvedinthemiddleandbottompanels. The second harmonics (k=2) are seen around ω=18. 4 ofthe 2-folddegeneratefamilies ofdiameter andcircular with repetition numbers k ≥ 3 and can be included in orbits to all orders in ε. This is analogous to uniform the trace formula using standard techniques [18, 19, 20]. trace formulae obtained earlier for U(1) [16] and U(2) As mentioned above, in a Fermi gas of atoms with re- symmetry breaking [17]. pulsive interaction (a > 0), atoms can be lost through three-body recombination events. Two atoms with opposite spin form a molecule while the third takes up energy. Having a low recombination rate, and thus a long 30 lifetimeofthesystem,isdesired. Petrov[11]madeanes- timate of the loss rate of particles, n˙/n≈111 na3 2ǫ¯/~ 15 where ǫ¯ is the average kinetic energy of ato(cid:0)ms. (cid:1)Tak- ing ǫ¯≈ 10µK, a realistic energy scale in current experi- c ments, we estimate the life time of atoms in the trap to Eos 0 be 10−6s, 10−3s, 10−2s for g = 2, 0.4, 0.2 respectively, when the number of particles is so large that the first node of the super-shell is reached. Hence the life time is −15 longer when g (or a) is smaller, reflecting that the loss rate is proportional to a6. The temperature regime of −30 this super-shell structure is below 0.1µK. 25 30 35 40 45 50 55 In conclusion, we have seen that the shell structure of N1/3 fermions with weak,repulsive interactions in a harmonic FIG. 4: (Color online) An enlarged part of Fig. 2 for g = trapshowsapronouncedbeatingpattern,withthesingle 2. The circles mark the harmonic oscillator magic numbers shell positions changing by half a period length between N . the different beat nodes. A Fourier analysis of the os- mag cillating shell-correctionpartof the Hartree-Fockenergy The beat structure in Eosc has some similarities with shows clear peaks at two slightly different frequencies. thatfoundinnuclei[1]andmetalclusters[2]. Thereare, This is interpreted semiclassically by the interference of however, two essential differences. 1. Those systems are the shortest periodic orbits generated by the breaking self-saturating and have steep mean-field potentials that of the U(3) symmetry of the non-interacting HO sys- canbemodeledbyasphericalcavity[6]. Thepresentsys- tem, which arethe families of diameter andcircle orbits, tem, in contrast, has a mean field with much smoother through a uniform trace formula given fully in [13] and, walls that are dominated at large distances by the con- in the perturbative limit, in Eq. (6). fining harmonic potential. 2. The super-shells in the Forvery weakinteractions,the splittings ofthe highly cavity model come from the interference of the shortest degenerateHOlevelshaveearlierbeencalculatedpertur- periodic orbit families with three-fold degeneracy, as is batively within the WKB approximation [21]. However, usual in spherical systems [18, 19]. Here, however, the the perturbative results do not apply for the interaction gross-shell structure comes from the diameter and cir- strengths where super-shell structure appears visible. cle orbits which are only 2-fold degenerate, whereas the WeacknowledgediscussionswithA.Bulgac,S.Creagh, fully3-folddegeneratefamiliesoftoriwithrationalratios S. Keppeler, B. Mottelson and C. Pethick. This work ω :ω =n:mofradialandangularfrequencyonlycon- r ϕ wasfinanciallysupportedby the SwedishFoundationfor tributetothefinerquantumstructuresathigherenergies. Strategic Research and the Swedish Research Council. 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