Level crossing in the three-body problem for strongly interacting fermions in a harmonic trap
aa r X i v : . [ c ond - m a t . o t h e r] S e p Level crossing in the three-body problem for strongly interacting fermionsin a harmonic trap
J. P. Kestner, L.-M. Duan
FOCUS center and MCTP, Department of Physics, University of Michigan, Ann Arbor, MI 48109
We present a solution of the three-fermion problem in a harmonic potential across a Feshbachresonance. We compare the spectrum with that of the two-body problem and show that it isenergetically unfavorable for the three fermions to occupy one lattice site rather than two. We alsodemonstrate the existence of an energy level crossing in the ground state with a symmetry change ofits wave function, suggesting the possibility of a phase transition for the corresponding many-bodycase.
I. INTRODUCTION
Ultracold atoms, tuned with Feshbach resonance, offera great opportunity to study strongly correlated many-body physics in a controlled fashion. For such stronglyinteracting systems, in general there is no well-controlledapproximation method to solve the many-body physics.Exact solution of few-body problems plays an importantrole in understanding the corresponding many-body sys-tems. Few-body (three- or four-body) problems havebeen solved for strongly interacting bosons or fermionsin free space [1], in a quasi-one-dimensional configura-tion [2], and in a three-dimensional (3D) harmonic trapfor bosons [3] or fermions in the unitary limit [4, 5].In this paper, we add to this sort of examples by ex-actly solving the three-body problem for strongly inter-acting two-component fermions in a 3D harmonic trapacross resonance. This work has two main motivations:Firstly, the situation considered here is relevant for ex-periments where one loads strongly interacting two com-ponent fermions into a deep 3D optical lattice [6]. Foreach site that can be approximated with a harmonic po-tential, one could have two identical fermions (spin- ↑ )strongly interacting with another distinct fermion (spin- ↓ ). The three-body problem for equal mass fermionsturns out to be very different from the corresponding casefor bosons. Instead of a hierarchy of bound Efimov statesfor bosons [3], we show that there is always a significantenergy penalty for three fermions to occupy the samelattice site ( ↑↑↓ ) instead of two ( ↑↓ + ↑ ). This resultjustifies an important assumption made in the derivationof an effective many-body Hamiltonian for this system[7]. Secondly, we analyze the ground-state structure ofthe three-body problem and show that as one scans the3D scattering length, there is a level crossing betweenthe lowest-lying three-fermion energy eigenstates, whichhave s - or p -wave symmetries respectively in the limitas two atoms are contracted to form a dimer. This levelcrossing with a symmetry change may correspond a quan-tum phase transition in the many-body case where onehas spin-polarized fermi gas loaded into an optical lat-tice. The latter system with polarized fermions has raisedstrong interest recently in both theory and experiments[8, 9]. II. METHODS
The method here is based on manipulation of aLippmann-Schwinger equation for the wavefunction, withthe formalism similar to the one presented in Ref. [2].As is standard, we separate the center-of-mass degree offreedom via an orthogonal transformation of variables,leaving the trapping potential diagonal in the new coordi-nates. The Schr¨odinger equation for the relative degreesof freedom is then (cid:20) − ~ m (cid:0) ∇ x + ∇ y (cid:1) + 14 m ω (cid:0) x + y (cid:1) − E (cid:21) Ψ ( x , y )= − X ± V ( r ± ) Ψ ( x , y ) , (1)where m is the atomic mass, ω is the trap frequency, y is the vector between the two ↑ fermions, √ x / ↑ fermions tothe ↓ fermion, and r ± = √ x / ± y / ↓ fermion to each of the two ↑ fermions. We ap-proximate the short-range interaction between fermionswith the usual zero-range pseudopotential [10] V ( r ) = π ~ am δ ( r ) ∂∂r ( r · ), where a is the 3D s-wave scatteringlength tunable through the Feshbach resonance.The above contact interaction is equivalent to impos-ing boundary conditions Ψ ( x , y ) ≃ ∓ f ( r ⊥ , ± )4 π r ± (cid:0) − r ± a (cid:1) for r ± →
0, where the r ⊥ , ± = x / ∓ √ y / ↑↓ pair and an ↑ fermion. The overall ∓ sign ensuresthe antisymmetry of the wavefunction upon swapping theidentical fermions. The undetermined function f ( r ⊥ , ± ),after a rescaling of the argument, is the relative atom-dimer wavefunction that results when two of the fermionsform a tightly bound pair. Solving for this asymptoticwavefunction f ( r ⊥ , ± ) fully determines Ψ ( x , y ).Since V ( r ± ) only acts at r ± = 0, we use the asymp-totic form of Ψ when computing V ( r ± ) Ψ. The formalsolution can then be written asΨ ( x , y ) = Z d x ′ d y ′ G (2) E ( x , y ; x ′ , y ′ ) × X ± ∓ ~ f ( r ′⊥ , ± ) m δ ( r ′± ) (2)where the two-particle Green’s function is given by G (2) E ( x , y ; x ′ , y ′ ) = X λ λ ψ λ ( x ) ψ λ ( y ) ψ ∗ λ ( x ′ ) ψ ∗ λ ( y ′ ) E λ + E λ − E (3)The ψ λ i ( x ) are the single-particle eigenfunctions witheigenenergies E λ i for the reduced mass m /
2. Here weuse spherical coordinates for the three-dimensional har-monic trap, so the quantum numbers are λ = ( n, l, m ), n = 0 , , , ... ; l = 0 , , , ... ; m = − l, − l + 1 , ..., l − , l .The eigenenergies are E λ = (2 n + l + 3 / ~ ω and theeigenfunctions are ψ λ ( r ) = R nl ( r ) Y ml ( θ, φ ), where the Y ml ( θ, φ ) are the standard spherical harmonics. The ra-dial wavefunction is given by [11] R nl ( r ) = s n ! /d ( n + l + 1 / e − r / d (cid:16) rd (cid:17) l L l +1 / n (cid:0) r /d (cid:1) (4)where d = q ~ m ω is the length scale of the trap and the L kn ( r ) are associated Laguerre polynomials.We can make use of the invariance of G (2) E and theintegration measure under an orthogonal transformationof variables to rewrite Eq. (2) in terms of r ≡ r − and r ⊥ ≡ r ⊥ , − ,Ψ ( r , r ⊥ ) = ~ m Z d r ′⊥ f ( r ′⊥ ) (cid:20) G (2) E ( r , r ⊥ ; 0 , r ′⊥ ) − G (2) E r √ r ⊥ , √ r − r ⊥ , r ′⊥ !(cid:21) (5)We can also decompose the asymptotic atom-dimer wave-function in terms of the complete set of single-particlewavefunctions, f ( r ⊥ ) = P λ f λ ψ λ ( r ⊥ ). Then Eq. (5)becomesΨ ( r , r ⊥ ) = d ~ ω X λ (cid:20) G E − E λ ( r , ψ λ ( r ⊥ ) − G E − E λ r √ r ⊥ , ! ψ λ √ r − r ⊥ !(cid:21) f λ (6)where G E is the single-particle Green’s function, G E ( r ,
0) = X λ ψ λ ( r ) ψ ∗ λ (0) E λ − E = e − r / d π / d ~ ω Γ (cid:18) − E/ ~ ω (cid:19) U (cid:18) − E/ ~ ω , , r d (cid:19) (7)[12], and U is the confluent hypergeometric function.Note that the three-fermion wavefunction is fully deter-mined by f ( r ⊥ ) and that if we consider Eq. (6) in thelimit as r →
0, we obtain a self-consistent equation for f ( r ⊥ ) by using the boundary conditions. After some −5 0 5−505 d/a E ( un i t s o f ¯ h ω ) FIG. 1: Energy vs. inverse scattering length. −15 −10 −5 0 5 10 1500.511.522.53 d/a E − E − / ( un i t s o f ¯ h ω ) spds FIG. 2: Difference between three-fermion energy and two-fermion energy plus one-fermion energy vs. inverse scatteringlength. work, we obtain X λ ′ A λλ ′ f λ ′ = da − (cid:16) / E λ / ~ ω − E/ ~ ω (cid:17) Γ (cid:16) / E λ / ~ ω − E/ ~ ω (cid:17) f λ (8)where A λλ ′ = Z d r ⊥ πd ~ ω G E − E λ ′ √ r ⊥ , ! ψ ∗ λ ( r ⊥ ) ψ λ ′ (cid:18) − r ⊥ (cid:19) (9) III. RESULTS
We anticipate that the low-energy physics should becontained in a truncated Hilbert space containing onlythe lowest few asymptotic atom-dimer energy levels.Then Eq. (8) is easily solved numerically. We havechecked that, indeed, the solution for the ground stateand the first excited manifold become insensitive to thecutoff, as long as the first four or five atom-dimer energy
FIG. 3: (Color online) Relative coordinates for the three-fermion problem. In terms of variables used in Eq. (6), r = | r | , R = √ | r ⊥ | / −5 (a) n = 0, l = 0, m = 0 −5 (b) n = 0, l = 1, m = ± −5 (c) n = 0, l = 1, m = 0 FIG. 4: Contour plots of r | Ψ ( r, R, θ ) | r → as a function of R and θ for r/d ≃ d/a ≃ levels are included. For all results presented in this pa-per, we have kept the first five energy levels. Because ofthe degeneracy of the excited levels, Eq. (8) becomes a35 ×
35 matrix equation. For a given energy, we solve nu-merically to obtain the corresponding scattering lengthand eigenstate. By sweeping through a range of energies,we map out the spectrum shown in Fig. 1. At unitarity,our result for the low-energy spectrum agrees with theanalytic result of Ref. [4]. Upon careful inspection, onecan discern the presence of level crossings. More usefully,in Fig. 2 we display the difference between the energy ofthree fermions in a single lattice site and the energy of −5 (a) n = 0, l = 0, m = 0 −7 (b) n = 0, l = 1, m = ± −4 (c) n = 0, l = 1, m = 0 FIG. 5: Contour plots of r R | sin θ Ψ ( r, R, θ ) | as a functionof R and θ for r/d = 2 and d/a ≃ two fermions ( ↑↓ ) in the site and the extra fermion alonein a separate site. Here we have used the well-knownexact solution for the two-body energy E [12], da = 2 Γ (cid:16) / − E / ~ ω (cid:17) Γ (cid:16) / − E / ~ ω (cid:17) (10)Figure 2 is our main result. There are two main fea-tures we would like to point out. First, it is clear that itis energetically favorable for atoms in a lattice to arrangethemselves such that there are less than three atoms persite, regardless of the scattering length. This has alreadybeen assumed in the derivation of an effective many-bodyHamiltonian for atoms in an optical lattice across a Fes-hbach resonance [7], and is confirmed by Fig. 2. Second,the level crossing in the ground state is now quite evi-dent. On the positive scattering length side of the cross-ing (the BEC side, where the many-body system forms aBose-Einstein condensate of bound dimers), the groundstate is nondegenerate. On the other side (the BCS side,where the many-body system forms a Bardeen-Cooper-Schrieffer superfluid of atomic Cooper pairs), it is triplydegenerate. Other crossings appear in the excited spec-trum, although to obtain quantitatively accurate results d/a ≃ +3 d/a ≃ d/a ≃ − d/a ≃ − d/a ≃ +14 (a)Normalized probability density vs. r/d (b)Normalized probability density vs. R/d π π π π π (c)Normalized probability density vs. θ FIG. 6: (Color online) Normalized probability density distribution functions of variables defined in Fig. 3 for various scatteringlengths. The solid line is for the n = 0, l = 0, m = 0 state; the dashed line is for the n = 0, l = 1, m = ± n = 0, l = 1, m = 0 state. for these one should include higher modes when solvingEq. (8).The origin of the level crossing is the differing symme-tries of the eigenstates. For very small, positive scatter-ing length (deep BEC side), formation of tightly bounddimers is favorable, so the state should behave as theground state of the relative atom-dimer motion, whichhas s-wave symmetry. For very small, negative scatter-ing length (deep BCS side), the atoms are essentiallynon-interacting, so the ground state comprises two atoms( ↑↓ ) in the ground state of the trap plus the third in thefirst excited state of the trap (which is triply degener-ate) because of Pauli exclusion. So, on the deep BCSside, the ground state has p-wave symmetry. Due to therotational symmetry of a spherical harmonic trap, thetotal angular momentum of the three particles should bea conserved quantity. However, from the above analysis,this quantity has different values for the ground state in the deep BEC and deep BCS limits. Therefore, theremust be a ground-state level crossing for this system asone scans the scattering length. If one considers multiplelattice sites with each site having on average two spin ↑ and one spin ↓ atoms (which could be realized withpolarized fermions in an optical lattice with appropriatefilling number and population imbalance), as the three-body problem has a level crossing with different groundstate degeneracies in the BCS and the BEC limits, therecould be a corresponding quantum phase transition forthis many-body system (with small tunneling betweenlattice sites) as one scans the scattering length.The wavefunction given in Eq, (6) does not generallyhave definite relative angular momentum for any twofermions. However, in the limit as the distance betweentwo distinguishable fermions goes to zero, the wavefunc-tion takes on the symmetry of the asymptotic atom-dimerwavefunction in the remaining coordinates. This is a rel-ative angular momentum eigenstate due to the sphericalsymmetry of the limiting case. With the relative coordi-nates defined by Fig. 3, the symmetry of the wavefunc-tion as a function of R and θ in the limit as r goes to zerois shown by Fig. 4. For finite r , the m = 0 wavefunc-tions are affected by the asymmetry and take nontrivialshapes. We have plotted an example in Fig. 5. Notethat in this figure we include a factor of sin θ since thewavefunction itself diverges at R = r/ θ = π accordingto the boundary conditions we have imposed. On theBEC side, the lobes are tightly bunched near θ = π , buton the BCS side they spread around as expected.In general, the eigenstates are rather difficult to vi-sualize, since they depend nontrivially on three spatialvariables as well as the scattering length. However, onecan get some idea of the evolution of the eigenstates fromFig. 6, which shows the normalized probability densityas a function of the variables introduced in Fig. 3 for var-ious scattering lengths. In each subplot we have numeri-cally integrated over the other two variables to obtain aone-dimensional function. The dimer size, r , clearly de-creases in size as one enters the BEC regime, developinga strong peak at the origin. However, as is clear fromFig. 3, there are two ways to form a tightly bound dimer(due to the two identical spin ↑ atoms) and we have arbi-trarily chosen one to define the origin r = 0. So it is notsurprising that we see a more diffuse second peak at large distance r , corresponding to the dimer forming betweenthe spin ↓ and the other spin ↑ atom. This is also themeaning of the spike at θ = π on the BEC side. IV. SUMMARY
We have found the low-lying energy levels of threefermions in a harmonic trap and examined the corre-sponding wavefunctions. The ground state has s-wavesymmetry on the BEC side of Feshbach resonance andhas p-wave symmetry on the BCS side. In the resonanceregion there is a level crossing, which may indicate aphase transition in the corresponding many-body case.We also note that, in the vicinity of resonance, the en-ergy of three atoms in a single site is greater than theirenergy if they are in two sites, with a gap on the orderof the trap spacing, validating the approximation in Ref[7].This work was supported by the MURI, the DARPA,the NSF award (0431476), the DTO under ARO con-tracts, and the A. P. Sloan Fellowship.Note added: After completion of this work, we becameaware of a recent work [13] which treats the trappedthree-fermion problem with a different approximationmethod. [1] D.S. Petrov, Phys. Rev. A , 010703(R) (2003); D.S.Petrov, C. Salomon, G.V. Shlyapnikov, Phys. Rev.Lett. , 090404 (2004); E. Braaten, H.-W. Hammer,Phys.Rept. , 259-390 (2006).[2] C. Mora, R. Egger, A.O. Gogolin, and A. Komnik, Phys.Rev. Lett. , 170403 (2004); C. Mora, A. Komnik, R.Egger, and A. O. Gogolin, Phys. Rev. Lett. 95, 080403(2005).[3] S. Jonsell, H. Heiselberg and C.J. Pethick, Phys. Rev.Lett. , 250401 (2002); M. Stoll and T. K¨ohler, Phys.Rev. A , 022714 (2005).[4] F. Werner and Y. Castin, Phys. Rev. Lett. , 150401(2006).[5] S. Y. Chang and G. F. Bertsch, Phys. Rev. A ,021603(R) (2007).[6] T. St¨oferle, H. Moritz, K. G¨unter, M. K¨ohl, T. EsslingerPhys. Rev. Lett. 96, 030401 (2006); J.K. Chin et al.,Nature London 443, 961 (2006). [7] L.-M. Duan, Phys. Rev. Lett. , 243202 (2005);arXiv:0706.2161.[8] M.W. Zwierlein, A. Schirotzek, C.H. Schunck, and W.Ketterle, Science , 492 (2006); G.B. Partridge, W.Li, R.I. Kamar, Y. Liao, and R.G. Hulet, Science ,503 (2006).[9] D. E. Sheehy, L. Radzihovsky, cond-mat/0607803; W.Yi, L.-M. Duan, Phys. Rev. A , 031604(R) (2006); T.N. De Silva, E. J. Mueller, Phys. Rev. Lett. 97, 070402(2006).[10] Kerson Huang and C.N. Yang, Phys. Rev. , 767(1957).[11] J.L. Powell and B. Crasemann, Quantum Mechanics (Addison-Wesley, Reading, MA, 1961), Chapter 7.[12] T. Busch, B.-G. Englert, K. Rzazewski, M. Wilkens,Found. Physics28