aa r X i v : . [ phy s i c s . a t m - c l u s ] J a n Efimov physics in bosonic atom-trimer scattering
A. Deltuva
Centro de F´ısica Nuclear da Universidade de Lisboa, P-1649-003 Lisboa, Portugal (Received August 10, 2010)Bosonic atom-trimer scattering is studied in the unitary limit using momentum-space equationsfor four-particle transition operators. The impact of the Efimov effect on the atom-trimer scatteringobservables is explored and a number of universal relations is established. Positions and widths oftetramer resonances are determined. Trimer relaxation rate constant is calculated.
PACS numbers: 31.15.ac, 34.50.-s, 34.50.Cx
Few-particle systems with resonant interactions, char-acterized by the two-particle scattering length a beingmuch larger than the range of the interaction, werepredicted to have a number of universal (interaction-independent) properties and correlations between observ-ables [1–6]. Such a behavior was recently confirmed incold atom physics experiments [7, 8], but can be seenqualitatively also in few-nucleon systems [1, 2]. Oneof the best-known examples is the existence of the in-finite number of weakly bound three-boson states (Efi-mov trimers) in the unitary limit a = ∞ [1]. In thatlimit, depending on the available energy, an infinite num-ber of atom-trimer channels may be present in the four-boson system. Due to complexity of the multichannelfour-particle scattering problem the universal propertiesof such a system, i.e., the atom-trimer continuum, are notknown yet; we therefore aim to study them for the firsttime in the present work. Furthermore, the existence ofa pair of four-boson states (tetramers) associated witheach Efimov trimer was predicted [3, 5]. However, onlythe two tetramers associated with the trimer ground stateare true bound states that have been studied in all thedetails using standard bound state techniques [3–5, 9].Higher tetramers are resonances above the atom-trimerthreshold and for this reason their properties are far lessknown; they will be determined in the present work usingproper scattering calculations.Our description of the four-boson scattering is basedon the Alt, Grassberger and Sandhas (AGS) equations[10] for the transition operators; they are equivalent tothe Faddeev-Yakubovsky equations [11] for the wave-function components. Symmetrized form of AGS equa-tions [12] is appropriate for the system of four identicalbosons, U = P ( G tG ) − + P U G tG U + U G tG U , (1a) U = (1 + P )( G tG ) − + (1 + P ) U G tG U , (1b)where U βα are the four-particle transition operators, G is the four free particle Green’s function, and P is thepermutation operator of particles 3 and 4 that ensurescorrect permutation symmetry of the system. The dy-namic input is the two-boson potential v from which thetwo-particle transition-matrix t and the AGS transition operators U α are derived, with α = 1 and 2 correspond-ing to the 1 + 3 and 2 + 2 subsystems, respectively. Asexplained in Ref. [12], the atom-trimer scattering ampli-tudes are given by the on-shell matrix elements of U calculated between the Faddeev amplitudes of the corre-sponding initial and final states.We solve AGS equations using momentum-spacepartial-wave representation [12] where they are a sys-tem of coupled three-variable integral equations that af-ter the discretization of momentum variables becomesa very large system of linear algebraic equations. Inthe case of the four-nucleon scattering those equationshave been successfully solved with realistic nuclear andCoulomb interactions [13, 14]. In the present calculationswe take the numerical techniques for the treatment offour-particle permutations and trimer bound-state polesfrom Refs. [12–14]. However, an important difference ascompared to the four-nucleon system is the presence ofsharp resonances in the four-boson system, manifestingthemselves as poles of the AGS operators (1). Since theconvergence of the multiple scattering series in the vicin-ity of the pole is very slow, in this work we solve sys-tems of linear equations by the direct matrix inversioninstead of the iterative double Pad´e summation method[12]. Since we are interested in the universal propertiesthat must be independent of the interaction details, weuse rank 1 separable two-boson potentials v = | g i λ h g | acting in S -wave only and thereby reducing Eqs. (1) toa small system of two-variable integral equations. Al-though the two-boson interaction is limited to S -wave,i.e., l x = 0 in the notation of Ref. [12], higher angularmomentum states with l y , l z ≤ h k | g i = e − ( k/ Λ) .The strength λ is chosen to reproduce infinite two-bosonscattering length. In that limit all observables scale withΛ; e.g., the binding energy of the n -th excited trimer b n ∼ Λ . It therefore makes no sense to specify partic-ular value of Λ as well as boson mass. Instead, we willuse dimensionless ratios. As the length scale associatedwith the n -th excited trimer we will use l n = ~ / √ µ b n where µ is the reduced atom-trimer mass.We do not include explicit three-body force, however,many-body forces are simulated by a different off-shell n b n − /b n κ n − /κ n b n /κ n . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE I. Trimer properties obtained with one-term (top)and two-term form factor potentials (bottom). behavior of the two-body potential. To prove that thisdoesn’t change universal properties of the four-boson sys-tem we also use the potential II with two-term form fac-tor h k | g i = [1 + c ( k/ Λ) ] e − ( k/ Λ) ; large negative valueof c = − .
17 ensures very different off-shell behavior.In the configuration space representation the potential IIhas several attractive and repulsive regions much like theone of Ref. [15] and supports a deeply bound trimer thatis non Efimov state in contrast to excited states and allstates of the standard potential. This can be seen in Ta-ble I which collects ratios for calculated trimer bindingenergies b n : all b n − /b n are quite close to the character-istic Efimov value of 515.035, the exception being b /b for the potential II that is much larger. However, the Efi-mov value is reached with good accuracy only for highexcited states n ≥
3. This is not surprising since theEfimov condition R n >> ρ ensuring truly universal be-havior, where R n is the size of the n -th trimer and ρ the range of the interaction, is only well satisfied for n large enough whereas for the ground states R < ρ maytake place [15, 16]. Simultaneously κ n − /κ n convergestowards 22.694 where κ n is the expectation value of the n -th excited trimer internal kinetic energy. The ratio b n /κ n is a measure for the high-momentum componentsin the trimer wave function; as Table I demonstrates, itdiffers significantly for the two employed potentials.In Table II we present results for the scattering length a n and effective range r n for the atom scattering fromthe n -th excited trimer up to n = 5. For both potentials a n /l n and r n /l n converge towards universal values as n increases, i.e., a n /l n ≈ . − . i, (2a) r n /l n ≈ . − . i, (2b)however, significant potential-dependent deviations areobserved for n ≤
2. Including strong repulsive three-body force that enforces the Efimov condition R n >> ρ (but with additional numerical complications) probablycould speedup the convergence with n but even in such acase n = 0 would be insufficient since it doesn’t accountfor inelasticities. n Re( a n ) /l n Im( a n ) /l n Re( r n ) /l n Im( r n ) /l n -6 -5 -4 -3 -2 -1 δ n S ( deg ) E n /b n n=1n=2n=3n=4n=4 (II)n=5-10-5050.0 0.5 1.0 δ nL ( deg ) E n /b n P waveD wave n=1n=3n=4n=4 (II)
FIG. 1. (Color online) S -, P - and D -wave phase shift for theatom scattering from the n -th excited trimer. It turns out that the universal limit exists for all scat-tering observables. In Figs. 1 - 2 we show all relevantphase shifts δ Ln and S -wave inelasticity parameter η Sn for the atom scattering from the n -th excited trimer asfunctions of the relative kinetic energy E n divided bythe respective b n ; the elastic S -matrix is parametrizedas s Ln = η Ln e iδ Ln . Again, results with n ≥ n . Elastic scattering is de-termined by relative atom-trimer S -, P -, and D -waves.In fact, S -wave dominates at lower energies but close to E n /b n = 1 the individual contributions of S -, P -, and -6 -5 -4 -3 -2 -1 η n S E n /b n n=1n=2n=3n=4n=4 (II)n=5 FIG. 2. (Color online) S -wave inelasticity parameter for theatom scattering from the n -th excited trimer. -7 -6 -5 -4 -3 -2 -1 -4 -3 -2 -1 σ ( n → n ’ ) / l n2 E n /b n n → n n → n - → n - → n - → n - → n - n=3n=4n=4 (II)n=5 FIG. 3. (Color online) Elastic and inelastic cross sections forthe atom scattering from the n -th excited trimer. D -waves to the elastic cross section are 31, 58, an 11%,respectively, while the F -wave with | δ Fn | < . ◦ yields lessthan 0.1% and therefore is negligible. Situation is differ-ent in the inelastic scattering where only S -wave con-tributes significantly: only η Sn clearly deviates from 1 asshown in Fig. 2 while in higher partial waves, due to verydifferent size of trimers and the angular momentum bar-rier, transitions to other trimers are strongly suppressedand η Ln are very close to 1, e.g., 1 − η Pn < − .The elastic and inelastic cross sections σ ( n → n ′ ) forthe atom scattering from the n -th excited trimer leadingto the lower-lying n ′ -th trimer are presented in Fig. 3. Wedo not show cross sections for n < n ′ that can be obtainedby time reversal as σ ( n ′ → n ) = ( E n /E n ′ ) σ ( n → n ′ ). Within the resolution of the plot σ ( n → n ′ ) /l n for n, n ′ ≥ n − n ′ are independent of n andemployed potential and thereby represent the universalvalues. Furthermore, for n > n ′ large enough the ratios σ ( n → n ′ ) σ ( n → n ′ − ≈ . σ ( n → n ′ ) results to any n and n ′ that aresufficiently large.Perhaps most interesting energy regions, namely thosecontaining S -wave four-boson resonances just slightly be-low E n /b n = 1, are not displayed in Fig. 3. There aretwo ( k = 1 ,
2) four-boson resonances associated with the N -th Efimov trimer. Resonances are poles of the AGStransition operators in the complex plane; thus, in thevicinity of the four-boson resonance U βα ≈ ˆ U ( − βα ( E − E r ) − + ˆ U (0) βα + ˆ U (1) βα ( E − E r ) , (4)where E = E n − b n and E r = − B N,k − i Γ N,k / − B N,k being the (
N, k )-th resonance position relativeto the four-body breakup threshold and Γ
N,k its width.These parameters were determined by fitting the on-shellmatrix elements of U into Eq. (4). Again, for N largeenough, i.e., N ≥
3, the results with good accuracy be-come independent of potential and N , B N, /b N ≈ . , Γ N, / b N ≈ . , (5a) B N, /b N ≈ . , Γ N, / b N ≈ . × − . (5b)We note that B N,k but not Γ
N,k have already been cal-culated in Ref. [5]. While B N, /b N ≈ .
58 of Ref. [5]is quite close to our number, the k = 2 resonance with B N, /b N ≈ .
01 was predicted in Ref. [5] to be signifi-cantly further from the atom-trimer threshold than ourresult. In contrast to S -wave, there is no four-boson res-onances in higher angular momentum states as can beseen in the bottom panel of Fig. 1.Unlike in nuclear physics, the direct measurement ofthe atom-trimer cross sections in cold atom physics ex-periments is not possible yet. Instead, one may be ableto create an ultracold mixture of atoms and excited Efi-mov trimers in a trap and observe the trimer relaxation,i.e., the inelastic collision of an atom and trimer in the n -th excited state leading to the atom and trimer in thelower-lying n ′ -th state. The kinetic energy ∆ K ≈ b n ′ released in this process is shared between the atom andtrimer with the ratio 3:1. Thus, if b n ′ / ρ n ( t ) of the n -th excited state trimers in the trap is given by dρ n ( t ) dt = − β n ρ a ( t ) ρ n ( t ) , (6) ρ a ( t ) being the atom density and β n the relaxation rateconstant [2]. The alternative way of the trimer loss, -5 -4 -3 -2 -1 β n / β n0 k B T/b n FIG. 4. (Color online) Temperature dependence of the trimerrelaxation rate constant. i.e., inelastic trimer-trimer collisions, is suppressed if ρ n (0) << ρ a (0). Under this condition ρ a ( t ) ≈ ρ a (0)and Eq. (6) has a simple solution ρ n ( t ) = ρ n (0) e − β n ρ a (0) t . (7)Thus, in this case the lifetime of the mixture is sim-ply given by 1 /β n ρ a (0). The relaxation rate constant β n = P n ′ β n → n ′ has contributions β n → n ′ = h v n σ ( n → n ′ ) i from transitions to all trimers n ′ < n , where v n = p E n /µ is the relative atom-trimer velocity and h . . . i denotes the thermal average. Thus, the trimer relaxationrate constant is determined by the atom-trimer inelasticcross sections calculated in the present work. In par-ticular, Eq. (3) implies that for n and n ′ large enough β n → n ′ /β n → n ′ − ≈ . n → n − β n = − π ~ µ Im( a n ) . (8)The results at finite temperature T are given in Fig. 4 as- suming the Boltzmann distribution for the relative atom-trimer energy; k B is the Boltzmann constant. Figure 4indicates that the use of T = 0 limit is inappropriate attemperatures above k B T /b n > − .In summary, we studied bosonic atom-trimer scatter-ing in the unitary limit. It is a complicated multichannelfour-particle scattering problem involving, in the presentcalculations, up to six open channels with the Efimovtrimer binding energies differing by a factor larger than515 ≈ × . Exact AGS equations were solved inmomentum-space framework with some important tech-nical modifications compared to previous calculations ofthe four-nucleon system. The results for reactions withhighly excited trimers (at least 2nd excited state) in theinitial and final channels were found to be independent ofthe used potential and thereby represent universal valuesfor atom-trimer scattering length, effective range, phaseshifts, elastic and inelastic cross sections and four-bosonresonance parameters. On the other hand, results forlower trimers demonstrate that significant quantitativedeviations from the universal behavior are possible. Thecomparison with the experimental data could not be per-formed yet, but the obtained atom-trimer scattering re-sults were related to the trimer relaxation rate constantthat hopefully will be measured in the future experimentswith ultracold mixtures of atoms and Efimov trimers.The developed technique is applicable also to dimer-dimer scattering. 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