Production of Efimov molecules in an optical lattice

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Production of Efimov molecules in an optical lattice Thorsten Köhler Still at: Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, United Kingdom Leaving to: Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, United Kingdom Santa Barbara 27 p.1

Outline Quantum halos Feshbach molecules in cold gases Efimov states Three-body recombination in 87 Rb and 133 Cs gases Excited 133 Cs 3 Efimov states Production of excited 85 Rb 3 Efimov states Conclusions Santa Barbara 27 p.2

Quantum Halos The average separation between constituent particles exceeds the distance scale set by the classical motion in the potential well! Potential Energy [a.u. 6 1-5 4 1-5 2 1-5 -2 1-5 4 He 4πr 2 φ b (r) 2.1.5 -.5 Radial Probability Density [1/a 1 1 1 1 2 1 3 r [a Classic examples: 4 He 2 and deuteron Review: A.S. Jensen, K. Riisager, D.V. Fedorov, and E. Garrido, RMP 76, 215 (24) Santa Barbara 27 p.3

Quantum Halos Universal properties of weakly bound diatomic molecules Hamiltonian: H 2B = 2 m 2 + V (r) van der Waals potential: V (r) C 6 /r 6 r Bound-state energy: E b = 2 /(ma 2 ) Bound-state wave function: φ b (r) = 1 e r/a 2πa r Potential Energy [a.u. 6 1-5 4 1-5 2 1-5 -2 1-5 4 He 4πr 2 φ b (r) 2 1 1 1 1 2 1 3 r [a.1.5 -.5 Radial Probability Density [1/a Santa Barbara 27 p.3

Quantum Halos Universality: Physical properties are determined by a single length scale, the s-wave scattering length, a! Hamiltonian: H 2B = 2 m 2 + V (r) van der Waals potential: V (r) C 6 /r 6 r Bound-state energy: E b = 2 /(ma 2 ) Bound-state wave function: φ b (r) = 1 e r/a 2πa r Potential Energy [a.u. 6 1-5 4 1-5 2 1-5 -2 1-5 4 He 4πr 2 φ b (r) 2 1 1 1 1 2 1 3 r [a.1.5 -.5 Radial Probability Density [1/a Santa Barbara 27 p.3

Quantum Halos Characteristic length scales Bond length: r = a/2 Classical turning point: r classical = [a(2l vdw ) 2 1/3 van der Waals length: l vdw = 1 2 (mc 6/ 2 ) 1/4 Characteristic property: Potential Energy [a.u. 6 1-5 4 1-5 2 1-5 -2 1-5 4 He r classical 4πr 2 φ b (r) 2 1 1 1 1 2 1 3 r [a.1.5 -.5 Radial Probability Density [1/a a r classical Santa Barbara 27 p.3

Quantum Halos Properties of the helium dimer Bond length: r = 5.2 ±.4 nm Classical turning point: r classical = 1.4 nm Characteristic property: a = 1.4 nm r classical 1 1 1 1 2 1 3 Potential Energy [a.u. 6 1-5 4 1-5 2 1-5 -2 1-5 4 He r classical 4πr 2 φ b (r) 2.1.5 -.5 Radial Probability Density [1/a r [a R.E. Grisenti, W. Schöllkopf, J.P. Toennies, G.C. Hegerfeldt, TK, and M. Stoll, PRL 85, 2284 (2) Santa Barbara 27 p.3

Feshbach molecules in cold gases a(b) [a 4 2-2 -4 85 Rb Scattering length goes to ±. A new vibrational bound state emerges at the collision threshold. E/h [Ghz -.1 -.2 -.3 -.4 1 15 2 25 3 B [G Theory: E. Tiesinga, B.J. Verhaar, and H.T.C. Stoof, PRA 47, 4114 (1993) Experiment: S. Inouyé et al., Nature 392, 151 (1998) Review: TK, K. Góral, and P.S. Julienne, RMP 78, 1311 (26) Santa Barbara 27 p.4

Feshbach molecules in cold gases Near-resonant bound-state energies become universal! B -.2 85 Rb E b (B)/h [MHz -.4 -.6 -.8 -h - 2 / ma 2 two channels coupled channels Claussen et al. -1 156 158 16 162 B [G Experiment: N.R. Claussen, S.J.J.M.F. Kokkelmans, S.T. Thompson, E.A. Donley, E. Hodby, and C.E. Wieman, PRA 67, 671(R) (23) Review: TK, K. Góral, and P.S. Julienne, RMP 78, 1311 (26) Santa Barbara 27 p.4

Feshbach molecules in cold gases Efimov plot: K = m E b / 2 vs. 1/a -.1 a< a> universal coupled channels Claussen et al. K [1/a -.2 -.3 -.4 a(b) [a 5-5 a bg a= -.5 1 15 2 B [G -.1.1.2.3 1/a [1/a V. Efimov, Phys. Lett. 33B, 563 (197) Santa Barbara 27 p.4

Feshbach molecules in cold gases Near-resonant Feshbach molecules are halo states! K [1/a -.1 -.2 -.3 -.4 -.5 a(b) [a a< a> 5-5 a bg a= 1 15 2 B [G universal coupled channels Claussen et al. -.1.1.2.3 1/a [1/a (4π) 1/2 rφ b (r) (4π) 1/2 rφ b (r).4.2 -.2.15.1.5 l vdw r classical closed channel entrance channel B=16 G r classical B=155.5 G -.5 1 2 1 3 1 4 r [a TK, T. Gasenzer, and K. Burnett, PRA 67, 1361 (23) TK, T. Gasenzer, P.S. Julienne, and K. Burnett, PRL 91, 2341 (23) Santa Barbara 27 p.4

Feshbach molecules in cold gases 85 Rb (F = 2,mF = 2) is an excited state 85 Rb2 Feshbach molecules decay due to spin relaxation! 1 coupled channels Lifetime [ms 1 1.1 156 158 16 162 164 B [G Santa Barbara 27 p.4

Feshbach molecules in cold gases The lifetime depends on the size of the molecule! Average volume of a halo state: V = 4π r 3 /3 = πa 3 Rate equation for spin relaxation: Ṅ(t) = K 2 n(t) N(t) Lifetime of the halo state: τ = V/[K 2 (B)/4 = 4πa 3 (B)/K 2 (B) Lifetime [ms 1 1 1.1 4πa 3 (B)/K 2 (B) coupled channels 156 158 16 162 164 B [G Santa Barbara 27 p.4

Feshbach molecules in cold gases Experiment and theory agree! Average volume of a halo state: V = 4π r 3 /3 = πa 3 Rate equation for spin relaxation: Ṅ(t) = K 2 n(t) N(t) Lifetime of the halo state: τ = V/[K 2 (B)/4 = 4πa 3 (B)/K 2 (B) Lifetime [ms 1 1 1.1 4πa 3 (B)/K 2 (B) coupled channels experiment 156 158 16 162 164 B [G Experiment: S.T. Thompson, E. Hodby, and C.E. Wieman, PRL 94, 241 (25) Theory: TK, E. Tiesinga, and P.S. Julienne, PRL 94, 242 (25) Santa Barbara 27 p.4

Efimov states Three-body Hamiltonian in Jacobi coordinates 2 H = 2(3m) 2 R + 1 2 (3m)ω2 ho R2 2 2( 2 3 m) 2 ρ + 1 ( 2 )ω 2 3 m 2ho ρ2 2 2( m 2 ) 2 r + 1 ( m ) 2 2 + V (r) + V (ρ + 12 ) r + V (ρ 12 ) r ω 2 ho r2 3 r=r 23 2 ρ r 31 r 12 1 =H + V 1 + V 2 + V 3 L.D. Faddeev, Soviet Phys. - JETP 12, 114 (1961) E.O. Alt, P. Grassberger, and W. Sandhas, Nucl. Phys. B 2, 167 (1967) Santa Barbara 27 p.5

Efimov states The ground and first excited states of the helium trimer molecule Scattering length of helium: a = 1.4 nm Effective range: r eff =.7 nm Number of Efimov trimer states as a function of the scattering length: N Efimov 1 π log ( a /r eff) =.8 G.C. Hegerfeldt and M. Stoll, PRA 71, 3366 (25) L.H. Thomas, Phys. Rev. 47, 93 (1935) V. Efimov, Phys. Lett. 33B, 563 (197) Helium case: T.K. Lim, Sister K. Duffy, and W.C. Damer, PRL 38, 341 (1977) Santa Barbara 27 p.5

Efimov states The ground and first excited states of the helium trimer molecule Scattering length of helium: a = 1.4 nm Effective range: r eff =.7 nm Number of Efimov trimer states as a function of the scattering length: N Efimov 1 π log ( a /r eff) =.8 G.C. Hegerfeldt and M. Stoll, PRA 71, 3366 (25) Detection of the ground state: W. Schöllkopf and J.P. Toennies, Science 266, 1345 (1994) Measurement of its bond length: R. Brühl, A. Kalinin, O. Kornilov, J.P. Toennies, G.C. Hegerfeldt, and M. Stoll, PRL 95, 632 (25) Santa Barbara 27 p.5

Efimov states Santa Barbara 27 p.5

Efimov states Efimov plot 1/a= zero-energy resonances dimer K= E n K E n-1 atom-dimer resonance 1/a V. Efimov, Phys. Lett. 33B, 563 (197) Santa Barbara 27 p.5

Efimov states zero-energy resonances Reality is often different! E n dimer zero-energy resonance K E n-1 1/a atom-dimer resonance Only ground levels have been experimentally observed which inevitably connect to non-universal physics. dimer K dimer level ground trimer level first excited trimer level L.W. Bruch and K. Sawada, PRL 3, 25 (1973) V. Efimov, Nucl. Phys. A 362, 45 (1981) 1/a Santa Barbara 27 p.5

Efimov states Efimov spectrum of 4 He -.2 -.4 4 He Schȯ. llkopf & Toennies dimer K [1/a -.6 -.8 -.1 -.2 -.1.1.2.3 1/a [1/a W. Schöllkopf and J.P. Toennies, Science 266, 1345 (1994) Santa Barbara 27 p.5

Efimov states The excited 4 He 3 level has not been found! -.2 Schȯ. llkopf & Toennies 4 He? dimer -.4 K [1/a -.6 -.8 -.1 -.2 -.1.1.2.3 1/a [1/a R. Brühl, A. Kalinin, O. Kornilov, J.P. Toennies, G.C. Hegerfeldt, and M. Stoll, PRL 95, 632 (25) Santa Barbara 27 p.5

Efimov states The 4 He 3 system is universal! -.2 -.4 4 He Schȯ. llkopf & Toennies dimer K [1/a -.6 -.8 -.1 -.2 -.1.1.2.3 1/a [1/a Interpretation of predictions on 4 He 3 : E. Braaten and H.-W. Hammer, PRA 67, 4276 (23) See also triton case: M.L.E. Oliphant, P. Harteck, and E. Rutherford, Proc. Roy. Soc. A 144, 692 (1934) Santa Barbara 27 p.5

Efimov states The ground level is never universal throughout! -.2 -.4 4 He Schȯ. llkopf & Toennies dimer K [1/a -.6 -.8 -.1 -.2 -.1.1.2.3 1/a [1/a L.W. Bruch and K. Sawada, PRL 3, 25 (1973) V. Efimov, Nucl. Phys. A 362, 45 (1981) Santa Barbara 27 p.5

Efimov states Efimov spectrum of 133 Cs at low magnetic fields dimer -.1 133 Cs B> B< K [1/a -.2 -.3 a(b) [a 5-5 a= a bg K [1/a -.2 -.4 -.6 -.8 -.1 4 He -.2 -.1.1.2.3 1/a [1/a Schȯ. llkopf & Toennies dimer -.4-5 5 1 B [G -.1 -.5.5 1/a [1/a Santa Barbara 27 p.5

Efimov states The first resonance belongs to a Borromean ground level! Kraemer et al. dimer -.1 133 Cs B> B< K [1/a -.2 -.3 a(b) [a 5-5 a= a bg K [1/a -.2 -.4 -.6 -.8 -.1 4 He -.2 -.1.1.2.3 1/a [1/a Schȯ. llkopf & Toennies dimer -.4-5 5 1 B [G -.1 -.5.5 1/a [1/a T. Kraemer, M. Mark, P. Waldburger, J.G. Danzl, C. Chin, B. Engeser, A.D. Lange, K. Pilch, A. Jaakkola, H.-C. Nägerl, and R. Grimm, Nature 44, 315 (26) Santa Barbara 27 p.5

Efimov states Efimov-state story to date Kraemer et al. dimer -.1 133 Cs B> B< K [1/a -.2 -.3 a(b) [a 5-5 a= a bg K [1/a -.2 -.4 -.6 -.8 -.1 4 He -.2 -.1.1.2.3 1/a [1/a Schȯ. llkopf & Toennies dimer -.4-5 5 1 B [G -.1 -.5.5 1/a [1/a RMP colloquium: TK, R. Grimm, P.S. Julienne, H.-C. Nägerl, W. Schöllkopf, and J.P. Toennies Santa Barbara 27 p.5

Three-body recombination in 87 Rb Three-body problem in 87 Rb Rate equation for three-body recombination: N(t) = K 3 (B) n 2 (t) N(t) Interferences in K 3 (B) occur due to the distortion of the two-body energy spectrum by the Feshbach resonance. The Efimov spectrum is largely inaccessible due to the narrow magnetic-field range. K 3 [cm 6 /s 1-23 1-24 1-25 1-26 1-27 1-28 1-29 1-3 1-31 1-18 1-2 1-22 1-24 B 1-26 17.3 17.4-29 14 16 995 1 15 B [G Theoretical approach: E.O. Alt, P. Grassberger, and W. Sandhas, Nucl. Phys. B 2, 167 (1967) Calculations for 87 Rb gases: G. Smirne, R.M. Godun, D. Cassettari, V. Boyér, C.J. Foot, T. Volz, N. Syassen, S. Duerr, G. Rempe, M.D. Lee, K. Góral, and TK, PRA 75, 272(R) (27) E -1 d E -2 d -27-28 B -2-4 -6-8 Energy/h [MHz Santa Barbara 27 p.6

Three-body recombination in 87 Rb Near-resonant three-body recombination rate constants K 3 [cm 6 /s 1-23 1-24 1-25 1-26 1-27 a(b) [a 9 6 3 a bg -1 1-28 1-29 -1 -.1 B-B [G -.1 G. Smirne, R.M. Godun, D. Cassettari, V. Boyér, C.J. Foot, T. Volz, N. Syassen, S. Duerr, G. Rempe, M.D. Lee, K. Góral, and TK, PRA 75, 272(R) (27) Santa Barbara 27 p.6

Three-body recombination in 87 Rb Comparison with experiment K 3 [cm 6 /s 1-23 1-24 1-25 1-26 1-27 1-28 a(b) [a 9 6 3 a bg -1 Oxford Garching 1-29 -1 -.1 B-B [G -.1 G. Smirne, R.M. Godun, D. Cassettari, V. Boyér, C.J. Foot, T. Volz, N. Syassen, S. Duerr, G. Rempe, M.D. Lee, K. Góral, and TK, PRA 75, 272(R) (27) Santa Barbara 27 p.6

Three-body recombination in 133 Cs Three-body recombination rate constants in 133 Cs K 3 [m 6 /s 1-31 1-32 1-33 1-34 1-35 1-36 1-37 1-38 1-39 1-4 1 nk experiment 2 nk experiment 5-5 -5 5 1 1 1 1 B [G Experiment: T. Kraemer, M. Mark, P. Waldburger, J.G. Danzl, C. Chin, B. Engeser, A.D. Lange, K. Pilch, A. Jaakkola, H.-C. Nägerl, and R. Grimm, Nature 44, 315 (26) a(b) [a B a bg Santa Barbara 27 p.7

Three-body recombination in 133 Cs Three-body zero-energy resonance in 133 Cs K 3 [m 6 /s 1-31 1-32 1-33 1-34 1-35 1-36 1-37 1-38 1-39 1-4 nk theory 1 nk experiment 2 nk experiment a(b) [a 5-5 -5 5 1 1 1 1 B [G B a bg General concept: E. Nielsen and J.H. Macek, PRL 83, 1566 (1999) B.D. Esry, C.H. Greene, and J.P. Burke, PRL 83, 1751 (1999) Quantitative approach: G. Smirne et al., PRA 75, 272(R) (27) Santa Barbara 27 p.7

Excited 133 Cs 3 Efimov states Measurement of excited-state resonances of the mirror-spin-channel might be hampered by spin relaxation! K 3 [m 6 /s 1-24 1-26 1-28 1-3 1-32 1-34 1-36 1-38 1-4 1-42 1-44 resonance B< B> highest excited second highest total a(b) [a 5-5 -5 5 1-1 -5-1 -4-1 -3-1 -2-1 -1-1 2-1 1 2 3 B [G E -1 d a= a bg E -2 d -1-1 Energy/h [MHz J.L. Roberts, N.R. Claussen, S.L. Cornish, and C.E. Wieman, PRL 85, 728 (2) Santa Barbara 27 p.8

Excited 133 Cs 3 Efimov states K 3 [m 6 /s Alternative: Measurement of three-body zero-energy resonances in the vicinity of 8 G 1-26 1-28 1-3 1-32 1-34 1-36 1-38 1-4 1-42 1-44 resonance highest excited second highest total resonance 82 84 86 88 B [G E -1 d E -2 d -1-5 -1-4 -1-3 -1-2 -1-1 -1-1 1-1 2 Energy/h [MHz M.D. Lee, TK, and P.S. Julienne, e-print cond-mat/72178 Santa Barbara 27 p.8

Excited 133 Cs 3 Efimov states K 3 [m 6 /s The loss minimum in the vicinity of 891 G might allow Bose-Einstein condensation of cesium! 1-26 1-28 1-3 1-32 1-34 1-36 1-38 1-4 1-42 1-44 resonance highest excited second highest total resonance loss minimum 82 84 86 88 B [G E -1 d E -2 d -1-5 -1-4 -1-3 -1-2 -1-1 -1-1 1-1 2 Energy/h [MHz T. Weber, J. Herbig, M. Mark, H.-C. Nägerl, and R. Grimm, Science 299, 232 (23) Santa Barbara 27 p.8

Excited 133 Cs 3 Efimov states K 3 [m 6 /s The theory does not account for the full complexity of cesium interactions! 1-26 1-28 1-3 1-32 1-34 1-36 1-38 1-4 1-42 1-44 resonance highest excited second highest total resonance 82 84 86 88 B [G E -1 d E -2 d -1-5 -1-4 -1-3 -1-2 -1-1 -1-1 1-1 2 Energy/h [MHz a(b) [a 1 5-5 -1 7 75 8 85 9 B [G M.D. Lee, TK, and P.S. Julienne, e-print cond-mat/72178 B -1-2 -3-4 Energy/h [MHz Santa Barbara 27 p.8

Excited 133 Cs 3 Efimov states The diatomic target states at negative scattering lengths are not uniform in the magnetic field! K 3 [m 6 /s 1-24 1-26 1-28 1-3 1-32 1-34 1-36 1-38 n=2 1-3 1-32 1-34 1-36 1-38 high field low field 1 nk experiment 2 nk experiment n=1-2 -1 1 n=1-2 -15-1 -5 a [a M.D. Lee, TK, and P.S. Julienne, e-print cond-mat/72178 Santa Barbara 27 p.8

Production of excited 85 Rb 3 Efimov states Efimov spectrum of 85 Rb -.2 85 Rb dimer K [1/a -.4 -.6 -.8 B= a(b) [a 5-5 a bg a= K [1/a -.2 -.4 -.6 -.8 -.1 4 He Schȯ. llkopf & Toennies dimer -.1 1 15 2 B [G -.2 -.1.1.2.3 1/a [1/a M. Stoll and TK, PRA 72, 22714 (25) K [1/a -.1 -.2 -.3 -.4 -.2 -.1.1.2.3 1/a [1/a Kraemer et al. a(b) [a 133 Cs 5-5 a= B> a bg -5 5 1 B [G B< -.1 -.5.5 1/a [1/a dimer Santa Barbara 27 p.9

Production of excited 85 Rb 3 Efimov states 85 Rb gases allow easy access to the resonance! -.2 85 Rb E2 resonance dimer K [1/a -.4 -.6 -.8 B= a(b) [a 5-5 a bg a= K [1/a -.2 -.4 -.6 -.8 -.1 4 He Schȯ. llkopf & Toennies dimer -.1 1 15 2 B [G -.2 -.1.1.2.3 1/a [1/a M. Stoll and TK, PRA 72, 22714 (25) K [1/a -.1 -.2 -.3 -.4 -.2 -.1.1.2.3 1/a [1/a Kraemer et al. a(b) [a 133 Cs 5-5 a= B> a bg -5 5 1 B [G B< -.1 -.5.5 1/a [1/a dimer Santa Barbara 27 p.9

Production of excited 85 Rb 3 Efimov states Association of dimer molecules in an optical lattice site Energy [h - ω ho 1 5-5 -1-15 -2 87 Rb E b (B) bound atoms B separated atoms 16.8 17 17.2 17.4 17.6 17.8 18 B [G v=6 v=5 v=4 v=3 v=2 v=1 v= G. Thalhammer, K. Winkler, F. Lang, S. Schmid, R. Grimm, and J.H. Denschlag, PRL 96, 542 (26) Santa Barbara 27 p.9

Production of excited 85 Rb 3 Efimov states Association of trimer molecules in an optical lattice site Energy [h - ω ho 1 5-5 -1-15 -2 87 Rb E b (B) bound atoms B separated atoms 16.8 17 17.2 17.4 17.6 17.8 18 B [G v=6 v=5 v=4 v=3 v=2 v=1 v= Energy/h [khz.5 -.5-1 separated atoms E 4 E 3 B + 3_ 2 hν ho -1.5 152 153 154 155 156 B [G trimer B 85 Rb G. Thalhammer, K. Winkler, F. Lang, S. Schmid, R. Grimm, and J.H. Denschlag, PRL 96, 542 (26) M. Stoll and TK, PRA 72, 22714 (25) See also: S. Jonsell, H. Heiselberg, and C.J. Pethick, PRL 89, 2541 (22) Santa Barbara 27 p.9

Production of excited 85 Rb 3 Efimov states Efimov trimer molecules in an optical lattice site The magnetic-field dependent binding energies and spatial extents of -trimer molecules are comparable to those of the associated diatomic Feshbach molecules. The lifetimes of Efimov-trimer molecules increase with increasing bond length. Hyperradial Probability Density P(R) [1-3 a -1 ν 6 ho = 1 MHz 4138 khz 483 khz + 1887 khz 4 2 6 4 2 4 ν ho = 3 khz 2411 khz 425 khz + 298 khz 6 ν ho = 5 khz 1692 khz 163 khz 33 khz 2 M. Stoll and TK, PRA 72, 22714 (25) See also: S. Jonsell, Europhys. Lett. 76, 8 (26) 1 1 Hyperradius R [a Santa Barbara 27 p.9

Conclusions Near-resonant diatomic Feshbach molecules are halo states. The halo nature of diatomic Feshbach molecules implies Efimov s effect in the associated three-body problem. Measurements of near resonant three-body recombination in cold gases of 133 Cs to date prove the existence of a Borromean ground state of the Efimov spectrum. Excited-state Efimov zero-energy resonances might be detected in cold gases of 133 Cs at magnetic fields in the vicinity of 8 G. Metastable excited-state Efimov trimers might be produced in cold gases of, e.g., 85 Rb using magnetic-field sweep techniques in analogy to the association of diatomic Feshbach molecules. Santa Barbara 27 p.1

Thanks to DPhil students: Thomas Hanna and Hugo Martay Postdocs: Marzena Szymańska, Mark Lee, and Martin Stoll Group leader: Keith Burnett Coupled-channels calculations: Paul Julienne (NIST) Experimental groups at: Oxford, JILA, Innsbruck, and Munich Discussions with: Wieland Schöllkopf, Peter Toennies, and Vitaly Efimov Santa Barbara 27 p.11

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