Universality in Four-Boson Systems

Size: px

Start display at page:

Download "Universality in Four-Boson Systems"

Rosemary Norris
6 years ago
Views:

1 Universality in Four-Boson Systems M. R. Hadizadeh INPP, Department of Physics & Astronomy, Ohio University Work done in collaboration with: M.T. Yamashita & L. Tomio (UNESP), A. Delfino (UFF), T. Frederico (ITA) INT Program: Universality in few-body systems: theoretical challenges and new directions Institute for Nuclear Theory, University of Washington, 9 Apr. 214

2 1 Introduction 2 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Numerical solution Computational approach Tetramer binding energies B & 4B scaling plots Four-atom resonances 4 On the range corrections 2B scattering amplitude B & 4B scaling functions Position of the 4-atom resonances 5 The stability of numerical results Numerical convergence Yakubovsky components & 4B wave function Momentum distribution functions 6 Conclusion

3 Introduction Outline 1 Introduction 2 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Numerical solution Computational approach Tetramer binding energies B & 4B scaling plots Four-atom resonances 4 On the range corrections 2B scattering amplitude B & 4B scaling functions Position of the 4-atom resonances 5 The stability of numerical results Numerical convergence Yakubovsky components & 4B wave function Momentum distribution functions 6 Conclusion

4 Introduction Main Goal Very accurate solution for the 4B problem - scaling and universality Motivation: why are weakly bound state problems interesting? Short-range interactions & large quantum systems Universality & model independence Reality: Cold atom physics close to a Feshbach resonance! Different techniques for few-body bound state calculations NCSM (Navratil et al. PRC62), HH (Viviani et al. PRC71) and EIHH (Barnea et al. PRC67) GFMC (Viringa et al. PRC62), CRCGV (Hiyama et al. PRL85), SV (Usukura et al. PRB59), FY (Hadizadeh et al. FBS4, EPJA21, PTP12, PRC8)

7 For finite a: discrete scaling is exact when range Efimov effect: If two bosons interact in such a way that a two-body bound state is exactly on the verge of being formed, then in a three-boson

5 Introduction Efimov Physics (197): Nuclear Physics Vitaly Efimov an infinite sequence of weakly bound -body states as a ± Unitary limit (a ± ): E n+1 /E n 1/22.72 = discrete scaling with scaling factor 22.7 For finite a: discrete scaling is exact when range Efimov effect: If two bosons interact in such a way that a two-body bound state is exactly on the verge of being formed, then in a three-boson system one should observe an infinite number of bound states. This phenomenon appears in a three-dimensional formalism for the three-body systems, and does not exist in one or two dimensions. This effect, predicted by Vitaly Efimov in 197 [Phys. Lett. B (197) 56; Sov. J. Nucl. Phys. 12 (1971) 589], have been recently verified in ultracold atom laboratories, with the increasing number of three-body bound-state levels, as the two-body scattering length goes to infinity.

in an ultracold gas of Cesium atoms [Nature 44 (26) 15], was confirmed by several other atomic

6 Introduction The observation of Efimov effect The observation of this effect, first reported by Kraemer et al. in an ultracold gas of Cesium atoms [Nature 44 (26) 15], was confirmed by several other atomic experimental groups, which are looking for the properties of such states. Zaccanti et al., Nature Phys. 5 (29) 586 Ferlaino et al, PRL 12 (29) 1441 Pollack et al., Science 26 (29) 168

7 Introduction Recent studies on four-boson system - Theory The addition of one more particle to the quantum three-body system has long challenged the Efimov picture. The need of an independent four-body parameter is discussed, considering the observational possibilities in cold-atom experiments. L. Platter et al., PRA7 (24); H.-W. Hammer and L. Platter, EPJA2 (27) Four-boson system with short-range interactions Universal properties of the four-body system with large scattering length M. T. Yamashita et al., EpL75 (26) Four-boson scale near a Feshbach resonance. R. Lazauskas and J. Carbonell, PRA 7 (26) Description of 4 He tetramer bound and scattering states J. von Stecher et al., Nature Physics 5 (29); J. von Stecher, JPB4 (21) Signatures of universal four-body phenomena and their relation to the Efimov physics. Weakly bound cluster states of Efimov character Y. Wang, B. D. Esry, PRL12 (29) Efimov trimer formation via ultracold four-body recombination A. Deltuva, PRA82 (21) Efimov physics in bosonic atom-trimer scattering M. R. Hadizadeh et al., PRL17 (211) Scaling properties of universal tetramers Gattobigio et al., PRA86 (212) Energy spectra of small bosonic clusters having a large two-body scattering length Hiyama et al. PRA85 (212) Linear correlations between 4 He trimer and tetramer energies calculated with various realistic 4 He potentials

2] We address this point with high-precision calculations using a zero-range model

8 Introduction Hammer et al., EPJA2 (27) [5, 1.1] Deltuva, PRA82 (21) [4.61, 1.227] No independent four-body parameter! Gattobigio et al., PRA86 (212) [4.5, 1.2] We address this point with high-precision calculations using a zero-range model and searching the excited tetramer states and their dependences on short-range parameters.

9 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Outline 1 Introduction 2 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Numerical solution Computational approach Tetramer binding energies B & 4B scaling plots Four-atom resonances 4 On the range corrections 2B scattering amplitude B & 4B scaling functions Position of the 4-atom resonances 5 The stability of numerical results Numerical convergence Yakubovsky components & 4B wave function Momentum distribution functions 6 Conclusion

10 Formalism: 4B bound state in Faddeev-Yakubovsky scheme ( ) ] K12, 4 = G t 12 P[ 1 + P 4 K12, 4 + H 12,4 ( ) ] H 12,4 = G t 12 P[ 1 + P 4 K12, 4 + H 12,4 Ψ = ( 1 + P + P 4 P + P ) [ ) ] (1 + P 4 K12, 4 + H 12,4 P = P 12 P 2 + P 1 P 2 P = P1 P 24 Kamada et al., NPA548 (1992)

11 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Definition of 4B basis states 1 1 u 1 u 2 u v 1 v 2 v u 1 = 1 ( ) k1 k 2 2 u 2 = 2 (k ( ) ) k 1 + k 2 u = 4 ( k 4 1 ( ) ) k1 + k 2 + k v 1 = 1 ( ) k1 k 2 2 v 2 = 1 2 ( ) 1 ( ) k1 + k 2 k + k 4 2 v = 1 ( ) k k 4 2

12 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Explicit representation of Yakubovsky equations Hadizadeh and Bayegan, FBS4 (27)

13 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Two-body transition amplitude for zero-range interaction V (r) = (2π) λ δ(r) p V p = λ p χ χ p ; p χ = d r e ip.r δ(r) = 1 τ(ɛ) = [ λ 1 λ 1 = d p d p ] 1 1 ɛ p 2 1 B 2 p 2 τ(ɛ) = 1 ( B2 2π 2 ) ( ɛ = 2π 2 a ) 1 ɛ Amorim et al., PRC46 (1992)

14 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Coupled Yakubovsky integral equations K ( u 2, u ) [ G () H ( v 2, v ) = 4π τ(ɛ) du 2 u 2 2 ( Π ( ) u 2, u ) 2, u 2, u dx ( ) K u 2, u ( dx G (4) Π ( u 2, u ) ( 2, Π2 u 2, u, x ) (, Π u 2, u, x )) ( ( K Π 2 u 2, u, x ) (, Π u 2, u, x )) ( dx G (4) Π ( u 2, u ) ( 2, Π4 u 2, u, x ) (, Π 5 u 2, u, x )) ( ( H Π 4 u 2, u, x ) (, Π 5 u 2, u, x ))] = 4π τ(ɛ ) dv v 2 [ ( dx G (4) ( v, Π 6 v2, v, x) (, Π 7 v2, v, x)) ( K (Π 6 v2, v, x) (, Π 7 v2, v, x)) ( ) ( ) ] +G (4) v, v 2, v H v 2, v Hadizadeh et al., PRA85 (212)

15 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Subtracted Green s Functions: G (N) 1 1 = E H µ 2 N H with µ (RED): B scale & µ 4 (BLUE): 4B scale

16 Numerical solution Outline 1 Introduction 2 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Numerical solution Computational approach Tetramer binding energies B & 4B scaling plots Four-atom resonances 4 On the range corrections 2B scattering amplitude B & 4B scaling functions Position of the 4-atom resonances 5 The stability of numerical results Numerical convergence Yakubovsky components & 4B wave function Momentum distribution functions 6 Conclusion

17 Numerical solution Computational approach Numerical solution algorithm standard eigenvalue problem ( ) ] K = G t 12 P[ 1 + P 4 K + H ( ) ] H = G t 12 P[ 1 + P 4 K + H λ(e).ψ = K (E).ψ; ψ = ( K H Searching E to get the solution of coupled Yakubovsky integral equations with λ = 1. )

18 Numerical solution Computational approach Integration: Gaussian quadrature Jacobi momenta: 8-16 mesh points; tricky mapping polar angles: 4 mesh points Dimension of Kernel after discretization: Eigenvalue problem: Lanczos type method Iterative orthogonal vectors (IOV) (Stadler PRC44 219) ARPACK Fortran library ( Dimension of eigenvalue problem after using Lanczos technique: # of iterations-1 1! Multidimensional interpolations: Cubic-Hermit Splines high computational speed and accuracy (Huber FBS22 17)

19 Numerical solution Tetramer binding energies Tetramer ground and excited state binding energies for B 2 = Within a renormalized zero-range model, where the relevant three- and four-body momentum scales (µ and µ 4 ) are introduced in a subtractive regularization procedure, we have the following results: µ 4 /µ B () 4 /B B (1) 4 /B B (2) 4 /B B () 4 /B

20 Numerical solution Tetramer binding energies Tetramer ground and excited state binding energies for B 2 =.2 B µ 4 /µ B () 4 /B B (1) 4 /B

21 Numerical solution Tetramer binding energies Tetramer ground and excited state binding energies for B2 virtual =.2 B µ4/µ B () 4 /B B (1) 4 /B

22 Numerical solution Tetramer binding energies Three- and Four-body Binding Energies B 4 (N) N= N=1 B 2 = N= N=1 B (N) 1-6 N=2 (a) N=2 (b) B () B 2

23 Numerical solution B & 4B scaling plots B Scaling Plot Yamashita et al., PRA66 (22) B (N-1) (N) 2 (B - B )/B (N+2) B 2 = B (N+1) B 2 = B B (N) B 2 B (N) B 2 a < B 2 = B (N) B virtual 2 a > 1/a = ~ B B (N+1) B B(N+1) 6.61 B B (N+2) B 2 2 B(N) = 515 B (N+1) 184 B (N+1) = [ a (N) /a (N+1) (N) B 2 / B ] = 22.7

24 Numerical solution B & 4B scaling plots 4B Scaling Plot Hadizadeh et al., PRL 17, 154 (211).25.2 (N+2) B = B4 B (N) 4 B (B 4 (N+1) -B )/B 4 (N).15 (N+) B = B4.15 B 2 =.2 B (a > ) B =.44 B (a > ) 2 B 2 =.2 B (a > ) B 2 = B 2 =.2 B (a < ).1.1 Stecher JPB4 (21) Stecher et al. Nature (29). Stecher et al. Nature Supp. (29) + ++ x Hammer-Platter EPJA2 (27) Hammer-Platter EPJA2 (27) (a > ) + Hammer-Platter EPJA2 (27) (a < ).5.5 Interpolation to 1/a = of and + + x. Platter et al. PRA7 (24) + + Lazauskas-Carbonell PRA7 (26) Deltuva, arxiv (21) B (N+1) B 4 (N) B / B 4 (N+1) B = B4 B (N) 4 B (N+1) 1.8 B(N+1) 4.6 B 4B (N+2) B 4 4 B (N) 4 B (N+1) 4 For ground trimer: B(N+1) 4 B (N+2) B(N+2) 4.6 B 4B (N+) B 4 B B (N) (with B 2 = ): an infinite tetramer levels with 4 B (N+1) 4.85 B (N). 4

25 Numerical solution B & 4B scaling plots 4B Scaling Plot For B (N+1) (N) (B -B 4 ) / B 4 B (N) B 2 = Hiyama et al. PRA85 (212) Gattobigio et al. PRA84 (212) Deltuva PRA82 (21) & arxiv: Stecher JPB4 (21) Stecher et al., Nat. Phy. 5 (29) Stecher et al., Nat. Phys. 5 Suppl. (29) Hammer et al. EPJA2 (27) Hammer et al. EPJA2 (27) [a>] Hammer et al. EPJA2 (27) [a<] Hammer et al. EPJA2 (27) 1/a=] Lazauskas et al. PRA7 (26) Platter et al. PRA7 (24) Blume et al. JCP112 (2) (N) B / B 4 B /B 2 =5.1 (a<) B /B 2 =498.6 (a>) B /B 2 = (a>) B /B 2 =5.1 (a>) > 1 =.44, at most three tetramers fit between two consecutive Efimov trimers. 22.7

Numerical solution Four-atom resonances 4-atom recombination losses Locations of four-atom loss features (a 4b1, a 4b2 < ) where two successive tetramers become unbound (blue-solid line for r ). 1.

26 Numerical solution Four-atom resonances 4-atom recombination losses Locations of four-atom loss features (a 4b1, a 4b2 < ) where two successive tetramers become unbound (blue-solid line for r ). 1.1 _ / a b _ a 4b Hadizadeh et al. PRL17 (211) Deltuva PRA85 (212) Berninger et al. PRL17 (211) Ferlaino et al. FBS51 (211) Stecher et al. Nature Phys. 5 (29) Pollack et al. Science 26 (29) Ferlaino et al. PRL12 (29) a 4b1 / a b

27 On the range corrections Outline 1 Introduction 2 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Numerical solution Computational approach Tetramer binding energies B & 4B scaling plots Four-atom resonances 4 On the range corrections 2B scattering amplitude B & 4B scaling functions Position of the 4-atom resonances 5 The stability of numerical results Numerical convergence Yakubovsky components & 4B wave function Momentum distribution functions 6 Conclusion

28 On the range corrections Our goal: Derive range corrections to the tetramer scaling functions calculated with the zero-range model Compare with potential models results Extract the effective range from cold atom experiments for the four-body recombination data close to a Feshbach resonance.

29 On the range corrections 2B scattering amplitude LO range correction in the two-body scattering amplitude: Two-body scattering amplitude: τ(ɛ) = 1/(2π2 ) k cot(δ ) ik effective range low energies: k cot(δ ) = 1 a + r 2 k 2 + O(r k 4 ) the bound (+) or virtual states (-): 1 a = ± ɛ 2 r 2 ɛ 2 Up to first order in r, τ(ɛ) (2π 2 ) 1 [ ɛ ± 1 + r ( ) ] ɛ ± ɛ2. ɛ 2 2 The subtracted form of the STM and FY are able to consider the higher momentum power in the effective range expansion.

30 On the range corrections B & 4B scaling functions Range Correction to B Scaling Plot Cornelius and Glöckle, JCP85 (1986) - realistic HFDHE2 interatomic potential of Aziz et al. ] 1/2 [(B (N+1) -B2 ) / B (N) r = r =.5/µ r =1./µ r =2./µ Cornelius et al. JCP85 (1986): s-wave Cornelius et al. JCP85 (1986): s+d waves [B 2 / B (N) ] 1/2 (N+1) [(B -B2 ) / B (N)] 1/ r = r =1.5/µ Cornelius et al. JCP85 (1986): s-wave (N) [B 2 / B ] 1/2

31 On the range corrections B & 4B scaling functions Range correction to 4B Scaling Plot (N+1) (N) (B -B 4 ) / B B 2 = B Hiyama et al. PRA85 (212) /B 2 =5.1 (a>) Gattobigio et al. PRA84 (212) Deltuva PRA82 (21) & arxiv: Stecher JPB4 (21) Stecher et al., Nat. Phy. 5 (29) Stecher et al., Nat. Phys. 5 Suppl. (29) Hammer et al. EPJA2 (27) Hammer et al. EPJA2 (27) [a>] Hammer et al. EPJA2 (27) [a<] Hammer et al. EPJA2 (27) 1/a=] Lazauskas et al. PRA7 (26) Platter et al. PRA7 (24) Blume et al. JCP112 (2) r =.1/µ 4 r =.5/µ 4 (N) B / B 4 B /B 2 =5.1 (a<) B /B 2 =498.6 (a>) B /B 2 = (a>)

32 On the range corrections B & 4B scaling functions Comparison to other calculations.12 (N+1) (N) (B -B 4 ) / B a< r =.1/µ 4 ( 2 =) r =.5/µ 4 ( 2 1/a= a> Hammer et al. EPJA2 (27) (N+1) (N) (B -B 4 ) / B Stecher JPB4 (21) Stecher et al., Nat. Phy. 5 (29) Stecher et al., Nat. Phys. 5 Suppl. r =.1/µ 4 ( 2 =) r =.5/µ 4 ( 2 =) (N) B / B (N) B / B 4 (N+1) (N) (B -B 4 ) / B Deltuva PRA82 (21) r =.1/µ 4 ( 2 =) (N) B / B 4 Even at unitary limit, a very small value of range parameter can shift the scaling plot to right.

33 On the range corrections Position of the 4-atom resonances Range correction to the scaling function for the positions of successive 4-atom resonances: a T 1,2 a 1 = A(x, y) = where x a T 1,1/a 1, y r / a 1. m+n 2 c mn (x.45) m y n ; m, n.95 Coefficients of the parametrization c c 1 c c 11 c 2 c a 1Ț /a, a T 1, /a, r /a 1.4.5

34 On the range corrections Position of the 4-atom resonances Range correction to the position of 4-atom resonance; _ a T 1,2 /a Hadizadeh et al. PRL17 (211) [r /a 1 _ = ] Gattobigio et al. PRA86 (212) Deltuva PRA85 (212) Berninger et al. PRL17 (211) Ferlaino et al. FBS51 (211) von Stecher et al. Nature Phys. 5 (29) Ferlaino et al. PRL12 (29) T a /a 1,1 1 _ -r /a 1 _ [.,.1] [.1,.2] [.2,.] [.,.4] [.4,.5] Hadizadeh et al PRA 87, 162 (21) von Stecher (priv. comm.).8 vs..6 Deltuva (priv. comm.). vs..29

35 On the range corrections Position of the 4-atom resonances r from the shift of the peaks of the four-atom losses Ref. a T 1,1/a 1 a T 1,2/a 1 a 1 [R vdw ] r [R vdw ] Ferlaino et al PRL (1) > 5 Berniger et al PRL (4).9(1) -9.54(28) 2.5 ± 1.7 Ferlaino et al FBS 11.47(1).87(1) ± 1. Ferlaino et al FBS 11.46(2).91() ± 2 R Cs 2 vdw = 11. a [Chin et al RMP82(21)] ā Cs R Cs 2 vdw = 96.5 a..5 < r < 4. R vdw Weighted average for the fitted r values:.9±.8 R vdw [ ( ) 2 ( ) ] 2 ā ā r ā + a a 1

36 The stability of numerical results Outline 1 Introduction 2 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Numerical solution Computational approach Tetramer binding energies B & 4B scaling plots Four-atom resonances 4 On the range corrections 2B scattering amplitude B & 4B scaling functions Position of the 4-atom resonances 5 The stability of numerical results Numerical convergence Yakubovsky components & 4B wave function Momentum distribution functions 6 Conclusion

37 The stability of numerical results Numerical convergence Example Convergence of 1 st and 2 nd exited tetramer energies for µ 4 µ = : u i = 1 + x i c 1 (1 x i ) + c 2 x i ; c 1 µ 4 µ, c 2 =.4 x i [ 1, +1] = u i [,.] + [., 5] B 4 (1) / B / B B 4 (2) Number of mesh points Number of mesh points

38 The stability of numerical results Yakubovsky components & 4B wave function Example Ψ(, u 2, u ) for µ 4 µ = 5: where β N = E (N) 4, N =, 1, 2 Ψ (,u 2,u ) *β u /β Ψ (,u 2,u ) *β u /β u 2 /β u 2 /β Ψ 1 (,u 2,u ) *β u /β Ψ 1 (,u 2,u ) *β u /β 1 u 2 /β 1 u 2 /β 1 Ψ 2 (,u 2,u ) *β u /β u 2 /β 2.1 u /β Ψ 2 (,u 2,u ) *β u 2 /β

39 The stability of numerical results Momentum distribution functions n(u i ) = u 2 i du j u 2 j du k u 2 k Ψ2 (u i, u j, u k ); (i, j, k) (1, 2, ) Example n(u 1 ), n(u 2 ) and n(u ) for µ 4 µ = 5: 25 N=2 2 N=2 15 N=2 n(u 1 ) [µ 4 1 ] N=1 N= n(u 2 ) [µ 4 1 ] N=1 N= n(u ) [µ 4 1 ] 1 5 N=1 N= u 1 [µ 4 ] u 2 [µ 4 ] u [µ 4 ].4 (N) n(u 1 ). B N= N=1 N=2 (N) n(u 1 ). B N= N=1 N=2 (N) n(u ). B N= N=1 N= (N) B /u (N) B /u (N) B /u 4

40 The stability of numerical results Momentum distribution functions Universal momentum distribution functions at unitary (B 2 = ) n(u i ). (N) B i=2 i= i=1 µ 4 /µ =5; Ν=1 µ 4 /µ =5; Ν=2 µ 4 /µ =5; Ν= µ 4 /µ =5; Ν= (N) B /ui 4

41 Conclusion Outline 1 Introduction 2 Formalism: 4B bound state in Faddeev-Yakubovsky scheme Numerical solution Computational approach Tetramer binding energies B & 4B scaling plots Four-atom resonances 4 On the range corrections 2B scattering amplitude B & 4B scaling functions Position of the 4-atom resonances 5 The stability of numerical results Numerical convergence Yakubovsky components & 4B wave function Momentum distribution functions 6 Conclusion

42 Conclusion Conclusion & Summary In our study on a general four-boson problem, within a renormalized zero-range model, we verify that it is not enough only two parameters (which determines trimer properties) to describe the the four-boson system. Four-body scale moving two successive tetramers below a given trimer Model independence of the limit cycle - comparison with other models No more than tetramers between successive Efimov trimers for a ± range correction r > : widens the B and 4B scaling functions range correction shifts the position of the four-atom losses Shift of the four-atom loss peaks found in a trapped cold Cesium gas close to a FB resonance can be associated with range effects Interwoven trimer-tetramer states? Scattering atom-trimer, dimer-dimer...? More bosons?

Conclusion More bosons Six-body problem in Yakubovsky scheme 5 coupled Yakubovsky integral equations for 6 identical particles 5 Jacobi momenta for each Yakubovsky component First attempt for

43 Conclusion More bosons Six-body problem in Yakubovsky scheme 5 coupled Yakubovsky integral equations for 6 identical particles 5 Jacobi momenta for each Yakubovsky component First attempt for solution of 6B Yakubovsky equations: two-neutron Halo nucleus 6 He Two coupled equations for the halo structure of the two loosely bound neutrons with respect to the core nucleons separable s wave nucleon-nucleon interaction Work in progress in collaboration with E. Ahmadi Pouya, M. Harzchi and S. Bayegan (Tehran few-body group)

44 Thanks Thank you :) 1 1 Work supported by the Brazilian agencies FAPESP and CNPq

45 Supporting slides Three- and Four-body scaling plots (N+1) (N) (B -BM-1 M ) / B M.2.1 M=4 M= B 2 = (N) B M-1 / B M

46 Supporting slides 4B limit cycle: scaling plot 4B Limit Cycle: Scaling Plot (a) (b) B = B (N+2) 2 N = 2 N = 1 N = a < a > (N) E = - B (N+1) B = (N) (N+1) B = 515 B B 2 = B (N+1) (c) (N+2) B = B4 (d) N = 2 N = 1 (a < ) ( a > ) N = (N) E 4 = - B4 (N+) B = B4 Ref.[21] Ref.[16] Ref.[18] Ref.[15] From Fig.2 of [15] (interpolated to 1/a=) Ref.[22] Ref.[1] (N+1) B = B 4

47 Supporting slides 4B limit cycle: scaling plot B and 4B Interwoven Cycles 5 4 B 4 (N,1) /B (1) (N,) () B /B 4 The fate of tetramers: cross the trimers thresholds...

48 Supporting slides 4B limit cycle: scaling plot New analysis of the peaks of the four-atom losses for 7 Li Dike, Pollack, Hulet arxiv:12.281; Pollack, Dries, Hulet, Science (29) / a 1 a 1, Dyke et al. arxiv:12.281v1(21) Gattobigio et al. PRA86 (212) Deltuva PRA85 (212) _ Hadizadeh et al. PRL17 (211) [r /a 1 = ] Berninger et al. PRL17 (211) Ferlaino et al. FBS51 (211) von Stecher et al. Nature Phys. 5 (29) Pollack et al. Science 26 (29) Ferlaino et al. PRL12 (29) Hadizadeh et al. PRA87 (21) a 1,1 / a 1 7 Li resonance is not open-channel dominated; -r / a 1 _ [.,.1] [.1,.2] [.2,.] [.,.4] [.4,.5] Effective range expansion in lowest order fails for coupled-channels Relative change of the three- and four-body short-range scales (induced few-body forces from the coupled-channel potential).

Universality in Four-Boson Systems

Few-Body Syst (201) 5:559 568 DOI 10.1007/s00601-012-06-6 T. Frederico A. Delfino M. R. adizadeh Lauro Tomio M. T. Yamashita Universality in Four-Boson Systems Received: 0 January 2012 / Accepted: 0 May