Photo Reactions in Few-Body Systems - From Nuclei to Cold-Atoms

Transcription

1 Photo Reactions in Few-Body Systems - From Nuclei to Cold-Atoms Nir Barnea The Racah institute for Physics The Hebrew University, Jerusalem, Israel TRIUMF 14 September 2012

2 Collaboration Jerusalem, Israel B. Bazak, D. Gazit, E. Liverts, N. Nevo Trento, Italy W. Leidemann, G. Orlandini Moscow, Russia V. Efros TRIUMF, Canada S. Bacca

3 What can we learn from photo reactions?

4 What can we learn from photo reactions? 1. Understanding of the systems at hand.

5 What can we learn from photo reactions? 1. Understanding of the systems at hand. 2. A test of the Hamiltonian at regimes not accessible by elastic reactions.

6 What can we learn from photo reactions? 1. Understanding of the systems at hand. 2. A test of the Hamiltonian at regimes not accessible by elastic reactions. 3. Reaction rates as input for experiments or applications (e.g. astrophysics).

7 What can we learn from photo reactions? 1. Understanding of the systems at hand. 2. A test of the Hamiltonian at regimes not accessible by elastic reactions. 3. Reaction rates as input for experiments or applications (e.g. astrophysics). 4. Underlying degrees of freedom.

8 What can we learn from photo reactions? 1. Understanding of the systems at hand. 2. A test of the Hamiltonian at regimes not accessible by elastic reactions. 3. Reaction rates as input for experiments or applications (e.g. astrophysics). 4. Underlying degrees of freedom. 5. The transition from single particle to collective behavior.

9 Photo Reactions The Interaction Hamiltonian between the photon field A(x) and the atomic/nuclear system H I = e c dxa(x) J(x) ω (E, ) f P f (E, P ) 0 0

10 Photo Reactions The Interaction Hamiltonian between the photon field A(x) and the atomic/nuclear system H I = e c dxa(x) J(x) ω (E, ) f P f The current is a sum of convection and spin currents (E, P ) J(x) = J c(x) + µ(x) 0 0

11 Photo Reactions The Interaction Hamiltonian between the photon field A(x) and the atomic/nuclear system H I = e c dxa(x) J(x) ω (E, ) f P f The current is a sum of convection and spin currents J(x) = J c(x) + µ(x) (E, P ) 0 0 H I = e c dx {A(x) J c(x) + B(x) µ(x)}

12 Photo reactions - Theoretical considerations

13 Photo reactions - Theoretical considerations The Wave Functions We solve the A-body non-realtivistic Schroedinger equation. The Hamiltonian H = T + ij V (2) ij + V (3) ijk +... ijk High precision two-nucleon potentials, well constraint by NN phaseshifts Less established 3NF EFT provides a solid theoretical framework for construction of the potentials. Phenomenological potential models are not that bad either.

14 Photo reactions - Theoretical considerations (II) The Electro-Magnetic Current The EM current is a sum of convection and spin currents J(x) = J c(x) + J s(x) = J c(x) + µ(x) Classicaly, the convection current J c = i Z iv i is the flow of the charged particles. In nuclei J c(x) is mainly due to proton movement. Meson exchange between nucleons leads to 2, 3,...-body currents J = J 1 + J Cold atoms are neutral J c(x) = 0 and the current µ(x) is dominated by the electronic spins.

15 Nuclear Physics - A tale of two potentials The nuclear Hamiltonian H = i 2 2m N 2 i + i<j V ij + V ijk +... i<j<k

16 Nuclear Physics - A tale of two potentials The nuclear Hamiltonian H = i 2 2m N 2 i + i<j V ij + V ijk +... i<j<k The Potential is composed of EM and NUCLEAR terms.

17 Nuclear Physics - A tale of two potentials The nuclear Hamiltonian H = i 2 2m N 2 i + i<j V ij + V ijk +... i<j<k The Potential is composed of EM and NUCLEAR terms. The NUCLEAR force cannot be derived from QCD and must be modeled.

18 Nuclear Physics - A tale of two potentials The nuclear Hamiltonian H = i 2 2m N 2 i + i<j V ij + V ijk +... i<j<k The Potential is composed of EM and NUCLEAR terms. The NUCLEAR force cannot be derived from QCD and must be modeled. The AV18 NN-Force

19 Nuclear Physics - A tale of two potentials The nuclear Hamiltonian H = i 2 2m N 2 i + i<j V ij + V ijk +... i<j<k The Potential is composed of EM and NUCLEAR terms. The NUCLEAR force cannot be derived from QCD and must be modeled. The AV18 NN-Force The 2-body potential V ij = v s ij + v2π ij + vπ ij

20 Nuclear Physics - A tale of two potentials The nuclear Hamiltonian H = i 2 2m N 2 i + i<j V ij + V ijk +... i<j<k The Potential is composed of EM and NUCLEAR terms. The NUCLEAR force cannot be derived from QCD and must be modeled. The AV18 NN-Force The 2-body potential V ij = v s ij + v2π ij + vπ ij v π ij - A Yukawa type interaction e µr /r, v 2π ij e 2µr /r.

21 Nuclear Physics - A tale of two potentials The nuclear Hamiltonian H = i 2 2m N 2 i + i<j V ij + V ijk +... i<j<k The Potential is composed of EM and NUCLEAR terms. The NUCLEAR force cannot be derived from QCD and must be modeled. The AV18 NN-Force The 2-body potential V ij = v s ij + v2π ij + vπ ij vij π - A Yukawa type interaction e µr /r, vij 2π e 2µr /r. vij s is expanded into a series of operators dictated by the symmetries.

22 Nuclear Physics - A tale of two potentials The nuclear Hamiltonian H = i 2 2m N 2 i + i<j V ij + V ijk +... i<j<k The Potential is composed of EM and NUCLEAR terms. The NUCLEAR force cannot be derived from QCD and must be modeled. The AV18 NN-Force The 2-body potential V ij = v s ij + v2π ij + vπ ij vij π - A Yukawa type interaction e µr /r, vij 2π e 2µr /r. vij s is expanded into a series of operators dictated by the symmetries. NNN force must be supplemented to reproduce 3,4-body binding-energies.

23 Nuclear Physics - A tale of two potentials The nuclear Hamiltonian H = i 2 2m N 2 i + i<j V ij + V ijk +... i<j<k The Potential is composed of EM and NUCLEAR terms. The NUCLEAR force cannot be derived from QCD and must be modeled. The AV18 NN-Force The 2-body potential V ij = v s ij + v2π ij + vπ ij vij π - A Yukawa type interaction e µr /r, vij 2π e 2µr /r. vij s is expanded into a series of operators dictated by the symmetries. NNN force must be supplemented to reproduce 3,4-body binding-energies. The JISP16 Potential

24 Nuclear Physics - A tale of two potentials The nuclear Hamiltonian H = i 2 2m N 2 i + i<j V ij + V ijk +... i<j<k The Potential is composed of EM and NUCLEAR terms. The NUCLEAR force cannot be derived from QCD and must be modeled. The AV18 NN-Force The 2-body potential V ij = v s ij + v2π ij + vπ ij vij π - A Yukawa type interaction e µr /r, vij 2π e 2µr /r. vij s is expanded into a series of operators dictated by the symmetries. NNN force must be supplemented to reproduce 3,4-body binding-energies. The JISP16 Potential A formal expansion of the potential V ij = (ls)jn V (ls)j nn (ls)jn lsjnn

25 Nuclear Physics - A tale of two potentials The nuclear Hamiltonian H = i 2 2m N 2 i + i<j V ij + V ijk +... i<j<k The Potential is composed of EM and NUCLEAR terms. The NUCLEAR force cannot be derived from QCD and must be modeled. The AV18 NN-Force The 2-body potential V ij = v s ij + v2π ij + vπ ij vij π - A Yukawa type interaction e µr /r, vij 2π e 2µr /r. vij s is expanded into a series of operators dictated by the symmetries. NNN force must be supplemented to reproduce 3,4-body binding-energies. The JISP16 Potential A formal expansion of the potential The HO basis is used, V (ls)j nn V ij = (ls)jn V (ls)j nn (ls)jn lsjnn fitted to reproduce NN scattering data.

26 A tale of two potentials AV18+UBIX Argonne V18 NN force + Urbana IX NNN force JISP16 J-matrix Inverse Scattering Potential, Shirokov et al. Binding Energies AV18+UBIX JISP16 Nature D H He He

27 A tale of two potentials Photodisintegration cross-section for A=2,3,4 JISP vs AV18+UBIX 3 AV18+UBIX Argonne V18 NN force + Urbana IX NNN force JISP16 J-matrix Inverse Scattering Potential, Shirokov et al. Binding Energies AV18+UBIX JISP16 Nature D H He He σ tot [mbarn] σ tot [mbarn] σ tot [mbarn] H He E-E threshold [MeV] 2 D

28 The Experimental Verdict! D. Gazit, S. Bacca, N. Barnea, W. Leidemann, and G. Orlandini, PRL 96, (2006) S. Quaglioni, and P. Navratil PLB 652, 370 (2007) R. Raut et al., PRL 108, (2012) W. Tornow et al., PRC85, (2012)

29 The Experimental Verdict? D. Gazit, S. Bacca, N. Barnea, W. Leidemann, and G. Orlandini, PRL 96, (2006) S. Quaglioni, and P. Navratil PLB 652, 370 (2007) R. Raut et al., PRL 108, (2012) W. Tornow et al., PRC85, (2012)

30 Ultra Cold atoms Bose systems, short range force, energy scale 10 9 ev A 3-body bound state E 3 < 0 exists even if the 2-body systems is unbound E 2 > 0. Physics Today, March 2010

31 Ultra Cold atoms Bose systems, short range force, energy scale 10 9 ev A 3-body bound state E 3 < 0 exists even if the 2-body systems is unbound E 2 > 0. When E 2 = 0, a s. Physics Today, March 2010

32 Ultra Cold atoms Bose systems, short range force, energy scale 10 9 ev A 3-body bound state E 3 < 0 exists even if the 2-body systems is unbound E 2 > 0. When E 2 = 0, a s. In 1970 V. Efimov found out that if E 2 = 0 the 3-body system will have an infinite number of bound states. Physics Today, March 2010

33 Ultra Cold atoms Bose systems, short range force, energy scale 10 9 ev A 3-body bound state E 3 < 0 exists even if the 2-body systems is unbound E 2 > 0. When E 2 = 0, a s. In 1970 V. Efimov found out that if E 2 = 0 the 3-body system will have an infinite number of bound states. The 3-body spectrum is E n = E 0 e 2πn/s 0 with s 0 = Physics Today, March 2010

34 Ultra Cold atoms Bose systems, short range force, energy scale 10 9 ev A 3-body bound state E 3 < 0 exists even if the 2-body systems is unbound E 2 > 0. When E 2 = 0, a s. In 1970 V. Efimov found out that if E 2 = 0 the 3-body system will have an infinite number of bound states. The 3-body spectrum is E n = E 0 e 2πn/s 0 with s 0 = In atomic traps a s can be manipulated through the Fesbach resonance. Physics Today, March 2010

35 Ultra Cold atoms Bose systems, short range force, energy scale 10 9 ev A 3-body bound state E 3 < 0 exists even if the 2-body systems is unbound E 2 > 0. When E 2 = 0, a s. In 1970 V. Efimov found out that if E 2 = 0 the 3-body system will have an infinite number of bound states. The 3-body spectrum is E n = E 0 e 2πn/s 0 with s 0 = In atomic traps a s can be manipulated through the Fesbach resonance. Particle losses in traps are closely related to Efimov s physics through the 3-body recombination process A + A + A A 2 + A Physics Today, March 2010

36 Few-Body Universality in a Bosonic 7 Li system

37 Photoassociation of Atomic Molecules The quest for the Efimov Effect RF-induce atom loss resonaces for different values of bias magnetic fields. O. Machtey, Z. Shotan, N. Gross and L. Khaykovich PRL 108, (2012)

38 The Static Response - Inelastic Reactions The response of an A-particle system is closely related to the static moments of the charge density A ρ(x) = Z i δ(x r i ) i

39 The Static Response - Inelastic Reactions The response of an A-particle system is closely related to the static moments of the charge density A ρ(x) = Z i δ(x r i ) i The Fourier Transform A ρ(q) = dxρ(x)e iq x = Z i e iq r i i

40 The Static Response - Inelastic Reactions The response of an A-particle system is closely related to the static moments of the charge density A ρ(x) = Z i δ(x r i ) i The Fourier Transform A ρ(q) = dxρ(x)e iq x = Z i e iq r i i In the long wavelength limit q 0 ρ(q) A A Z i + i Z i q r i 1 2 i i A Z i (q r i ) 2 i

41 The Static Response - Inelastic Reactions The response of an A-particle system is closely related to the static moments of the charge density A ρ(x) = Z i δ(x r i ) i The Fourier Transform A ρ(q) = dxρ(x)e iq x = Z i e iq r i i In the long wavelength limit q 0 A A ρ(q) Z i + i Z i q r i 1 2 i i For a system of identical particles A Z i (q r i ) 2 i ρ(q) AZ 1 + iz 1 R cm 1 2 Z 1 A i ( q 2 r 2 i 6 + 4π q2 r 2 i 15 Y 2 m (ˆq)Y 2m (ˆr i ) ) m

42 The Static Response - Inelastic Reactions The response of an A-particle system is closely related to the static moments of the charge density A ρ(x) = Z i δ(x r i ) i The Fourier Transform A ρ(q) = dxρ(x)e iq x = Z i e iq r i i In the long wavelength limit q 0 A A ρ(q) Z i + i Z i q r i 1 2 i i For a system of identical particles A Z i (q r i ) 2 i ρ(q) AZ 1 + iz 1 R cm 1 2 Z 1 A i ( q 2 r 2 i 6 + 4π q2 r 2 i 15 Y 2 m (ˆq)Y 2m (ˆr i ) ) m Conclusion A: In general the Dipole is the leading term.

43 The Static Response - Inelastic Reactions The response of an A-particle system is closely related to the static moments of the charge density A ρ(x) = Z i δ(x r i ) i The Fourier Transform A ρ(q) = dxρ(x)e iq x = Z i e iq r i i In the long wavelength limit q 0 A A ρ(q) Z i + i Z i q r i 1 2 i i For a system of identical particles A Z i (q r i ) 2 i ρ(q) AZ 1 + iz 1 R cm 1 2 Z 1 A i ( q 2 r 2 i 6 + 4π q2 r 2 i 15 Y 2 m (ˆq)Y 2m (ˆr i ) ) m Conclusion A: In general the Dipole is the leading term. Conclusion B: For identical particles the leading terms are ˆR 2 and ˆQ.

44 Photo Reactions with Cold-Atoms For RF photons in the MHz region the wave length is meters so qr 1.

45 Photo Reactions with Cold-Atoms For RF photons in the MHz region the wave length is meters so qr 1. The Atoms reside in a strong magnetic field, thus spins are frozen Ψ 0 = Φ 0 (r i ) m 1 F m2 F... ma F

46 Photo Reactions with Cold-Atoms For RF photons in the MHz region the wave length is meters so qr 1. The Atoms reside in a strong magnetic field, thus spins are frozen Ψ 0 = Φ 0 (r i ) m 1 F m2 F... ma F In the final state the photon can either change one of the spins or leave them untouched.

47 Photo Reactions with Cold-Atoms For RF photons in the MHz region the wave length is meters so qr 1. The Atoms reside in a strong magnetic field, thus spins are frozen Ψ 0 = Φ 0 (r i ) m 1 F m2 F... ma F In the final state the photon can either change one of the spins or leave them untouched. Spin-flip reaction m 1 F m2 F... ma F m1 F m2 F ± 1... ma F

48 Photo Reactions with Cold-Atoms For RF photons in the MHz region the wave length is meters so qr 1. The Atoms reside in a strong magnetic field, thus spins are frozen Ψ 0 = Φ 0 (r i ) m 1 F m2 F... ma F In the final state the photon can either change one of the spins or leave them untouched. Spin-flip reaction m 1 F m2 F... ma F m1 F m2 F ± 1... ma F Frozen-Spin reaction m 1 F m2 F... ma F m1 F m2 F... ma F

49 Photo Reactions with Cold-Atoms For Spin-flip reactions we get the Fermi operator R(ω) = Ck Φf Φ 0 2 δ(e f E 0 ω) f,λ

50 Photo Reactions with Cold-Atoms For Spin-flip reactions we get the Fermi operator R(ω) = Ck Φf Φ 0 2 δ(e f E 0 ω) f,λ For Frozen-Spin reactions we get a sum of the monopole operator ˆM = R 2 = ri 2 and the Quadrupole operator ˆQ = ri 2Y 2(ˆr i ) O = α ˆM + β ˆQ

51 Photo Reactions with Cold-Atoms For Spin-flip reactions we get the Fermi operator R(ω) = Ck Φf Φ 0 2 δ(e f E 0 ω) f,λ For Frozen-Spin reactions we get a sum of the monopole operator ˆM = R 2 = ri 2 and the Quadrupole operator ˆQ = ri 2Y 2(ˆr i ) The response is given by R(ω) = k 5 f,λ O = α ˆM + β ˆQ Φ f O Φ 0 2 δ(e f E 0 ω)

52 Photoassociation of The Atomic Dimer For the dimer case the response function can be written as [ 1 R(ω) = Cω ϕ 0(q) ˆM ψ ] ϕ 2(q) ˆQ ψ 0 2

53 Photoassociation of The Atomic Dimer For the dimer case the response function can be written as [ 1 R(ω) = Cω ϕ 0(q) ˆM ψ ] ϕ 2(q) ˆQ ψ 0 2 Where the G.S. wave function is given by ψ 0 = Y 0 2κe κr /r ; κ 1/a s

54 Photoassociation of The Atomic Dimer For the dimer case the response function can be written as [ 1 R(ω) = Cω ϕ 0(q) ˆM ψ ] ϕ 2(q) ˆQ ψ 0 2 Where the G.S. wave function is given by ψ 0 = Y 0 2κe κr /r ; κ 1/a s The continuum state is given by ϕ l (q) = Y l (ˆr)χ l (r)/r χ l (r) = 2qr[cos δ l j l (qr) sin δ l n l (qr)]

55 Photoassociation of The Atomic Dimer For the dimer case the response function can be written as [ 1 R(ω) = Cω ϕ 0(q) ˆM ψ ] ϕ 2(q) ˆQ ψ 0 2 Where the G.S. wave function is given by ψ 0 = Y 0 2κe κr /r ; κ 1/a s The continuum state is given by ϕ l (q) = Y l (ˆr)χ l (r)/r The l = 0 matrix element ( ϕ 0 (q) ˆM ψ 0 2 = 1 4π χ l (r) = 2qr[cos δ l j l (qr) sin δ l n l (qr)] 4q 2κ (q 2 + κ 2 ) 3 ) 2 [ ] cos δ 0 (3κ 2 q 2 κ 2 ) sin δ 0 q (3q2 κ 2 )

56 Photoassociation of The Atomic Dimer For the dimer case the response function can be written as [ 1 R(ω) = Cω ϕ 0(q) ˆM ψ ] ϕ 2(q) ˆQ ψ 0 2 Where the G.S. wave function is given by ψ 0 = Y 0 2κe κr /r ; κ 1/a s The continuum state is given by ϕ l (q) = Y l (ˆr)χ l (r)/r The l = 0 matrix element ( ϕ 0 (q) ˆM ψ 0 2 = 1 4π χ l (r) = 2qr[cos δ l j l (qr) sin δ l n l (qr)] 4q 2κ (q 2 + κ 2 ) 3 The l = 2 matrix element, assuming δ 2 = 0 ) 2 [ ] cos δ 0 (3κ 2 q 2 κ 2 ) sin δ 0 q (3q2 κ 2 ) [ ϕ 2 (q) ˆQ ψ 0 2 = 5 16q 3 ] 2 2κ 4π (q 2 + κ 2 ) 3

57 Photoassociation of The Atomic Dimer The s-wave and d-wave components in the response function upper panel a/r eff = 2 lower pannel a/r eff = 200 red - r 2 monopole blue - quadrupole Normalized transition M.E. [n.d.] q/κ [n.d.]

58 Photoassociation rates Photoassociation of 7 Li atoms a s = 1000a 0 T = 5µK (lower panel), T = 25µK (upper panel) red - r 2 monopole, blue - quadrupole Normalized photoassociation rate [n.d.] hω/e B [n.d.] hω/e B [n.d.]

59 Photoassociation rates Photoassociation of 7 Li atoms a s = 1000a 0 T = 5µK (lower panel), T = 25µK (upper panel) red - r 2 monopole, blue - quadrupole Relative Contribution [n.d.] k B T/E B [n.d.] The relative contribution to the peak Normalized photoassociation rate [n.d.] hω/e B [n.d.] hω/e B [n.d.]

60 Photoassociation of The Atomic Dimer Comparison to the Khaykovich group data Normalized 2-body rate [n.d.] as = 847a0 T = 2.6µK Khaykovich Normalized 2-body rate [n.d.] as = 640a0 T = 1.8µK Khaykovich RF frequency [MHz] RF frequency [MHz] The fitted values of a s and T are in reasonable agreement with the estimates of the experimental group. Effect of RF field on dimers not included. Finite time effect Disagreement are due to 3-body (4-body?) association. Effects of δ 2 0 are negligible.

61 The Atomic Trimer The quadrupole response of the bosonic trimer 1e+06 1e+05 ε 3 = ε 3 =0.047 ε 3 = R(ω) [a.u.] ω ω th [h 2 2 /mr 0 ] E. Liverts, B. Bazak, and N. Barnea Phys. Rev. Lett. 108, (2012).

62 Photodisintegration Sum Rules S n dω ω n R(ω) ω th

63 Photodisintegration Sum Rules S n dω ω n R(ω) ω th The sum rule S n Exists if R(ω) 0 faster than ω n 1. Can be expressed as GS observable utilizing the closure of the eigenstates of H.

64 Photodisintegration Sum Rules S n dω ω n R(ω) ω th The sum rule S n Exists if R(ω) 0 faster than ω n 1. Can be expressed as GS observable utilizing the closure of the eigenstates of H. S 1 = 0 [O, [H, O]] 0 = 0 O (H E 0 ) O 0 S 0 = 0 OO 0 S 1 = 0 O 1 H E 0 O 0

65 Naive Scaling Using simple dimensional arguments we expect that r 1/ E The Quadrupole operator behaves as r 2 so R(ω) r 4 /E 1/E 3 It follows that the sum rules should fulfill S n 1/E 2 n or S 0 1/E 2 S 1 1/E 3 S 0 /S 1 E

66 Calculated Sum Rules 1e+08 1e+06 S -1 S 0 S 1 Squares Gauss Potential Triangles Yukawa Potential Sum Rule [a.u.] E T [ h 2 2 /m r 0 ] Fitted lines S 1 = A 1 E 2.13 S 0 = A 0 E 1.34 S 1 = A 1 E 0.55

67 A For S 1 we got a power of 0.55 instead of 1. B For S 0 we got a power of 1.33 instead of 2. C For S 1 we got a power of 2.13 instead of 3. D The ration S n/s n 1 E 0.8 instead of S n/s n 1 E. E The results seems to be independent of the short range specifications of the potential. Naive scaling doesn t work!!!

68 Summary and Conclusions 1. The new RF experiments in Cold-Atoms systems carry much in common with photo-reactions and charged current reactions in nuclei.

69 Summary and Conclusions 1. The new RF experiments in Cold-Atoms systems carry much in common with photo-reactions and charged current reactions in nuclei. 2. For spin-flip reaction a Fermi type operator is the leading contribution to the cross-section, and R(ω) ω.

70 Summary and Conclusions 1. The new RF experiments in Cold-Atoms systems carry much in common with photo-reactions and charged current reactions in nuclei. 2. For spin-flip reaction a Fermi type operator is the leading contribution to the cross-section, and R(ω) ω. 3. For frozen-spin reactions the monopole R 2 and the Quadrupole are the leading terms, and R(ω) ω 5.

71 Summary and Conclusions 1. The new RF experiments in Cold-Atoms systems carry much in common with photo-reactions and charged current reactions in nuclei. 2. For spin-flip reaction a Fermi type operator is the leading contribution to the cross-section, and R(ω) ω. 3. For frozen-spin reactions the monopole R 2 and the Quadrupole are the leading terms, and R(ω) ω We have studied Dimer formation and found that the reaction mechanism change from monopole to quadrupole with increasing gas temprature.

72 Summary and Conclusions 1. The new RF experiments in Cold-Atoms systems carry much in common with photo-reactions and charged current reactions in nuclei. 2. For spin-flip reaction a Fermi type operator is the leading contribution to the cross-section, and R(ω) ω. 3. For frozen-spin reactions the monopole R 2 and the Quadrupole are the leading terms, and R(ω) ω We have studied Dimer formation and found that the reaction mechanism change from monopole to quadrupole with increasing gas temprature. 5. We have studied the Quadrupole response of a bosonic trimer near threshold.

73 Summary and Conclusions 1. The new RF experiments in Cold-Atoms systems carry much in common with photo-reactions and charged current reactions in nuclei. 2. For spin-flip reaction a Fermi type operator is the leading contribution to the cross-section, and R(ω) ω. 3. For frozen-spin reactions the monopole R 2 and the Quadrupole are the leading terms, and R(ω) ω We have studied Dimer formation and found that the reaction mechanism change from monopole to quadrupole with increasing gas temprature. 5. We have studied the Quadrupole response of a bosonic trimer near threshold. 6. We have seen that the response function diverge as E 3 0.

74 Summary and Conclusions 1. The new RF experiments in Cold-Atoms systems carry much in common with photo-reactions and charged current reactions in nuclei. 2. For spin-flip reaction a Fermi type operator is the leading contribution to the cross-section, and R(ω) ω. 3. For frozen-spin reactions the monopole R 2 and the Quadrupole are the leading terms, and R(ω) ω We have studied Dimer formation and found that the reaction mechanism change from monopole to quadrupole with increasing gas temprature. 5. We have studied the Quadrupole response of a bosonic trimer near threshold. 6. We have seen that the response function diverge as E The sum-rules S 1, S 0, S 1 diverge as well.

75 Summary and Conclusions 1. The new RF experiments in Cold-Atoms systems carry much in common with photo-reactions and charged current reactions in nuclei. 2. For spin-flip reaction a Fermi type operator is the leading contribution to the cross-section, and R(ω) ω. 3. For frozen-spin reactions the monopole R 2 and the Quadrupole are the leading terms, and R(ω) ω We have studied Dimer formation and found that the reaction mechanism change from monopole to quadrupole with increasing gas temprature. 5. We have studied the Quadrupole response of a bosonic trimer near threshold. 6. We have seen that the response function diverge as E The sum-rules S 1, S 0, S 1 diverge as well. 8. We note that the divergence pattern of the sum-rules doesn t follow the simple dimensional analysis.

76 Summary and Conclusions 1. The new RF experiments in Cold-Atoms systems carry much in common with photo-reactions and charged current reactions in nuclei. 2. For spin-flip reaction a Fermi type operator is the leading contribution to the cross-section, and R(ω) ω. 3. For frozen-spin reactions the monopole R 2 and the Quadrupole are the leading terms, and R(ω) ω We have studied Dimer formation and found that the reaction mechanism change from monopole to quadrupole with increasing gas temprature. 5. We have studied the Quadrupole response of a bosonic trimer near threshold. 6. We have seen that the response function diverge as E The sum-rules S 1, S 0, S 1 diverge as well. 8. We note that the divergence pattern of the sum-rules doesn t follow the simple dimensional analysis.

77 Summary and Conclusions 1. The new RF experiments in Cold-Atoms systems carry much in common with photo-reactions and charged current reactions in nuclei. 2. For spin-flip reaction a Fermi type operator is the leading contribution to the cross-section, and R(ω) ω. 3. For frozen-spin reactions the monopole R 2 and the Quadrupole are the leading terms, and R(ω) ω We have studied Dimer formation and found that the reaction mechanism change from monopole to quadrupole with increasing gas temprature. 5. We have studied the Quadrupole response of a bosonic trimer near threshold. 6. We have seen that the response function diverge as E The sum-rules S 1, S 0, S 1 diverge as well. 8. We note that the divergence pattern of the sum-rules doesn t follow the simple dimensional analysis.

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