Range eects on Emov physics in cold atoms

Transcription

1 Range eects on Emov physics in cold atoms An Eective eld theory approach Chen Ji TRIUMF in collaboration with D. R. Phillips, L. Platter INT workshop, November Canada's national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules Range eects on Emov physics in cold atoms 1 / 22

2 Universal features at low energies Separation of scales: a l 2-body S-wave universality: B d = 1/Ma 2 Range eects on Emov physics in cold atoms 2 / 22

3 Universal features at low energies Separation of scales: a l 2-body S-wave universality: B d = 1/Ma 2 Physics at large distance is insensitive to physics at short distance Large-distance physics is studied in l/a expansion Eects from SR-dynamics can be included in perturbation theory Range eects on Emov physics in cold atoms 2 / 22

4 Large-scattering-length physics Universal Physics exists in systems with l a Nuclear Physics Few-nucleon systems (NN, NNN): i.e., l t np 1.7 fm, a t np 5.4 fm Halo nuclei ( 6 He, 11 Li) E sep E core [ [ Range eects on Emov physics in cold atoms 3 / 22

5 Large-scattering-length physics Universal Physics exists in systems with l a Nuclear Physics Few-nucleon systems (NN, NNN): i.e., l t np 1.7 fm, a t np 5.4 fm Halo nuclei ( 6 He, 11 Li) E sep E core Unit of a B a r 0 a B (G) 7 Li atoms [Gross et al. '09 Atomic Physics Cold atomic gases ( 133 Cs, 7 Li, 39 K): r 0 and a varies near Feshbach resonance 4 He atoms (dimer, trimer, tetramer): l vdw 7Å, a 100Å 6 He tetramer [D. Blume Range eects on Emov physics in cold atoms 3 / 22

6 Eective eld theory An approach to systems with a separation of scales Systems with l a an EFT with contact interactions Few-nucleon systems pionless EFT Halo nuclei halo EFT Atomic systems short-range EFT Physical quantities are expanded in powers of r 0 /a Contact interactions at LO 2-body contact interaction (LO) introduce a dimer eld = ic 0 C 0 determined by a 2-body observable 3-body contact interaction (LO) = id 0 D 0 determined by a 3-body observable Bedaque, Hammer, van Kolck '99 Range eects on Emov physics in cold atoms 4 / 22

7 Eective eld theory An approach to systems with a separation of scales Systems with l a an EFT with contact interactions Few-nucleon systems pionless EFT Halo nuclei halo EFT Atomic systems short-range EFT Physical quantities are expanded in powers of r 0 /a Contact interactions at LO 2-body contact interaction (LO) introduce a dimer eld = ic C 0 = i 0 =g 2 / ============= 2g C 0 determined by a 2-body observable 3-body contact interaction (LO) = id D 0 0 =3hg 2 / ================= 2 = ih D 0 determined by a 3-body observable Bedaque, Hammer, van Kolck '99 Range eects on Emov physics in cold atoms 4 / 22

8 EFT for 3 identical bosons LO (r 0 /a) 0 EFT Lagrangian for 3 identical bosons ( ) ) L = ψ i ψd (i m 4m d 2 g ( ) d ψψ + h.c +hd dψ ψ+ terms with more derivatives are at higher orders (r 0 /a) n Non-perturbative features at LO atom-atom (dimer) scattering (tune g) atom-dimer scattering (tune h) = + 1 1/a+ik t 0 = + + t 0 + t 0 Bedaque, Hammer, van Kolck '99 Range eects on Emov physics in cold atoms 5 / 22

9 LO renormalization 10 3 Without 3BF: 3-body spectrum: cuto dependent (Λ 1/l) Platter '09 B 3 [B LO 3BF h: tune H(Λ) = Λ 2 h/2mg 2 : x one 3-body observable limit cycle: H(Λ) periodic for Λ Λ(22.7) n Bedaque et al. '00 scaling invariance Emov physics Emov ' Λ [B 2 1/2 Range eects on Emov physics in cold atoms 6 / 22

10 Universal physics at LO Universal features in three-body systems (Emov eects) 3-body spectrum: a function of scattering length a geometric spectrum E n = (22.7) 2 E n1 in the limit a universal relation of recombination features a = a + /4.5 = a /21.3 i.e. a (n) = 22.7a (n1) Zaccanti et al. '09 Range eects on Emov physics in cold atoms 7 / 22

11 Recombination features of cold atoms loss rates of free atoms in ultracold atomic gases 39 K atoms Zaccanti et al. '09 7 Li atoms F = 1, m F = 0 Gross et al. '09 7 Li atoms F = 1, m F = 1 Pollack et al. '09 Pic. credit: Ferlaino, Grimm '10 Range eects on Emov physics in cold atoms 8 / 22

12 Beyond universality: range eects range eects on universal physics 2-body observable: k cot δ 0 = 1 a + r0 2 k2 + 3-body observables: in r 0 /a expansion Previous EFT calculations of r 0 /a corrections (xed a): Hammer, Mehen '01 (NLO) Bedaque, Rupak, Griesshammer, Hammer '03 (N 2 LO, partial iteration) Platter, Phillips '06 (N 2 LO, partial iteration) We perform a rigorous perturbative caluclation NLO (systems with variable a) ultracold atomic gases (r 0/a varies near Feshbach resonance) 3-body recombination N 2 LO (systems with a xed a) 4 He atoms (r 0/a 0.07) 4 He trimer (bound state), atom-dimer phase shift (scattering state) Range eects on Emov physics in cold atoms 9 / 22

13 NLO (r 0 /a) range eects Calculate r 0 /a correction to atom-dimer amplitude NLO correction to atom-atom propagator (dimer): = ir 2π 1/a+ik 0 mg 2 1/a+ik NLO correction to atom-dimer contact term (3BF): = i 2mg2 Λ 2 H 1 (a, Λ) Contribution in 1 st order perturbation theory: NLO dimer: t 0 t 0 NLO 3BF: t 0 t 0 t 0 t 0 NLO 3-body force: H 1 (Λ) = r 0 Λ h 10 (Λ) + r 0 /a h 11 (Λ) a xed: h 11 is absorbed (no additional 3-body input is needed) a varies: one additional 3-body input is needed Range eects on Emov physics in cold atoms 10 / 22

14 NLO three-body 3BF H 1 (Λ) = r 0 Λ h 10 (Λ) + r 0 /a h 11 (Λ) 1 / h 10 (Λ) / h 11 (Λ) Λ/κ * Λ/κ * 1/h 10 (Λ) 1/h 11 (Λ) Range eects on Emov physics in cold atoms 11 / 22

15 In ultracold atomic gases a and r 0 are functions of the magnetic eld 7 Li F = 1, m F = 0 > 7 Li F = 1, m F = 1 > Gross et al. '09 Range eects on Emov physics in cold atoms 12 / 22

16 Recombination of 7 Li Atoms F = 1, m F = 0 10 LO: rn to a NLO: rn to a & a + Gross et al. data 3-body recombination rate of 7 Li atoms NLO (r 0/a) range eects to recombination positions are not shifted by K 3 [1000 a B deep-dimer at LO [Braaten et al. ' a [1000 a B 1 1 Experiment LO LO NLO 3A res a (-) [a B rec min a + [a B Ad res a [a B (44) Gross et al. '09 Machtey et al. '12 Ji, Phillips, Platter '10 Range eects on Emov physics in cold atoms 13 / 22

17 Recombination of 7 Li Atoms F = 1, m F = 0 10 LO: rn to a NLO: rn to a & a + Gross et al. data 3-body recombination rate of 7 Li atoms NLO (r 0/a) range eects to recombination positions are not shifted by K 3 [1000 a B deep-dimer at LO [Braaten et al. ' a [1000 a B 1 1 Experiment LO LO NLO 3A res a (-) [a B rec min a + [a B Ad res a [a B (44) Gross et al. '09 Machtey et al. '12 Ji, Phillips, Platter '10 Range eects on Emov physics in cold atoms 13 / 22

18 Recombination of 7 Li Atoms F = 1, m F = 0 10 LO: rn to a NLO: rn to a & a + Gross et al. data 3-body recombination rate of 7 Li atoms NLO (r 0/a) range eects to recombination positions are not shifted by K 3 [1000 a B deep-dimer at LO [Braaten et al. ' a [1000 a B 1 1 Experiment LO LO NLO 3A res a (-) [a B rec min a + [a B Ad res a [a B (44) Gross et al. '09 Machtey et al. '12 Ji, Phillips, Platter '10 Range eects on Emov physics in cold atoms 13 / 22

19 Recombination of 7 Li Atoms F = 1, m F = 0 10 LO: rn to a NLO: rn to a & a + Gross et al. data 3-body recombination rate of 7 Li atoms NLO (r 0/a) range eects to recombination positions are not shifted by K 3 [1000 a B deep-dimer at LO [Braaten et al. ' a [1000 a B 1 1 Experiment LO LO NLO 3A res a (-) [a B rec min a + [a B Ad res a [a B (44) Gross et al. '09 Machtey et al. '12 Ji, Phillips, Platter '10 Range eects on Emov physics in cold atoms 13 / 22

20 Recombination of 7 Li Atoms 10 2 F = 1, m F = 1 7 Li(mF = 1) recombination [Pollack et al. '09 data disagree with universality by a factor of 2 EFT analysis at NLO agrees with universality K 3 [1000 a B LO: rn to a 1 NLO: rn to a 1 & a a [1000 a B Experiment LO LO NLO [Pollack et al. '09 [Ji et al. '10 3A res a (-) [a B A res a [a B rec min a + [a B (151) Ad res a [a B (55) Range eects on Emov physics in cold atoms 14 / 22

21 Recombination of 7 Li Atoms 10 2 F = 1, m F = 1 7 Li(mF = 1) recombination [Pollack et al. '09 data disagree with universality by a factor of 2 EFT analysis at NLO agrees with universality K 3 [1000 a B LO: rn to a 1 NLO: rn to a 1 & a a [1000 a B Experiment LO LO NLO [Pollack et al. '09 [Ji et al. '10 3A res a (-) [a B A res a [a B rec min a + [a B (151) Ad res a [a B (55) Range eects on Emov physics in cold atoms 14 / 22

22 Recombination of 7 Li Atoms 10 2 F = 1, m F = 1 7 Li(mF = 1) recombination [Pollack et al. '09 data disagree with universality by a factor of 2 EFT analysis at NLO agrees with universality K 3 [1000 a B LO: rn to a 1 NLO: rn to a 1 & a a [1000 a B Experiment LO LO NLO [Pollack et al. '09 [Ji et al. '10 3A res a (-) [a B A res a [a B rec min a + [a B (151) Ad res a [a B (55) Range eects on Emov physics in cold atoms 14 / 22

23 Recombination of 7 Li Atoms 10 2 F = 1, m F = 1 7 Li(mF = 1) recombination [Pollack et al. '09 data disagree with universality by a factor of 2 EFT analysis at NLO agrees with universality K 3 [1000 a B LO: rn to a 1 NLO: rn to a 1 & a a [1000 a B Experiment LO LO NLO [Pollack et al. '09 [Ji et al. '10 3A res a (-) [a B A res a [a B rec min a + [a B (151) Ad res a [a B (55) Range eects on Emov physics in cold atoms 14 / 22

24 Universal relations at NLO 1 a = γ 1 2 r 0γ 2 Ji, Phillips, Platter '12 Range eects on Emov physics in cold atoms 15 / 22

25 Universal relations at NLO 1 a = γ 1 2 r 0γ 2 NLO corrections γ 0 = γ = 0.210γ () 0.939γ () + r 0 (a 0 ) I () 0 γ () + r 0 (a ) I () γ () 2 2 Ji, Phillips, Platter '12 Range eects on Emov physics in cold atoms 15 / 22

26 Universal relations at NLO 1 a = γ 1 2 r 0γ 2 NLO corrections γ 0 = γ = 0.210γ () 0.939γ () + r 0 (a 0 ) I () 0 γ () + r 0 (a ) I () γ () 2 2 Linear correlation at NLO I () = I () 0 Ji, Phillips, Platter '12 Range eects on Emov physics in cold atoms 15 / 22

27 Universal relations at NLO 1 a = γ 1 2 r 0γ 2 NLO corrections γ 0 = γ = 0.210γ () 0.939γ () + r 0 (a 0 ) I () 0 γ () + r 0 (a ) I () γ () 2 2 Linear correlation at NLO I () = I () 0 Universal relation at NLO γ = 0.939γ () r 0 (a )γ () 2 ( r 0(a ) r 0(a 0) γ 0 ) γ () Ji, Phillips, Platter '12 Range eects on Emov physics in cold atoms 15 / 22

28 Universal relations at NLO 1 a = γ 1 2 r 0γ 2 NLO corrections γ 0 = γ = 0.210γ () 0.939γ () + r 0 (a 0 ) I () 0 γ () + r 0 (a ) I () γ () 2 2 Linear correlation at NLO I () = I () 0 Universal relation at NLO γ = 0.939γ () r 0 (a )γ () 2 ( r 0(a ) r 0(a 0) γ 0 ( ) γ = 4.47γ r 0 (a )γ r0(a ) γ () r 0(a () ) γ 0 ) γ () Ji, Phillips, Platter '12 Range eects on Emov physics in cold atoms 15 / 22

29 N 2 LO (r 0 /a) 2 range eects N 2 LO corrections to atom-dimer scattering amplitude: in 2 nd order perturbation theory ( r 2 0/a 2 ): N 2 LO dimer: t 0 t 0 N 2 LO 3BF: t 0 t 0 t 0 t 0 two NLO terms: t 0 t 0 t 0 t 0 t 0 t 0 + t 0 t 0 t 0 + N 2 LO 3-body force: H 2 (E, Λ) = r 2 0 Λ2 h 20 (Λ) + r 2 0 me 3 h 22 (Λ) one additional 3-body input is needed (even when a is xed) c.f. Bedaque et al. '03 & Platter, Phillips '06 Range eects on Emov physics in cold atoms 16 / 22

30 N 2 LO renormalization H 2 = r 2 0 Λ2 h 20 H 2 = r 2 0 Λ2 h 20 + r 2 0 me t h 22 B n (1) [γ n B d (1) B (1) B 1 (1) B Λ [γ B n (0) [γ n B d B 0 (0) B 1 (0) 0.1B 2 (0) Λ [γ a ad LO/NLO/N 2 LO B (1) t = B (1) 0 + r 0 B (1) 1 + r 2 0B (1) 2 a ad LO/NLO/N 2 LO B (1) t N 2 LO B (0) t = B (0) 0 + r 0 B (0) 1 + r 2 0B (0) 2 Range eects on Emov physics in cold atoms 17 / 22

31 NLO three-body 3BF H 2 (Λ) = r 2 0 Λ2 h 20 (Λ) + r 2 0 me t h 22 (Λ) 1 / h 20 (Λ) Λ [γ 1 / h 22 (Λ) Λ [γ 1/h 20 (Λ) 1/h 22 (Λ) [partial version Range eects on Emov physics in cold atoms 18 / 22

32 4 He trimer Input B (1) t [B d B (0) t [B d a ad [γ 1 r ad [γ 1 TTY potential a ad LO a ad NLO a ad, B (1) t N 2 LO B (1) t LO B (1) t NLO B (1) t, a ad N 2 LO Range eects on Emov physics in cold atoms 19 / 22

33 4 He trimer Input B (1) t [B d B (0) t [B d a ad [γ 1 r ad [γ 1 TTY potential a ad LO a ad NLO a ad, B (1) t N 2 LO B (1) t LO B (1) t NLO B (1) t, a ad N 2 LO Range eects on Emov physics in cold atoms 19 / 22

34 4 He trimer Input B (1) t [B d B (0) t [B d a ad [γ 1 r ad [γ 1 TTY potential a ad LO a ad NLO a ad, B (1) t N 2 LO B (1) t LO B (1) t NLO B (1) t, a ad N 2 LO Range eects on Emov physics in cold atoms 19 / 22

35 4 He trimer Input B (1) t [B d B (0) t [B d a ad [γ 1 r ad [γ 1 TTY potential a ad LO a ad NLO a ad, B (1) t N 2 LO B (1) t LO B (1) t NLO B (1) t, a ad N 2 LO Dierence in 2 renormalization schemes (LO NLO N 2 LO): atom-dimer eective range r ad : 5% 0.9% 0.02% ground-state trimer B (0) t : 2% 0.07% 0.9% Range eects on Emov physics in cold atoms 19 / 22

36 4 He trimer Input B (1) t [B d B (0) t [B d a ad [γ 1 r ad [γ 1 TTY potential a ad LO a ad NLO a ad, B (1) t N 2 LO B (1) t LO B (1) t NLO B (1) t, a ad N 2 LO Dierence in 2 renormalization schemes (LO NLO N 2 LO): atom-dimer eective range r ad : 5% 0.9% 0.02% ground-state trimer B (0) t : 2% 0.07% 0.9% Compare with TTY: B (0) t 20% r0b 2 (0) t 0.5 EFT expansion needs to be corrected for deep trimer with U vdw [Gao '98 Range eects on Emov physics in cold atoms 19 / 22

37 He-4 trimer phase shift at N 2 LO a ad B (1) t LO/NLO/N 2 LO N 2 LO B (1) t a ad LO/NLO/N 2 LO N 2 LO kcotδ ad [γ LO (a ad ) NLO (a ad ) N 2 LO (a ad, B t (1) ) kcotδ ad [γ LO (B t (1) ) NLO (B t (1) ) N 2 LO (B t (1), a ad ) k [γ k [γ Range eects on Emov physics in cold atoms 20 / 22

38 Dierence btw 2 renormalization LO NLO N 2 LO kcotδ ad [γ k [γ Range eects on Emov physics in cold atoms 21 / 22

39 Conclusion We studied eective-range corrections on Emov physics in perturbation theory NLO for varying a: recombination in cold atoms H 1 (Λ) = r 0 Λ h 10 (Λ) + r 0 /a h 11 (Λ) one additional 3-body input is needed for NLO renormalization N 2 LO for xed a: He-4 trimer H 2 (E, Λ) = r 2 0Λ 2 h 20 (Λ) + r 2 0mE 3 h 22 (Λ) one additional 3-body data is needed for N 2 LO renormalization Range corrections are also important in nuclear physics Range eects on Emov physics in cold atoms 22 / 22