Efimov Physics and Universality

Transcription

1 Efimov Physics and Universality Eric Braaten Ohio State University Bonn University support DOE High Energy Physics DOE Basic Energy Sciences Air Force Office of Scientific Research Army Research Office Alexander von Humboldt Foundation

2 Efimov Physics and Universality 2-Body Universality 3-Body Universality 3-Body Parameters Range corrections 4-Body Universality Search for Efimov States Efimov Physics in Cold Atoms Outlook

3 Particle physics/atomic physics Dictionary particle atom 2-body bound state dimer (diatomic molecule) 3-body bound state trimer (triatomic molecule)

4 2-Body Universality 2 atoms interacting through short-range potential V r Dimer: energy depends on depth, range

5 2-Body Universality Zero-Range limit: range 0 depth binding energy fixed V r

6 2-Body Universality Zero-Range limit is universal! independent of shape of potential determined by scattering length a only Cross section low energy: 8π a 2 high energy: 8πħ 2 /(m E) Dimer if a > 0 binding energy: ħ 2 /(m a 2 ) mean radius: a/2 halo dimer, giant dimer a/2

7 2-Body Universality Unitary Limit: a ± Cross section 8πħ 2 /(m E) saturates unitarity bound! Dimer if a > 0 binding energy: 0 mean size: Scale Invariance! invariance under λ t λ 2 t

8 2-Body Universality scale invariance at a = ± scaling at finite a exhibit scaling with Efimov plot energy variable: K = ± m E / 2 interaction variable: 1/a K 2-atom threshold 1 a dimer

9 3-Body Universality Is there a well-behaved Zero-Range limit? 3 identical fermions: free particles 3 fermions with two spin states: Yes! depends only on scattering length 3 identical bosons? No! Thomas (1935) Yes! Efimov (1970)

10 3-Body Universality Is there a well-defined Zero-Range limit for 3 identical bosons? Thomas (1935): No! 3 particles interacting with scattering length much larger than range R can make trimer of size R with binding energy of order ħ 2 /mr 2 as R 0, energy is unbounded from below

11 3-Body Universality Efimov obtained crucial insights into the Zero-Range limit by considering the Unitary limit: range fixed tune depth binding energy 0 V r

12 3-Body Universality Efimov Effect Vitaly Efimov (1970) In the unitary limit a ± (with fixed range) there are infinitely many trimers accumulation point at the threshold binding energies differ by factors of 1/515.0 sizes differ by factors of 22.7

13 3-Body Universality Russian nesting dolls with discrete scaling factor 1.3 In the unitary limit a ± Efimov trimers have same structure but with discrete scaling factor 22.7

14 3-Body Universality Efimov Effect is characterized by discrete scale invariance: invariance under r λ n 0 r t λ 2n 0 t for integer powers of discrete scaling factor λ0 discrete scaling factor depends on mass ratios and symmetries for 3 identical bosons, discrete scaling factor is λ λ 0 = exp(π/s 0 ), s 0 cosh(πs 0 /2) = 8 3 sinh(πs 0 /6)

15 3-Body Universality Is there a well-defined Zero-Range limit for 3 identical bosons? Efimov (1970): Yes! Zero-Range limit is universal! independent of shape of potential determined by two parameters: scattering length a 3-body parameter on which dependence can only be log-periodic with discrete scaling factor λ spectrum is unbounded from below (but don t worry about it)

16 3-Body Universality convenient choice of 3-body parameter: κ * = binding momentum of one of the Efimov trimers in the unitary limit a ± binding energy: E (n ) T = 2 κ 2 /m ( 1 spectrum: E (n) T = ) n n 2 κ 2 m

17 3-Body Universality Efimov plot for 3-atom sector energy variable: K = ± m E / 2 interaction variable: K 1/a 3-atom threshold trimers atom-dimer threshold 1/a trimers

18 3-Body Universality What happens to Efimov trimers away from the unitary limit? for a < 0, they disappear thru 3-atom threshold at values of a that differ by factors of 22.7 K 1/a trimers

19 3-Body Universality What happens to Efimov trimers away from the unitary limit? for a > 0, they disappear thru atom-dimer threshold at values of a that differ by factors of 22.7 K 1/a trimers

20 3-Body Universality Problem: spectrum is unbounded from below! Solution: ignore it! For physical system with nonzero range R universal behavior holds only inside window of universality of radius 1/R on Efimov plot K 1/a

21 Search for Efimov States 4 He atoms Predictions from potential models scattering length: 190 a0 >> range ~ 10 a0 one dimer: binding energy 1.3 mk two trimers: binding energies 2.3 mk 126 mk Are the 4 He trimers Efimov states? Lim, Duffy & Damert (1977) 1 mk = ev

22 Search for Efimov States Is the excited 4 He trimer an Efimov state? Yes! Is the ground state 4 He trimer an Efimov state? Efimov: Yes! It lies within the window of universality window of universality K trimers 1/a

23 Search for Efimov States experiments with 4 He atoms 4 He atoms escaping into the vacuum form a jet that can reach comoving temperature as low as 0.3 mk Luo, Giese, Gentry, et al. (Minnesota) 1992, 1995 electron impact ionization of He beam Schollkopf, Toennies, et al. (Gottingen) 1995, 2000 diffraction of He beam by nanoscale transmission grating dimer observed mean radius: 98±8 a0 universal prediction: a/2 = 95 a0 ground state trimer observed excited state trimer not yet observed

24 Search for Efimov States Nucleons proton (p), neutron (n) scattering lengths nn: -24 fm >> range ~ 2 fm pn isosinglet: 5.5 fm stable nucleus with 2 nucleons deuteron (pn) binding energy: 2.2 MeV stable nuclei with 3 nucleons triton (pnn) binding energy: 7.2 MeV 3 He nucleus (ppn) 9.0 MeV triton, 3 He nucleus: are they Efimov states?

25 Search for Efimov States 3 He nucleus (ppn): complicated by long-range Coulomb force between protons Is the triton (pnn) an Efimov state? Efimov: Yes! It lies within the window of universality window of universality K triton 1/a trimers

26 Search for Efimov States If 3 He nucleus is excluded (to avoid complications from Coulomb force), the simplest 3-body observables affected by Efimov physics are triton binding energy neutron-deuteron scattering length in spin-½ S-wave channel If the triton binding energy is used to determine κ * the neutron-deuteron scattering length is correctly postdicted

27 Search for Efimov States Efimov effect in QCD There are critical values of the up, down quark masses (pion mass 180 MeV) at which the low-energy 3-nucleon system has discrete scale invariance with scaling factor 22.7 Braaten and Hammer 2003 deuteron (pn) has zero binding energy dineutron (nn) has zero binding energy triton (pnn) has infinitely many excited states, ratio of successive binding energies is 1/515.0 Real QCD (pion mass = 140 MeV) is not far from this critical point

28 Search for Efimov States Halo Nuclei nucleus + 2 neutrons 14 Be 12 Be + n + n? 20 C 18 C + n + n? see Tomio Universality and halo nuclei see Fedorov Structure and reactions for quantum halos see Mazumdar Efimov effect in 2-neutron halo nuclei

29 Cold Atom Physics Atoms trapped and cooled using lasers Nobel Prize 1997: Chu, Cohen-Tannoudji, Phillips

30 Cold Atom Physics Bose-Einstein condensation of atoms! 87 Rb atoms JILA (Cornell, Wieman) Li atoms Rice (Hulet) Na atoms MIT (Ketterle) 1995 Nobel Prize 2001: Cornell, Wieman, Ketterle

31 Cold Atom Physics Cooling of fermions to quantum degeneracy! 40 K atoms JILA (Jin) Jan Li atoms Ecole Normale (Salomon) July Li atoms Rice (Hulet) Aug 2001

32 a(b) 0 a bg B res B

33 Efimov Physics in Cold Atoms Atoms can be lost from trapping potential by 3-body recombination into deep dimers three atoms collide two of them bind to form a deep dimer atom and dimer recoil with large kinetic energy

34 Efimov Physics in Cold Atoms Efimov trimers disappear through 3-atom threshold at values of a differing by factors of 22.7 give resonant enhancement of losses from 3-body recombination into deep dimers 3-atom loss resonances K trimers 1 a

35 Efimov Physics in Cold Atoms effects of deep dimers can be taken into account indirectly by analytically continuing 3-body parameter κ * to a complex value κ * exp(i η * /s0), where s0 = Braaten & Hammer 2003 Efimov spectrum in the unitary limit ( 1 λ 2 0 ) n n κ 2 m ( 1 λ 2 0 ) n n κ 2 m [cos(2η /s 0 )+i sin(2η /s 0 )] imaginary part is half the width of the Efimov trimer

36 Efimov Physics in Cold Atoms universal line shape for 3-body recombination into deep dimers as a function of the scattering length Braaten & Hammer 2003 K (deep) 3 = 4590 sinh(2η ) a 4 sin 2 [s 0 log(a/a )] + sinh 2 η m scales like a 4 coefficient is log-periodic function of a with discrete scaling factor λ resonant enhancement at a * -1.5/κ * and at scattering lengths differing by λ0 n

37 Efimov Physics in Cold Atoms first observation of 3-atom loss resonance from Efimov trimer near 3-atom threshold 133 Cs atoms Kraemer et al (Innsbruck) Nov Recombination length (1000 a 0 ) Scattering length (1000 a 0 ) resonance in 3-body recombination rate near -850 a0 agrees with universal line shape

38 Efimov Physics in Cold Atoms Atoms can be lost from trapping potential by 3-body recombination into the shallow dimer three atoms collide two of them bind to form shallow dimer atom and dimer recoil with substantial kinetic energy

39 Efimov Physics in Cold Atoms 3-body recombination rate into the shallow dimer has total destructive interference at values of a differing by factors of 22.7 Nielsen & Macek 1999, Esry, Greene & Burke 1999, Braaten & Hammer 2000 recombination K zeroes trimers 1 a

40 Efimov Physics in Cold Atoms universal line shape for 3-body recombination into shallow dimer as a function of the scattering length Macek, Ovchinnikov & Gasaneo 2005 Petrov K (shallow) 3 = 128π2 (4π 3 3) sin 2 [s 0 log(a/a 0 )] a 4 sinh 2 (πs 0 ) + cos 2 [s 0 log(a/a 0 )] m scales like a 4 coefficient is log-periodic function of a with discrete scaling factor λ zero at a *0 0.32/κ * and at scattering lengths differing by λ0 n

41 Efimov Physics in Cold Atoms first observation of interference minimum in 3-body recombination 133 Cs atoms Kraemer et al (Innsbruck) Nov Recombination length (1000 a 0 ) Scattering length (1000 a 0 ) minimum of 3-body recombination rate near +200 a0

42 Efimov Physics in Cold Atoms Atoms can be lost from trapping potential by dimer relaxation atom and shallow dimer collide dimer relaxes into deep dimer atom and deep dimer recoil with large kinetic energy

43 Efimov Physics in Cold Atoms Efimov trimers disappear through atom-dimer threshold at values of a that differ by factors of 22.7 give resonant enhancement of losses from dimer relaxation K atom-dimer loss resonances 1 a trimers

44 Efimov Physics in Cold Atoms universal line shape for dimer relaxation as a function of the scattering length Braaten & Hammer 2003 β = 20.3 sinh(2η ) a sin 2 [s 0 log(a/a )] + sinh 2 η m scales like a coefficient is log-periodic function of a with discrete scaling factor λ resonant enhancement at a * 0.08/κ * and at scattering lengths differing by λ0 n

45 Efimov Physics in Cold Atoms first observation of atom-dimer loss resonance from Efimov trimer near atom-dimer threshold 133 Cs atoms+dimers Knoop et al (Innsbruck) July 2008 ( ) ( ) ( ) resonance in dimer relaxation rate near +400 a0

46 Efimov Physics in Cold Atoms First Efimov features in fermionic atoms 6 Li atoms: 3 lowest hyperfine states, 3 scattering lengths 3-atom loss resonance near 130 G (relatively large negative scattering lengths) Ottenstein et al (Heidelberg) June 2008 Huckans et al (Penn State) Oct atom loss resonance near 895 G (very large negative scattering lengths) Williams et al (Penn State) Aug 2009 Jochim et al (Heidelberg) see O Hara Stability of a Fermi gas with 3 spin states see Ottenstein Towards a finite ensemble of ultracold fermions

47 Efimov Physics in Cold Atoms First multiple Efimov features related by discrete scaling 39 K atoms (identical bosons) discrete scaling factor 22.7 two 3-atom recombination minima at large positive a Zaccanti et al (Florence) Oct 2008 recombination K minima trimers 1 a

48 Efimov Physics in Cold Atoms First Efimov features in mixture of atoms mixture of 87 Rb and 41 K atoms Rb-K scattering length controlled by magnetic field discrete scaling factor 131 for Rb-Rb-K 350,000 for Rb-K-K 3-atom loss resonances at large negative scattering lengths from Rb-Rb-K Efimov trimers Rb-K-K Efimov trimers Barontini et al (Florence) Jan 2009 See Helfrich The heteronuclear Efimov effect

49 Efimov Physics in Cold Atoms First Efimov features at positive and negative a in same universal region 7 Li atoms (identical bosons) 3-atom loss resonance at large negative a 3-atom recombination minimum at large positive a Gross et al (Bar-Ilan) June atom loss resonance trimers K recombination minimum 1 a

50 3-Body Parameters Universal results for the zero-range limit depend on two parameters: scattering length a 3-body parameter κ * Effects of deep dimers can be taken into account by analytically continuing κ * to κ * exp(i η * /s0) 3-body parameters: κ * and η * sensitive to details of atomic physics at short distances can be determined by measuring position and width of one Efimov loss feature

51 Range Corrections Universal results correspond to the zero-range limit Calculate corrections to universal results as systematic expansion in powers of range/a 1 st order in range/a: NLO (next-to-leading order) 2 nd order in range/a: NNLO = N 2 LO (next-to-next-to-leading order) Non-integer powers of range/a (anomalous dimensions) enter first at order N 3.1 LO Griesshammer 2005

52 Range Corrections Universal results depend on two parameters: scattering length a 3-body parameter κ * In 2-body sector, what new parameters enter at each order in range/a? NLO: effective range N 2 LO: none N 3 LO: shape parameter In 3-body sector, what new 3-body parameters enter at each order in range/a?

53 Range Corrections In 3-body sector, how many new 3-body parameters enter at each order in range/a? Efimov (1991) NLO: none Hammer & Mehen (2001) NLO: none Bedaque, Griesshammer, Hammer & Rupak (2002) N 2 LO: one new 3-body parameter all assumed fixed scattering length a K 1 a

54 Range Corrections to relate Efimov features, must consider different scattering lengths K 1 a In 3-body sector with variable a, how many new 3-body parameters enter at each order in range/a? NLO: one N 2 LO: two see Phillips Universality and beyond Ji, Phillips & Platter (2009?)

55 4-Body Universality Is there a well-behaved Zero-Range limit for 4 identical bosons? If so, it must respect discrete scaling with discrete scaling factor λ it must be determined by at least two parameters: scattering length a 3-body parameter κ * Is a 4-body parameter required?

56 4-Body Universality April 2004 Meissner, Hammer & Platter no 4-body parameter is required first evidence for two universal tetramers associated with each Efimov trimer identification of 4 He tetramers as universal tetramers September 2004 Meissner, Hammer & Platter identification of alpha particle ( 4 He nucleus) as universal tetramer associated with triton October 2006 Hammer & Platter stronger evidence for two universal tetramers associated with each Efimov trimer calculation of their binding energies near unitary limit E 4 E 3 5 E 4 E

57 4-Body Universality August 2008 von Stecher, D Incao & Greene confirmed existence of two universal tetramers associated with each Efimov trimer mapped out their binding energies for all scattering lengths signature: resonance in 4-body recombination a 4 a a 4 a K 1 a

58 4-Body Universality April 2009 Ferlaino et al (Innsbruck) experimental discovery of universal tetramers resonant enhancement of 4-body loss rate in 133 Cs atoms from universal tetramers near 4-atom threshold 4-atom loss resonances K 1 a

59 4-Body Universality dimer-dimer scattering atom-trimer scattering D Incao, von Stecher & Greene March 2009 D Incao see Ferlaino From Feshbach molecules to universal four-body states: quantum leaps in few-body physics see D Incao Universal properties in the four-body system

60 Outlook Experimental progress in ultracold atoms: dramatic! discoveries of Efimov loss features 3-body 4-body ? 2+? Efimov loss features have been observed in 133 Cs, 6 Li, 39 K, 87 Rb+ 41 K, 7 Li Outlook: many more to come

61 Outlook Theoretical outlook 4-body universality: dramatic progress! D Incao, von Stecher & Greene range corrections: possible breakthrough Ji, Phillips & Platter? goal: given positions of some Efimov features and effective range, predict positions of others 3-body parameters? goal: estimates of κ * and η * for new universal region universal calculations: many more are needed (preferably in form easily used by experiments)