Few-body problems in ultracold alkali-earth atoms and superfluid Boson-Fermion mixture

Transcription

1 Few-body problems in ultracold alkali-earth atoms and superfluid Boson-Fermion mixture Peng Zhang Department of Physics, Renmin University of China 中国人民大学物理系 ( ,INT 15-1,Seattle)

2 Outline Orbital Feshbach Resonance in Alkali-Earth Atoms. Ren Zhang, Yanting Cheng, Hui Zhai and PZ, arxiv: Calibration of Interaction Energy between Bose and Fermi Superfluids Ren Zhang, Wei Zhang, Hui Zhai and PZ, PRA, 90, (2014).

3 Outline Orbital Feshbach Resonance in Alkali-Earth Atoms. Ren Zhang, Yanting Cheng, Hui Zhai and PZ, arxiv: Calibration of Interaction Energy between Bose and Fermi Superfluids Ren Zhang, Wei Zhang, Hui Zhai and PZ, PRA, 90, (2014).

4 alkali-earth (like) atoms: long-lifetime excited state energy levels of Yb atom 1 P 1 5.5ns 3 P 2 3 P 1 3 P 0 ~15s 875ns ~23s long-lifetime excited state optical coupling 1 S 0 ground state Yoshiro Takahashi s ppt M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. Blatt, T. Zanon-Willette, S. M. Foreman, and J. Ye, PRA, 76, , (2007).

5 lkali-earth (like) atoms: SU(N) symmetry of fermions 1 P 1 5.5ns 173 Yb atom: I=5/2 3 P 2 ~15s 3 P 1 875ns 3 P 0 ~23s m I -5/2-3/2-1/2 1/2 3/2 5/2 1 S 0 ground state m I -5/2-3/2-1/2 1/2 3/2 5/2 interactions between atoms in 1S0 or 3P0 states are independent on m I : SU(N=6) symmetry ( 87 Sr : N=10, 171 Yb: N=2) precision of SU(N): 10-9 for atoms in 1S0 and 10-3 for atoms in 3P0 X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S. Safronova, P. Zoller, A. M. Rey, J. Ye., Science, 345, 1467 (2014). (Sr87, with long supplementary material)

6 Quantum degenerate gases of alkali-earth (like) atoms 2003: 174 Yb 2006: 170 Yb, 176 Yb, 173 Yb 2009: 84 Sr 2010: 86 Sr, 88 Sr, 87 Sr, 171 Yb- 173 Yb mixture Yoshiro Takahashi s ppt, S. Stellmer, F. Schreck, T. C. Killian, arxiv: (in Annual Review of Cold Atoms and Molecules (World Scientific Publishing Co.), volume 2, chapter 1 (2014)) S. Sugawa, Y. Takasu, K. Enomoto, and Y. Takahashi, Annual Review of Cold Atoms and Molecules (World Scientific Publishing Co., 2013), chap. Ultracold Ytterbium: Generation, Many-Body Physics, and Molecules.

7 Our motivation 1 P 1 5.5ns 173 Yb atom: I=5/2 3 P 2 ~15s 3 P 1 875ns 3 P 0 ~23s m I -5/2-3/2-1/2 1/2 3/2 5/2 1 S 0 ground state m I -5/2-3/2-1/2 1/2 3/2 5/2 Our problem: how to control interaction between fermonic alkali-earth atoms in 1S0 or 3P0 states?

8 Our result two atoms in different electronic orbital and different nuclear spin states two atoms in different electronic orbital and same nuclear spin states two atoms in same electronic orbital and different nuclear spin states 3 P 0 3 P 0 3 P 0 1 S 0 1 S 0 1 S 0 m I -5/2-3/2 m I -5/2-3/2 m I -5/2-3/2 interaction can be controlled by B-field via orbital Feshbach resonance (~60G for Yb173) so far no good approach Ren Zhang, Yanting Cheng, Hui Zhai and PZ, arxiv:

9 Our result two atoms in different electronic orbital and different nuclear spin states 3 P 0 1 S 0 m I -5/2-3/2 interaction can be controlled by B-field via orbital Feshbach resonance (~60G for Yb173) Ren Zhang, Yanting Cheng, Hui Zhai and PZ, arxiv:

10 Orbital FR spin-exchange interaction Zeeman effect orbital Feshbach resonance

11 Orbital FR spin-exchange interaction Zeeman effect orbital Feshbach resonance

12 Interaction between alkali atoms S: total spin of two electrons e1 e2 V S=1 (R 1 -R 2 ) V S=0 (R 1 -R 2 ) R 1 R 2 R 1 -R 2 Two potential curves: necessary condition for FR T. Köhler, K. Góral and P. S. Julienne, RMP, 78, 1311 (2006).

13 One 1S0 and one 3P0 atom with different nuclear spin 173 Yb atom: I=5/2 3 P 0 (e) 1 S 0 (g) -5/2 ( ) -3/2 ( ) total electronic spin is one: intuitive speaking, there is only one short-range interaction potential curves in fact, NO. There are two interaction potential curves

14 consideration I: exchange symmetry in s-wave scattering two atoms in different electronic orbital and different nuclear spin states initial internal state for s-wave scattering of identical fermionic atoms e g scattering processes with SU(N) symmetry (nuclear spin is not changed) scattering length: a eg (+) scattering length: a eg (-) experiments: a eg (+) a eg (-) 173 Yb: a eg (-) =3300a 0 +i0.78a 0 ;a eg (+) =219.5a 0 87 Sr: a eg (-) =169a 0 ;a eg (+) =68a 0 There are two different potential curves for +> and ->. G. Cappellini, M. Mancini, G. Pagano, P. Lombardi, L. Livi, M. S. de Cumis, P. Cancio, M. Pizzocaro, D. Calonico, F. Levi, C. Sias, J. Catani, M. Inguscio, L. Fallani, Phys. Rev. Lett. 113, (2014) (Yb173). F. Scazza, C. Hofrichter, M. Höfer, P. C. De Groot, I. Bloch and S. Fölling, Nature Physics, 10, 779 (2014). (Yb173) X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S. Safronova, P. Zoller, A. M. Rey, J. Ye., Science, 345, 1467 (2014). (Sr87)

15 consideration II: total parity of out-shell electrons Schroedinger equation of four electrons e1 e3 e2 e4 R 1 R 2 Total parity P of the spatial motion of all four electrons is conserved.

16 consideration II: total parity of out-shell electrons e> g> V P=-1 (R 1 -R 2 ) V P=1 (R 1 -R 2 ) R 1 -R 2 R1-R2 P=1 P=-1 Two different potential curves for

17 Spin-exchange interaction ineraction potential convenient basis e> V-induced coupling e> g> g>

18 Spin-exchange interaction e> e> g> g> G. Cappellini, M. Mancini, G. Pagano, P. Lombardi, L. Livi, M. S. de Cumis, P. Cancio, M. Pizzocaro, D. Calonico, F. Levi, C. Sias, J. Catani, M. Inguscio, L. Fallani, Phys. Rev. Lett. 113, (2014) (Yb173). F. Scazza, C. Hofrichter, M. Höfer, P. C. De Groot, I. Bloch and S. Fölling, Nature Physics, 10, 779 (2014). (Yb173) X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S. Safronova, P. Zoller, A. M. Rey, J. Ye., Science, 345, 1467 (2014). (Sr87)

19 Orbital FR spin-exchange interaction Zeeman effect orbital Feshbach resonance interaction

20 Zeeman effect 173 Yb: M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. Blatt, T. Zanon-Willette, S. M. Foreman, and J. Ye, PRA, 76, , (2007).

21 Free Hamiltonian for two-atom relative motion Free Hamiltonian

22 Orbital FR spin-exchange interaction Zeeman effect orbital Feshbach resonance interaction free Hamitltonian

23 Orbital Feshbach resonance Total two-atom Hamiltonian Incident channel: o> Orbital Feshbach resonance: scattering length of channel o> can be controlled by B

24 Physical realization Yb173 atoms Free Hamiltonian Interaction ( +>, -> basis) Incident channel: o> two-atom loss Scattering length: F. Scazza, C. Hofrichter, M. Höfer, P. C. De Groot, I. Bloch and S. Fölling, Nature Physics, 10, 779 (2014). (Yb173) δ<<e vdw (~10 6 Hz): Huang-Yang potential is applicable.

25 Physical realization Yb173 atoms Free Hamiltonian Interaction ( c>, o> basis) Incident channel: o> Binding energy of bound-state in c>:

26 Physical realization Yb173 atoms Re[a s ] Im[a s ] Incident channel: o> Orbital Feshbach resonance: δ~10 4 Hz<<E vdw 2-body loss: dn/dt=-βn 2 ; β=8πim[a s ]

27 Physical realization Re[a s ] and two-body loss rate β of Yb173 atoms Re[a s ] (a 0 ) Re[a s ] β(cm 3 /s) β B(G) Lifetime~1ms-10ms for density /cm 3

28 Physical realization Re[a s ] and two-body loss rate β of Yb173 atoms β β(cm 3 /s) Re[a s ] (a 0 ) Re[a s ] B(G) Lifetime~1ms-10ms for density /cm 3

29 Zero-range V.S. finite-range Yb173 atoms Re[a s ] Im[a s ] Huang-Yang interaction Re[a s ] Im[a s ] Finite-range calculation

30 Summary Orbital Feshbach resonance can occur between one 1S0 and one 3P0 alkali-earth (like) atoms with different nuclear spin. Two alkali atoms: Two alkali-earth (like) atoms: In orbital FR, the electronic orbital degree of freedom plays the same role as the electronic spin in magnetic FR. May occur for Yb173 when B~60G.

31 Outline Orbital Feshbach Resonance in Alkali-Earth Atoms. Ren Zhang, Yanting Cheng, Hui Zhai and PZ, arxiv: Calibration of Interaction Energy between Bose and Fermi Superfluids Ren Zhang, Wei Zhang, Hui Zhai and PZ, PRA, 90, (2014).

32 Mixture of Bose and Fermi Superfluids Li6 (2-component )+ Li7 (1-component ) atom number Li6: Li7: I. Ferrier-Barbut, M. Delehaye, S. Laurent, A. T. Grier, M. Pierce, B. S. Rem, F. Chevy, C. Salomon, Science, 345, 1035 (2014).

33 Mixture of Bose and Fermi Superfluids Li6 ( 1>) Li7 Li7 Li6 ( 2>) I. Ferrier-Barbut, M. Delehaye, S. Laurent, A. T. Grier, M. Pierce, B. S. Rem, F. Chevy, C. Salomon, Science, 345, 1035 (2014).

34 BEC region of Fermi Superfluid Li6 ( 1>) Li7 (BEC region) Li7 Li6 ( 2>) How large is Boson-Fermion interaction energy?

35 Calculation I: simple mean-field (MF) theory interaction energy field operators of bosonic atoms mean-field approximation field operators of fermonic atoms (with internal state σ>) Interaction energy density given by simple MF

36 Calculation II: weakly-interacting Bose gas interaction energy density given by weakly-interacting Bose gas theory Li6 ( 1>) Li7 Li7 : atom-dimer scattering length Li6 ( 2>) dimer : atom-dimer reduced mass

37 Comparison Interaction energy density given by simple MF interaction energy density given by weakly-interacting Bose gas theory E MF =E IB only when (Li6/Li7 system) atom-dimer scattering length given by mean-field theory

38 a ad and MF Error of simple mean-field approximation is determined by We calculate the exact value of a ad the accurate boson-fermion interaction energy the validity of the simple MF treatment

39 System dimer Fermion (atom 1) Boson (atom 3) Fermion (atom 2) 2-body interaction (Huang-Yang potential) a bf <0 or a bf >>l vdw : Xiaoling Cui, arxiv:

40 STM equation for our system M=m b /m f Λ: momentum cutoff, or the boundary condition for region where all 3 atoms are close. G.V. Skorniakov, K.A. Ter-Martirosian, Sov. Phys. JETP 4, 648 (1957). P. Naidon, and M. Ueda, Comptes Rendus Physique, 12, 13 (2011).

41 a ad for Li6/Li7 Li6( 1>) Li7 Li6( 2>) simple MF: Fix Λ, a bf, change simple MF a ad from STM Equation Error of MF is on the order Error 10% of MF when is about a bf /a s 10% ~1% when a s ~100a bf a s a s decreasing BEC region

42 a ad for Li6/Li7 Li7 Li6( 1>) Li6( 2>) simple MF a ad from STM Equation simple MF: a ad is almost Λ- independent when Error of MF is about 10% BEC region when a s ~100a bf a s decreasing a ad =2.74a bf when a s = ; (2.74a bf -a ad )/a ad ~10% when a bf /a s ~1%

43 Influence of mass ratio Fermion ( 1) 30% (a ad0 -a ad )/a ad 0 (relative error of simple MF) Boson 20% heavy boson Mass ratio Fermion (2) M=m boson / m fermion 10% light boson Error of MF is large when Boson is heavy.

44 Frequency of effective potential for Bosons without fermions with fermions

45 Experimental observation Li6( 1>) Li7 Li6( 2>) I. Ferrier-Barbut, M. Delehaye, S. Laurent, A. T. Grier, M. Pierce, B. S. Rem, F. Chevy, C. Salomon, arxiv:

46 Frequency shift: BEC region Li6( 1>) 15 simple MF Li7 10 BEC region Li6( 2>) 5 STM calculation 0 1

47 Van der Waals physics separable potential with van der Waals physics V=ξ χ><χ simple MF separable potential with van der Waals physics zero-range potential can give good results (relative error~2%) Huang-Yang potential P. Naidon, S. Endo, and M. Ueda, Phys. Rev. Lett. 112, (2014).

48 Summary: Error of simple MF approximation rapidly increases with a bf /a s. Effects beyond simple MF can be experimentally observed. Boson Fermion ( 1) Fermion (2)

49

50 Formal Expansion with a bf Boson 3 Fermion 1 scattering length Fermion 2 scattering state Formal expansion with a bf G3: Green s function for free boson and interacting fermions

51 Born approximation a ad from Born approximation is nothing but the one from simple MF Beyond simple MF: high-order processes should be included

52 Physical realization Yb173 atoms Huang- Yang potential Finiterange calculation Orbital Feshbach resonance: δ~10 4 Hz<<E vdw