Ultracold Fermi gases in quasi low dimensions

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1 Conference on Research Frontiers in Ultra-Cold Atoms 4-8 May 2009 Ultracold Fermi gases in quasi low dimensions ZHANG Wei Renmin University of China Physics Department 59, Zhong Guan Cun Da Jie Hai Dian Beijing PEOPLE'S REPUBLIC OF CHINA

2 Ultracold Fermi Gases in Quasi Low Dimensions Wei Zhang Physics Department, Renmin University of China, Beijing ICTP, Trieste, Italy, May 2009 Luming Duan (Umich) Guin-Dar Lin Jason Kestner Congjun Wu (UCSD) Qijin Chen (UChicago) Hui Hu (RUC) Yingmei Liu (NIST) Xuong Chen (PKU) Support: NSF, DARPA, RUC, NSF-China 1

3 Outline Introduction Ultracold Fermi gases in quasi low dimensions Feshbach resonance BCS-BEC crossover in 3D Fermions in quasi-low dimensions: two-body problem Binding energy DOF in strongly confined transverse directions Fermions in quasi-low dimensions: many-body problem Effective low D Hamiltonian and dressed molecules BCS-BEC crossover in Q2D: an example The significance of dressed molecules Summary 2

4 Fermi systems in low dimensions Semiconductors QHE Science 306, 666 (2004) HTC Nat. Mater. 6, 183 (2007) Nature. 447, 565 (2007) 3

5 Fermi gases in optical lattice V V k x y w OL 0 cos( ) exp[ 2( ) / ] + = 4

Quasi-low Dimensional Fermi system m V V V r 2 with 2 2 2 2 OL ( ) Quasi 2D Conditions: Trapping potential

6 Quasi-low Dimensional Fermi system m V V V r 2 with OL ( ) Quasi 2D Conditions: Trapping potential EF, k B T a /( m ) 3D interaction e R e is interaction potential range (~nm) a ~um R Q1D geometry: 2D lattice 5

7 Feshbach Resonance 40 K Regal, et. al., PRL,

8 Two-body Problem Two-channel Model H3D d x Vext d 2Vext b, 2m x 4m g d x H.C. U d b x 3 3 b K:202G Li:834G Binding Energy: Wide FR a a s a bg / 1 m W B B 0 Universality 7

9 BCS-BEC Crossover in 3D a s 0 BEC limitweakly int. Bose gas a s 0 BCS limitweakly int. Fermi gas 8

10 BCS-BEC Crossover in 3D k, /2 b 2 H a a b b k k k q q q q 3/2 3 b q b + g / L a a b H.C. U / L a a a a kq, k, kq, k, k', k' q, kq, kk, ', q E F E b /2 T=0 9

11 Outline Introduction Fermi systems in low dimensions Feshbach resonance BCS-BEC crossover in 3D Fermions in quasi-low dimensions: two-body problem Binding energy Excited states in strongly confined directions Fermions in quasi-low dimensions: many-body problem Effective Hamiltonian and dressed molecules BCS-BEC crossover in Q2D: an example The significance of dressed molecules Summary 10

12 Two-body Problem in Quasi-2D Two-channel Model H3D d x Vext d 2Vext b, 2m x 4m g d x H.C. U d b x 3 3 b Strong confinement along -direction Homogeneous in x-y plane x y 0 11

13 First Trial: Push to 2D Excited states along direction are completely froen. fermions 1/4 2 m m exp 2 molecules: 1/4 2 2m m exp H2D d d, 2m 2 4m r r b 1/4 1/2 2 m 2 r H.C. b r m gb d U d

14 Results v.s 3D Loss of Universality? Excitations along direction are froen 13

15 More General Description Expanded by Harmonic Oscillators fermions: molecules a ( ) mm m b ( ) mm m H a a b b 00 2D mk mk mk 0 b 0 m, k, g b L m 1/4 mn,, k 1/2 Ub m 2 L mn,, k, m', n', k ' a a b mn m, k, n, k, 00 H.C. a a a a mn m' n' m, k, n, k, n', k', m', k', 14

16 Two-body Problem Revisited Binding Energy Universality Excited states are important 15

17 Population of Molecules and Excitations 16

18 Excitation fraction cannot be suppressed 17

19 Two-body problem in Q1D Kestner et. al., PRA, 74,

20 Discussion BCS side E, a b BEC side E, b Excitation is important a 19

21 No matter how strong the confinement is, excited states can still be populated to a siable value. Quasi low D will never be low D? Are these excited states important in a many-body problem?? If yes, how to deal with them? 20

22 BCS-BEC Crossover of a Q2D system: a 2D model w/o excited states Assume ground state along -direction-> 2D model 1/2 Ueff m 2D 0k 0k 0k 2 0, kq, 0, k, 0, k', 0, k' q, k, 2 L 2 k, k', q H a a a a a a MF Approach: homogeneous case, T=0 E / 0.01 F 21

23 Inhomogeneous case Harmonic trap LDA 1 V() r m r r V () r m r Total Particle No. N Thomas-Fermi Radius TF TF 2 ( ) 0 R nr ( ) 0 n r rdr 22

24 A constant TF Radius? BCS limitweakly Int. Fermi Gas R TF RBCS 0 BEC limit: Weakly Int. Bose Gas RTF 0 23

25 The Failure of the pure 2D model 1 Fa ( s, a) Equation of State nr () () r 2 a an Fa ( s, a) Chemical Pot. of Eff. Bosons binding energy Constant interaction between effective bosons. The failure of the pure 2D model: cannot predict a weakly interacting Bose gas in the BEC limit. 24

26 The excited states seem to be important for many-body problems? How to incorporate them? 25

27 H2D a a b 2 b mq mk mk mk mq b mq m, k, 2 m, q 2 2 1/4 gb m + mn a a b H.C. m,, n,, l kq k q L mnl,,, kq, U b 2 L m 1/2 mnm' n' a a a a m, kq, n, k, n', k', m', k' q, mn,, k, m', n', k', q Hard to solve, even in MF level A detour is needed 26

28 The detour turns out to be an effective Hamiltonian Can mimic the original Hamiltonian, at least around the regime of interest. Easy to solve. 27

29 The Effective Hamiltonian Feshbach Mol. Excited Fermion Fermions in ground state dressed molecule: d q H a a d 2 d eff q 2D 0, k, k 0, k, q b q k, 2 q 2 2 b L kq a a d V L 0, kq, 0, k, b 2 kk ' q q H.C. a a a a 0, kq, 0, k, 0, k', 0, k' q, Kestner et. al., PRA, 76, WZ et. al., PRA, 77,

30 Matching parameters How to deal with excited states? Excited states is populated when pair sie is small Short-range physics is dominated by two-body process due to diluteness condition Ground state Excited state Matching cond. Original H Effective H Bg. Scattering Binding E. Ub E b Vb E b Population excitation+ Feshbach mol. dressed mol. 29

31 Validity In the T-matrix representation, the matching conditions actually match two quantities: The position of singular points of T(x) The first derivative of 1/T(x) around singular points Thus, the effective H is approximately identical to the original H around the binding energy, with error to the order of E /2 E Eb /2 V O Since b F the validity of this effective H is guaranteed by the quasi-low-d or the diluteness conditions 2 2 EF; na 1 30

32 BCS-BEC Crossover in Q2D revisited Homogeneous case E / 0.01 F WZ et. al., PRA, 77,

33 TF radius In a harmonic trap WZ et. al., PRA, 77,

34 BKT transition MF+phase fluctuation rt, 0 exp i rt, S S S eff 0 fluc 1 Sfluc d d ij K 2 / kt B s r BKT transition temperature TBKT 2 R s E 2 2 E k k k s nmol 1 tanh k sech 2 2 4m L k Ek 2kBT 4mkBTL k 2kBT 33

35 BKT transition WZ et. al., PRA, 78,

36 Summary In quasi low D conditions, no matter how strong the transverse confinement is, fermions can still be populated to excited states, and these populations do matter. These degrees of freedom can be incorporated by introducing an effective Hamiltonian. The excited states are described by dressed molecules. The effective H takes the form of a 2-channel model, with parameters given by matching 2-body physics. As an example, this model has been used to analye BCS-BEC crossover in Q2D, and predicted the trend of TF radius as expected. 35

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Untitled-3 SEC.. Separable Equations In each of problems 1 through 8 solve the given differential equation : ü 1. y ' x y x y, y 0 fl y - x 0 fl y - x 0 fl y - x3 3 c, y 0 ü. y ' x ^ y 1 + x 3 x y 1 + x 3, y 0 fl