Reference for most of this talk:

Transcription

1 Cold fermions

2 Reference for most of this talk: W. Ketterle and M. W. Zwierlein: Making, probing and understanding ultracold Fermi gases. in Ultracold Fermi Gases, Proceedings of the International School of Physics "Enrico Fermi", Course CLXIV, Varenna, June 2006, edited by M. Inguscio, W. Ketterle, and C. Salomon (IOS Press, Amsterdam) 2008, pp ; e-print, arxiv: ; Rivista del Nuovo Cimento 31, (2008).

3 Lithium Sodium

4 At absolute zero temperature Bosons Particles with an even number of protons, neutrons and electrons Bose-Einstein condensation atoms as waves superfluidity Fermions Particles with an odd number of protons, neutrons and electrons Fermi sea: Atoms are not coherent No superfluidity

5 Pairs of fermions Particles with an even number of protons, neutrons and electrons Two kinds of fermions Fermi sea: Atoms are not coherent No superfluidity

6 At absolute zero temperature Pairs of fermions Particles with an even number of protons, neutrons and electrons Bose-Einstein condensation atoms as waves superfluidity Two kinds of fermions Particles with an odd number of protons, neutrons and electrons Fermi sea: Atoms are not coherent No superfluidity

7 Weak attractive interactions Cooper pairs larger than interatomic distance momentum correlations BCS superfluidity Two kinds of fermions Particles with an odd number of protons, neutrons and electrons Fermi sea: Atoms are not coherent No superfluidity

8 E Free atoms Molecule Magnetic field Feshbach resonance

9 Disclaimer: Drawing is schematic and does not distinguish nuclear and electron spin. E Free atoms Molecule Magnetic field Feshbach resonance

10 Two atoms. E Free atoms Molecule Magnetic field Feshbach resonance

11 form a stable molecule E Free atoms Molecule Magnetic field Feshbach resonance

12 Atoms attract each other E Free atoms Molecule Magnetic field Feshbach resonance

13 Atoms repel each other Atoms attract each other E Free atoms Molecule Magnetic field Feshbach resonance

14 Atoms repel each other Atoms attract each other Force between atoms Scattering length Magnetic field Feshbach resonance

15 Observation of a Feshbach resonance S. Inouye, M.R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn, WK, Nature 392 (1998).

16 Energy Atoms Molecules Magnetic field Atoms form stable molecules Atoms repel each other a>0 BEC of Molecules: Condensation of tightly bound fermion pairs Molecules are unstable Atoms attract each other a<0 BCS-limit: Condensation of long-range Cooper pairs

17 Atom pairs Electron pairs Bose Einstein condensate of molecules BCS Superconductor

18 BEC BCS supe

19 Magnetic field BEC BCS supe

20 BEC Crossover superfluid BCS supe

21 How do atoms pair?

22 Two-body bound states in 1D, 2D, and 3D 1D, 2D: bound state for arbitrarily small attractive well 3D: Well depth has be larger than threshold

23 Connection to the density of states In momentum space

24 Connection to the density of states In momentum space Short range potential: V(q)=V 0 for q<1/r Integrate over q, divide by common factor

25 Bound state for arbitrarily small V 0 only if integral diverges for E 0

26 Bound state for arbitrarily small V 0 only if integral diverges for E 0 In 2D (constant density of states): logarithmic divergence

27 The Cooper problem:

28 Two fermions with weak interactions on top of a filled Fermi sea Total momentum zero Total momentum non-zero 2q

29 Pauli blocking Compare with previous result for single particle bound state

30 Pauli blocking After replacing the bare interaction V 0 by the scattering length a

31 Cooper Pairing Consider two particles,, on top of a filled, inert Fermi sea Total momentum zero Total momentum non-zero Reduced density of states Much smaller binding energy The important pairs are those with zero momentum

32 BCS Wavefunction How can we find a state in which all fermions are paired in a self-consistent way? John Bardeen Leon N. Cooper John R. Schrieffer

33 BCS Wavefunction Many-body wavefunction for a condensate of Fermion Pairs: Spatial pair wavefunction Spin wavefunction Second quantization: Fourier transform: Pair wavefunction: Operators: Pair creation operator: Many-body wavefunction: a fermion pair condensate

34 is not a Bose condensate Commutation relations for pair creation/annihilation operators Occupation of momentum k pairs do not obey Bose commutation relations, unless BEC limit of tightly bound molecules

35 BCS Wavefunction Introduce coherent state / switch to grand-canonical description: and commute because Normalization: BCS wavefunction: with and

36 BCS Wavefunction and are identical!

37 Many-Body Hamiltonian Second quantized Hamiltonian for interacting fermions: Contact interaction: Fourier transform via BCS Approximation: Only include scattering between zero-momentum pairs Solve via 1) Variational Ansatz, 2) via Bogoliubov transformation

38 Variational Ansatz: Insert BCS wavefunction into many-body Hamiltonian. Minimize Free Energy: Result: E k Gap equation:

39 Solution via Bogoliubov Transform BCS Hamiltonian is quartic: Introduce pairing field (mean field or decoupling approximation): small fluctuations (assumption) Neglect products (correlations) of those small fluctuations Define This plays the role of the condensate wavefunction

40 Solution via Bogoliubov Transform Rewrite Hamiltonian, drop terms quadratic in C s: Hamiltonian is now bilinear Solve via Bogoliubov transformation to quasiparticle operators: With the choice we get and Ground state energy Non-interacting gas of fermionic quasi-particles

41 Critical temperature Can be derived from Bogoliubov Hamiltonian with fluctuations T C /T F Conventional SC: Superfluid 3 He: 10-3 High-T C SC: 10-2 High-T C Superfluid: 0.15 BEC-limit: BCS-limit:

42 Experimental realization of the BEC-BCS Crossover

43 Preparation of an interacting Fermi system in Lithium-6 Electronic spin: S = ½, Nuclear Spin: I = 1 (2I+1)(2S+1) = 6 hyperfine states Optical 1064 nm ν axial = Hz ν radial = Hz E trap = µk

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46 BEC of Fermion Pairs (Molecules) T > T C T < T C T 6T C These days: Up to 10 million condensed molecules Boulder Nov 03 Innsbruck Nov 03, Jan 04 MIT Nov 03 Paris March 04 Rice, Duke M.W. Zwierlein, C. A. Stan, C. H. Schunck, S.M.F. Raupach, S. Gupta, Z. Hadzibabic, W. Ketterle, Phys. Rev. Lett. 91, (2003)

47 JILA, Nature 426, 537 (2003). MIT, PRL 91, (2003) Innsbruck, PRL 92, (2004). ENS, PRL 93, (2004).

48 Observation of Pair Condensates BEC-Side Resonance BCS-Side (above dissociation limit for molecules) Thermal + condensed pairs First observation: C.A. Regal et al., Phys. Rev. Lett. 92, (2004) M.W. Zwierlein, C.A. Stan, C.H. Schunck, S.M.F. Raupach, A.J. Kerman, W. Ketterle, Phys. Rev. Lett. 92, (2004).

49 How can we show that these gases are superfluid?

50 Vortex lattices in the BEC-BCS crossover Establishes superfluidity and phase coherence in gases of fermionic atom pairs B M.W. Zwierlein, J.R. Abo-Shaeer, A. Schirotzek, C.H. Schunck, W. Ketterle, Nature 435, (2005)

51 Cold atomic gases provide the building blocks of quantum simulators Quantum engineering of interesting Hamiltonians Ultracold Bose gases: superfluidity (like 4 He) Ultracold Fermi gases (with strong interactions near the unitartiy limit): pairing and superfluidity (BCS, like superconductors) Optical lattices: crystalline materials Soon: magnetism in strongly correlated systems