Fermions in cold atom gases
Magnetic Feshbach resonance & scattering length M. Greiner et al. arXiv:cond-mat/0502539
Fabian Heidrich-Meisner Institut fur ¨ Theoretische Physik C, RWTH Aachen
RWTH Aachen – Jan 17 & Jan 18 (2008)
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Fermions in cold atom gases 1. General remarks (a) Quantum degeneracy, hyperfine states 2. The BEC-BCS crossover (a) (b) (c) (d)
Feshbach resonance: molecular bound state BEC condensate of composite bosons BCS theory Models for the BEC-BCS crossover
3. Pairing in spin-imbalanced Fermi gases (a) Fulde-Ferrel-Larkin-Ovchinikov states Giorgini, Pitaevskii, Stringari, Theory of ultracold atomic Fermi gases, to appear in RMP, arxiv:0706.3360
RWTH Aachen – Jan 17 &Jan 18 (2008)
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Quantum degenerate Fermi gases
Kinetic energy vs temperature DeMarco, Jin, Science 285, 1703 (1999); O’Hara et al., Science 298, 2179 (2002)
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Feshbach resonance: Open and closed channel
Energy
ε
Closed Channel
Open Channel
Internuclear Separation
Plot taken from: Holland et al PRL 86, 1915 (2001)
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Molecules in the repulsive regime of a Fermi gas
a [nm]
200 100
2
04 -100 -200 0
3
0.5
1
1
1.5
Magnetic Field [kG]
2
: two (!) resonances (at 834 G and 543 G)
Li atoms: Cubizolles et al. PRL 91, 240401 (2003)
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Molecules in the repulsive regime of a Fermi gas
Potassium atoms: Regal et al Nature 424, 47(2003)
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Spin changing radio-frequency transitions
Plots taken from Greiner et al. PRL 92, 140404 (2004)
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BEC-BCS crossover
Greiner et al. arXiv:cond-mat/0502539
RWTH Aachen – Jan 17 &Jan 18 (2008)
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Molecule Number Per Unit Length [ 1/mm ]
Observation of BEC of molecules formed of fermions
0.9 mW
2.4 mW
2.7 mW
3.3 mW
4.5 mW 6
1x10
8.9 mW
0 0.0
0.5
1.0
Position [ mm ]
Greiner, Regal, Jin Nature 426, 537 (2003), Zwierlein et al. PRL 92, 250401 (2003) Jochim et al Science 302, (2003)
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Evidence for Cooper pairs in the BCS regime
Regal et al. PRL 92, 040403 (2004)
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Condensation of Fermionic atom pairs near the resonance & in the strongly interacting regime 204
B (gauss)
N0 / N
0.15 0.10
202 200 198 -9 -6 -3 0 3 16 18 t (ms)
0.05 0 -0.5
0
0.5
1.0
DB (gauss) Regal et al. PRL 92, 040403 (2004)
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Condensation of Fermionic atom pairs near the resonance & in the strongly interacting regime
Zwierlein et al. PRL 92, 120403 (2004)
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Collapse of quantum degenerate Fermi gases
Front: fermions (potassium); back: bosons (rubidium) Mudugno et al Science 297, 2240 (2002)
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BEC-BCS crossover
Greiner et al. arXiv:cond-mat/0502539
RWTH Aachen – Jan 17 &Jan 18 (2008)
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Phase-diagram of the BEC-BCS crossover
Plot taken from: Ketterle, Zwierlein, arxiv.org/0801.2500
RWTH Aachen – Jan 17 &Jan 18 (2008)
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Summary: Experiments 1. Realization of Quantum Degeneracy in Fermi gases 2. Interactions tuned by magnetic Feshbach resonances: Molecular bound state on the repulsive side (a > 0, BEC regime) 3. BEC-BCS crossover: from tightly bound molecules to pairs formed in momentum state; condensation of composite bosons changes/ influence of quantum statistics 4. Condensates in the BEC limit observed 5. Condensation in the crossover regime observed nature of condensed objects under debate 6. Theory: model with fermion-boson conversion 7. Collapse of Fermi gases, Fermion-boson mixtures
RWTH Aachen – Jan 17 &Jan 18 (2008)
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Theoretical model for the crossover need to model conversion between molecules (bk) and atoms (ak)
H
= 2ν
X
b†kbk
+
X
k
k a†k↑
ak↑ +
† ak↓ak↓
k
a†k1↑ a†k2↓ak3↓ak4↑
X
+ Ubg
k1 ...k3
+g
X
b†q a q +k↑ a q −k↓ + bq a†q −k↓a†q +k↑
k,q
2
2
2
2
Particle number N conserved: N =
X k
a†k↑
ak↑ +
† ak↓ak↓
+2
X
b†kbk
k
Two-channel model (or Bose-Fermi model): for narrow Fashbach resonances Timmermans et al. Phys. Lett. A 285, 228 (2001); Holland et al. Phys. Rev. Lett. 87, 120406 (2001) Compare: Sheehy & Radzihovsky Ann. of Physics 332, 1790 (2007)
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BCS: Gap and chemical potential
Plot taken from Ketterle, Zwierlein, arxiv.org/0801.2500
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BCS: Spatial extenion of Cooper pairs
Plot taken from Ketterle, Zwierlein, arxiv.org/0801.2500
RWTH Aachen – Jan 17 &Jan 18 (2008)
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MDF: One-channel model vs QMC
Red: BCS; solid lines: QMC
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Plot taken from Giorgini et al, arxiv.org/0706.3360
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Condensate fraction: One-channel model vs QMC
Green: BCS; Symbols: QMC Plot taken from Giorgini et al, arxiv.org/0706.3360
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Transition temperature
From BCS theory; Plot taken from: Giorgini et al, arxiv.0706.3360
RWTH Aachen – Jan 17 &Jan 18 (2008)
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Theoretical model for the crossover need to model conversion between molecules (bk) and atoms (ak)
H
= 2ν
X
b†kbk
+
X
k
k a†k↑
ak↑ +
† ak↓ak↓
k
a†k1↑ a†k2↓ak3↓ak4↑
X
+ Ubg
k1 ...k3
+g
X
b†q a q +k↑ a q −k↓ + bq a†q −k↓a†q +k↑
k,q
2
2
2
2
Particle number N conserved: N =
X k
a†k↑
ak↑ +
† ak↓ak↓
+2
X
b†kbk
k
Two-channel model (or Bose-Fermi model): for narrow Fashbach resonances Timmermans et al. Phys. Lett. A 285, 228 (2001); Holland et al. Phys. Rev. Lett. 87, 120406 (2001) Compare: Sheehy & Radzihovsky Ann. of Physics 332, 1790 (2007)
RWTH Aachen – Jan 17 &Jan 18 (2008)
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Chemical potential from the two-channel model 1
0.9
µ/E
F
0.8
0.7
0.6
0.5 0
0.1
0.2
0.3
T/TF
0.4
0.5
0.6
0.7
Holland et al. PRL 87, 120406 (2001)
RWTH Aachen – Jan 17 &Jan 18 (2008)
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Two-channel model: Molecular field and gap 0.025
∆ / EF
0.6 0.4 0.2
0.015
0 0
m
2|φ |2 / N
0.02
0.2
0.01
0.4 T/TF
0.6
0.005
0 0
0.1
0.2
0.3
T/TF
0.4
0.5
0.6
0.7
Holland et al. PRL 87, 120406 (2001)
RWTH Aachen – Jan 17 &Jan 18 (2008)
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