Bogoliubov-de Gennes Study of Trapped Fermi Gases Han Pu Rice University (INT, Seattle, 4/14/2011) Leslie Baksmaty Hong Lu Lei Jiang Randy Hulet Carlos Bolech

Imbalanced Fermi mixtures

BCS Cooper pairs have zero momentum Population imbalance leads to finite-momentum pairs FFLO instability results in textured states Fulde-Ferrel-Larkin-Ovchinnikov instability

Rice (Hulet Group) –Science 311, 503 (2006) –PRL 97, (2006) –Nuclear Phys. A 790, 88c (2007) –J. Low. Temp. Phys. 148, 323 (2007) –Nature 467, 567 (2010) MIT (Ketterle Group) –Science 311, 492 (2006) –Nature 442, 54 (2006) –PRL 97, (2006) –Science 316, 867 (2007) –Nature 451, 689 (2008) ENS (Salomon Group) –PRL 103, (2009) Experiments on spin-imbalanced Fermi gas

Observation: Phase separation Superfluid core with polarized halo MW. Zwierlein, A. Schirotzek, C.H. Schunck, and W, Ketterle: Science 311, (2006)

High T Low T Hulet Ketterle n↑n↑ n↓n↓ n ↑ - n ↓ MIT/Paris data are consistent with Local Density Approximation (LDA) Rice data (low T) strongly violates LDA. Experimental results Salomon

Surface Tension Phase Coexistence -> Surface Tension 1 mm 60  m Aspect Ratio of Cloud: 50:1 Aspect Ratio of Superfluid: 5:1 Data: Hulet Surface tension causes density distortion Effects of surface tension more important in smaller sample.

De Silva, Mueller, PRL 97, (2006) Data points from Rice experiment. P=0.14 P=0.53 P=0.72 LDA LDA + surface tension Breakdown of LDA

Optimal value that fits data: However, from microscopic theoretical calculation: PRA 79, (2009) Surface tension

Choose T and Solving BdG equations

AR=1AR=5AR=50 Density along z-axis Density along r-axis Gap along z-axis Effect of trap anisotropy: N=200, P=0.4

Density along z-axis Density along r-axis Gap along z-axis P=0.2P=0.4P=0.7 Quasi-1D system: N=200, AR=50

Gap along z-axis Gap along r-axis LDA BdG n↑n↑ n↓n↓ n ↑ - n ↓ N~200,000 BdG vs. LDA: N=200, AR=50, P=0.6

BdG equation is very nonlinear, it may support many stationary states. Complicated energy landscape For large N, starting from different initial configurations, the BdG solver may converge to different final states. Going to higher N

SF LO NN 3 classes of states

SFLO NN Increasing energy Density profiles (N=50,000)

n↑n↑ n↓n↓ Upclose on the LO state

Pei, Dukelsky and Nazarewicz, PRA 82, (2010) Bulgac and Forbes, PRL 101, (2008) Robustness of the density oscillation

homogeneoustrapped Orso, PRL (2007); Hu et al., PRL (2007) FFLO in 1D

Liao et al., Nature 467, 567 (2010) Experiment in 1D (Hulet group)

3D t 1D … … t X 3D 1D Dimensional crossover: 3D – 1D

Model for single impurity in Fermi superfluidity H 0 is BCS mean field Hamiltonian

BdG and T matrix methods T-matrix gives exact solutions for localized contact impurity without trap. Contact potential: T matrix only depends on energy. BdG method gives numerical results for single impurity in harmonic trap. BdG solves self-consistently a set of coupled equations

Localized non-magnetic impurity in 1D trap Bound state occurs when T -1 (w)=0 BdG results T matrix results with impurity without impurity

Localized magnetic impurity Bound state energy inside the gap BdG results T matrix results This bound state is below the bottom of quasiparticle band.

Density and gap profiles for localized magnetic impurity What if we increase impurity width and strength ? Spin up Spin down

Spin up Spin down Magnetic impurity induced FFLO state Impurity: Gaussian potential Impurity strength

Magnetic impurity induced FFLO state (3D)

Two component Fermi gas offers very rich physics. Effects of trapping confinement. Flexibility of atomic system provides opportunities of studying exotic pairing mechanisms. Conclusion

“Concomitant modulated superfluidity in polarized Fermi gases”, Phys. Rev. A (2011) “Single impurity in ultracold Fermi superfluids”, arXiv: “Bogoliuvob-de Gennes study of trapped spin-imbalanced unitary Fermi gases”, arXiv: References