Course in 3 lectures on Superfluidity in Ultracold Fermi Gases FF..MM.. MMaarrcchheettttii Physics by the Lake, Ambleside, 11, 12, 13 September 2007 DDuurriinngg lleeccttuurree IIII.. …… 1) Tuneable interaction strength in two-component Fermi mixtures 2) BEC-BCS crossover BEC BCS III. Polarised Fermi gases BBaallaanncceedd ppooppuullaattiioonnss BEC BCS PPooppuullaattiioonn iimmbbaallaannccee BEC BCS Control over the atom number in different spin species Can superfluidity persist in presence of a population imbalance? TThhee mmaannyy--bbooddyy vveerrssiioonn ooff GGuunnnnaarr ddiilleemmmmaa WWhhyy iinntteerreessttiinngg?? Magnetised superconductors (Zeeman) Quantum Chromodynamics (and neutron stars) Electron-hole bilayers conduction c band e- + h valence v band position BBEECC--BBCCSS ccrroossssoovveerr iinn eelleeccttrroonn--hhoollee ssyysstteemmss!! c e- h+ v + e- h+ h BEC ‘BCS’ e- BEC of excitons! exciton insulator [L.V. Keldysh & Yu V. Kopaev, Sov. Phys. Solid State 6, 2219 (1965)] TThhee mmooddeell ffoorr ppoollaarriisseedd FFeerrmmii ggaasseess Allow for different densities of atoms in the two spin states Averaged chemical potential & ‘Zeeman’ term (total density & population imbalance [or ‘magnetisation’]) MMeeaann--fifieelldd && ggrroouunndd ssttaattee No change in the symmetry of the Hamiltonian! …i.e., same ground state but different way to fill the states up! RReemmeemmbbeerr ssppeeccttrruumm ooff eexxcciittaattiioonnss wwhheenn hh==00 MMeeaann--fifieelldd && ggrroouunndd ssttaattee E.g., on the BCS side… …one start depleting paired states when We can expect the BCS superfluid phase to be very fragile! On the BEC side requires much higher h (much more robust superfluid phase to imbalance!) WWhhyy iiss hh ccaalllleedd ZZeeeemmaann tteerrmm?? Mean-field Hamiltonian AAnnaallooggyy wwiitthh mmaaggnneettiisseedd ssuuppeerrccoonndduuccttoorrss A population imbalance like a Zeeman term in a superconductor Neglect the orbital effect? B B SC SC N TT==00 mmaaggnneettiisseedd ssuuppeerrccoonndduuccttoorrss ((BBCCSS ssiiddee)) Gap equation (problem 4) Two branches: BCS h<Δ Sarma h>Δ Which solution shall we choose when h/ΔBCS>0.5? 11st oorrddeerr pphhaassee ttrraannssiittiioonn Free energy Sarma solution ZZeerroo tteemmppeerraattuurree pphhaassee ddiiaaggrraamm [G. Sarma, J. Phys. Chem. Solids 24 1029 (1963)] 1st SF N (BCS) FFiinniittee tteemmppeerraattuurree pphhaassee ddiiaaggrraamm [G. Sarma, J. Phys. Chem. Solids 24 1029 (1963)] 2nd N tricritical point SF 1st WWhhaatt tthhiiss iimmpplliieess ffoorr tthhee ppoollaarriisseedd FFeerrmmii ggaass?? 11st oorrddeerr pphhaassee ttrraannssiittiioonn N SF (oil&water) phase separation in density space! PPhhaassee ddiiaaggrraamm iinn ddeennssiittyy ssppaaccee 2nd N 2nd N tricritical tricritical SF point SF point 1st PS 1st population imbalance AAnnaallooggyy wwiitthh 3HHee--4HHee mmiixxttuurreess But 3He-4He is a Bose-Fermi mixture! … and the polarised Fermi gas is a Bose-Fermi mixture on the BEC side of the resonance Expect the same structure on the BEC side? PPhhaassee ddiiaaggrraamm aaccrroossss tthhee rreessoonnaannccee [M. Parish, F.M. Marchetti et al., Nature Physics 3, 124 (2007)] TTrraappppeedd FFeerrmmii GGaasseess LDA decreases V(r) r PPhhaassee DDiiaaggrraamm ffoorr TTrraappppeedd GGaasseess N SF N 2nd T1/h1 r r T /h <T /h 1st 2 2 1 1 [M. Parish, F.M. Marchetti et al., Nature Physics 3, 124 (2007)] EExxppeerriimmeennttss oonn IImmbbaallaanncceedd FFeerrmmii CClloouuddss In-situ imaging of phase separation (3D density distribution ) N SF N r [Y. Shin et al., PRL 97, 030401 (2006)] EExxppeerriimmeennttss oonn IImmbbaallaanncceedd FFeerrmmii CClloouuddss Sharp phase boundary at low temperatures (1st order transition) T < 0.05 TF m/n = 0.35 [G. B. Partridge et al., PRL 97, 190407 (2006)] TTeemmppeerraattuurree DDeeppeennddeennccee ooff PPhhaassee SSeeppaarraattiioonn [M. Parish, F.M. Marchetti et al., Nature Physics 3, 124 (2007)] T/T <0.05 2nd F T1/h1 r T /h <T /h 1st 2 2 1 1 1st T/TF=0.2 [G. B. Partridge et al., PRL 97, 190407 (2006)] 2nd PPhhaassee sseeppaarraattiioonn:: ssppiinnooddaall ddeeccoommppoossiittiioonn?? … from Mike Evans lectures! Temperature quenches in, e.g., polymers in solutions, homogenised colloids, … [Courtesy of Nigel Clarke, Polymer IRC] …and also many examples in solid state SSppiinnooddaall RReeggiioonn PS PS metastable unstable metastable DDyynnaammiiccss ooff PPhhaassee SSeeppaarraattiioonn Spinodal: phase separation starts via a linear instability [A. Lamacraft & F.M. Marchetti, cond-mat/0701692].