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

[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

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]