Ingrid BausmerthIngrid BausmerthAlessio RecatiAlessio Recati

Sandro StringariSandro Stringari

OutlineOutline

Introduction and Motivation:Feshbach Resonances

Normal State of a unitary Fermi gas with equal masses:

Normal State with unequal masses

T=0: μ-h phase diagram of the system :What happens for unequal masses?

Trapped System: Local Density Approximation (LDA)

How does the trapped configuration depend on the mass ratio and trapping parameters?

BEC of molecules: strong coupling, kFas<<1,interaction is repulsive

condensation oftightly bound fermions,

size of molecules much smaller than average

distance between pairs: BEC gas of

molecules

aass ±∞±∞

BCS-limit: weak coupling, kF|as|<<1,interaction is attractive,

condensation of long-range Cooper Pairs in momentum

space,

negative values of as, size of pairs is larger

than interparticle distance

BECBEC

a>0a>0

BCSBCS

a<0a<0

T=0 and 3D Fermions: BCS-BEC Crossover

system is strongly correlated, but its properties do not depend on value of scattering length as (independent even of sign of as) everything is expressed in terms of kF

normal to superfluid transition: n↓/n↑ = xc = 0.44

Normal State of a Fermi gas at Unitarity (Lobo et al. , ‘06)

Recati Recati et al., et al., PRA ’08, PRA ’08, exp: MITexp: MIT

Carlson ‘03 Carlson ‘03 Giorgini ‘04Giorgini ‘04

Pilati Pilati et al. et al. ‘07‘07

A, m* and B are now functions of mA, m* and B are now functions of m↓↓/m/m↑↑== κ κ : :

A(A(κκ) ) and and F(F(κκ)) ≡ ≡ m*/m*/ m m↓↓ from diagrammatic many body techniques from diagrammatic many body techniques (Combescot (Combescot et al., et al., ‘07)‘07)

B(B(κκ) ) fromfrom requirement requirement E(1, E(1, κ κ) = E) = ENN((κκ))

Carlson ‘03 Carlson ‘03 Giorgini ’04;Giorgini ’04;

Astracharchik ‘07 Astracharchik ‘07

Normal state of a Fermi Gas with unequal masses

variation with respect to nS, n↑ , and n↓ yields

with

we can write the energy of the system at T=0

Carlson ‘03 Carlson ‘03 Giorgini ’04;Giorgini ’04;

Astracharchik ‘07 Astracharchik ‘07

Equilibrium Conditions

pressures are the same:pressures are the same:

BCS mean-fieldBCS mean-field

density jump/drop in trapdensity jump/drop in trap

BCS: Wu BCS: Wu et al. ‘06et al. ‘06

Equilibrium Conditions

μμ-h phase diagram:-h phase diagram: chemical potential chemical potential μ μ = ½(= ½(μμ↑↑++μμ↓↓)) effective magnetic field h = effective magnetic field h = ½(½(μμ↑↑- - μμ↓↓)) μμ-h phase diagram:-h phase diagram: chemical potential chemical potential μ μ = ½(= ½(μμ↑↑++μμ↓↓)) effective magnetic field h = effective magnetic field h = ½(½(μμ↑↑- - μμ↓↓))

from from xxcc((κκ) ) we are able to determine (we are able to determine (μμ↓↓//μμ↑↑)|)|xxcc((κκ) ) = = ηηcc((κκ) ) for sf to norm transfor sf to norm trans

for x=0 crossover from partially to fully polarized : (for x=0 crossover from partially to fully polarized : (μμ↓↓//μμ↑↑)|)|x=0 x=0 = -3/5 A(= -3/5 A(κκ))

NN↑↑>N>N↓↓NN↓↓>N>N↑↑

κκ =1 =1κκ =1.5 =1.5κκ =2 =2

8787Sr-Sr-4040KK

κκ* ~2.72* ~2.72

BCS :BCS :κκ* ~ 3.95* ~ 3.95

κκ =6.7 =6.7

4040K-K-66LiLi

κκ >1: superfluid moves clockwise, >1: superfluid moves clockwise, partially polarized anticlockwisepartially polarized anticlockwise

What happens with the phase diagram if κ ≠ 1?

different species with unequal masses have different magnetic and optical properties: restoring forces as additional parameters

,2

1

2

1)( 222 withrrmrV

Configuration in the trap: use μσ= μ0σ - ½ασr2 in μ = ½(μ↑+μ↓) and h = ½(μ↑- μ↓)

centre imbalancecentre imbalance

trapping anisotropytrapping anisotropy

note, that for equal ↑ and ↓ trapping note, that for equal ↑ and ↓ trapping αα↓↓== α α↑ ↑ δδ=1, and h does not depend on position! =1, and h does not depend on position!

Trapped System – Local Density Approximation

fix mass ratio fix mass ratio κκ=2.2 (e.g. =2.2 (e.g. 8787Sr-Sr-4040K) and see what happens in dependence on K) and see what happens in dependence on ηη0 0 and and αα↓↓,,αα↑ ↑

ηη0 0 =1=1 αα↓↓== α α↑↑

ηη0 0 == η ηcc ((κκ)) αα↓↓== α α↑↑

ηη0 0 >>>>ηηcc (1/(1/κκ)) αα↓↓ >> α α↑↑

μμlocloc

superfluid sandwiched betweensuperfluid sandwiched between two normal shells! two normal shells!

μμlocloc

μμlocloc

Trapped System - Results

κ=2.2 (e.g. 87Sr-40K) η0 ~2.1 and α↓ ~ 8α ↑

normal phases with opposite polarization, so that trapped system is globally unpolarized !

P=0P=0

Trapped System - Results

• BCS mean-field leads to quantitatively different results at Unitarityfor the Chandrasekhar-Clogston limit and the critical polarization

• different species with κ≠1 and different restoring forcespermits to engineer novel exotic configurations ,

as e.g. sandwiched superfluid:can be best understood by studying the phase diagram

with trap (an)isotropy

ConclusionsConclusions