1. Ingrid Bausmerth Alessio Recati Sandro Stringari Ingrid Bausmerth Alessio Recati Sandro Stringari Chandrasekhar-Clogston limit in Fermi mixtures with unequal masses at Unitarity

2. 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?

3. BEC of molecules: strong coupling, k F a s <<1, interaction is repulsive condensation of tightly bound fermions, BEC of molecules: strong coupling, k F a s <<1, interaction is repulsive condensation of tightly bound fermions, size of molecules much smaller than average distance between pairs: BEC gas of molecules a s  ±∞ BCS-limit: weak coupling, k F |a s |<<1, interaction is attractive, condensation of long-range Cooper Pairs in momentum space, BCS-limit: weak coupling, k F |a s |<<1, interaction is attractive, condensation of long-range Cooper Pairs in momentum space, negative values of a s, size of pairs is larger than interparticle distance BEC a>0 a>0BCSa<0 T=0 and 3D Fermions: BCS-BEC Crossover system is strongly correlated, but its properties do not depend on value of scattering length a s (independent even of sign of a s )  everything is expressed in terms of k F

4. normal to superfluid transition: n ↓ /n ↑ = x c = 0.44 Normal State of a Fermi gas at Unitarity (Lobo et al., ‘06) Normal State of a Fermi gas at Unitarity (Lobo et al., ‘06) Recati et al., PRA ’08, exp: MIT Carlson ‘03 Giorgini ‘04 Giorgini ‘04 Pilati et al. ‘07

5. A, m* and B are now functions of m ↓ /m ↑ = κ : A( κ) and F(κ) ≡ m*/ m ↓ from diagrammatic many body techniques (Combescot et al., ‘07) B(κ) from requirement E(1, κ) = E N (κ) Carlson ‘03 Giorgini ’04; Astracharchik ‘07 Normal state of a Fermi Gas with unequal masses

6. variation with respect to n S, n ↑, and n ↓ yields with we can write the energy of the system at T=0 Carlson ‘03 Giorgini ’04; Astracharchik ‘07 Equilibrium Conditions

7. pressures are the same: BCS mean-field density jump/drop in trap BCS: Wu et al. ‘06 Equilibrium Conditions

8. μ -h phase diagram: chemical potential μ = ½(μ ↑ +μ ↓ ) μ -h phase diagram: chemical potential μ = ½(μ ↑ +μ ↓ ) effective magnetic field h = ½(μ ↑ - μ ↓ ) effective magnetic field h = ½(μ ↑ - μ ↓ ) μ -h phase diagram: chemical potential μ = ½(μ ↑ +μ ↓ ) μ -h phase diagram: chemical potential μ = ½(μ ↑ +μ ↓ ) effective magnetic field h = ½(μ ↑ - μ ↓ ) effective magnetic field h = ½(μ ↑ - μ ↓ ) from x c (κ) we are able to determine ( μ ↓ /μ ↑ )| x c (κ) = η c (κ) for sf to norm trans from x c (κ) we are able to determine ( μ ↓ /μ ↑ )| x c (κ) = η c (κ) for sf to norm trans for x=0 crossover from partially to fully polarized : ( μ ↓ /μ ↑ )| x=0 = -3/5 A(κ) for x=0 crossover from partially to fully polarized : ( μ ↓ /μ ↑ )| x=0 = -3/5 A(κ) N ↑ >N ↓ N ↓ >N ↑

9. κ =1 κ =1.5 κ =2 87 Sr- 40 K κ* ~2.72 BCS : κ* ~ 3.95 κ =6.7 40 K- 6 Li κ >1: superfluid moves clockwise, partially polarized anticlockwise What happens with the phase diagram if κ ≠ 1?

10. different species with unequal masses have different magnetic and optical properties: restoring forces as additional parameters Configuration in the trap: use μ σ = μ 0 σ - ½α σ r 2 in μ = ½(μ ↑ +μ ↓ ) and h = ½(μ ↑ - μ ↓ ) centre imbalance trapping anisotropy note, that for equal ↑ and ↓ trapping α ↓ = α ↑  δ=1, and h does not depend on position! Trapped System – Local Density Approximation

11. fix mass ratio κ=2.2 (e.g. 87 Sr- 40 K) and see what happens in dependence on η 0 and α ↓, α ↑ η 0 =1 η 0 =1 α ↓ = α ↑ α ↓ = α ↑ η 0 = η c (κ) η 0 = η c (κ) α ↓ = α ↑ α ↓ = α ↑ η 0 >>η c (1/κ) η 0 >>η c (1/κ) α ↓ > α ↑ α ↓ > α ↑ μ loc superfluid sandwiched between two normal shells! two normal shells! μ loc Trapped System - Results

12. κ=2.2 (e.g. 87 Sr- 40 K) η 0 ~ 2.1 and α ↓ ~ 8α ↑ normal phases with opposite polarization, so that trapped system is globally unpolarized ! P=0 Trapped System - Results

13. BCS mean-field leads to quantitatively different results at Unitarity for the Chandrasekhar-Clogston limit and the critical polarization different species with κ≠1 and different restoring forces permits to engineer novel exotic configurations, as e.g. sandwiched superfluid: can be best understood by studying the phase diagram with trap (an)isotropy Conclusions