Chandrasekhar- Clogston limit in Fermi mixtures with unequal masses at Unitarity

1. Chandrasekhar-Clogston limit in Fermi mixtures with unequal masses at Unitarity Ingrid Bausmerth AlessioRecati SandroStringari

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

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

5. Normal state of a Fermi Gas with unequal masses Carlson ‘03 Giorgini ’04; Astracharchik ‘07 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) fromrequirement E(1, κ) = EN(κ)

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

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

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

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

10. Trapped System – Local Density Approximation different species with unequal masses have different magnetic and optical properties: restoring forces as additional parameters Configuration in the trap: use μσ= μ0σ - ½ασr2in μ = ½(μ↑+μ↓)andh= ½(μ↑- μ↓) centre imbalance trapping anisotropy note, that for equal ↑ and ↓ trapping α↓= α↑  δ=1, and h does not depend on position!

11. Trapped System - Results fix mass ratio κ=2.2 (e.g. 87Sr-40K) and see what happens in dependence on η0 and α↓,α↑ η0 = ηc(κ) α↓= α↑ η0 =1 α↓= α↑ η0 >>ηc(1/κ) α↓> α↑ μloc μloc μloc superfluid sandwiched between two normal shells!

12. 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=0

13. Conclusions • 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