In this supplementary material, we discuss the calculation of the radio-frequency spectra arising from confinement-induced dimers and polarons in a quasi-two-dimensional Fermi gas. We determine the dimer binding energies, including both the tight axial confinement and the nonzero transverse confinement. We provide the probabilities for dimer-to-dimer transitions and the shape of the dimerto-scattering state spectrum. We also find the energy and quasi-particle weights for polarons in the two-dimensional gas and the corresponding spectra for polaron to polaron transitions.PACS numbers: 03.75.SsWe begin by reviewing briefly in § I the radio-frequency spectrum arising from confinement-induced pairs, including final state interactions, but ignoring manybody effects, using the method employed for the threedimensional case by Chin and Julienne [1]. We consider mixtures of the three lowest hyperfine states of 6 Li, denoted |1 , |2 , |3 . For the conditions of our experiments in a 12 mixture at 720 G, the observed 2 → 3 threshold spectrum is well described by a 12-dimer-to-13-scattering-state transition. In contrast, at 834 G, the predicted dimer spectrum is in marked disagreement with the data. In particular, we find that the difference between the ground and excited state dimer energies is too small. In § II we determine the energies for noninteracting confinement-induced polarons. We find that the locations of the observed resonances for a 12 mixture near 834 G are well modeled by the predicted energy difference between isolated state 2 polarons and state 3 polarons, in a bath of atoms in state 1. I. CONFINEMENT-INDUCED DIMERSA simple golden rule calculation gives the radiofrequency-induced transition rate out of the initial state to all possible final states R i (ω rf ) = F R f ←i , whereHere, Ω f i is the Rabi frequency for changing the hyperfine state of a single atom from the chosen populated state (i) to the initially unpopulated state (f ) and F |I is the overlap between the initial and final wave-functions for the relative motion of the atompair. Since the center of mass energy does not change in the rf transition, E f − E i is the total change in the atomic hyperfine energy (≡hω f i ) plus the change in the energy of the relative motion of the pair E F − E I . Since F | F |I | 2 = 1, dω rf R i (ω rf ) = (π/2)Ω 2 f i . We define a normalized spectrum I(ω) where R i (ω rf ) = (π/2)Ω 2 f i I(ω) and ω rf = ω f i + ω, with ω the frequency relative to the (unshifted) free-atom hyperfine transition frequency. Then,and dωI(ω) = 1.To determine the spatial wavefunctions and the pair binding energies, we note that the range of the two-body interaction is small compared to the interparticle spacing as well as to the harmonic oscillator confinement scale l z ≡ h/(mω z ). In this case, interactions between atoms in two different spin states are well described by the s-wave pseudopotential in three dimensions [2], V (r) = (4πh 2 a/m) δ(r)∂ r (r...), where r is the distance between the atoms, m is the mass of a single atom an...

We measure the density profiles for a Fermi gas of 6 Li containing N1 spin-up atoms and N2 spin-down atoms, confined in a quasi-two-dimensional geometry. The spatial profiles are measured as a function of spin-imbalance N2/N1 and interaction strength, which is controlled by means of a collisional (Feshbach) resonance. The measured cloud radii and central densities are in disagreement with mean-field Bardeen-Cooper-Schrieffer (BCS) theory for a true two-dimensional system. We find that the data for normal-fluid mixtures are reasonably well fit by a simple two-dimensional polaron model of the free energy. Not predicted by the model is a phase transition to a spin-balanced central core, which is observed above a critical value of N2/N1. Our observations provide important benchmarks for predictions of the phase structure of quasi-two-dimensional Fermi gases.

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