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 tune the dimensionality of pancake-shaped strongly-interacting 6 Li Fermi gas clouds from twodimensional (2D) to quasi-2D, by controlling the ratio of the radial Fermi energy EF to the harmonic oscillator energy hνz in the tightly confined direction. In the 2D regime, where EF << hνz, the measured radio frequency resonance spectra are in agreement with 2D-BCS theory. In the quasi-2D regime, where EF ≃ hνz, the measured spectra deviate significantly from 2D-BCS theory. For both regimes, the measured cloud radii disagree with 2D-BCS mean field theory, but agree approximately with predictions using a free energy derived from the Bethe-Goldstone equation.

Weakly interacting Fermi gases exhibit rich collective dynamics in spin-dependent potentials, arising from correlations between spin degrees of freedom and conserved single atom energies, offering broad prospects for simulating many-body quantum systems by engineering energy-space "lattices," with controlled energy landscapes and site to site interactions. Using quantum degenerate clouds of 6 Li, confined in a spin-dependent harmonic potential, we measure complex, time-dependent spin-density profiles, varying on length scales much smaller than the cloud size. We show that a one-dimensional mean field model, without additional simplifying approximations, quantitatively predicts the observed fine structure. We measure the magnetic fields where the scattering lengths vanish for three different hyperfine state mixtures to provide new constraints on the collisional (Feshbach) resonance parameters.

We study the pairing of fermions in a one-dimensional lattice of tunable double-well potentials using radio-frequency spectroscopy. The spectra reveal the coexistence of two types of atom pairs with different symmetries. Our measurements are in excellent quantitative agreement with a theoretical model, obtained by extending the Green's function method of Orso et al. [Phys. Rev. Lett. 95, 060402 (2005)PRLTAO0031-900710.1103/PhysRevLett.95.060402] to a bichromatic 1D lattice with nonzero harmonic radial confinement. The predicted spectra comprise hundreds of discrete transitions, with symmetry-dependent initial state populations and transition strengths. Our work provides an understanding of the elementary pairing states in a superlattice, paving the way for new studies of strongly interacting many-body systems.