High-Temperature Superfluidity of Fermionic Atoms in Optical Lattices

Fermionic atoms confined in a potential created by standing wave light can undergo a phase transition to a superfluid state at a dramatically increased transition temperature. Depending upon carefully controlled parameters, a transition to a superfluid state of Cooper pairs, antiferromagnetic states or $d$-wave pairing states can be induced and probed under realistic experimental conditions. We describe an atomic physics experiment that can provide critical insight into the origin of high-temperature superconductivity in cuprates.

Figure 1

(a) Attractive fermionic atoms in optical lattices. (b) Critical temperature for the SF transition of $\mathrm{L}{\mathrm{i}}^{\mathrm{6}}$ atoms (circles) as a function of the optical lattice depth in a 3D $\mathrm{C}{\mathrm{O}}_{\mathrm{2}}$ lattice. Li atoms in the states $|\downarrow (\uparrow )⟩=|F=1/2,m_F=\mp 1/2⟩$ are considered at a magnetic field of $\sim 0.1\mathrm{T}$ corresponding to $a_s\sim -2.5\times {10}^3{a}_0$. The absolute energy scale is given by $t/\hslash \approx 0.5\mathrm{k}\mathrm{H}\mathrm{z}$ at the phase transition. For the same density and scattering length, the free-space BCS formula yields $T_c^0=1.6\times {10}^{-12}{E}_F^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}/k_B$ for $\mathrm{L}{\mathrm{i}}^{\mathrm{6}}$. Inset: analogous plot for ${\mathrm{K}}^{\mathrm{40}}$ atoms in a $\mathrm{N}\mathrm{d}:\mathrm{Y}\mathrm{A}\mathrm{G}$ lattice at half filling at a magnetic field above a Feshbach resonance and $a_s\sim -2.\times {10}^2{a}_0$. Dashed curve: the adiabatic cooling effect described in the text.

Figure 2

In the case of repulsive interactions one finds either antiferromagnetic (AFM, upper left) or $d$-wave SF phases (lower left), depending upon the filling fraction $n$. Right: phase diagram for repelling $\mathrm{L}{\mathrm{i}}^{\mathrm{6}}$ atoms in a 2D lattice (obtained in the fluctuation-exchange approximation). Adiabatic cooling due to switching on of the lattice has been taken into account. Note: the repulsive Hubbard model may also have phase separation for some filling factors, which corresponds to immiscibility of the two spin species of the atoms.

Figure 3

Central part of the atomic interference pattern after a free expansion for a time $t=500\times (\hslash /E_R)$ and a $10\times 10$ optical lattice (with lattice constant $a$). Left: normal state at half filling. The sharp edges of the interference peaks reflect the atomic momentum distribution. Right: BCS state at half filling, with a gap $\Delta /t=0.6$ corresponding to $U/t\approx -2.5$.

Figure 4

Left: Bragg scattering of atoms off laser beams with frequency difference $\delta \omega$ and wave-vector difference $q_x=q_y=0.1\pi$ for attractive fermions at filling $n=0.6$ and $U/t=-2.5$, corresponding to an $s$-wave gap $\Delta /t\approx 0.45$. Inset: dispersion of the collective mode (solid line) and the onset frequency of the continuum (dashed line). Right: schematic picture of the two-photon process involved with ${\Omega }_{1,2}$ Rabi frequencies.

Figure 5

Probing $d$-wave pairing at filling $n<1$ via Bragg scattering. Left: schematic diagram of the Fermi surface (solid line) and the $q$ dependence of the gap $\Delta (q_x,q_y)$ (dashed line). At the four nodal points shown by black dots, the wave function and the quasiparticle excitation energy vanishes. For the special wave vectors connecting these points, the density response is gapless (black spots in the right figure). Right: onset frequency ${\omega }_{\mathrm{m}\mathrm{i}\mathrm{n}}(q_x,q_y)$ of the quasiparticle continuum, dark regions corresponding to low frequencies (vanishing gap).