Lattice introduction

The elevator pitch…

The prospect of using cold neutral atoms to perform a quantum simulation of condensed matter systems has garnered much excitement in recent years. Quantum simulation experiments may be able to shed light on the outstanding issue of high temperature superconductivity (HTSC) by experimentally investigating the predictions of the Fermi-Hubbard model (FHM)- a theoretical model which shows promise in explaining HTSC, yet has rejected analytical and computational solutions to date.

Consequently, the predictions of the FHM remain ambiguous. Because of the high degree of control over experimental parameters, a quantum simulation experiment using fermions trapped in an optical lattice could be able to produce an atomic physics analogue of the situation described by the FHM. Specifically, one could hope to observe the predicted Neel phase - a phase in which single spins populate each site on a 2D square lattice in a checkerboard, or antiferromagnetic, pattern - as a first step towards observing the superfluid HTSC phase.

Therefore, we are developing a quantum simulator using $^{40}K$ fermions with site-resolved imaging in order to measure the entropy and temperature of atoms in an optical lattice undergoing a phase transition into the Neel state. Site-resolved imaging will be a powerful tool in determining the lattice occupation, domain structure and spatial inhomogeneity of the system, as well as what role the background trapping potential of an optical lattice has on this phase transition.

The CN tower elevator pitch…

The single-band Fermi-Hubbard model is given by:

(1)
\begin{align} H_{FH}=-t\sum_{\sigma}\sum_{\langle\mathit{i,j}\rangle}\psi^{\dag}_{i,\sigma}\psi_{j,\sigma}+U\sum_{i}\psi^{\dag}_{i,\uparrow}\psi^{\dag}_{i,\downarrow}\psi_{i,\uparrow}\psi_{i,\downarrow} \end{align}

where $\psi^{\dag}$ and $\psi$ are the fermionic creation and annihilation operators, $i$ and $j$ label the lattice sites, $\sigma$ labels the two hypefine spin states $\uparrow$ or $\downarrow$ of the atoms, and the second sum in the first term is over all adjacent sites. The first term, proportional to $t$, represents the energy contribution of an atom tunnelling between adjacent lattice sites whereas the second term, proportional to $U$, represents the interaction energy of two atoms on a single site. It is these two parameters, $t$ and $U$, that we can tune experimentally. This is done by changing the lattice potential $V_0$ and making use a Feshbach resonance to vary the s-wave scattering length $a_s$. However, this model is only valid for the situation in which $U$ is much less than the gap energy $\Delta$. Furthermore, because the optical lattice is created from Gaussian laser beams, a background harmonic potential will be introduced to the FHM:

(2)
\begin{align} H_{trap}=\sum_{i}\epsilon_i\psi^{\dag}_{i,\sigma}\psi_{i,\sigma} \end{align}

where $\epsilon$ is the energy offset of site $i$ with respect to the center of the trap.

This system undergoes a phase transition to a Mott insulating state when $U/zt$ approaches unity, where $z$ is the number of nearest neighbours. In this phase, the energy of the many-body state is minimized when atoms are localized on single sites of the lattice. This phenomenon was recently observed in a 3D optical lattice here and here. If the on-site interaction energy is increased such that $U$ is much greater than both the tunnelling energy $t$ and the temperature $k_BT$ then the FHM may be represented by the Heisenberg Hamiltonian:

(3)
\begin{align} H_H=\frac{J}{2}\sum_{\langle\mathit{i,j}\rangle}\bf{S}_i\cdot\bf{S}_j \end{align}

where $J=4t^2/U$ is the superexchange energy and $\bf{S}$ is equal to one half of the Pauli matrices. The ground state of this Hamiltonian is the antiferromagnetically ordered Neel state.

The phase transition from a paramagnetic Mott insulating state to the Neel state has been mapped out according to the parameters $V_0$ and $a_s$. However, we also require that the temperature $T$ be small compared to $t$ and $U$ such that we remain faithful to the FHM and Heisenberg models. With these constraints, it is predicted that the maximum Neel transition temperature $T_N$ for a 3D FHM should be:

(4)
\begin{align} k_BT_N^{max}\approx.03E_R \end{align}

occurring when $V_0\approx3E_R$ and $a_s\approx0.15d$, where $E_R=\left(\pi\hbar\right)^2/\left(2md^2\right)$ is the recoil energy and $d$ is the lattice spacing. For example, for $^{40}K$ with $d=760$, we have $E_R/k_B=95nK$ and $T_N\approx3nK$. For comparison, in a recent experiment observing the Mott-insulating phase of fermions in a 3D lattice, a temperature of $T=0.15T_F=1.3E_R$ was achieved, a factor of more than forty times $T_N^{max}$.

Therefore, not only do new techniques need to be imagined for cooling fermions in a lattice, but new ways to measure their temperature and assess "cold-loading" techniques need to be developed as well. We propose to solve the latter problem by measuring the entropy of the system via site-resolved imaging.

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