Simulating high-temperature superconductivity model Hamiltonians with atoms in optical lattices

We investigate the feasibility of simulating different model Hamiltonians used in high-temperature superconductivity. We briefly discuss the most common models and then focus on the simulation of the so-called $t\text{−}J\text{−}U$ Hamiltonian using ultra-cold atoms in optical lattices. For this purpose, previous simulation schemes to realize the spin interaction term $J$ are extended. We especially overcome the condition of a filling factor of exactly one, which otherwise would restrict the phase of the simulated system to a Mott-insulator. Using ultra-cold atoms in optical lattices allows simulation of the discussed models for a very wide range of parameters. The time needed to simulate the Hamiltonian is estimated and the accuracy of the simulation process is numerically investigated for small systems.

Figure 1

Lower bounds for the calculated shifting and holding times ${\tau }_{sh}$ [cf. Eq. (18)] for lattices with ${}^{87}\mathrm{Rb}$ atoms $(\lambda =826\mathrm{nm}$, $a_s=5.1\mathrm{nm}$), ${}^{40}\mathrm{K}$ atoms $(\lambda =826\mathrm{nm},a_s=5.5\mathrm{nm})$ and ${}^6\mathrm{Li}$ atoms $(\lambda =670\mathrm{nm}$, $a_s=2.4\mathrm{nm})$. The values given in brackets are typical values used for the calculation of ${\tau }_{sh}$.

Figure 2

Simulation of Hamiltonian Eq. (3) using first- and second-order Trotter expansions, cf. Eqs. (B1, B2). Upper bounds for the anti-fidelity $\mathcal{F}$ are shown. The number of lattice sites is $M=5$, the parameters for the simulation are $J=0.3t$, $U_s=5t$, $U_{\mathrm{eff}}^{\prime }=-2t$, $m=1$ leading to $U=-t$.

Figure 3

Upper bounds for the anti-fidelity $\mathcal{F}$ for first- and second-order Trotter expansions, cf. Eq. (B4). The simulated time is fixed to $\tau =100\hslash ∕t$, the number of lattice sites is $M=5$. We have used $J=0.3t$, $U_s=10t$, and $\varphi =0$.

Figure 4

Upper bounds for the anti-fidelity $\mathcal{F}$ vs number of lattice sites $M$ for first- (pluses) and second-order (stars) Trotter expansion. The simulated time was $\tau =0.01\hslash ∕t$ and we have chosen $J=0.3t$, $U_s=10t$, and $\varphi =0$.