Two Fermions in a Double Well: Exploring a Fundamental Building Block of the Hubbard Model

We have prepared two ultracold fermionic atoms in an isolated double-well potential and obtained full control over the quantum state of this system. In particular, we can independently control the interaction strength between the particles, their tunneling rate between the wells and the tilt of the potential. By introducing repulsive (attractive) interparticle interactions we have realized the two-particle analog of a Mott-insulating (charge-density-wave) state. We have also spectroscopically observed how second-order tunneling affects the energy of the system. This work realizes the first step of a bottom-up approach to deterministically create a single-site addressable realization of a ground-state Fermi-Hubbard system.

Figure 1

Experimental realization and eigenenergies of the two-site Hubbard model. (a) Experimental setup: The double-well potential is created by focusing two laser beams with a high-resolution objective. By independently controlling the intensity and position of the two laser beams with an acousto-optic deflector (AOD) we can tune the tunnel coupling $J$ and the tilt $\mathrm{\Delta }$ between the two wells. (b) Energies of the four lowest two-particle eigenstates in a symmetric double-well potential as a function of the on-site interaction energy $U$. (c),(d) Tunneling of two particles in the double well. The data show the time evolution of the particle number in the right well after initializing the system with both particles in the left well and abruptly switching on the tunnel coupling $J$ between the two sites. For $U\approx 0$ and $\mathrm{\Delta }\approx 0$ (c), we can extract the value of the tunnel coupling by fitting the data with a damped sine wave. For intermediate interaction strength ($U\approx J$) (d), we observe correlated tunneling of the two particles, which shows good agreement with the prediction from the Hubbard model (solid line). The error bars denote the $1\sigma$ statistical uncertainty.

Figure 2

Single-particle and pair tunneling as a function of the tilt $\mathrm{\Delta }$ at an interaction energy of $U/J=10.05\pm 0.19$. The data points show the time-averaged probability of finding a single particle (green circles) or a pair of particles (blue triangles) in the right well after initializing the system with two atoms in the left well and switching on the tunneling. Pair tunneling is resonant in a symmetric double well, while conditional single-particle tunneling occurs for a tilt of $-2\mathrm{\Delta }=U$. The error bars denote the $1\sigma$ statistical uncertainty.

Figure 3

Occupation statistics as a function of interaction strength. The relative probabilities of measuring both particles in the same well ($P_2$, blue squares) or in different wells ($P_1$, green circles) are shown as a function of the on-site interaction energy $U$. Open (filled) symbols indicate a tunnel coupling of $J/h\simeq 142\mathrm{Hz}$ ($J/h\simeq 67\mathrm{Hz}$), the solid lines show the prediction of the Hubbard model. (a) For the ground state $|a⟩$, double occupancy is suppressed for increasing repulsive interactions. This indicates the crossover from a metallic to a Mott-insulating regime. For attractive interactions, double occupancy is enhanced, which we interpret as the onset of a charge-density-wave regime. (b) For the excited state $|c⟩$, we observe the crossover to the charge-density-wave regime for strong repulsive interactions. For both measurements, the data have been corrected for the effect of the finite fidelities of preparation and detection (Sec. VII and Fig. S5 of [21]). The error bars denote the $1\sigma$ statistical uncertainty.

Figure 4

Transition from first-order to second-order tunneling. To directly measure the influence of the tunnel coupling on the energy of the system, we determine the energy difference $E_{bc}=E_c-E_b$ (shaded region) between state $|c⟩$ (orange) and state $|b⟩$ (green) using trap modulation spectroscopy. To compare our results to the eigenenergies of the Hubbard model [Fig. 1], we plot the sum of $E_{bc}$ and the energy of state $|b⟩$ ($E_b=U$). The error bars for the $1\sigma$ statistical uncertainties (Sec. VIII of [21]) are smaller than the symbols of the data points. For $U=0$, we observe that $E_{bc}\approx 2J$, which is consistent with single-particle tunneling. For increasing repulsion, first-order tunneling is suppressed and $E_{bc}$ converges to the superexchange energy $4J^2/U$ (see dashed line).