Quantum Simulation with a Trilinear Hamiltonian

Interaction among harmonic oscillators described by a trilinear Hamiltonian $\hslash \xi (a^{†}bc+a{b}^{†}{c}^{†})$ is one of the most fundamental models in quantum optics. By employing the anharmonicity of the Coulomb potential in a linear trapped three-ion crystal, we experimentally implement it among three normal modes of motion in the strong-coupling regime, where the coupling strength is much larger than the decoherence rate of the ion motion. We use it to simulate the interaction of an atom and light as described by the Tavis-Cummings model and the process of nondegenerate parametric down-conversion in the regime of a depleted pump.

Figure 1

Trilinear mode coupling. (a) One axial and two radial modes of motion involved in the experiment. Two ions (black) are optically pumped into the metastable ${}^2{F}_{7/2}$ state and remain there during the experiment. (b) Probability to drive an axial-mode blue sideband transition as a function of Raman detuning (vertical axis) and three-mode detuning $\delta$ (horizontal axis), after all three motional modes are prepared in the ground state. Solid lines are the eigenvalues of the Hamiltonian Eq. (3), and $|n_a,n_b,n_c⟩$ are the eigenstates in the far-detuned limit with $n_a$, $n_b$, and $n_c$ phonons in mode $a$, $b$, and $c$, respectively. Inset: The on-resonance vacuum Rabi splitting measured in the spectrum of the axial-mode blue sideband. (c) The coherent energy exchange between three modes of motion. The purple, red, and blue dots represent the probabilities to measure the ion in state $|\uparrow ⟩$ after applying the corresponding red sideband $\pi$ pulse to the axial zigzag (${\omega }_a=1414\mathrm{kHz}$), the radial tilt (${\omega }_b=878\mathrm{kHz}$), and the radial zigzag (${\omega }_c=536\mathrm{kHz}$) modes, respectively, as functions of the interaction time $\tau$. The purple, red, and blue lines are the corresponding sinusoidal or cosinusoidal fits. (d) The oscillation amplitude determined by fitting the three oscillation curves in (c) with a sinusoidal (cosinusoidal) fit function and averaging their amplitudes.

Figure 2

Simulations of the Tavis-Cummings model. Before the system is brought into resonance for interaction, the axial zigzag and the radial tilt modes are initially both prepared in a vacuum state and the radial zigzag mode in (a) the Fock state $|1⟩$ and (b) the coherent state with the average phonon number $\overline{n}=1.8(1)$. The red (blue) dots represent the probabilities to find the ion in $|\uparrow ⟩$ after applying a red sideband $\pi$ pulse to the axial zigzag (radial tilt) mode as a function of interaction time $\tau$, while the red (blue) lines show the corresponding fits. The inset in (a) shows the extracted Rabi oscillation frequency from the fits as a function of the phonon numbers in the radial zigzag mode Fock states. The orange dots are the experimental data, and the orange line is the theoretical prediction ${\mathrm{\Omega }}_n/2\pi =2\sqrt{n+1}\xi /2\pi$. The extracted $\overline{n}=1.8(1)$ in (b) from the fits, assuming a Poisson phonon number distribution, is consistent with an independent phonon number calibration [38, 45]. The oscillations in this figure correspond to the Rabi oscillations of the initially excited atom interacting with a light field (see the main text).

Figure 3

Simulation of the nondegenerate parametric down-conversion in the depleted-pump regime. The upper, middle, and lower figure panels show the temporal evolutions of the reconstructed phonon number distributions (see the main text) of the axial zigzag, the radial tilt, and the radial zigzag modes, respectively. The solid lines are the calculated phonon number distributions using Hamiltonian Eq. (3) with the axial zigzag mode initially in a coherent state with $\overline{n}=3.7$ and the radial tilt and the radial zigzag modes both in a vacuum state.