Experimental Realization of the Rabi-Hubbard Model with Trapped Ions

Quantum simulation provides important tools in studying strongly correlated many-body systems with controllable parameters. As a hybrid of two fundamental models in quantum optics and in condensed matter physics, the Rabi-Hubbard model demonstrates rich physics through the competition between local spin-boson interactions and long-range boson hopping. Here, we report an experimental realization of the Rabi-Hubbard model using up to 16 trapped ions and present a controlled study of its equilibrium properties and quantum dynamics. We observe the ground-state quantum phase transition by slowly quenching the coupling strength, and measure the quantum dynamical evolution in various parameter regimes. With the magnetization and the spin-spin correlation as probes, we verify the prediction of the model Hamiltonian by comparing theoretical results in small system sizes with experimental observations. For larger-size systems of 16 ions and 16 phonon modes, the effective Hilbert space dimension exceeds $2^{57}$, whose dynamics is intractable for classical supercomputers.

Figure 1

Schematic of the experiment. (a) We use two global Raman laser beams to create a Rabi-Hubbard model Hamiltonian on an ion chain. Two frequency components are used to drive the blue and the red phonon sidebands (BSB and RSB) simultaneously. The frequency of the Raman pairs is locked by a phase-locked loop (PLL). The hopping rates among different sites $t_{ij}$ are determined by the interion spacings, and the local quantum Rabi model Hamiltonian is controlled by the amplitudes and the frequencies of the driving lasers. (b) We use 355 nm pulsed laser beams with a frequency comb structure to bridge the Raman transitions of the qubits [31]. The large energy splitting of ${\omega }_{\mathrm{hf}}=2\pi \times 12.6\mathrm{GHz}$ is covered by about 107 teeth of the frequency comb so we only need frequency shifts on the order of tens of MHz to set suitable detuning for the bichromatic Raman beams.

Figure 2

Quantum phase transition under slow quench. We start from the ground state $|{\mathrm{\Psi }}_0⟩\equiv |\downarrow ,0{⟩}^{\otimes N}$ with the on-site spin-phonon coupling $g=0$, and then slowly tune up the coupling across the predicted critical point. (a)–(e) Spin-spin correlations $C_{ij}\equiv ⟨{\sigma }_x^i{\sigma }_x^j⟩-⟨{\sigma }_x^i⟩⟨{\sigma }_x^j⟩$ for various ion pairs in an $N=10$ chain versus the coupling $g$ after slow quench. For small $g$, the correlation remains close to zero apart from small detection errors; once $g$ is tuned across a critical point (indicated by the vertical dashed line as the numerically computed value $g_c\approx 1.03g_c^{\mathrm{mf}}$), the spin-spin correlation increases rapidly, which indicates a quantum phase transition. The solid line is the theoretical ground-state value from the DMRG calculation, and the shaded region between the dashed lines represents the theoretical results under a shift of $\pm 300\mathrm{Hz}$ in the trap frequency. (f) The nearest-neighbor spin-spin correlation for two central ions in a chain of 2–16 ions (dots with error bars representing one standard deviation) and the corresponding theoretical ground-state values from the DMRG calculation (solid lines). Here, we normalize the horizontal axis by the mean-field transition point $g_c^{\mathrm{mf}}$, and scale the vertical axis by $N^{2\beta /\nu }$ where $\beta =1/8$ and $\nu =1$ are two critical exponents. Theoretically, the rise of the curves becomes sharper near the predicted transition point as $N$ increases. Although this is less clear from the experimental data due to the noise and errors including the violation of adiabaticity and decoherence, the overall tendency between the theoretical and the experimental results still agrees with each other for different system sizes (see Supplemental Material for individual plots of each ion number $N$ [26]).

Figure 3

Generic spin dynamics. We initialize the system in $|\uparrow ,0{⟩}^{\otimes N}$, immediately turn on the RH Hamiltonian to evolve the system, and measure $⟨{\sigma }_z^i(t)⟩$ of individual spins. (a) Measured data (dots with error bars representing one standard deviation) and theoretical results from numerically integrating the Schrödinger equation (solid lines) for $N=2$ at $g=2\pi \times 2\mathrm{kHz}$ (red) and $2\pi \times 7\mathrm{kHz}$ (blue). The two ions are symmetric and hence only one is plotted. (d),(e) Similar results for $N=4$ at $g=2\pi \times 1\mathrm{kHz}$ (red) and $2\pi \times 6\mathrm{kHz}$ (blue) for an ion on the edge and in the center, respectively. (b),(f) The corresponding evolution of entanglement entropy $S(t)$ for $N=2$ and $N=4$ between the left and the right halves of the chain (with both spin and phonon states included) under the two coupling strengths. (c),(g) Corresponding theoretical results for local phonon numbers $⟨a_i^{†}(t)a_i(t)⟩$. For $N=4$ the solid and the dashed lines are for the central and the edge sites, respectively. (h),(i) The measured dynamics for the edge and the central spins of an $N=16$ chain at $g=2\pi \times 1\mathrm{kHz}$ (red) and $2\pi \times 6\mathrm{kHz}$ (blue).