Quantum Rabi model in the Brillouin zone with ultracold atoms

The quantum Rabi model describes the interaction between a two-level quantum system and a single bosonic mode. We propose a method to perform a quantum simulation of the quantum Rabi model, introducing an implementation of the two-level system provided by the occupation of Bloch bands in the first Brillouin zone by ultracold atoms in tailored optical lattices. The effective qubit interacts with a quantum harmonic oscillator implemented in an optical dipole trap. Our realistic proposal allows one to experimentally investigate the quantum Rabi model for extreme parameter regimes, which are not achievable with natural light-matter interactions. When the simulated wave function exceeds the validity region of the simulation, we identify a generalized version of the quantum Rabi model in a periodic phase space.

Figure 1

Band structure for an optical lattice potential. Comparison of the dispersion relation for the first and second bands for a lattice-potential depth of $V=2E_r$ (solid blue line) and a free particle (dashed red line). The gap between the first and second bands corresponds to the effective qubit energy splitting.

Figure 2

Comparison between the full cold-atom Hamiltonian (red continuous line) and the quantum Rabi model (blue dashed line). The momentum is shown in units of $\hslash k_0$, while the position is shown in units of $1/k_0$ and the coupling strength $g/{\omega }_0=5.18$. For the dDSC, the ratio of frequencies is given by ${\omega }_q/{\omega }_0=28.7$, while for the DSC regime, ${\omega }_q/{\omega }_0=0$.

Figure 3

Numerically evaluated real-momentum distribution of the cold-atom cloud during the dynamics shown in Fig. 2, at different evolution times. For the dDSC regime (upper panel), the Rabi parameters are given by $g/{\omega }_0=7.7$ and $g/{\omega }_q=0.43$. In this case, the initial wave function is transformed back and forth between two distributions centered on the states $\left|p=\pm 2\hslash k_0\right.〉$. For the DSC regime (lower panel), $g/{\omega }_0=10$ and ${\omega }_0={\omega }_q$. In this case, the system is continuously displaced in momentum space until the maximum value of the momentum is reached.

Figure 4

Comparison between the full cold-atom Hamiltonian (red continuous line) and the QRM (blue dashed line). (a) The plot shows collapses and revivals of the initial-state population $P_{\mathrm{in}}=|\langle {\psi }_{\mathrm{in}}\left|\psi \left(t\right)\right.〉{|}^2$. The initial state is given by $\left|{\psi }_{\mathrm{in}}\right.〉=\left|q=0\right.〉\left|n_{\text{b}}=1\right.〉$. The coupling strength is given by $g/{\omega }_0=5.18$, while the qubit energy spacing vanishes, ${\omega }_q=0$. In this limit, collapses and revivals correspond to harmonic oscillations of the atoms in the trap potential. (b) Temporal evolution of the quasimomentum in units of $2\hslash k_0$, as in the right column of Fig. 2, for the long-time dynamics. Notice that the value of the quasimomentum is bound at $\left|q\right|\le 2\hslash k_0$ and that the different behavior between the Rabi and the periodic quantum Rabi model appears at the boundary of the Brillouin zone.