Quantum Simulation of Quantum Field Theories in Trapped Ions

We propose the quantum simulation of fermion and antifermion field modes interacting via a bosonic field mode, and present a possible implementation with two trapped ions. This quantum platform allows for the scalable add up of bosonic and fermionic modes, and represents an avenue towards quantum simulations of quantum field theories in perturbative and nonperturbative regimes.

Figure 1

Feynman diagrams for a number of terms in a finite-mode Dyson expansion associated with the interaction Hamiltonian in Eq. (11). Note that loops do not imply sum over all QFT modes.

Figure 2

(a) Setup for the trapped-ion simulation. (b) $|⟨f,0,0|\psi (t{)}_1⟩{|}^2$ as a function of $t$ in units of ${\omega }_0$ (red or upper curves), where $|\psi (t{)}_1⟩$ is the evolved state from $|\psi (0{)}_1⟩=|f,0,0⟩$, and average number of virtual bosons (blue or lower curves), $⟨a_0^{†}{a}_0⟩$, for $g_1=0.15{\omega }_0$, $0.1{\omega }_0$, $0.05{\omega }_0$, $0.01{\omega }_0$, $g_2=0$. The largest amplitudes correspond to the largest couplings. (c) $|⟨f,\overline{f},0|\psi (t{)}_2⟩{|}^2$ as a function of $t$ in units of ${\omega }_0$ (red or upper left curve), where $|\psi (t{)}_2⟩$ is the evolved state from $|\psi (0{)}_2⟩=|f,\overline{f},0⟩$, and average number of virtual bosons (blue or lower left curve), $⟨a_0^{†}{a}_0⟩$, for $g_1=0.01{\omega }_0$, $g_2=0.21{\omega }_0$, ${\sigma }_t=3/{\omega }_0$, $T=30/{\omega }_0$, $\delta =0$. (d) The same as (c) for $g_1=0.1{\omega }_0$, $g_2={\omega }_0$, ${\sigma }_t=4/{\omega }_0$, $T=30/{\omega }_0$, $\delta =0$.