Dirac Equation in ($1+1$)-Dimensional Curved Spacetime and the Multiphoton Quantum Rabi Model

We introduce an exact mapping between the Dirac equation in ($1+1$)-dimensional curved spacetime (DCS) and a multiphoton quantum Rabi model (QRM). A background of a ($1+1$)-dimensional black hole requires a QRM with one- and two-photon terms that can be implemented in a trapped ion for the quantum simulation of Dirac particles in curved spacetime. We illustrate our proposal with a numerical analysis of the free fall of a Dirac particle into a ($1+1$)-dimensional black hole, and find that the Zitterbewegung effect, measurable via the oscillatory trajectory of the Dirac particle, persists in the presence of gravity. From the duality between the squeezing term in the multiphoton QRM and the metric coupling in the DCS, we show that gravity generates squeezing of the Dirac particle wave function.

Figure 1

Dynamics of a massive Dirac particle near a black hole and the multiphoton QRM. Properties of the simulated Dirac particle (a) are related to those of the ion (b), under the mapping (3). In both cases the upper plot shows the initial (green) and final (blue) probability density profiles, while the lower plots show the expectation values of the corresponding position operators. For the simulated Dirac particle, the position of the horizon is indicated by a vertical line labeled $r_s$. The initial state of the ion is ${\mathrm{\Psi }}_0={\varphi }_{X_0/\lambda =8}(X)\otimes |+{⟩}_x$, where ${\varphi }_{X_0/\lambda =8}(X)$ is a Gaussian wave function centered at $X_0$, and $|+{⟩}_x$ is the eigenstate of operator ${\sigma }_x$ with positive eigenvalue. The simulation is performed under the Hamiltonian in Eq. (6), for the case where $m\lambda =0.3$ and $M\lambda =0.01$, with $c=1$. The scale of the oscillations of the simulated particle is smaller than that of the oscillations of the ion, as the position of the simulated particle is rescaled by $1/r_s$ under mapping in Eq. (3).

Figure 2

Squeezing of the dynamics by curvature of spacetime. The continuous line corresponds to the variance of the position of the ion $(\mathrm{\Delta }X/\lambda {)}^2$ while the dashed line corresponds to the variance of the momentum of the ion $(\lambda \mathrm{\Delta }P{)}^2$, both being dimensionless. A logarithmic scale is used. The simulation regime is the same as that in Fig. 1. As the ion evolves in time a clear trend towards its position getting localized accompanied with an exponential growth of the uncertainty of its momentum can be observed.