Collective dynamics of accelerated atoms

We study the collective dynamics of accelerated atoms interacting with a massless field via an Unruh-deWitt–type interaction. We first derive a general Hamiltonian describing such a system and then, employing a Markovian master equation, we study the corresponding collective dynamics. In particular, we observe that the emergence of entanglement between two-level atoms is linked to the building up of coherences between them and to superradiant emission. In addition, we show that the derived Hamiltonian can be experimentally implemented by employing impurities in Bose-Einstein condensates.

Figure 1

Schematic figure showing atoms interacting with a massless scalar field. The ground state of the field is entangled between regions $\text{I}$ and $\text{II}$. Therefore the reduced vacuum state in either of the regions is given by a thermal state. In (a), the two atoms are in antiparallel accelerated motion and interact with the massless field supported in causally disconnected regions of spacetime. In (b), the two atoms are accelerated in parallel and therefore only interact with the field in region $\text{I}$.

Figure 2

Evolution of the atomic population (top panel) and emission rate (bottom panel) for $N=6$ atoms at the same acceleration. The different curves correspond to increasing values of the acceleration $\alpha =2,4,6,8,10$ in a rainbow scale going from blue to red curves (in direction of the arrow). In the top panel, higher accelerations also have higher values of the long-time limit population, while in the lower panel, higher accelerations correspond to a higher maximum in the emission rate.

Figure 3

Evolution of the atomic coherences (top panel) and concurrence (bottom panel) for $N=6$ atoms at the same acceleration. The different curves correspond to increasing values of the acceleration $\alpha =2,4,6,8,10$ (blue to red curves, in direction of the arrow). In both panels, the curves with higher acceleration have lower maxima than the ones with lower acceleration.

Figure 4

Evolution of the emission rate (top panel) and concurrence (bottom panel) for different acceleration distributions and considering all atoms initially excited. Solid red and solid black curves correspond, respectively, to the cases (a) where all atoms have the same acceleration $\alpha =2$ and (b) where all atoms have different accelerations given by Eq. (35) with $\mathrm{\Delta }\alpha =0.6$ and ${\omega }_j={\omega }_1=1$ for all $j$. In the latter case, no entanglement is generated. The dashed green and dotted blue curves correspond to the case (c) where atoms have different accelerations, as given by Eq. (35), with $\mathrm{\Delta }\alpha =0.6$ and $\mathrm{\Delta }\alpha =0.03$, respectively. In this case the atomic frequencies are chosen such that ${\mathrm{\Omega }}_j={\omega }_1=1$.

Figure 5

(a) Width of the bound states as a function of the optical tweezer potential depth. (b) Dependence of the atom energy on the bound state width. In both panels, the shadowed region corresponds to the two-level condition in Eq. (B9).

Figure 6

(a) Dispersion relation of the Bogoliubov modes and the choice of the resonance value of the atoms. The horizontal (vertical) dashed line indicates the frequency (wave number) resonant with the central atom transition $\mathrm{\Omega }$. The shadowed horizontal (vertical) stripes indicate the range of near-resonant frequencies (wave vectors) for a centered distribution of atom proper accelerations. (b) Strength of the couplings $|{\mathcal{G}}_{ik}^{00}|$ (dashed line), $|{\mathcal{G}}_{ik}^{11}|$ (dot-dashed line), and $|{\mathcal{G}}_{ik}^{10}|$ (solid line) near resonance. For illustration, we have used $a_0=2.2w$.