Resonance interaction energy between two entangled atoms in a photonic bandgap environment

Abstract

We consider the resonance interaction energy between two identical entangled atoms, where one is in the excited state and the other in the ground state. They interact with the quantum electromagnetic field in the vacuum state and are placed in a photonic-bandgap environment with a dispersion relation quadratic near the gap edge and linear for low frequencies, while the atomic transition frequency is assumed to be inside the photonic gap and near its lower edge. This problem is strictly related to the coherent resonant energy transfer between atoms in external environments. The analysis involves both an isotropic three-dimensional model and the one-dimensional case. The resonance interaction asymptotically decays faster with distance compared to the free-space case, specifically as 1/r² compared to the 1/r free-space dependence in the three-dimensional case, and as 1/r compared to the oscillatory dependence in free space for the one-dimensional case. Nonetheless, the interaction energy remains significant and much stronger than dispersion interactions between atoms. On the other hand, spontaneous emission is strongly suppressed by the environment and the correlated state is thus preserved by the spontaneous-decay decoherence effects. We conclude that our configuration is suitable for observing the elusive quantum resonance interaction between entangled atoms.

Figure 1

The figure shows the position of the atomic frequency ω_a, in relation to the lower and upper edge of the first photonic gap at wavenumber k₀, given by ${\omega }_{\ell }$ and ω_u respectively. We assume that the width of the photonic gap, $\mathrm{\Delta }\omega ={\omega }_u-{\omega }_{\ell }$, is such that ${\omega }_a-{\omega }_{\ell }\ll \mathrm{\Delta }\omega$.

Figure 2

Plot of the resonance interaction energy ΔE_3D in arbitrary units for distances of the atoms between 10⁻⁷ m and 10⁻⁵ m obtained by a numerical integration using the exact dispersion relation of the photonic crystal (blue continuous curve). The atomic dipoles are assumed equal and oriented perpendicularly to the atomic distance. Parameters are such that n = 3, a = 2 · 10⁻⁸ m, ω_a = 2.65 · 10¹⁵ s⁻¹, and thus k₀ = 1.96 · 10⁷ m⁻¹. The red dashed line represents the function r⁻². The figure clearly shows the asymptotic power-law decay as r⁻², in agreement with the analytical results obtained using our approximate dispersion relation in the first Brillouin zone of the photonic crystal.

Figure 3

Plot of r²ΔE_3D in arbitrary units for distances of the atoms between 10⁻⁷ m and 10⁻⁵ m obtained by a numerical integration using the exact dispersion relation of the photonic crystal and the same parameters of Fig. 2. The figure clearly shows that, after the transition region between the near and far zone at $r\sim 5\cdot {10}^{-6}\mathrm{m}$, the function r²ΔE_3D asymptotically settles to a constant value. This confirms that asymptotically the resonance interaction energy ΔE_3D scales as r⁻².