Exact decoherence dynamics of a detector coupled to vacuum fluctuations in bounded geometries

Abstract

The properties of quantum vacuum are highly sensitive to the underlying spacetime geometry and boundary conditions. We investigate in this paper the decoherence dynamics of a two-level system (detector) coupled to a vacuum scalar field through the $\hatσ_z$-coupling scheme under various boundary environments. By rigorously evaluating the reduced density matrix of the detector, we derive an exact general expression for the detector's decoherence function, $γ(Δt)$, and use it to characterize the dynamical decoherence of a detector in three scenarios: free space, a perfectly reflecting boundary, and a cavity formed by two parallel reflecting boundaries. We find that in free space the detector's coherence decays monotonically with time, indicating that vacuum fluctuations alone inevitably lead to a complete loss of coherence. In the single-boundary case, by contrast, the detector's coherence displays non-monotonic time evolution: $γ(Δt)$ increases with time up to a maximum and then settles to a steady value, indicating that the initial coherence can be partially preserved due to the presence of the boundary. When the detector is placed inside a cavity, its coherence evolution exhibits oscillatory behavior, and a significant fraction of the initial coherence survives. Moreover, for the same detector-boundary distance, more coherence is preserved in the cavity case than in the single-boundary case. These results provide a viable route for engineering quantum coherence via boundary conditions.

The paper asks how reflecting boundaries modify vacuum-induced pure dephasing of a static two-level detector. It considers a massless scalar field in three spatial dimensions, coupled longitudinally through $H_I\propto\sigma_z\phi$, in free space, near one infinite Dirichlet plane, and between two parallel Dirichlet planes. The initial state is a detector superposition uncorrelated with the field vacuum. A spatial profile of width $l$ regularizes ultraviolet contributions. The longitudinal coupling permits an exact reduced-state calculation without Born–Markov or weak-coupling approximations: populations remain fixed and coherences acquire a factor $e^{-\gamma(t)}$, with $\gamma$ determined by an integrated, spatially averaged field two-point function. In free space, $\gamma_0(t)=\lambda^2\ln[1+t^2/(4l^2)]/(2\pi^2)$, giving complete asymptotic dephasing and an $e^{-1}$ coherence time $t_c=2l\sqrt{e^{2\pi^2/\lambda^2}-1}$. For one boundary, an analytic boundary correction produces a dephasing exponent that rises to a maximum and then approaches a finite constant, leaving residual coherence. The cavity calculation gives a mode-sum expression and numerically exhibits damped oscillations toward finite residual coherence. Smaller nearest-boundary distances preserve more coherence; the reported numerical comparisons find greater asymptotic coherence in the cavity than near a single plane at the same nearest-boundary distance. The boundary calculations assume detector–boundary distances much larger than $l$, neglecting boundary-induced changes to the detector profile; the simplified cavity expression assumes separations at least an order of magnitude larger than $l$. The treatment is restricted to ideal reflecting planes and longitudinal coupling to a vacuum scalar field.