Entanglement harvesting in the presence of cavities

TL;DR

This work extends entanglement harvesting to cavity environments by modeling two identical Gaussian detectors on the symmetry axis of a cylindrical cavity, interacting adiabatically with the cavity’s EM vacuum. It combines analytic expressions for the negativity with extensive numerical analysis across microcavity to optical-cavity regimes, revealing distinct scaling with cavity length versus radius and highlighting parity-driven control of correlations. The key contributions include closed-form expressions for the local and non-local contributions to entanglement, a detailed map of how cavity geometry and detector parameters shape timelike versus spacelike harvesting, and the identification of parity effects that can enhance entanglement and widen operational parameter ranges. The findings illuminate how cavity boundaries and mode structure can be engineered to optimize quantum correlations, with implications for cavity QED experiments and quantum information protocols involving vacuum-mediated entanglement.

Abstract

So far, entanglement harvesting has been extensively studied in free space setups. Here, we provide a detailed analytical and numerical analysis of entanglement harvesting in cavities. Specifically, we adiabatically couple the quantized electromagnetic field to two identical Gaussian detectors located on the symmetry axis of a cylindrical cavity. Our numerical investigations reveal a strong dependence on the cavity length, while showing invariance under changes in the cavity radius in regimes of maximal entanglement. Moreover, we identify different scalings of the detector system parameters for entanglement inside and outside the light cone. Finally, we uncover a strong dependence of the harvested correlations on the cavity induced parity of the electromagnetic field.

Figures (6)

• Figure 1: Example of the rotation of a Gaussian state in position representation. For the Euler angles we have $\vartheta$ defining rotations from $z$ to $z'$, $\psi$ defining the rotation from $x$ to N and $\varphi$ for rotations from N to $x'$. Here N is the line of nodes where the $x$, $y$ surfaces of primed and unprimed coordinate system intersect. With these 3 angles all possible rotations between two coordinate systems can be described.
• Figure 2: The four cavity regimes, including the microcavity regime (blue), the waveguide regime (purple), the disc cavity regime (red) and the optical cavity regime (green). The four cases (a)-(d) chosen in the discussion of the detector system parameters in Figs. \ref{['Fig2']} and \ref{['Fig3']} are sketched and marked with crosses. The dimensionless cavity radius $R/\sigma$ and dimensionless cavity length $L/\sigma$ are scaled on the left and lower axis. The maximal number of modes necessary for convergence is scaled on the upper axis and the right axis for $l$ and $m$, respectively. For the smallest cavity length $L/\sigma = 20$, already longitudinal modes with $\max(l) = 10^2$ suffice, while for the maximum $L/\sigma = 10^3$ longitudinal modes up to $\max(l) = 4\cdot 10^3$ have to be chosen to guarantee convergence. For the cavity radii stronger convergence is obtained with transversal modes up to $\max(m) = 10$ sufficient for $R/\sigma = 5$, while $\max(m) = 10^3$ is needed for $R/\sigma = 500$.
• Figure 3: Variation of local correlations $\mathcal{L}$ and non-local correlations $|\mathcal{M}|$ for $D/\sigma = 5$, $\Omega T = 1$ and different detector interaction times in the timelike $t_{BA}/T = 3$ and spacelike regime $t_{BA}/T = 0.5$. In (a) we plot $R/\sigma$ on the $x$-axis while the color gradient scales with $L/\sigma$. The upper and lower curve for each $\mathcal{L}$ and $|\mathcal{M}|$ are for the minimal cavity length $L/\sigma= 10$ and the maximal length $L/\sigma = 10^3$, respectively. The color gradient in between gives the stability of the respective correlations, i.e., for large opacity strong variation in $L/\sigma$ gives little variation in correlations and vice versa for small opacity. In (b) we scale $R/\sigma$ in the interval from $R/\sigma = 5$ to $R/\sigma = 10^3$ on the $x$-axis and the color gradient is scaled with $L/\sigma$ in the same interval.
• Figure 4: Negativity estimator $\mathcal{N}$ for $D/ \sigma = 5$ in the timelike regime, $t_{BA}/T = 3$ (a), and the spacelike regime, $t_{BA}/T = 0.5$ (b). We choose different detector energies with $\Omega T= 1$ (a.1, b.1) and $\Omega T = 3$ in (a.2, b.2). To cover all the cavity regimes logarithmic axis are chosen. Additionally we include in (a) a zoomed plot showing solely the microcavity regime.
• Figure 5: Negativity estimator $\mathcal{N}$ for the four different cavity regimes. Each row resembles a different cavity regime: micro cavity ($L/\sigma = 20,\,R/\sigma = 10$) in (a), waveguide ($L/\sigma = 10^3,\,R/\sigma = 10$) in (b), disc cavity ($L/\sigma = 20,\,R/\sigma = 500$) in (c) and a optical cavity ($L/\sigma = 10^3,\,R/\sigma = 500$) in (d). In the first column the dimensionless equivalent of the detector distance $D/\sigma$ is varied with respect to the dimensionless separation time $t_{BA}/T$ of the detectors. Here we choose for the detectors energy gap $\Omega/T = 1$. In the second column $t_{BA}/T$ is varied for different $\Omega/T$. To obtain maximal negativity we choose the minimal detector separation of $D/\sigma = 5$. In the third column different $D/\sigma$ for different $\Omega/T$ are investigated for $t_{BA}/T = 2.5$. The red line marks the separation of timelike and spacelike regime.