Quantum vacuum emission from a refractive-index front

A moving boundary separating two otherwise homogeneous regions of a dielectric is known to emit radiation from the quantum vacuum. An analytical framework based on the Hopfield model, describing a moving refractive-index step in $1+1$ dimensions for realistic dispersive media has been developed by S. Finazzi and I. Carusotto [Phys. Rev. A 87, 023803 (2013)]. We expand the use of this model to calculate explicitly spectra of all modes of positive and negative norms. Furthermore, for lower step heights we obtain a unique set of mode configurations encompassing black-hole and white-hole setups. This leads to a realistic emission spectrum featuring black-hole and white-hole emission for different frequencies. We also present spectra as measured in the laboratory frame that include all modes, in particular a dominant negative-norm mode, which is the partner mode in any Hawking-type emission. We find that the emission spectrum is highly structured into intervals of emission with black-hole, white-hole, and no horizons. Finally, we estimate the number of photons emitted as a function of the step height and find a power law of 2.5 for low step heights.

Figure 1

Sellmeier dispersion relation, Eq. (2), with three resonances in the laboratory frame. There are eight branches (black lines). A contour of ${\omega }^{\prime }$ is shown in blue (dark gray). Their intersection points indicate the modes of propagation in the medium (red circles).

Figure 2

Sketch of the RIF in the moving frame: there are two homogeneous regions of uniform refractive index on the left and right of a dielectric boundary of height $\delta n$. For suitable moving-frame frequencies, light on the left (right) of the RIF may not (may) propagate away to the right as the step moves at superluminal (subluminal) speeds. Light on the left is trapped behind an analog horizon at $\zeta =0$.

Figure 3

Sellmeier dispersion relation of fused silica in a frame moving at a velocity $u=0.66c$. Part of the optical branch is shown: branches with positive (negative) laboratory frequencies are represented by thick (thin) curves. A curve for zero refractive-index change $\delta n$ is shown in black, and that for a high change, $\delta n=0.02$, is in orange. Frequency intervals corresponding to black- and white-hole analog horizons are shaded in orange $\left([{\omega }_{\max L}^{\prime },{\omega }_{\max R}^{\prime }]\right)$, and blue $\left([{\omega }_{\min L}^{\prime },{\omega }_{\min R}^{\prime }]\right)$, respectively.

Figure 4

Diagrammatic explanation of the possible mode configurations for positive-frequency optical modes for various comoving frequencies. Configurations change if two of the modes share their group velocity with the RIF (green lines), which occurs at the frequencies indicated above and in Fig. 3.

Figure 5

Spectrum of emission into the uniquely right propagating mode moR on the right side of the RIF. Emission is calculated in the moving frame of velocity $u=0.66c$. The number of particles per time and bandwidth, the flux density, is displayed for increasing values of $\delta n$ ($1\times {10}^{-3}$, $2\times {10}^{-3}$, $4\times {10}^{-3}$, $1\times {10}^{-2}$, $2\times {10}^{-2}$ (orange dot-dashed line), $3\times {10}^{-2}$, $4\times {10}^{-2}$, $5.6\times {10}^{-2}$). The spectrum is confined to a finite interval over which the mode moR exists.

Figure 6

Spectrum of emission as in Fig. 5. To compare the shapes of the traces, spectral densities are scaled such that all traces line up with the $\delta n=0.02$ trace (orange dot-dashed line). Values of $\delta n$ range from $4\times {10}^{-5}$ to $5.2\times {10}^{-2}$.

Figure 7

Emission spectra of each optical mode in the comoving frame ($u=0.66c$, $\delta n=0.02$): emission into mode noL, purple solid line; loL, blue dashed line; uoL, green dotted line; moR, orange dot-dashed line.

Figure 8

Total emission over the subluminal (black-hole) interval into the unique right moving mode moR. (a) Estimated photon number for different velocities $u$ and (b) size of the interval in the moving frame as a function of index change $\delta n$.

Figure 9

Emission spectral density in the laboratory frame. At each wavelength the total spectral flux density, the number of photons emitted per unit time and unit bandwidth, is the sum of contributions from all modes. Emission is concentrated in the UV in a narrow spectral peak generated from mode noR. Outside the peak the emission decreases except for spectral intervals corresponding to black- and white-hole analog horizons. Spectra are calculated for wavelengths above the violet line, beyond which there are no contributions from the uppermost dispersion branch. The red line corresponds to ${\omega }^{\prime }=0$ (phase velocity horizon). The black and orange dashed lines indicate the interval of the black-hole (white-hole) mode configuration for the moR (loL) mode at short (long) wavelengths.