Spontaneous quantum emission from analog white holes in a nonlinear optical medium

We use a microscopic quantum optical model to compute the spectrum of quantum vacuum emission from strong laser pulses propagating in nonlinear optical media. Similarities and differences with respect to the emission of analog white holes as predicted by quantum field theory in curved spacetime are highlighted. Conceptual issues related to the role played by the material dispersion and to the presence or absence of the horizon are clarified. Critical comparison with available experimental data is made.

Figure 1

Top panel: Sketch of the refractive index profile for a right-going pulse generating an analog white-hole horizon at $x=0$. Bottom row: single-resonance dispersion relation seen from the comoving reference frame in the external (left) and internal (right) regions. Solid (dashed) curves represent positive (negative) norm branches. Parameters of the single resonance: ${\Omega }_0={\Omega }_2$, ${\beta }_0={\beta }_2$, $\epsilon =0.3$, and $v=0.83c$.

Figure 2

Emission spectrum from an analog white hole in a nonpolar dielectric such as diamond. Left panel: comoving frame emission spectrum for the two positive norm outgoing modes: The solid (dashed) line corresponds to the mode indicated by an open (filled) dot on the dispersion shown in Fig. 1. Right panel: total emission spectrum in the laboratory frame. The arrow indicates the wavelength at which the phase velocity equals the pulse velocity, i.e., $\omega =0$. Same parameters as in Fig. 1.

Figure 3

Upper panels: Optical branch (${\Omega }_1<\Omega <{\Omega }_2$) of the dispersion relation of fused silica [Eq. (1)] as seen from the comoving frame. The left (right) panels refer to the external (internal) region of an analog white hole. Solid (dashed) curves represent positive (negative) norm branches. Parameters: $v=0.66c$ and $\epsilon =0.3$, giving a strong refractive index modulation $\delta n\approx 0.12$ in the visible range. Lower left panel: Emission spectrum into the optical branch as seen from the comoving frame. The solid (dashed) line refers to the emission into the positive norm modes indicated by open (filled) dots in the dispersion shown in the upper left panel. The emission on the other upper and lower branch is much weaker. Bottom right panel: Total emission spectrum in the laboratory frame. The arrow indicates the wavelength ${\Lambda }_0$ at which the phase velocity equals the pulse velocity, i.e., $\omega =0$.

Figure 4

Optical branch (${\Omega }_1<\Omega <{\Omega }_2$) of the dispersion relation of fused silica [Eq. (1)] as seen from the comoving frame for a Bessel pulse with $v=0.6885c=v_{\delta =6.7^{\circ }}$ and small refractive index modulation of $\delta n\approx 0.0016$. The thick (thin) lines refer to the external (internal) region of an analog white hole. Solid (dashed) curves represent positive (negative) norm branches.

Figure 5

Solid line: Emission spectrum in the comoving frame into the positive norm mode indicated by the dot in the dispersion relation of Fig. 4, for a horizonless geometry formed by a Bessel pulse with $v=0.6885c=v_{\delta =6.7^{\circ }}$. Dashed line: Sum of the emission spectra into the positive norm mode indicated by the closed and open dots in the dispersion relation of the upper left panel of Fig. 3 for a geometry with white-hole horizon formed by a Gaussian pulse with $v=v_0\approx 0.684c$. In both configurations $\delta n\approx 0.0016$, corresponding to an input energy of $E=1280\mu \text{J}$ in the experiment of Ref. [4].

Figure 6

Laboratory frame emission spectra in the case of a weak refractive index modulation obtained with a 1055-nm laser pulse. Upper panels: Gaussian pulse propagating at $v=v_0\approx 0.684c$ (left) and Bessel pulse propagating at $v=0.6885c=v_{\delta =6.7^{\circ }}$ (right). The different curves refer to different values of the refractive index modulation $\delta n\approx 1.6,1.3,0.88,0.33\times {10}^{-3}$ (solid, dashed, dotted, dot-dashed line), corresponding to input energies of $E=1280,1080,720,270\mu \text{J}$ in the experiment of Ref. [4]. Lower panels: fixed value of $\delta n=1.6\times {10}^{-3}$ and different pulse velocities. Left: $v=0.684c,0.685c,0.686c$ (solid, dashed, dotted line, corresponding to $\delta =0$ Gaussian and $\delta =3,5$ Bessel pulses). Right: $v=0.688c,0.689c,0.690c$ (solid, dashed, dotted line, corresponding to $\delta =6,7,8$ Bessel pulses). In each panel, the arrows indicate the wavelength ${\Lambda }_0$ at which the phase velocity equals the pulse velocity. The vertical lines indicate the wavelengths ${\Lambda }_l^{\left(i\right)}$ at which resonant radiation can occur [26]. The line style coding follows the one of the spectra. In the upper panels, ${\Lambda }_0$ and ${\Lambda }_l^{\left(i\right)}$ are the same for all curves. In the left upper panel, ${\Lambda }_0=497$ nm and the corresponding arrow falls outside the field of view.

Figure 7

Graphical representation of the Sellmeier dispersion relation, as seen from the laboratory reference frame $(\Omega ,K)$, for $x<0$ (left panels) and $x>0$ (right panels). The $(\omega ,k)$ axes of the comoving reference frame are obtained through a boost of velocity $v$. As sketched in the top panel, the refractive index in the right region is larger than in the left region. This difference in the refractive index is obtained by properly changing the parameters ${\beta }_i$ and ${\Omega }_i$ appearing in the Lagrangian Eq. (A1). In this plot the values of the velocity $v$ and of the refractive index change $\delta n$ have been arbitrarily chosen for illustrative purposes. The bottom panels are enlargements of the gray dot-bordered squared of the respective upper panels. The dispersion relation is graphically solved for a fixed comoving frequency $\omega$, chosen in the frequency window in which the black-hole horizon is present. Solutions appear both on the positive norm positive-$\Omega$ branches (solid curves) and on the negative norm negative-$\Omega$ branches (dashed curves). The empty dots denote solutions on the optical branches with positive (o) and negative ($\tilde{\mathrm{o}}$) frequency $\Omega$. The arrows indicate the direction of propagation (group velocity in the comoving frame) of the associated modes $V_{\omega ,L/R}^{\alpha /\tilde{\alpha }}$. In the right region (right panels), the dispersion relation has only six real-$k$ solutions. In the left region (left panels), the number of real-$k$ solutions is 8, and the two extra solutions are one left-going ($V_{\omega ,L}^{\mathrm{o}2}$) and one right-going ($V_{\omega ,L}$).