Paraxial Theory of Direct Electro-optic Sampling of the Quantum Vacuum

Direct detection of vacuum fluctuations and analysis of subcycle quantum properties of the electric field are explored by a paraxial quantum theory of ultrafast electro-optic sampling. The feasibility of such experiments is demonstrated by realistic calculations adopting a thin ZnTe electro-optic crystal and stable few-femtosecond laser pulses. We show that nonlinear mixing of a short near-infrared probe pulse with the multiterahertz vacuum field leads to an increase of the signal variance with respect to the shot noise level. The vacuum contribution increases significantly for appropriate length of the nonlinear crystal, short pulse duration, tight focusing, and a sufficiently large number of photons per probe pulse. If the vacuum input is squeezed, the signal variance depends on the probe delay. Temporal positions with a noise level below the pure vacuum may be traced with subcycle resolution.

Figure 1

Setup and geometry for free-space electro-optic sampling. (a) The incoming near-infrared (NIR) probe and multi-THz signal fields mix in the electro-optic crystal (EOX). The NIR (blue) spatial mode amplitude is depicted by the contour plot, whereas a THz (red) spatial mode is indicated by wave fronts. Bottom left corner: temporal profiles of the NIR intensity envelope $I_{\mathrm{NIR}}(t)$ and a representative multi-THz vacuum field $E_{\mathrm{THz}}(t)$. After collimating with a lens ($L$), the modified NIR field is analyzed using a quarter-wave plate ${(\lambda /4)}_a$, a Wollaston prism (WP), and balanced detectors ($D_s$, $D_z$) measuring the difference in photon flux of the split components. (b) Spatial directions determining the electro-optic effect in a zinc-blende-type EOX and the following ellipsometry analysis.

Figure 2

(a) Calculated integrand function $\mathrm{\Omega }(n/n_{\mathrm{\Omega }})|R(\mathrm{\Omega }){|}^2$ entering Eq. (13). (b) Double-logarithmic plot of the ratio $\mathrm{\Delta }\mathcal{S}/N$ in dependence on $N$. The black dotted (red dashed) line shows the bare SN (multi-THz vacuum) contribution. (c) Increase of $(\mathrm{\Delta }\mathcal{S}-\mathrm{\Delta }{\mathcal{S}}_{\mathrm{SN}})/\mathrm{\Delta }{\mathcal{S}}_{\mathrm{SN}}$ with $N$. Parameters are defined in the main text.

Figure 3

(a) Frequency dependence of the squeeze factor $r$. Squeezing correlates $\mathrm{\Omega }$ and $2{\mathrm{\Omega }}_c-\mathrm{\Omega }$ modes (as indicated by arrows). (b) Error contour in the complex-amplitude plane for PV (gray circle) and SV with $\theta =0$ ($\theta =\pi$) [red (green) ellipse with reduced uncertainty in the phase (amplitude) quadrature $Y$ ($X$)]. (c) Normalized (with respect to the constant PV level, solid black line) EOS variance in dependence on the time delay $\tau$ of the probe NIR pulse for SV with $\theta =0$ ($\theta =\pi$) [solid red line (dashed green line)].