Figure 1

Illustration of multiple scattering leading to spatial quantum correlations. The nonclassical light source that is used in the experiment is created by overlapping a squeezed vacuum light beam (SQZ) and a coherent light beam (C) on a beam splitter with two input ports, 1 and 2, respectively. The relative phase, $\Delta \phi$, between C and SQZ can be tuned continuously. The resultant light beam in direction $a$ is incident onto a medium with randomly distributed scatterers. Light is split into a multitude of different trajectories, and the number of photons exiting the medium in a specific direction, $b$, can be correlated with the number of photons in another direction, $b^{\prime }$, to a degree dependent on the quantum state of the illuminating light source. In the diffusive regime, the spatial quantum correlation function can be determined from total transmission quantum noise measurements. A similar experimental scheme is applied for reflection measurements.Reuse & Permissions

Figure 2

(a) Sketch of the experimental setup. MC: mode cleaner, PPKTP: periodically poled potassium titanyl phosphate nonlinear crystal as part of an optical parametric oscillator (gray shaded box), SQZ: vacuum squeezed light beam, C: coherent light beam with a variable optical phase, $\Delta \phi$, relative to the squeezed vacuum, BS: beam splitter, FM: flip mirror, S: multiple scattering sample, SA: spectrum analyzer, D: detector. The detection detector, $D2$, has a bandwidth of 315 kHz. The multiple scattered light is collected with a microscope objective featuring a numerical aperture of $\text{NA}=0.63$. In reflection geometry, the incident light illuminates the sample under an angle $\alpha ={69}^{\circ }$. A detailed explanation of the setup can be found in the main text. (b) Measured Fano factor, $F({\omega }_s,\delta \omega )$, at an electrical sideband frequency of ${\omega }_s=3.93$ MHz of the bright squeezed light source without insertion of the multiple scattering sample (black circles) depending on the phase $\Delta \phi$ of the coherent beam, as schematically shown in the inset. The measured classical limit is plotted with gray triangles and is obtained by blocking the squeezed beam. As the resolution bandwidth, we used $\delta \omega /2\pi =300$ kHz. The uncertainty in the Fano factor is estimated over 18 full periods of $\Delta \phi =0,...,36\pi$. (c) Reconstructed Wigner function of the squeezed vacuum state (SQZ) produced by the optical parametric oscillator. The Wigner function is obtained from time-resolved measurements using a homodyne setup with a 50:50 beam splitter (see inset) and quantum state reconstruction.Reuse & Permissions

Figure 3

(a) Scanning electron microscope image of the multiple scattering medium consisting of TiO${}_2$ that has been grinded into strongly scattering particles with a typical size of $200\text{nm}$ and deposited on a glass substrate. (b) Scanning of a sample at different positions with the tip of a micrometer screw to obtain the average sample thickness. The scanned positions are shown as squares and their color represents the measured thickness. The average thickness for this sample is determined to be $L=(6\pm 2)\mu$m. (c) Measured inverse total transmission through the multiple scattering samples vs sample thickness obtained using an integrating sphere. The line represents a linear fit to the data whereby the transport mean free path and the effective refractive index are obtained.Reuse & Permissions

Figure 4

Overall collection efficiency for each sample thickness as measured in transmission (${\eta }_T$, gray triangles) and reflection (${\eta }_R$, black circles). The error bars on the sample thickness are included only on the reflection data.Reuse & Permissions

Figure 5

Measured Fano factor $F_{T,R}({\omega }_s,\delta \omega )$ after multiple scattering of a squeezed state, as recorded in (a) reflection geometry and (b) transmission geometry for $L=16\mu \text{m}$ (black circles). The classical limit is depicted with gray triangles. (c) Ensemble-averaged transmitted (filled gray symbols) and reflected (empty black symbols) Fano factor depending on the incident Fano factor, $F$. The inset shows a zoom in of the data that have been taken for Fano factors smaller than $F<1$, highlighting the quantum regime of multiple scattering. The sample thicknesses are $6\mu$m (triangles) and $20\mu$m (circles), respectively. The theory expressed by Eq. (9) is shown by the straight lines, and the classical limited is given by the dashed line. The ensemble average has been performed for six different sample positions, and the result has been verified 18 times per sample position.Reuse & Permissions

Figure 6

Measured spatial quantum correlation function depending on the incident Fano factor for different sample thicknesses (blue triangles, $6\mu$m; black circles, $20\mu$m). The spatial quantum correlation function has been measured in transmission (filled symbols) and reflection (empty symbols), respectively. The solid gray line displays the theoretical prediction while the dashed gray line represents the quantum limit of $\overline{C_{b{b}^{\prime }}^Q}({\omega }_s,\delta \omega )=0$. For visibility, we only show selected data points for each measurement.Reuse & Permissions