Observation of spatial quantum correlations induced by multiple scattering of non-classical light
We present the experimental realization of spatial quantum correlations of photons that are induced by multiple scattering of squeezed light. The quantum correlation relates photons propagating along two different light trajectories through the random medium and is infinite in range. Both positive and negative spatial quantum correlations are observed when varying the quantum state incident to the multiple scattering medium, and the magnitude of the correlations is controlled by the number of photons. The experimental results are in excellent agreement with recent theoretical proposals by implementing the full quantum model of multiple scattering.
Multiple scattering of light waves propagating through a disordered medium can be described as a random walk of the direction of propagation (1), whereby the medium becomes opaque (2). Such random scattering determines the performance of nanophotonic devices based on, e.g., photonic crystals (3) or surface plasmon polaritons (4), and can be applied for enhancing the information capacity for optical communication (5). In multiple scattering, the different possible light trajectories through the random medium will interfere and a complex spatial intensity distribution of light is generated, cf. Fig. 1(b). Usually interference effects tend to vanish after averaging over all possible realizations of the disorder and the light transport is described by diffusion theory. Under conditions where multiple scattering is strong, light diffusion is modified leading to spatial correlations such as mesoscopic fluctuations (6); (7); (8). These correlations imply that the light intensity observed at one position after the medium depends on the intensity at a different position even after averaging over all ensembles of disorder.
The significance of the quantum nature of light in a multiple scattering context has only been addressed recently (9); (10); (11); (12); (13); (14). Pioneering work has provided the theoretical framework for a quantum optical description of multiple scattering (9) that was used to predict the existence of spatial quantum correlations induced by multiple scattering of non-classical light (11). These quantum correlations exist even in the diffusive regime of multiple scattering and add to the traditional mesoscopic correlations that are of classical origin (6). Here we present the experimental demonstration of spatial quantum correlations induced by multiple scattering of squeezed light.
Due to the Heisenberg uncertainty principle intrinsic quantum fluctuations are always present. Although always of inherent quantum origin, photon fluctuations are classified as either classical or non-classical if similar fluctuations can be induced by classical light sources or not. The classical boundary corresponds to Poissonian photon statistics where the variance of the photon number fluctuations equals the mean number of photons. A purely quantum regime exists where photon fluctuations are reduced beyond the classical limit leading to sub-Poissonian photon statistics.
The number of photons exiting a multiple scattering medium in a specific direction can be correlated or anti-correlated with the number of photons in another direction, and this spatial correlation depends on the quantum state of light illuminating the medium, c.f. Fig 1(a). The degree of correlation is quantified by the spatial quantum correlation function (15)
where labels the direction of the incident light and and the two different output directions. The operator represents the number of photons in output mode . denotes the quantum mechanical expectation value while the bars refer to an average over all realizations of disorder. The need for two averages is due to the fact that quantum fluctuations and random multiple scattering are both stochastic processes. is the quantum optical generalization of the classical intensity correlation function that has been intensely investigated in multiple scattering experiments (6). Note that the quantum correlations are induced despite the fact that the response of the multiple scattering medium is fully linear. Thus, a non-classical resource combined with linear optics is quite generally sufficient for many quantum optics experiments, and even enable quantum computing (16).
The spatial quantum correlation function defined in Eq. (1) depends sensitively on the number and fluctuations of photons incident on the sample through direction . Restricting to a theory for discrete spatial and temporal degrees of freedom, it is given by
is the average number of incident photons and the Fano factor gauges the variance in the photon number fluctuations specified within the spectral bandwidth relevant for the measurement. and are classical spatial correlation functions (6) that can be neglected in the diffusive regime of multiple scattering that is of concern here, but it would certainly be exciting to study their quantum optical equivalents as well. The size of the Fano factor determines the transition from the classical to the non-classical regime Using non-classical light opens the door to a genuine quantum regime where corresponding to spatially anti-correlated photons. Such spatial quantum correlations are of infinite range in the sense that the magnitude is independent of the angular difference of the two output directions, which translates into a spatial separation in the far field. This is an example of the fundamentally new phenomena that arise in quantum optical descriptions of multiple light scattering. So far, quantum optical variables have been measured in very few multiple scattering experiments that only have focused on the regime of classical photon fluctuations (12); (13), while spatial quantum correlations were not demonstrated.
which holds in the diffusive regime of multiple scattering where the spatial correlation function is independent of the output direction . A similar equation applies to the case of total reflection measurements. Thus, the spatial quantum correlation function is directly accessible by measuring in addition to the photon fluctuations also the average number of photons entering the sample and the ratio of the transport mean free path to the sample thickness Note that the latter ratio is modified by so-called extrapolation lengths in order to account for the effects of the interface of the multiple scattering medium (17). In the experiment we extract the spatial correlation function in the frequency domain, which has dimensions of an inverse bandwidth. The complete continuous mode theory will be described elsewhere (18).
As a non-classical resource we use squeezed light generated in a second-order nonlinear process at a wavelength of (19). This versatile source features continuous tuning between classical (super-Poissonian) and non-classical (sub-Poissonian) photon fluctuations by varying the relative phase between the squeezed vacuum state generated in the nonlinear process and a bright displacement beam. We are able to reduce photon fluctuations () below the classical limit and induce excess fluctuations above the classical limit . The multiple scattering experiment is conducted by focusing the non-classical light source onto either the front surface or the back surface of the sample, cf. Fig. 2(a), to perform total transmission and total reflection measurements, respectively. We record the total number of photons and the photon fluctuations transmitted through or reflected from the sample. The multiple scattering samples consist of TiO that has been grounded into strongly scattering particles with a typical size of . A range of samples with different thicknesses has been fabricated, and the total transmission of each sample was recorded with an integrating sphere. From these measurements the transport mean free path is found to be and all samples are in the diffusive regime where
Figure 2(b) shows the measured photon fluctuations, recorded in reflection, for squeezed light illumination of the multiple scattering medium. The classical limit was recorded by blocking the squeezed beam. The phase of the displacement beam is scanned in order to continuously tune the photon fluctuations of the incident light below and above the classical limit. We record photon fluctuations that are reduced below the classical limit after the photons have been multiple scattered, which corresponds to a Fano factor for the total reflection of . Correcting for the overall detection efficiency of , we infer for light exiting the multiple scattering sample. This reduction is the direct experimental proof that non-classical properties of light survive the complex stochastic process of multiple scattering even after ensemble averaging over realizations of disorder despite the common belief that quantum properties of light are fragile. We calculate the average number of scattering events in the multiple scattering process (20), which is given by . For the particular sample used for the measurements of Fig. 2(b) we get , including effects of the interface, which quantifies the complexity of the multiple scattering process. The observed reduction in the photon fluctuations shows that the information capacity associated with the multiple scattering channels can be enhanced beyond the classical limit, as predicted theoretically (10).
We have carried out a detailed investigation of the transport of non-classical and classical photon fluctuations through the multiple scattering medium for a range of different sample thicknesses. Figure 3 displays the detected photon fluctuations after multiple scattering relative to the fluctuations associated with the classical limit, which are plotted versus the sample thickness. Using light sources with classical fluctuations, the multiple scattered light always displays excess photon fluctuations corresponding to the classical regime (Fig. 3(a)). Non-classical light allows entering the quantum regime where the photon fluctuations are reduced below the classical limit, see Fig. 3(b). Our experimental results can be compared to the predictions from the full quantum theory for multiple scattering of photons (11); (14). Excellent agreement between experiment and theory is apparent from Fig. 3 both in the classical and quantum regime. It should be stressed that the comparison to theory requires no adjustable parameters, and only depends on the measured photon fluctuations of the incident light source, the intensity transmission coefficient, and the total collection efficiency of the setup.
The theory predicts that the multiple scattered photon fluctuations depend on two terms varying as and , respectively, cf. Eq. (3). The latter term is only observable for non vanishing spatial quantum correlations, i.e. the comparison between experiment and theory allows extracting the spatial quantum correlation function, cf. Fig. 4. We observe negative (positive) spatial correlations in the case where the transmitted photon fluctuations are in the quantum (classical) regime. According to Eq. (2) the strength of the spatial quantum correlation is expected to increase when reducing the number of photons of the incident light. This pronounced behavior is clearly demonstrated in Fig. 4(a) and is in excellent agreement with theory. Controlling the power of the non-classical light source thus provides an efficient way of tuning the strength of the spatial quantum correlations. Furthermore, the spatial correlation function is predicted to be independent of the sample thickness, which holds in the diffusive regime of multiple scattering. This behavior is experimentally confirmed as well, see Fig. 4(b).
We presented the first experimental demonstration that photon fluctuations below the classical limit can be observed after multiple scattering using squeezed light as non-classical resource. Multiple scattering was found to induce infinite range spatial correlations that are of purely quantum origin. Our experimental results were in excellent agreement with the full quantum optics theory for multiple scattered light. We believe that our results will inspire further explorations of the multitude of new phenomena encoded in the quantum optical properties of multiple scattered light.
We thank Jirí Janousek for help with the non-classical light source and Elbert G. van Putten, Ivo M. Vellekoop, and Allard P. Mosk for providing the samples. We gratefully acknowledge the Danish Research Agency for financial support (project FNU 645-06-0503). A.H. and U.L.A. acknowledge support from the Danish Research Agency (project FTP 274-07-0509) and the EU project COMPAS for construction of the squeezed light source.
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