Ensemble-Averaged Quantum Correlations between Path-Entangled Photons Undergoing Anderson Localization

We measure ensemble-averaged quantum correlations of path-entangled photons, propagating in a disordered lattice and undergoing Anderson localization. These result in intriguing patterns, which show that quantum interference leads to unexpected dependencies of the location of one particle on the location of the other. These correlations are shared between localized and nonlocalized components of the two-photon wave function, and, moreover, yield information regarding the nature of the disorder itself. Such effects cannot be reproduced with classical waves, and are undetectable without ensemble averaging.

Figure 1

Propagation of light in a disordered lattice. (a) Schematic of a disordered evanescently coupled waveguide array. Light is inserted into one or more input sites, tunneling between waveguides along propagation, and detected at the output. For a full description of the experimental setup, see Supplemental Material [24]. (b)–(e) Experimentally obtained particle density distributions after propagation, with light input at site 0. The area around the origin is highlighted in light green, while the lobe regions are highlighted in darker blue. (b) No disorder ($\mathrm{\Delta }=0$), single realization—a perfectly periodic lattice displaying discrete diffraction. (c) Moderate disorder ($\mathrm{\Delta }=0.4C$), single realization. The discrete diffraction pattern is lost, and the output does not show any discernible structure. (d) Moderate disorder ($\mathrm{\Delta }=0.4C$), averaged over 30 realizations. The light is now partially localized, leading to approximately 50% staying near the origin and 50% reaching the lobes. This is the regime in which the quantum correlation measurements were performed. (e) Stronger disorder ($\mathrm{\Delta }=0.67C$), averaged over 30 realizations. Most of the light localizes in the center, while the lobes nearly disappear.

Figure 2

Measured correlation matrices of single realizations. (a) Experimentally obtained correlation matrix (${\mathrm{\Gamma }}_{q,r}$) for the input state $\frac{1}{2}(a_0^{{†}^2}+a_1^{{†}^2})|0⟩$, measured in a single realization of an ordered lattice. Antibunching between the lobes is evident. (b) The same matrix measured in a single realization of a disordered lattice. No clear pattern is apparent, demonstrating the need for ensemble averaging to obtain meaningful results.

Figure 3

Measured correlation matrices for path-entangled photons propagating in a disordered lattice, for an input state of $\frac{1}{2}(a_0^{{†}^2}+e^{i\varphi }{a}_1^{{†}^2})|0⟩$. (a) Measured and (b) simulated correlation matrix ${\mathrm{\Gamma }}_{q,r}$ for $\varphi =0$. Compare with data of a single realization presented in Fig 2. (c) Measured (blue, solid line) and simulated (red markers) mean interparticle distance for $\varphi =0$. (d) Measured and (e) simulated correlation matrix ${\mathrm{\Gamma }}_{q,r}$ for $\varphi =\pi$. (f) Measured (blue, solid line) and simulated (red markers) mean interparticle distance for $\varphi =0$. Experiments are averaged over 12 realizations, while simulations were performed using 1000 realizations of disorder. Error bars of the measured mean distance plots (c),(f) are smaller than the data point markers. The simulated data in (c),(f) represent the expected results of a 12-realization set; the error bars are derived by taking 83 simulated 12-realization sets drawn from the initial 1000, and calculating the standard deviation among the results from those 83 sets.