Quantum and Classical Correlations in Waveguide Lattices

We study quantum and classical Hanbury Brown–Twiss correlations in waveguide lattices. We develop a theory for the propagation of photon pairs in the lattice, predicting the emergence of nontrivial quantum interferences unique to lattice systems. Experimentally, we observe the classical counterpart of these interferences using intensity-correlation measurements. We discuss the correspondence between the classical and quantum correlations, and consider path-entangled input states which do not have a classical analogue. Our results demonstrate that waveguide lattices can be used as a robust and highly controllable tool for manipulating quantum states, and offer new ways of studying the quantum properties of light.

Figure 1

(a) A schematic view of the waveguide lattice used in the experiment. The red arrow represents the input light beam. (b) The calculated probability distribution $⟨n_q(z)⟩$ of a single photon injected into the central waveguide of a periodic lattice, as a function of the propagation distance. The photon couples coherently from each waveguide to its neighbors, and the probability distribution concentrates at two outer lobes. (c) The calculated correlation matrix ${\Gamma }_{q,r}$, representing the probability to detect at the output of the lattice exactly one photon at waveguide $r$ and one photon at waveguide $q$, when both photons are coupled to a single waveguide at the center of the lattice, i.e., $|{\phi }_0⟩\equiv 1/\sqrt{2}{a}_0^{†2}|0⟩$. This matrix is a simple product of two single-photon distributions, thus showing no quantum interference. The grey bars are obtained by summing the matrix along one axis, representing the results of a single photon measurement.

Figure 2

Quantum and classical correlations in waveguide lattices. (a) The correlation matrix ${\Gamma }_{q,r}$ when the photons are coupled to two adjacent waveguides, i.e., $|{\phi }_1⟩\equiv a_0^{†}{a}_1^{†}|0⟩$. The two photons exhibit bunching, and will emerge from the same side of the lattice. (b) The correlation matrix when the two photons are coupled to two waveguides separated by one waveguide, $|{\phi }_2⟩\equiv a_{-1}^{†}{a}_1^{†}|0⟩$. Here the two photons will both emerge either from the lobes or from the center. (c) Measured classical intensity correlations ${\Gamma }_{q,r}^{(c)}$, corresponding to the $|{\phi }_1⟩$ input state. (d) Measured classical intensity correlations ${\Gamma }_{q,r}^{(c)}$, corresponding to the $|{\phi }_2⟩$ input state.

Figure 3

Quantum correlation maps ${\Gamma }_{q,r}$ for path-entangled input states. (a) The two photons are injected together to either of two neighboring waveguides, $|{\psi }^{(+)}⟩=\frac{1}{2}(a_1^{†2}+a_0^{†2})|0⟩$. The correlation is significant only in the off-diagonal peaks, which indicates that the two photons will emerge from opposite sides of the lattice. (b) The correlation map for an input state with two photons in either of two waveguides with one waveguide separation, where there is a $\pi$ phase between the two possibilities, $|{\psi }^{(-)}⟩=\frac{1}{2}(a_1^{†2}-a_{-1}^{†2})|0⟩$.