Figure 1

(a) The photon density $n_{\mu }(z)$ for a single photon initiated at the waveguide ${\mu }^{\prime }=0$. The photon is mostly localized in two narrow peaks at the edges of the distribution. The path of each peak (red dashed line) follows a sinusoidal trajectory with a period ${\lambda }_B$, and marks a branch of the BO. In this example the sinusoidal trajectory has an amplitude of 24 waveguides. (b) The photon density $n_{\mu }(z)$ for a $N00N$ state input with $N=1$ coupled to waveguides ${\mu }^{\prime }=0$ and ${\nu }^{\prime }=1$. The photon is located at a single branch which oscillates around the input waveguide with a period ${\lambda }_B$. (c) The photon density $n_{\mu }(z)$ for a $N00N$ input state with $N=2$ coupled to waveguides ${\mu }^{\prime }=0$ and ${\nu }^{\prime }=1$. The photons from the ${\mu }^{\prime }=0$ and ${\nu }^{\prime }=1$ inputs add up incoherently, showing double-branch oscillations.Reuse & Permissions

Figure 2

Bloch oscillations of $N00N$ states with $N=2$ coupled to two adjacent waveguides $|\psi ⟩=\frac{1}{\sqrt{2}}(|2{⟩}_0|0{⟩}_1+e^{-i\phi }|0{⟩}_0|2{⟩}_1)$. (a) The multiple detection probability ${\Gamma }_{\mu ,\nu }^{(1,1)}$ at several propagation distances, for $\phi =0$. At the beginning of the propagation the two photons exhibit antibunching and are located at the two different branches of the oscillations. As the photons approach the turning point ($z={\lambda }_B/4$), they bunch and are found with the highest probability in the same branch. (b) Same as (a) for $\phi =\frac{\pi }{2}$. The photons show bunching-antibunching cycle, but in this case start the oscillation partially bunched. (c) Same as (a) and (b), for $\phi =\pi$. Here the photons start the bunching-antibunching cycle bunched. (d) The normalized coincidence rate ${\gamma }^{(1,1)}(z)$ as a function of the lattice length for $\phi =0$ (blue solid line), $\phi =\frac{\pi }{2}$ (green dash-dotted line), and $\phi =\pi$ (red dotted line). The coincidence rate is calculated between the positions of the central waveguide in each branch, showing oscillations with a period ${\lambda }_B/2$. See 25 for a movie visualizing the propagation of (a).Reuse & Permissions

Figure 3

Bloch oscillations of $N00N$ states with sub-${\lambda }_B/2$ correlation-oscillation periods. (a) The multiple detection probability ${\Gamma }_{\mu ,\nu }^{(1,1)}$ at several propagation distances for the input state $|\psi ⟩=\frac{1}{\sqrt{2}}(|2{⟩}_{-1}|0{⟩}_1+|0{⟩}_{-1}|2{⟩}_1)$. The photons exhibit bunching-antibunching oscillations (see text) with a period ${\lambda }_B/4$. (b),(c) The multiple detection probability ${\Gamma }_{\mu ,\nu }^{(N/2,N/2)}$ for a $N00N$ state with $N=6$ (b) and $N=10$ (c), injected to adjacent waveguides $|\psi ⟩=\frac{1}{\sqrt{2}}(|N{⟩}_0|0{⟩}_1+|0{⟩}_0|N{⟩}_1)$. The oscillations of the correlation matrix are much faster, hence the probability matrix is calculated for five lattice lengths close to the turning point $z={\lambda }_B/4$. (d) The normalized coincidence rate ${\gamma }^{(N/2,N/2)}(z)$ as a function of the lattice length for the above three cases. The period of the oscillations are ${\lambda }_B/4$ [(a), blue solid line], ${\lambda }_B/6$ [(b), green dash-dotted line], and ${\lambda }_B/10$ [(c), red dotted line]. See 25 for a movie visualizing the propagation of (a) and (b).Reuse & Permissions