Discretely observable continuous-time quantum walks on Möbius strips and other exotic structures in three-dimensional integrated photonics

We theoretically analyze the dynamical evolution of photonic quantum walks on Möbius strips and other exotic structures in three-dimensional integrated photonics. Our flexible design allows discrete observations of continuous-time quantum walks of photons in a variety of waveguide arrays. Furthermore, our design allows one to inject photons during the evolution, allowing the possibility of interacting with the photons as they are “walking.” We find that nontrivial array topologies introduce time-dependent symmetries of the two-photon correlations. These properties allow a large degree of control for quantum state engineering of multimode entangled states in these devices.

Figure 1

(a) General schematic of experimental proposal. Photons are injected into directional couplers (denoted by $\widehat{B}$), which couple into a waveguide array and evolve under a local Hamiltonian $\widehat{H}$ for time $\tau$. Due to the geometry of the array, the waveguides then undergo a permutation denoted by $\widehat{P}$. After the permutation the photons enter the directional couplers again and either exit the device to be detected or remain in the waveguide array for another loop. (b)–(d) Three specific implementation of the proposal: (b) cylindrical array, (c) Möbius strip waveguide array, and (d) twisted circular waveguide array. For clarity, the directional coupler array on the twisted circular array has been omitted.

Figure 2

Two-photon correlation functions ${\Gamma }_{r,s}^{1,7}\left(n\tau \right)$ [Eq. (12)] for cylindrical array (top row) (a) $n=0$, (b) $n=1$, (c) $n=2$, (d) $n=3$, and for the Möbius strip array (bottom row) (e) $n=0$, (f) $n=1$, (g) $n=2$, (h) $n=3$. Here we set $\tau =1$, measure time in units of $g^{-1}$, and for clarity we rescale the vertical axis by ${\cos }^{4(n-1)}\left(\theta \right){\sin }^4\left(\theta \right)$. Note the mirror flipping of the photon statistics at odd time steps in the Möbius strip array on the bottom row.

Figure 3

Two-photon correlation functions ${\Gamma }_{r,s}^{1,7}(n\tau +n_d\tau )$ [Eq. (14)] for cylindrical array with delayed two-photon input. (Top row) $n_d=0$ (simultaneous two-photon input) (a) $n=0$, (b) $n=1$, (c) $n=2$, (d) $n=3$. (Middle row) $n_d=1$ (input the second photon after one traversal) (e) $n=0$, (f) $n=1$, (g) $n=2$, (h) $n=3$. (Bottom row) $n_d=2$ (input the second photon after two traversals) (i) $n=0$, (j) $n=1$, (k) $n=2$, (l) $n=3$. Here we set $\tau =1$ and measure time in units of $g^{-1}$ and for clarity we rescale the vertical axis by ${\cos }^{4(n-1)}\left(\theta \right){\sin }^4\left(\theta \right)$.

Figure 4

Two-photon correlation functions ${\Gamma }_{r,s}^{1,7}\left(n\tau \right)$ [Eq. (12)] for circular array with no twists (top row) (a) $n=0$, (b) $n=1$, (c) $n=2$, (d) $n=3$, and for the twisted circular array (bottom row) (e) $n=0$, (f) $n=1$, (g) $n=2$, (h) $n=3$. For clarity we rescale the vertical axis by ${\cos }^{4(n-1)}\left(\theta \right){\sin }^4\left(\theta \right)$. On the twisted array (bottom row) there is a $4\times 2\pi /12$ rotation each loop.