Figure 1

(a) Unidimensional quantum walk: depending on the result of the coin toss the walker moves upward ($U$) or downward ($D$). (b) Scheme of an array of beam splitters (BSs) for a four-step quantum walk. Vertical dashed lines indicate the steps $T$ of the quantum walk and the horizontal stripes represent the position $|j⟩$ of the walker. In an array with an even (odd) number of steps the output ports $J$ are grouped into the even (odd) final positions $|j⟩$ of the walker. (c) Different behaviors of bosons and fermions on a BS.Reuse & Permissions

Figure 2

Integrated optical circuits. (a) Measured intensity profile for the guided modes with polarization $V$ and $H$, at 806 nm wavelength. The $1/e^2$ dimensions are reported. (b) Ratio of the estimated coupling coefficient for polarization $V$ ($C_V)$ and polarization $H$ ($C_H$) in directional couplers fabricated with different angles $\theta$ between the waveguides (see inset), but fixed interaction length (3 mm) and distance ($11\mu \mathrm{m}$). The fitting line is a guide to the eye. (c) Schematic of the network of directional couplers fabricated for implementing a four-step quantum walk. The color coding indicates the writing depth of the waveguides, which is varying from point to point. (d) 3D representation of the basic cell of the network, which acts as a Mach-Zehnder interferometer.Reuse & Permissions

Figure 3

The experimental setup can be divided into three parts. (i) The source: polarization entangled photon pairs at wavelength $\lambda =806\mathrm{nm}$ generated via spontaneous parametric down conversion in a 1.5 mm $\beta$-barium borate crystal (BBO) cut for type-II noncollinear phase matching, pumped by a cw diode laser with power $P=50\mathrm{mW}$ [35]. Wave plates (WPs) allow generation of any single photon state and the Bell states. A delay line is inserted to control the temporal superposition of the photons, which are injected into the integrated device through single-mode fibers (SMFs). Interference filters determine the photon bandwidth $\Delta \lambda =6\mathrm{nm}$. (ii) Integrated quantum walk circuit realized by ultrafast laser writing technique (see inset on the left). (iii) Measuring apparatus: the chip output is divided by a beam splitter (BS) and magnified through a set of two lenses. The photons coupled to multimode fibers (MMFs) are then delivered to single photon counting modules. The MMFs are mounted on motorized translation stages in order to select an arbitrary combination of two output ports and measure two-photon coincidences. Polarization controllers (PCs) are used before the chip to compensate for the polarization rotations induced by the fibers.Reuse & Permissions

Figure 4

Two-particle quantum walks: ideal (left) and measured (right) distributions of (a) bosonic, (b) fermionic and (c) anyonic (with $\varphi =\pi /2$) quantum walks.Reuse & Permissions