Automated quantum operations in photonic qutrits

We report an experimental implementation of automated state transformations on spatial photonic qutrits following the theoretical proposal made by Baldijão et al. [Phys. Rev. A 96, 032329 (2017)]. A qutrit state is simulated by using three Gaussian beams, and after some state operations, the transformed state is available in the end in terms of the basis state. The state transformation setup uses a spatial light modulator and a calcite-based interferometer. The results reveal the usefulness of the operation method. The experimental data show a good agreement with theoretical predictions, opening possibilities for explorations in higher dimensions and in a wide range of applications. This is a necessary step in qualifying spatial photonic qudits as a competitive setup for experimental research in the implementation of quantum algorithms which demand a large number of steps.

Figure 1

Experimental setup. State preparation consists of wave plates, calcite $\widehat{x}$-beam displacers, and a polarizer, in order to prepare $\left|\mathrm{\Psi }\right.〉$ shown in (1). The transformation apparatus consists in actuating in each path with a specific $\widehat{y}$-diffraction grating, using the programmable phase-only reflection SLM (Hamamatsu, model LCOS-SLM X1046801), and a sequence of wave plates, calcite $\widehat{x}$-beam displacers, PBS, and spatial filters in order to recombine the beam transformed as $\left|{\mathrm{\Psi }}^{\prime }\right.〉$ shown in (2). In the right-hand panel, the effect of the spatial filter is highlighted, with each of the three $\widehat{x}$ paths generating three $\widehat{y}$ orders of diffraction. After a collimating lens, the final readout is accomplished by a CCD.

Figure 2

Integrated transverse optical profile (ITOP) of resulting states after performing the operations. (a)–(c) Projections onto computational basis states ${\left(0,1,0\right)}^T$, ${\left(0,0,1\right)}^T$, and ${\left(0,0,1\right)}^T$, respectively. The initial state is fixed as $\left|{\psi }_i\right.〉={\left|1\right.〉}_x$. The continuous red line represents a Gaussian fit. In (d) and (e), the black circles register the projection onto ${\left(1,0,1\right)}^T$ for the initial states (d) $\left|{\psi }_i\right.〉={\left|0\right.〉}_x$ and (e) $\left|{\psi }_i\right.〉\propto {\left|0\right.〉}_x+{\left|1\right.〉}_x$, while the continuous red line represents a two-Gaussian fit. In (f) and (g), projection onto ${\left(1,0,-1\right)}^T$ for the initial states (f) $\left|{\psi }_i\right.〉={\left|0\right.〉}_x$ and (g) $\left|{\psi }_i\right.〉\propto {\left|0\right.〉}_x+{\left|1\right.〉}_x$, while the continuous red line represents a two-Gaussian fit. In (h), projection onto ${\left(-1,1,1\right)}^T$ for the initial state $\left|{\psi }_i\right.〉={\left|0\right.〉}_x$, with a three-Gaussian fit (continuous red line).

Figure 3

Interference patterns at the Fourier plane. (a) Interference fringes after projection onto ${\left(1,0,-1\right)}^T$ (blue square) and ${\left(1,0,1\right)}^T$ (red dot). (b) Interference fringes after projection onto ${\left(-1,1,1\right)}^T$. In all cases, the solid lines just link the points.

Figure 4

Integrated transverse optical profiles (ITOPs) for identity and cyclic right permutation applied to the state $\left|{\psi }_i\right.〉=A_0{\left|0\right.〉}_x+A_2{\left|2\right.〉}_x$, in which $|A_0|>|A_2|$. Blue squares stand for the data from the identity operation, with the red dotted line a two-Gaussian fit, while black circles show the permutation operation, with the green dashed line the corresponding two-Gaussian fit.

Figure 5

Experimental setup to prepare and operate over a qutrit state encoded at the transverse paths of a single-photon heralded source (spontaneous parametric down-conversion).