Experimental Engineering of Arbitrary Qudit States with Discrete-Time Quantum Walks

The capability to generate and manipulate quantum states in high-dimensional Hilbert spaces is a crucial step for the development of quantum technologies, from quantum communication to quantum computation. One-dimensional quantum walk dynamics represents a valid tool in the task of engineering arbitrary quantum states. Here we affirm such potential in a linear-optics platform that realizes discrete-time quantum walks in the orbital angular momentum degree of freedom of photons. Different classes of relevant qudit states in a six-dimensional space are prepared and measured, confirming the feasibility of the protocol. Our results represent a further investigation of quantum walk dynamics in photonics platforms, paving the way for the use of such a quantum state-engineering toolbox for a large range of applications.

Figure 1

Setup for the quantum state engineering toolbox. (a) Conceptual scheme of the protocol. At each step of the QW the coin operator is changed to obtain a target state in the output. (b) A single-photon source, composed of a periodically poled potassium titanyl phosphate (PPKTP) crystal, generates pairs of photons that are coupled in a single-mode fiber. One photon acts as trigger while the other is prepared in $|{\psi }_0⟩=|+⟩\otimes |0⟩$ through polarization controllers and a polarizing beam splitter. Five sets of quarter (QWP) and half (HWP) wave plates implement operators $\{C_i\}$ for each step. Five $Q$ plates implement the shift operator of the QW $\{S_i\}$. The detection stage consists of a PBS followed by a spatial light modulator, a SMF and an avalanche photodiode detector (APD), for projection onto $|+⟩\otimes |\psi ⟩$. (c) Pictures of OAM modes of the output states after PBS, obtained with coherent light. From right: OAM eigenstate corresponding to $m=5$; balanced superposition of $m=\pm 5$; balanced superposition of all OAM components covered by the five-step QW $m=\{\pm 5,\pm 3,\pm 1\}$.

Figure 2

Experimental results for the engineering of SLOS states. (a) Representation on a Bloch-like ball of the four target states corresponding to the superposition of $|\pm 5⟩$, corresponding to OAM states with maximum and minimum projection of the angular momentum along the quantization axis. (b) Population of the OAM components after five-step QW for states $|{\psi }_i⟩(i=1,2,3,4)$ in panel (a). Odd-$m$ position states (bold numbers on the $x$ axis) should be the only ones involved in the state engineering. However, we report also the populations of even-$m$ position states (light-black numbers on the $x$ axis) to illustrate possible imperfections at generation and detection stages. The error bars associated with experimental populations are shown by the transparent areas on top of each histogram. (c)–(d) Distributions of probabilities $P_i=⟨B_i^{(j)}|{\rho }_{\exp }|B_i^{(j)}⟩(j=1,2)$ that the experimental walker state ${\rho }_{\exp }$ is found to be one of the elements of the bases $B^{(j)}=\{|{\psi }_p⟩,|{\psi }_{p+1}⟩,|\pm 4⟩,|\pm 3⟩,|\pm 2⟩,|\pm 1⟩,|0⟩\}$ with $p=1$ for $j=1$ and $p=3$ for $j=2$. All the error bars are due to Poissonian uncertainties, propagated through Monte Carlo methods.

Figure 3

Experimental results for the engineering of SCSs and their coherent superposition: (a) Bloch-sphere representation for the mutually orthogonal SCSs $|S_1⟩$ and $|S_2⟩$. (b) Probability distributions associated to the projection of $|S_1⟩$ onto the computational basis. As previously explained, we also consider the contribution of even OAM components. (c) Probability distribution corresponding to the basis that contains the target state itself $|S_1⟩$, generated with the fidelity reported in the panel. Such orthonormal basis, $S_i$ with $i=1,\dots ,6$, contains eigenstates of ${\widehat{S}}_x$ for a particle with spin $s=5/2$ that are in turn all spin-coherent states. (d) Experimental probability distribution on computational basis for $|{\psi }_2⟩=(1/\sqrt{2})(|S_1⟩-|S_2⟩)$. Only components $\{-5,-1,3\}$, corresponding to logical states $\{1,3,5\}$, have nonzero probabilities. (e) Quantum state fidelity evaluated measuring state $|{\psi }_2⟩$ on the orthonormal basis that contains state $|{\psi }_1⟩$, as described in the main text. (f) Summary of quantum state fidelities for the 32 states generated in the experiment. Magenta area: average fidelity $\overline{\mathcal{F}}=0.954\pm 0.001$.