Engineering of maximally entangled orbital angular momentum states via path identity

Abstract

Cutting-edge quantum technologies lean on sources of high-dimensional entangled states (HDES) that reliably prepare high-fidelity target states. The idea to overlap photon paths from distinct but indistinguishable sources was recently introduced for the creation of HDES, known as entanglement by path identity. In this regard, the use of orbital angular momentum (OAM) modes is promising, as they offer a high-dimensional and discrete Hilbert space to encode information. While entanglement by path identity with OAM has been verified experimentally, a detailed investigation of how the OAM distribution of photon pairs can be engineered to maximize the entanglement is lacking. We address this gap and identify an optimal dimensionality for maximally entangled states (MESs) when the spatial engineering of the pump beam and the path identity approach are combined. Our theoretical study reveals notable limitations for the fidelity of high-dimensional target states. We also establish the equivalence of entangled biphoton states pumped by a spatially engineered beam and generated via path identity. These findings constitute a valuable step toward the optimized preparation of MESs in high dimensions.ngineered beam. These findings constitute a valuable step toward the optimized preparation of MESs in high dimensions.

Figures (13)

• Figure 1: Sketch of a SPDC setup and the corresponding normalized OAM spectrum. A pump beam carrying OAM $\ell_{\mathop{\mathrm{\mathrm{p}}}\nolimits}=0$ triggers the generation of a photon pair along paths $\mathcal{A}$ and $\mathcal{B}$. The pump is blocked afterwards. The generated OAM distribution of signal and idler photons represents a high-dimensional entangled quantum state. Since the OAM is conserved, the OAM spectrum shows an anti-diagonal line of possible modes whose OAMs add up to zero. The projection probability is in general non-uniformly distributed, as can be seen from the bars of different heights.
• Figure 2: Sketch of a SPDC setup and the corresponding normalized OAM spectra. Suppose a superposition of different OAM modes pumps the nonlinear crystal. In that case, the OAM spectrum of the photon pair spans multiple anti-diagonals, ensuring that the conservation law is fulfilled for each term in the superposition. The OAM spectrum can be adjusted by the additional parameters $a_{\ell_{p}}$. Here, for example, a superposition of OAM indices $\ell_{\mathop{\mathrm{\mathrm{p}}}\nolimits} = -2, 0, 2$ is chosen such that the biphoton OAM modes $\ket{-1,-1}, \ket{0,0}$ and $\ket{1,1}$ have the same detection probability. The fidelity for the state $\ket{\psi_{\mathrm{tar}}} =1/\sqrt{3} \Bigl( \ket{-1,-1}+ \ket{0,0}+\ket{1,1} \Bigl)$ in the subspace $\ell=\{-1,0,1\}$ (shown in the inset) is $\mathcal{F}_{\mathrm{sub}}=0.876$. However, a large amount of modes are concentrated outside this subspace, which is quantified by the fidelity $\mathcal{F}=0.401$ for the whole space.
• Figure 3: Mode transformation via OAM shifts. After generation, the OAM of photons traveling along paths $\mathcal{A}$ and $\mathcal{B}$ can be manipulated by adding or subtracting $\Delta\ell_{A}$ or $\Delta\ell_{B}$ quanta of OAM, respectively. This is achieved using components such as spiral phase plates. In collinear configurations, the transformation is typically $\Delta\ell_{A}=\Delta\ell_{B}$ as both photons pass through the same device. Non-identical shifts $\Delta\ell_{A} \neq \Delta\ell_{B}$ are possible when the components are placed after the photon pair has been separated. The action of applying $\Delta\ell_{A}=4$ and $\Delta\ell_{B}=5$ is illustrated on the right for an OAM spectrum generated by an initial pump mode $\ell_{\mathop{\mathrm{\mathrm{p}}}\nolimits}=-4$. These mode shifts affect all OAM modes simultaneously, displacing the entire anti-diagonal.
• Figure 4: Phase transformations via phase shifters. Very similar to the mode shift concept in Fig. \ref{['fig03']}, the optical phase can be altered along paths $\mathcal{A}$ and $\mathcal{B}$. (a) Shifters add a phase factor $\varphi_A$ along path $\mathcal{A}$ and $\varphi_B$ along $\mathcal{B}$, which effectively adds a global phase factor of $\varphi := \varphi_A + \varphi_B$ for the two-photon state. (b) Such a shift is mathematically equally obtained by a shift $\varphi$ only in one of the paths, or (c) inherited from the pump beam, if the pump beam undergoes a phase shift as well, e.g., via an SLM.
• Figure 5: Two-crystal path identity setup. The paths from crystal $\mathsf{1}$ are made identical (perfectly overlapped) with the paths $\mathcal{A}$ and $\mathcal{B}$ from crystal $\mathsf{2}$. Both crystals are coherently pumped with OAM values $\ell_{p_1}$ and $\ell_{p_2}$. The setup can be built by using a single pump beam consecutively (i.e., without blocking the pump). Alternatively, it can be realized interferometrically, where the initial pump beam is split and its OAM is modified differently on each path before being directed to the crystals. Any distinguishability between the photon pairs generated in crystals $\mathsf{1}$ and $\mathsf{2}$, such as misalignment of the paths, will reduce the quantum interference of different generation events. By incorporating the transformation components discussed earlier for the OAM and phase, we can directly engineer high-dimensional quantum states.