An all-digital approach for versatile hybrid entanglement generation

We outline our method for tailoring hybrid entangled states through spatial-to-polarisation conversion (SPC). It is instructive to first introduce the concept of polarisation-spatial mode hybrid entanglement. A typical hybrid entangled state of this kind has the form

$$\begin{align} \vert{\Psi}\rangle_{AB} = \sqrt{a} \vert{H}\rangle_A\vert{\psi_1}\rangle_B + \sqrt{1-a}\vert{V}\rangle_A\vert{\psi_2}\rangle_B , \end{align} \tag{ 1 }$$

where $\vert{H}\rangle$ and $\vert{V}\rangle$ are the horizontal and vertical polarisation states of photon A, while $\vert{\psi_1}\rangle$ and $\vert{\psi_2}\rangle$ are the orthogonal transverse spatial modes of photon B. Here the factor a varies the state from being completely separable (a = 0 and a = 1) to maximally entangled (a = 0.5), provided $\langle{\psi_1\vert\psi_2}\rangle$ = 0.

Each photon occupies an independent DoF and the corresponding Hilbert spaces, $\{\vert{H}\rangle, \vert{V}\rangle\}_\textrm{polarisation}$ and $\{\vert{\psi_1}\rangle, \vert{\psi_2}\rangle\}_\textrm{spatial}$, are two-dimensional and can be represented on independent Bloch-spheres as shown in figure 1. Photon A is restricted to a two-dimensional subspace, however, multiple subspaces can be formed for photon B. Consequently, one can construct various hybrid subspaces by tailoring the spatial modes of photon B. In this paper, we demonstrate the adaptive control of the spatial DoF of photon B.

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To demonstrate the versatility of our approach and the corresponding diversity in hybrid states we can create, we use two mode families as exemplars: the Laguerre–Gaussian (LG) and HG modes, having Cylindrical and Cartesian symmetry, respectively. At the waist plane the LG basis modes have the field profiles

$$\begin{align} LG_{\ell,p}(r,\phi) & = \mathcal{C}_{\ell p} \ \left[\sqrt{2}\frac{r}{w}\right]^{|\ell|}{L_p}^{|\ell|} \left(\frac{2r^{\,2}}{w^2}\right) \nonumber \\ & \quad \times \exp\left(-\frac{r^{\,2}}{w^2}\right)\exp(i\ell\phi). \end{align} \tag{ 2 }$$

Here the factor $\mathcal{C}_{\ell p}$ is a normalisation constant, the function $L_p^{|\ell|}(\cdot)$ is the Associated Laguerre polynomial with radial number $p \geqslant 0$, $w\sqrt{2p+\vert \ell \vert+ 1 }$ is the second-moment radius of the mode and w is the size of the Gaussian envelope ($\exp(-r^{\,2}/w^2)$). The azimuthally dependent factor, $\exp(i\ell\phi)$, is associated with photon fields that carry an orbital angular momentum of $\ell\hbar$ per photon where $\ell \in \mathbb{Z}$ is an unbounded integer, called the topological charge, characterising the helicity of the photon field. The LG basis forms a complete set over the azimuthal ($\ell$) and radial indexes ($p\geqslant0$). We will select the subspaces of photon B as $\psi_1(\textbf{r}) = LG_{\ell, p} (\textbf{r})$ and $\psi_2(\textbf{r}) = LG_{-\ell, p} (\textbf{r})$, for the LG basis, and use the shorthand notation $\{\ell, -\ell\}$ to refer to the individual subspaces. Note that we will select fixed radial indices for each subspace.

The HG basis modes can be expressed in Cartesian coordinates, $\textbf{r} = (x,y)$,

$$\begin{align} \textrm{HG}_{mn}(x,y) & = \mathcal{D}_{nm} H_m\left(\sqrt{2}\frac{x}{w}\right) H_n\left(\sqrt{2}\frac{y}{w}\right) \nonumber \\ & \quad\times \exp\left(-\frac{x^2+y^2}{w^2}\right), \end{align} \tag{ 3 }$$

indexed by the positive integers $m\geqslant0$ and $n\geqslant0$, where $H_m(\cdot)$ denotes the Hermite polynomials and $w\sqrt{m+n+1}$ is the second-moment radius of the mode and, once again, w is the size of the Gaussian envelope, while $\mathcal{D}_{nm}$ is a normalisation constant. We will select the subspaces of photon B as $\psi_1(\textbf{r}) = \textrm{HG}_{m, n} (\textbf{r})$ and $\psi_2(\textbf{r}) = \textrm{HG}_{n, m} (\textbf{r})$, and adopt the notation $\{m,n\}$ to refer to the HG subspaces.

To create the hybrid modes in equation (1), we first begin by generating entangled photons from SPDC using a NC that is pumped with a high-energy photon, thereafter producing two lower-energy photons that conserve the energy and momentum of the input photon. The schematic for achieving this is shown in figure 2(a). Crucially, the SPDC photons are correlated in the position basis and their state can be described as

$$\begin{align} \vert{\Phi}\rangle_{AB} = \int \Phi(\textbf{r}) \vert{\textbf{r}}\rangle_A \vert{\textbf{r}}\rangle_B \vert{H}\rangle_A\vert{H}\rangle_B \ d^2~r, \end{align} \tag{ 4 }$$

where $\Phi(\textbf{r})$ is the two-photon probability amplitude due to the down-conversion process, while the position basis vectors, $\vert\textbf{r}\rangle$, are complete and satisfy $\langle\textbf{r}^{^{\prime}}|\textbf{r}\rangle = \delta(\textbf{r}-\textbf{r}^{^{\prime}})$, where $\delta(\textbf{r}-\textbf{r}^{^{\prime}})$ is the Dirac delta function.

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While the assumptions we will make simplify the calculations that follow, in general, the two photon field amplitude of the SPDC state is determined by the phase matching conditions and pump photon profile [41]. This can have an impact on the spatial bandwidth of the SPDC state [42]. In practise, it is crucial to ensure that the spatial bandwidth of source is high enough. The theory [43], and experimental tools [44, 45], for achieving this are well established.

Next, we define an operator, $\hat{S}_{\textrm{sp} \otimes \textrm{pol}}$, that maps the SPDC state to a tailored hybrid state corresponding to equation (1) by coupling the internal DoF of photon A using spatial-to-polarisation coupling (SPC) as illustrated in figure 2 (b). The mapping due to the SPC transformation takes a single position basis vector, $\vert{\textbf{r}}\rangle_A$, and transforms it following

$$\begin{align} \vert{\textbf{r}}\rangle_A \vert{V}\,\rangle_A \xrightarrow{\hat{S}_{\textrm{sp} \otimes \textrm{pol}}} \psi_{1}(\textbf{r}) \vert{\textbf{r}}\rangle_A\vert{H}\rangle_A, \end{align} \tag{ 5 }$$

$$\begin{align} \vert{\textbf{r}}\rangle_A \vert{H}\rangle_A \xrightarrow{\hat{S}_{\textrm{sp} \otimes \textrm{pol}}} \psi_{2}(\textbf{r}) \vert{\textbf{r}}\rangle_A \vert{V}\,\rangle_A, \end{align} \tag{ 6 }$$

where $\psi_{1,2}(\textbf{r}) = \langle\textbf{r}, \psi_{1,2}\rangle$ and $\vert{\psi_{1,2}}\rangle = \int \psi_{1,2}(\textbf{r})\vert{\textbf{r}}\rangle d^{\,2}\textrm{r}$, so that the final output states are imprinted with independent spatial modes depending on the input polarisation states. The operator can therefore be summarised as

$$\begin{align} \hat{S}_{\textrm{sp} \otimes \textrm{pol}} & = \int d^2\textrm{r}\, \psi_1(\textbf{r}) \vert{\textbf{r}}\rangle_A\langle\textbf{r}\vert_A \otimes \vert{H}\rangle_A\langle V\vert_A \nonumber \\ &\quad+\int d^2\textrm{r}\, \psi_2(\textbf{r})\vert{\textbf{r}}\rangle_A\langle\textbf{r}\vert_A \otimes \vert{V}\,\rangle_A\langle H\vert_A. \end{align} \tag{ 7 }$$

To show that the resulting state of the SPDC photon can be mapped to a hybrid state using the operator $\hat{S}_{\textrm{sp} \otimes \textrm{pol}}$, we will first assume that the two-photon amplitude is uniform and normalised over a bounded domain, and therefore photon A and B have non-local correlations that approximate a maximally entangled state, i.e. $\Phi(\textbf{r}) \propto 1$. While this approximation simplifies the calculation, it is imperative to include the phase-matching function and pump profile for a general treatment [43].

Before applying the operator on the SPDC state, polarisation of photon A in equation (4) is first rotated to the diagonal polarisation state, $\vert{D}\rangle = \frac{1}{\sqrt{2}} \left( \vert{H}\rangle + \vert{V}\,\rangle\right)$, using a half-wave plate. As a result, the SPDC state is now

$$\begin{align} \vert{\Phi}\rangle_{AB} \propto \frac{1}{\sqrt{2}} \int d^2\textrm{r} \, \left( \vert{\textbf{r}}\rangle_A\vert{H}\rangle_A + \vert{\textbf{r}}\rangle_A \vert{V}\,\rangle_A \right) \vert{\textbf{r}}\rangle_B \vert{H}\rangle_B . \end{align} \tag{ 8 }$$

After applying our operator, $\hat{S}_{\textrm{sp} \otimes \textrm{pol}}$, on photon A, we obtain the state

$$\begin{align} \hat{S}_{\textrm{sp} \otimes \textrm{pol}} \otimes \mathbb{I}_B \vert{\Phi}\rangle_{AB} &\propto \frac{1}{\sqrt{2}} \int \big[ \psi_2(\textbf{r}) \vert{\textbf{r}}\rangle_A \vert{V}\,\rangle_A \nonumber \\ &\quad+ \psi_1(\textbf{r}) \vert{\textbf{r}}\rangle_A \vert{H}\rangle_A \big] \vert{\textbf{r}}\rangle_A \vert{H}\rangle_B d^2\textrm{r}. \end{align} \tag{ 9 }$$

To complete the transformation, we have to integrate over the position vectors in photon A. To achieve this, we use a single-mode fibre (SMF) to perform a partial inner product, enabling us to remove the spatial information of photon A. The SMF filters the fundamental mode, $\vert{G}\rangle = \sqrt{\frac{2}{\pi w_0^2}} \int \exp \left(- \frac{r^{^{\prime} 2}}{{w_0}^2} \right) \vert{\textbf{r}^{^{\prime}}}\rangle d^{\,2}~\textrm{r}^{^{\prime}}$, resulting in

$$\begin{align} \vert{\Phi^{^{\prime}}}\rangle_{AB} & = \frac{1}{\sqrt{2}} \mathcal{M} \int \int \exp \left( - \frac{r^{^{\prime} 2}}{{w_0}^2} \right) \delta(\textbf{r} - \textbf{r}^{^{\prime}}) \big( \psi_2(\textbf{r}) \vert{V}\,\rangle_A \nonumber \\ &\quad+\psi_1(\textbf{r}) \vert{H}\rangle_A \big) \vert{\textbf{r}}\rangle_B d^2~\textrm{r}^{^{\prime}} d^2~\textrm{r}, \end{align} \tag{ 10 }$$

where $\vert{\Phi^{^{\prime}}}\rangle_{AB} = \langle G|\Phi\rangle_{AB}$ and $\mathcal{M} = \sqrt{2/\pi w_0^2}$. Next, we integrate over the spatial coordinates of photon A and ignore the polarisation of photon B because it is factorisable and subsequently arrive at the state

$$\begin{align} \vert{\Phi^{^{\prime}}}\rangle_{AB} = \frac{1}{\sqrt{2}} \big( \vert{H}\rangle_A \vert{\tilde{\psi_1}}\rangle_B + \vert{V}\,\rangle_A \vert{\tilde{\psi_2}}\rangle_B \big). \end{align} \tag{ 11 }$$

Here $\vert{\tilde{\psi_{1,2}}}\rangle_B = \mathcal{M} \int \psi_{1,2}(\textbf{r}) \exp \left( \frac{r^{\,2}}{{w_0}^2} \right) \vert{\textbf{r}}\rangle d^2~\textrm{r}$ is the modulated version of the spatial modes in our hybrid state from equation (1) (with a = 0.5). If the fields, $\psi_{1,2}$, are sufficiently smaller than the fundamental mode, then $\tilde{\psi}_{1,2}(\textbf{r}) \rightarrow \psi_{1,2}(\textbf{r})$ and therefore equations (11) and (1) are equivalent (for a = 0.5). Next, we show how adaptive control of the operator $\hat{S}_{\textrm{sp} \otimes \textrm{pol}}$ can be achieved using an SLM and linear optical elements.

The transition from the SPDC state to the hybrid entangled state in equation (1) requires the construction of the operator $\hat{S}_{\textrm{sp} \otimes \textrm{pol}}$. We achieve this with a SLM and linear polarisation optics (PO) as illustrated in figure 2(b). We rewrite the operator using Jones calculus and project $\hat{S}_{\textrm{sp} \otimes \textrm{pol}}$ onto the position basis. Assuming that the polarisation basis vectors are $\vert{H}\rangle = \begin{pmatrix} 1 &0~\end{pmatrix}^\intercal$ and $\vert{V}\,\rangle = \begin{pmatrix} 0 &1~\end{pmatrix}^\intercal$, the corresponding Jones matrix can be written as

$$\begin{align} S_{\textrm{sp} \otimes \textrm{pol}}(\textbf{r}) & = \int \langle\textbf{r}\vert\hat{S}_{\textrm{sp} \otimes \textrm{pol}}\vert{\textbf{r}^{^{\prime}}}\rangle d^2~\textrm{r}^{^{\prime}} \end{align} \tag{ 12 }$$

$$\begin{align} & = \int \delta(\textbf{r}-\textbf{r}^{^{\prime}})\begin{pmatrix} 0 & \psi_1(\textbf{r}^{^{\prime}}) \\ \psi_2(\textbf{r}^{^{\prime}}) & 0 \end{pmatrix} d^2~\textrm{r}^{^{\prime}}, \,\,\,\,\, \end{align} \tag{ 13 }$$

$$\begin{align} & = \begin{pmatrix} 0 & \psi_1(\textbf{r}) \\ \psi_2(\textbf{r}) & 0 \end{pmatrix}.\qquad \qquad\qquad \qquad \end{align} \tag{ 14 }$$

Conveniently, we can decompose the operator into three independent transformations

$$\begin{align} S_{\textrm{sp} \otimes \textrm{pol}}(\textbf{r}) = \hat{A}_1~\sigma_x \hat{A}_2, \end{align} \tag{ 15 }$$

where

$$\begin{align} \hat{A}_{1,2} = \begin{pmatrix} \psi_{1,2}(\textbf{r}) & 0 \\ 0 & 1 \end{pmatrix}, \ \sigma_x = \begin{pmatrix} 0 &1 \\ 1 & 0~\end{pmatrix}. \end{align} \tag{ 16 }$$

Here, $\hat{A}_{1,2}$ encodes the field profiles that map onto the states $\vert{\psi}\rangle_{1,2}$, by modulating the horizontal components of the incoming field, while the vertical component is unaffected. Then, the X Pauli gate (σ_x ) flips between the linear polarisation basis states so that the field profile encoded by $\hat{A}_2$ is transferred to the vertical polarisation component while $\hat{A}_1$ only acts on the horizontal component.

We use an SLM separated into two halves to perform the SPC conversion, as shown in figure 2(b). Since the SLM is birefringent and transfers the desired phase onto the horizontal polarisation component, while the vertical component is unaffected, we can apply the operation $\hat{A}_{2}$ on the first reflection and $\hat{A}_{1}$ on the second reflection. Importantly, the photon field is modulated on axis and there is no grating. For this reason, we encode the phase of the desired modes, although an inverted grating can be applied to achieve the desired amplitude modulation. We apply the σ_x gate by transmitting the field through the quarter-wave plate (QWP) at $\pi/4$ twice, but with a mirror in between. This is because the QWP maps between the linear polarisation basis and the circular polarisation (CP) basis, while the mirror flips the handedness of the incident CP state. The subsequent transmission through the same QWP maps the polarisation state back to the linear polarisation state, but with the initial polarisation now orthogonal to the input. This allows us to apply $\hat{A}_{1}$ to the remaining photon field, on the second side of the SLM screen, enabling us to perform polarisation to spatial mode coupling. The lens (L) is used to image the plane of the first half of the SLM to second half.

We illustrate the working principle of the technique graphically in figures 2(c) and (d) for the LG ($\psi_{1,2}(\textbf{r}) \rightarrow \textrm{LG}_{\ell = \pm1, 0}(\textbf{r})$) and HG ($\psi_{1,2}(\textbf{r}) \rightarrow \textrm{HG}_{2, 3}(\textbf{r}) \ \textrm{and} \ \textrm{HG}_{3, 2}(\textbf{r})$) basis. We can think of photon A in the retroactive picture (back-projection) as though it were emanating from the detection SMF fibre and passing through the SPC system to produce the desired target state. The reverse process filters the desired spatial and polarisation mode, therefore converting spatial information to polarisation information. This means that we can couple the photon fields $\psi_{1,2}(\textbf{r})$ to desired polarisation states by controlling the modes encoded on each half of SLM A.

In this work we reconstruct each generated hybrid entangled state, ρ, using QST. To achieve this, we perform spatially separated measurements $M_{ij} = P^{A}_{i} \otimes P^{B}_{j}$, where $P_{i,j}^{A,B}$ are local projections of photon A and photon B, respectively. The detection probabilities on a system with a corresponding density matrix (ρ) are

$$\begin{align} p_{ij} = \textrm{Tr}\left(M_{ij}\rho \left( M_{ij} \right)^\dagger\right), \end{align} \tag{ 17 }$$

where $\textrm{Tr}(\cdot)$ represents the trace operation. In the experiment, photon A is projected onto the spin basis states $\vert{R}\rangle$ and $\vert{L}\rangle$, along with their equally weighted superpositions of linear anti-diagonal, diagonal, horizontal, vertical polarisation states, i.e. $\vert{A}\rangle, \vert{D}\rangle, \vert H \rangle$ and $\vert V\, \rangle$, respectively. Similarly, photon B is locally projected onto the spatial eigenstates $\vert\psi_{1,2}\rangle$ along with superpositions

$$\begin{align} \vert\theta\rangle = \frac{1}{\sqrt{2}} \big( \vert\psi_1\rangle + e^{i\theta}\vert\psi_2\rangle \big), \end{align} \tag{ 18 }$$

for relative phase $\theta = 0, \pi, \frac{\pi}{2},$ and $\frac{3\pi}{2}$. We perform these projections digitally using SLMs and linear optical elements only.

The measurement outcomes are then used to reconstruct the state by employing a maximum likelihood algorithm [46]. In this work, the density matrix reconstruction assumes the decomposition given by

$$\begin{align} \rho = \frac{1}{2} \bigg( \mathbb{I} + \sum_{3}^{m,n = 1} b_{mn} \sigma_{A,m} \otimes \sigma_{B,n} \bigg), \end{align} \tag{ 19 }$$

where $\mathbb{I}$ is the four dimensional identity matrix and $\sigma_{A,m}$ and $\sigma_{B,n}$ are the Pauli matrices that span the two-dimensional hybrid space for polarisation and spatial DoF respectively.

We quantify the degree of entanglement of our reconstructed density matrices by calculating their concurrences [47]. For a reconstructed density matrix ρ, the concurrence is given by

$$\begin{align} C(\rho) = \textrm{max}\left\{0,\lambda_1-\sum_{i = 2}^4\lambda_i\right\}, \end{align} \tag{ 20 }$$

where ρ is the density matrix, λ are the eigenvalues of the operator $R = \sqrt{\sqrt{\rho} \tilde{\rho} \sqrt{\rho }}$ in descending order with $\tilde{\rho} = \Theta \rho^* \Theta$ and $^*$ denotes a complex conjugation. The operator Θ represents any arbitrary anti-unitary operator satisfying $\langle\psi\vert\Theta\vert\phi\rangle$ = $\langle\phi\vert\Theta^{-1}\vert\psi\rangle$ for any state $\vert\phi\rangle$ and $\vert\psi\rangle$, if $\Theta^{-1} = \Theta^{\dagger}$.

For pure states, the concurrence can be simplified as

$$\begin{align} C(\rho) = \sqrt{2\left( 1 - \textrm{Tr}(\rho_A^2) \right)}, \end{align} \tag{ 21 }$$

where ρ_A is the reduced density matrix found by computing the partial trace of ρ. For the state in equation (1), the concurrence is given by

$$\begin{align} C(\vert\Psi_{AB}\rangle\langle\Psi_{AB}\vert) = 2|\sqrt{a(1-a)}|, \end{align} \tag{ 22 }$$

where C = 0 when a = 0 or 1, and C = 1 for a = 0.5.