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= Orbital Angular Momentum in Quantum Optics = The total angular momentum of light can be decomposed into two components of orbital angular momentum (OAM) and spin orbital angular momentum. OAM of photons plays an important role in quantum optics. Photons with OAM can be used for testing high dimensional Bell's inequalities and quantum key distribution (QKD).

Introduction
The total angular momentum of a paraxial field can be split into two components, spin and orbital angular momentum (OAM).

$$\begin{align} &\mathbf{J}=\mathbf{L}+\mathbf{S}\\ &\mathbf{L} = \varepsilon_0\sum_j\int_Vd^3\mathbf{r}E^\perp_j(\mathbf{r}\times\nabla)A^\perp_j\\ &\mathbf{S}=\varepsilon_0\int_Vd^3\mathbf{r}\mathbf{E}_\perp\times\mathbf{A}_\perp \end{align}$$

Paraxial fields which have OAM exhibit helical phase structure and have a phase singularity at the center. Typically Laguerre-Gaussian modes are used when working with OAM.

Quantizing the field
The starting point to quantize the field in terms of OAM is the vector potential in the Coulomb gauge expanded in terms of plane-waves.

$$\mathbf A (\mathbf r,t)=\sum_{\sigma}\int \frac{d^3\mathbf{k}}{(16 \pi\varepsilon_0c|\mathbf{k}|)^{1/2}}[\mathbf{\epsilon}_\sigma(\mathbf{k})\alpha_\sigma(\mathbf{k})e^{i(\mathbf{k\centerdot r}-c|\mathbf{k}|t)}+c.c]$$

$$\alpha_\sigma$$ is the complex amplitude for either right-handed or left-handed circularly polarized light and $$\epsilon_\sigma$$ is the unit polarization vector. Typically the paraxial approximation yields a field which is a plane wave modified with an envelope. The wave vector of the carrier field, $$\mathbf{k}_0$$, makes a small angle with the optical axis. For the rest of the derivation, it is assumed that $$\mathbf{k}_0$$ has positive magnitude and propagates along the z-axis. Multiplying vector potential by the identity

$$\int_{0}^{\infin}{dk_0 \frac{e^{ik_0(z-ct)}}{e^{ik_0(z-ct)}}}\delta [k_0-f(\mathbf{k})]=1$$

where $$k_0 > 0$$, the vector potential can be written as

$$\mathbf{A}(\mathbf{r},t) = \int_{0}^{\infin}{dk_0 e^{ik_0(z-ct)}\mathbf{A}_{k_0}(\mathbf{r},t)}$$

with $$\mathbf{A}_{k_0}(\mathbf{r},t)=\sum_{\sigma}\int \frac{d^3\mathbf{k}}{(16 \pi\varepsilon_0c|\mathbf{k}|)^{1/2}}[\mathbf{\epsilon}_\sigma(\mathbf{k})\alpha_\sigma(\mathbf{k})e^{i(\mathbf{k\centerdot r}-c|\mathbf{k}|t)}+c.c]\frac{1}{e^{ik_0(z-ct)}}$$.

$$\mathbf{A}_{k_0} $$ is the envelope which modifies the carrier plane wave. Therefore $$\mathbf{A}_{k_0} $$must obey the paraxial wave equation $\frac{\partial^2 \mathbf{A}}{\partial x^2} + \frac{\partial^2 \mathbf{A}}{\partial y^2}+2ik_0\frac{\partial \mathbf{A}}{\partial z}=0$. After plugging in $\mathbf{A}_{k_0} $ into the paraxial wave equation the resulting formula is

$$\begin{align} -k^2_x-k^2_y+2if(\mathbf{k})(-if(\mathbf{k})+ik_z)&=0 \\ -k^2_x-k^2_y +2f(\mathbf{k})^2-2f(\mathbf{k})k_z &=0 \\ -2k_x^2-2k_y^2+4f(\mathbf{k})^2-4f(\mathbf{k})k_z &=0 \\ 4f(\mathbf{k})^2-4f(\mathbf{k})k+k_z^2 &= 2q^2+k_z^2 \\ (2f(\mathbf{k})-k_z)^2 &= 2q^2+k_z^2q\\ f(\mathbf{k}) &= \frac{k_z+\sqrt{k_z^2+2q^2}}{2}

\end{align}$$

where $$\mathbf{q} = k_x\mathbf{\hat{x}}+k_y\mathbf{\hat{y}}$$, the wavevector in the transverse plane. Then the relation between $$k_0$$ and $$k_z$$ can be found using the identity above. It yields the result $\delta [k_0-f(\mathbf{k})] = \delta[k_z-(k_0-(q^2/2k_0))](1+\vartheta)$ where $\vartheta = q/\sqrt{2k_0^2}$. This allows the vector potential to be written in terms of a strictly positive variable and a parameter which quantifies the 'paraxial-ness ' of the beam.

$$\mathbf{A}(\mathbf{r},t)=\sum_\sigma\int_{0}^{\infin}dk_0\int d^2 \mathbf{q}\left [ \frac{(1+\vartheta^2)^2}{16\pi^3\varepsilon_0ck_0\sqrt{1+\vartheta^4}} \right ]^{1/2} \mathbf{\epsilon}_\sigma[\mathbf{q},k_0(1-\vartheta^2)]\alpha_\sigma[\mathbf{q},k_0(1-\vartheta^2)] e^{ik_0(z-ct)}e^{i\mathbf{q\cdot r_\perp}-ik_0\vartheta^2z - ick_0(\sqrt{1+\vartheta^4}-1)t} +c.c.$$

Defining the polarization as is done in and taking the paraxial approximation, $$\vartheta \ll1$$ keeping quadratic terms of $$\vartheta $$, the vector potential becomes

$$\mathbf{A}(\mathbf{r},t)=\sum_\sigma \int_0^\infin \frac{dk_0}{(16\pi\varepsilon_0ck_0)^{1/2}} \int d^2\mathbf{q}[\epsilon_\sigma \alpha_\sigma(\mathbf{q},k_0)e^{ik_0(z-ct)} e^{i(\mathbf{q \cdot r_\perp}-k_0\vartheta^2z)}+c.c]$$

Now going to the quantum regime, $\mathbf{A}(\mathbf{r},t)\longrightarrow \hat \mathbf{A}(\mathbf{r},t)$ and $\alpha_\sigma(\mathbf{q},k_0)\longrightarrow \hat a_\sigma(\mathbf{q},k_0)$. The vector potential and the complex amplitudes become operators. The operators $\hat a_\sigma(\mathbf{q},k_0)$ and $\hat a_\sigma^\dagger(\mathbf{q},k_0)$  satisfy the commutation relation $\left [\hat a_\sigma(\mathbf{q},k_0), \hat a_\sigma^\dagger(\mathbf{q'},k_0') \right ] = \delta_{\sigma \sigma'}\delta^{(2)}(\mathbf{q}-\mathbf{q'})\delta(k_0-k_0')$. Now the vector potential takes the form

$$\mathbf{\hat A}(\mathbf{r},t)=\sum_\sigma \int_0^\infin dk_0\left ( \frac{\hbar}{16\pi\varepsilon_0ck_0} \right )^{1/2} \int d^2\mathbf{q}[\epsilon_\sigma \hat a_\sigma(\mathbf{q},k_0)e^{ik_0(z-ct)} e^{i(\mathbf{q \cdot r_\perp}-k_0\vartheta^2z)}+H.c.]$$

In order to write the vector potential in terms of quantized Laguerre-Gaussian modes the closure relations of the fourier transformed Laguerre-Gaussian modes is used. Noting that $\sum_{l,p}\mathcal{LG}_{l,p}^*(\mathbf{q})\mathcal{LG}_{l,p}^*(\mathbf{q'})\equiv \delta^{(2)}(\mathbf{q}-\mathbf{q'})$ ,

$$\begin{align} e^{i(\mathbf{q \cdot r_\perp}-k_0\vartheta^2z)} &= \int d^2\mathbf{q}e^{i(\mathbf{q' \cdot r_\perp}-k_0\vartheta^2z)}\delta^{(2)}(\mathbf{q}-\mathbf{q'}) \\ & = \int d^2\mathbf{q'}e^{i(\mathbf{q \cdot r_\perp}-k_0\vartheta^2z)}\left [ \sum_{l,p}\mathcal{LG}_{l,p}^*(\mathbf{q})\mathcal{LG}_{l,p}(\mathbf{q'})\right]\\ & = \sum_{l,p}\mathcal{LG}_{l,p}^*(\mathbf{q})\int d^2\mathbf{q'}e^{i(\mathbf{q \cdot r_\perp}-k_0\vartheta^2z)}\mathcal{LG}_{l,p}(\mathbf{q'})\\ & = \sum_{l,p}\mathcal{LG}_{l,p}^*(\mathbf{q})LG_{l,p}(\mathbf{r_\perp},z;k_0) \end{align}$$

The quantized vector potential can be written in terms of the LG modes.

$$\hat \mathbf{A}(\mathbf{r},t) = \sum_{\sigma, l, p}\int_0^\infin dk_0\left ( \frac{\hbar}{16\pi\varepsilon_0ck_0} \right )^{1/2}[\mathbf{ \epsilon_\sigma}\hat a_{\sigma,l,p}(k_0)e^{ik_0(z-ct)}LG_{l,p}(\mathbf{r_\perp},z;k_0)+H.c.]$$

where $\hat a_{\sigma,l,p}(k_0) = \int d^2\mathbf{q}\mathcal{LG_{l,p}^*(\mathbf{q})\hat a_\sigma(\mathbf{q},k_0)}$. The quantized vector potential is now in terms of a continuous, paraxial plane wave expansion with a transverse field given by the LG modes. The operator $$\hat a_{\sigma,l,p}^\dagger(k_0)$$ is the creation operator of a quantized LG transverse mode with polarization $$\sigma$$, radial quantum number $$p$$, and orbital quantum number $$l$$ carried by a plane wave with wavevector $$k_0$$. A single photon state can be written as $|\psi\rangle = \sum_{\sigma,l,p}\int_0^\infin dk_0C_{\sigma,l,p}(k_0)\hat a^\dagger_{\sigma, l, p}k_0|0\rangle$ where $C_{\sigma,l,p}(k_0)$ is the probability amplitude of finding a photon in a specific LG mode.

The quantum number $$l$$ contains information about the orbital angular momentum of the state. In the paraxial approximation two quantities of interest are the orbital angular momentum and spin angular momentum about the z-axis. This can be found using the operators

$$\begin{align} \hat L_z =\hbar\sum_{\sigma, l,p}l\int_0^\infin dk_0\hat a^\dagger_{\sigma ,l,p}(k_0)\hat a_{\sigma ,l,p}(k_0)\\ \hat S_z =\hbar\sum_{\sigma, l,p}\sigma\int_0^\infin dk_0\hat a^\dagger_{\sigma ,l,p}(k_0)\hat a_{\sigma ,l,p}(k_0) \end{align} $$

These operators 'count' how many photons are in a given mode with OAM (polarization) $$l $$ ($$\sigma $$) and sum those values for all the modes in the field together.

Bell's Inequalities with Qudits
Photons that are entangled in OAM are entangled qudits compared to photons that are entangled in polarization which are entangled qubits. This allows Bell's inequalities to be tested in higher dimensions.

Theory
Typically with entangled qubits the CHSH inequality is used and has the simple expression

$$S = E(a, b) - E(a, b^\prime) + E(a^\prime, b) + E(a^\prime, b^\prime) $$

However the expression $I \equiv P(A_1=B_1)+P(B_1=A_2+1)+P(A_2=B_2)+P(B_2=A_1) $ is equally valid with $P(A_a=B_b+k) $ is the probability that measurement outcomes $A_a $ and $$B_b $$ differ by $k $  modulo $$d $$, the dimension of the qudits. This equation is upper bounded by 3 for local variable theories as the maximum number of constraints that can be satisfied is 3. However quantum, nonlocal correlations can satisfy all constraints the upper bound for the quantity $I $ is 4. $I $ can be generalized for higher dimensional where $$d>2 $$. This is given by the expression

$$\begin{align} I_d \equiv \sum_{k=0}^{[d/2]-1}\biggl (1-\frac{2k}{d-1} \biggl)\biggl(&[P(A_1=B_1+k) +P(B_1=A_2+k+1)+P(A_2=B_2+k)+P(B_2=A_1+k)] \\ &-[P(A_1=B_1-k-1)+P(B_1=A_2-k)+P(A_2=B_2-k-1)+P(B_2+A_1-k-1)]\biggl) \end{align} $$

Experiment
Violation of Bell's inequalities in higher dimensions has previously been done in the group of Anton Zeilinger using OAM qudits with $$d=3$$. They use photon's resulting from parametric down conversion with type-1 phase matching. The photons from this process are entangled in OAM, if one photon is measured to have $l=+1\hbar$ then the other photon will be measured to have $l=-1\hbar$. The quantity of interest is Bell's expression for dimension of 3 given by

$$\begin{align} I_3 \equiv &P(A_1=B_1) + P(B_1=A_2+1) + P(A_2=B_2) \\ & +P(B_2=A_1)-P(A_1=B_1-1)-P(B_1=A_2) \\ & -P(A_2=B_2-1) -P(B_2=A_1-1) \end{align}$$

It is said that the correlation are local if $I_3 \leq 2$ and non-local otherwise. The maximum value that $I_3$ may take is 4 in a non-local variable theory. Zeilinger's experiment is a generalization of the two channel Bell test. There are three possible measurement outcomes for each arm of the experiment corresponding to $l=-1,0,1$. The two channel polarizers are generalized using phase holograms to 'rotate' the photon state to an arbitrary superposition of LG modes. The photon then is projected onto the $l=-1,0,1$ states using 3 separate holograms and single mode fibers each with its own single photon detector. The maximum value of $$I_3$$ achieved was $2.9045\pm0.0517$ which violates the inequality set by local correlations by more than 18 standard deviations.

More recent work in the group of Miles Padgett showed Bell's inequality violations for dimension up to $d=11$.

Mutually Unbiased Basis
OAM has also been used to demonstrate QKD in high dimensions by the group of Robert Boyd. In order to perform QKD a second basis, called the angular (ANG) basis, is used. Modes in the ANG basis are superpositions of OAM modes which are given by

$$\Psi^n_{\mathrm{ANG}} = \frac{1}{\sqrt{d}} \sum_{l=-N}^N\Psi_{\mathrm{OAM}}^l e^{\frac{i2\pi nl}{d}}$$

Note that the ANG modes and the OAM modes are mutually unbiased bases, namely, $|\langle \Psi^n_{\mathrm{ANG}}|\Psi_{\mathrm{OAM}}^L \rangle | ^2 = 1/d $ for all $n,l$. In addition, the OAM modes used in the experiment are not the LG modes but a flattop beam with a phase singularity.

Procedure
In the experiment of Boyd's group, Alice sends a state in either a OAM or ANG mode to Bob. Bob performs a log-polar to cartesian transformation. This transformation takes OAM modes to plane waves with a wave vector proportional to $l$ which can be focused to a point with a lens. In addition ANG modes are localized to a spot conditioned on $n$. Next Bob sends these modes through a mode sorter with 50% probability that a measurement is made in the OAM basis and 50% probability that a measurement is made in the ANG basis. Alice and Bob then publicly compare preparation and measurement bases and keep the results where they are the same.

Security
The mutual information between Alice and Bob is given by

$$I_{AB}=\sum_{i,j}P(x_i,y_j)\mathrm{log}_2\left [ \frac{P(x_i,y_j)}{P(x_i)P(y_j)}\right]$$

where $x_i$ the event of sending $i$  and $y_j$  the event of measuring $j$. When probability of error of detection of the modes is included into the model the mutual information can be written as

$$I_{AB}=\mathrm{log}_2(d)+F\mathrm{log}_2(F)+(1-F)\mathrm{log}_2\left ( \frac{1-F}{d-1}\right)$$

where $F$ is the probability that the correct mode is detected. For their experiment it is estimated that the mutual information is 2.05 bits per photon. In addition a secret key rate is calculated and found to be 6.8 secure bits per second. Three attacks are considered: intercept-resend, coherent attacks, and photon number splitting. It was shown that this QKD protocol was secure against intercept-resend and coherent attacks but was not secure against photon number splitting.