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Transfer of angular spectrum in parametric down-conversion with structured light

Antonio Z. Khoury1, Paulo H. Souto Ribeiro2, and Kaled Dechoum1 Instituto de Física, Universidade Federal Fluminense, 24210-346 Niterói, RJ, Brasil Departamento de Física, Universidade Federal de Santa Catarina, 88040-900 Florianópolis, SC, Brasil
Abstract

We develop the formal approach to the angular spectrum transfer in parametric down-conversion that allows pumping with a structured beam. The scheme is based on an entangled photon source pumped by a laser beam structured with a vector vortex polarization profile. This creates a two-photon quantum state exhibiting polarization-dependent transverse correlations that can be accessed through coincidence measurements on the spatially separated photons. The calculated correlations are shown to present a spin-orbit profile typical of vector beams, however, distributed on separate measurement regions. Our approach allows the design of the pump beam vector spatial structure and measurement strategies for potential applications of these entangled states, such as in quantum communication.

pacs:
03.65.Vf, 03.67.Mn, 42.50.Dv

I introduction

Quantum technologies based on photonic devices require coherent control of different optical degrees of freedom. The interplay between polarization and transverse modes of a laser beam has been successfully exploited in a number of quantum and classical experiments. In the quantum domain we can quote different applications, such as simulations of quantum algorithms de Oliveira et al. (2005); Souza and Khoury (2010); Pinheiro et al. (2013), quantum random walks Goyal et al. (2013), environment-induced entanglement Hor-Meyll et al. (2009); Passos et al. (2018), decoherence Obando et al. (2019), non-Markovianity Passos et al. (2019), quantum communications Milione et al. (2015) and quantum sensing Töppel et al. (2014); Berg-Johansen et al. (2015). The structural non-separability between polarization and transverse modes has been approached from different points of view Simon et al. (2010); Holleczek et al. (2011); Qian and Eberly (2011); Pereira et al. (2014); Ghose and Mukherjee (2014); Aiello et al. (2015); Mc Laren et al. (2015); Qian et al. (2016) and has been used for investigating important properties of entangled states Souza et al. (2007); Borges et al. (2010); Kagalwala et al. (2013). Quantum inspired experiments in classical optics has led to exciting applications, such as the mode transfer between different degrees of freedom using the teleportation algorithm Hashemi Rafsanjani et al. (2015); da Silva et al. (2016); Guzman-Silva et al. (2016). Tripartite non-separability has also been studied in classical optics using polarization, transverse and longitudinal modesBalthazar et al. (2016). All these developments were considerably favored by the development of important tools for spin-orbit coupling in laser beams Nagali et al. (2009); Barreiro et al. (2010); Karimi et al. (2010); Cardano et al. (2012). These developments made possible the implementation of alignment-free quantum cryptography with vector beams Souza et al. (2008); D’Ambrosio et al. (2012); Guo et al. (2020). The orbital angular momentum can be also combined with other degrees of freedom like optical path to generate hyper-entanglement between two quantum memories Zhang et al. (2016).

In the quantum domain, the interplay between polarization and spatial coherence of entangled photon pairs was approached and has demonstrated quantum image control through polarization entanglement in spontaneous parametric down-conversion (SPDC)Santos et al. (2001); Caetano et al. (2003). SPDC is a reliable source of photon pairs entangled in different degrees of freedom Walborn et al. (2010). The phase-matching conditions fulfilled by parametric interaction impose time-energy, space-momentum and polarization constraints that are at the origin of multiple quantum correlations characterizing entanglement between the generated photons. This multiple entanglement in different degrees of freedom is sometimes referred to as hyper-entanglement Barreiro et al. (2005).

In this work we present the theoretical description of photon pairs, simultaneously entangled in spin and orbital angular momentum, generated by the two-crystal-sandwich SPDC source Kwiat et al. (1999) pumped by a vector vortex beam. The approach is an extension of the theory that describes the angular spectrum transfer in parametric down-conversion Monken et al. (1998), for the case of structured beams. The polarization dependent spatial correlations between the down-converted beams are calculated using the formal approach developed in Ref. Caetano et al. (2003), where a setup with similar characteristics was implemented. While the individual intensity distributions of the down-converted beams do not carry the pump spin-orbit properties, the quantum correlations between them exhibit the typical polarization dependent spatial distribution of a vector beam, as expected and experimentally observed in Ref. Jabir et al. (2017). We explore and illustrate our approach by calculating the polarization-dependent transverse spatial correlations of the two-photon quantum state for a few interesting cases, including the correlations in the orbital angular momentum (OAM) basis.

II Experimental Scheme

Let us consider a frequently used source of polarization entangled photon pairs also known as two-crystal-sandwich SPDC source Kwiat et al. (1999). It is composed by two identical non-linear crystals placed close together with their optical axes rotated by 90o90^{o} relative to each other, as shown in Fig. 1. A laser beam at frequency ωp\omega_{p} and wavector 𝐤p\mathbf{k}_{p} is used to pump the crystals and generate photon pairs by spontaneous parametric down-conversion (SPDC). The down-converted signal (ss) and idler (ii) photons are generated with frequencies ωs\omega_{s} and ωi\omega_{i}\,, and wave vectors 𝐤s\mathbf{k}_{s} and 𝐤i\mathbf{k}_{i}\,, constrained by ωs+ωi=ωp\omega_{s}+\omega_{i}=\omega_{p} and 𝐤s+𝐤i=𝐤p\mathbf{k}_{s}+\mathbf{k}_{i}=\mathbf{k}_{p}\,, that express energy and momentum conservation, respectively.

The pump beam is assumed to be prepared in a vector-vortex mode of the kind

Ψ​(𝐫)=ψ1​(𝐫)​𝐞^H+ψ2​(𝐫)​𝐞^V2,\Psi(\mathbf{r})=\frac{\psi_{1}(\mathbf{r})\mathbf{\hat{e}}_{H}+\psi_{2}(\mathbf{r})\mathbf{\hat{e}}_{V}}{\sqrt{2}}\;, (1)

where Ψ​(𝐫)\Psi(\mathbf{r}) is the classical field amplitude, 𝐞^H\mathbf{\hat{e}}_{H} and 𝐞^V\mathbf{\hat{e}}_{V} are the horizontal and vertical polarization unit vectors, respectively, and ψ1​(𝐫)\psi_{1}(\mathbf{r}) and ψ2​(𝐫)\psi_{2}(\mathbf{r}) are two orthonormal functions that are solutions of the paraxial wave equation Yariv (1989). Those can be either Hermite-Gaussian (HG) or Laguerre-Gaussian (LG) modes, for example. The LG modes are eigenstates of orbital angular momentum (OAM), that can be carried in multiples of ℏ\hbar by each single photon. The OAM carried by each photon is given by l​ℏl\hbar\,, where l∈ℤl\in\mathbb{Z} is the mode topological charge. The internal non-separability between the spin-orbit degrees of freedom can be evidenced by measuring the spatial mode after polarization filtering. For example, if the vector-vortex mode given by (1) passes through a polarization analyzer (a sequence of a quarter waveplate (QWP) and a half waveplate (HWP)followed by a horizontal polarizer, for example) characterized by angles γ\gamma (QWP) and θ/2\theta/2 (HWP) with respect to the horizontal, the transmitted beam will exhibit a spatial function that is the linear combination

ψγ​θ​(𝐫)=cos⁡θ​ψ1​(𝐫)+ei​γ​sin⁡θ​ψ2​(𝐫).\psi_{\gamma\theta}(\mathbf{r})=\cos\theta\,\psi_{1}(\mathbf{r})+e^{i\gamma}\sin\theta\,\psi_{2}(\mathbf{r})\;. (2)

Therefore, a variable spatial profile is manifested after polarization projection. Interestingly, this feature can be transferred to the spatial quantum correlations between signal and idler photons generated by SPDC.

Under type-I phase-matching, the vertically polarized component of the pump beam generates a pair of horizontally polarized photons in the first crystal and transfers its accompanying transverse mode ψ1\psi_{1} to the spatial quantum correlations of the down-converted photons Monken et al. (1998). In the same way, the transverse mode ψ2\psi_{2} is transferred to the spatial quantum correlations of the vertically polarized down-converted photons generated in the second crystal. If the coherence length of the pump laser is larger than the length of the two-crystal source, vertically and horizontally polarized signal-idler modes will add coherently and produce a two-photon vector vortex state. A subtle and interesting effect takes place when the polarization information of the down-converted photons is erased by means of a variable polarization measurement. Two polarizers are used to set the measurement bases before the photocounts are acquired by single-photon counting modules (SPCM), which can be scanned to register the polarization-dependent spatial correlations. As we will show, these correlations exhibit typical features of vector beams, where polarization filtering is accompanied by a variable spatial profile. However, in our proposal this spin-orbit cross-talk is nonlocal. Note that these correlation images are generated by fixing either the signal or the idler position and scanning the other. Alternatively, the spatially dependent coincidence counts can be registered by currently available single-photon counting cameras.

Refer to caption
Figure 1: Setup for the generation of two-photon vector vortex states. Two non-linear crystals cut for type-I phase-matching are placed close together with their optical axes rotated with respect to each other. A pump beam prepared in a vector-vortex mode can generate either horizontally polarized photons in the first crystal or vertically polarized photons in the second. Before detection, a polarization analysis (PA) is performed in each detection arm. The polarization analyzers are set to (θs/2​(H​W​P),γs​(Q​W​P))(\theta_{s}/2(HWP),\gamma_{s}(QWP)) for the signal beam and (θi/2​(H​W​P),γi​(Q​W​P))(\theta_{i}/2(HWP),\gamma_{i}(QWP)) for the idler. After polarization analysis, the down-converted photons hit the single-photon counting modules (SPCM) that can be scanned to register the polarization-dependent spatial correlations.

III Theoretical model

Following the sketch of section II, we now develop a theoretical approach for the spin-orbit quantum correlations between signal and idler fields. Our strategy will be first to obtain the quantum state produced by the SPDC process and then use it to evaluate the spatial distribution of signal-idler intensity correlations. Let us start by writing the positive and negative frequency parts of the electric field operator of pump, signal and idler beams as a superposition of plane waves with vertical and horizontal polarization

𝐄^j+​(𝐫,t)\displaystyle\mathbf{\hat{E}}^{+}_{j}(\mathbf{r},t) =\displaystyle= (E^j​H+​(𝐫)​𝐞^H+E^j​V+​(𝐫)​𝐞^V)​e−i​ωj​t,\displaystyle\left(\hat{E}^{+}_{jH}(\mathbf{r})\,\mathbf{\hat{e}}_{H}+\hat{E}^{+}_{jV}(\mathbf{r})\,\mathbf{\hat{e}}_{V}\right)\,e^{-i\omega_{j}t}\;,
𝐄^j−​(𝐫,t)\displaystyle\mathbf{\hat{E}}^{-}_{j}(\mathbf{r},t) =\displaystyle= [𝐄^j+​(𝐫,t)]†,\displaystyle\left[\mathbf{\hat{E}}^{+}_{j}(\mathbf{r},t)\right]^{\dagger}\;, (3)
E^j​μ+​(𝐫)\displaystyle\hat{E}^{+}_{j\mu}(\mathbf{r}) =\displaystyle= i​ℰj​∫a^μj​(𝐤j)​ei​𝐤j⋅𝐫​d3​𝐤j,\displaystyle i\,\mathcal{E}_{j}\int\hat{a}^{j}_{\mu}(\mathbf{k}_{j})\,e^{i\mathbf{k}_{j}\cdot\mathbf{r}}\,d^{3}\mathbf{k}_{j}\;,

where j=p,s,ij=p,s,i\,; a^μj​(𝐤j)\hat{a}^{j}_{\mu}(\mathbf{k}_{j}) is the annihilation operator of photons with wave vector 𝐤j\mathbf{k}_{j} and polarization μ=H,V\mu=H,V\,, and ℰj\mathcal{E}_{j} is a constant resulting from the quantization process and having units of electric field.

The non-linear coupling between pump, signal and idler is described by the interaction Hamiltonian

HI​(t)\displaystyle H_{I}(t) =\displaystyle= χ​e−i​Δ​ω​t​OI+χ​ei​Δ​ω​t​OI†,\displaystyle\chi\,e^{-i\Delta\omega\,t}\,O_{I}+\chi\,e^{i\Delta\omega\,t}\,O_{I}^{\dagger}\;, (4)
OI\displaystyle O_{I} =\displaystyle= ∫𝒱(E^p​V+​E^s​H−​E^i​H−+E^p​H+​E^s​V−​E^i​V−)​d3​𝐫,\displaystyle\int_{\mathcal{V}}\left(\hat{E}^{+}_{pV}\,\hat{E}^{-}_{sH}\,\hat{E}^{-}_{iH}+\hat{E}^{+}_{pH}\,\hat{E}^{-}_{sV}\,\hat{E}^{-}_{iV}\right)d^{3}\mathbf{r}\;,

where χ\chi is the non-linear susceptibility, 𝒱\mathcal{V} is the crystal volume and Δ​ω=ωp−ωs−ωi\Delta\omega=\omega_{p}-\omega_{s}-\omega_{i}\,. The first term inside the integral describes the annihilation of a VV polarized photon of the pump and creation of HH polarized signal and idler photons in the first crystal. The second term describes the annihilation of an HH polarized photon of the pump and the creation of VV polarized signal and idler photons in the second crystal.

III.1 Two-photon spin-orbit quantum state

Up to first-order in perturbation theory, the time evolution operator in the interaction picture is

U(1)​(τ)=𝟙−iℏ​∫0τHI​(t)​𝑑t\displaystyle U^{(1)}(\tau)=\mathbb{1}-\frac{i}{\hbar}\int_{0}^{\tau}H_{I}(t)\,dt (5)
=𝟙−i​χℏ​sin⁡(Δ​ω​τ/2)Δ​ω/2​(e−i2​Δ​ω​τ​OI+ei2​Δ​ω​τ​OI†),\displaystyle=\mathbb{1}-\frac{i\chi}{\hbar}\,\frac{\sin\left(\Delta\omega\,\tau/2\right)}{\Delta\omega/2}\,\left(e^{-\frac{i}{2}\Delta\omega\,\tau}\,O_{I}+e^{\frac{i}{2}\Delta\omega\,\tau}\,O_{I}^{\dagger}\right)\;,

where τ\tau is the interaction time, which is assumed to be the same for all three fields, pump, signal, and idler, under phase-matching conditions. As we can see, the longer the interaction time, the tighter the energy conservation condition (Δ​ω=0\Delta\omega=0). At this point we can make the monochromatic approximation for the pump laser and assume the interaction time long enough to impose practically perfect energy conservation. In this case, the time evolution operator becomes

U(1)​(τ)=𝟙−i​χ​τℏ​(OI+OI†).\displaystyle U^{(1)}(\tau)=\mathbb{1}-\frac{i\chi\tau}{\hbar}\,\left(O_{I}+O_{I}^{\dagger}\right)\;. (6)

After passing through the crystals, the quantum state of the interacting beams is given by

|Ψ​(τ)⟩=U(1)​(τ)​|Ψ​(0)⟩,\displaystyle\ket{\Psi(\tau)}=U^{(1)}(\tau)\ket{\Psi(0)}\;, (7)

where |Ψ​(0)⟩=|ψ0⟩p​|0⟩s​|0⟩i\ket{\Psi(0)}=\ket{\psi_{0}}_{p}\ket{0}_{s}\ket{0}_{i} is the input state of pump, signal and idler fields. Since signal and idler are initially in the vacuum state, no contribution to the time evolution can appear from the OI†O_{I}^{\dagger} term because its action involves the annihilation of signal and idler photons. The pump laser will be treated as a monochromatic beam, described by a multimode coherent state

|ψ0⟩p\displaystyle\ket{\psi_{0}}_{p} =\displaystyle= ∏𝐤p|vH​(𝐤p)⟩H​|vV​(𝐤p)⟩V,\displaystyle\prod_{\mathbf{k}_{p}}\ket{v_{H}(\mathbf{k}_{p})}_{H}\ket{v_{V}(\mathbf{k}_{p})}_{V}\;, (8)

where

a^μ​(𝐤)​|vμ​(𝐤)⟩μ\displaystyle\hat{a}_{\mu}(\mathbf{k})\ket{v_{\mu}(\mathbf{k})}_{\mu}\ =\displaystyle= vμ​(𝐤)​|vμ​(𝐤)⟩μ,\displaystyle v_{\mu}(\mathbf{k})\ket{v_{\mu}(\mathbf{k})}_{\mu}\;, (9)

and vμ​(𝐤)v_{\mu}(\mathbf{k}) is the coherent state amplitude associated with wave vector 𝐤\mathbf{k} and polarization μ\mu\,. After parametric interaction, the quantum state of the pump, signal and idler modes is given by

|Ψ​(τ)⟩=|Ψ​(0)⟩−i​χ​τℏ​OI​|Ψ​(0)⟩.\displaystyle\ket{\Psi(\tau)}=\ket{\Psi(0)}-\frac{i\chi\tau}{\hbar}\,O_{I}\ket{\Psi(0)}\;. (10)

Let us work out the interaction term:

OI=−i​ℰp​ℰs​ℰi​∫d3​𝐤p​∫d3​𝐤s​∫d3​𝐤i​F​(𝐤p,𝐤s,𝐤i)\displaystyle O_{I}=-i\,\mathcal{E}_{p}\mathcal{E}_{s}\mathcal{E}_{i}\int d^{3}\mathbf{k}_{p}\int d^{3}\mathbf{k}_{s}\int d^{3}\mathbf{k}_{i}\,F(\mathbf{k}_{p},\mathbf{k}_{s},\mathbf{k}_{i})
[a^Vp​(𝐤p)​a^Hs⁣†​(𝐤s)​a^Hi⁣†​(𝐤i)+a^Hp​(𝐤p)​a^Vs⁣†​(𝐤s)​a^Vi⁣†​(𝐤i)],\displaystyle\left[\hat{a}^{p}_{V}(\mathbf{k}_{p})\hat{a}^{s\,\dagger}_{H}(\mathbf{k}_{s})\hat{a}^{i\,\dagger}_{H}(\mathbf{k}_{i})+\hat{a}^{p}_{H}(\mathbf{k}_{p})\hat{a}^{s\,\dagger}_{V}(\mathbf{k}_{s})\hat{a}^{i\,\dagger}_{V}(\mathbf{k}_{i})\right], (11)

where we defined the phase-matching function as

F​(𝐤p,𝐤s,𝐤i)\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!F(\mathbf{k}_{p},\mathbf{k}_{s},\mathbf{k}_{i})\! =\displaystyle= ∫𝒱ei​(𝐤p−𝐤s−𝐤i)⋅𝐫​d3​𝐫\displaystyle\!\!\int_{\mathcal{V}}e^{i\left(\mathbf{k}_{p}-\mathbf{k}_{s}-\mathbf{k}_{i}\right)\cdot\mathbf{r}}\,d^{3}\mathbf{r} (12)
≈\displaystyle\approx Lz​∏l=x,y2​sin⁡[(kp​l−ks​l−ki​l)​Ll/2]kp​l−ks​l−ki​l,\displaystyle\!\!\!L_{z}\!\!\prod_{l=x,y}\frac{2\sin\left[\left(k_{pl}-k_{sl}-k_{il}\right)L_{l}/2\right]}{k_{pl}-k_{sl}-k_{il}},

with LlL_{l} being the crystal width along the ll direction. Note that we have assumed a longitudinally thin crystal satisfying (kp​z−ks​z−ki​z)​Lz≪1(k_{pz}-k_{sz}-k_{iz})L_{z}\ll 1\,. This results in

OI​|Ψ​(0)⟩=\displaystyle O_{I}\ket{\Psi(0)}= (13)
−i​ℰp​ℰs​ℰi​∫d3​𝐤p​∫d3​𝐤s​∫d3​𝐤i​F​(𝐤p,𝐤s,𝐤i)​|ψ0⟩p\displaystyle-i\,\mathcal{E}_{p}\mathcal{E}_{s}\mathcal{E}_{i}\int d^{3}\mathbf{k}_{p}\int d^{3}\mathbf{k}_{s}\int d^{3}\mathbf{k}_{i}\,F(\mathbf{k}_{p},\mathbf{k}_{s},\mathbf{k}_{i})\,\ket{\psi_{0}}_{p}
⊗[​vVp​(𝐤p)​|1𝐤s,H⟩​|1𝐤i,H⟩+vHp​(𝐤p)​|1𝐤s,V⟩​|1𝐤i,V⟩],\displaystyle\otimes\left[\frac{}{}v^{p}_{V}(\mathbf{k}_{p})\ket{1_{\mathbf{k}_{s},H}}\ket{1_{\mathbf{k}_{i},H}}+v^{p}_{H}(\mathbf{k}_{p})\ket{1_{\mathbf{k}_{s},V}}\ket{1_{\mathbf{k}_{i},V}}\right],

where |1𝐤,μ⟩\ket{1_{\mathbf{k},\mu}} is a single-photon Fock state with wave vector 𝐤\mathbf{k} and polarization μ\mu\,. We assume that the pump beam comes from a collimated and monochromatic laser propagating along the zz direction, so that we can approximate

vμp​(𝐤p)≈vμp​(𝐪p)​δ​(kp​z−k0),\displaystyle v^{p}_{\mu}(\mathbf{k}_{p})\approx v^{p}_{\mu}(\mathbf{q}_{p})\,\delta(k_{pz}-\,k_{0})\,, (14)

where 𝐪p\mathbf{q}_{p} is the transverse and k0k_{0} is the longitudinal wave vector component. Moreover, signal and idler photons are detected at small solid angles along specific directions compatible with the phase-matching condition. This geometric configuration, together with interference filters placed before the detectors, fix the selected wavelengths of signal and idler. This also restricts their detected wave vectors to a small neighborhood around their respective solid angles Ωj\Omega_{j} (j=s,ij=s,i). It will be useful to decompose the wavectors into longitudinal (𝐤j∥\mathbf{k}^{\parallel}_{j}) and transverse (𝐪j\mathbf{q}_{j}) parts 𝐤j=𝐤j∥+𝐪j\mathbf{k}_{j}=\mathbf{k}^{\parallel}_{j}+\mathbf{q}_{j}\,, and apply the paraxial approximation |𝐪j|≪|𝐤j∥||\mathbf{q}_{j}|\ll|\mathbf{k}^{\parallel}_{j}|\,. In this case,

kj∥=kj2−qj2≈kj−ql22​kj.\displaystyle k^{\parallel}_{j}=\sqrt{k_{j}^{2}-q_{j}^{2}}\approx k_{j}-\frac{q_{l}^{2}}{2k_{j}}\,. (15)

In the paraxial regime and with fixed wavelengths for pump, signal and idler, the longitudinal wave vector components are fixed and the relevant plane wave modes can be labeled by the transverse wave vector 𝐪\mathbf{q}\,. The amplitude distribution vμp​(𝐪)v^{p}_{\mu}(\mathbf{q}) represents the angular spectrum carried by the pump polarization mode μ\mu\,. The triple integrals in Eq. (13) are reduced to double integrals over a small domain Ωj\Omega_{j} around the main longitudinal component, as illustrated in Fig. 2.

Refer to caption
Figure 2: Representation of the transverse momentum domains for the pump, signal and idler beams.

Under these assumptions, the two-photon quantum state can be written in terms of the transverse wavector distributions,

OI​|Ψ​(0)⟩=\displaystyle O_{I}\ket{\Psi(0)}= (16)
−i​ℰp​ℰs​ℰi​∫Ωpd2​𝐪p​∫Ωsd2​𝐪s​∫Ωid2​𝐪i​F​(𝐤p,𝐤s,𝐤i)​|ψ0⟩p\displaystyle-i\,\mathcal{E}_{p}\mathcal{E}_{s}\mathcal{E}_{i}\int_{\Omega_{p}}\!\!\!d^{2}\mathbf{q}_{p}\int_{\Omega_{s}}\!\!\!d^{2}\mathbf{q}_{s}\int_{\Omega_{i}}\!\!\!d^{2}\mathbf{q}_{i}\,F(\mathbf{k}_{p},\mathbf{k}_{s},\mathbf{k}_{i})\ket{\psi_{0}}_{p}
⊗[​vVp​(𝐪p)​|1𝐪s,H⟩​|1𝐪i,H⟩+vHp​(𝐪p)​|1𝐪s,V⟩​|1𝐪i,V⟩].\displaystyle\otimes\left[\frac{}{}v^{p}_{V}(\mathbf{q}_{p})\ket{1_{\mathbf{q}_{s},H}}\ket{1_{\mathbf{q}_{i},H}}+v^{p}_{H}(\mathbf{q}_{p})\ket{1_{\mathbf{q}_{s},V}}\ket{1_{\mathbf{q}_{i},V}}\right]\,.

In most experiments for producing spatial quantum correlations in SPDC, one deals with transversely wide [(kp​l−ks​l−ki​l)​Ll≫1(k_{pl}-k_{sl}-k_{il})L_{l}\gg 1 for l=x,yl=x,y] and longitudinally thin [(kp​z−ks​z−ki​z)​Lz≪1(k_{pz}-k_{sz}-k_{iz})L_{z}\ll 1] crystals. In this case, a tight phase-matching condition is imposed on the transverse components of the wave vectors and a loose condition applies to the longitudinal component. This approximation can be expressed as

F​(𝐤p,𝐤s,𝐤i)≈Lz​δ​(𝐪p−𝐪s−𝐪i).\displaystyle F\left(\mathbf{k}_{p},\mathbf{k}_{s},\mathbf{k}_{i}\right)\approx L_{z}\,\delta\left(\mathbf{q}_{p}-\mathbf{q}_{s}-\mathbf{q}_{i}\right)\;. (17)

In this case, the integration over 𝐪p\mathbf{q}_{p} can be readily performed and the final expression for the quantum state produced by the SPDC process is

|Ψ​(τ)⟩=|ψ0⟩p⊗(|0⟩s​|0⟩i+|Φ⟩s​i),\displaystyle\ket{\Psi(\tau)}=\ket{\psi_{0}}_{p}\otimes\left(\ket{0}_{s}\ket{0}_{i}+\ket{\Phi}_{si}\right)\;, (18)

where

|Φ⟩s​i\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\ket{\Phi}_{si}\!\! =\displaystyle= κ​∫d2​𝐪s​∫d2​𝐪i​vVp​(𝐪s+𝐪i)​|1𝐪s,H⟩​|1𝐪i,H⟩\displaystyle\!\!\kappa\int d^{2}\mathbf{q}_{s}\int d^{2}\mathbf{q}_{i}\,v^{p}_{V}(\mathbf{q}_{s}+\mathbf{q}_{i})\ket{1_{\mathbf{q}_{s},H}}\ket{1_{\mathbf{q}_{i},H}} (19)
+\displaystyle+ κ​∫d2​𝐪s​∫d2​𝐪i​vHp​(𝐪s+𝐪i)​|1𝐪s,V⟩​|1𝐪i,V⟩,\displaystyle\!\!\,\kappa\int d^{2}\mathbf{q}_{s}\int d^{2}\mathbf{q}_{i}\,v^{p}_{H}(\mathbf{q}_{s}+\mathbf{q}_{i})\ket{1_{\mathbf{q}_{s},V}}\ket{1_{\mathbf{q}_{i},V}}\!,

and κ≡−(χ/ℏ)​τ​Lz​ℰp​ℰs​ℰi\kappa\equiv-(\chi/\hbar)\tau L_{z}\mathcal{E}_{p}\mathcal{E}_{s}\mathcal{E}_{i}\,. The expression in (19) encompasses the transverse momentum constraint between signal and idler that gives rise to spatial quantum correlations.

III.2 Spatial quantum correlations

In the paraxial regime, we can adopt a scalar diffraction theory to study the propagation of the interacting beams after leaving the crystals. The relevant plane wave modes are labeled by the transverse wave vector 𝐪\mathbf{q} and the pump amplitude distribution vμp​(𝐪)v^{p}_{\mu}(\mathbf{q}) represents the pump angular spectrum carried by the polarization mode μ\mu\,. At the crystals’ center (z=0z=0), the spatial distribution of the pump beam in each polarization component is given by the following Fourier decomposition,

Wμ​(𝝆,0)=∫vμp​(𝐪)​ei​𝐪⋅𝝆​d2​𝐪,\displaystyle W_{\mu}(\bm{\rho},0)=\int v^{p}_{\mu}(\mathbf{q})\,e^{i\mathbf{q}\cdot\bm{\rho}}\,d^{2}\mathbf{q}\,, (20)

and the propagated field distribution is given by the Fresnel integral

Wμ​(𝝆,z)=ei​k​z​∫vμp​(𝐪)​ei​(𝐪⋅𝝆−q22​k​z)​d2​𝐪.\displaystyle W_{\mu}(\bm{\rho},z)=e^{ikz}\int v^{p}_{\mu}(\mathbf{q})\,e^{i\left(\mathbf{q}\cdot\bm{\rho}-\frac{q^{2}}{2k}z\right)}\,d^{2}\mathbf{q}\,. (21)

The multimode coherent state |ψ0⟩p\ket{\psi_{0}}_{p} carries the spatial properties of the pump beam in the Fourier domain through the angular spectra vH​(𝐪)v_{H}(\mathbf{q}) and vV​(𝐪)v_{V}(\mathbf{q})\,, independently imprinted in each pump polarization component. This will be crucial for the polarization-dependent spatial correlations between signal and idler photons.

The longitudinal positions of signal and idler detectors are fixed and the spatial quantum correlations are measured as a function of their transverse position 𝝆j\bm{\rho}_{j} (j=s,ij=s,i). We recall that a polarization analyzer is placed before each detector. If we assume that these analyzers are set at angles γj\gamma_{j} and θj/2\theta_{j}/2\,, then the electric field operator in each detector is

E^j′​(𝝆j)=cos⁡θj​E^j​H​(𝝆j)+sin⁡θj​ei​γj​E^j​V​(𝝆j).\displaystyle\hat{E}_{j}^{\,\prime}(\bm{\rho}_{j})=\cos\theta_{j}\,\hat{E}_{jH}(\bm{\rho}_{j})+\sin\theta_{j}\,e^{i\gamma_{j}}\,\hat{E}_{jV}(\bm{\rho}_{j})\,. (22)

The intensity distribution in each detection arm is

I​(𝝆j)\displaystyle I(\bm{\rho}_{j}) =\displaystyle= ⟨E^j′⁣−​(𝝆j)​E^j′⁣+​(𝝆j)⟩\displaystyle\langle\hat{E}^{\,\prime-}_{j}(\bm{\rho}_{j})\hat{E}^{\,\prime+}_{j}(\bm{\rho}_{j})\rangle (23)
=\displaystyle= ∥E^j′⁣+​(𝝆j)​|Ψ​(τ)⟩∥2,\displaystyle\lVert\hat{E}^{\,\prime+}_{j}(\bm{\rho}_{j})\ket{\Psi(\tau)}\rVert^{2}\,,

and the intensity correlations between the two detection arms are

C​(𝝆s,𝝆i)\displaystyle C(\bm{\rho}_{s},\bm{\rho}_{i}) =\displaystyle= ⟨E^s′⁣−​(𝝆s)​E^i′⁣−​(𝝆i)​E^i′⁣+​(𝝆i)​E^s′⁣+​(𝝆s)⟩\displaystyle\langle\hat{E}^{\,\prime-}_{s}(\bm{\rho}_{s})\hat{E}^{\,\prime-}_{i}(\bm{\rho}_{i})\hat{E}^{\,\prime+}_{i}(\bm{\rho}_{i})\hat{E}^{\,\prime+}_{s}(\bm{\rho}_{s})\rangle (24)
=\displaystyle= ∥E^i′⁣+​(𝝆i)​E^s′⁣+​(𝝆s)​|Ψ​(τ)⟩∥2.\displaystyle\lVert\hat{E}^{\,\prime+}_{i}(\bm{\rho}_{i})\hat{E}^{\,\prime+}_{s}(\bm{\rho}_{s})\ket{\Psi(\tau)}\rVert^{2}\,.

In the monochromatic and paraxial approximations, the electric field operators can be written as

E^j​μ+​(𝝆)\displaystyle\hat{E}^{+}_{j\mu}(\bm{\rho}) =\displaystyle= i​ℰj​∫a^μj​(𝐪j)​ei​[𝐪j⋅𝝆+φ​(𝐪j)]​d2​𝐪j,\displaystyle i\,\mathcal{E}_{j}\int\hat{a}^{j}_{\mu}(\mathbf{q}_{j})\,e^{i\left[\mathbf{q}_{j}\cdot\bm{\rho}+\varphi(\mathbf{q}_{j})\right]}\,d^{2}\mathbf{q}_{j}\;,
φ​(𝐪j)\displaystyle\varphi(\mathbf{q}_{j}) =\displaystyle= kj2−qj2​zj≈kj​zj−qj2​zj2​kj,\displaystyle\sqrt{k_{j}^{2}-q_{j}^{2}\,}\,\,z_{j}\approx k_{j}\,z_{j}-\frac{q_{j}^{2}z_{j}}{2k_{j}}\;, (25)

where zjz_{j} is the longitudinal distance between the crystals’ center and detector jj .

Note that no contribution comes from the vacuum component in |Ψ​(τ)⟩\ket{\Psi(\tau)}\,, so that we only need to care about contributions coming from |Φ⟩s​i\ket{\Phi}_{si}\,. The calculation of the intensity distributions and correlations will be significantly simplified by the definition of the following vectors

|αj​μ⟩\displaystyle\ket{\alpha_{j\mu}} =\displaystyle= E^j​μ+​(𝝆j)​|Φ⟩s​i,\displaystyle\hat{E}^{+}_{j\mu}(\bm{\rho}_{j})\ket{\Phi}_{si}\;,
|βμ⟩\displaystyle\ket{\beta_{\mu}} =\displaystyle= E^i​μ+​(𝝆i)​E^s​μ+​(𝝆s)​|Φ⟩s​i.\displaystyle\hat{E}^{+}_{i\mu}(\bm{\rho}_{i})\hat{E}^{+}_{s\mu}(\bm{\rho}_{s})\ket{\Phi}_{si}\;. (26)

We can easily workout these auxiliary vectors using

a^μ​(𝐪′)​|n𝐪,ν⟩=δμ​ν​δ​(𝐪−𝐪′)​n𝐪,ν​|n𝐪,ν−1⟩.\displaystyle\hat{a}_{\mu}(\mathbf{q}^{\prime})\ket{n_{\mathbf{q},\nu}}=\delta_{\mu\nu}\delta(\mathbf{q}-\mathbf{q}^{\prime})\sqrt{n_{\mathbf{q},\nu}}\ket{n_{\mathbf{q},\nu}-1}\;. (27)

As detailed in Appendix A, the result for the individual intensities of signal and idler is

Ij=cos2⁡θj​ℐH+sin2⁡θj​ℐV,I_{j}=\cos^{2}\theta_{j}\mathcal{I}_{H}+\sin^{2}\theta_{j}\mathcal{I}_{V}\,, (28)

where j=s,ij=s,i and

ℐH​(V)=κ2​ℰs2​∫d2​𝝆​|WV​(H)​(𝝆,z)|2.\mathcal{I}_{H(V)}=\kappa^{2}\mathcal{E}_{s}^{2}\,\int d^{2}\bm{\rho}\lvert W_{V(H)}\left(\bm{\rho},z\right)\rvert^{2}\,. (29)

Note that the pump’s spatial properties are washed out in the individual intensities. In contrast, the coincidence count distribution carries the spatial profile of the pump beam, distributed in the joint coordinates of signal and idler. The result derived in Appendix B is

C​(𝝆s,𝝆i)\displaystyle\!\!\!\!C(\bm{\rho}_{s},\bm{\rho}_{i}) =\displaystyle= K2|​cos⁡θs​cos⁡θi​WV​(𝝆+,z)\displaystyle K^{2}\left|\frac{}{}\!\cos\theta_{s}\cos\theta_{i}\,W_{V}(\bm{\rho}_{+},z)\right. (30)
+\displaystyle+ sinθssinθiei​(γs+γi)WH(𝝆+,z)|2,\displaystyle\left.\sin\theta_{s}\sin\theta_{i}\,e^{i(\gamma_{s}+\gamma_{i})}\,W_{H}(\bm{\rho}_{+},z)\right|^{2},

where we assume degenerate SPDC (ks=ki=kp/2k_{s}=k_{i}=k_{p}/2) and equidistant signal and idler detectors (zs=zi=zz_{s}=z_{i}=z). In this case, we have

𝝆+=𝝆s+𝝆i2.\displaystyle\bm{\rho}_{+}=\frac{\bm{\rho}_{s}+\bm{\rho}_{i}}{2}\,. (31)

Note that Eq. (30) shows simultaneous dependence on the joint coordinates of signal and idler detectors and on the joint orientations of their respective polarization analyzers.

IV Nonlocal Vector Vortex Beam

We can now investigate the polarization dependent spatial correlations when the pump beam is prepared in a vector mode of the kind expressed in Eq.(1). For example, let the pump beam be prepared in a superposition of the Hermite-Gaussian mode (0,1)(0,1) with horizontal polarization and mode (1,0)(1,0) with vertical polarization so that

WV​(𝝆)=ψ10​(𝝆)2=xπ​w2​e−(x2+y2)/w2,\displaystyle W_{V}(\bm{\rho})=\frac{\psi_{10}(\bm{\rho})}{\sqrt{2}}=\frac{x}{\sqrt{\pi}\,w^{2}}\,e^{-(x^{2}+y^{2})/w^{2}}\,,
WH​(𝝆)=ψ01​(𝝆)2=yπ​w2​e−(x2+y2)/w2.\displaystyle W_{H}(\bm{\rho})=\frac{\psi_{01}(\bm{\rho})}{\sqrt{2}}=\frac{y}{\sqrt{\pi}\,w^{2}}\,e^{-(x^{2}+y^{2})/w^{2}}\,. (32)

The longitudinal dependence has been made implicit in the variation of the mode width w​(z)=w0​1+(z/zR)2w(z)=w_{0}\sqrt{1+(z/z_{R})^{2}}\,, where w0w_{0} is the mode waist and zR=π​w02/λz_{R}=\pi w_{0}^{2}/\lambda is the Rayleigh distance.

Refer to caption
Figure 3: Hermite-Gaussian coincidence pattern as a function of the signal coordinates with the idler detector fixed at the origin xi=yi=0x_{i}=y_{i}=0 and the signal polarization analyzer fixed at γs=0\gamma_{s}=0 and θs=π/4\theta_{s}=\pi/4\,. The HG pattern follows the rotation of the idler polarization.

Then, the expression given in (30) brings us to

C(𝝆s,𝝆i)=K2π​w4e−2​(xs+xi)2w2e−2​(ys+yi)2w2×\displaystyle C(\bm{\rho}_{s},\bm{\rho}_{i})=\frac{K^{2}}{\pi\,w^{4}}\,e^{-\frac{2(x_{s}+x_{i})^{2}}{w^{2}}}e^{-\frac{2(y_{s}+y_{i})^{2}}{w^{2}}}\times (33)
|cos⁡θs​cos⁡θi​(xs+xi)+sin⁡θs​sin⁡θi​ei​(γs+γi)​(ys+yi)|2.\displaystyle\left|\cos\theta_{s}\cos\theta_{i}\,(x_{s}+x_{i})+\sin\theta_{s}\,\sin\theta_{i}\,e^{i(\gamma_{s}+\gamma_{i})}(y_{s}+y_{i})\right|^{2}.

This expression shows simultaneous nonlocal behavior on position and polarization settings. The spatial distribution of the coincidence counts depends on the joint orientations of the detection polarizers and on the joint transverse positions of the detectors. The physical consequence of this double nonlocal behavior can be revealed by a simple measurement strategy. Let us set the signal polarization analyzer at γs=0\gamma_{s}=0 and θs=π/4\theta_{s}=\pi/4\,, so that the polarization information of the signal photons is erased, and the position of the idler detector kept at its origin xi=yi=0x_{i}=y_{i}=0\,. In this case, the resulting coincidence pattern becomes

C​(𝝆s,𝟎)=K22​π​w4​e−2​(xs2+ys2)w2​(cos⁡θi​xs+sin⁡θi​ei​γi​ys)2.C(\bm{\rho}_{s},\mathbf{0})=\frac{K^{2}}{2\pi\,w^{4}}\,e^{-\frac{2(x_{s}^{2}+y_{s}^{2})}{w^{2}}}\!\left(\cos\theta_{i}\,x_{s}+\sin\theta_{i}\,e^{i\gamma_{i}}y_{s}\right)^{2}. (34)

We can see that the resulting coincidence pattern corresponds to the intensity distribution of a first-order Hermite-Gaussian mode function of the signal position, transformed according to the parameters of the idler polarization analyzer. In Fig. 3 we plot this coincidence pattern as a function of the signal coordinates 𝝆s=(x,y)\bm{\rho}_{s}=(x,y), indicating that it follows the rotation of the idler polarization analyzer. This situation is similar to the one exhibited in Eq.(1), where the spatial profile after transmission of a single vector beam through a polarizer depends on the transmission angle. However, here we have the orientation of the spatial pattern determined by the angle of a remote polarizer. This effect can be useful for remote alignment of quantum cryptography stations or as a gyroscope. For example, in Refs. Souza et al. (2008); D’Ambrosio et al. (2012) it was demonstrated that the internal non-separability between the spin and orbital degrees of freedom can be used to implement alignment-free quantum cryptography, thanks to the rotational invariance of spin-orbit modes. However, this method has never been considered in connection with quantum cryptography protocols employing non-local polarization correlations Ekert (1991). The non-local spin-orbit correlations described here can be useful in this context.

V Pumping with Laguerre-Gaussian beams

It is also interesting to see how orbital angular momentum affects the nonlocal correlations. For this end we assume the pump mode to be prepared in a superposition of Laguerre-Gaussian modes with zero radial order and OAM −l-l with horizontal polarization and +l+l with vertical polarization so that

WV​(H)​(𝝆)=2|l|π​w2​(x±i​yw)|l|​e−(x2+y2)/w2.\displaystyle W_{V(H)}(\bm{\rho})=\sqrt{\frac{2^{\lvert l\rvert}}{\pi w^{2}}}\left(\frac{x\pm iy}{w}\right)^{\!\lvert l\rvert}e^{-(x^{2}+y^{2})/w^{2}}\,. (35)

Then, the coincidence counts are given by

C(𝝆s,𝝆i)=K2π​w4e−2​(xs+xi)2w2e−2​(ys+yi)2w2×\displaystyle C(\bm{\rho}_{s},\bm{\rho}_{i})=\frac{K^{2}}{\pi\,w^{4}}\,e^{-\frac{2(x_{s}+x_{i})^{2}}{w^{2}}}e^{-\frac{2(y_{s}+y_{i})^{2}}{w^{2}}}\times (36)
|cosθscosθi[(xs+xi)+i(ys+yi)]|l|+\displaystyle\left|\cos\theta_{s}\cos\theta_{i}\,\left[(x_{s}+x_{i})+i(y_{s}+y_{i})\right]^{\lvert l\rvert}+\right.
sinθssinθiei​(γs+γi)[(xs+xi)−i(ys+yi)]|l||2.\displaystyle\left.\sin\theta_{s}\,\sin\theta_{i}\,e^{i(\gamma_{s}+\gamma_{i})}\left[(x_{s}+x_{i})-i(y_{s}+y_{i})\right]^{\lvert l\rvert}\right|^{2}.

As before, the measurement strategy to evidence the nonlocal spin-orbit correlations will be to fix the idler detector at its origin xi=yi=0x_{i}=y_{i}=0 and to fix the signal polarization analyzer at θs=π/4\theta_{s}=\pi/4 and γs=0\gamma_{s}=0\,. Then, the coincidence pattern as a function of the signal coordinates becomes

C(𝝆s,𝝆i)=K22​π​w4e−2​(xs2+ys2)w2×\displaystyle C(\bm{\rho}_{s},\bm{\rho}_{i})=\frac{K^{2}}{2\pi\,w^{4}}\,e^{-\frac{2(x_{s}^{2}+y_{s}^{2})}{w^{2}}}\times (37)
|cos⁡θi​(xs+i​ys)|l|+sin⁡θi​ei​γi​(xs−i​ys)|l||2.\displaystyle\left|\cos\theta_{i}\,\left(x_{s}+iy_{s}\right)^{\lvert l\rvert}+\sin\theta_{i}\,e^{i\gamma_{i}}\left(x_{s}-iy_{s}\right)^{\lvert l\rvert}\right|^{2}.

This result shows that any mode in the OAM Poincaré sphere Padgett and Courtial (1999) can be produced in the coincidence pattern by scanning of the signal detector and varying the idler polarization settings. A few examples are shown in Fig. 4 for l=3l=3\,, as measured in Ref.Jabir et al. (2017). As the idler polarization settings are changed, the coincidence pattern is modified.

Refer to caption
Figure 4: Laguerre-Gaussian coincidence pattern for an OAM pump beam with l=3l=3 as a function of the signal coordinates. The idler coordinates are fixed at the origin xi=yi=0x_{i}=y_{i}=0 and the signal polarization analyzer is set to γs=0\gamma_{s}=0 and θs=π/4\theta_{s}=\pi/4\,. The LG coincidence pattern is modified as the idler polarization settings are changed.

V.1 OAM quantum correlations in two-photon vector vortex beams

The two-photon spatial distribution can be written in the Laguerre-Gaussian basisTorres et al. (2003). However, it is interesting to extend this description for the case of the scheme with the two-photon-sandwich source. It will be useful for dealing with vector vortex pump beams. The LG modes are solutions of the paraxial wave equation in cylindrical coordinates. At the beam waist plane (z=0z=0), their mathematical expression in polar coordinates is

ψlp​(𝝆)=R|l|p​(ρ)​ei​l​ϕ,\displaystyle\psi^{p}_{l}(\bm{\rho})=R^{p}_{\lvert l\rvert}(\rho)\,e^{il\phi}\,,
R|l|p​(ρ)=(2/π)​p!(p+|l|)!​ρ~|l|w0​Lp|l|​(ρ~2)​e−ρ~2/2,\displaystyle R^{p}_{\lvert l\rvert}(\rho)=\sqrt{\frac{\left(2/\pi\right)\,p!}{\left(p+\lvert l\rvert\right)!}}\,\frac{\tilde{\rho}^{\lvert l\rvert}}{w_{0}}\,L_{p}^{\lvert l\rvert}(\tilde{\rho}^{2})\,e^{-\tilde{\rho}^{2}/2}\,,
ρ~=2​ρ/w0,\displaystyle\tilde{\rho}=\sqrt{2}\,\rho/w_{0}\,, (38)

where pp is the radial order, ll is the topological charge, (ρ,ϕ)(\rho,\phi) are the transverse coordinates in the polar system, w0w_{0} is the mode waist and Lp|l|L^{\lvert l\rvert}_{p} are the associated Laguerre polynomials. Pump, signal and idler beams are assumed to be mode matched, so that their wavefront radii are equal along the interaction length. This imposes a common Rayleigh distance zR​p=zR​s=zR​iz_{Rp}=z_{Rs}=z_{Ri}\,, which requires a different waist for each interacting beam according to zR=π​w0​j2/λjz_{R}=\pi w_{0j}^{2}/\lambda_{j}\,.

The two-photon quantum state generated by the SPDC process can be cast as a superposition of different partitions of the pump OAM between signal and idler. First, let us derive the LG expansion of the correlated transverse momentum distribution of signal and idler,

v​(𝐪s+𝐪i)=∑p1,l1p2,l2Al1​l2p1​p2​ψ~l1p1​(𝐪s)​ψ~l2p2​(𝐪i),\displaystyle v(\mathbf{q}_{s}+\mathbf{q}_{i})=\sum_{\begin{subarray}{c}p_{1},l_{1}\\ p_{2},l_{2}\end{subarray}}\,A^{p_{1}p_{2}}_{l_{1}l_{2}}\,\tilde{\psi}^{p_{1}}_{l_{1}}(\mathbf{q}_{s})\,\tilde{\psi}^{p_{2}}_{l_{2}}(\mathbf{q}_{i})\,, (39)

where {ψ~lp​(𝐪)}\{\tilde{\psi}^{p}_{l}(\mathbf{q})\} are the LG mode functions in Fourier domain,

ψ~lp​(𝐪)\displaystyle\tilde{\psi}^{p}_{l}(\mathbf{q}) =\displaystyle= 12​π​∫ψlp​(𝝆)​e−i​𝐪⋅𝝆​d2​𝝆,\displaystyle\frac{1}{2\pi}\int\psi^{p}_{l}(\bm{\rho})\,e^{-i\mathbf{q}\cdot\bm{\rho}}\,d^{2}\bm{\rho}\;,
ψlp​(𝝆)\displaystyle\psi^{p}_{l}(\bm{\rho}) =\displaystyle= 12​π​∫ψ~lp​(𝐪)​ei​𝐪⋅𝝆​d2​𝐪.\displaystyle\frac{1}{2\pi}\int\tilde{\psi}^{p}_{l}(\mathbf{q})\,e^{i\mathbf{q}\cdot\bm{\rho}}\,d^{2}\mathbf{q}\;. (40)

Both in the Fourier and position domains, the LG modes satisfy the following orthonormality

∫ψlp⁣∗​(𝝆)​ψl′p′​(𝝆)​d2​𝝆\displaystyle\int\psi^{p\,*}_{l}(\bm{\rho})\,\psi^{p^{\prime}}_{l^{\prime}}(\bm{\rho})\,d^{2}\bm{\rho} =\displaystyle= δp​p′​δl​l′,\displaystyle\delta_{pp^{\prime}}\,\delta_{ll^{\prime}}\;,
∫ψ~lp⁣∗​(𝐪)​ψ~l′p′​(𝐪)​d2​𝐪\displaystyle\int\tilde{\psi}^{p\,*}_{l}(\mathbf{q})\,\tilde{\psi}^{p^{\prime}}_{l^{\prime}}(\mathbf{q})\,d^{2}\mathbf{q} =\displaystyle= δp​p′​δl​l′,\displaystyle\delta_{pp^{\prime}}\,\delta_{ll^{\prime}}\;, (41)

and completeness

∑p,lψlp⁣∗​(𝝆)​ψlp​(𝝆′)\displaystyle\sum_{p,l}\psi^{p\,*}_{l}(\bm{\rho})\,\psi^{p}_{l}(\bm{\rho}^{\prime}) =\displaystyle= δ​(𝝆−𝝆′),\displaystyle\delta(\bm{\rho}-\bm{\rho}^{\prime})\;,
∑p,lψ~lp⁣∗​(𝐪)​ψ~lp​(𝐪′)\displaystyle\sum_{p,l}\tilde{\psi}^{p\,*}_{l}(\mathbf{q})\,\tilde{\psi}^{p}_{l}(\mathbf{q}^{\prime}) =\displaystyle= δ​(𝐪−𝐪′),\displaystyle\delta(\mathbf{q}-\mathbf{q}^{\prime})\;, (42)

relations. By using them, the LG expansion coefficients in Eq. (39) are given by

Al1​l2p1​p2=∫v​(𝐪s+𝐪i)​ψ~l1p1⁣∗​(𝐪s)​ψ~l2p2⁣∗​(𝐪i)​d2​𝐪s​d2​𝐪i.\displaystyle\!\!\!\!A^{p_{1}p_{2}}_{l_{1}l_{2}}=\int v(\mathbf{q}_{s}+\mathbf{q}_{i})\,\tilde{\psi}^{p_{1}\,*}_{l_{1}}(\mathbf{q}_{s})\,\tilde{\psi}^{p_{2}\,*}_{l_{2}}(\mathbf{q}_{i})\,d^{2}\mathbf{q}_{s}\,d^{2}\mathbf{q}_{i}\;. (43)

These coefficients are more easily calculated in the position domain. This can be achieved by plugging the inverse Fourier transform of ψ~lp​(𝐪)\tilde{\psi}^{p}_{l}(\mathbf{q}) and v​(𝐪)v(\mathbf{q}) into Eq. (43) and using the Fourier representation of the Dirac delta function. The resulting expression is

Al1​l2p1​p2=∫W​(𝝆)​ψl1p1⁣∗​(𝝆)​ψl2p2⁣∗​(𝝆)​d2​𝝆.\displaystyle\!\!\!\!A^{p_{1}p_{2}}_{l_{1}l_{2}}=\int W(\bm{\rho})\,\psi^{p_{1}\,*}_{l_{1}}(\bm{\rho})\,\psi^{p_{2}\,*}_{l_{2}}(\bm{\rho})\,d^{2}\bm{\rho}\,. (44)

With the expansion coefficients in hands, we can rewrite the two-photon state (19) in the Fock basis of OAM modes

|Φ⟩s​i=κ∑p1,l1p2,l2[Al1​l2p1​p2|p1,l1,H⟩s|p2,l2,H⟩i\displaystyle\ket{\Phi}_{si}=\kappa\sum_{\begin{subarray}{c}p_{1},l_{1}\\ p_{2},l_{2}\end{subarray}}\left[A^{p_{1}p_{2}}_{l_{1}l_{2}}\ket{p_{1},l_{1},H}_{s}\ket{p_{2},l_{2},H}_{i}\right.
+Bl1​l2p1​p2|p1,l1,V⟩s|p2,l2,V⟩i],\displaystyle\left.+\,\,B^{p_{1}p_{2}}_{l_{1}l_{2}}\ket{p_{1},l_{1},V}_{s}\ket{p_{2},l_{2},V}_{i}\right]\!, (45)

where

|p,l,μ⟩=∫d2​𝐪​ψ~lp​(𝐪)​|1𝐪,μ⟩(μ=H,V),\displaystyle\ket{p,l,\mu}=\int d^{2}\mathbf{q}\,\,\tilde{\psi}^{p}_{l}(\mathbf{q})\,\ket{1_{\mathbf{q},\mu}}\qquad(\mu=H,V)\,,
⟨p,l,μ|p′,l′,μ′⟩=δp​p′​δl​l′​δμ​μ′,\displaystyle\braket{p,l,\mu}{p^{\prime},l^{\prime},\mu^{\prime}}=\delta_{pp^{\prime}}\delta_{ll^{\prime}}\delta_{\mu\mu^{\prime}}\,, (46)

are single-photon OAM states with polarization μ\mu and the coefficients Al1​l2p1​p2A^{p_{1}p_{2}}_{l_{1}l_{2}} and Bl1​l2p1​p2B^{p_{1}p_{2}}_{l_{1}l_{2}} are given by Eq. (44) with WVW_{V} and WHW_{H}\,, respectively.

Let us assume that the pump beam is prepared in a vector mode of the kind considered in the second example of section IV, a superposition of the LG mode (0,−l)(0,-l) with horizontal polarization and mode (0,+l)(0,+l) with vertical polarization

𝚿​(𝝆)=ψ−l0​(𝝆)​𝐞^H+ψ+l0​(𝝆)​𝐞^V2.\displaystyle\bm{\Psi}(\bm{\rho})=\frac{\psi^{0}_{-l}(\bm{\rho})\,\mathbf{\hat{e}}_{H}+\psi^{0}_{+l}(\bm{\rho})\,\mathbf{\hat{e}}_{V}}{\sqrt{2}}\;. (47)

In this case, we have set WV=ψ+l0/2W_{V}=\psi^{0}_{+l}/\sqrt{2} and WH=ψ−l0/2W_{H}=\psi^{0}_{-l}/\sqrt{2} and the OAM expansion coefficients are

Al1​l2p1​p2=2​π​δl,l1+l2​∫R|l|0​(ρ)​R|l1|p1⁣∗​(ρ)​R|l−l1|p2⁣∗​(ρ)​ρ​𝑑ρ,\displaystyle A^{p_{1}p_{2}}_{l_{1}l_{2}}=\sqrt{2}\pi\delta_{l,l_{1}+l_{2}}\!\!\int\!\!R^{0}_{\lvert l\rvert}(\rho)\,R^{p_{1}\,*}_{\lvert l_{1}\rvert}(\rho)\,R^{p_{2}\,*}_{\lvert l-l_{1}\rvert}(\rho)\,\rho\,d\rho,
Bl1​l2p1​p2=2​π​δ−l,l1+l2​∫R|l|0​(ρ)​R|l1|p1⁣∗​(ρ)​R|l+l1|p2⁣∗​(ρ)​ρ​𝑑ρ,\displaystyle B^{p_{1}p_{2}}_{l_{1}l_{2}}=\sqrt{2}\pi\delta_{-l,l_{1}+l_{2}}\!\!\int\!\!R^{0}_{\lvert l\rvert}(\rho)\,R^{p_{1}\,*}_{\lvert l_{1}\rvert}(\rho)\,R^{p_{2}\,*}_{\lvert l+l_{1}\rvert}(\rho)\,\rho\,d\rho, (48)

where the Kronecker deltas, δl,l1+l2\delta_{l,l_{1}+l_{2}} and δ−l,l1+l2\delta_{-l,l_{1}+l_{2}}\,, result from the angular integration. They impose the OAM conservation condition in the two-photon state (45). Moreover, the following symmetry relations hold

Am​l−mp1​p2\displaystyle A^{p_{1}p_{2}}_{m\,l-m} =\displaystyle= B−m​m−lp1​p2,\displaystyle B^{p_{1}p_{2}}_{-m\,m-l}\;,
Am​l−mp1​p2\displaystyle A^{p_{1}p_{2}}_{m\,l-m} =\displaystyle= Al−m​mp2​p1.\displaystyle A^{p_{2}p_{1}}_{l-m\,m}\;. (49)

They allow us to rewrite the two-photon state in a more convenient way that makes more evident the simultaneous OAM and polarization entanglement

|Φ⟩s​i=κ∑p1,p2,mAm​l−mp1​p2×\displaystyle\ket{\Phi}_{si}=\kappa\!\sum_{p_{1},p_{2},m}\!A^{p_{1}p_{2}}_{m\,l-m}\times (50)
(​|p1,m,H⟩​|p2,l−m,H⟩+|p1,−m,V⟩​|p2,m−l,V⟩).\displaystyle\left(\frac{}{}\!\!\ket{p_{1},m,H}\ket{p_{2},l-m,H}+\ket{p_{1},-m,V}\ket{p_{2},m-l,V}\right).

This form of the two-photon vector beam quantum state exhibits explicitly the simultaneous OAM and polarization entanglement. It is useful for measurement schemes where OAM sorting is implemented in each detection arm, as depicted in Fig. 5. While this representation of the OAM sorter is idealized for pedagogic purposes, there are several types of architectures being developed to this end, meaning that the OAM sorting opeartion is already viable with increasing efficiency and resolutionKishikawa et al. (2018).

Refer to caption
Figure 5: Setup for OAM correlation measurements. An OAM analyzer (OA) is used in each detection arm, allowing for correlation measurements between different outputs of signal and idler, with topological charges constrained by ms+mi=±lm_{s}+m_{i}=\pm l\,.

It is instructive to obtain the OAM decomposition of the spatial correlations. We start by writing the positive frequency component of the electric field operator in terms of annihilation operators ap​lja^{j}_{pl} and bp​ljb^{j}_{pl} (j=s,ij=s,i) of photons in Laguerre-Gaussian modes (p,l)(p,l) with horizontal and vertical polarizations, respectively,

E^j​H+\displaystyle\hat{E}^{+}_{jH} =\displaystyle= ∑p​lap​lj​ψlp​(𝝆),\displaystyle\sum_{pl}a^{j}_{pl}\,\psi^{p}_{l}(\bm{\rho})\,,
E^j​V+\displaystyle\hat{E}^{+}_{jV} =\displaystyle= ∑p​lbp​lj​ψlp​(𝝆).\displaystyle\sum_{pl}b^{j}_{pl}\,\psi^{p}_{l}(\bm{\rho})\,. (51)

These annihilation operators are related to those in the transverse momentum basis through

aHj​(𝐪)\displaystyle a^{j}_{H}(\mathbf{q}) =\displaystyle= 12​π​i​ℰj​∑p​lap​lj​ψ~lp​(𝐪),\displaystyle\frac{1}{2\pi i\mathcal{E}_{j}}\sum_{pl}a^{j}_{pl}\,\tilde{\psi}^{p}_{l}(\mathbf{q})\,,
ap​lj\displaystyle a^{j}_{pl} =\displaystyle= 2​π​i​ℰj​∫aHj​(𝐪)​ψ~lp⁣∗​(𝐪)​d2​𝐪,\displaystyle 2\pi i\mathcal{E}_{j}\int a^{j}_{H}(\mathbf{q})\,\tilde{\psi}^{p\,*}_{l}(\mathbf{q})\,d^{2}\mathbf{q}\,, (52)

and the equivalent relations for aVj​(𝐪)a^{j}_{V}(\mathbf{q}) and bp​ljb^{j}_{pl}\,. The action of the OAM annihilation operators on the corresponding Fock states is given by

ap​lj​|p′,l′,H⟩\displaystyle a^{j}_{pl}\ket{p^{\prime},l^{\prime},H} =\displaystyle= bp​lj​|p′,l′,V⟩=δp​p′​δl​l′​|vac⟩,\displaystyle b^{j}_{pl}\ket{p^{\prime},l^{\prime},V}=\delta_{pp^{\prime}}\delta_{ll^{\prime}}\ket{\mathrm{vac}}\,,
ap​lj​|p′,l′,V⟩\displaystyle a^{j}_{pl}\ket{p^{\prime},l^{\prime},V} =\displaystyle= bp​lj​|p′,l′,H⟩=0.\displaystyle b^{j}_{pl}\ket{p^{\prime},l^{\prime},H}=0\,. (53)

In terms of the OAM eigenfunctions, the auxiliary vectors defined in Eq. (26) for calculating the quantum correlations become

|βH⟩\displaystyle\ket{\beta_{H}} =\displaystyle= E^i​H+​(𝝆i)​E^s​H+​(𝝆s)​|Φ⟩s​i\displaystyle\hat{E}^{+}_{iH}(\bm{\rho}_{i})\hat{E}^{+}_{sH}(\bm{\rho}_{s})\ket{\Phi}_{si}
=\displaystyle= κ​∑p1,p2,mAm​l−mp1​p2​ψmp1​(𝝆s)​ψl−mp2​(𝝆i)​|vac⟩,\displaystyle\kappa\!\sum_{p_{1},p_{2},m}\!A^{p_{1}p_{2}}_{m\,l-m}\,\psi^{p_{1}}_{m}(\bm{\rho}_{s})\psi^{p_{2}}_{l-m}(\bm{\rho}_{i})\ket{\mathrm{vac}}\,,
|βV⟩\displaystyle\ket{\beta_{V}} =\displaystyle= E^i​V+​(𝝆i)​E^s​V+​(𝝆s)​|Φ⟩s​i\displaystyle\hat{E}^{+}_{iV}(\bm{\rho}_{i})\hat{E}^{+}_{sV}(\bm{\rho}_{s})\ket{\Phi}_{si}
=\displaystyle= κ​∑p1,p2,mAm​l−mp1​p2​ψ−mp1​(𝝆s)​ψm−lp2​(𝝆i)​|vac⟩.\displaystyle\kappa\!\sum_{p_{1},p_{2},m}\!A^{p_{1}p_{2}}_{m\,l-m}\,\psi^{p_{1}}_{-m}(\bm{\rho}_{s})\psi^{p_{2}}_{m-l}(\bm{\rho}_{i})\ket{\mathrm{vac}}\,.

Finally, the coincidence counts can be calculated from Eq. (64) giving

C​(𝝆s,𝝆i)=κ2​|∑p​1,p​2,mAm​l−mp1​p2​ℱm​l−mp1​p2​(𝝆s,𝝆i)|2,\displaystyle C(\bm{\rho}_{s},\bm{\rho}_{i})=\kappa^{2}\,\left|\sum_{p1,p2,m}A^{p_{1}p_{2}}_{m\,l-m}\mathcal{F}^{p_{1}p_{2}}_{m\,l-m}(\bm{\rho}_{s},\bm{\rho}_{i})\right|^{2}\,, (55)

where

ℱm​l−mp1​p2​(𝝆s,𝝆i)\displaystyle\!\!\!\!\!\!\!\!\mathcal{F}^{p_{1}p_{2}}_{m\,l-m}(\bm{\rho}_{s},\bm{\rho}_{i})\! =\displaystyle= cos⁡θs​cos⁡θi​ψmp1​(𝝆s)​ψl−mp2​(𝝆i)+\displaystyle\!\cos\theta_{s}\cos\theta_{i}\,\psi^{p_{1}}_{m}(\bm{\rho}_{s})\psi^{p_{2}}_{l-m}(\bm{\rho}_{i})+
+\displaystyle+ ei​γ+​sin⁡θs​sin⁡θi​ψ−mp1​(𝝆s)​ψm−lp2​(𝝆i),\displaystyle\!e^{i\gamma_{+}}\,\sin\theta_{s}\sin\theta_{i}\,\psi^{p_{1}}_{-m}(\bm{\rho}_{s})\psi^{p_{2}}_{m-l}(\bm{\rho}_{i})\,,

and γ+≡γs+γi\gamma_{+}\equiv\gamma_{s}+\gamma_{i}\,. Note that Eq. (55) is the Schmidt decomposition of the coincidence pattern derived in Eq. (36) in terms of factorized OAM eigenfunctions for signal and idler. The different components ℱm​l−mp1​p2​(𝝆s,𝝆i)\mathcal{F}^{p_{1}p_{2}}_{m\,l-m}(\bm{\rho}_{s},\bm{\rho}_{i}) can be accessed by mode sorting before each detector, as indicated in Fig. 6a. First, the HH and VV polarizations of signal and idler are separated by polarizing beam splitters (PBS) and pass through OAM sorters. Then, the mm component with HH polarization is recombined with the −m-m component with VV polarization in a second PBS at the signal arm. In the same way, the l−ml-m component with HH polarization and m−lm-l with VV polarization are recombined at the idler arm. Finally, the polarization information is erased in each arm by analyzers (PA).

Note that two correlation channels are involved. The OAM components of H−H-polarized photons add up to ll while those of V−V-polarized photons add up to −l-l\,. After polarization erasure, the two channels interfere and the resulting coincidence pattern is |ℱm​l−mp1​p2​(𝝆s,𝝆i)|2|\mathcal{F}^{p_{1}p_{2}}_{m\,l-m}(\bm{\rho}_{s},\bm{\rho}_{i})|^{2}. They differ fundamentally from those calculated in section IV (see Fig. 4). There, similar patterns are obtained when either 𝝆s\bm{\rho}_{s} or 𝝆i\bm{\rho}_{i} is scanned while the other remains fixed. Here, due to transverse mode analysis before detection, the resulting coincidence patterns will present different shapes, depending on which detector is scanned. In Fig. 6b we show different OAM correlated images |ℱm​l−m00​(𝝆s,𝝆i)|2|\mathcal{F}^{00}_{m\,l-m}(\bm{\rho}_{s},\bm{\rho}_{i})|^{2} for a pump topological charge l=2l=2 and polarization analysis set to θs=θi=π/4\theta_{s}=\theta_{i}=\pi/4\,, γs=γi=0\gamma_{s}=\gamma_{i}=0\,.

Refer to caption
Figure 6: (a) Setup for measuring the OAM correlated images |ℱm​l−m00​(𝝆s,𝝆i)|2|\mathcal{F}^{00}_{m\,l-m}(\bm{\rho}_{s},\bm{\rho}_{i})|^{2}. PBS: polarizing beam splitter, PA: polarization analyzer, OA: OAM analyzer. (b) Correlation images between different OAM outputs of signal and idler with topological charges constrained by ms+mi=±lm_{s}+m_{i}=\pm l and p1=p2=0p_{1}=p_{2}=0\,. In this example we have set the pump topological charge l=2l=2 and polarization analysis is set to θs=θi=π/4\theta_{s}=\theta_{i}=\pi/4 and γs=γi=0\gamma_{s}=\gamma_{i}=0\,. The images at the signal (idler) plane are calculated with the idler (signal) detector fixed at xi=w0​|l−m|/2x_{i}=w_{0}\,\sqrt{\lvert l-m\rvert/2}\,, yi=0y_{i}=0 (xs=w0​|m|/2x_{s}=w_{0}\,\sqrt{\lvert m\rvert/2}\,, ys=0y_{s}=0) for optimal coincidence amplitude.

VI conclusion

In conclusion, we present the quantum theory of the angular spectrum transfer in spontaneous parametric down-conversion, generalized for structured light beams. The scheme studied is based on a source composed by two non-linear crystals to produce simultaneous nonlocal correlations in the spin and orbital angular momentum of entangled photon pairs. A structured pump beam, carrying internal non-separability between its polarization and transverse mode, generates photon pairs that are quantum correlated in both polarization and transverse spatial degrees of freedom. The two-photon spin-orbit quantum state generated by the process is derived and the corresponding quantum correlations are calculated both in position and OAM domains. The results were obtained in the weak interaction regime that allows us to safely neglect higher order terms, keeping only two-photon events. In cases where the interaction strength increases, the presence of four-photon events may become significant, leading to the reduction of the purity of the two-photon states and reduction of the entanglement in both polarization and transverse spatial degrees of freedom.

We show how the spatial correlations between signal and idler fields can be shaped by different settings of remote polarizers placed before their respective detectors. As an example, we show that Hermite-Gaussian and Laguerre-Gaussian coincidence distributions over the signal coordinates can be transformed by the remote control of the idler polarization settings. This resembles the behavior of a vector beam and, in this sense, we interpret this type of two-photon spin-orbit structure as a non-local vector beam. One natural follow up of the present work is the application of this structure to alignment-free quantum cryptography with non-local correlations Souza et al. (2008); D’Ambrosio et al. (2012); Ekert (1991). Another promising application concerns the use of machine-learning-based protocols to classify non local vector vortex beams. One potential route to this application is the combination of the classification scheme for vector vortex beams introduced in Ref. Giordani et al. (2020) with the single photon wave front correction scheme demonstrated in Ref. Bhusal et al. (2020), where a triggering photon heralds the single photon populating a Laguerre Gaussian mode. In the extension of the approach to non local vector vortex beams, polarization measurements in the triggering photon would allow the classification of the two-photon state. This is a natural resource for quantum communication systems. Moreover, the Schmidt decomposition in terms of Laguerre-Gaussian modes gives rise to different types of polarization dependent spatial correlations that can be accessed with mode filtering techniques.

Acknowledgments

The authors would like to thank the Brazilian Agencies, Conselho Nacional de Desenvolvimento Tecnológico (CNPq), Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ), Fundação de Amparo à Pesquisa do Estado de Santa Catarina (FAPESC) and the Brazilian National Institute of Science and Technology of Quantum Information (INCT/IQ). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.

Appendix A Individual Intensities

We will workout explicitly the intensity of horizontally polarized signal photons. The extension to vertical polarization and to idler photons is straightforward.

Ij\displaystyle I_{j} =\displaystyle= ∥cos⁡θj​|αj​H⟩+sin⁡θj​ei​γj​|αj​V⟩∥2,\displaystyle\lVert\cos\theta_{j}\ket{\alpha_{jH}}+\sin\theta_{j}\,e^{i\gamma_{j}}\ket{\alpha_{jV}}\rVert^{2}\;, (57)

The corresponding auxiliary vector becomes

|αs​H⟩=i​κ​ℰs​∫d2​𝐪i​GV​(𝐪i,𝝆s)​|1𝐪i,V⟩,\displaystyle\ket{\alpha_{sH}}=i\kappa\mathcal{E}_{s}\int d^{2}\mathbf{q}_{i}\,G_{V}(\mathbf{q}_{i},\bm{\rho}_{s})\ket{1_{\mathbf{q}_{i},V}}\,, (58)

where

GV​(𝐪i,𝝆s)\displaystyle G_{V}(\mathbf{q}_{i},\bm{\rho}_{s}) ≡\displaystyle\equiv ∫d2​𝐪s​vVp​(𝐪s+𝐪i)​ei​[𝐪s⋅𝝆s+φ​(𝐪s)]\displaystyle\int d^{2}\mathbf{q}_{s}\,v^{p}_{V}(\mathbf{q}_{s}+\mathbf{q}_{i})\,e^{i\left[\mathbf{q}_{s}\cdot\bm{\rho}_{s}+\varphi(\mathbf{q}_{s})\right]}\qquad
=\displaystyle= e−i​(𝐪𝐢⋅𝝆s+qi22​ks​zs)​WV​(𝝆s+zs2​ks​𝐪i,zs).\displaystyle e^{-i\left(\mathbf{q_{i}}\cdot\bm{\rho}_{s}+\frac{q_{i}^{2}}{2k_{s}}z_{s}\right)}\,W_{V}\left(\bm{\rho}_{s}+\frac{z_{s}}{2k_{s}}\mathbf{q}_{i}\,,z_{s}\right)\,.

Using the orthonormality condition for the Fock states ⟨n𝐪,μ|m𝐪′,ν⟩=δn​m​δμ​ν​δ​(𝐪−𝐪′)\braket{n_{\mathbf{q},\mu}}{m_{\mathbf{q}^{\prime},\nu}}=\delta_{nm}\,\delta_{\mu\nu}\,\delta(\mathbf{q}-\mathbf{q}^{\prime})\,, we have ⟨αs​H|αs​V⟩=0\braket{\alpha_{sH}}{\alpha_{sV}}=0 and

⟨αs​H|αs​H⟩\displaystyle\braket{\alpha_{sH}}{\alpha_{sH}} =\displaystyle= κ2​ℰs2​∫d2​𝐪i​|WV​(𝝆s+zs2​ks​𝐪i,zs)|2\displaystyle\kappa^{2}\mathcal{E}_{s}^{2}\,\int d^{2}\mathbf{q}_{i}\lvert W_{V}\left(\bm{\rho}_{s}+\frac{z_{s}}{2k_{s}}\mathbf{q}_{i}\,,z_{s}\right)\rvert^{2} (60)
=\displaystyle= κ2​ℰs2​∫d2​𝝆​|WV​(𝝆,0)|2,\displaystyle\kappa^{2}\mathcal{E}_{s}^{2}\int d^{2}\bm{\rho}\,\lvert W_{V}\left(\bm{\rho},0\right)\rvert^{2}\,,

assuming WVW_{V} to be normalizable. Along the same lines we can easily arrive at

⟨αs​V|αs​V⟩\displaystyle\braket{\alpha_{sV}}{\alpha_{sV}} =\displaystyle= κ2​ℰs2​∫d2​𝐪i​|WH​(𝝆s+zs2​ks​𝐪i,zs)|2\displaystyle\kappa^{2}\mathcal{E}_{s}^{2}\,\int d^{2}\mathbf{q}_{i}\lvert W_{H}\left(\bm{\rho}_{s}+\frac{z_{s}}{2k_{s}}\mathbf{q}_{i}\,,z_{s}\right)\rvert^{2} (61)
=\displaystyle= κ2​ℰs2​∫d2​𝝆​|WH​(𝝆,0)|2.\displaystyle\kappa^{2}\mathcal{E}_{s}^{2}\int d^{2}\bm{\rho}\,\lvert W_{H}\left(\bm{\rho},0\right)\rvert^{2}\,.

Moreover,the same deduction can be applied to the idler individual intensity and we finally get

Ij=cos2⁡θj​ℐH+sin2⁡θj​ℐV,I_{j}=\cos^{2}\theta_{j}\mathcal{I}_{H}+\sin^{2}\theta_{j}\mathcal{I}_{V}\,, (62)

with j=s,ij=s,i and

ℐH​(V)=κ2​ℰs2​∫d2​𝝆​|WV​(H)​(𝝆)|2.\mathcal{I}_{H(V)}=\kappa^{2}\mathcal{E}_{s}^{2}\,\int d^{2}\bm{\rho}\lvert W_{V(H)}\left(\bm{\rho}\right)\rvert^{2}\,. (63)

Note that after integration, no spatial dependence is left in the individual intensities. This means that the pump spatial properties are washed out in the individual intensity of the signal beam.

Appendix B Coincidence Counts

The coincidence count can be obtained from the following norm,

C​(𝝆s,𝝆i)=\displaystyle C(\bm{\rho}_{s},\bm{\rho}_{i})= (64)
‖​cos⁡θs​cos⁡θi​|βH⟩+sin⁡θs​sin⁡θi​ei​(γs+γi)​|βV⟩‖2.\displaystyle\left|\left|\!\!\frac{}{}\cos\theta_{s}\cos\theta_{i}\ket{\beta_{H}}\right.\right.+\left.\left.\sin\theta_{s}\sin\theta_{i}\,e^{i(\gamma_{s}+\gamma_{i})}\ket{\beta_{V}}\right|\right|^{2}.

This norm can be calculated by writing the auxiliary vectors in terms of the vacuum state as follows,

|βμ⟩\displaystyle\!\!\!\!\!\!\!\!\!\!\!\!\!\ket{\beta_{\mu}} =\displaystyle= −κ​ℰs​ℰi​∫d2​𝐪s​∫d2​𝐪i​vμ′p​(𝐪s+𝐪i)\displaystyle-\kappa\mathcal{E}_{s}\mathcal{E}_{i}\int d^{2}\mathbf{q}_{s}\int d^{2}\mathbf{q}_{i}\,v^{p}_{\mu^{\prime}}(\mathbf{q}_{s}+\mathbf{q}_{i}) (65)
×\displaystyle\times ei​[𝐪s⋅𝝆s+𝐪i⋅𝝆i+φ​(𝐪s)+φ​(𝐪i)]​|vac⟩\displaystyle\,e^{i\left[\mathbf{q}_{s}\cdot\bm{\rho}_{s}+\mathbf{q}_{i}\cdot\bm{\rho}_{i}+\varphi(\mathbf{q}_{s})+\varphi(\mathbf{q}_{i})\right]}\,\ket{\mathrm{vac}}
=\displaystyle= −κ​ℰs​ℰi​|J|2​∫d2​𝐪−​ei​[𝐪−⋅𝝆−+φ​(𝐪−)]\displaystyle-\kappa\mathcal{E}_{s}\mathcal{E}_{i}\lvert J\rvert^{2}\int d^{2}\mathbf{q}_{-}\,e^{i\left[\mathbf{q}_{-}\cdot\bm{\rho}_{-}+\varphi(\mathbf{q}_{-})\right]}
×\displaystyle\times ∫d2​𝐪+​vμ′p​(𝐪+)​ei​[𝐪+⋅𝝆++φ​(𝐪+)]​|vac⟩,\displaystyle\int d^{2}\mathbf{q}_{+}\,v^{p}_{\mu^{\prime}}(\mathbf{q}_{+})\,e^{i\left[\mathbf{q}_{+}\cdot\bm{\rho}_{+}+\varphi(\mathbf{q}_{+})\right]}\,\ket{\mathrm{vac}}\,,

where μ≠μ′\mu\neq\mu^{\prime} and we defined

𝐪+=𝐪s+𝐪i,𝐪−=ki​zsks​zi​𝐪s−𝐪i,\displaystyle\mathbf{q}_{+}=\mathbf{q}_{s}+\mathbf{q}_{i}\,,\qquad\qquad\;\;\mathbf{q}_{-}=\frac{k_{i}z_{s}}{k_{s}z_{i}}\,\mathbf{q}_{s}-\mathbf{q}_{i}\,,
𝝆+=ks​zi​𝝆s+ki​zs​𝝆iks​zi+ki​zs,𝝆−=ki​zs​(𝝆s−𝝆i)ks​zi+ki​zs,\displaystyle\bm{\rho}_{+}=\frac{k_{s}z_{i}\,\bm{\rho}_{s}+k_{i}z_{s}\,\bm{\rho}_{i}}{k_{s}z_{i}+k_{i}z_{s}}\,,\;\;\bm{\rho}_{-}=\frac{k_{i}z_{s}\,\left(\bm{\rho}_{s}-\,\bm{\rho}_{i}\right)}{k_{s}z_{i}+k_{i}z_{s}}\,,\qquad
φ​(𝐪+)=ks​zs+ki​zi−(ks​ziks​zi+ki​zs)​zs2​ks​q+2,\displaystyle\varphi(\mathbf{q}_{+})=k_{s}z_{s}+k_{i}z_{i}-\left(\frac{k_{s}z_{i}}{k_{s}z_{i}+k_{i}z_{s}}\right)\,\frac{z_{s}}{2k_{s}}\,q_{+}^{2}\,,
φ​(𝐪−)=−(ks​ziks​zi+ki​zs)​zi2​ki​q−2,\displaystyle\varphi(\mathbf{q}_{-})=-\left(\frac{k_{s}z_{i}}{k_{s}z_{i}+k_{i}z_{s}}\right)\,\frac{z_{i}}{2k_{i}}\,q_{-}^{2}\,,
J=−ks​ziks​zi+ki​zs.\displaystyle J=-\frac{k_{s}z_{i}}{k_{s}z_{i}+k_{i}z_{s}}\,. (66)

The integrals in (65) are considerably simplified by assuming zs=zi=zz_{s}=z_{i}=z and using ks+ki=kpk_{s}+k_{i}=k_{p}\,,

∫d2​𝐪−​ei​[𝐪−⋅𝝆−+φ​(𝐪−)]=e−i​kp​kiks​ρ−22​z,\displaystyle\int d^{2}\mathbf{q}_{-}\,e^{i\left[\mathbf{q}_{-}\cdot\bm{\rho}_{-}+\varphi(\mathbf{q}_{-})\right]}=e^{-i\frac{k_{p}k_{i}}{k_{s}}\frac{\rho_{-}^{2}}{2z}}\,,
∫d2​𝐪+​vμ′p​(𝐪+)​ei​[𝐪+⋅𝝆++φ​(𝐪+)]=Wμ′​(𝝆+,z).\displaystyle\int d^{2}\mathbf{q}_{+}\,v^{p}_{\mu^{\prime}}(\mathbf{q}_{+})\,e^{i\left[\mathbf{q}_{+}\cdot\bm{\rho}_{+}+\varphi(\mathbf{q}_{+})\right]}=W_{\mu^{\prime}}\left(\bm{\rho}_{+},z\right)\,.\qquad (67)

The integration on 𝐪−\mathbf{q}_{-} gives an irrelevant phase factor, while that on 𝐪+\mathbf{q}_{+} brings the pump spatial properties as a function of the joint positions of signal and idler. Thus, we arrive at the following result for the auxiliary vectors

|βμ⟩=−κ​ℰs​ℰi​|J|2​e−i​kp​kiks​ρ−22​z​Wμ′​(𝝆+,z)​|vac⟩.\displaystyle\ket{\beta_{\mu}}=-\kappa\mathcal{E}_{s}\mathcal{E}_{i}\lvert J\rvert^{2}\,e^{-i\frac{k_{p}k_{i}}{k_{s}}\frac{\rho_{-}^{2}}{2z}}\,W_{\mu^{\prime}}\left(\bm{\rho}_{+},z\right)\,\ket{\mathrm{vac}}\,.\qquad (68)

Finally, the intensity correlations are given by

C​(𝝆s,𝝆i)\displaystyle\!\!\!\!\!\!\!C(\bm{\rho}_{s},\bm{\rho}_{i}) =\displaystyle= K2|​cos⁡θs​cos⁡θi​WV​(𝝆+,z)\displaystyle K^{2}\left|\!\!\frac{}{}\cos\theta_{s}\cos\theta_{i}W_{V}\left(\bm{\rho}_{+},z\right)\right. (69)
+\displaystyle+ sinθssinθiei​(γs+γi)WH(𝝆+,z)|2.\displaystyle\left.\sin\theta_{s}\sin\theta_{i}\,e^{i(\gamma_{s}+\gamma_{i})}W_{H}\left(\bm{\rho}_{+},z\right)\right|^{2}.

where K≡κ​ℰs​ℰi​|J|2K\equiv\kappa\mathcal{E}_{s}\mathcal{E}_{i}\lvert J\rvert^{2}\,.

References