Entangled photons are generally collected by detection systems that select their certain spatial modes, for example using single-mode optical fibers. We derive simple and easy-to-use expressions that allow us to maximize the coupling efficiency of entangled photons with specific orbital angular momentum (OAM) correlations generated by means of spontaneous parametric downconversion. Two different configurations are considered: one in which the beams with OAM are generated by conversion from beams without OAM, and the second when beams with OAM are generated directly from the nonlinear medium. Also, an example of how to generate a maximally entangled qutrit is presented.
©2011 Optical Society of America
Spontaneous parametric downconversion (SPDC), a nonlinear optical process in which two lower-frequency photons (signal and idler) are generated when a strong pump interacts with the atoms of a nonlinear material, is a reliable source for generating pairs of entangled photons. When SPDC is properly engineered, the photon pairs can be entangled in any degree of freedom: polarization, frequency, space or time. The entanglement can reside in one degree of freedom, or can also be shared between them [1,2]. In the case of the spatial degree of freedom, spatial entanglement can be depicted as residing in the quantum correlations between spatial modes that bear orbital angular momentum (OAM) . The use of the OAM of photons allows the generation of quantum states in a high-dimensional Hilbert space .
In general, one of the drawbacks of SPDC is that the flux of generated paired photons is very low. For instance, Dayan et al.  generated an ultra-high flux of downconverted photons (∼ 0.3μW), that even while being orders of magnitude greater than what is typically utilized in quantum optics experiments, it still shows an efficiency of only ∼ 10−7. In order to increase the flux of paired photons, one has to choose longer nonlinear materials or materials with higher nonlinear coefficients.
Moreover, any experimental detection system selects only part of the total number of generated photon pairs. Therefore, to increase the flux of paired photons, one should also add to the optimization toolkit the use of the most appropriate configuration that collects as many photons as possible without disturbing the sought-after quantum correlations between them. For instance, in most SPDC sources that generate photons entangled in polarization, the signal and idler photons are collected with single-mode optical fibers before being detected. In this case, the goal is then to maximize the coupling efficiency of paired photons into the fundamental Gaussian mode of the fiber. A number of studies have established some basic rules to optimize the efficiency of these particular SPDC configurations [6–10] and several experiments have confirmed some of the predictions . However, a similar study to optimize the collection efficiency of entangled photons with OAM has not been reported.
In this paper, we address the question of what is the optimum SPDC configuration, i.e., the pump beam waist and the size of the collection mode for a given crystal length, that maximizes the flux of entangled photons that exhibit specific OAM correlations. For this, it is necessary to calculate the spiral spectrum of the photons generated in SPDC, i.e., the decomposition of the biphoton mode function in a basis of spatial modes with OAM. Even though the decomposition has been calculated  and experimentally demonstrated , the implications of the engineering of the spiral spectrum for enhancing the flux of paired photons in selected SPDC configurations have not yet been fully explored.
Along these lines, we will consider two scenarios. On the one hand, the case where OAM correlations are generated by making use of spin-orbital coupling devices that starting from polarization entangled photons with a Gaussian spatial shape, generate photon pairs with OAM and polarization correlations . In this case, to increase the flux it is necessary to maximize the generation of pairs of photons with a Gaussian spatial shape. On the other hand, we can make use of the OAM correlations that are directly harvested at the output face of the nonlinear crystal . This is necessarily the case if the goal is to generate quantum states in multidimensional Hilbert spaces. Here the aim is to maximize the generation of photons with specific non-Gaussian spatial shapes.
The paper is organized as follows. In Section 2, we derive the main equations that lay the foundations of our analysis. In Section 3, we obtain the optimum configuration that maximizes the generation and coupling efficiency of photon pairs into single-mode optical fibers. In Section 4, we address the enhancement of the flux when downconverted photons are projected into Laguerre-Gaussian spatial modes.
2. General equations
Let us consider a periodically-poled nonlinear crystal of length L and nonlinear coefficient χ (2) illuminated by a continuous wave (CW) pump beam, with central frequency ωp. All the interacting waves (the pump, signal and idler) propagate along the same direction (collinear configuration), and the signal-idler pairs can be distinguished because either they show orthogonal polarizations or because they have different central frequencies. We consider a non-critical configuration, i.e., neither of the interacting waves experiences a Poynting-vector walk-off. The absence of spatial walk-off allows one to employ longer nonlinear media. Narrowband filters are located in front of the single-photon counting modules that detect in coincidence the arrival of a pair of photons.
The spatial distribution of the pump beam at the center of the nonlinear crystal, in the transverse wavevector domain, writes
The signal and idler photons are projected into specific spatial modes before detection. The use of Laguerre-Gaussian modes (Um,p) allows one to describe the spatial quantum correlations of the paired photons in a straightforward and clear way. The Laguerre-Gaussian modes are characterized by two integer indices, p and m. The positive index p is the radial index, and the winding number m, which can be any integer number, determines the azimuthal phase dependence of the mode, which is of the form ∼ exp(imφ). The functions Um,p are normalized, i.e., ∫ d q|Ump(q)|2 = 1.
The use of a collinear configuration and the absence of spatial walk-off makes hold the selection rule mp = m 1 + m 2. Here mp is the optical vortex winding number of the pump beam, and m 1 and m 2 are the winding numbers of the modes into which the quantum states of the signal and idler photons are projected, respectively. The configuration considered here allows to establish clear spatial correlations between the photons in terms of modes with OAM. For instance, if the pump beam has a Gaussian spatial shape (mp = 0), only paired photons with m 1 = −m 2 can be detected in coincidence. It is important to remark that this might not be the case for other non-collinear or critical SPDC configurations .
Most analyses of the spatial characteristics of photons entangled in OAM usually follow the Schrödinger picture, where it is the quantum state that evolves in time. The quantum theoretical analysis presented in this paper will use the Heisenberg picture, where it is the signal and idler field operators, a 1 and a 2, that evolve as a function of the propagation distance inside the nonlinear medium. In the Heisenberg picture, all the coherence functions of interest can be easily calculated, even when a first-order approximation of the quantum evolution equations is used, as it is the case here .
In the framework of the Heisenberg picture, the propagation equations for the signal and idler operators are given by 
The quantities and designate the spectral brightness (photons/s/Hz) of signal and idler photons, respectively, generated with transverse wavenumbers between q 0 – Δq/2 and q 0 + Δq/2. Therefore, the overall spectral brightness is
The second-order correlation function given by Eq. (5) can be written in terms of the first-order correlation functions , and as19], so that the first-order correlation functions can be written as
If we make use of the paraxial approximation (i = p, 1, 2), where is the wavenumber at the corresponding central frequency, and of the condition np ≅ n 1 ≅ n 2, we obtain Eqs. (4) and (10), the total spectral brightness F can be calculated Eq. (12), we obtain
The role of as the spectral brightness of interest here is more clearly revealed when we notice thatEq. (16), one has to make use of the completeness relationship .
Regarding the spatial shape of the projection modes Um,p, in what follows we will restrict ourselves to the case p 1 = p 2 = 0. The spatial shape of the LG modes in which the signal and idler photons will be projected, Um ≡ Um,p =0 writes
3. Generation of two-photon entangled states with correlations in polarization and OAM
Let us consider the generation of a quantum state entangled in the polarization and OAM degrees of freedom of the form 21]: Eq. (18) makes use of a non-collinear SPDC configuration with a type-II BBO crystal. The use of a such a non-collinear configuration, which also shows spatial walk-off, prevents from using long crystals to enhance the spectral brightness, since in that case the degree of polarization entanglement would decrease.
A possibility to overcome such a limitation is to use a non-critical collinear SPDC configuration with a Sagnac interferometer ( , see Fig. 1). In this case, the pump beam induces the generation of pairs of photons with orthogonal polarizations that propagate clock-wise or counter-clockwise in the Sagnac interferometer. Before any polarization or OAM transformation takes place, the quantum state of the generated paired photons can be written as
Paired photons generated in the quantum state given by Eq. (20) are projected into Gaussian modes. The coupling efficiency of interest in this case is . With the help of Eqs. (11), (13) and (15), we obtainEq. (23) is easily found to be α = 1.39. This solution gives us the optimum value of the pump beam waist, Equations (22) and (24) give us the optimum values of the pump beam waist and the collection system that maximizes the number of generated photons with the desired spatial shape.
Some authors [9, 11, 23] make use of the parameters ξp = L/zp and ξs = L/zs, where and are the Rayleigh ranges of the pump and signal waves. With these parameters, the result given by Eq. (24) can be rewritten in terms of ξp and ξs, so that the optimum values are .
Figure 2(a) shows the optimum value of the waist of the collection beam, namely w̄s, that yields the maximum value of coupling efficiency P 0,0 for a given value of the pump beam waist wp. Figure 2(b) shows the maximum value of P 0,0 for each value of the pump beam waist. We see that for all lengths, the global maximum of P 0,0 is . However, as expected from Eq. (24), the optimum value of the pump beam waist depends on the length of the crystal. In Fig. 2(a) we also plot the condition . In both figures, we consider two different values of the nonlinear crystal length: L = 10 mm and L = 20 mm.
As we have demonstrated above, the global maximum of P 0,0 is obtained for and . However, if we fix a value of the pump beam different from the optimum value, i.e., , the optimum size of the signal collection system that maximizes the efficiency (w̄s), no longer fulfills the condition . We can observe this in Fig. 2(a). For , the condition is fulfilled (open circles), and it corresponds to a global maximum of the coupling efficiency (open circles) in Fig. 2(b). But for larger values of the pump beam (wp > ), w̄s is indeed smaller than , whereas for smaller values (wp < ), w̄s is larger than . In other words, w̄s yields a local maximum of the coupling efficiency for a fixed value of wp. To highlight the different coupling efficiency achieved when ws = w̄s and when , we plot both cases in Fig. 2(b).
The maximum value of collection efficiency P 0,0 that can be achieved with a pump beam waist given by Eq. (24) and collection system , is
This maximum value is obtained because we are approximating the exact field of the fiber’s fundamental mode with a Gaussian function. It is expected that when considering the exact field, the coupling efficiency would decrease. This is analogical to what happens with single-mode fiber-coupled receivers in optical free-space communications : When the optical input field is focused in a single-mode fiber, the maximum coupling efficiency is ηmax = 81.5% if the field of the fundamental mode of the fiber is approximated by a Gaussian function. On the other hand, if the exact form of the fundamental mode is used , the maximum coupling efficiency turns out to be ηmax = 78.6%.
As an example of the usefulness of the results obtained in this section, let us estimate the spectral brightness that can be achieved in the considered SPDC configuration. If η accounts for all the losses of the experimental set-up, i.e., efficiency of the diffractive elements, losses of the singlemode-fibers and quantum efficiency of the detectors, the spectral brightness of generated paired photons is14], we see that by properly choosing the optimum value of the pump beam waist wp and the optimum size of the collection mode ws, one could observe a noteworthy enhancement of the spectral brightness of the source.
4. Generation of two-photon entangled states with OAM correlations
In the previous section, we considered the maximization of the spectral brightness of photons entangled in the polarization degree of freedom with a Gaussian spatial shape. Even though the entanglement in the polarization degree of freedom is transformed to entanglement in the polarization and OAM degrees of freedom, the dimension of the working Hilbert space is still d = 2. If the goal is to generate entanglement in a multidimensional Hilbert space (d > 2), we can take advantage of the OAM correlations directly generated in the process of SPDC (see Fig. 3).
In this section, first a general calculation is performed to show how to maximize the coupling efficiency of photons with OAM and then an example of how to employ it to generate a maximally entangled qutrit is presented.
Let us consider an SPDC process pumped by a beam with OAM winding number mp. The two-photon state, which is entangled in the OAM degree of freedom, can be written as
When projecting the paired photons into non-Gaussian modes, two cases should be considered. Firstly, the case when the winding numbers m 1 and m 2 have the same sign, i.e., sgn(m 1) · sgn(m 2) = 1, and secondly, the case with sgn(m 1) · sgn(m 2) = −1. The first situation corresponds, for instance, to cases where the pump beam, and the signal and idler modes are described by Laguerre-Gaussian modes with positive indices mp, m 1 and m 2. Making use of Eqs. (13) and (15), we obtainEq. (21).
Inspection of Eq. (28) shows that the optimum signal and idler mode widths are obtained when and , as in Section 3. The maximum coupling efficiency is now
Results are different for the case sgn(m 1) · sgn(m 2) = −1. For a Gaussian pump beam (mp = 0), the coupling efficiency is given by a somehow more cumbersome expression that writesEq. (30) can be written as Fig. 4(b), which shows the maximum coupling efficiency that can be achieved as a function of the pump beam waist, when the optimum value of ws, given by Fig. 4(a), is chosen. Figure 4(b) also shows the v optimum value of P 0,0 (red dash-dotted line), and P 0,0 when setting (red dash-dot-dotted line).
As an example of the application of the results depicted in Fig. 4(b), let us consider a detection system that projects the generated photons into a particular form of the state given by Eq. (27) in a three-dimensional Hilbert subspaceFig. 3. To detect a specific spatial mode with winding number m 1,2 = −1,+1, we can make use of a properly designed hologram of order M = +1,−1, or a spatial light modulator. The hologram projects the incoming photons into outgoing photons (in reflection or transmission) that propagate along the first diffraction order of the hologram, and have a winding number m + M. After the hologram, a single-mode fiber detects photons with a Gaussian shape, i.e., with winding number m + M = 0. For instance, if M = +1, detection of a photon after traversing the single-mode fiber implies the presence of an incoming photon with winding number m = −1, since any other incoming photon with a different winding number would not be allowed to propagate inside the single-mode fiber.
Notice that the beam waists and of the modes U 0 and U 1 in Eq. (33) can be different to achieve a maximum efficiency. In the detection arrangement described above, the beam waist of the modes detected can be controlled by modifying the optical coupling system of light into the single-mode fiber. The weights of each mode, P 0,0 = |C 0,0|2 and P 1,−1 = P −1,1 = |C −1,1|2 can be read in Fig. 4(b). To generate a maximally entangled qutrit, i.e., a quantum state that fulfills the condition P 1,−1 = P 0,0, one has to use specific values of wp and ws. For instance, from Fig. 4(b), we can see that if we make use of the condition , the value of wp ∼ 55μm where the curves intersect corresponds to a high-flux configuration for a maximally-entangled qutrit with P 0,0 = P −1,1. For this value of wp, the quantum state given by Eq. (32) represents roughly a 46% of the whole parameter space, i.e., , which is the coupling efficiency of this particular SPDC configuration. That means that by choosing appropriate values of wp, and , the coupling efficiency of the projection into a maximally-entangled quantum state of the form given by Eq. (32) can reach a value of 46 %.
We have presented an analysis of how to design optimum SPDC configurations that maximize the coupling efficiency of entangled photon pairs with specific OAM correlations in detection systems sensitive to the spatial shape of photons. Collinear non-critical SPDC configurations have been considered, as they facilitate the use of longer nonlinear crystals with the corresponding enhancement of the flux of the generated photons. Furthermore, they allow for a simpler description of the quantum spatial correlations of the two-photon states in terms of spatial modes that bear orbital angular momentum.
The optimization consists in shaping the spiral spectrum of the two-photon state by choosing appropriate values of the pump beam waist (wp) and the waist of the collection mode (ws).
If the aim is to generate a quantum state that bears quantum correlations in the OAM and polarization degrees or freedom, the optimum approach is to maximize the flux of polarization entangled photons projected into Gaussian modes and then transform them into modes with OAM using diffractive elements, such as holograms. We have found that fraction of photon pairs that can be detected under ideal conditions can approach a value of ∼ 82%.
On the other hand, if the aim is to generate higher-dimensional quantum states, we have to maximize the spectral brightness of photons with a Laguerre-Gaussian shape directly coming out of the nonlinear material. In the case of a maximally-entangled qutrit, a coupling efficiency of ∼ 46% can be achieved.
This work was supported by the Government of Spain (Consolider Ingenio CSD2006-00019, FIS2010-14831). The project PHORBITECH acknowledges the financial support of FET programme, under FET-Open grant number: 255914. This work was also supported in part by FONCICYT project 94142.
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