Abstract
Photonics has become a mature field of quantum information science, where integrated optical circuits offer a way to scale the complexity of the setup as well as the dimensionality of the quantum state. On photonic chips, paths are the natural way to encode information. To distribute those highdimensional quantum states over large distances, transverse spatial modes, like orbital angular momentum possessing Laguerre Gauss modes, are favourable as flying information carriers. Here we demonstrate a quantum interface between these two vibrant photonic fields. We create threedimensional path entanglement between two photons in a nonlinear crystal and use a mode sorter as the quantum interface to transfer the entanglement to the orbital angular momentum degree of freedom. Thus our results show a flexible way to create highdimensional spatial mode entanglement. Moreover, they pave the way to implement broad complex quantum networks where highdimensionally entangled states could be distributed over distant photonic chips.
Introduction
The progress in technological developments over the last decades has enabled the realization of highly complex photonic quantum experiments^{1}. A popular approach to increase the complexity of the setup and the dimensionality of the observed system uses integration of optical elements on photonic chips. On these chips the most convenient way of encoding information is the path of the photons, which is inherently extendable to higherdimensional systems^{2,3}. Another possibility of encoding highdimensional quantum information is the transverse spatial mode of light^{4}. The fact that beams encoded with orbital angular momentum (OAM) can copropagate along the same optical axes makes their use, especially over long distances, advantageous^{5,6,7,8,9}. Various modes, such as OAMcarrying LaguerreGauss (LG)^{10,11}, InceGauss^{12} and BesselGauss^{13}, have been used to demonstrate experimentally highdimensionally entangled states and to implement quantum informational tasks. In a wide quantum network^{14}, both degrees of freedom, local onchip path encoding and longdistance bridging spatial modes, will have to be matched to each other. Recently, first experiments of an integrated OAM beam emitter demonstrated an interface between both fields, although not yet in the quantum regime^{15}. In another approach, qplates have been demonstrated to interface the OAM with the polarization of photons^{16,17}.
In this Article we demonstrate a quantum interface between two approaches to highdimensional photonic quantum information: path encoding for complex onchip experiments and OAM carrying light modes to transmit the information over large distances. At the same time, we investigate a flexible way to create higherdimensional OAM entanglement that does not rely on angular momentum correlation. The essential tool in our experiment is a mode sorter (MS) that was invented^{18,19} to convert the OAM content of an incident light beam to lateral positions of the output beam. By using this device in reverse, different spatial positions can be transformed to the cylindrically symmetric LG modes. We demonstrate that a highdimensional pathentangled state can be transferred to highdimensional OAM entanglement with the help of the reversed MS.
Results
Generation of OAM modes with a MS in reverse
Transverse spatial LG modes have a helical phase front e^{ilθ}, where l can take any integer value and represents the quanta of OAM each individual photon possesses^{20}. If l≠0, such modes exhibit a vortex along the beam axis and show a ringshaped intensity structure; consequently they are also called ‘doughnut modes’. LG modes can be used to transmit more classical information^{5,6,7,8,9} or realize a higherdimensional quantum state^{11,21,22}. To access the encoded information efficiently a MS was developed^{18,19}, which consists of two freeform refractive optical elements that convert the OAM content of an incident light field to lateral positions of the output. The first element maps the azimuthal to the lateral coordinates. Thereby, the ringshaped intensity is transferred to a straight intensity line and the ldependent helical phase structure to a transverse phase gradient. The second element corrects for phase distortion due to optical path length differences. A lens after the second element Fouriertransforms the transverse phase gradient to specific spatial position, that is, finishes the sorting of the modes. Recently, the operation of the MS in reverse was demonstrated by converting a light field with a transverse phase gradient, which has been created with a spatial light modulator (SLM), into an LG mode^{23}.
Here we investigate a different, simple way of using the MS operated in reverse as a source for OAM states. A narrow slit that diffracts the light is positioned in the focal plane of the lens (Fig. 1 inset). Thus, behind the lens, a parallel beam emerges with a phase profile given by the position of the slit relative to the optical axis. That phase distribution can be adjusted to be a multiple of 2π by adjusting the lateral position of the slit. A subsequent MS in reverse transforms the state’s amplitude distribution into a circle and thus into an OAM state (Fig. 1 inset). A continuous lateral movement of the slit leads to integer OAM modes at multiples of 2π and fractional OAM in intermediate positions (see Fig. 1 and Supplementary Movie 1). To define the lateral position states, we overfill the slit with a Gaussian beam. Hence, the specific parameters of the slit define the characteristics of the OAM state generated. The width of the slit was chosen such that it corresponds approximately to the diffractionlimited spot size of a beam being focused by the lens just before the MS. The height of the slit defines the radial characteristics, but is not specifically controlled and larger than the beam radius in our experiment. As height is mapped to the radius (logarithmically) of the OAM beam generated, this will result in an OAM state with a corresponding radial extent. Additionally, hard apertures are present in the optical elements of the MS (see the inset in Fig. 1), which lead to diffraction of the transformed beam. Both, the unrestricted slit height and the apertures, result in an observation of higherorder radial modes containing the same OAM. A similar effect has been used in an experiment where OAMcontaining Bessel beams have been separated by the MS in their azimuthal and radial components^{24}. With our simple approach we were able to create LG modes up to order l=±10. The order of the generated modes was only limited by the size of the Gaussian laser beam used to illuminate the slit. Importantly, by using multiple slits superpositions of LG modes could also be generated^{25} (Fig. 2).
To confirm that the generated annularshaped modes have the correct helical phase, we modulate the generated modes with the oppositehanded phase holograms displayed on a SLM and investigate the resulting mode. First, we image the modulated mode with a chargecoupled device (CCD) camera. The recorded modal structure shows an intensity along the beam axis that resembles a fundamental Gauss mode or higherorder radial mode containing no OAM (see three exemplary tests in Fig. 3a). In a second step, we unambiguously determine the OAM mode purity by adding a grating to the phase pattern on the SLM. We then couple the modulated first diffraction order into a singlemode fibre, thereby filtering higherorder components^{10,26}. By additionally changing the phase holograms on the SLM, we are able to measure the efficiency with which the expected mode is generated and thereby the modal overlap between this mode and its neighbouring modes (see Fig. 3b). Similar to the overlap if the interface is used for sorting^{18}, we find an average overlap of 17±8% between the closest neighbouring modes, 5±4% between the second closest neighbours and 0.7±0.6% between the third closest neighbouring modes. Note that this performance might be improved by a recently introduced version of the MS scheme^{27}.
Interfacing highdimensional path and OAM entanglement
After characterization of the reversed MS as an interface between the lateral position and OAM, we demonstrate its use in quantum optical experiments and transform a bipartite pathentangled state to an OAMentangled state, even for higherdimensional states. We create pairs of orthogonally polarized, pathentangled photons from position correlation in a spontaneous parametric downconversion process^{2,28,29} (Fig. 4a and Methods). By placing a triple slit directly behind (less than 1 mm) the crystal we filter out only three positions, which leads to an expected twophoton qutrit pathentangled state
where S_{1/2/3} denote the slit that both photons pass through and the subscripts H/V label their polarization. The amplitudes a, b and c (a^{2}+b^{2}+c^{2}=1) as well as the phases ϕ_{1} and ϕ_{2} are described by real numbers and depend on the pump beam behind each slit. Therefore, they are flexibly adjustable by modulating the intensity and phase of the pump beam. If only two slits are used a state is created that consists of the first two terms.
The slits are placed in the focal plane of a lens followed by the reversed MS, which leads to a transformation of the path entanglement to the OAM degree of freedom. Thus, the total transformation acts as an interface between the highdimensional path and spatial mode entanglement (Fig. 4b). In our experiment the distance between the three (two) slits corresponds to the 0th, −3rd and +3rd order LG modes (0th and −3rd order). We chose these three orders to reduce the earlierdemonstrated modal overlap of the MS to <1%. Hence the expected state can be written as
where −3, 0 and 3 label the order of the mode or OAM quanta. The flexibility in adjusting the amplitudes, phases, OAM values and dimensionality of the state (via transmittance, positions and number of slits) implies a general method to customtailor highdimensional OAM entanglement. To verify the OAM entanglement we split the transferred photon pair with a polarizing beam splitter and analyse each photon with a spatial mode filter. Again, we realize the filter by a combination of a phase hologram on the SLM and a singlemode fibre^{10,26}. Singlephoton detectors (avalanche diodes) together with a coincidence logic are used to register correlations between the two spatial modes of a pair (Fig. 4c).
Measuring highdimensional OAM entanglement
We quantitatively demonstrate entanglement by using a simple, powerful entanglement witness that compares the extracted fidelity (overlap between an ideal state of equation (2) and the measured data) and the maximally expected fidelity for ddimensional entangled states. If the measured fidelity exceeds the known bound for a ddimensional entangled state, our results prove at least (d+1)dimensional entanglement (see Methods).
In a first experiment, we test for qubit entanglement by measuring correlations between the 0th order (Gauss mode) and the −3rd order LG mode (Fig. 5). From the measured maxima and minima of these correlation fringes, the visibilities can be deduced. The highest fidelity calculated from the visibilities in all mutually unbiased bases was 97±2% with an ideal state where a=0.54 and b=0.84. The maximum fidelity that would be achievable for separable states is 71%.
In addition to the results of the entanglement witness, we used the measurements shown in Fig. 5 to test for twodimensional entanglement with the popular criterion of a CHSHBell inequality^{30}. For local realistic theories the following bound holds:
where α, α′, β and β′ denote different measurement settings (phases of the measured superpositions) and E stands for the normalized expectation value for photon pairs to be found with this combination of modes. In our measurement we achieve a value of 2.47±0.04, which violates the classical bound by more than 10 standard deviations (Poissionian count statistics assumed). Both results confirm our observation of twodimensional entanglement.
In a second experiment we take advantage of all three implemented slits and test for OAM qutrit entanglement. Similar to earlier results^{12,22}, the entanglement witness enables us to draw conclusions about the global dimensionality of the entanglement while restricting ourselves to qubitsubspace measurements. In our experiment this approach corresponds to the measurements of all visibilities in all twodimensional subspaces that is for lvalues of 0/−3, 0/+3 and −3/+3. The best statistical significance for genuine qutrit entanglement was found for an ideal state where a=c=0.48, b=0.73. Here, the fidelity obtained from the measured visibilities is 89±4%, which exceeds the upper bound for any twodimensional entangled state (77%) by more than three standard deviations. Note that in both cases (qubit and qutrit entanglement) no background subtraction has been applied. Also, the largest amplitude was found for the Gauss mode because it corresponds to the central slit where the pump intensity and thus the downconversion rate is maximal.
Discussion
Our setup is not limited to qutrits but can be naturally extended to ddimensional entanglement. A broader pump beam and a wider crystal in combination with a dslits arrangement would lead to d possible paths and thus ddimensional entanglement. A further way to increase the dimensionality of the path entanglement as well as the efficiency would be an arrangement with many integrated downconversion crystals^{31} in waveguide structures or fibrecoupled downconversion crystals pumped in parallel^{3}. In that case the waveguides or a fibre groove array could replace the multiple slits. Furthermore, it has been shown recently that the MS works for up to 50 states^{32} and that the MS can be improved to reduce the overlap between neighbouring modes^{27}. These improvements in MS design suggest that our approach can be readily extended to higher qudit entanglement. Although outside of the scope of the present work, it will be interesting to further investigate the detected higherorder radial structures, their potential as an additional dimension to encode information and their relation to the slit height. In addition, a suppression of the additional rings, by adapting the shape of the slit and improving the MS performance, might be crucial for the efficient interconnection of waveguide structures over long distances.
In conclusion, we have shown that the MS in combination with a slit can be used, in reverse, to generate LG modes and their superpositions up to at least l=±10. Using fast switching techniques for optical paths the method can increase the switching rate between different LG beams in classical information technologies or LGmodebased quantum cryptography applications. In addition, our results demonstrate a flexible way to create highdimensional OAM entanglement. Most importantly, they show a way to implement an quantum interface between two approaches to highdimensional quantum information: path encoding in waveguide structures for scalable, complex photonic quantum circuits including arbitrary unitary operation for higherdimensional states^{1,3}, and OAM encoding to distribute those highdimensional quantum states^{5,6,7,8,9} in a broad quantum network scenario.
Methods
Creation of pathentangled photons
To create correlated photons we used a cw 405nm laser diode to pump a periodically poled potassium titanyl phosphate crystal (1 × 2 × 5 mm^{3}) with ~30 mW pump power. The pump beam diameter was ~1 mm (FWHM), which leads to a broad region where the two orthogonally polarized photons (Type II downconversion) with 810 nm wavelength can be created. In the nearfield of the crystal—directly behind it—the photons are correlated in the transverse spatial position^{28,29}, that is, if one photon is found to be at a specific transverse spatial location the partner photon will be found ideally at exactly the same location too. This stems from the downconversion process itself, where the pump photon generates two downconverted photons at some transverse spatial position of the crystal. Because the actual location of the downconversion process is not determined, that is, it can happen everywhere within the pump beam spread, the generated photon pairs are in a superposition of all possible positions. A double or triple slit placed behind the crystal selects only two or three lateral positions; thus the photon pairs that pass the slits are in a superposition of two or three locations. In other words, they are in pathentangled qubit or qutrit states. For the multislit arrangement, the slits had a width of 150 μm and were separated by 250 μm, which resemble the dimensions typically used for telecomstandard fibre arrays. A schematic of the setup can be seen in Fig. 4a. The presented scheme is readily extendable to more than threedimensional entanglement, by increasing the number of slits. Furthermore, our source could be directly connected to complex integrated waveguide structures.
Bipartite witness for ddimensional entanglement
We develop a witness framework requiring only measurements in twodimensional subspaces. From these subspace measurements we compute the fidelity F of the experimentally produced state with a specially chosen highdimensional state. We then use the techniques developed below to bound the maximal overlap of states with a bounded Schmidt rank d and the chosen highdimensional state. If the subspace measurements reveal a higher overlap than this bound, the production of at least d+1dimensional entanglement is proven.
Construction of the witness proceeds as follows. The Schmidt decomposition of the assumed highdimensional state can be written as , where i labels the different states (in our experiment the spatial modes), D denotes the dimension of the Hilbert space and the coefficients are chosen in a decreasing order, that is, λ_{1}≥λ_{2}≥(...)≥λ_{d} (in the main text λ_{i} corresponds to the amplitudes a, b and c). This choice has the advantage that , where all appearing matrix elements can be determined by three visibility measurements in twodimensional subspaces, the number of which scales as the square root of those required for a fullstate tomography. We can now construct a witness for ddimensional entanglement (Schmidt number witness) by comparing the two fidelities
and
where ρ labels the density matrix related to our measurements and ϕ_{d}› denotes states with a bounded Schmidt rank d. If equation F>f_{d} holds, the measurements cannot be explained by a ddimensional entangled state; thus, the generated bipartite system was (at least) genuinely (d+1)dimensionally entangled.
Calculation of the maximal fidelity for dentangled states proceeds in the following manner. The maximization in equation (5) runs over all states with at most ddimensional entanglement (Schmidt rank d). This maximal overlap between and the guessed state ψ› (expressed in terms of the Schmidt coefficients) can be rewritten as
By rearranging the terms and introducing the operator B=c_{kl}k›‹ l
and the rank dprojector P_{d}B*=B* (which always exists if B* is of rank d, as ϕ_{d}› is of Schmidt rank d), we get
and since the trace is invariant under cyclic permutations we can write
Using the Cauchy–Schwarz inequality for the Hilbert–Schmidt inner product (for the inner product ‹ A,B›:=Tr(AB^{†}) it reads ‹ A,B›^{2}≤‹ A,A›‹ B,B›), we get the inequality
Because and choosing the obviously maximizing we get the upper bound for the fidelity of ddimensional entangled states of
By choosing we find that , thus proving that our bound is indeed tight, that is,
The inequality for the witness of ddimensional entanglement follows from the above. Hence, from the visibility measurements in all twodimensional subspaces we can now reveal information about the global dimensionality of the bipartite entanglement. If the following inequality holds, we have proven that our measurement results can only be explained by an at least (d+1)dimensionally entangled state:
Note that although we have derived the proof by assuming a pure state the witness holds even for mixed states because they would only lower the bound (due to the convexity of the fidelity). Thus, the presented witness is a stateindependent test for highdimensional entanglement.
Additional information
How to cite this article: Fickler, R. et al. Interface between path and orbital angular momentum entanglement for highdimensional photonic quantum information. Nat. Commun. 5:4502 doi: 10.1038/ncomms5502 (2014).
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Acknowledgements
We thank Sven Ramelow and Mario Krenn for fruitful discussions, Milan Mosonyi for helping with the proof of the witness and Otfried Gühne for giving valuable insight into the solution, by pointing out that the overlap in equation (13) was already proved (in a different way) in his PhD thesis. This work was supported by the Austrian Science Fund (FWF) through the Special Research Program (SFB) Foundations and Applications of Quantum Science (FoQuS; Project No. F4006N16), and the European Community Framework Programme 7 (SIQS, collaborative project, 600645). RF and RL are supported by the Vienna Doctoral Program on Complex Quantum Systems (CoQuS, W12102). MH would like to acknowledge the MarieCurie IEF grant QuaCoCoS—302021. MPJL and MJP are supported by the EPSRC.
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RL and RF conceived the experiment. RF designed and built the experiment, collected and analysed the data. RL and MPJL participated in designing and building of the experiment. MH developed the entanglement witness. The project was supervised by MJP and AZ. All authors contributed to discussing the results and the writing of the manuscript.
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Supplementary information
Supplementary Movie 1
Orbital angular momentum (OAM) mode generation from lateral positions of a slit. The movie was recorded with a CCD camera (false colour) placed after the reversed mode sorter (see Fig. 1 in the main text for a sketch of the setup). In front of the mode sorter, a slit is placed in the focal plane of a lens. This leads to a transformation of lateral position to transverse phase gradient states, which in turn are transferred to OAM states by the reversed mode sorter. Hence, a continuous lateral movement of slit leads to a generation of OAM modes with continuously varying quanta of OAM (fractional and integer OAM quanta). In the movie, the slit is placed first on the optical axis, which corresponds to the generation of Gauss modes with higher order radial components. Then, it is continuously displaced approximately 900 μm in the lateral direction, thereby generating higher order OAM modes up to the 6th order. Afterwards, the slit is shifted in the opposite direction up to 3rd order (passing the optical axis) and then moved back to the original position on the optical axis. (AVI 5181 kb)
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Fickler, R., Lapkiewicz, R., Huber, M. et al. Interface between path and orbital angular momentum entanglement for highdimensional photonic quantum information. Nat Commun 5, 4502 (2014). https://doi.org/10.1038/ncomms5502
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DOI: https://doi.org/10.1038/ncomms5502
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