Results

In a first series of experiments we test the Michelson-type interferometric transfer setup with a diagonally polarized laser (810-nm wavelength) and show that the SPMs are indeed able to generate light fields with unprecedented quanta of OAM. Note that we also refer to quanta of OAM in the following first experiment that only involve light from a laser although it might be perfectly described in the classical Maxwell-theory of light and large topological charges. However, we demonstrate in the experiments investigating quantum entanglement afterward that the description with quantum numbers or quanta of OAM is fully justified.

We start by transferring photons from a Gaussian mode with a beam diameter of around 1 in. to a mode with 500 or 1,000 quanta of OAM. Typical conversion efficiencies, that is, percentages of light transferred from the Gaussian mode to OAM carrying light modes, significantly larger than 50% are observed. This becomes apparent in Fig. 1 due to no visible intensity along the optical axis. In both cases the complex superposition structure formed by 2l maxima, which were counted directly from the recordings, agrees exactly with the expected values (Fig. 1 A and B). The maxima were counted by a computer program after contrast enhancement. The proper functioning of the code was tested by recounting the maxima by hand for the ±500ħ structure.

However, an incident 0th-order Gaussian mode is unable to realize the transfer to modes carrying 10,000ħ. This is due to a difficulty in machining the complex phase profile in the central region of the SPMs. The surface structure corresponding to quantum number of 10,000 is too intricately complex and is therefore not cut (Fig. 1C). To obtain sufficient mode fidelity and efficiency of conversion to this very-high-order OAM mode, we first transfer the photons to l = ±10 quanta of OAM (described in the legend for Fig. 2). Because the radial increase of the amplitude for higher-order OAM modes scales approximately with ∼r|l|, we prevent the light from illuminating the uncut central region and increase the transformation efficiency and mode fidelity. This first transfer is realized with a folded Sagnac-like interferometric setup and a spatial light modulator as described in detail in ref. 11; however, the polarization information is not yet erased with a polarizer. We then transfer the photons in a second step with the above-mentioned Michelson-type setup including the SPMs. We complete the transfer with a polarizer at 45°, such that the photons are in a superposition between +10,010 and −10,010 quanta of OAM. Due to limited resolution of the camera the whole ring-shaped mode had to be imaged by stitching together 20 single images (Fig. 1C). Moreover, the superposition structure became visible only after further propagation and recording of an even smaller part of the mode, and thus simple counting of all maxima to verify the modal order was no longer feasible. Instead, we measured the change of intensity at a fixed angular position while rotating one SPM, thus modulating the phase between + l and – l. To account for misalignments and vibrations of the high-precision rotation stage, we recorded five intensity fringes at 12 different angular positions of the SPM (see Supporting Information for more details). The average over the 12 slightly different OAM quanta corresponds to an equally weighted superposition with ±l = 9,737 with a 1-sigma SD of 327. Thus, we are demonstrating that the SPMs are indeed able to generate light modes with the expected, very high order.

Before testing the OAM transfer in the quantum regime, it is worth checking whether the paraxial approximation and with it the description of OAM per photon is still valid for the beams we generate. It is known that if optical beams can be decomposed into plane wave components that are only tilted up to around 0.5 rad with respect to the optical axis, corrections to the paraxial approximations are negligible (23). This condition is fulfilled in all our measurements, that is, we focus the beams only weakly with lenses (2-in.) of focal distances of minimal 200 mm to ring-shaped beams with minimum diameters of 15 mm (500ħ/1,000ħ) and 25 mm (10,010ħ), respectively. The degree to which the paraxial approximation may be considered valid can be quantified using the paraxial estimator, P∼ (24), which has a value of 1 in the limit of a perfectly paraxial beam. All our beams have a P∼ > 0.99999. Thus, we are confident of the validity of the paraxial approximation in our experiments.

After having thus demonstrated the transfer classically, we show that an entangled quantum state with these high OAM quanta can be generated and measured. At first, we generate polarization entangled pairs by pumping a type-II down-conversion process (15-mm-long periodically poled KTP crystal) with a laser operating at 405 nm (∼30 mW power) in a Sagnac-type arrangement (25, 26). For frequency and spatial filtering we implement a 3-nm bandpass filter and couple the generated photons into single mode fibers, after which we detect ∼1.2 MHz of polarization entangled photon pairs with a wavelength of around 810 nm. Then, we transfer Bob’s photon with the schemes described above to ±500ħ, ±1,000ħ, or ±10,010ħ of OAM while keeping the entangled partner photon of Alice unchanged. To verify the successful generation of this state (1), we used two recently introduced methods, coincidence imaging (27) for ±500 quanta and coincidence detection after masking (11) for ±1,000 or ±10,010 quanta.

In the first method we triggered an intensified CCD camera (ICCD; Andor iStar A-DH334T-18U-73, ∼20% quantum efficiency, effective pixel size 13 μm × 13 μm, maximum 500-kHz triggering rate, 5-ns gating/coincidence window) to image in coincidence the transverse profile of the OAM photon at Bob’s location. Here, the polarization measurement of Alice’s photon, that is, the signal of a single photon avalanche photo diode (APD), serves as the trigger. Note that because of the intrinsic delay of the ICCD between the trigger detection and the actual gating of the ICCD, we delay the imaged photon by a 35-m-long fiber before it is transferred and recorded. The successful generation of the hybrid-entangled state (1) can be seen in Fig. 3, where the measurements of different polarization superpositions trigger the camera to image different OAM mode superpositions and thus slightly rotated petal-wheel structures on the ICCD. To more quantitatively verify the nonseparable nature of our measurements, we evaluate a witness that corresponds to the sum of the two visibilities in two mutually unbiased bases. Because separable states can only have a perfect visibility of 1 in one of these bases, the witness upper bounds the sum of the two visibilities to 1 for separable states. Values above this classical bound demonstrate quantum entanglement (27, 28). We measured accidental coincidences by shifting the delay of the trigger signal at the ICCD of around 100 ns and subtracted them from the count rates (see Supporting Information for more details). A witness value of 1.626 ± 0.022 (Poissonian counts statistics assumed) demonstrates a successful generation of quantum entanglement with ±500ħ of OAM.

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Fig. 3.

Coincidence images recorded with a triggered ICCD camera to demonstrate hybrid entanglement of polarization and 500ħ of OAM. Black letters depict Alice’s trigger polarization with which the images were recorded (Diagonal, Anti-diagonal, and Right- and Left-handed circular polarization). The images, recorded at Bob’s location, are built by summing over 60 images each exposed for 10 s with a triggering rate of around 500 kHz. A zoom shows the expected shift of the fringes (dashed line to guide the eye) corresponding to different OAM superpositions and depending on the trigger polarization. The opposing location of the extrema of A and D or R and L, respectively, are a direct observable signature of quantum entanglement.

Although the coincidence imaging scheme works very well for SPMs imprinting 500 quanta of OAM, the low triggering rate and detection efficiency of the ICCD prevents its use for higher quantum numbers. Thus, for measurements of entanglement with ±1,000ħ and ±10,010ħ of OAM we measure the polarization of the unchanged photons of Alice in coincidence with the detections of Bob’s transferred photons, which are transmitted through a mask that mimics the superposition pattern (11, 29). The ratio between slit width and the distance between the slits was ∼1/7, which reduces the maximal measurable visibility to around 97%. Because the masks are fabricated with a low-precision laser cutter (resolution approximately 100 µm) and normal paper, we were only able to fabricate and use a small fraction of the whole slit mask (around 60 slits). This allowed us to approximate the curvature of slit arrangement by a linear distribution of slits over a ∼50-mm length. In a similar way we approximated the rotation of the mask during the measurements with a linear translation (Figs. 4 and 5). To fit the beam of photons to the dimensions of the mask, we had to enlarge the beam size with lenses and free propagation over many meters. The transmitted photons are detected by a bucket detector (APD, active area with a diameter of 500 µm) and recorded in coincidence with four polarization measurements of the partner photons (diagonal, antidiagonal, and left- and right-handed circular). From the shifted extrema of the four measured coincidence fringes the aforementioned entanglement witness can be evaluated.

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Fig. 4.

Coincidence detections between Alice’s polarization measurements and Bob’s OAM measurements. The OAM photons were measured in different superposition states by detecting transmitted photons behind different positions of a slit mask that mimics the OAM superpositions structure. Because we only used 3% of the full mode and scanned fewer than two fringes, we approximated the circular rotation of 0.33° by a lateral translation. As expected for an entangled state, extrema of the opposing fringes belong to orthogonal polarizations and cannot be explained by a separable state. The corresponding entanglement witness verifies this observation by more than 10 SDs (Poissionian count statistics assumed; error bars are too small to be seen). Although unnecessary for the demonstration of entanglement, we also show the evaluated accidental coincidence detections (dashed lines) for the sake of completeness. For measurements in R/L-polarization bases the count rate was smaller, which stemmed from misalignments of the additional quarter wave plate.

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Fig. 5.

Quantum entanglement between polarization and 10,010ħ of OAM. We recorded correlations in the coincidence counts between Alice’s polarization measurements and Bob’s OAM measurements of superposition states. Again, the OAM superpositions are measured by detecting transmitted photons behind a slit mask that mimics the superposition structure. The circular rotation of the mask (<0.018°) is approximated by a linear translation. The colored region below each fringe corresponds to the accidental coincidence detections, derived from single counts and a measured coincidence window of 4.69 ± 0.34 ns. The error bars show Poissionian error estimation. The lines correspond to the best sin²-fit function with the condition to be above the accidentals to have nonnegative coincidence rates. From the visibilities of the fitted functions we derived entanglement witness value above the classical bound.

For entanglement between polarization and 1,000 quanta of OAM we find a value of 1.128 ± 0.013 (Poissonian count statistics assumed) without any subtraction of background or accidental coincidences (Fig. 4), which verifies the nonseparable nature of the state. We note that if accidental coincidence detection are subtracted (procedure explained in more detail below) the witness value increases to 1.228 ± 0.013.

As is visible in the laser images, the efficiency is drastically reduced for our final step, the generation and verification of entangled quantum states with 10,010 OAM quanta. Various sources of loss, such as the first transfer to ±10 OAM quanta, the bright unused higher-order radial modes (Fig. 1C), the measurement of only 0.3% of the whole beam, and the lower detection efficiency when focusing on the bucket detector after the mask all lead to a coincidence rate that is more than 10 times smaller than the accidental coincidence detections (Fig. 5). Thus, subtraction of accidentals is indispensable. We evaluate them from the measured coincidence window of 4.68 ± 0.34 ns, which is in agreement with the one given by electronics of coincidence logic and single photon detections (Supporting Information). We test for signatures of quantum entanglement by evaluating the witness in two slightly different ways. In the first data analysis, we derive the required visibilities from fitting a sin²-function, thereby taking into account all measured data points and increasing the statistical significance. For fitting we use a least-squares method including weighting of data points with respect to the errors and upper bounding the visibility to 1. We evaluate a witness value of 1.43 ± 0.25, which exceeds the classical bound with 1.7 sigma significance. In a second method of analyzing the data we split the whole dataset into 10 equally long measurement intervals and evaluate for each the witnesses separately. Again we correct for accidental coincidences by evaluating accidentals from single detections and only take values above the accidentals into account. Here, the mean value and the standard error of the mean over all 10 measured witnesses is found to be 1.41 ± 0.13 (see Supporting Information for more details, e.g., the 10 witness values).

Interestingly, both results show an even higher witness value than the measurements of 1,000ħ, which might be caused by the larger error margin. Because all results show values above the classical bound of 1, our findings can be seen as a clear signature that we successfully generated entanglement between polarization and up to 10,010 quanta of angular momentum.