Introduction

Quantum-information science has seen rapid progress in recent years. Quantum entanglement is one of the driving forces behind quantum-information processing, and its generation and distribution are therefore of utmost importance. Among many other applications, quantum key distribution (QKD)1,2,3, quantum computation4,5 and superdense coding6 heavily rely on the presence of entanglement. A key feature of photons is that they can be simultaneously entangled in multiple of their degrees of freedom (DOF), thus creating hyper-entangled states7,8,9,10,11. By exploiting hyper-entanglement one can greatly increase the dimensionality of the resulting Hilbert space, effectively increasing the quantum information per transmitted photon12.

Another way of increasing the transmission rate has been a standard in the classical telecommunication industry for quite some time, namely multiplexing. Here, many signals are combined (multiplexed) into one transmission channel and separated (de-multiplexed) at the receiver site. In the quantum regime, multiplexing can be used to overcome the timing limitations imposed by the timing precision and dead time in the detection module, and hence significantly increase the secure key rate13.

In recent years, the interest in multiplexing of entangled photons in different DOFs has grown13,14,15,16,17. Achieving coherence and entanglement in the multiplexed DOFs offers another exciting avenue for high-rate quantum communication: directly encoding in energy-time or spatial DOFs also allows for high-dimensional protocols, which offer increased capacity in high-dimensional QKD protocols18,19,20 as well as improved noise resilience21,22,23, essentially only limited by the spatial or temporal resolution of the detection scheme.

In this article, we present an experiment separated into two parts, in which we prove successful transmission of spatial, energy-time as well as polarization entanglement through more than 400 m of multicore fiber (MCF). First, we certified hyper-entanglement in polarization and energy-time and combined it with the technique of space-division multiplexing. The entangled photons were spatially multiplexed using a multicore fiber, which consists of 19 single-mode fibers inside a single cladding, connected to a fan-in/fan-out device. We verified entanglement for four randomly chosen opposite core pairs of the MCF and achieved visibilities up to 94% in both energy-time and polarization.

In a second step, we verified the transmission of path entanglement through the MCF by non-locally interfering two neighboring core pairs on two separate beamsplitters. By varying the phase between the photon pairs, visibility fringes with a visibility up to 96% were measured. As the core pairs are identical, we can conclude that the MCF is indeed capable of transmitting hyper-entanglement in all DOFs faithfully.

The presented approach opens up the possibility of using a single source for generating hyper-entangled photons and efficiently distribute them through a MCF. Furthermore, because the wavelength of the entangled photons is centered around 1560 nm, the presented approach can be readily integrated into existing telecommunication infrastructure, thus paving the way for higher transmission rates in existing and future QKD protocols as well as other quantum-information applications. To the best of our knowledge, this is the first demonstration of transmitting a hyper-entangled state in three DOFs through a multicore fiber.

Results

Energy-time and polarization entanglement analysis

In the first part of the experiment, we simultaneously studied the entanglement in the energy-time and polarization domain. The experimental setup consists of a high-dimensional photon pair source and different analysis setups both at Alice’s and Bob’s site (see Fig. 1). The entangled photon pairs were produced via spontaneous parametric down-conversion (SPDC)24.

Fig. 1: Experimental setup.
figure 1

a Hyper-entanglement source: A MgO:ppLN non-linear crystal is bi-directionally pumped in a Sagnac configuration by a continuous wave laser at 780.24 nm. Because of this special configuration a hyper-entangled state in polarization, energy-time and position-momentum with a center wavelength of 1560.48 nm is produced. A dichroic mirror (DM) splits the pump light and the SPDC signal. Through the use of a lens configuration the momentum anti-correlations of the entangled photons are imaged onto the front facet of a 19 core MCF. A longpass filter further filters out the pump light while a bandpass filter with a FWHM of 1560 ± 2.4 nm selects only the photons centered around the center wavelength. This filtering procedure greatly increases the momentum anti-correlations, since a filtering in wavelength is directly proportional to a filtering on the k-vectors of the photons. Since the opening angle of the emission cone is highly dependent on the crystals’ temperature30 a temperature of 80.5 °C was chosen for the inner ring (cores 1, 1′, 2, 2′) while for the outer ring (cores 3, 3′, 4, 4′) the temperature was 77.5 °C. b Energy-time analysis: Two opposite core pairs (either 1 and 1′ or 2 and 2′) are coupled to two separate Franson interferometers. The mirrors of the long arm of Bob’s interferometer are mounted on a translation stage. By moving the stage one can vary the phase ϕ in Eq. (1) between the long and short arms. c Polarization analysis: Consisting of a HWP and a PBS. By rotating the HWPs one can specify the measurement basis, either HV or DA. d Path analysis: Two neighboring core pairs non-locally interfere on two separate 50/50 BS. Through the thusly created path indistinguishability a superposition basis is generated. A key aspect in achieving this is ensuring equal path length differences of all cores within the coherence length of the entangled photons. By varying the distance of one core to one BS with a translation stage the phase θ in Eq. (2) is tuned. In order to compensate for polarization drifts in the fiber a PBS was inserted before every input port of the BS, which for simplicity is not shown in the figure. All optical elements in (b), (c) and (d) are bulk optics.

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Because of momentum conversation in the SPDC process, the signal and idler photons are emitted with opposite transverse momenta, resulting in qs = − qi, where qs (qi) refers to the transverse momentum of the signal (idler) photon. Entanglement in position-momentum for this particular source has already been shown in a previous work25. By utilizing a lens system in the far-field plane of the crystal, we imaged the momentum correlations onto the front facet of a 411-m-long MCF. Our MCF consists of 19 single-mode fibers (SMF) in a hexagonal pattern (see Fig. 1). Because of the anti-correlation in momentum, the partner photon of a collected photon in any core can be found in the diametrically opposite core. Collecting the photons in opposing cores enables the separation of the two entangled photons despite its wavelength-degenerate spectrum. For a more detailed description of the lens setup and MCF the reader is referred to15.

Entanglement in the energy-time DOF arises from energy conservation in the SPDC process. After the fan-out of the MCF, the entangled photon pairs are transmitted to two imbalanced Mach–Zehnder interferometers, one for each user, which constitutes a Franson interferometer26.

The created hyper-entangled wave function in each core is close to

$$\left\vert {{{\Psi }}}_{E,P}\right\rangle =\frac{1}{2}\left[\left(\left\vert S\right\rangle \left\vert S\right\rangle +{{{{\rm{e}}}}}^{{{{\rm{i}}}}\phi }\left\vert L\right\rangle \left\vert L\right\rangle \right)\otimes \left(\left\vert H\right\rangle \left\vert H\right\rangle +\left\vert V\right\rangle \left\vert V\right\rangle \right)\right]$$
(1)

where \(\left\vert S\right\rangle\) and \(\left\vert L\right\rangle\) refer to short and long paths and \(\left\vert H\right\rangle\) and \(\left\vert V\right\rangle\) refer to horizontal and vertical polarization. The phase ϕ can be adjusted via a piezo actuator in the long arm of Bob’s interferometer.

After the energy-time analysis the photons are transmitted to a polarization analysis module, consisting of a half-wave plate and a polarizing beamsplitter. They are subsequently measured by superconducting nanowire single-photon detectors (SNSPD) with an efficiency of ~80% and dark-count rates of ~102 Hz.

The results of the visibility fringe measurements for different polarization settings and cores are shown in Tables 1, 2.

Table 1 Obtained visibilities from scanning the long arm ob Bob’s interferometer for the core combination 1-1′.
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Table 2 Obtained visibilities from scanning the long arm ob Bob’s interferometer for the core combination 2-2′.
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As can be seen in Tables 1, 2 all visibilities are greater than 81%, which is the limit for QKD protocols for which a key can be extracted.

Path entanglement analysis

As a second step the path entanglement transmitted through the MCF was analyzed. The experimental setup for the source is identical to the one explained in Sec. Energy-time and polarization entanglement analysis. In order to analyze path entanglement two core pairs in the outer ring of the MCF were non-locally interfered on two BS, as is shown in Fig. 1. By guaranteeing an equal distance from the front facet of the MCF to the interfering point on the BS a superposition basis is created27. The wave function projected on these two cores would ideally read as

$$\left\vert {\psi }_{S}\right\rangle =\frac{1}{\sqrt{2}}\left(\left\vert 3\right\rangle \left\vert {3}^{{\prime} }\right\rangle -{{{{\rm{e}}}}}^{{{{\rm{i}}}}\theta }\left\vert 4\right\rangle \left\vert {4}^{{\prime} }\right\rangle \right)$$
(2)

where \(\left\vert 3\right\rangle\), \(\left\vert {3}^{{\prime} }\right\rangle\), \(\left\vert 4\right\rangle\) and \(\left\vert {4}^{{\prime} }\right\rangle\) refer to the different cores of the MCF. By adjusting the path length difference of one core with a piezo electrical actuator, effectively varying the phase θ, visibility fringes between different output ports of the BS are observed.

Before we move on to results let us briefly define the overall target state, which we would expect to feed into the MCF and hope to certify at the output:

$$\begin{array}{ll}\left\vert {{{\Psi }}}_{T}\right\rangle := \frac{1}{4}(\left\vert 1{1}^{{\prime} }\right\rangle +\left\vert 2{2}^{{\prime} }\right\rangle +\left\vert 3{3}^{{\prime} }\right\rangle +\left\vert 4{4}^{{\prime} }\right\rangle )\\\qquad\quad\otimes (\left\vert SS\right\rangle +\left\vert LL\right\rangle )\otimes (\left\vert HH\right\rangle +\left\vert VV\right\rangle )\end{array}$$
(3)

The obtained results are shown in Fig. 2. The calculated visibilities are listed in Table 3. As can be seen visibilities as high as 97.6% were achieved, demonstrating a high degree of entanglement.

Fig. 2: Measured path visibility fringes between different detector combinations with an integration time of 1s.
figure 2

The detector combinations are the following: blue D1/D3, red D2/D4, green D1/D4 and yellow D2/D3. For the errors we assumed Poissonian statistics. The difference in the absolute coincidence counts is most likely due to different losses for each channel, such as coupling and detection efficiencies at the SNSPDs.

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Table 3 Obtained visibilities from varying the length of one core.
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In Fig. 2 a slight offset between different detector combinations is noticeable. This most likely stems from a slight length-offset between the non-interfering core pairs. This fact, however, does not compromise the visibility results.

The visibility implies a verified off-diagonal element in the path basis of \(| \left\langle 3{3}^{{\prime} }\right\vert \rho \left\vert 4{4}^{{\prime} }\right\rangle | =0.24\)28. In this non-adversarial scenario one can safely assume that this value is representative for all the elements \(| \left\langle i{i}^{{\prime} }\right\vert \rho \left\vert j{j}^{{\prime} }\right\rangle |\) (when utilizing a multicore fiber in a real-world scenario, it is of course of utmost importance to experimentally verify the equivalency of all cores). This, together with the coincidences across the cores, i.e. the measured diagonal basis elements in path basis, gives a final fidelity with the maximally path entangled state of Fpath = 0.953, which is well above the bound for qutrit entanglement of 0.75, thus proving a Schmidt number of four or in other words genuine four-dimensional entanglement.

Discussion

The presented approach offers multiple advantages for applications which need to be highlighted. While we connected separate measurement setups to verify entanglement in different DOFs, the source (except for crystal temperature) and the fiber remained unperturbed, proving simultaneous presence of all hyper-entangled DOFs after MCF transmission. First, we demonstrated the successful transmission of a spatially multiplexed polarization and energy-time hyper-entangled state through the inner cores of the 411-m-long MCF. By using the technique of multiplexing, timing limitations in present detector systems are circumvented and the secure key rate is increased13. Since the photons’ momenta are anti-correlated, one can distribute multiple pairs to different users by using opposing cores of a multicore fiber. Furthermore by using two MCFs, one for each user, one can use the intrinsic phase-stability of MCFs29 for efficiently distributing high-dimensional entanglement. One can also use the spatial entanglement across cores to maximize rates for a single pair of users. In this context, and as a second step, we verified entanglement in the path DOF by non-locally interfering two opposite core pairs of the outer ring of the MCF on two beamsplitters. We achieved visibilities as high as 97%, underlining the high-dimensional nature of the path entanglement. Note that the temperatures for the first and second measurement differed by 3 °C, maximizing the count rates either in the inner or outer rings of the MCF. This is because the spatial crosstalk between different cores is reduced by about 1% in the outer ring, as is shown in30. When combining all measurement setups sequentially, the coincidence count rates were in the order of 20 Hz. Since no active phase stabilization was used, the phase fluctuations of all setups were rendering a measurement impossible. In a real-world scenario one could use active spatial phase-stabilization31 and active polarization stabilization32 for a continuous QKD-protocol. Another possible application is superdense coding33 which has recently been realized with qu-quarts in Ref. 34 as well as superdense teleportation35 and single-copy entanglement distillation36. In recent years, another interesting possibility has been studied, namely the transmission of high-dimensional hybrid orbital angular momentum (OAM) and polarization states, both for prepare and measure37 as well as entanglement-based QKD schemes38. By utilizing the high-dimensional OAM modes of the photons, reliable QKD schemes with a high secure key rate can be implemented, as is shown in Ref. 39. To further improve the transmission rate in future applications, one could utilize few-mode MCF for the distribution of high-dimensional entanglement, as is shown in Ref. 40. For a thorough review on previous works in the field of transmission of OAM through fibers, the reader is referred to Ref. 41.

Since our entanglement source naturally produces wavelength-correlated photon pairs, the presented approach offers high scalability by additionally integrating dense wavelength-division multiplexing, as is done for example in Ref. 14. The used wavelength of 1560 nm makes our approach readily integrable into current telecommunication infrastructure, thus paving the way for highly efficient quantum-information protocols over deployed fiber networks.

Methods

Source design

By pumping a type-0 non-linear SPDC crystal in a Sagnac configuration42 with a CW-laser at a wavelength of 780.24 nm, polarization-entangled photons with a center wavelength of 1560.48 nm are produced. The created photons are inherently entangled in the energy-time and position-momentum domain. Because of the special Sagnac configuration the produced photons also exhibit polarization entanglement. The photon-pair spectrum is filtered by a ±2.4 nm (FWHM) bandpass filter to select degenerate photons. By decreasing the spectral width of the photons, the momentum correlations are greatly enhanced in return, which limits the crosstalk between different cores inside the MCF15. While SPDC might be the most commonly used way of generating photonic entanglement, it has several limitations which are worth mentioning, such as the low efficiency of roughly 10−5 pairs per pump photon43 as well as the probability to generate multi-pairs.

Franson Interferometer

At the first beamsplitter of the interferometer, the photons are either transmitted into the short arm or reflected into the long arm. The propagation difference between the long and the short arm amounts to a delay of 1.2 ns. They are then recombined at a second beamsplitter. The cases where both photons take the same paths in both Mach-Zehnder interferometers (either short-short or long-long) are temporally indistinguishable. Therefore, post-selection on the coincidences reveals non-local interference44. It is important to note that the path length difference ΔL between the short and long arms of both interferometers must be smaller than the coherence time τcc of the entangled photons. Furthermore, the path length difference has to be greater than the detector’s timing jitter, which is about 15 ps for SNSPDs. Due to polarization mode dispersion inside the MCF a half-wave plate at 0° was inserted in the long arm of Bob’s interferometer. This acts as a phase retarder and compensates for the phase difference of one photon due to transmission inside the MCF. On average, the single-photon count rate was about 125 kHz and the coincidence count rate was about 300 Hz. Accidental coincidences in the order of 5 Hz were subtracted from all coincidence counts. The low heralding efficiency, i.e. coincidence-to-single-ratio, is a result of the low-efficiency coupling of the SPDC light into the MCF. A micro-lens array45 could greatly increase the heralding efficiency of our source.

Path interferometer

In order to create a superposition basis for the Path DOF one has to ensure an equal distance from the front facet of the MCF to the interfering point on the BS27. Additionally, the path length difference between all cores has to be smaller than the coherence length of the entangled photons. Since only photons with the same polarization state interfere, a PBS was inserted in front of the BS. By this measure polarization drifts in one core will not affect the path visibility but rather just reduce the heralding. Accidental coincidences in the order of 10 Hz were subtracted from all coincidence counts.

Energy-time and polarization visibility measurements

In order to verify the transmission of hyper-entanglement through the MCF, at first, both HWPs at the polarization modules were set to H, inducing no phase shift on the polarization state of the photons, thus corresponding to a measurement in the HV-basis. To analyze the hyper-entangled state in its entirety, the long arm of Bob’s interferometer was scanned, effectively varying the phase ϕ in the first part of Eq. (1). The obtained visibility fringes are shown in Fig. 3. At every step Δx of Bob’s interferometer the integration time was 30s and the coincidence counts for the detector combinations HH (D1/D3), VV (D2/D4), HV (D1/D4) and VH (D2/D3) were recorded. Thereby a polarization visibility at every Δx is obtained. The polarization visibilities are shown in Fig. 4, where the green area represents a visibility of higher than 81%. Since a \(\left\vert {{{\Phi }}}^{+}\right\rangle\) Bell state was produced, correlations only arise between HH (DD) and VV (AA), while there are no correlations between HV (DA) and VH (AD).

Fig. 3: Measured time visibility fringes by varying Bob’s phase.
figure 3

Blue represents the coincidences for HH (DD), red for VV (AA), green for HV (DA) and yellow for VH (AD). a Core pair 1-1′ HV Basis, (b) Core pair 1-1′ DA Basis, (c) Core pair 2-2′ HV Basis, (d) Core pair 2-2′ DA Basis. For each data point the integration time was 30 s. We assumed Poissonian statistics, note that the errors are smaller than the data points.

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Fig. 4: Measured polarization visibility fringes by varying Bob’s phase.
figure 4

The green area represents a visibility of greater than 81%, representing the limit for QKD protocols. a Core pair 1-1′ HV Basis, b Core pair 1-1′ DA Basis, (c) Core pair 2-2’s HV Basis, (d) Core pair 2-2′ DA Basis. At multiples of 2π the polarization visibility drops. This stems from the fact that destructive interference for the coincidences occurs, thus fewer coincidences are measured. For each data point the integration time was 30 s. We assumed Poissonian statistics, note that the errors are smaller than the data points.

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The polarization visibility is calculated with

$${V}_{{{{\rm{pol}}}}}=\frac{{{{{\rm{CC}}}}}_{HH}+{{{{\rm{CC}}}}}_{VV}-{{{{\rm{CC}}}}}_{HV}-{{{{\rm{CC}}}}}_{VH}}{{{{{\rm{CC}}}}}_{HH}+{{{{\rm{CC}}}}}_{VV}+{{{{\rm{CC}}}}}_{HV}+{{{{\rm{CC}}}}}_{VH}},$$
(4)

where CCsignal,idler corresponds to the coincidence counts for each output port of the polarizing beamsplitter. In order to calculate the visibilities in the energy-time DOF, the obtained fringes were fitted with a sine function and calculated with

$${V}_{{{{\rm{time}}}}}=\frac{{{{{\rm{CC}}}}}_{\max }-{{{{\rm{CC}}}}}_{\min }}{{{{{\rm{CC}}}}}_{\max }+{{{{\rm{CC}}}}}_{\min }}.$$
(5)

The same procedure was repeated for a measurement in the DA basis, meaning that both HWPs in the polarization modules were rotated by 22.5°.

Path visibility measurements

In order to verify the transmission of path entanglement through the MCF the path lengths of all cores have to be identical in the order of the coherence length of the photons. By varying the distance to the BS of one core via a piezo electrical actuator the phase θ in Eq. (2) is tuned. Coincidence counts between all four detector combinations were recorded with an integration time of 1 s.

In order to obtain the visibilities the obtained fringes were fitted with a sine function and subsequently calculated with

$${V}_{{{{\rm{path}}}}}=\frac{{{{{\rm{CC}}}}}_{\max }-{{{{\rm{CC}}}}}_{\min }}{{{{{\rm{CC}}}}}_{\max }+{{{{\rm{CC}}}}}_{\min }}$$
(6)