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We directly generate three entangled photons using the nonlinear process of cascaded spontaneous parametric downconversion (C-SPDC; ref. 13). In downconversion, a pump photon, with frequency ωp, inside a nonlinear material will occasionally fission into a pair of daughter photons with frequencies ω0 and ω1. The total energy in the process is conserved14 with ℏωp = ℏω0+ℏω1. The daughter photons share strong energy and time correlations that are the hallmark of entanglement15,16. The SPDC process is repeated with one of these daughter photons, at ω0, now serving as the pump, creating a pair of granddaughter photons simultaneously at ω2 and ω3. Again energy is conserved, and the total energy of the the three photons created in C-SPDC must sum to the energy of the pump: ℏωp = ℏω1+ℏω2+ℏω3. The simplified representation of our three-photon state in frequency space, assuming a monochromatic pump, has the form

where G1(ω1,ωp−ω1) and G2(ω2,ωp−ω1−ω2) are the joint-spectral functions resulting from the phase-matching conditions of the first and second SPDC crystals respectively17. The three photons, consequently, share strong spectral correlations and exhibit genuine tripartite energy–time entanglement.

To verify the tripartite entanglement of the photons generated in our C-SPDC process we use continuous-variable entanglement criteria, which we derive based on the work in ref. 18, for position and momentum. Consider three separable particles each described by the dimensionless observables xk, pk (k = 1,2,3) fulfilling the commutation relations [xk,pl] = i δk l, where δk l is the Kronecker delta (note: in ref. 18 a different commutation relation is used). Each individual particle must satisfy the uncertainty relationship ΔxiΔpi≥1/2. Together, all three particles must satisfy the following position–momentum uncertainty inequalities (see Supplementary Information for details):

Violating any one of these inequalities is sufficient to demonstrate that a state contains some entanglement. Violating any two inequalities demonstrates that the state is fully inseparable19. For pure states, full inseparability implies genuine tripartite entanglement20. However, full inseparability and genuine tripartite entanglement are not, in general, the same thing. Mixtures of bipartite entangled states that are fully inseparable but not genuinely tripartite entangled are also capable of violating two of the above inequalities. A more general entanglement criterion is therefore required to detect genuine tripartite entanglement. In the Supplementary Information we provide an overview of the definitions of full inseparability and genuine tripartite entanglement and derive the following inequalities:

Violating any one of them is sufficient to demonstrate genuine tripartite entanglement.

The position and momentum operators x and p are well-defined for narrow-band photons21, such as those generated by our C-SPDC process, with the usual commutation relation [x,p] = i. Because photons propagate at the speed of light, c, measuring the arrival time, t, of a photon at a single-photon detector is equivalent to measuring its longitudinal position x (t = x/c), and measuring its frequency, ω, is equivalent to measuring its longitudinal momentum p (ℏω = c p). Using this correspondence it is possible to write down the energy–time equivalents to the inequalities in equations (5)–(8):

States of the form in equation (1) can violate all the inequalities maximally, namely the left-hand side goes to zero, and thus exhibit genuine tripartite entanglement.

Measuring the difference in arrival times of the three photons using fast single-photon detectors gives the required timing uncertainties for testing the inequalities. However, directly measuring the frequencies of each individual photon with the precision needed (sub-gigahertz resolution over a bandwidth of several terahertz) to violate the inequalities is infeasible with current count rates. Instead we rely on the fact that energy is conserved in the process of downconversion. The energy of the pump is equal to the energy of the three daughter photons created in C-SPDC (ℏωp = ℏω1+ℏω2+ℏω3); measuring the frequency of the pump provides a direct measurement of the total frequency of the three daughter photons required by the inequalities. We experimentally verify that energy is conserved in downconverion in our high-efficiency periodically-poled lithium niobate (PPLN) waveguide, the central component of our experiment, using an unbalanced interferometer (see Supplementary Information for more details). Furthermore, energy (frequency) conservation in second-order nonlinear processes, such as downconversion, has been extensively verified using techniques such as Franson interferometry15,16, and been measured down to a line width of 200 kHz, an uncertainty much smaller than the scale considered in our experiment, in second-harmonic generation22,23 (the time-reversed process of SPDC).

To create our three entangled photons using C-SPDC (see Fig. 1), first a narrowband pump laser at 404 nm is used to produce a pair of non-degenerate SPDC photons at 776 and 842 nm. The photon at 776 nm is then sent through a second SPDC crystal, where a pair of granddaughter photons at 1530 and 1570 nm are generated (see Methods for more details). This process leaves the 842, 1530 and 1570 nm photons entangled in energy and time. Our set-up detects an average of 7 triples h−1, from which we can infer the generation of 45 triples min−1 accounting for losses due to coupling and detection. To obtain sufficient photon counts with small statistical fluctuations, data was collected for a total of 72.6 hours.

Figure 1: Our three entangled photons are created using C-SPDC.
figure 1

A narrowband pump laser at 404 nm downconverts into a pair of orthogonally polarized photons at 842 and 776 nm inside a periodically-poled KTP crystal (PPKTP). A filter (FP) removes the remaining pump light. A polarizing beamsplitter (PBS) is used to separate the two photons, and narrowband filters, F0 and F1, are used to block stray light. The photon at 842 nm is coupled into a single-mode fibre (SMF) and sent to the single-photon detector D1. The photon at 776 nm is coupled into single-mode polarization maintaining fibre (PMF) and sent to a PPLN waveguide, where it downconverts into a pair of photons at 1530 and 1570 nm. The photons are outcoupled into free space, where a dichroic mirror (DM) is used to split the photons. The photons are then coupled back into single-mode fibre and sent to single-photon detectors D2 and D3 (see Methods for more information about the detectors). The signals from all three detectors are sent to a time-tagging unit, and a computer (PC1) is used to process coincidence events. A wave plate (WP) and PBS send part of the 404 nm pump to a Fabry-Perot Interferometer (FPI), controlled by a second computer (PC2), to continuously monitor its spectrum throughout the run.

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The timing information from the detections was analysed, and the triple coincidence counts binned into a two-dimensional (2D) histogram based on t2−t1 and t3−t2, as shown in Fig. 2. From the histogram it is clear that the photon arrivals are tightly correlated in time. The uncertainty in the arrival time between any pair of photons can be found by integrating over the arrival time of the other photon, removing its dependence, as shown in Fig. 3. From these integrated histograms we find that Δ(t2−t1) = 0.37±0.02 ns, Δ(t3−t2) = 0.162±0.004 ns, and Δ(t3−t1) = 0.31±0.02 ns. Our measurements are limited by the timing jitter in our detectors and the resolution of our time-tagging unit (156 ps). The effect of the jitter can be clearly seen in the elliptical shape of the 2D arrival time histogram—the jitter on the detector used to detect the 842 nm photon is a factor of two larger than the jitter of the two telecom detectors. This is reflected in the uncertainty Δ(t3−t2), which is approximately a factor of two smaller than either Δ(t2−t1) or Δ(t3−t1). Alternatively, we can study the two-photon coincidences between detectors D1 and D2, and D2 and D3, independent of the third detector (see Supplementary Information) to verify that integrating over the third photon yields the correct two-photon timing histograms. The need for gating with the 1570 nm detector (t3) prevents the coincidences between t3 and t1 from being analysed independently of t2, but the 50 ns gate width is much larger than the uncertainty in the arrival time of a photon, and approximates the response of a free-running detector.

Figure 2: 2D histogram of the difference in arrival times for the measured triple coincidences over 72.6 hours.
figure 2

The triple events are all localized to a small region of the histogram, indicating strong correlations in the arrival times of the three photons.

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Figure 3: Histograms of the difference in arrival times between two of the three photons measured over 72.6 hours.
figure 3

Each histogram is obtained by integrating the triples counts over the arrival time of the third photon to remove its dependence on the results. From this we find the uncertainty in the arrival times of the photons to be Δ(t2−t1) = 0.37±0.02 ns (a), Δ(t3−t2) = 0.162±0.004 ns (b) and Δ(t3−t1) = 0.31±0.02 ns (c). The timing uncertainties were verified using two-fold coincidence data that was obtained at the same time as the three-fold coincidence data (see Supplementary Information).

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It is possible for the first downconversion crystal to create two pairs of photons. To prevent two pump photons at 776 nm from reaching detectors D2 and D3 and creating a false triple, a spectral filter is used that blocks the pump light. Furthermore, the telecom detectors have a negligible efficiency at the pump wavelength. By tuning the temperature of the PPLN waveguide away from phasematching, we verified that the primary source of accidental triples is due to coincidences with detector dark counts13.

Owing to energy conservation, the energy uncertainty of the photon triplets is given by the energy uncertainty in the 404 nm pump photons. To measure the uncertainty in the pump energy, a scanning Fabry–Perot interferometer (FPI) was used to continuously monitor the bandwidth of the 404 nm laser throughout the experiment. Owing to instabilities caused by temperature fluctuations, the measured bandwidth fluctuates over time, as shown in Fig. 4a, leading to the distribution in Fig. 4b. The average value and standard deviation of this distribution yield a pump bandwidth of Δωp/2π = (6±2) MHz.

Figure 4: Bandwidth of the pump photons as measured by a Fabry–Perot interferometer at five-minute intervals during the 72.6 hour run.
figure 4

a, The measured bandwidth as a function of time. The fluctuations in the measured bandwidths are the result of thermal drifts in the apparatus over the course of the run. b, Histogram of the measured pump bandwidths over the duration of the run. The average bandwidth is measured to be Δωp/2π = 6 MHz with a standard deviation of 2 MHz (illustrated by the shaded regions in the graphs).

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The four measured time–bandwidth products for our three photons are

Our three photons strongly violate inequalities (9)–(12) and are genuinely tripartite entangled. The state exhibits energy–time correlations close to the ideal state described in equation (1) where the time–bandwidth products are exactly zero. This state is the continuous-variable analogue to the famous Greenberger, Horne and Zeilinger entangled state18,24, and the natural extension of the two-party continuous-variable Einstein–Podolsky–Rosen state15,16,25. The limiting factor in our measurements is the several hundred picosecond timing jitter of our detectors. On the basis of the bandwidth of our downconverted photons, the arrival times of the three photons should have a fundamental uncertainty on the order of a picosecond. With the development of faster detectors we should be able to lower our measured values of the inequalities, which are already close to ideal, by over two orders of magnitude.

Recent improvements in telecom wavelength detectors26 and advances in nonlinear materials promise to greatly increase our detected triples rate27,28. Furthermore, new techniques to enhance the strength of nonlinear effects29,30 mean that our scheme can in principle be scaled up to larger photon numbers. A major advantage of our states is that the continuous-variable entanglement is distributed amongst three individual photons, each at a different, tunable, wavelengths, enabling the creation of hyper-entangled states that simultaneously take advantage of both discrete- and continuous-variable quantum correlations. This multiplexing of entanglement over multiple discrete and continuous degrees of freedom may have important applications in quantum communication tasks. For example, a slight modification to our set-up would enable a photon at 776 nm to be interfaced with an atomic storage medium such as rubidium while the remaining two photons are transmitted over telecom fibres to remote quantum nodes. This would open up new possibilities in the storage and distribution of quantum information needed for quantum computing, cryptography, and secret sharing, and could lead to new fundamental tests of quantum mechanics.

Methods

In our set-up (shown if Fig. 1) we use a grating-stabilized pump laser with a wavelength of 404 nm and a bandwidth of 5 MHz (Toptica Bluemode) to pump a 30 mm potassium titanyl phosphate (PPKTP) crystal phase matched for Type-II SPDC. A pair of orthogonally polarized signal and idler photons, at 842 nm and 776 nm respectively, are generated co-linearly, and a polarizing beam splitter (PBS) is used to separate them. The signal photon at 842 nm is coupled into an optical fibre and sent to a Si single-photon detector. With 12 mW of pump power, 106 signal photons s−1 are detected. The idler photon is fibre-coupled and sent to a second SPDC crystal, a 30 mm Type-I phase matched PPLN waveguide (HC Photonics), where it splits into a pair of granddaughter photons at 1570 and 1530 nm. These granddaughter photons are outcoupled into free space and then split using a dichroic mirror. The photon at 1530 nm is sent to a self-built free-running InGaAs/InP-Avalanche Photo Diodes31 (Princeton Lightwave, Negative Feedback Avalanche Diode - NFAD) detector cooled to 193 K, operating at 10% efficiency with approximately 100 dark counts s−1. This detector is used to gate a second InGaAs/InP detector (iD Quantique, id201-SMF-ULN) operating at 25% detection efficiency with a 50 ns gate window to detect the granddaughter photon at 1570 nm. The gated detector had a much higher dark count rate of approximately 5×10−5 dark counts/(ns of gate). The arrival times of each photon in the three detectors are recorded by a time-tagging system (DotFast/UQDevices) with 156 ps resolution. In this way all the timing statistics from the two-fold and three-fold coincidence events generated by the C-SPDC process can be measured.