Nature Physics  Letter
Threephoton energy–time entanglement
 L. K. Shalm^{1}^{, }
 D. R. Hamel^{1}^{, }
 Z. Yan^{1}^{, }
 C. Simon^{2}^{, }
 K. J. Resch^{1}^{, }
 T. Jennewein^{1}^{, }
 Journal name:
 Nature Physics
 Volume:
 9,
 Pages:
 19–22
 Year published:
 DOI:
 doi:10.1038/nphys2492
 Received
 Accepted
 Published online
Entangled quantum particles have correlations stronger than those allowed by classical physics. These correlations are at the heart of deep foundational questions in quantum mechanics^{1, 2, 3}, and form the basis of many emerging quantum technologies^{4, 5, 6, 7, 8, 9}. Although the discrete variables of up to 14 ions^{10} and the continuous variables between three intense optical beams^{11, 12} have been entangled, it has remained an open challenge to entangle the continuous properties of three or more individual particles. Here we experimentally demonstrate genuine tripartite continuousvariable entanglement between three separated particles. In our setup the three particles are photons created directly from a single input photon; the creation process leads to quantum correlations between the energies and emission times of the photons. The entanglement between our photons is the threeparty generalization of the Einstein–Podolsky–Rosen^{1} correlations for continuous variables, and could serve as a valuable resource in a wide variety of quantum information tasks.
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At a glance
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Main
We directly generate three entangled photons using the nonlinear process of cascaded spontaneous parametric downconversion (CSPDC; ref. 13). In downconversion, a pump photon, with frequency ω_{p}, inside a nonlinear material will occasionally fission into a pair of daughter photons with frequencies ω_{0} and ω_{1}. The total energy in the process is conserved^{14} with ω_{p} = ω_{0}+ω_{1}. The daughter photons share strong energy and time correlations that are the hallmark of entanglement^{15, 16}. The SPDC process is repeated with one of these daughter photons, at ω_{0}, now serving as the pump, creating a pair of granddaughter photons simultaneously at ω_{2} and ω_{3}. Again energy is conserved, and the total energy of the the three photons created in CSPDC must sum to the energy of the pump: ω_{p} = ω_{1}+ω_{2}+ω_{3}. The simplified representation of our threephoton state in frequency space, assuming a monochromatic pump, has the form
where G_{1}(ω_{1},ω_{p}−ω_{1}) and G_{2}(ω_{2},ω_{p}−ω_{1}−ω_{2}) are the jointspectral functions resulting from the phasematching conditions of the first and second SPDC crystals respectively^{17}. 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 CSPDC process we use continuousvariable 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 x_{k}, p_{k} (k = 1,2,3) fulfilling the commutation relations [x_{k},p_{l}] = iδ_{kl}, where δ_{kl} is the Kronecker delta (note: in ref. 18 a different commutation relation is used). Each individual particle must satisfy the uncertainty relationship Δx_{i}Δp_{i}≥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 inseparable^{19}. For pure states, full inseparability implies genuine tripartite entanglement^{20}. 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 welldefined for narrowband photons^{21}, such as those generated by our CSPDC 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 singlephoton 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 (ω = cp). 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 lefthand side goes to zero, and thus exhibit genuine tripartite entanglement.
Measuring the difference in arrival times of the three photons using fast singlephoton detectors gives the required timing uncertainties for testing the inequalities. However, directly measuring the frequencies of each individual photon with the precision needed (subgigahertz 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 CSPDC (ω_{p} = ω_{1}+ω_{2}+ω_{3}); 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 highefficiency periodicallypoled 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 secondorder nonlinear processes, such as downconversion, has been extensively verified using techniques such as Franson interferometry^{15, 16}, and been measured down to a line width of 200 kHz, an uncertainty much smaller than the scale considered in our experiment, in secondharmonic generation^{22, 23} (the timereversed process of SPDC).
To create our three entangled photons using CSPDC (see Fig. 1), first a narrowband pump laser at 404 nm is used to produce a pair of nondegenerate 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 setup 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.
The timing information from the detections was analysed, and the triple coincidence counts binned into a twodimensional (2D) histogram based on t_{2}−t_{1} and t_{3}−t_{2}, 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(t_{2}−t_{1}) = 0.37±0.02 ns, Δ(t_{3}−t_{2}) = 0.162±0.004 ns, and Δ(t_{3}−t_{1}) = 0.31±0.02 ns. Our measurements are limited by the timing jitter in our detectors and the resolution of our timetagging 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(t_{3}−t_{2}), which is approximately a factor of two smaller than either Δ(t_{2}−t_{1}) or Δ(t_{3}−t_{1}). Alternatively, we can study the twophoton 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 twophoton timing histograms. The need for gating with the 1570 nm detector (t_{3}) prevents the coincidences between t_{3} and t_{1} from being analysed independently of t_{2}, 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 freerunning detector.
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 counts^{13}.
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 Δω_{p}/2π = (6±2) MHz.
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 continuousvariable analogue to the famous Greenberger, Horne and Zeilinger entangled state^{18, 24}, and the natural extension of the twoparty continuousvariable Einstein–Podolsky–Rosen state^{15, 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 detectors^{26} and advances in nonlinear materials promise to greatly increase our detected triples rate^{27, 28}. Furthermore, new techniques to enhance the strength of nonlinear effects^{29, 30} mean that our scheme can in principle be scaled up to larger photon numbers. A major advantage of our states is that the continuousvariable entanglement is distributed amongst three individual photons, each at a different, tunable, wavelengths, enabling the creation of hyperentangled states that simultaneously take advantage of both discrete and continuousvariable 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 setup 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 setup (shown if Fig. 1) we use a gratingstabilized 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 TypeII SPDC. A pair of orthogonally polarized signal and idler photons, at 842 nm and 776 nm respectively, are generated colinearly, 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 singlephoton detector. With 12 mW of pump power, 10^{6} signal photons s^{−1} are detected. The idler photon is fibrecoupled and sent to a second SPDC crystal, a 30 mm TypeI 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 selfbuilt freerunning InGaAs/InPAvalanche Photo Diodes^{31} (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, id201SMFULN) 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 timetagging system (DotFast/UQDevices) with 156 ps resolution. In this way all the timing statistics from the twofold and threefold coincidence events generated by the CSPDC process can be measured.
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Acknowledgements
We thank the Ontario Ministry of Research and Innovation ERA, QuantumWorks, NSERC, OCE, Industry Canada, Canadian Institute for Advanced Research (CIFAR) and CFI for financial support. C.S. acknowledges support by an Alberta Innovates Technology Futures (AITF) New Faculty Award. We thank B. Boulanger and A. M. Steinberg for useful discussions.
Author information
Affiliations

Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, N2L 3G1, Canada
 L. K. Shalm,
 D. R. Hamel,
 Z. Yan,
 K. J. Resch &
 T. Jennewein

Institute for Quantum Information Science and Department of Physics and Astronomy, University of Calgary, Calgary, Alberta, T2N 1N4, Canada
 C. Simon
Contributions
L.K.S. and D.R.H. carried out the experiment. C.S., K.J.R. and T.J. conceived the experiment. Z.Y. developed the detectors and electronics used in the experiment. Data was analysed by L.K.S., D.R.H., K.J.R., and T.J. All authors contributed to the writing of the manuscript.
Competing financial interests
The authors declare no competing financial interests.
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L. K. Shalm
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D. R. Hamel
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Z. Yan
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C. Simon
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K. J. Resch
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T. Jennewein
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Supplementary information
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Supplementary Information