We present a theoretical model for the distribution of polarization-entangled photon-pairs produced via spontaneous parametric down-conversion within a local-area fiber network. This model allows an entanglement distributor who plays the role of a service provider to determine the photon-pair generation rate giving highest two-photon interference fringe visibility for any pair of users, when given user-specific parameters. Usefulness of this model is illustrated in an example and confirmed in an experiment, where polarization-entangled photon-pairs are distributed over 82 km and 132 km of dispersion-managed optical fiber. Experimentally observed visibilities and entanglement fidelities are in good agreement with theoretically predicted values.
© 2008 Optical Society of America
In future, quantum communication applications such as multi-party quantum cryptography , quantum secret sharing , distributed quantum computing , and quantum teleportation  would require distant users to consume shared entanglement as a resource . One interesting and non-classical feature of shared entanglement is that it can neither be created nor increased by local operations and classical communications , and so sharing of entanglement among distant parties must definitely involve some physical means of distributing the entanglement. One practical approach is to produce telecom-band entangled photon-pairs locally, followed by distribution of the generated photon-pairs over fiber-optic transmission lines to users who wish to share entanglement.
While entanglement distribution over very long distances is still far from reach due to the difficulty of implementing quantum repeaters [7, 8], a local-area entanglement distribution fiber network covering a distance of the order of 100 km seems feasible in the near future. Despite that on-demand sources of entangled photon-pairs are still hard to realize [9, 10], several groups have demonstrated entanglement distribution over 100 km of optical fiber, utilizing either time-bin-entangled or polarization-entangled photon-pairs produced via spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing (SFWM), as well as a variety of single-photon detectors [11–14]. Although all of them reported observation of a two-photon interference fringe visibility high enough for violating the Clauser-Horne-Shimony-Holt (CHSH) inequality  after distribution, it is not explicitly stated if the photon-pair generation rates had been chosen for observing the highest visibility. Most had relied on simple models that neglected multiple-pair emissions [13, 14]. In general, entanglement produced via SPDC is plaqued by multiple-pair emissions at high generation rates (in units of per pump pulse), and this leads to a tradeoff relation between entanglement distribution rate and entanglement quality. Since the two-photon interference fringe visibility is a measure of entanglement quality in this case, it should be maximized using a model that includes the effect of multiple-pair emissions as well.
Moreover, in a local-area fiber network where fiber transmission lines leading to different users could have different losses, and where different users might employ different types of single-photon detectors, a theoretical model that allows an entanglement distributor who plays the role of a service provider to determine the photon-pair generation rate giving highest visibility for any two users within the network, would be very useful. This is of course provided that user-specific parameters such as transmission losses and detector dark count rates are available. Depending on user requirement, there might also be times when the service provider wishes to maximize the photon-pair generation rate (and thus final distribution rate) for a given desired visibility. This would no doubt require a model that takes into account the effect of multiple-pair emissions.
It should be mentioned here that the undesirable effects of multiple-pair emissions in SPDC have been studied previously in the context of two-photon Bell experiments  and quantum key distribution (QKD) . In this work, we develop a theoretical model for calculating the two-photon interference fringe visibility at the entanglement distributor’s location, taking into full account multiple-pair emissions and detector dark-counts. We consider only polarization-entangled photon-pairs produced by combining two orthogonally-polarized SPDC processes [18–22], a method first demonstrated by Kwiat et al. using a two-crystal-geometry . Since polarization-entanglement is generally affected by polarization drifts in transmission fiber, we also assume that the transmission fiber lines are actively polarization stabilized by methods demonstrated in [24, 25], so that we can safely ignore detrimental effects caused by such polarization drifts. Usefulness of this model is illustrated in an example and confirmed in an experiment, where we distributed polarization-entangled photon-pairs over 82 km and 132 km of dispersion-managed optical fiber.
This paper is organized as follows. In Section 2, we describe the concept of a local-area entanglement distribution network and explain the role of a service provider. In Section 3, we outline the theoretical model. Section 4 illustrates the application of our model in an example and Section 5 describes an entanglement distribution experiment which confirms the usefulness of the model. Section 6 concludes this paper.
2. Local-area entanglement distribution network
Figure 1 shows our concept of a local-area entanglement distribution fiber network. An entanglement distributor playing the role of a service provider distributes entanglement to many application users scattered within a local-area network via fiber-optic transmission lines. As application users in general do not need to share entanglement all the time, there is no need for a constant supply of entangled photon-pairs to every single user. The requirement to share entanglement arises only when two users wish to perform quantum cryptography or send quantum information via quantum teleportation, and this suggests that the service provider needs only a small number of sources despite a large number of application users located within the network. In addition, application users need only to receive photons using single-photon detectors (without having to possess a source of entangled photon-pairs) and this lowers the overall implementation cost of the system.
As application users are not served with dedicated sources of entangled photon-pairs in this kind of network, the service provider must be able to react flexibly to the requirements of different pairs of users. To illustrate this point more clearly, let us look at three users, Alice, Bob and Charlie in the figure. Distance between each of them and the service provider is not the same, so photons transmitted to them should experience different losses. We can also assume that they use different types of detectors. For example, Alice uses commercially available peltier-cooled InGaAs single-photon detectors [26, 27], while Bob uses up-conversion detectors [28–30], and Charlie uses ultra-low dark-count superconducting single-photon detectors (SSPDs) [31, 32]. We thus expect that the condition for attaining a high visibility after distribution would be different for different pairs of users. A theoretical model that allows one to calculate visibility as a function of photon-pair generation rate and user-specific parameters would enable a service provider to quickly determine the optimal photon-pair generation rate. As the photon-pair generation rate depends only on pump power, it is an easily adjustable parameter at the service provider, unlike other parameters which are more or less fixed by the equipment in use, or beyond the service provider’s control.
3. Theoretical model
Figure 2 shows the generic scheme for producing polarization-entangled photon-pairs that is particularly of our interest [18–23]. Horizontally- and vertically-polarized laser pulses having a fixed phase relation are used to pump two identical nonlinear optical waveguides/crystals such as periodically-poled lithium niobate (PPLN) so as to produce correlated photon-pairs having the same polarization as the pump via SPDC. After combination at the polarization beam-splitter (PBS), if one is unable to distinguish the two processes, output photon-pairs would be entangled in the polarization degree of freedom. If the photon-pairs produced from the two waveguides combine in-phase at the PBS, the output photon-pairs would be in the maximally entangled state described as
where the subscripts s and i denote signal and idler, respectively. Time-bin-entangled or polarization-entangled photon-pairs produced via other methods are beyond the scope of this work.
For entanglement distribution, the signal photon and the idler photon of each pair are separated and sent to different destinations. It is a common practice to filter the photon-pairs either before or after transmission using narrow-band filters, in order to avoid undesirable effects caused by polarization-mode dispersion (PMD) of the transmission fiber, and to remove background noise photons. The photon coherence time is therefore lengthened considerably, and when it becomes much longer than the pump pulse duration, the photon-pair number distribution can be well described by a thermal distribution .
We shall first model the polarization-entangled photon-pair production process, writing visibility as a function of physical parameters, before going on to outline a method for obtaining the highest visibility. Let us assume that the SPDC process in each waveguide produces the ideal two-mode squeezed vacuum state expressed as 
n is photon number, r is a parameter dependent on pump power, and φ is the pump phase. The subscripts s and i denote signal and idler, respectively. If we let µ≡sinh2 r, we can write
The photon-pair number from each waveguide is thus thermally distributed with the probability of detecting n photon-pairs given by
where µ λ is the mean photon-pair number for polarization state λ=H,V. If we assume that both waveguides have the same photon-pair generation rate, we can let µ H=µ V≡µ/2, and the overall photon-pair number distribution P(µ, n) can be expressed as
This expression looks different from Eq. (4) because one needs to account for the combination of two independent polarization modes at the PBS. It can be derived easily if one starts from the interaction Hamiltonian of the SPDC process [17, 35]. In Appendix A, we give a more intuitive derivation of Eq. (5). Here, µ is the overall photon-pair generation probability per pump pulse, or simply, photon-pair generation rate. It is straight-forward to check that
To obtain a two-photon interference fringe, one usually places a polarizer before each of the two single-photon detectors and records coincidence count rates for various polarizer settings. Maximum and minimum coincidence count rates are observed when the polarizer settings are aligned parallel and orthogonal (for example, HH and HV), respectively, and are given by
assuming non-photon-number-resolving photon detectors. F is the clock rate, p i is the lumped photon collection and detection efficiency for channel i (1: signal, 2: idler), which includes coupling loss, transmission loss and quantum efficiency of the photon detector, but excludes the 3 dB loss due to the presence of a polarizer. If we perform the summations, Eq.s (7) and (8) become
The accidental coincidence count rate is
where d i are detector dark-count probabilities per gate, and s i are single-channel count rates given by
The two-photon interference fringe visibility can therefore be expressed as a function of photon-pair generation rate and the photon collection and detection efficiencies
Now we outline the method for obtaining the highest visibility for any pair of users labelled by i=1, 2. The idea is to first calculate p 1 and p 2 from given user-specific parameters, and then substitute them into Eq. (15) so that the visibility is expressed as function of the photon-pair generation rate µ. Once this is done, the entanglement distributor can choose µ for obtaining the highest visibility.
It is an important point to note that the lumped photon collection and detection efficiency p i of user i can be decomposed into
where η ο is the photon transmittance at the source, which includes coupling and component losses, η i is the quantum efficiency of the single-photon detector used, and t i is the channel transmittance. If α is the attenuation coefficient of the transmission fiber, and L i is the transmission distance, then . η ο and α are fixed parameters known to the service provider, whereas η i and L i are user-specific parameters. The other important user-specific parameter is the detector dark count probability per gate, d i.
The visibility calculated from Eq. (15) using user-specific parameters can be plotted as a function of photon-pair generation rate µ. Figure 3 shows a typical plot of visibility versus photon-pair generation rate and distance. Usually it is not necessary to plot against distance since user distances from the service provider are fixed, but we plot it here to show that the photon-pair generation rate giving highest visibility is a function of total distribution distance. From this plot, the service provider can easily determine the photon-pair generation rate (per pump pulse) giving the highest visibility at distribution distance.
For illustration, let us apply the method in an example. Table 1 shows the quantum efficiency and dark count probability per gate for three different types of single-photon detectors that application users might use. The numerical values given are representative values chosen for convenience.
We shall let η ο be 0.1, and fiber loss be 0.2 dB/km. Using the values given in Table 1, we can find the maximum distribution distance for two users using the same type of detector, on the condition that the two-photon interference fringe visibility exhibited by the transmitted photon-pairs must have a value higher than 0.71 (only above this value that a violation of the CHSH inequality can be demonstrated). For two users employing peltier-cooled InGaAs single-photon detectors, the maximum distribution distance is 100 km. For up-conversion detectors, the maximum distribution distance is 270 km, while for SSPDs, the maximum distribution distance is 400 km. Do note that the results given in this example will differ for a different visibility requirement. This local-area fiber network can thus be divided into three zones, as shown in Fig. 4. All users located within the central 100-km-diameter Zone A can use any type of single-photon detector including peltier-cooled InGaAs detectors. On the other hand, users in Zone B cannot use InGaAs detectors and must use either up-conversion detectors or SSPDs. Users in Zone C can only use SSPDs.
In Fig. 4, we have also shown the distance from the service provider for Alice, Bob and Charlie, which is 40 km, 100 km and 170 km, respectively. Let us now find the photon-pair generation rate giving highest visibility for entanglement distribution to Alice, who uses an InGaAs single-photon detector, and Charlie, who uses an SSPD. Figure 5 shows a plot of visibility versus photon-pair generation rate at the total distribution distance of 210 km. The line in bold is the exact result. We also show for comparison the broken line which does not include multiple-pair emissions, and the thin line which includes only contributions from double- and triple-pairs.
In the case where multiple-pair emissions are totally ignored, visibility decreases with increasing photon-pair generation rate beyond 0.1 per pump pulse because of a larger number of detector-dark-count-induced accidental coincidences at high detection rates, but the visibility value is overestimated. If detector dark-count is neglected as well, the visibility becomes unity regardless of photon-pair generation rate. It is also observed that for photon-pair generation rates below 0.2 per pump pulse, the thin line gives a good approximation to within 1 percent accuracy. This shows that it should be sufficient to consider up to triple-pair emission in the low generation rate regime . As is clearly shown in the figure, the highest visibility of 0.86 is obtained for a photon-pair generation rate of 0.04 per pump pulse. This example shows how the method can be applied in a practical situation.
5. Entanglement distribution experiment
To further confirm the usefulness of the model, we have performed an entanglement distribution experiment. Figure 6 shows a schematic of the experiment. The source that we use to produce polarization-entangled photon-pairs is a single 1-mm-long MgO-doped periodically-poled lithium niobate (PPLN) waveguide (HC Photonics) in a polarization-diversity loop configuration pumped by a Ti:Sapphire femtosecond laser. Details on this source can be found in . The source is located between two users, Alice and Bob, who wish to share entanglement, and is optimized for producing the maximally entangled state
The total transmission length is 132.4 km between Alice and Bob. If the two outer spans of dispersion-shifted fibers (DSFs) are removed, the transmission length becomes 82.4 km. Residual dispersion is negligible in this experiment. Transmission losses are 0.25 dB/km for both spans. Span A has an additional loss of 3 dB due to a non-ideal splice at the DCF, which can be taken as a realistic scenario where two fiber spans leading to different users have different losses. The optical coupling and component loss at the source is 10 dB. The transmitted photon-pairs are filtered with 3-nm band-pass filters (center wavelengths at 1542 nm for Alice and 1562 nm for Bob) followed by detection using InGaAs single-photon detectors (id Quantique) gated at 4.06 MHz. The quantum efficiencies of the detectors were 0.19 and the dark count probabilities were 10-5 per gate for Alice and 10-4 per gate for Bob. For quantum state tomography , we have used a combination of polarizer and quarter-wave plate placed before both detectors to realize the sixteen polarization settings required .
Figure 7 (a) shows the reconstructed density matrix and two-photon interference fringe after 82.4 km of transmission. Each coincidence count measurement took 100 seconds. Because of randomly fluctuating fiber birefringence, we had to manually compensate for polarization drifts using a monitoring laser and polarization controllers after every 100 seconds. At a photon-pair generation rate of 0.06 per pump pulse, the entanglement fidelity was found to be 0.86, and the two-photon interference fringe visibility was 0.77 for H/V basis and 0.72 for diagonal basis. The slightly lower visibility for the diagonal basis could be due to imperfect combination of the two orthogonally-polarized SPDC processes at the source. Next we increased the photon-pair generation rate to 0.25 per pump pulse and distributed the photon-pairs over a total fiber length of 132.4 km. Figure 7 (b) shows the results. The two-photon interference fringe visibility was 0.50 for H/V basis and 0.42 for diagonal basis. The observed visibilities in this case were insufficient for demonstrating a violation of the CHSH inequality but the entanglement fidelity was found to be 0.60, which suggests that subsequent purification of the transmitted photon-pairs into higher-fidelity-pairs is still possible . Note that all our results are obtained without subtracting accidental coincidences. The photon-pair generation rates in this experiment were determined by a method similar to that outlined in the appendix of , however, using a thermal photon-pair number distribution instead of Poissonian distribution.
The coherence time of the filtered photons is estimated to be 1.2 ps, which is longer than the pump pulse duration of 300 fs, and therefore we can assume the photon-pair number to be thermally distributed, and apply the model given in Section 3. The theoretical predictions are shown in Fig. 8. For 82 km transmission, a photon-pair generation rate of 0.06 per pump pulse maximizes the visibility V. For calculation of entanglement fidelity F, we have used the relation F=(3V+1)/4 . As shown in Fig. 8 (a), the predicted visibility and fidelity values after 82 km transmission are 0.79 and 0.85, respectively. These values are very close to the experimentally observed visibility value of 0.77 in the H/V basis and entanglement fidelity value of 0.86. For 132 km transmission, the model predicted that a photon-pair generation rate of 0.25 per pump pulse is required to achieve a visibility of 0.50 and an entanglement fidelity of 0.62, as shown in Fig. 8 (b). This is also in excellent agreement with the experimental result.
In conclusion, we have presented a theoretical model for distribution of polarization-entangled photon-pairs produced via SPDC within a local-area fiber network. With this model, a service provider can easily determine from his location the photon-pair generation rate giving highest visibility for any pair of users, provided that user-specific parameters are available. Usefulness of the model has been illustrated and confirmed in an experiment, where we successfully distributed polarization-entangled photon-pairs over 82 km and 132 km of optical fiber, and observed experimental results in good agreement with theoretical predictions.
Appendix A: Derivation of Eq. (5)
We derive Eq. (5) in the main text. As we have mentioned, this expression differs from the usual thermal distribution because one has to account for the combination of two independent polarization modes at the PBS. This is done by considering independent two-mode squeezed vacuum states from the two waveguides, given by
where the subscripts H and V denote the horizontal and vertical polarization mode, respectively. Combining these two independent polarization modes at a PBS produces the following state
If we assume the same mean photon-number for both polarizations
which can be re-expressed as
The photon-pair number here is m+n≡n ′, and as a result, the photon-pair number distribution is
The authors thank H. Takesue (NTT) for helpful discussions and for pointing out to them Eq.s (9) and (10) . Before that, they had relied on a counting technique similar to that outlined in , which gave theoretical expressions valid in the low photon-pair generation rate regime. H. C. Lim was on a postgraduate scholarship awarded by DSO National Laboratories.
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