Remote temporal wavepacket narrowing

Quantum communication protocols can be significantly enhanced by careful preparation of the wavepackets of the utilized photons. Following the theoretical proposal published recently by our group, we experimentally demonstrate the effect of remote temporal wavepacket narrowing of a heralded single photon produced via spontaneous parametric down-conversion. This is done by utilizing a time-resolved measurement on the heralding photon which is frequency-entangled with the heralded photon. We then investigate optimal photon pair source characteristics to minimize heralded wavepacket width.

The phenomenon of entanglement can certainly be seen as the essence of quantum theory. It leads to a plethora of counterintuitive effects, allowing for substantial improvement of numerous applications, particularly in the field of quantum communication 1 . Interestingly, entanglement of different physical properties of light can be utilized in different ways. For example, polarization entanglement is an important resource for protocols of secure information transfer [2][3][4] , while spatial entanglement is essential for ghost imaging 5,6 and spatial quantum state encoding [7][8][9] . Spectral entanglement, on the other hand, can be utilized to improve quantum communication (QC) security 4,10 or the accuracy of remote clock synchronization 11 .
Photonic implementations of QC protocols suffer from many device imperfections that plague realistic single-photon sources, communication channels and detection systems. Their combination limits the maximal secure distance of information transfer. One of the most effective ways to reduce this limitation is to use temporal filtering to decrease the amount of noise registered by single-photon detectors 12 . Unfortunately, the full potential of this method cannot be used if the single-photon wavepacket is affected by temporal broadening. This turns out to be a common problem in fiber-based QC systems, where the signals propagate through dispersive media.
In order to counter this problem one may try to control the temporal properties of photons emitted by the source. For sources based on the spontaneous parametric down-conversion (SPDC) process, which are particularly useful for quantum communication applications, there are several methods to do so. However, most of them rely on sending at least one of the photons from every SPDC pair through some kind of spectral or temporal modulator, which acts as a filter [13][14][15][16] . Therefore, the efficiency of those schemes is significantly reduced, which can be seen as a serious drawback from the perspective of practical quantum communication. Furthermore, some of the methods for temporal shaping of SPDC photons require using specifically designed setup elements 16,17 . For this reason their utilization in broad applications does not seem to be probable in the foreseeable future.
However, a new method to significantly reduce the problem of temporal broadening was proposed recently by our group 10 . It is based on appropriate preparation of spectrally correlated photon pairs [18][19][20][21][22][23] and their subsequent time-resolved detection. This method does not require using any highly-specific setup elements. Furthermore, it does not introduce any type of additional filtering to the SPDC photons. This means that the heralding efficiency of temporally-narrowed photons is in principle the same as in the case when our method is not used.
In this work, we experimentally investigate the task of remote preparation of a single-photon wavepacket by a SPDC source. Following the theoretical proposal published in ref. 10 , we show how adjusting the spectral entanglement and applying a time-resolved heralding procedure can substantially narrow the wavepacket of the propagated photons. We also discuss the problem of optimizing the SPDC source for applications utilizing telecommunication fibers of a given length and dispersion coefficient. We discuss our results in the context of improving the performance of quantum key distribution (QKD) schemes.
The experimental setup utilized to measure the arrival time distribution of the photon pairs is depicted in Fig. 1. The parametric down conversion process takes place in a type-II 10 mm-long PPKTP crystal with a poling period of 46.2 μm, designed for collinear phase matching 780 nm → 2 × 1560 nm. The crystal is pumped by a pulsed Ti:Sapphire laser coupled to a single-mode fiber, with central wavelength 780.1(1) nm and repetition rate 80.14(2) MHz. A dichroic mirror is used to separate the pump beam and a photon pair. The photodiode where τ 1 , τ 2 are temporal widths of those photons and ρ t accounts for temporal correlation between them. In our analysis, we acquire three data sets, consisting of approximately 82 × 10 3 pairs of photon arrival times, for the three pump settings with different spectral widths, Δλ, specified in Table 1. For each data set, we compute a histogram, to which we subsequently fit the distribution given in Eq. (1). We take into account the background noise in the model, which turned out to be negligible. The parameters are fitted using standard nonlinear model fitting functions. The best fit parameters, ρ t , τ 1 and τ 2 are gathered in Table 1. The measured statistics and fitted   Table 1. Values of the main parameters. The three pump bandwidths, Δλ, utilized in the experiment (standard deviation), the corresponding pulse duration, τ p , the best fit parameters ρ t , τ 1 , τ 2 for the statistics of arrival times of SPDC photons to the detectors, and the ratio of reduced temporal width of the heralded photon to the respective temporal width of non-heralded photon, τ 1h /τ 1 .
www.nature.com/scientificreports www.nature.com/scientificreports/ functions are depicted with blue dots and solid lines, respectively, in Fig. 2, where the columns correspond to the three data sets specified in Table 1. Note that the error bars are smaller than the markers and the same situation is in Figs 3-5.
Subsequently, we choose two 100 ps-wide detection windows for photon number two (heralding photon) and computed histograms for the measured arrival times of corresponding photon number one (heralded photon). The probability distribution function fitted to the obtained data sets is given by the formula (12) in the Methods section. The results of this fitting procedure are plotted with dashed yellow and green lines on the panels (d)-(f) in Eq. (1). Comparing them with the corresponding blue solid lines we observe that the temporal width of the heralded photon is reduced as compared to the situation when there is no information on the heralding photon arrival time. It implies narrower time distribution of the heralded photon.
The strength of this narrowing effect depends on the temporal correlation between the SPDC photons, which in turn depends on the pump settings. In order to quantify it we introduce the quantity τ 1h (ΔT), which denotes the temporal width of the heralded photon provided that the detection window for the heralding photon has ΔT width. Its theoretical value is simply the standard deviation of the Gaussian function given by formula (12). In general, the higher the absolute value of ρ t , the lower the ratio of τ 1h (ΔT)/τ 1 . This can be seen in Fig. 3(a), where we plot the experimental values of this ratio, obtained through the fitting procedure, as a function of ΔT. This data is also compared with the results of the theoretical model depicted with solid lines. The value of τ 1h (ΔT)/τ 1 is reduced when ΔT decreases. However, below a certain threshold value of ΔT it approaches a limit, which in theory is given by (see the Methods section) In our experiment, this threshold value can be estimated to be ΔT ≈ 300 ps. The strength of the wavepacket narrowing that we were able to observe, calculated in terms of the ratio of τ 1h (ΔT)/τ 1 , is given in Table 1.
The average arrival time of the heralded photon, T 1h , depends on the average detection time of the heralding photon, T 2 . The experimental results are depicted in Fig. 3(b). In general there is no analytical formula to express this dependence. However, if one chooses a detection window close to the limiting value ΔT → 0, the theoretical dependence is linear. It can be derived from Eq. (14), yielding To illustrate this, for every T 2 we took fit parameters from Table 1 and numerically calculated T 1h . The results of this calculation are depicted by solid black lines in Fig. 3(b).
Temporal widths of photons as functions of the SPDC source parameters. Now we wish to find the optimal source parameters that could provide us with the narrowest possible wavepackets for a given length of transmission links, L, characterized by the dispersion coefficient, β. In order to achieve this goal, we first calculate  12)). Each column corresponds to one of the three data sets specified in Table 1. The orange and green colored areas on panels (a)-(c) denote arbitrarily chosen, 100 ps wide detection time windows for the heralding photon. The measured (calculated) distribution of heralded photon arrival times corresponding to those selected areas are plotted on panels (d)-(f) with dots (dashed lines) of the the same colors. The numbers of coincidences in the heralded photons peaks are scaled for convenient presentation of the wavepacket narrowing effect. On panels (d)-(f), the total number of coincidences in peaks of dashed yellow (green) plots are 125 (48), 865 (368) and 1396 (562). www.nature.com/scientificreports www.nature.com/scientificreports/  It is not surprising that the value of τ 1 grows to infinity when the pump pulse duration is infinitely short, τ p → 0, or infinitely long, τ p → ∞ (CW laser). On the other hand its minimum In the asymptotic case, in which the arrival time of the heralding photon is perfectly known, one gets the following expression for the temporal width of the heralded photon: This quantity also reaches its minimum for τ p opt . It is given by On the other hand, for the situation when the emission time of the pump pulse is unknown but the detection time of the heralding photon is available the temporal width of the heralded photon reads: It is interesting to note that, for a given nonlinear crystal with fixed σ parameter, this does not depend on the pump settings.
Optimizing an SPDC source over the pump pulse duration. Let us first analyze the scenario where the crystal parameter, σ, is fixed and the experimenter can modify the pulse duration. The relation between τ 1 , τ 1h (0) and τ 1h,Δt (0) can be clearly seen in Fig. 4, where the theoretically calculated and experimentally measured values of these functions are plotted. It can be seen that when the additional information about the photon pair generation time is available, the respective temporal width is small compared to the case when there is no such information, τ τ < . This can be seen by comparing the solid red and the dashed green curves in Fig. 4. Furthermore, one can observe that the temporal width of the non-heralded wavepacket is never smaller than the heralded one, τ 1h (0) ≤ τ 1 . The strength of the narrowing effect, τ 1h (0)/τ 1 , asymptotically goes to zero for τ p → 0 www.nature.com/scientificreports www.nature.com/scientificreports/ and τ p → ∞. On the other hand, for τ p = 2/σ and τ p = |β|Lσ it reaches the maximal value of one. The first maximum can be attributed to the crystal producing spectrally decorrelated photon pairs. The second one is the consequence of the propagation in the fiber. As shown in ref. 10 , the temporal correlation changes with the propagation distance. Therefore the second maximum corresponds to the point where there is no temporal correlation. The local minimum of τ 1h (0)/τ 1 is reached for τ p opt . It can be calculated using the formulas (5) and (7). For the case illustrated in Fig. 4, this central minimum within our experimental setting takes the value of approximately 11.3%, meaning that the lowest value of τ 1h (0)/τ 1 measured in our experiment, 29.49% (see Table 1), could be further reduced by setting the pulse duration to τ p = 15.2 ps.
In connection with our previous work 10 , it is interesting to ask what type of spectral correlation we have for the optimal pump settings, for which both τ 1 and τ 1h (0) reach their minima. Through simple mathematical calculations, one can derive the following formula for the optimal value of spectral correlation coefficient: This implies that for sufficiently short propagation distance the optimal SPDC photons are always positively correlated, while for very long distance communication schemes negative correlation is best. However, the distance at which the type of optimal spectral correlation changes from positive to negative depends on the parameter σ.
Designing the optimal SPDC source. It is also important to know the optimal photon pair source design when one can arbitrarily choose both the crystal and the pump laser settings, τ p and σ, for a given pair of symmetric transmission links characterized by the parameters L and β. The dependence of the temporal widths τ 1 , τ 1h (0) and τ 1h,Δt (0) on the pump laser settings, plotted for fixed values of L and β, can be seen in Fig. 5. A simple calculation shows that both τ 1 and τ 1h (0) reach their absolute minima for τ β = | |L 2 p opt and σ β = | |L 2/ opt . Those minima are identical and read: This means that for an optimal SPDC source, the narrowing effect is not present. Therefore, spectrally decorrelated pairs are optimal for quantum communication in this case. For our 10 km-long SMF fiber links, we get τ ≈ . 15 2 ps p opt and σ opt ≈ 132 GHz. Alternatively the absolute minimal value of τ 1h,Δt (0) equals h t 1 , abs Again, it can be reached for σ β = | |L 2/ opt , but for arbitrary τ p .

Discussion
The possibility for narrowing temporal widths of photons by performing time-resolved measurements and optimizing the settings of the photon pair source, discussed above, could have multiple applications. As an example, one can consider the problem of long-distance fiber-based QKD. While the first experimental realizations of QKD protocols with pairs of photons created in SPDC process were presented almost 20 years ago [25][26][27] , very few QKD implementations utilizing long-distance telecommunication fibers and SPDC sources have been demonstrated to date. Furthermore, according to our knowledge, the parameters of the source were not optimized in any way for these implementations. For example, ref. 28 used pairs of photons created utilizing a CW laser while ref. 29 used relatively long pump pulses of 1.4 ns temporal width. The detection gates for SPDC photons that were defined in the aforementioned works (2.5 ns-long and 2 ns-long, respectively) were much longer than they could have been if the photon-pair sources were properly optimized. Such optimization would then lead to much higher signal-to-noise ratios and enable generation of secure keys for considerably longer distances separating the participants of the protocol.
In fact, all of the record-breaking long-distance fiber-based QKD implementations reported in the literature in recent times [33][34][35][36] have been realized with weak coherent pulses (WCP) and decoy-pulse method [30][31][32] , which allows the trusted parties to neutralize multiphoton events. However, there are many papers suggesting that using a heralded single-photon source can be more beneficial for decoy-based QKD, both in the standard 37,38 and in the measurement-device-independent 39-42 regimes. One of the limiting factors of WCP sources in this context is that even in the ideal decoy case the amount of truly single-photon pulses that can be obtained from a WCP source cannot be larger than approximately 37% of all the emitted signals. Meanwhile, the heralded single-photon sources based on SPDC process can provide the participants of a QKD protocol with pulses that have much better photon statistics and heralding efficiency exceeding 80% 43-46 . Moreover, by optimizing the settings and performing temporal filtering in the way we demonstrated in our work, one can reduce the temporal width of the emitted photons to a level comparable to the temporal width of WCP photons, without substantially decreasing the efficiency of the source. Contrary to WCP sources, photon-pair sources can also be placed in the middle of the distance separating the trusted parties, forming a symmetric QKD setup configuration, analogous to the scheme analyzed in this paper. The superiority of such a scheme over the standard, highly asymmetric setup configuration in terms of the maximal security distance between the participants of a QKD protocol was theoretically demonstrated in ref. 47 . Taking all of the above considerations into account, one can expect SPDC sources to provide very attractive alternative to WCP for long-distance fiber-based QKD, if optimized properly.
In summary, we investigated the problem of wavepacket shaping of a single photon heralded by the time-resolved detection of the other photon from an SPDC pair. We showed how the strength of the wavepacket www.nature.com/scientificreports www.nature.com/scientificreports/ narrowing depends on the parameters of the photon pair source and the width of the detection window for the heralding photon. Our theoretical predictions 10 were compared with the experimental results, showing very good agreement. We experimentally observed the reduction of the width of the heralded wavepacket to approximately 29% as compared to the non-heralding scenario. It should be stressed here that the detection windows used in our experiment were established only for the purpose of showing the wavepacket narrowing. In general case, using free-running detectors with good time-resolving capability, one can detect all of the heralding photons and subsequently perform temporal narrowing of the heralded photons as a postprocessing procedure. In this way it is possible to increase signal-to-noise ratio of the measurement results without introducing any filtering of SPDC photons, which always lead to lower heralding efficiency. Moreover, in this work we performed optimization over the pump laser pulse duration in order to find the minimal possible wavepacket temporal width. For the case of our experimental setup, further narrowing to 11.4 % is feasible for τ p = 15.2 ps. Finally, we derived formulas for optimal parameters of the SPDC source, minimizing the aforementioned temporal widths. In our work we focused on the scenario, in which the SPDC photons propagate through a pair of identical single-mode fibers. A generalization to the case of fibers with non-equal lengths or different dispersion characteristics is currently under consideration. Further extension of this framework, which would allow for the optimization of quantum communication protocols relying on interference between two photons, e.g. quantum teleportation 48,49 or entanglement swapping 50,51 , is also possible.

Methods
Heralded photon's arrival time in realistic situations. In practice, the detection window for the heralding photon always has finite width. Let us assume that the heralding photon was detected at the time T 2 with uncertainty ΔT, which is the width of detection window. In this case, the probability distribution for the arrival time of the heralded photon can be calculated from (1) by utilizing the following expression: Note that ΔT → ∞ when the information on the arrival time of the heralding photon is not available to the experimenter. In this case: On the other hand, for very short detection window we get: Therefore, the temporal width of the heralded photon in the asymptotic case of ΔT → 0 reads From this expression, we can easily obtain Eq. (2). Furthermore, Eq. (14) allows us to calculate the average detection time of the heralding photon in the case of ΔT → 0, which is given by the formula (3).

Calculation of the temporal widths of SPDC photons.
In order to express the temporal widths of SPDC photons in terms of the source parameters σ and τ p , we start by assuming a simplified biphoton wavefunction in the following form 23,52 : This expression is similar to the formula (3) in ref. 10 , where we utilized parameters directly describing the properties of the biphoton state, namely its spectral correlation coefficient ρ and spectral widths σ 1 , σ 2 . Those two formulas are equivalent to each other if the crystal produces pairs of photons with equal spectral widths, i.e. σ 1 = σ 2 ≡ σ 0 . In this case the comparison between them gives the following transformation: www.nature.com/scientificreports www.nature.com/scientificreports/ This allows us to rewrite the expressions (8), (10) and (12) for the temporal widths of the heralded and non-heralded wavepackets derived in 10 in terms of the parameters describing the photon pair source, σ and τ p , and transmission links, β and L. In this way we obtain the formulas (4), (6) and (8).