3.2 Heralded Amplification of Single Photon Path Entanglement
3.2.4 Experimental implementation of path entanglement amplifi-
left: forτ ={0.5, 0.8} and right: for the optimalτ.
state but at the cost of a lower heralding probability. To experimentally verify the amplification and the preservation of the path entanglement, we will analyse the path entangled states before and after amplification using two displacement-based measurements and compute the adapted fidelity given in eq.3.13. The following section presents how to implement this.
3.2.4 Experimental implementation of path entanglement amplification
Displacement-based verification of path-entanglement requires a good control of the relative phase between the two displacements. This is easily achieved when the path entangled state and the coherent state co-propagate since the BS on which the path-entangled state is created, as previously explained in section3.1.2. Figure 3.9 shows a set-up to amplify path-entanglement in which the path entangled state and coherent state travel together in each mode. The BS with transmission (1−τ) is now replaced by a PBS, preceded by a HWP on each input port, such that the coherent state and the path entangled state propagate together on pathbandcbut with orthogonal polarizations. The photon state in path c is sent to the amplifier via a lossy channel. At the input of the amplifier, the photon and coherent states are separated on PBS**. The coherent state is sent towards Bob and the photon state to the BSM. On the same PBS**, the auxilliary photon is transmitted towards Bob with probability t and reflected to the BSM with probability (1-t). The heralded output state in path e will then travel together with the coherent state but with orthogonal polarization.
One disadvantage of this set-up is that any leakage of the coherent state through PBS** in path f, will induce undesired detections at the BSM and false-positive
50:50
BSM
Loss
a b
c
d t
e
f n
m
Alice Bob HWP
PBS
**
HPA
Figure 3.9: Set-up of the path entanglement amplifier modified to implement the displacement-based detection. The coherent state is injected via PBS-1 with a polarization orthogonal to ρin. At PBS** the coherent state is reflected towards Bob while Bob’s part of the path-entangled state is transmitted to the BSM.
heralding. This effect could be suppressed with an attenuation of the coherent state through PBS** of∼10−5, which is calculated for the optimal value of coherent state needed for the displacements. However, current cubic and plate PBSs are generally limited to an attenuation of ∼ 10−4. One solution to this problem is to add a temporal delay between the path entangled state and the coherent state and to gate the detector at the BSM accordingly. This delay can then be compensated at Alice and Bob by phase-locked unbalanced interferometers, as shown in Figure3.10 (a).
One goal of this experiment is to compare path-entanglement before and after amplification. Hence, the set-up should allow to easily by-pass the amplifier. Fig-ure 3.10 (b) shows an implementation of the previous set-up, which can by-pass the amplifier with the rotation of the HWP labelled 2. To understand how this is done, we should first look at the implementation of the interferometers. The interferometers at Alice and Bob are both made of: a PBS to separate the photon state and the coherent state, two QWPs and a HWP in the long arm to rotate the polarization, and a 99:1 BS which closes the interferometer. The photon state tak-ing the short arm and the coherent state taktak-ing the long arm arrive together at the 99:1 BS, now with the same polarization. A CW reference laser for phase-locking is inserted through the unused port of the 99:1 BS. It counter-propagates along the two arms and exits the PBS with two modes orthogonally polarized. After passing through a HWP and a PBS, where they interfere, the two modes are detected with linear photo-diodes. The signal is used to correct the phase fluctuations with a PZT mounted on a mirror in the long arm of the interferometer. At Alice, a liquid crystal (LC) is placed just before the HWP and the PBS on which the two modes of the reference laser interfere. The change in phase introduced by the LC, ∆φLC will then be compensated by the locking system to recover the phase at which it was initially locked. The phase correction made by the PZT will be equal to ∆φPZT =−∆φLC. Importantly, this correction will also change the phase difference between the pho-ton state (short arm) and the coherent state (long arm), even though they never
3.2. Heralded Amplification of Single Photon Path Entanglement 51
Source DFG PPKTP Source (Auxiliary)
HWP QWP Liquid Crystal SPD FBG Filter PBS
Heralding Photon Coherent State Stabilization LASER Auxillary Photon Intensity Detector
Figure 3.10: (a) Implemented set-up of path entanglement amplifier. A delay is introduced between the path-entangled state and the coherent state to avoid false positive detection at the BSM. (b) detailed implementation of the set-up above, slightly modified to be able to easily by-pass the amplifier. The interferometers used to compensate the delay between the photon state and coherent state were phase-locked in a way to control also the relative phase between the two displacements.
pass the LC. This method avoids extra losses on the path entangled state. If the interferometer at Bob is also locked, then the relative phase between Alice’s and Bob’s displacements can be adjusted by varying ∆φLC.
By placing the first PBS** of the amplifier in the short arm of Bob’s interfer-ometer, we can easily by-pass the amplifier. Indeed, the HWP-2 can be set to send the photon state either to the 99:1 BS for displacement or to the amplifier.
The heralded single photons and coherent state sources were fully described in section3.1.3. The high coupling efficiency (≈80%) and purity (≈90%) of the PP-KTP source are essential for this experiment. The coupling efficiency is important because, as we saw in section3.2.2, the gain of the amplifier is mainly affected by the coupling and transmission losses on the auxiliary photon. The high purity, on the other hand, is an advantage to avoid additional filtering stages, as discussed below.
The amplifier relies on the interference between Bob’s part of the path entangled state and the auxiliary photon. A motorized delay line on the second-pass of the PPKTP source is used to ensure that these two arrive at the BSM at the same time. The exact position of the delay line is set by performing a Hong-Ou-Mandel (HOM) interference experiment. With a 200-GHz DWDM filter placed before the detector, the HOM visibility was about 90% [116]. This visibility could be increased close to unity by filtering narrower either the photons at the BSM or the heralding photons. However, this will further decrease the detection rate at the BSM, which is already small due to the probabilistic nature of the two SDPC sources. With a 76 MHz pulsed pump laser, the SPDC sources simultaneously generated about 30 path entangled states and auxiliary photons per second. To compromise between high visibility interference and very low rate of the experiment, we decided not to filter narrower at the heralding of the signal photon nor at the BSM. The heralding of the amplification at the BSM was achieved with rates 0.25 Hz, 0.19 Hz and 0.15 Hz, respectively, for decreasing transmission ηL of the lossy channel of approximately 0.28, 0.17 and 0.09, corresponding to equivalent transmission distances in standard Telecom fibre (0.2 dB km−1) of around 30, 40 and 50 km. Without the amplifier, the rate of experiment was given by the heralding rate of the signal photon, which was about 50 kHz. The choice of not filtering further also meant that the spectrum overlap between the path entangled state and the coherent state was not higher than 0.9±0.02, value measured without any filtering. Although not ideal, this value was enough to perform the displacement-based measure of the path entanglement.
3.2.5 Measurements and Results
The aim of the experiment is to verify the amplification and the preservation of the entanglement. The amplification can be tested by comparing the fidelity of eq.3.13 before and after amplification. These fidelities are obtained from measurements of the joint probabilities p00, p0c, pc0 and pcc for both no displacement |α|A,B = 0 and the optimal displacement |α|A,B ≈ 0.707. To verify the preservation of entangle-ment, these fidelities are then compared to the maximum valueFsepthat the fidelity can reach for separable states. If F > Fsep, the entanglement is preserved during the amplification. The value Fsep is computed from the measurements p00, p0c, pc0 and pcc with |α|A,B = 0 and the double pair probabilities p2c at Alice and Bob.
Measurements of the joint probabilities were performed for |α|A,B = 0 and
|α|A,B ≈ 0.707, both with and without amplifier. The transmission τ = 0.6 was used to create the initial path entangled state to be amplified, while the reflection of the auxiliary photon through the PBS** was set to t = 0.93. These two values were maintained for all measurements, as they provide a reasonable trade-off be-tween measurement time and the fidelity after amplification for all tested cases. The voltage applied to the liquid crystal was adjusted to set the relative phase between Alice and Bob displacements to zero.
The measured fidelities before and after amplification are plotted in Figure3.11
3.2. Heralded Amplification of Single Photon Path Entanglement 53 Equivalent Fibre Distance (km)
a)
Figure 3.11: Measured fidelities before and after amplification for the three channel transmissions ηL. The right axis shows the expected fidelities we would obtain in the case Alice and Bob use single photon detectors with unit efficiency. The top axis shows the fibre distance that has equivalent transmission of our lossy channel.
ηlink ≈0.09 ≈0.17 ≈0.28
F −Fsep 0.034 0.031 0.018
Confidence level that F > Fsep 2.4 s.d. 2.6 s.d. 1.2 s.d.
Table 3.1: CalculatedF−Fsepfor each amplified state and its respective confidence level in standard deviation (s.d.). The probability that F < Fsep for all measure-ments is ≈ 5× 10−6, showing with a high confidence level that the amplifier is preserving entanglement.
for the three channel transmissions. The blue and pink bands are the theoretically expected fidelities with and without amplification, respectively. They were calcu-lated usingF =hψ|ρi,f|ψi, where ρi is the expected state before amplification and ρf is the expected state after amplification taking into account the coupling efficien-cies of photon sources and the detection and transmission efficienefficien-cies for Alice and Bob, see Appendix 5.0.3. We can see that the the error bars for the experimental data are much larger for the amplified case. They are mainly dominated by mea-surements of the off-diagonal coherence term d, which is a function of all the joint probabilities and terms exponential in|α|2, see eq. 3.16. Thus, small changes of|α| during the measurements, for example due to polarization drifts, will result in larger errors indthan just the statistical case. Nevertheless, the measured fidelities agree with the model, and we observe a clear increase in the fidelity after amplification.
To ensure that entanglement was also preserved, the amplified fidelities were
compared to the maximum fidelityFsepthat can be obtained with a separable state, see Table 3.1. By combining all results it is possible to calculate a probability of
≈5×10−6 that F < Fsep, i.e that entanglement is not preserved. We can therefore conclude with a high confidence level that the amplifier is preserving entanglement.
The joint probabilities were further used to reconstruct the density matrices for the initial and final states, assuming unit detection efficiencies. Figure 3.12 shows the density matrices for the transmission ηL = 0.09. We first observe that the vacuum component |00i h00|has decreased after amplification. We can also see that the amplifier has enhanced the click probability at Bob (|01i h01|) while slightly decreasing the one at Alice (|10i h10|). Importantly, the coherence terms |01i h10| and|10i h01|have also increased after amplification, and we recover a state close to the maximally entangled state.
Figure 3.12: Reconstructed density matrices of the state before amplification (left) and after amplification (right), forηL= 0.09.
3.2.6 Discussion
The results showed a clear increase of the fidelity with respect to the maximally entangled state after amplification. What are the limitations preventing us to ap-proach a fidelity of one?
First of all, we are greatly limited by the low efficiency of the InGaAs detectors.
Replacing them by high efficiency superconducting nanowire single photon detectors (SNSPD) is not an option in our case, as the arrival of the coherent state at 76 MHz will saturate the detector and greatly reduce the probability of detecting events heralded by the BSM. One possibility would be to reduce the clock rate of the experiment to match the relatively slow recovery time of the SNSPDs, as done in [105]. This is obviously not ideal given the low rate of our experiment. This detection efficiency issue could be solved with the development of SNSPDs with fast recovery time or means to gate them.
Secondly, the parametersτ and tcould be optimized to reach higher fidelities at the cost of much lower rates. One would need to find means to increase the rates or to achieve long time stability. On the stability side, one clear improvement would be to implement polarization and power stabilization of the coherent state. However, achieving stability over long periods would still be very challenging. Finding means
3.3. Heralded Distribution of Path Entanglement via Swapping 55