High fidelity field stop collection for polarization-entangled photon pair sources
Alexander Lohrmann, Aitor Villar, Arian Stolk, Alexander Ling
HHigh fidelity field stop collection for polarization-entangled photon pairsources
Alexander Lohrmann, Aitor Villar, Arian Stolk,
1, 2 and Alexander Ling
1, 3 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, S117543,Singapore Currently with QuTech, Delft University of Technology, PO Box 5046, 2600 GA Delft,The Netherlands Physics Department, National University of Singapore, 2 Science Drive 3, S117542,Singapore (Dated: 26 October 2018)
We present an experimental demonstration of a bright and high fidelity polarization entangledphoton pair source. The source is constructed using two critically phase matched β -BariumBorate crystals with parallel optical axes and photon pairs are collected after filtering witha circular field stop. Near-unity fidelities are obtained with detected pair rates exceeding100 000 pairs / s / mW, approaching the brightness of practical quasi-phase matched entangledphoton sources. We find that the brightness scales linearly with the crystal length. We presentmodels supporting the experimental data and propose strategies for further improvement.The source design is a promising candidate for emerging quantum applications outside oflaboratory environments.Keywords: Entangled photon sources, spatial phase, critical phase matchingEntangled photon pairs lie at the heart of many emerg-ing quantum technologies, such as quantum communica-tion, key distribution and teleportation . An on-goingresearch problem is to develop sources of entangled pho-tons that are bright and, at the same time, robust enoughto be suitable for long term operation outside of labo-ratory environments. The main method of generatingentangled photons is based on spontaneous parametricdownconversion (SPDC) in nonlinear crystals.One of the important considerations in the design ofentangled photon sources is the effective collection an-gle of SPDC photons. This is critical because the an-gle dependent phase can degrade the desired entangledstate. To negate this effect, many source designs andapplications use single-mode collection or spatial filter-ing. This, however, strongly reduces the source bright-ness. Collection through a field stop, on the other hand,may allow for high brightness and fidelity, if the angledependent phase is taken into consideration . Previ-ous work in this direction has been restricted to the useof thin crystals . In this paper, we report on the col-lection of high fidelity entangled photons through a fieldstop when using thick crystals.This result utilizes a source based on type-I, collinear,non-degenerate critical phase matching using two β -Barium Borate (BBO) crystals with parallel opticalaxes . The parallel axes approach leads to an almostperfect spatial overlap of the emission modes enabling thedetection of maximally entangled photon pairs withoutsingle-mode filtering. We demonstrate pair rates of upto 100 000 pairs / s / mW with near-unity fidelity. Further-more, by relaxing the collection conditions, we achieverates of 400 000 pairs / s / mW with QKD compatible fi-delity.The source design is sketched in Fig. 1(a). The pumpwith a vertical (extraordinary) polarization undergoes awalk-off within the first BBO crystal. Due to the type-Iphase matching, photon pairs with horizontal (ordinary) polarization ( | H s H i (cid:105) ) are generated. An achromatichalf-wave plate rotates the polarization of these photonpairs to the orthogonal state ( | H s H i (cid:105) → | V s V i (cid:105) ) whileleaving the pump polarization unaffected.In the second crystal the pump again generates | H s H i (cid:105) photons. The photon pair from the first crys-tal has extraordinary polarization in the second crystaland therefore undergoes a walk-off in the same directionas the pump. This is the reason for the almost perfectspatial overlap of the SPDC emission modes from thetwo crystals. A residual spatial mismatch between thetwo modes exists, due to the different walk-off angles ofpump and SPDC photons (walk-off angle difference:∆ ρ ≤ .In general, the state generated after the two BBO crys-tals can be written as, | Φ (cid:105) = 1 √ (cid:0) | H s H i (cid:105) + e i ∆ ϕ | V s V i (cid:105) (cid:1) , (1)where ∆ ϕ denotes the phase difference between the twoemission processes. In order to observe one of the twomaximally entangled Bell states, Φ ± , (from now on weassume Φ − for convenience) in type-I phase matching,the phase difference ∆ ϕ of all collected photon pairs mustbe constant in all spatial and spectral degrees of freedom.To determine the total phase difference, we considerthe phase accumulated by the individual photons (e.g.,the signal photon) as they pass through the elements ofthe setup,∆ ϕ ( λ, (cid:126)x, (cid:126)α ) = (cid:88) j ϕ V ( λ, (cid:126)x, (cid:126)α ) − (cid:88) j ϕ H ( λ, (cid:126)x, (cid:126)α ) . (2) a r X i v : . [ qu a n t - ph ] O c t (c) BBO I HWP Δ φ Δ φ BBO IIL LL
HWP k s , k s , k ' s , k ' s , BBO I BBO IIHWP YVO ϑ = 28.8°(a) ϑ = 28.8°(1) (2) (3) (i) (ii) (iii) (b) (1/i)(2/ii)(3/iii)Exit face view0.0 1.0-1.00.01.0-1.0 Polar (°) A z i m u t h ( ° ) -1 -2 Δ φ ( r a d ) (e) ϑ = 28.8° ϑ = 28.8° M g F Q u a r t z (d) 051015 Azimuth (°)0.0 1.0-1.0 F i d e li t y Δ φ ( r a d ) FIG. 1. (a) Basic principle of the parallel crystal configuration. The numbers 1-3 and i-iii indicate three possible SPDCgeneration positions for each crystal. The colors indicate the different photon polarizations (green: | V V (cid:105) , red: | HH (cid:105) ). (b)Sketch of the SPDC spatial distribution at the exit face of the second crystal. The numbers 1-3 and i-iii indicate the photonorigin as depicted in (a). (c) Signal (idler omitted for clarity) ray propagation diagram for collinear (green) and non-collinearemission (black). The rays in the figure indicate the light propagation (Poynting vector) for the initial k -vectors ( k s, and k s, from the first crystal and k (cid:48) s, and k (cid:48) s, from the second crystal). Pump indicated in blue. The angle dependence of the phasedifference originates from the effective path length difference. (d) ∆ ϕ as a function of the polar opening angles for the signalray in air for every point in (b) for a source based on 6 mm BBO crystals. The area inside the red circle indicates a region ofapproximately constant fidelity. (e) Phase difference (dashed line) and fidelity of | Φ − (cid:105) (solid line) as a function of the verticalpolar emission angle (upwards in (a)) extracted from (d) when the horizontal polar angle is 0 ◦ . The state transitions between | Φ − (cid:105) ( F = 1) and | Φ + (cid:105) (F=0). The red dotted lines indicate a region of approximately constant phase equivalent to (d). (Coloronline) Here λ denotes the photon wavelength, (cid:126)x the posi-tion within the crystal where the downconversion oc-curred, (cid:126)α the signal emission angle and j an index foreach traversed medium (BBOs, air gaps, HWP, andpost compensation crystal). The superscripts H and V denote the final photon polarization and also includethe phase of the pump photons before downconversion(see supplementary material). The wavelength depen-dence of ∆ ϕ is sufficiently compensated with a bire-fringent element (yttrium orthovanadate, YVO inFig. 1(a)). This leaves only angular and position depen-dencies, ∆ ϕ = ∆ ϕ ( (cid:126)x, (cid:126)α ), which are usually negated byeither using thin crystals or single-mode fiber collection.Regarding the position dependencies of ∆ ϕ , the design &BBO I BBO IIHWP YVO FF LD Iris 1
Lens pair
Detection
Iris 2
PolarizerLP DM FIG. 2. Sketch of the experimental setup. LD: laser diode,FF: fluorescence filter, HWP: half-wave plate, LP: long-passfilter, DM: dichroic mirror. An iris is used to control theopening angle of the SPDC light that reaches the detectors.It can be placed either before splitting signal and idler (Iris1) or after the dichroic mirror in the signal arm (Iris 2). TheYVO (length: 3.6 mm) is placed in the collimated beam toavoid an additional angle dependence of ∆ ϕ . (Color online) used in the present work has an advantage over previousconfigurations, e.g. using crossed crystals. In general,the SPDC pair origin of each emission process is mappedonto the transverse emission profile by the walk-off. Forexample, pairs originating from the entry face of a crys-tal are emitted from the top of the emission profile (seefigure 1(b), 1/i). These pairs traverse more nonlinearmaterial than those originating from near the exit faceof the crystal. Only if the phase difference for each pointon the combined final transverse emission profile (at theexit face of the second crystal) is constant, the desiredentangled state can be produced.In the present configuration, phase compensation isachieved for each photon pair irregardless of the down-conversion location within the crystals. For example,photon pairs generated at the entry face of the firstcrystal are spatially indistinguishable from photon pairsborn at the corresponding position in the second crystal(see Fig. 1(b)). This results in a constant phase differ-ence across the emission profile. The parallel axes con-figuration therefore allows the collection of SPDC pho-tons through a field stop without single-mode filtering orseverely restricting the collection region. This is a par-ticular feature of the parallel axes configuration. In theanti-parallel case, it is straightforward to determine that∆ ϕ is not constant over the emission profile.The angular dependency of ∆ ϕ is linked to the pathlengths experienced by SPDC photons emitted under dif-ferent angles (see Fig. 1(c)). This path length effect isshown in Fig. 1(d) as a function of the polar emissionangles (see supplementary material for further details).To collect the maximally entangled Bell state one can -45 0 45 90 1350.01.02.03.04.05.0 Polarizer angle (degree) P a i r r a t e ( k / s / m W ) HH DD VV AA-3 -2 -1 0 1 2 30.00.20.40.60.81.0 Iris position (mm) F i de li t y -0.75 -0.5 -0.25 0 0.25 0.5 0.75Opening angle (degree)(a) F = 0.975(b) FIG. 3. (a) Detected photon pair rate when using a single po-larizer acting simultaneously on signal and idler photons. Inthis case the iris was centered in the signal arm. The diameterof the iris was 0 . ± .
10 mm. The fidelity was 97 . ± . (cid:126)α isassumed to be linear in iris position. The gray shaded areaindicates the uncertainty of 1 σ due to the finite iris diameter.Squares are experimental data. restrict the SPDC collection angle appropriately using acircular field stop. Figure 1(e) illustrates how the relativephase impacts the entangled state fidelity.The implementation of the entangled photon source isshown in Fig. 2. The output from a collimated, narrow-band 405 nm laser diode (∆ ν ≤
160 MHz) is used togenerate SPDC in the 6 mm BBO crystals (optical axisangle θ = 28 . ◦ ) set for type-I, collinear phase match-ing. The photons are non-degenerate at 785 nm and837 nm. The downconverted photons are collimated us-ing a lens pair (see supplementary material), split accord-ing to their wavelengths and detected using Geiger-modeavalanche photo diodes (GM-APD). The spectral band-width of the photons is 25 nm for signal and idler photonsand no additional spectral filtering is used.To restrict the opening angles of signal and idler pho-tons, a field stop can be placed in the collimated beam.We use a circular, adjustable iris as a field stop (Iris1 in Fig. 2). This allows the fidelity and brightness ofthe source to be actively controlled by the iris diame-ter. Alternatively, the iris can be placed in the signalor idler arm and scanned to access individual emissionangles (Iris 2 in Fig. 2).To characterize an unknown quantum state, full quan-tum state tomography is usually employed. However,with some reasonable assumptions regarding the gener-ated state, the measurement can be greatly simplified. As the phasematching prevents the generation of | HV (cid:105) and | V H (cid:105) components, the generated state is impactedonly by the imbalance between the polarization compo-nents and the mixing of pure states with different valuesof ∆ ϕ . The fidelity of the photon pairs can be estimatedby partial quantum state tomography using a single po-larizer after the temporal compensation crystal. Thisprojects signal and idler into the same linear polarizationbasis and, when the polarizer is rotated, this leads to aunique signature for the maximally entangled Bell statesΦ + (zero contrast curve) and Φ − (full contrast curve) asdiscussed elsewhere .The angle dependent phase model can be validatedby translating the iris in the signal arm while employ-ing bucket detection in the idler arm. At each positionof the iris, this setup enables the fidelity of the photonpairs emitted at different angles to be experimentally de-termined. Figure 3(a) shows the characteristic signatureof a state close to | Φ − (cid:105) when the iris is in the center ofthe signal beam (iris diameter d = 0 . ± .
10 mm). Thecontrast of the curve indicates a high entanglement qual-ity and the fidelity towards | Φ − (cid:105) extracted from the fitis F = 0 . ± . | Φ − (cid:105) can be attributed to residual spatial and spec-tral phase components that impact the measured fidelity.Note that a fidelity of above 99 % can be achieved as pre-sented below.The fidelity when the iris is translated across the signalbeam in vertical direction is shown in Fig. 3(b). In thiscase, the phase was set for maximum fidelity when the iriswas at 0 mm and not adjusted for different data points.Ideally, the state transitions between | Φ − (cid:105) and | Φ + (cid:105) (seeFig. 1(e)). However, the fidelity is limited by the finiteiris size and the consequent mixing of states with differentvalues of ∆ ϕ . A model of the convolution of ∆ ϕ withthe finite iris size is constructed and shown in Fig. 3(b)(solid black line). One standard deviation derived fromthe uncertainty of the iris diameter is shown as the grayshaded area. The experimental results and the modelare in good agreement. In particular, the revival of thefidelity as the iris is translated in the signal arm confirmsthe relationship between phase and emission angle. Thisresult holds for both, horizontal and vertical, polar angles(see supplementary material).In the next step, we placed the iris in the collimatedbeam before splitting signal and idler photons and mea-sured the pair rate as a function of the emission angleby gradually opening the aperture. The photon pair ratewas measured at low pump power of P ≤ µ W to avoidsaturating the passively quenched single photon detec-tors. The source brightness increased as expected forlarger collection angles; with the fully open aperture, weobserved a pair rate of 321 ±
11 k / s / mW (see Fig. 4(a)).As the angular far-field emission profile of the SPDCphotons approximates a Gaussian function, the curve ofthe pair rate takes the shape of an error function. Theobserved pair-to-singles ratios were approximately con-stant over this range with 18 . ± .
9% and 20 . ± . ≤ − F i de li t y P a i r r a t e ( k / s / m W ) F i de li t y RateFidelity0.9 1.8 2.6 3.5 4.4Iris diameter (mm) (a) (b)
Emission half angle (degree) Pair rate (k/s/mW)
FIG. 4. (a) Observed pair rate (open triangles) and fidelity (filled triangles) as a function of emission angle for a source using6 mm BBO crystals. The emission angle was calculated from the iris diameter. An error function was fitted to the pair ratedata points. The solid line describes the calculated fidelity obtained from the phase map (Fig. 1(d), and is not fitted using afree parameter). (b) Correlation between fidelity and brightness for multiple crystal lengths. The brightness is controlled bythe iris diameter. The dashed line indicates the QBER limit for the Ekert protocol of QBER ≤ . (Color online) reflection and absorption at optical surfaces.For each iris diameter, we measured the fidelity. Byopening the iris, we are effectively integrating over thevariation in ∆ ϕ as shown in Fig. 1(d). The relation-ship between fidelity and opening angle is shown inFig. 4(a). When the collection angle is below 0.1 ◦ thefidelity reached near-unity values of F = 0 . +0 . − . anddegrades for greater collection angles due to increasedmixing. Such a degradation has been observed beforefor thin crystals . When applied to thick crystals withcollinear emission, our results show a high Bell state fi-delity of F ≈ .
99 at detected pair rates of more than100 000 pairs / s / mW without pump shaping or spectralfiltering. This is particularly interesting for applica-tions outside of laboratory environments, where criticallyphase matched sources are thought to be advantageousdue to their relative temperature stability. For com-parison, a high-fidelity single-mode fiber coupled sourcebased on a comparable crystal length and configurationyields pair rates of only up to 65 000 pairs / s / mW .To further improve the brightness, one cannot simplyincrease the crystal length. This is because the angledependent phase difference scales linearly with the in-teraction length (see supplementary material). The ex-perimentally observed correlation between pair rate andfidelity is shown in Fig. 4(b) for different crystal lengths.As expected, higher pair rates can be collected fromlonger crystals under the same collection conditions, butthere is always an upper limit for the pair rate at accept-able fidelity.One possible strategy to increase the pair rate whilemaintaining near-unity fidelity is to compensate the angledependent phase difference. Due to the approximatelycircular symmetry of ∆ ϕ (see Fig. 1(d)), this can beachieved conveniently by placing a spherical birefringentlens in the collimated beam. Such a lens could be usedto simultaneously compensate any remaining spatial dis-tinguishability caused by the different spatial origins ofthe | HH (cid:105) and | V V (cid:105) pairs.In conclusion, we have shown that the beneficial spa-tial overlap in the parallel axes geometry enables the useof free-space detection which greatly enhances the rate of detected photon pairs for type-I, critically phase matchedsources. This result was achieved for crystal lengths of upto 12 mm in contrast to earlier works which only focusedon thin crystals . Similar to these previous works, thepair rate can be controlled by restricting the SPDC emis-sion angle at the expense of entanglement quality. If highfidelity is needed, the pair rate may be reduced to ensurea near-unity fidelity, while for non-critical applicationspair rates of more than 0 . / s / mW are possible.Such detected pair rates are comparable to those in quasiphase-matched sources .In future work, the angle dependent phase may becompensated by an additional birefringent lens or a spa-tial light modulator to maintain the entanglement qualitywhen the SPDC emission angle is not restricted. More-over, an additional birefringent crystal may be added tothe source to compensate the residual spatial mismatchbetween the | HH (cid:105) and | V V (cid:105) pairs. SUPPLEMENTARY MATERIAL
The supplementary material contains further informa-tion about the phase calculations, the single polarizermeasurement and the experimental setup.
ACKNOWLEDGMENTS
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