The biaxial nonlinear crystal BiB3O6 as a polarization entangled photon source using non-collinear type-II parametric down-conversion
A. Halevy, E. Megidish, L. Dovrat, H.S. Eisenberg, P. Becker, L. Bohatý
aa r X i v : . [ qu a n t - ph ] O c t The biaxial nonlinear crystal BiB O asa polarization entangled photon sourceusing non-collinear type-II parametricdown-conversion A. Halevy, E. Megidish, , L. Dovrat, H. S. Eisenberg, P. Becker, and L. Bohat´y Racah Institute of Physics, Hebrew University of Jerusalem,Jerusalem 91904, Israel Institute of Crystallography, University of Cologne,50939 Cologne, [email protected]
Abstract:
We describe the full characterization of the biaxial nonlinearcrystal BiB O (BiBO) as a polarization entangled photon source usingnon-collinear type-II parametric down-conversion. We consider the relevantparameters for crystal design, such as cutting angles, polarization of thephotons, effective nonlinearity, spatial and temporal walk-offs, crystalthickness, and the effect of the pump laser bandwidth. Experimental resultsshowing entanglement generation with high rates and a comparison to thewell investigated b -BaB O (BBO) crystal are presented as well. Changingthe down-conversion crystal of a polarization entangled photon sourcefrom BBO to BiBO enhances the generation rate as if the pump power wasincreased by 2.5 times. Such an improvement is currently required for thegeneration of multiphoton entangled states. © 2011 Optical Society of America OCIS codes: (190.4400) Nonlinear optics, materials; (190.4410) Nonlinear optics, parametricprocesses; (260.1180) Crystal optics; (270.0270) Quantum optics; (270.5585) Quantum infor-mation and processing.
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1. Introduction
For more than two decades, parametric down-conversion (PDC) is a central tool for the gen-eration of entangled photons. This is a second order nonlinear process, where a photon from pump beam splits into two photons, known as signal and idler , while conserving energyand momentum. The down-converted photons exhibit strong correlations in various degrees offreedom, such as wavelength, time of emission, polarization, momentum, and position [1]. Thedown-conversion pair generation rate depends linearly on the pump beam power, and quadrati-cally on the crystal thickness and on its nonlinear coefficients [2]. Since the first demonstrationof an efficient PDC polarization entangled photon source [3], the most commonly used nonlin-ear birefringent crystal for this purpose is the uniaxial crystal b -BaB O (BBO). The reasonsfor this are its relatively high nonlinear coefficients, high transparency, and the possibility forphase-matching over a broad spectral window [4, 5].In the last decade, many quantum optics experiments have used two consequent PDC events[6–8]. Others have used second order events of PDC [9–11]. These events occur when two in-distinguishable pump photons split into four, during the same coherence time (or pulse durationfor pulsed pump sources). Both approaches require high efficiency of the PDC process as suc-cess probability is quadratic with the single pair generation probability. Later, the third orderPDC event as well as three consequent first order events have been used to create entangledstates of six photons [12–15]. Recently, four consequent first order PDC events were used todemonstrate an eight photon entangled state [16].One possibility to enhance the PDC generation probability is to use thicker nonlinear crys-tals. The crystal length is limited by the non-collinearity of the process, that spatially separatesthe pump beam and the down-converted photons. In addition, there is the spatial walk-off ef-fect between the two polarizations that degrades the entanglement quality (see Sec. 2.5). Thus,higher pump intensity is required. Usually, the pump beam is generated by frequency doublingthe radiation of a Ti:Sapphire laser in another nonlinear crystal [6–16]. Reported typical in-tensities are above 1 W, but as the doubling crystal is damaged by the high power, it has to betranslated continuously in order to maintain stable operation [12]. Additionally, the pump beamintensity can be enhanced inside a synchronized external cavity. Such a setup has been shownto pump a BBO crystal with about 7 W [17].In this work, we suggest and demonstrate the use of a novel crystal with higher nonlinearcoefficients than BBO for the generation of polarization entangled photons. It is the mono-clinic biaxial BiB O (BiBO) crystal that has been introduced [18] and characterized [19] asa nonlinear optical crystal about a decade ago. Since then, it was used in numerous frequencyconversion experiments (for example, see Ref. [20], and Refs. within). BiBO was also usedwith type-I PDC for generating photon pairs with a pulsed laser source [21] and for generatingpolarization entangled photons with a continuous pump source [22]. It has a very broad trans-parency window and its nonlinear coefficients are considerably higher than those of BBO [18].Nevertheless, the biaxiallity introduces many differences and difficulties, compared to BBO.This paper is organized as follows: in Sec. 2 we present the various considerations in choos-ing the crystal parameters. These parameters are affected by the phase-matching angles, thepolarization direction of the pump beam and the down-converted photons, the pump beam band-width, the spectral and angular properties of the down-converted photons, spatial and temporalwalk-off effects, and the effective second order nonlinear coefficient d e f f dependence on thepump beam direction. Section 3 describes the experimental validation of our theoretical resultsby demonstrating and quantifying the entanglement produced by using two known configura-tions.
2. Investigation of PDC parameters in BiBO
In order to lower reflections and to simplify the required calculations and alignment, it is de-sirable to cut the crystal facets perpendicular to the designed direction of the wave vector k f e e a || a a e e e = F e e e e e e T( y, r)ry (a) (b) (c) Fig. 1. (a)
Relative orientation of the crystallographic axes { a i } and the crystal physicalaxes { e i } . (b) Orientation of the crystal physical axes { e i } and the optical indicatrix mainaxes { e i } . F is the angle of orientational dispersion of the principal axes. (c) The propaga-tion direction of the pump beam T inside the crystal is defined using the two angles y and r in the wavelength independent { e i } system. of the fundamental (pump) wave. The phase-matching calculation for the k f direction scansa quadrant of space, according to the monoclinic symmetry of BiBO. In order to choose theoptimal phase-matching direction, the effective second order nonlinear coefficient d e f f is cal-culated for each k f direction. As an approximation for d e f f , we use the effective nonlinearcoefficient of collinear second-harmonic generation, d SHGe f f , calculated for any direction, eventhough the phase-matching condition is not fulfilled. The optimum direction of k f within therange of the highest values of d e f f should allow the two cones to intersect at 90 ◦ , which isoptimal for the photon collection efficiencies. For the selected direction of k f as well as forthe down-converted photons at the intersection points of the emission cones, the polarizationorientation is calculated. Finally, the temporal and the spatial walk-offs are calculated for thechosen crystal parameters. In order to find the spatial distribution of the cones of down-converted photons, we calculatednumerically the non-collinear type-II PDC process in BiBO. We are interested in the degen-erated case in which the down-converted photons share the same wavelength. The most basicreference system that we use is the crystal physical Cartesian system { e i } . It is linked to thecrystallographic system { a i } (see Ref. [23]) by e k a , e k a k e = e × e , seeFig. 1(a). The point group symmetry 2 of the monoclinic BiBO crystal structure allows theoccurrence of enantiomorphic (i.e., ”left-handed” and ”right-handed”) species. All our samplesfor optical investigations were prepared using crystals that were grown as descendants from thesame parent crystal and therefore posses the same handedness. For our crystals, the positivedirection of a (and e ) corresponds to a positive sign of the pyroelectric coefficient p s (atconstant stress) and to a negative sign of the longitudinal piezoelectric coefficient d [24, 25].In BiBO, the principal axes { e i } of the optical indicatrix coincide with the { e i } systemonly for e = e while e and e change their orientation with wavelength. This orientationaldispersion is illustrated by the angle F = ( e , e ) in Fig. 1(b). For the fundamental and thedown-converted wavelengths used in this work ( l f =
390 nm and l dc =
780 nm), F equals43 . ◦ and 46 . ◦ , respectively [19].Our calculations of the collinear and non-collinear PDC phase-matching cases [26], basicallyfollows the calculation strategy described by Ref. [27]. For a chosen direction T of the funda-mental wave vector k f , we define the propagation direction in spherical coordinates ( y , r ) with
10 20 30 40 50 60 70 80 900102030405060708090 ψ [deg.] ρ [ d e g . ]
60 70 5060ψ [deg.] ρ [ d e g . ] e e P TR (a) (b)
Fig. 2. (a)
Stereographic projection of collinear type-II phase-matching angles for l f =390 nm (black line), with several non-collinear down-converted circles for differentpropagation directions of the fundamental wave. The chosen working point for this workis marked with X. (b) Stereographic projection of PDC in BiBO for y ( T ) = . ◦ and r ( T ) = . ◦ . The vectors P and R are indicated, as defined in the text. respect to { e i } (see Fig. 1(c)). The phase-matching conditions are satisfied when D k = k signal + k idler − k f = , (1)where D k is the phase-mismatch vector, and k signal and k idler are the wave vectors of the down-converted waves. First, we find the collinear phase-matching angles, as in this case Eq. 1 be-comes scalar and simple to solve. Then, we use a search algorithm around the collinear direc-tion to find the non-collinear directions that correspond to the minimal values of D k . We havechosen a numerical threshold value of | D kk f | < × − . Photons are emitted into two coneswith different, and not necessary perpendicular, polarizations. The stereographic projections ofseveral down-converted emission cones onto the ( e , e ) plane are presented in Fig. 2(a). Thisprojection preserves angles and projects circles in three dimensions as circles on the plane [26].Each two tangent circles represent a non-collinear solution, where the direction of the funda-mental wave k f is their collinear intersection point. The down-converted photons experiencerefraction when they emerge from the crystal to air, which depends on their propagation di-rection and their polarization. The calculation results given in this work are of the photon’sproperties outside the crystal. For our wavelength parameters, the phase-matching calculationsresulted in a suitable direction T with spherical coordinates y = . ◦ and r = . ◦ . In thiscase, the two down-converted cones intersect at an angle of 90 ◦ and the intersection pointsare separated by 6 . ± . ◦ . In order to simplify the crystal alignment process, it is convenientto define a sample reference system according to the PDC emission results. The direction ofthe wave vector k f of the fundamental wave is parallel to T . T is also normal to the inputfacet of the sample. We define P to be the vector connecting the two cones intersection points(see Fig. 2(b)) and R the vector that connects the most distant points on each circle. Conse-quently, T , P , and R form an orthogonal set. The BiBO samples used in our PDC experimentshave spherical coordinates ( y , r ) of T = ( . ◦ , . ◦ ) , P = ( − . ± . ◦ , + . ± . ◦ ) ,and R = ( − . ± . ◦ , − . ± . ◦ ) . The errors result from the finite grid resolution of thecalculation for the intersection points. [deg.] r [ d e g . ] ° ° ° slow fastslow P -100 -50 0 50 100 150 200 2500.00.20.40.60.81.0 S i ng l e pho t on i n t e n s it y [ A . U . ] Polarization angle [deg.] (a) (b)
Fig. 3. (a)
The relevant photon polarization directions for the designed crystal. The ellipticalcross-sections of the wavelength dependant indicatrix are marked for the fundamental beamand the two cones intersection directions. Note that the ellipticity of the cross-sections isexaggerated for clarity reasons. The long and short semi-axes of the cross-sections indicatethe polarization directions of the slow and fast waves, respectively. (b)
Experimental resultsof the normalized intensity of the down-converted photons as a function of the fundamentalwave polarization angle.
For any light propagation direction inside a non-cubic crystal, there are two orthogonal modesof the dielectric displacement field D and D , each with a different corresponding refractiveindex. In uniaxial crystals, such as BBO, these two modes are known as the ordinary (o) and extraordinary (e) polarizations, while in biaxial crystals they are known as the fast (f) and slow (s) polarizations, both behaving in general as an extraordinary wave [28].When choosing the crystal parameters, we need to consider the polarization of the pumpbeam and the down-converted photons. It is possible to calculate the directions of the dielectricdisplacement vectors D and D of the two linearly polarized waves in respect to the physicalaxes { e i } . However, it is more convenient to define the photon polarizations with respect tothe P and R directions. In the BBO crystal, the pump beam is polarized along the R direction,one cone is polarized in the same direction, and the other cone is polarized in the P direction.In BiBO, the pump beam should be polarized in its fast polarization mode in order to achievemaximal conversion efficiency, which usually differs from these convenient directions. For thegeneral case, we define the cartesian coordinates of the propagation direction T by the unitvector ( x , y , z ) in the optical indicatrix system { e i } . Using the Sellmeier formula for BiBO [19],we calculated the wavelength dependant principal refractive indices ( n x < n y < n z ) . From themwe derived the slow and fast refractive indices [27]. Using these refractive indices, the ratiosbetween the components of the normal polarization modes (i.e., the components of the unitvectors along the displacement field vectors D i ) are given by [28] D i x : D i y : D i z = n x x ( n i − n x ) : n y y ( n i − n y ) : n z z ( n i − n z ) , (2)where i stands for ’fast’ or ’slow’. For our crystal parameters, the fast polarization mode of thepump beam was calculated to be 13 . ± . ◦ from P , as shown in Fig. 3(a). We also measuredthis value by rotating the pump polarization direction with a half-wave plate. At each rotation
10 20 30 40 50 60 70 80 900102030405060708090 y [deg.] r [ d e g . ] y [deg.] r [ d e g . ] e e -0.2502.00.251.751.51.251.00.750.5 e e (a) (b) Fig. 4. (a)
The calculated d e f f [pm/V] of BiBO. Each contour line marks a step of0.14 pm/V. The thick black line represents collinear type-II phase-matching directions for l f =390 nm. The X symbol marks the chosen pump direction in this work. (b) The Spatialwalk-off angle [deg.] for BiBO. Each contour line marks a step of 0 . ◦ . step we took a picture of the down-converted circles. In Fig. 3(b) we plot the normalized in-tensity of the down-converted photons vs the polarization angle. Setting 0 ◦ parallel to P , themaximal value was obtained at an angle of 11 . ± . ◦ from P , within the crystal fabricationerrors.In order to calculate the polarization of the down-converted photons at the cones intersectionpoints, we need to consider these two propagation direction inside the crystal. For the down-converted photons propagating at the top left (bottom right) intersection point in Fig. 3(a), thepolarization direction of the fast wave is at 14 . ◦ ( . ◦ ) from P . The slow polarization modesare perpendicular to the fast modes. The effective strength of the second order nonlinear coefficient d e f f is an important consid-eration for the crystal design. For the uniaxial BBO crystal, there is an analytical expressionthat appears in Ref. [29]. Using the BBO d matrix elements from Ref. [5] and our wavelengthparameters, a maximal value of d e f f = .
15 pm/V is calculated. The calculation assumes acollinear type-II phase-matching process.A rigorous treatment of biaxial crystals appears in Ref. [30]. We used the relevant formulafor d f s fe f f of collinear type-II phase-matching in the { e i } reference system (the f s f indices re-fer to the pump and the down-converted photon polarization modes). The calculation resultswere rotated to the { e i } reference system, where for the parameters used in this work we get d e f f = .
00 pm/V. The calculation considers a wavelength of 780 nm, although there is almostno wavelength dependency. For this calculation we used the four d matrix elements given inRef. [30]. The results for any T direction are shown in Fig. 4(a). Note that because the calcula-tion assumes collinear propagation, the results have significant meaning mainly in the vicinityof the collinear phase-matching curve. Furthermore, we have also removed the Kleinman sym-metry assumption of Ref. [30] and derived a formula containing the eight d matrix elementsgiven in Ref. [31]. This generalization resulted with a similar value ( d e f f = .
02 pm/V). Thealmost doubled value of the nonlinear parameter of BiBO compared to BBO promises a majoradvantage for the generation of entangled photons. .5. The spatial walk-off angle
During the propagation through a birefringent crystal, the Poynting vector may point awayfrom the direction defined by the k vector, depending on the beam polarization [32]. This phe-nomenon is called spatial walk-off. It should be taken into consideration when designing apolarization entangled photon source since it can create spatial labeling of the down-convertedphotons, which in turn will reduce the entanglement quality. The spatial walk-off angle q swo between the Poynting vector and the k vector, together with the crystal thickness L, determinesthe overall spatial walk-off. The pump beam spot-size at the crystal should be large comparedto the spatial walk-off in order to prevent the labeling effect [3].In uniaxial crystals, such as BBO, an ordinary photon’s k vector and Poynting vector havethe same direction while an extraordinary polarized photon deviates from that direction by anangle that can be calculated using a simple analytical expression [33]. In biaxial crystals, suchas BiBO, both the fast and slow polarized photons deviate from the direction defined by the k vector while passing through the crystal. The spatial walk-off angle in this case is the anglebetween the two down-converted photons’ Poynting vectors.We present here the results of a numerical approach for the walk-off calculation for BiBO.The direction of the Poynting vectors of the slow and fast down-converted photons are normal tothe surface of the corresponding indicatrix. For each photon, we calculated three wave vectorswith small deviations from their propagation direction k . We then found the plane that containsthese three vectors. The direction normal to this plane is the direction of the Poynting vector.The angle between the two Poynting vectors of the slow and fast photons is the required walk-off angle. Note that it is also possible to treat this problem analytically, but as our numericalresults are sufficiently accurate, we leave the rigorous treatment for a later work.We calculated numerically the spatial walk-off angle in BBO and BiBO for a wavelengthof l dc =
780 nm. We have validated our numerical approach by comparing its results to theanalytical expression for BBO [33]. The typical deviation between the numerical and analyticalcalculations is about 10 − degree. For collinear PDC in BBO the walk-off angle is q swo = . ◦ ,corresponding to an overall walk-off of 145 m m for a 2 mm thick crystal. The results for BiBOare presented in Fig. 4(b). For our crystal parameters, the calculated walk-off values are q swo = . ◦ for one of the cones’ intersection points and q swo = . ◦ for the other. These resultscorrespond to a deviation of about 95 m m for the 1.5 mm thick crystal used in our experiments. As their name suggests, the two polarization modes propagate through the birefringent crystalwith different group velocities. This may cause temporal distinguishability between the slowand fast photons. This phenomena is known as temporal walk-off. A birefringent crystal ofthickness L separates the photons by d T = Lv s − Lv f = L ( n sr c − n fr c ) = L D n r c , (3)where c is the speed of light in vacuum, and v s ( v f ) and n sr ( n fr ) are the group velocity and theray refractive index of the slow (fast) photon, respectively. The ray refractive index n r and therefractive index n are related via n r = n cos a , where a is the angle between the k vector and thecorresponding Poynting vector [32]. The problem is more significant when d T is comparableto or larger than the coherence time t c . We addressed this issue with two methods. The firstis to add two compensating crystals, cut at the same directions as the generating crystal but ofhalf the thickness, in each down-conversion path [3]. The second approach is to overlap the twophotons at a polarizing beam splitter (PBS) [34], as will be described later in Sec. 3.1.
388 nm 389 nm 390 nm 391 nm 392 nm (b)(a)
Fig. 5. (a)
Down-converted photon circles through a 3 nm bandpass filter, from a 2 mmthick BBO crystal. (b)
Down-converted photon circles through a 3 nm bandpass filter, froma 2.7 mm thick BiBO crystal with different pump wavelengths, as indicated. Several lowercircles are cropped due to the filter size.
60 70 5060ψ [deg.] ρ [ d e g . ]
60 70 5060ψ [deg.] ρ [ d e g . ] (a) (b) Fig. 6. (a)
Stereographic projection of non-collinear type-II PDC processes in BiBOwith different pump wavelengths. The two inner circles originate from a pump beam of l =
389 nm, while the two outer circles from l =
391 nm. The thicker ring (left, red) ispolarized slow, while the thinner one (right, green) is polarized fast. The X symbol marksthe pump direction for the BiBO crystal in this work. (b)
Stereographic projection of non-collinear type-II PDC processes in BiBO with a pump wavelength of 390 nm and differentdown-converted wavelengths. The two inner circles wavelength is 781.51 nm (left, red) and778.5 nm (right, green) and the two outer circles are of the opposite process.
For our crystal parameters, D n r is approximately 0.05 for BBO and 0.15 for BiBO, whichresults with d T =
330 fs for a 2 mm thick BBO and d T =
750 fs for a 1.5 mm thick BiBO.Compensation is required in both cases as these values are larger than t c =
180 fs, the coherencetime that corresponds to the used 3 nm filters.
One advantage of down-converting a pulsed source over a continuous source is its energy con-centration in a short coherence length which increases the probability of higher order PDCevents. Furthermore, its timing information is inherited by the down-converted photons. How-ever, the pulses broadband spectrum can cause a variety of undesired effects that decrease theentanglement quality.
88 389 390 391 3920.60.70.80.91.01.11.21.31.4 R e l a ti v e r a d i u s [ A . U . ] Wavelength [nm]
Fig. 7. A comparison of the measured (open circles) and calculated (solid circles) normal-ized down-converted circles radii of the slow (dashed red) and fast (solid blue) photons.Table 1. dnd l for l =
780 nm in nm − BBO BiBOslow 3 . × − . × − fast 2 . × − . × − Figure 5(a) presents a picture of the down-converted photons from BBO recorded by a sen-sitive CCD camera through a 3 nm bandpass filter. The pump wavelength is l f =
390 nm witha full width at half-maximum (FWHM) of ∼ f ast ( nm ) −→ slow ( nm ) + f ast ( . nm ) , f ast ( nm ) −→ slow ( . nm ) + f ast ( nm ) , f ast ( nm ) −→ slow ( nm ) + f ast ( . nm ) , f ast ( nm ) −→ slow ( . nm ) + f ast ( nm ) . We present on a stereographic projection only the circles of l dc =
780 nm (Fig. 6(a)). Thecircles angular radii are measured and normalized by the radius of 780 nm circles from down-converting 390 nm photons. Figure 7 presents a comparison between the numerically calculatedradii and those measured from Fig. 5(b). The calculated (measured) slopes for the two polar-izations differ by a factor of 3 . ± .
15 (2 . ± . . ± . S p ec t r u m [ A . U . ] (b)(a)
775 780 785775780785 l f [nm] l s [ n m ]
775 780 785775780785 l e [nm] l o [ n m ] Wavelength [nm]
Fig. 8. The BiBO ( a ) and BBO ( b ) collinear type-II PDC spectra. For BiBO (BBO), thespectrum of the fast (extraordinary) photons is presented by a solid blue line, while thatof the slow (ordinary) photons’ by a dashed red line. In both cases, the crystals’ thicknessis 2 mm, the filter bandwidth is 3 nm, and the pump bandwidth is 2 nm. Spectral overlapis 89 .
6% for BiBO and 98 .
2% for BBO.
Insets:
Phase-matching spectral dependency be-tween the slow (ordinary) and the fast (extraordinary) photons from BiBO (BBO). Thespectra aspect ratios are 1:3 and 2:3 for BiBO and BBO, respectively.
Filter bandwidth [nm] S p ec t r a ov e r l a p [ % ] Crystal thickness [mm] (b)(a)
Fig. 9. (a)
Spectra overlap as a function of the crystal thickness with a 3 nm bandpass filterfor BiBO (blue squares, solid line) and BBO (red circles, dashed line) crystals. (b)
Spectraoverlap as a function of the filter bandwidth for 2 mm thick BiBO (blue squares, solid line)and BBO (red circles, dashed line) crystals. we have also calculated the circle widths due to the filter bandwidth for a 390 nm pump. Weconsider the processes that result with photons at the FWHM of the 3 nm filters f ast ( nm ) −→ slow ( . nm ) + f ast ( . nm ) , f ast ( nm ) −→ slow ( . nm ) + f ast ( . nm ) . The results are presented in Fig. 6(b). There is no significant effect due to the filter’s width.Thus, the slow polarized circle larger width is attributed to its higher dispersion, that resultswith the asymmetry shown in Fig. 5(b). The filters bandwidth do not add asymmetry betweenthe circles. This conclusion suggests that a symmetric PDC picture may be obtained from BiBOusing a continuous pump source.
BSSHG SM fiberTi:sapphire FNL CCL SPDMHWPQWPBP
I II
Fig. 10. The experimental setup. See text for details.
In order to evaluate the effect of these results on the quality of the generated entangled state,we calculated the spectra of the down-converted photons form a pulsed source in BiBO andin BBO. Our calculation was based on the work of Grice et al. , that was previously applied toBBO [35]. As before, we used the collinear approximation. The normalized overlap betweenthe two down-converted photons’ spectra corresponds to the quantum state visibility. The pumpbandwidth and the crystal thickness also influence the down-converted spectra and thus, shouldbe considered when designing such a polarization entangled photon source. Bandpass filterswith the proper bandwidth can enhance the overlap between the two down-converted photons,and thus reduce the distinguishability between them. Figure 8 presents calculations of the down-converted spectra for a 2 mm thick BiBO and BBO crystals, assuming spectra with a FWHMof 2 nm for the pump photons and with 3 nm for the bandpass filters. The overlap between theintegrated spectra of the two photons for BiBO and BBO are 89 .
6% and 98 . .
3. Entanglement measurements
The setup used in this experiment is presented in Fig. 10. The radiation of a mode-lockedTi:sapphire laser at 780 nm is up-converted to 390 nm by second-harmonic generation (SHG).The beam is focused by a lens (L) on the BBO or BiBO crystal (NL). The spatial modes ofthe pump beam and the down-converted photons are matched to optimize the collection effi-ciency [36]. The photons are coupled into single mode fibers (SM), where their polarizationis adjusted by polarization controllers. The relative propagation delay between the two opticalpaths is adjusted by translating one of the fiber ends with a linear motor (M). A quarter-waveplate (QWP) and a half-wave plate at each path are used for the quantum state tomography.The photons are spectrally filtered by using 3 nm wide bandpass filters (F) and coupled intomultimode fibers that guide them to the single-photon detectors (SPD).We tried two configurations in order to remove the temporal and spectral distinguishability of V i s i b ilit y [ % ] Twofold coincidence [Hz] (a) T w o f o l d c o i n c i d e n ce [ H z ] Pump power [mW] (b)
Fig. 11. Results with configuration I. (a)
Visibilities vs the twofold coincidence rate in threepolarization bases: HV (black squares), PM (red circles), and RL (blue triangles). Straightlines represent linear fits, calculated without the last three points, where stimulation is moresignificant. (b)
Twofold coincidence rates vs pump power. The solid black line representsthe quadratic fit and the dashed red line the linear slope at low pump powers. -400 -300 -200 -100 0 100 200 300 4000250050007500100001250015000175002000022500 T w o f o l d c o i n c i d e n ce [ H z ] Delay [ µ m] (b) V i s i b ilit y [ % ] Twofold coincidence [Hz] (a)
Fig. 12. Results with configuration II. (a)
Visibilities vs the twofold coincidence rate inthree polarization bases: HV (black squares), PM (red circles), and RL (blue triangles).Straight lines represent linear fits. (b)
Twofold coincidence rates as a function of the opticalpath difference. The red circles correspond to a projection to the | f + i state and the blacksquares, a projection to the | f − i state. Blue triangles represent coincidence events from thesame side. the down-converted photons. The elements used in each configuration are labeled I and II in Fig.10. In the first configuration (I), the photon polarizations are 90 ◦ rotated by a half-wave plate(HWP), and temporal and spatial walk-offs are corrected by compensating crystals (CC) of halfthe thickness of the generating crystal. In the second configuration (II), two perpendicularlyoriented Calcite crystals (arrows indicate the optical axis direction) are used for aligning thebirefringent phase (BP). The photons are then overlapped at a PBS. We generated polarization entangled states with a 1.5 mm thick BiBO crystal and compensatedfor distinguishability effects with two configurations [3,34] (see Sec. 3.1). In order to character-ize the entanglement quality, we recorded visibilities [3] at three polarization bases (horizontaland vertical linear polarizations (HV), plus and minus 45 ◦ linear polarizations (PM), and rightand left circular polarizations (RL)). Full quantum state tomography was also performed. Com- H HV VH VV0.000.250.50 HHHVVHVV
HH HV VH VV0.000.250.50 HHHVVHVV
HH HV VH VV0.000.250.50 HHHVVHVV
HH HV VH VV0.000.250.50 HHHVVHVV (b)(a) (c) (d)
Fig. 13. Real parts of the measured density matrices for the two configurations. Imaginaryvalues are smaller than 0.08 and therefore not presented. (a)
Configuration I, 40 mW pump. (b)
Configuration I, 300 mW pump. (c)
Configuration II, 42 mW pump. (d)
ConfigurationII, 310 mW pump. parison is made with results obtained using a 2 mm thick BBO crystal, in a setup optimized forits parameters.Using configuration I, we generated the | y + i Bell state. The recorded visibilities were V HV = . ± . V PM = . ± . V RL = . ± . . ± . V HV = . ± . , V PM = ± . V RL = . ± . . ± .
01 and 0 . ± . et al. [34]. Using this configuration, we generated the | f + i Bell state. The recorded visibilities were V HV = . ± . , V PM = . ± . V RL = . ± . . ± . V HV = . ± . , V PM = . ± . V RL = . ± . . ± .
01 and 0 . ± . ◦ rotated base, simultaneously projects on the | f + i and | f − i states (see Fig. 12(b), pump poweris 320 mW). The dip visibility is V PM = . ± . V PM = . ± . | f + i Bell staterom a 2 mm thick BBO crystal with configuration I. The measured visibilities in the threepolarization bases V HV , V PM , and V RL were 95 ± ± . ± ± . ± . ±
1% loss, assuming the pump beam and the down-converted photons hit the crystal facets perpendicularly. It should also be considered that, dueto some technical issues, we pumped the two crystals with different powers. Thus, we calculatethe down-conversion efficiency as the number of detected pairs per second, per mW of pumppower, per mm of crystal thickness. The efficiency values for BiBO and BBO, as measuredin configuration I, are 58 ± − mm − and 23 ± − mm − , respectively. Thesevalues account for an improvement by 2 . ± .
15, compared to the 3.09 ratio predicted by thecalculated d e f f values of BiBO and BBO (see Sec. 2.4).
4. Conclusions
We have studied the various properties of the biaxial BiBO crystal, which are relevant forutilizing it as a polarization entangled photon source using non-collinear type-II PDC and apulsed pump source. Theoretical and numerical treatment of the relevant crystal parameters ispresented. We calculated the crystal cutting angles, the polarization directions, temporal andspatial walk-offs, and the effective nonlinear coefficient. We have also demonstrated the effectsof crystal dispersion and the broad spectrum of the pulsed pump on the angular and spectralproperties of the down-converted photons, and therefore on the entanglement quality. The ex-perimental results demonstrate the higher efficiency of BiBO compared to the commonly usedBBO, and the potential BiBO has as an ultra bright source of entangled photons. Although itfocuses on BiBO, our work can be considered as general guidelines for considering any otherbiaxial nonlinear crystal as a non-collinear type-II polarization entangled photon source. Asthere are a growing number of quantum optics experiments that require highly efficient PDCsources, we hope that this work will encourage the use of BiBO as a source for polarizationentangled photons.