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We present several novel sources of polarization-entangled states based on beam displacer interferometers. These sources generate a large number of high-quality entangled photon pairs in a compact alignment-free layout and can work with both cw and pulsed pumps.
©2008 Optical Society of America
1. Introduction
Polarization-entangled photons generated by spontaneous parametric down-conversion (SPDC) have been used in a wide range of experiments, from tests of Bell’s inequalities to the demonstration of various quantum information processing (QIP) protocols. A compact, robust, and intense source of polarization-entangled photons would be an important contribution to the development of practical applications of QIP.
A number of recent papers have explored novel approaches to the generation of polarization entangled photons. Following the original concept developed by Kwiat and coworkers [1] and the demonstration by Kim and coworkers [2], there have been several implementations of interferometric SPDC sources of polarization entangled photons [3, 4, 5]. Interferometric sources generate photons that are entangled independently of their wavelengths and angles of emission and work with both cw and pulsed pumps. However, because of their interferometric nature, these sources require careful alignment and, in some cases [3], active stabilization. In addition, these sources in their current embodiments are bulky and require a large number of optical elements and high precision opto-mechanical components. These are disadvantages for space- and weight-sensitive applications of polarization-entangled photons, such as space-based systems [6], that require adjustment-free long term stability.
Here we present a novel source geometry that improves previous interferometric sources in several key aspects: size, complexity of the alignment procedure, and stability. Our source is based on polarization displacers, periodically poled crystals, and violet laser diode pumps. We show the flexibility of our approach by building two sources: a quasi-degenerate source that emits two photons at≃800nm that can be used for atmospheric transmission, and a non-degenerate source that emits pairs at 586 and 1310 nm that can be used for heralded transmission over optical fibers.
2. Polarization displacer-based sources of entangled photons
All of our source designs rely on interferometers constructed using polarization beam displacers. A beam displacer is a piece of birefringent crystal cut so that the optical axis forms an angle with the direction of propagation of the optical beams. At the interface between air and the crystal the two polarization components of the incoming beam are refracted differently (double refraction) and the horizontally and vertically polarized components of the incoming beam emerge from the birefringent crystal as two separate parallel beams. Polarization beam displacers are commonly used in telecommunications devices to build compact fiber-coupled polarizers and isolators. Polarization interferometers based on beam displacers have been recently used in quantum optics to realize linear optics quantum computation [7]. The advantage of these types of interferometers is that they are not sensitive to vibrations and shifts of the optical components, since the two beams of the Mach-Zehnder interferometer essentially share the same path.
The simplest implementation of a beam-displacer-based source of entangled photon is shown in Fig. 1(a). The pump laser impinges on a beam displacer with a linear polarization forming an angle of 45° with respect to the horizontal direction (the half-wave plate HWP1 is used to adjust the polarization). Two orthogonally polarized beams emerge from the crystal and the polarization of the top beam is rotated by 90° using a half-wave plate. The two beams enter a crystal of periodically-poled potassium titanyl phosphate (PPKTP) which is phase-matched for collinear frequency quasi-degenerate type-I down-conversion: the down-converted beams are collinearly polarized and their wavelengths are separated by a few nanometers. One possible choice of wavelengths for a violet 405 nm laser diode pump is 800 nm and 820 nm: these wavelengths are well suited for free space transmission within 790 nm atmospheric transparency window. The polarization of both down-converted photons is rotated by 90° in the top beam using HWP3 and the down-converted beams are recombined using a second beam displacer. As we discuss in the following paragraph, because of the dispersion in the crystal birefringence the second crystal will need to have a different length than that of the first displacer in order to exactly compensate the beam displacement. The polarization-entangled beams are separated by a dichroic mirror and then fiber coupled. The lengths of the two interferometer arms need to be equalized in order to insure proper temporal overlap of the down-converted photons. Length equalization is accomplished by placing the wave plates that rotate the polarization inside the interferometer in opposite arms. According to our calculations no additional compensation is needed.
Fig. 1. Schematics various implementations of the compact sources of polarization-entangled photons. (a) Quasi degenerate source (b) Non-degenerate source (c) Degenerate source. The state generated by each source is written on the right, the phase factor θ can be adjusted by aligning the interferometer. HWP: half-wave plate. BD: polarization beam displacer. DM: dichroic mirror. PBS: polarization beam-splitter.
This layout has considerable flexibility. For example, Fig. 1(b) shows a schematic for a non-degenerate source of polarization-entangled photons. In this layout the pump is still split on a beam displacer but the PPKTP crystal is phase matched for non-degenerate type-II down conversion. (Type-I down conversion can also be accommodated with a slight modification in the arrangement of the wave plates.) A possible choice of the wavelengths of the down-converted photons for a violet 405 nm pump is 586–1310 nm, with one photon falling in the high detection efficiency window of Si APDs and the other in the low-loss window for telecom fibers. Because the wavelengths of the down-converted photons are very different, the birefringence of the beam-displacer crystal will also be different and therefore one will need crystals of different lengths to compensate the displacement between the down-converted beams. For this reason, we separate the down-converted photons using a dichroic beam-splitter. The two arms of the interferometer are then separately recombined using two beam displacers after the polarization of one of them is rotated by a half-wave plate. As shown in Fig. 1(b), to equalize the length of the interferometer arms for the 1310 nm photons, one needs to add 2 wave plates, each rotating the polarization by 90°. Here we exploit the fact that wave plates built for a given wavelength and order (zero or multiple) have the same thickness with a precision that exceeds the requirement of our experiment.
Another variation allows one to build a source that is frequency degenerate, as shown in Fig. 1(c). The pump laser is split by the first beam displacer and the polarization of the upper beam is rotated by 90°. The two beams impinge on a PPKTP crystal phase matched for type- II degenerate collinear down conversion. One possible choice of wavelengths is a laser diode pump at 405 nm and degenerate down converted photons at 810 nm that are within the atmospheric transparency window centered at 790 nm. The down converted photons are separated by a polarization beam-splitter (PBS). The two arms of the interferometer are then recombined using two separate beam displacers. As in the case of the previous source, to balance the length of the interferometer arms one needs an additional wave plate (HWP4) in the lower arm.
All the sources described here share a number of advantages. Because they use collinear down-conversion, they have a long interaction length between the pump and down converted photons. Therefore they are able to generate a much larger flux of photons than sources based on non-collinear down conversion [2, 8]. Because of the intrinsic stability of displacer-based interferometers they do not require active stabilization. Compared with other interferometric sources [3, 4], these sources have a smaller footprint and require fewer high-precision optomechanical mounts. Displacer-based interferometers are easy to align and can be mounted on micro-benches and sealed for long-term operation in extreme environments. These sources can operate robustly with both cw and pulsed pumps. In addition this scheme can be adapted to generate indistinguishable polarization-entangled photons using the techniques described in [9]. An effort to experimentally realize this scheme is currently under way [10].
3. Experimental setup and results
Key elements for the implementation of the compact sources described in the previous section are birefringent crystals to be used as beam displacers. Optimal crystals for the beam displacers are transparent at the wavelength of interest, have high birefringence so that a large beam separation can be obtained with short crystals, and are mechanically stable so that precise dimensional control is possible. Calcite (calcium carbonate) is typically used in optics as a highly birefringent crystal; however, this crystal has fairly poor mechanical qualities that make it unattractive for our application. A good alternative is α-barium-borate (α-BBO), which has much better mechanical stability with a large birefringence (albeit smaller than that of Calcite). The chemical and physical characteristics of α-BBO, the high-temperature phase of BBO, are similar to those of the more familiar low-temperature phase β -BBO; however, β-BBO is centrosymmetric and therefore has a zero χ (2) coefficient. In Fig. 2 we compare measured double refraction angles for a crystal of β-BBO obtained from CASIX with calculated values. The crystal was cut so that the optical axis forms an angle of 45° with respect to the entrance and exit faces. The double refraction angle θ can be calculated using the formula [11]
where n o and n e are the ordinary and extraordinary indexes of refraction, respectively. The indices of refraction were calculated using Sellmeier coefficients from two sources: a set provided by the vendor [12] and a set for β -BBO from [13]. The second set seem to be in better agreement with the data. The discrepancy between the calculated and measured values of the double refraction angles is probably due to variations in the crystal growth process. To calculate the crystal lengths for the various sources we used measured values of θ when possible and linear interpolations between two adjoining measured points otherwise.
Fig. 2. Measured and calculated double refraction angle in an α-BBO crystal with the optical axis forming an angle of 45° with respect to the beam propagation direction.
We have experimentally implemented the two sources of Fig. 1(a) and (b). For the quasi-degenerate source, the beam displacer for the 405 nm pump has a nominal length of 14.0 mm and the second beam displacer has a nominal length of 15.0 mm. For the non-degenerate source the nominal lengths of the beam displacers for the 405 nm pump, the 586 nm down converted photon, and the 1310 nm down-converted photon are 14.0 mm, 14.8 mm, and 15.5 mm, respectively. In both sources the beam separation is 1.1 mm.
Our first test of our design is conducted on the quasi-degenerate layout of Fig. 1(a) using a cw 405 nm grating-stabilized laser diode (Sacher Lasertechnik model TEC-100-0405-20) as pump. The nonlinear crystal is a 10×4×1 mm3 PPKTP crystal with a poling period of 3.45 µm. The crystal is phase matched for collinear non-degenerate phase matching of a 405 nm pump into 800/820 nm photon pairs at a temperature of 30°C. The photons are separated using an interference filter that transmits the signal photons at 800 nm (FWHM 1 nm) and reflects the idler photons at 820 nm. Additional filtering is used to remove the residual pump. Signal and idler photons are transmitted through polarization analyzers, are coupled into single mode fibers, and are detected using Si-APD single photon counters. Coincidences are detected with a “home-made” AND gate with a coincidence window of 3.2 ns.
Figure 3 present results for the measurement of the visibility of coincidence fringes. To obtain the fringes we set the analyzer angle in the signal arm to be 0± or 45± and rotate the analyzer in the idler arm. We observe visibilities of V 0=96.1±0.9% and V 45=92.3±0.4%; when we correct for accidental counts we obtain visibilities of V 0=97.6±0.9% and V 45=93.7±0.4%. We measure a coincidence rate of 4000 coincidences/s for a pump power of 2 mW, which gives a measured flux of 2000 entangled pairs/s/mW/nm. We also measured Bell’s inequalities’ [14, 15] S parameter to be S=2.630±0.005, a value that clearly violates the classical value of 2.
Fig. 3. Visibility results for the polarization-entangled state generated in the quasi-degenerate cw source. Plotted here are the coincidence counts versus idler analyzer angle when the signal analyzer angle is set to 0° (diamonds) and 45° (triangles). Each data point is the result of an average over 10 s and the curves are best fits to the data.
The same pump setup can be used to verify operation with a pulsed pump. In this experiment the pump is a 407 nm pulsed laser diode with a repetition rate of 40 MHz and a pulse width of approximately 50 ps (Picoquant PDL 800-B). The laser diode is free running with a spectral width of 1 nm. By heating the same PPKTP crystal used for the cw experiment to 90°C we can phase match near-degenerate down conversion of pairs with wavelengths of 805 nm and 815 nm.
Figure 4 present results for the measurement of the visibility of coincidence fringes. To obtain the fringes we set the analyzer angle in the signal arm to be 0° or 45° and rotate the analyzer in the idler arm. We observe visibilities of V 0=98±1% and V 45=81±1%. We measure a coincidence rate of 250 coincidences/s pairs/s/mW/nm. We also measured Bell’s inequalities’ S parameter to be S=2.54±0:01. We attribute the reduced visibility in the pulsed experiment when compared with the cw experiment to the wide bandwidth of the pump laser. A wide-band laser generates down converted photons that have a shorter coherence length, thus making the constraints on the balancing of the interferometer arms more stringent. We believe that using a pulsed laser with a transform-limited spectrum would significantly improve the quality of the entangled photons being generated. We also believe that the wide band of the laser causes the reduction in the brilliance of the source as a large number of the down converted photons are rejected by the narrow band interference filter used to separate signal and idler.
In a third experiment we built the non-degenerate source shown in Fig. 1(b). We use a cw grating stabilized 405 nm laser diode pump. The nonlinear crystal is a 10 mm long PPKTP crystal with a poling period of 10 µm phase matched for collinear down conversion of the pump into 586/1310 nm pairs at a temperature of 30°C. The two outputs are filtered using 10 nm bandpass filters to remove the residual pump. They are transmitted through polarization analyzers and are then coupled into single mode fibers and sent single photon detectors. The 586 nm photon is detected using one channel from a 4-channel Si avalanche photodiode detector (Perkin-Elmer SPCM-AQ4C). The output from this detector is used to gate a InGaAs avalanche photodiode detector (Princeton Lightwave Benchtop Receiver). This arrangement is used to limit the number of dark counts in the relatively noisy InGaAs detector.
Fig. 4. Visibility results for the polarization-entangled state generated in the quasidegenerate pulsed source. Plotted here are the coincidence counts versus idler analyzer angle when the signal analyzer angle is set to 0° (diamonds) and 45° (triangles). Each data point is the result of an average over 10 s and the curves are best fits to the data.
Fig. 5. Visibility results for the polarization-entangled state generated in the non-degenerate cw source. Plotted here are the coincidence counts versus idler analyzer angle when the signal analyzer angle is set to 0° (diamonds) and 45° (triangles). Each data point is the result of an average over 10 s and the curves are best fits to the data.
Figure 5 present results for the measurement of coincidence fringes visibility for the non degenerate source. We observe visibilities of V 0=98±4% and V 45=81±1%. The measured coincidence rate is 2500 pairs/s/mW/nm. We attribute the low visibility of this source to the large number of components in the interferometer and the fact that the paths for the two photons are separated.
A number of improvements in these sources would allow the generation of higher quality entangled states. The quality of the optical elements inside the interferometer is critical; in particular, one has to ensure the parallelism of the surfaces of the various optical elements such as the displacers, the nonlinear crystal, and the wave plates. A more accurate balancing of the interferometers’ arms would also improve visibility, especially for wide-band lasers. This would be critical if one wants to use sources with pulse durations shorter than those used here. Many quantum information processing experiments use pumps with pulse lengths of 100 fs or less. While we have not carried out any experiment using sources with femtosecond pulses, our calculations suggest that it would be extremely hard to balance the interferometer arms with the precision required for such short pulses.
4. Conclusions
We have presented a novel scheme for the generation of polarization entangled photons that use compact and stable interferometers based on polarization beam displacers. We showed how this scheme can be adapted to produce degenerate, non-degenerate, and quasi-degenerate entangled photon pairs. We experimentally verified that this type of source can be used with cw laser pumps as well as pulsed sources with moderate spectral bandwidth. We believe that sources such as the ones described here will be useful for applications requiring long-term deployment without adjustments.
References and links
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