We report on single photon frequency downconversion from the red part of the spectrum (738nm) to the telecommunications C-band. By mixing attenuated laser pulses with an average photon number per pulse < 1 with a strong continuous light field at 1403nm in a periodically poled Zn:LiNbO3 ridge waveguide an internal conversion efficiency of ∼ 73% is achieved. We further investigate the noise properties of the process by measuring the output spectrum. Our results indicate that by narrow spectral filtering a quantum interface should be feasible which bridges the wavelength gap between quantum emitters like color centers in diamond emitting in the red part of the spectrum and low-loss fiber-optic telecommunications wavelengths.
©2011 Optical Society of America
The ability to efficiently transfer quantum states between photons of different frequencies is a key requirement for the implementation of fiber-based quantum networks . It allows for the interchange of photons between dissimilar quantum systems which emit/absorb at different wavelengths. Furthermore, it provides the possibility to transfer quantum information carried by telecommunications-band photons (1310nm/1550nm) to the visible spectral range and vice versa. The first observation of quantum frequency conversion was reported in 1992  for quantum correlations of coherent states. About ten years later a number of experiments particularly focusing on single photon upconversion from telecommunications wavelengths to the red part of the spectrum [3–6] were demonstrated, all of them aiming at efficient single photon detection. Furthermore, it was experimentally proven that (time-bin) entangled pairs of photons stay entangled after one of them has been converted to another frequency . Only very recently an upconversion experiment using 1.3 μm single photons from an InAs quantum dot was presented  showing that the nonclassical state of light remains unchanged in the frequency translation process, i.e., that photon antibunching is preserved. While frequency upconversion was widely investigated, its complementary process, stimulated downconversion by difference frequency generation (DFG), came to interest only in the last two years where both theoretical [9, 10] and experimental work [11–16] on the subject has been published. So far three implementations [12, 15, 16] have used optical powers at the single photon level. Radnaev et al.  achieved a high conversion efficiency of 54% relying on four-wave mixing in a cold Rb gas while implementations based on three-wave mixing in a periodically poled LiNbO3 (PPLN) waveguide (WG) suffer from low conversion efficiencies < 2% [12, 15]. However, these low efficiencies are rather due to technical difficulties, i.e., limited pump power and/or bad spatial mode overlap in the WG, than fundamental limitations. Moreover, the solid state approach is very attractive because of its flexibility and its comparatively low experimental complexity. This is especially the case when combining the conversion module with a solid state quantum emitter, e.g., a quantum dot or a color center in diamond. For such systems the conversion process might also be enhanced by nano-structured nonlinear optical media .
Within the context of color centers in diamond an experimental scheme which is aimed at efficient frequency downconversion of single photons from a Nitrogen-vacancy (NV) center (zero phonon line at 637nm) to the telecom C-band has been demonstrated [13, 14]. However, due to strong electron-phonon coupling the emission spectrum of NV-centers is very broad (∼ 100nm) compared to the acceptance bandwidth of the required DFG process in a PPLN WG crystal which is < 0.5nm for typical interaction lengths. Recently, much progress has been made in the fabrication of single photon sources based on Silicon-vacancy (SiV) centers in diamond emitting around 738nm . Compared to NV-centers they feature higher emission rates and significantly narrower spectral linewidths (≲ 2nm). Motivated by these results we investigate the feasibility of stimulated downconversion of light emitted from a SiV center in diamond. To this end photons at λ a = 738nm are mixed with a strong classical pump field at λ p = 1403nm in a periodically poled Zn-doped LiNbO3 (Zn:PPLN) ridge waveguide to yield converted single photons in the telecom C-band (λ b = 1557nm). We study the Raman noise spectrum generated by the strong 1403nm pump light and demonstrate an internal conversion efficiency of ∼ 73% under realistic experimental conditions. We believe that our work is a significant advance in the field of quantum frequency downconversion as it features high internal conversion efficiency in combination with single photon level input power in the red. In previously reported experiments either single photon input was achieved but with modest conversion efficiency [12, 15] or high conversion efficiency was attained but the noise generated by the strong pump field precluded the experiment from being performed with input powers at the single photon level .
2. Experimental setup
The experimental setup is shown in Fig. 1(a). To simulate the single photon source at 738nm a continuous wave Ti:Sapphire laser (CW Ti:Sa; Matisse TX, Sirah GmbH, Germany; linewidth < 100kHz) together with a pulse picker and an attenuator is used. The 738nm light is guided to the experiment by a single-mode optical fiber. A home-built continuous wave optical parametric oscillator (CW OPO) is used to generate the strong field at 1403nm . The CW OPO is based on a 30mm long periodically poled LiTaO3 crystal and pumped at 532nm by a frequency-doubled solid-state laser (Coherent Verdi V10). Its idler wavelength is tunable from 1202-1564nm while it delivers more than 1W of single mode, single frequency output power. The wide tunability adds an enormous flexibility to the setup: the CW OPO can also be employed as a pump source for frequency downconversion of light emitted by other types of color centers in diamond (e.g., from a chromium-related color center ) or from a semiconductor quantum dot . The attenuated pulses at 738nm and the strong pump field at 1403nm are combined on a dichroic mirror and coupled into the Zn:PPLN WG (NTT Electronics Corp., Japan)  by a single aspheric lens which has an antireflection coating from 600-1050nm. The WG has a length of L = 40mm, a grating period of Λ = 16.20 μm and features antireflection coatings for λ a ,λ p and λ b. It is held at constant temperature of 21.3°C using a Peltier module to fulfill the quasi-phase-matching (QPM) condition. As the two wavelengths, λ a and λ p, are spectrally separated by 665nm it is necessary to use an additional telescope in the beam line of the pump light to compensate for chromatic aberration. In this way maximum coupling efficiency can be achieved for both wavelengths simultaneously: we yield 90% and 65% for 738nm and 1403nm, respectively. No efforts were made to discriminate between Fresnel losses at the facets, input coupling losses and propagation losses of the WG. Propagation losses in this context refer to any losses experienced by the light field while it is transmitted through the waveguide from the input to the output facet. It includes losses that may be induced by waveguide imperfections (e.g., by scattering due to finite surface roughness) or by absorption in the waveguide material. We can quote 0.08dB/cm as an upper bound for the propagation losses at 738nm since the highest transmission through the WG that could be achieved is 93% (input coupling at λ p not optimized in this case). The dimensions of the ridge waveguide are designed to support only the fundamental spatial mode at telecom wavelengths which usually implicates that higher order spatial modes can be excited in the visible. This leads to a poor mode overlap and thus reduces the conversion efficiency. Nevertheless, by carefully optimizing the mode matching it is possible to excite only the fundamental WG mode at λ a. This is proven by Fig. 1(b) which shows the mode profile of the collimated 738nm beam after being transmitted through the WG. The image was recorded using a CCD camera. For comparison Fig. 1(c) shows the calculated intensity distribution of the fundamental mode at 738nm for a ridge waveguide with 8 × 8 μm2 cross section. The calculations were performed following the method described in . Note that for the measurement shown in Fig. 1(b) the beam was collimated with an aspheric lens (effective focal length f eff = 11mm) and had to pass several neutral density filters before it hit the CCD chip of the camera (measurement was performed at mW power level). This may explain the slight distortion of the CCD image compared to the perfect mode profile shown in Fig. 1(c). After collimation behind the WG the three wavelengths λ a ,λ p and λ b are spatially separated by a Pellin-Broca prism in combination with a pinhole. By rotating the prism we can select which of the three beams is coupled into a single-mode optical fiber to guide it to a commercial InGaAs/InP single photon avalanche diode (SPAD, id Quantique id201) or a grating spectrometer (Princeton Instruments SP2500A with OMA V InGaAs linear array and Spec-10 CCD camera) for spectral analysis. For detection with the SPAD we eliminate residual pump light after the pinhole employing a 1450nm longpass filter and a fiber optic circulator together with a fiber Bragg grating (center wavelength: 1557.025nm, −1.0dB reflection bandwidth: 0.769nm). Insertion losses for the whole filtering system are 10% (prism and two mirrors) plus 30% (longpass filter). The input coupling efficiency into the circulator is ∼ 50%. All together this results in a total transmission coefficient of T tot ≈ 0.3 (−5.2dB), i.e., on the way from the WG exit to the active area of the detector 70% of the generated C-band photons are lost due to spectral filtering. We estimate the suppression of pump light at λ p = 1403nm by the filtering system to be better than 130dB.
3. Raman spectroscopy
For quantum frequency converters it is known that noise, i.e., unwanted photons, at the target wavelength λ b can be generated by the strong driving field at λ p. The reason is either spontaneous Raman scattering  (Stokes for λ p < λ b, anti-Stokes for λ p > λ b) or spontaneous parametric downconversion (SPDC) of the pump light . In the following we investigate the noise properties of our frequency converter in more detail. To record the noise spectrum we coupled P p = 100mW into the WG. Behind the collimation asphere two mirrors are used to directly couple the light into a SMF28-fiber which guides it to the grating spectrometer. In this case the whole spectral filtering system as described in the previous section was bypassed to prevent distortion of the spectrum. Instead two longpass filters (cut-on wavelength 1450nm) were mounted in front of the fiber entrance to attenuate the intense light at λ p. Thus we avoid saturation of the spectrometer’s InGaAs array and the amount of spontaneous Raman scattering generated in the fiber is kept negligibly low. We recorded the spectra at wavelengths > 1450nm for two different pump wavelengths, 1398.2nm and 1403.5nm respectively. The results are shown in Fig. 2(a). We see that all spectral features shift with the pump wavelength. Figure 2(b) shows a comparison of the two spectra with a Raman spectrum of a bulk MgO:LiNbO3 sample excited with laser light at 647nm along the x-axis of the crystal (note that the effect of different dopants, Zn or Mg, can be neglected here ). The agreement of the relative frequency shifts of all three spectra is excellent. From this observation and from comparison with Raman spectra of LiNbO3 that can be found in the literature , we infer that the dominating noise source in our experiment is spontaneous Stokes Raman scattering. We assume that SPDC of the pump light  can be excluded as a noise source in our case since the splitting of a 1403nm pump photon into a 1550nm and a 14.8 μm photon is extremely inefficient because of the strong absorption of the crystal at wavelengths > 5 μm.
So far, we have considered Stokes Raman scattering since it is the relevant noise process for our scheme where λ p < λ b. This short-wavelength pumping is inevitable if we want to perform frequency downconversion from 738nm to the C-band in a single conversion step. However, it is worthwile to briefly discuss the strength of anti-Stokes Raman scattering which can be the reason for noise in the case of long-wavelength pumping, i.e., λ p > λ b [12, 15]. As shown in Fig. 3 anti-Stokes Raman scattering is observed in our device as well. To obtain the spectrum of Fig. 3 the CW OPO was tuned to 1536nm and 185mW of excitation power were coupled into the WG at this wavelength. The filtering system for the anti-Stokes measurement consists of a shortpass filter (cut-off wavelength 1470nm) and a FBG (center wavelength: 1535.822nm, −0.5dB reflection bandwidth: ±10GHz) which suppress the intense excitation light. We notice that the peak at 630cm−1 does not seem to exist in the anti-Stokes Raman spectrum although it is present in the Stokes spectrum. A phenomenon which seems to be related can also be observed in the Stokes spectra of Fig. 2 where the relative heights of the peaks at 580cm−1 and 630cm−1 depend on the excitation wavelength. A variation of the relative amplitudes of the 580cm−1 and 630cm−1 Raman peaks, including a virtual suppression of the 630cm−1 peak has been previously observed by Sidorov et al. . A resonance between Raman scattering and the electronic polaron absorption band is supposed to be responsible for varying Raman line intensities .
According to theory the intensities I as() and I s() of anti-Stokes and Stokes Raman bands measured at Raman shifts (cm−1) are related by the Boltzmann factor Figs. 2 and 3. Moreover, the measurement was performed with excitation light at 735nm because at telecom wavelengths it is not possible to achieve proper alignment for anti-Stokes and Stokes Raman light simultaneously with our current setup. By comparing the integrated intensities (578-594cm−1) of anti-Stokes and Stokes Raman peaks we experimentally find I as /I s ≈ 0.11. This is in reasonable agreement with the theoretical prediction of I as /I s = 0.08 calculated from relation (1) for T = 295K and = 13605cm−1 (735nm). Assuming that similar conditions prevail for the case of telecom wavelength excitation we would gain one order of magnitude in signal to noise ratio by interchanging λ p and λ b in our experiment. In the present work, this does not seem desirable as an ultimate goal since wavelengths around 1400nm are critical in terms of fiber transmission losses (water peak). In general, however, we could exploit the reduced noise on the anti-Stokes side of the Raman spectrum and use a long-wavelength pumping scheme. In this regard it would be favorable to select 1310nm as the target wavelength (telecom O-band). With 738nm input photons this would require a 1690nm pump wavelength, meaning that the spectral separation between pump and target wavelength is > 1700cm−1.
As can be seen from Fig. 2 the Stokes Raman noise stretches over hundreds of nm and no particular spectral interval, especially not in the C-band, can be identified which is not affected by a certain noise background. It is obvious that, without any spectral filtering, the amount of generated noise photons is not tolerable for quantum frequency conversion. For experiments at the single photon level we thus employ the spectral filtering system introduced in section 2 which acts as a narrow bandpass filter. To illustrate its effect we recorded the spectrum of the light exiting port 2 and port 3 of the circulator-FBG arrangement. This is shown in Fig. 4. The red curve represents the part of the spectrum as “seen” by a single photon detector when connected to port 3 while the black curve is the part which is discarded at port 2. For this measurement the coupled pump power at 1403nm was 100mW while the signal light field at 738nm was attenuated to 2pW. In the next section we will investigate the influence of residual Raman noise on the performance of the frequency converter under realistic experimental conditions.
4. Efficiency of the Zn:PPLN frequency converter
The frequency downconversion will be most efficient if the spectral bandwidth Δλ a of the input photons satisfies the condition Δλ a ≤ Δλ DFG, where Δλ DFG is the spectral acceptance bandwidth  of the DFG process. Considering typical interaction lengths of 10-60mm we can calculate Δλ DFG to be on the order of 0.1nm in the case of our wavelength combination. For photons generated by a single color center in diamond at room temperature this is hard to achieve. Currently, the emission linewidths of the narrowest color centers (SiV centers) are on the order of 1nm at room temperature. Consequently, the mismatch between Δλ a and Δλ DFG reduces the conversion efficiency. One solution to this problem can readily be implemented by cooling of the diamond sample to temperatures below 30K where the emission linewidth of the SiV centers becomes as narrow as 0.17nm . Another promising approach which currently is subject to intensive research is the coupling of a color center to a cavity [28, 29]. In the future, this may allow for much narrower emission linewidths even at room temperature. However, for a proof of principle experiment our goal for the time being is to operate the SiV centers at room temperature and without any cavity coupling. Hence, to evaluate the influence of the bandwidth mismatch we experimentally determined Δλ DFG for our frequency converter in the following way: a constant pump power of 27mW at a fixed wavelength of 1403nm was launched into the WG together with continuous light from the Ti:Sa laser. The wavelength of the Ti:Sa was tuned from 737.816nm to 738.616nm while keeping the coupled power at a constant level of 0.8mW. At the same time we detected the generated power around 1557nm with an InGaAs photodiode. The result is plotted in Fig. 5(a). From a sinc2-fit we yield a spectral acceptance bandwidth of Δλ DFG = 0.16nm. The emission linewidths of the bright SiV centers investigated in  were measured to be 0.7-2.2nm at room temperature while count rates up to 4.8 × 106 s−1 were observed. Assuming a Lorentzian lineshape and perfect phase-matching we can estimate that the flux of photons lying within the measured 0.16nm phase-matching bandwidth is on the order of a few 105 s−1. Thus, we simulate realistic experimental conditions by setting the repetition rate of the pulse picker to ν rep = 500kHz and attenuating the 738nm light to an average photon number per pulse of 〈n a〉 ≈ 0.76 < 1 (this corresponds to an optical power of ∼ 100fW). The temporal width of the generated pulses, limited by the resolution of the pulse picker, was determined to be 9.4ns (FWHM) as shown in Fig. 5(b).
Using the setup as in Fig. 1(a) with the InGaAs/InP SPAD the conversion efficiency of the setup can be determined. We define N dc+R as the count rate of the detector at 1557nm when only pump light is present in the WG (signal light blocked). In this case a detection event can either be caused by a detector dark count (dc) or by a Raman photon (R) that was generated by the strong pump. We further define N dc+R+b as the count rate when both pump and signal light are coupled into the WG. In this situation a detection event can additionally be caused by a photon at λ b which was generated by DFG. We measured the count rates N dc+R and N dc+R+b as a function of the pump power. The result is shown in Fig. 6(a). The number of photons generated by DFG N b (net count rate) is easily obtained from N b = N dc+R+b – N dc+R and is also shown in the plot. In this measurement the parameters of the InGaAs/InP detector were set to the following values: quantum efficiency η qe = 0.25, trigger rate ν t = ν rep = 500kHz (external from pulse picker), gate width τ g = 5ns, dead time τ d = 1μs. With these settings the dark count rate of the detector is about N dc = 107s−1 corresponding to a dark count probability within a gate time of 2.14 × 10−4. From the count rates given in Fig. 6(a) the maximum total conversion efficiency of our setup is readily calculated to be . The internal conversion efficiency is given by η int = 〈n b(L)〉/〈n a(0)〉 where 〈n a(0)〉 is the average number of signal photons per pulse coupled into the WG and 〈n b(L)〉 is the average number of converted photons per pulse exiting the WG. Since the transmission of the attenuator is known, 〈n a(0)〉 can be determined by measuring the optical power before the attenuator. 〈n b(L)〉 is calculated from the measured net count rate N b using 〈n b(L)〉ν rep = N b/(T tot × η qe × 0.86 × 0.47), where the factor 0.86 × 0.47 takes into account the non-perfect extinction ratio of the pulse picker and the mismatch between the temporal width of the signal pulses and the gate width of the detector. Figure 6(b) shows η int of our frequency converter as a function of the coupled pump power P p. The data are fit according to the relation which can be obtained either from solving the classical coupled mode equations [5,6] or from a quantum mechanical approach [9,13] when WG losses and pump depletion are neglected. A maximum of at 240 mW of pump power is achieved with the normalized efficiency η nor = 61%/W/cm2.
It is clear that the above method of measuring the internal conversion efficiency may be affected by several experimental uncertainties. For example to calculate 〈n b(L)〉 we have to multiply a number of quantities that are subject to measurement errors themselves. Further, we have chosen a relatively short dead time of 1 μs for the InGaAs/InP detector. This might cause additional detection events generated by the afterpulsing effect  which could lead to an over-estimation of the conversion efficiency. To evaluate the potential impact of this effect we have independently measured the afterpulsing probability by means of an autocorrelation technique. The measured probability shows a nearly exponential decay in time. For the parameters of our experiment we yield an upper limit for the afterpulsing probability of 4.4% which could add a maximum error of 4.4% to the count rate N b and thus to the calculated conversion efficiency. To further verify that our results are reliable we additionally investigated the depletion of signal photons in two ways: first we performed a classical measurement (WG input of 1 mW at 738nm) of the signal depletion with a Si-based powermeter which yields . Second, the analogous experiment was conducted at the single photon level while the powermeter was substituted by a free running Si SPAD (Perkin Elmer SPCM-AQRH-14, ∼ 65% quantum efficiency at 738nm). Here we get . In both signal depletion measurements was reached at a pump power around 240mW. All together we find that, within measurement accuracy, the results obtained with the InGaAs/InP SPAD are confirmed by the depletion measurements.
In the absence of any propagation losses the internal conversion efficiency that is expected from theory (either classical coupled mode equations or quantum mechanical approach) is 1, i.e., perfect conversion. We have given 7% as an upper limit for the propagation losses of our device assuming an input coupling efficiency of 1 and zero Fresnel losses at the input coupling lens, at both facets of the waveguide and at the output coupling lens. These assumptions are obviously not fulfilled in practice. Consequently, pure propagation losses will be less than 7% and we would theoretically expect for the maximum internal conversion efficiency. In general, non-perfect spatial mode overlap within the WG might be a reason for reduced conversion efficiency. However, in our case we suppose another effect to be mainly responsible for the fact that the internal conversion efficiency is about 15-20% less than the theoretical prediction. From Fig. 5(a) we see that the measured data of the spectral acceptance bandwidth significantly deviate from the tails of the sinc2-fit that is expected from theory. In general, such behavior indicates that due to inhomogeneities (spatial fluctuations of the refractive index due to fluctuations in stoichiometry or waveguide imperfections, temperature variations along the crystal, etc.) the (quasi-)phase mismatch Δβ is not constant (Δβ = 0 in the ideal case) along the propagation direction. This was studied for the first time by Nash et al.  for the case of second harmonic generation in bulk LiNbO3. It was shown that the area under the phasematching scan devided by the height of the central peak can be considered a figure of merit for the nonlinear susceptibility of the crystal (while the phasematching curve can change its shape the area under the curve is constant). The relation between the shape of the phasematching curve and the conversion efficiency was also studied for quasi-phasematched second harmonic generation in periodically poled materials  and has been experimentally observed for various χ(2)-processes in different types of QPM waveguide devices [32–34]. In our case inhomogeneities could be caused, for example, by slightly imperfect waveguide structures or minimal variations in WG temperature.
To complete the discussion we consider the signal to noise ratio SNR = N b /N dc+R which is an important figure of merit for a quantum frequency conversion device. The data are plotted in Fig. 6(b). The SNR reaches its maximum at Pp ≈ 60mW yielding a value of about 6:1. This is comparable to what was achieved in another frequency downconversion experiment using long-wavelength pumping . Note that due to the linearly rising Raman background the SNR attains its maximum before the point of maximum conversion effciency is reached.
5. Summary and discussion
In conclusion we have demonstrated frequency downconversion at the single photon level from 738nm to 1557nm with an internal conversion efficiency of 73%. The wavelength combination was designed to match the emission wavelength of the SiV center in diamond, a bright, narrow-band quantum emitter that can be operated at room temperature. To this end we have built a continuous-wave optical parametric oscillator to deliver the strong pump field at ∼ 1400nm. The CW OPO is widely tunable and can, in principle, serve as a pump source in a variety of frequency conversion experiments with different input wavelengths. Using off-the-shelf optical and mechanical components we achieved a remarkably high input coupling efficiency of 90% at 738nm leading to a total conversion efficiency (including detection) of 2%. Maximizing the input coupling efficiency, though often disregarded in previous reports, is an important task since it dramatically increases the overall efficiency of the device. The dominating noise source in our short-wavelength pumped frequency converter was unambiguously identified to be spontaneous Stokes Raman scattering induced by the strong pump field at 1403nm. We also studied the noise spectrum that could potentially be generated by anti-Stokes Raman scattering. This is of interest for evaluation of the expected noise level in frequency downconversion experiments that use long-wavelength pumping (e.g., 87Rb D 1 line → C-band: 1/795nm – 1/1632nm = 1/1550nm as proposed in ). Despite the strong Stokes Raman noise background a signal to noise ratio of 6:1 was achieved using temporal and narrow spectral filtering. In principle, the lower limit for the filtering bandwidth is given by the frequency bandwidth Δν b of the downconverted light. Since for SiV centers currently Δν b ≥ Δν DFG the minimum filtering bandwidth is actually given by the frequency phasematching bandwidth Δν DFG ≈ 88GHz, corresponding to a bandwidth of 0.7nm at 1557nm. If we would consider a single photon source with a much narrower emission linewidth, e.g., a trapped atom coupled to a high-finesse cavity , we could employ even narrower spectral filtering achieving a much better signal to noise ratio. For comparison, table 1 gives a brief overview of selected frequency downconversion experiments including the results reported in this letter. We consider our work to be an important contribution to the field of quantum frequency downconversion as it unites the advantages of three recent experiments: high conversion efficiency  together with optical powers corresponding to the output of real single photon sources [12, 15]. However, as Ref.  and our own work indicate, it is not the limited conversion efficiency but the noise problems in quantum frequency downconverters that have to be overcome in order to make them useful devices for a future quantum network. One promising approach is the cascaded DFG scheme as proposed in . A less elaborate concept, which is yet attractive in the context of SiV centers, is to choose long-wavelength pumping [12, 15]. When aiming at the telecom O-band, i.e., λ b ≈ 1310nm, the required pump wavelength has to be around 1690nm. As our measurements clearly reveal, this should exclude any SPDC or (anti-Stokes) Raman noise at the expense of only slightly higher fiber transmission losses at 1310nm compared to 1550nm. In future experiments we plan to use true single photons from a SiV center in diamond and investigate the photon statistics after the conversion process.
We acknowledge support of the Bundesministerium für Bildung und Forschung within the funding programs “Optical Technologies” (contract 13N9461) and “QuOReP” (contract 01BQ1011). We would like to thank C. Hepp for assistance with the grating spectrometer as well as J. A. L’huillier and Y. Nishida for many helpful discussions.
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