FEASIBILITY STUDY OF FREE-SPACE QUANTUM KEY DISTRIBUTION IN THE MID-INFRARED

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1 Quantum Information and Computation, Vol. 8, No. 1&2 (2008) c Rinton Press FEASIBILITY STUDY OF FREE-SPACE QUANTUM KEY DISTRIBUTION IN THE MID-INFRARED GUILHERME TEMPORAO, HUGO ZBINDEN a, SEBASTIEN TANZILLI, NICOLAS GISIN Group of Applied Physics, University of Geneva 1211 Geneva, Switzerland THIERRY AELLEN, MARCELLA GIOVANNINI, JEROME FAIST Institute of Physics, University of Neuchatel 2000 Neuchatel, Switzerland JEAN PIERRE VON DER WEID Center for Telecommunication Studies, Pontificia Universidade Catolica do Rio de Janeiro Rio de Janeiro, Brazil Received September 19, 2006 Revised May 24, 2007 We report on a feasibility study of a free-space Quantum Key Distribution setup operating at a mid-infrared wavelength. Alice sends polarization-coded pseudo-single photons from a Quantum Cascade Laser at 4.6 μm to Bob, who uses a nonlinear crystal and a Silicon Avalanche Photodiode to perform the detection via Sum Frequency Generation. Theoretical predictions, based on a proof-of-principle experiment, show that, under certain foggy atmospheric conditions, the proposed system is barely affected, whereas other systems operating at standard near-infrared wavelengths would simply not work at all. Keywords: Quantum Key Distribution, Mid-Infrared, Free-Space Optics, Frequency Up- Conversion Communicated by: H-KLo&RHughes 1 Introduction Quantum Key Distribution (QKD) [1, 2] systems have already been successfully implemented and characterized, either in optical fiber [3] or free-space [4, 5] links. In both cases, much progress has been reported on the maximum distance over which the secret key can be exchanged, on how the effects of noise can be mitigated and on security analysis. However, there is a parameter of the quantum channel that is usually overlooked and rarely discussed: the wavelength of the single photons. It is true that, in optical fibers, the wavelength of 1550 nm is very convenient, as it experiences the lowest absorption losses of the whole spectrum [6], and can be detected with Indium-Gallium-Arsenide avalanche photodiodes (InGaAs APD) in the single photon counting regime [7]. For free-space quantum channels, the transmission wavelength is usually chosen around 780 nm, which corresponds to the quantum efficiency peak of Silicon Avalanche Photodiodes (Si APD); however, there is no clear evidence that this is the most suitable choice! a Corresponding author: hugo.zbinden@physics.unige.ch 1

2 2 Feasibility study of free-space quantum key distribution in the mid-infrared Figure 1, below, shows that there are many transmission windows in the atmosphere, all of which could, in principle, be used in free-space communications. Fig. 1. Typical curve of atmospheric transmission in clear weather conditions. Detector responsivity is considered wavelength-independent. Extracted from Ref. [9]. A mere glimpse at figure 1 suggests that a wavelength deeper into the infrared could potentially increase the bit rate and the maximum distance achieved by a QKD system; in fact, recent experimental investigations and simulations confirm that longer wavelengths, in the mid- and far-infrared (MIR, FIR) a, are less susceptible to atmospheric attenuation and scintillation compared to visible and near-infrared (NIR) wavelengths, especially in the presence of fog or haze [8, 9]. Besides, with the recent developments of Quantum Cascade Lasers (QCL) [10], it is possible to create faint laser pulses at any wavelength between 3.5 and 67 μm [11]. So, why not trying a different wavelength in the MIR or FIR? The answer is in the absence of single-photon counting devices at those wavelengths. In other words, the existing solid-state detectors of MIR/FIR radiation lack the necessary sensitivity and timing resolutions required by quantum communication applications. In principle, superconducting bolometers could be used, but there is no report of such detectors for wavelengths beyond the NIR. The only practical solution using present-day technology to overcome this problem is frequency up-conversion, which dates from the late 1950s [12], has been thoroughly discussed in the following decade [13] and has recently been experimentally demonstrated in the single-photon counting regime for a MIR wavelength in [14]; briefly, it consists in mixing the incoming (mid- or far-) infrared single photons with a strong local pump inside a nonlinear crystal such that, if the necessary phase-matching requirements are met, the infrared photons are upconverted via Sum Frequency Generation (SFG) to a shorter wavelength that can be detected with a Si APD. However, the so-called upconversion detectors may have low efficiency, depending on the nonlinear crystal being used, unless one is willing to use extremely powerful pump sources; a We assume that MIR encompasses all vavelengths between 3 and 13μm, whereas the FIR wavelengths go up to 50 μm.

3 G. Temporao, H. Zibinden, S. Tanzilli, N. Gisin, T. Aellen, M. Giovannini, J. Faist, and J. von der Weid 3 moreover, the upconversion stage may introduce additional noise to the detector, in many different possible ways [13, 14, 15, 16, 17]. In order for such a system to offer any advantage, the lower efficiency and greater noise count rate must be counterbalanced by the increase in atmospheric transparency, which will depend on the weather conditions. This paper concerns precisely this issue. Throughout the paper, we discuss the advantages and drawbacks of a QKD system operating in a MIR/FIR wavelength using QCLs and upconversion detectors. Figures of merit will be selected, calculated and the results compared to data from a proof-of-principle experiment, in order to determine, if possible, in which atmospheric conditions it would be useful. 2 System Parameters Every practical QKD system must be able to generate a secret key shared by two parties, henceforth called Alice and Bob, with a reasonably fast rate and over several kilometers. Thus, the performance evaluation of such a system has to consider the secret key generation rate (R secret ) and the maximum distance that still allows secure communication (L max )as figures of merit. Both figures of merit, however, depend on the probability of error that arises due to propagation losses in the atmosphere, imperfect optical components and detection noise. This is usually expressed as the Quantum Bit Error Rate (QBER), which is given by the ratio of the probability of getting a false detection to the total probability of detection per pulse. It is usually expressed as QBER QBER det +QBER opt, where the first term concerns detector noise and the second is related to the probability of detecting a photon in the wrong detector due to imperfect optical components, which will be considered negligible throughout this paper [18]. This way, we have: ( P (noise) QBER QBER det = P (photon)+2p(noise) = 2+ P (photon) ) 1. (1) P (noise) In the above expression, P(photon) μη atm η cpl η det is the probability of detecting a photon, where η det is the detector efficiency, η cpl is the coupling efficiency of the free-space link, η atm is the atmospheric transmission and μ is the mean number of photons per pulse. P(noise) is the probability of detecting a noise photon for the duration of a signal pulse, which will depend on the Si APD s dark counts and on the background noise, explained in section 3. The second figure of merit, L max, depends on the same parameters, and also on the weather conditions. It actually corresponds to the atmospheric transmission that generates the QBER security limit, which is about 10% for BB84 (for single-photon sources) [19]. The secret key generation rate, on the other hand, involves more parameters; it can be defined as R secret k R raw,wherek is a factor smaller than 1, depending on the protocol being used, on the QBER and on the error correction and privacy amplification algorithyms (which are beyond the scope of this paper), and R raw is the raw key generation rate, given by: where f is Alice s pulse rate in Hertz. R raw = fμη atm η cpl η det. (2)

4 4 Feasibility study of free-space quantum key distribution in the mid-infrared In order for a QKD system operating in an alternative wavelength to be feasible, thus, we establish the following natural set of criteria: i. Technical feasibility: the system must be constructed with present-day technology; ii. Loss budget: the key exchange must be secure over a link with considerable loss, i.e. the QBER must not exceed 10% for a loss of, say, more than 10dB; and iii. The achieved raw bit rates must considerably exceed, at least under certain conditions, the rates obtained with a much simpler, standard scheme working at 780 or 1550 nm. 3 Wavelength Selection Despite the wide range of transparency windows in the atmosphere, not all of them can be used as the quantum channel of a free-space QKD system. In fact, the photodetector restricts the number of practically useful wavelengths, for two main reasons. The first one concerns the nonlinear crystal, which must be able to provide phase-matching between the MIR/FIR wavelength, the pump laser s wavelength (commercially available with a reasonable output power) and an output wavelength inside the detection range of Si or InGaAs APDs. Moreover, the crystal must have a high nonlinear coefficient, such that the SFG efficiency, which is the probability that a MIR or FIR single photon is be up-converted, is not too low. The second and most important reason is the photodetector noise. In addition to the APD s dark counts, there is also the background noise, which affects all free-space optical systems. In the case of MIR and FIR wavelengths, background noise will be mostly comprised of up-converted thermal photons [13, 14], as the emission peak of blackbody radiation at room temperature (300K) is found inside the MIR wavelength range. The total probability of having at least one noise count per pulse, taking into account dark counts and background noise, can be written as the complementary probability of not having any noise counts: P (noise) =1 (1 p DC ) p BG (j)(1 η det ) j. (3) where p DC is the probability of dark counts per pulse and p BG (n)= N BG n /( N BG +1) n+1 is the probability distribution for the background noise [20]. Solving the infinite sum, one obtains the following result for the probability of noise: 1 p DC P (noise) =1 N BG η det +1 N BG η det + p DC. (4) where the approximation is valid if N BG is sufficiently small, which usually holds true for nanosecond pulses. Expression (4) clearly shows that the relative importance between dark counts and background noise counts will depend only on the detection efficiency. Now we must find expressions for the mean number of noise counts. For the background noise, we can write N BG n BG Δτ, where n BG is the mean blackbody radiation photon rate and Δτ is the pulse duration. For a single-mode upconversion detector, n BG will approximately be equal to [14, 21]: n BG j=0 Δν exp(hν 0 /kt ) 1. (5)

5 G. Temporao, H. Zibinden, S. Tanzilli, N. Gisin, T. Aellen, M. Giovannini, J. Faist, and J. von der Weid 5 where ν 0 is the signal optical frequency and Δν is the detector optical bandwidth, determined by a filter or by the phase-matching acceptance bandwidth of the nonlinear crystal. Figure 2 shows how this noise component depends on the wavelength, supposing T = 300K and assuming realistic values of Δτ = 1 ns and a constant ratio Δν/ν (that is, a constant filter quality factor) chosen to be This is a realistic value, corresponding to Δλ =0.1 nm at 800 nm. The effect of detection efficiency was neglected. Fig. 2. Solid line: probability of background noise as a function of wavelength, for unit detection efficiency. Dashed line: normalized solar radiance, expressed in photons per second per unit area per unit solid angle. It can be seen from figure 2 that a system operating in the FIR (beyond 20 μm) has a normalized probability of noise greater than 15% per pulse, which is an overwhelmingly high value. This simple observation leads us to the conclusion that QKD systems operating in the FIR are not feasible. Moving towards shorter wavelengths, figure 2 also shows that the probability of noise decreases almost exponentially, and it is already 5 orders of magnitude lower at 3 μm. The curve should not be extrapolated to the NIR or visible wavelengths, however, as scattered sunlight is the dominant source of background noise in these cases; for this reason, figure 2 also shows the relative importance of thermal photons from the sun in the background noise (dashed line), which does not change significantly within this wavelength range b. Even though one may feel tempted to say that lower wavelengths are better, several factors must be taken into account. The most important seems to be a compromise between background noise and atmospheric transmission, which will strongly depend on the kind of weather conditions (light haze, heavy fog, etc) one is trying to circumvent with a MIR wavelength. In this paper, we have selected 4.6 μm, which seems appropriate for certain atmospheric conditions. This choice is explained in detail in section 5. b The emission peak, found in the NIR, concerns the photon flux and not the intensity of sunlight radiation. If one multiplies the curve by the photon energy, the peak is moved to the visible.

6 6 Feasibility study of free-space quantum key distribution in the mid-infrared 4 The Experiment In a previous experiment, we have demonstrated detection of single photons at 4.6 μm withan overall efficiency η det = and a total noise count rate n tot n DC + n BG η det = 87.8 Hz [14]. The present goal is to evaluate the detection QBER of a practical QKD system operating in the MIR using the same detection technology. Figure 3 shows the setup. On the left side, Alice prepares her polarization-coded qubits using an attenuated QCL at 4.65 μm from Alpes Lasers, generating 1-ns light pulses at 750 khz, and a pair of teflon polarizers. While the second polarizer sets the single-photon polarization, the first one ensures that the overall transmission of the two polarizers is always the same, simulating a half-waveplate with losses. An attenuator is used such that each pulse contains 0.8 photons in average. Fig. 3. Schematic of experimental setup. QCL = Quantum Cascade Laser; Att = Attenuator; Pi = Polarizers; Li = Lenses; Fi = Filters; DM = Dichroic Mirror; PC = Polarization Controller; SMF = Single Mode Fiber; TAC = Time to Amplitude Converter; SCA = Single Channel Analyzer. At Bob s side, the incoming MIR single photons are combined with a 980 nm diode laser beam of 63 mw net power inside a 1-cm long Periodically Poled Lithium Niobate (PPLN) crystal, generating single photons at 809 nm by SFG if the phase-matching conditions are satisfied. In the case of PPLN, the polarization of both the single photons and pump beam must be vertical, which means that, if the pump is constantly vertically polatized, the crystal acts as a vertical polarizer with respect to the incoming single photons. After the up-conversion stage, the 809 nm photons are appropriately filtered and finally detected by a Si APD, and the generated electric pulse is sent to a Time-to-Amplitude Converter (TAC) together with the electric pulse in the QCL. This way, coincidence counts are made in order to synchronize the detection events with the single-photon arrival times. Furthermore, a detection window of exactly Δτ = 1 ns is selected with a Single Channel Analyzer (SCA), performing a post-selection of counting events that fall exclusively inside this window. In order to measure the QBER, the polarization of the single photons sent by Alice is varied and, for each step, the number of detection events in the SCA is recorded several times for a fixed integration time of 100 seconds each. This number of events is plotted against the angle of polarizer P 2 and a cosine function is used to fit the data points, as shown in Figure 4. The vertical bars indicate the standard deviation based on the five measurements that have been performed for each data point. From the raw visibility V of the cosine fit, found to be 92.6%, one can easily obtain the total QBER. Neglecting the 0.11% transmission of unwanted radiation in the teflon polarizers,

7 G. Temporao, H. Zibinden, S. Tanzilli, N. Gisin, T. Aellen, M. Giovannini, J. Faist, and J. von der Weid 7 Fig. 4. Visibility curve for detection of single MIR photons at the vertical-horizontal basis for different incoming polarization values at μ =0.8. we have QBER det =(1-V )/2 = 3.7%. Now we can compare this experimental value with the expected value from expressions (1), (4) and (5): considering η det = , n tot = 87.8 Hz, η atm =1,μ =0.8,Δτ = 1 ns and additional 20% losses in the post-selection stage (coincidence window), we obtain QBER = 3.7%, thus in excellent agreement with the experimental results. 5 Discussion The experimental results above confirm that a free-space QKD system operating at a MIR wavelength (in the case, 4.6 μm) can be secure, as the experimental QBER of 3.7% is considerably below the security limit, even at a very low probability of detection. Now we must discuss the technical requirements to implement such a system in a real-life application. At Alice s side, as there are no commercially available devices to actively change the polarization state of a MIR photon, a configuration with four identical QCLs can be used, each one polarized in one of the four BB84 states. We assume that a pulse rate f =10MHz can, in principle, be achieved. At Bob s side, the detector must have a reasonable efficiency, without the need for very powerful (and very expensive) pump sources. A possible solution to this problem uses resonator-enhanced SFG process, as shown in figure 5. Recent experiments in the NIR show that, using a 5 W pump laser diode, circulating powers of 25 W can be easily achieved with this kind of configuration [15, 16]. A Nd: YAG (or similar) crystal is pumped at 809 nm by an external laser diode such that a strong circulating power at 1064 nm is created. The upconverted wavelength would be shifted to 870 nm, but it can still be detected with a Si APD. The single photons at 4.6 μm are inserted by means of a CaF2 dichroic mirror, and focalized at a series combination of two PPLNs with perpendicularly oriented ordinary polarization planes, following the idea proposed in [22], such that the nonlinear process preserves the polarization state sent by Alice.

8 8 Feasibility study of free-space quantum key distribution in the mid-infrared Fig. 5. Sketch of intracavity SFG setup for Bob s detector. PBS: Polarizing Beam Splitter; D: Detector; NC: Nonlinear Crystal; LC: Laser Crystal; DM: Dichroic Mirror. Finally, a practical system would need a good filtering stage. According to expression (4), the background radiation noise increases linearly with the overall efficiency. It is straightforward from this expression that the only way to keep the noise counts as low as possible consists in using the smallest available filter bandwidth Δν. Considering the realistic values for the linewidths of commercially available QCLs (at 4.6 μm) and pump diode lasers (at 980 nm) of 1.5 nm and 0.07 nm FWHM, respectively, the spectrum of the up-converted photons will be lower than (810/4600) (810/980) nm, which is smaller than the 0.35 nm bandwidth used in the experiment. We can now estimate the performance of the system with the suggested configuration. Supposing a net intracavity pump power of 25 W, a 0.1 nm wide filter with 80% transmission (instead of the current value of 19%) and keeping all other parameters used in the experiment unchanged, the detection efficiency is increased to η det = Consequently, the background noise rate also increases, yielding n BG η det = 8900 counts per second, according to expression (5). Recent experimental investigations [23] show that the combined effect of any other contributions to the total noise rate, such as dark counts, non-filtered pump photons or third-order nonlinear effects, would still be negligible. Thus, replacing (4) and (5) on (1), we have: QBER [ 2+ μη ] cplη 1 atm ΔτΔν (exp(hν 1 0/kT ) 1) =. (6) μη cpl η atm Therefore, for security reasons, the product μη cpl η atm cannot be less than 10 2,which corresponds to a QBER of 10%. Therefore, once the values of μ and η cpl are known, the system s loss budget can be determined. For example, if Alice sends 1 photon per pulse in average and the coupling efficiency for a certain link length is 0.2, the loss budget is 13 db. Note that this value cannot be increased by further increasing the detection efficiency; it is a fundamental limit imposed by background noise. In order to calculate the raw key rate R raw and the maximum distance L max, one needs, thus, the information about propagation losses in the selected wavelength. It should be clear that the product μη cpl η atm will be a function of the weather conditions; what is often not so clear, however, is the dependence on the type of weather.

9 G. Temporao, H. Zibinden, S. Tanzilli, N. Gisin, T. Aellen, M. Giovannini, J. Faist, and J. von der Weid 9 The most important attenuation factor in fogs, for all wavelengths from the visible to the MIR, is scattering. Whenever the wavelength is within one order of magnitude from the mean fog particle diameter, Mie Scattering is dominant. It is a known fact that the Mie Scattering coefficient depends exactly on the ratio between these quantities, usually expressed as the size parameter α 2πr/λ, where r is the radius of the scattering particle. As different fogs have different particle size distributions, the atmospheric attenuation for a given wavelength will depend on the fog type. As there are not enough experimental results for the attenuation of 4.6 μm in any kind of fog, one must borrow results obtained for other wavelengths, provided that the size parameter remains constant. For instance, we can use the data obtained in ref. [8] for wavelengths of 9.1 and 1.55 μm propagating through Coastal Upslope Fog. As this fog has an average particle radius of 4μm, the size parameters (and, consequently, the scattering coefficients) are almost the same as those for 4.6 μm and 780 nm in a kind of fog which has particles of radii around 2μm. This is the case of Radiation Fog [24], a kind of fog which can be found, for instance, in the region of Neuchatel, Switzerland [25], or in any area in the vicinity of lakes where clear skies are frequent and the wind speed is low enough [26]. The results can be seen in figure 6, which shows the atmospheric attenuation as a function of the link visibility c. Fig. 6. Atmospheric attenuation in db/km for different visibility conditions in a Radiation Fog environment, at 780 nm (triangles) and 4.6 μm (squares). It is clear from figure 6 that the proposed system at 4.6 μm would be useful at least for all visibility conditions up to 2000 m. Moreover, the difference in attenuation increases with the link distance; for example, the attenuation factors for a 10 km link under visibility conditions of 600 m are 164 db at 780 nm and 1.9 db at 4.6 μm, which means a transmission gain of 16 orders of magnitude! However, in long-distance applications, atmospheric turbulence is more pronounced, and wavefront distortion effects become more important. This means c The visibility is defined as the distance corresponding to the minimum contrast, between object and background, that can be discerned with the human eye. Not to be confounded with interferometric visibility.

10 10 Feasibility study of free-space quantum key distribution in the mid-infrared that, if single-mode detectors are used, the coupling efficiency η cpl may become too low, such that the product μη cpl η atm may be well below For this reason, single-mode detection is a double-edged blade: the background noise is largely reduced, but the system is much more affected by turbulence. This means that our system is really useful in conditions of low visibility, where the maximum distance is short enough. Table 1 shows a comparison between QKD systems operating at the two wavelengths in the specific case where the visibility is 300 m. Detection at 780 nm was assumed to be performed using a single mode fiber-coupled Si APD with 70% quantum efficiency and dark count rate of 100 Hz; in both cases, Alice sends weak pulses at f =10MHzandμ = 1, and the coupling efficiency η cpl is 0.2, which is a realistic value for a 1.5 km link. It can be seen that, in this situation, the system at 780 nm would become completely unavailable, whereas a system at 4.6 μm would work with a reasonable QBER value. Table 1. Comparison between two Free-Space QKD systems operating at 4.6 μm and 780 nm over 1.5 km, under visibility conditions of 300 m and average fog particle radius of 2 μm (radiation fog). η atm η cpl η det P (photon) P (noise) R raw QBER 4.6μm khz 2.3% 780 nm Hz 43% One should note that, in the example above, all background noise from scattered sunlight was neglected for both wavelengths, which may be difficult to achieve in daylight operation, even if the detection is performed with single-mode detectors [27]. One could hope that the coupling of scattered sunlight is reduced when MIR wavelengths are used, because the probability of scattering is lower. However, one should also take into account the probability that a scattered photon changes its direction of propagation and goes toward the detector, which, in the case of Mie Scattering, increases with the wavelength. Depending on the fog s particle size distribution, these two concomitant effects may or may not compensate each other; this way, the robustness to sunlight is, actually, a function of the weather conditions. 6 Conclusion We have performed a feasibility study of a free-space Quantum Key Distribution operating at a mid-infrared wavelength. Theoretical expressions, which have been confirmed by a proofof-principle experiment, show that a practical system operating at 4.6 μm can be constructed with the currently available technology. Even though the single-photon detection efficiency at this wavelength can be made reasonable, by intracavity SFG with strong pump powers, the noise rate is largely increased by background radiation from up-converted thermal photons, which leads to a limited loss budget. For this reason, the system at 4.6 μm wouldnotbe useful under clear weather conditions. On the other hand, the proposed system was shown to be very robust in the presence of fog. In the case of fog with particle radius around 2 μm, a standard QKD system operating at 780 nm would become completely unavailable in situations of low visibility, whereas a

11 G. Temporao, H. Zibinden, S. Tanzilli, N. Gisin, T. Aellen, M. Giovannini, J. Faist, and J. von der Weid 11 system at 4.6 μm would be barely affected. Therefore, despite all limitations, a mid-infrared QKD system can be useful under certain conditions. It should be noted, however, that all long-distance single-mode detection systems are more vulnerable to atmospheric turbulence. Acknowledgements The authors would like to thank Alexios Beveratos and Claiton Colvero for the helpful discussions and Claudio Barreiro for the technical support. We also thank EXFO Inc. and the Swiss NCCR Quantum Photonics for the financial support. References 1. C. H. Bennett and G. Brassard (1984), Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, (Bangalore, India) pp N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden (2002), Rev. Mod. Phys. 74, D. Stucki, N. Gisin, O. Guinnard, G. Ribordy and H. Zbinden (2002), New Journal of Physics 4, R. J. Hughes, J. E. Nordholt, D. Derkacs and C. G. Peterson (2002), New Journal of Physics 4, K. Resch, M. Lindenthal, B. Blauensteiner, H. Bhm, A. Fedrizzi, C. Kurtsiefer, A. Poppe, T. Schmitt-Manderbach, M. Taraba, R. Ursin, P. Walther, H. Weier, H. Weinfurter, and A. ZeilingerM.A. Nielsen and J. Kempe (2005), Opt. Express 13, pp G. Keiser (1999), Optical Fiber Communications, McGraw-Hill 7. G. Ribordy, J.D. Gautier, H. Zbinden and N. Gisin (1998), Appl. Optics 37 (12), pp C. P. Colvero, M. C. R. Cordeiro, G. V. Faria and J P von der Weid, Microwave and Optical Technology Lett. 46, pp H. Manor and S. Arnon (2003), Appl. Optics 42, p A. Mller, M. Beck, J. Faist, U. Oesterle and M. Illegems (1999) Appl. Phys. Lett. 75, pp G. Scalari, L. Ajili, J. Faist, H. Beere, E. Linfield, D. Ritchie and G. Davies (2003) Appl. Phys. Lett. 82, pp N. Bloembergen (1959), Phys. Rev. Lett. 2, p D. A. Kleinman and G. D. Boyd (1969), J. Appl. Phys. 40, pp , and references therein. 14. G. Temporao, S. Tanzilli, H. Zbinden, N. Gisin, T. Aellen, M. Giovannini, and J. Faist (2006), Opt. Lett. 31, pp M. A. Albota and F. N. C. Wong (2004), Opt. Lett. 29, pp H. Pan and H. Zeng (2006), Opt. Lett. 31, pp R. T. Thew, S. Tanzilli, L. Krainer, S. C. Zeller, A. Rochas, I. Rech, S. Cova, H. Zbinden and N. Gisin (2006), New J. Phys., Vol 8, H. Zbinden, H. Bechmann-Pasquinucci, N. Gisin and G. Ribordy (1998), Appl. Phys. B 67, B. Kraus, N. Gisin and R. Renner (2005), Phys. Rev. Lett. 95, B. E. A. Saleh and M. C. Teich (1991), Fundamentals of Photonics, Wiley, pp G. D. Boyd and D. A. Kleinman (1968), J. Appl. Phys. 39, pp A. P. VanDevender and P. G. Kwiat (2004), Proc. SPIE 5551, pp G. Temporao, H. Zbinden, P. Eraerds and N. Gisin (2007), not yet published. 24. M. E. Thomas and D. D. Duncan (1993), Atmospheric transmission, in The Infrared and Electro- Optical Systems, vol. 2 (F.G.Smith,Ed.)SPIEOpticalEngineeringPress 25. S. Blaser, D. Hofstetter, M. Beck, and J. Faist (2001), Electronics Lett. 37, pp R. Krolak (2001), NAV EDTRA US Naval Education and Training Professional Development and Technology Center. 27. M. Aspelmeyer, H. R. Bohm, T. Gyatso, T. Jennewein, R. Kaltenbaek, M. Lindenthal, G. Molina- Terriza, A. Poppe, K. Resch, M. Taraba, R. Ursin, P. Walther and A. Zeilinger (2003), Science 301, pp