Atmospheric Channel Characteristics for Quantum Communication with Continuous Polarization Variables
Bettina Heim, Dominique Elser, Tim Bartley, Metin Sabuncu, Christoffer Wittmann, Denis Sych, Christoph Marquardt, Gerd Leuchs
AAtmospheric Channel Characteristics for Quantum Communicationwith Continuous Polarization Variables
Bettina Heim,
1, 2, ∗ Dominique Elser,
1, 2
Tim Bartley,
1, 2
Metin Sabuncu,
1, 2
ChristofferWittmann,
1, 2
Denis Sych,
1, 2
Christoph Marquardt,
1, 2 and Gerd Leuchs
1, 2 Institute of Optics, Information and Photonics, University of Erlangen-NurembergStaudtstr. 7/B2, 91058 Erlangen Max Planck Institute for the Science of LightG¨unther-Scharowsky-Str. 1, Bau 24, 91058 Erlangen
We investigate the properties of an atmospheric channel for free space quantum communicationwith continuous polarization variables. In our prepare-and-measure setup, coherent polarizationstates are transmitted through an atmospheric quantum channel of 100 m length on the roof of ourinstitute’s building. The signal states are measured by homodyne detection with the help of a localoscillator (LO) which propagates in the same spatial mode as the signal, orthogonally polarizedto it. Thus the interference of signal and LO is excellent and atmospheric fluctuations are auto-compensated. The LO also acts as a spatial and spectral filter, which allows for unrestrained daylightoperation. Important characteristics for our system are atmospheric channel influences that couldcause polarization, intensity and position excess noise. Therefore we study these influences in detail.Our results indicate that the channel is suitable for our quantum communication system in mostweather conditions.
PACS numbers: 03.67.Dd, 03.67.Hk, 42.68.BzKeywords: Quantum Communication; Quantum Key Distribution, Polarization Encoding, AtmosphericNoise, Atmospheric Optics, Quantum Optics
I. INTRODUCTION
Quantum communication describes the distribution ofquantum states between two parties, traditionally namedAlice and Bob. These states can for example be entan-gled [1] states, providing the basis for various protocolssuch as quantum teleportation [2] or quantum dense cod-ing [3]. Many of the initial research projects used discretequantum variables. Later also continuous variables haveproven suitable for quantum communication (for a reviewsee [4]).Quantum key distribution (QKD) [5, 6] is a further im-portant branch of quantum communication and concernsthe establishment of a secret key jointly between Aliceand Bob with the help of a quantum channel. The secu-rity is based on the laws of quantum mechanics. In prin-ciple unconditional security can be achieved. Any twonon-orthogonal quantum states suffice to ensure securekey distribution [7]. This holds as long as the detectionmatches the quantum state emitted by the source. Asingle photon detector e.g. matches weak coherent statesas long as the probability for multiphoton events is lowenough. For higher multiphoton probabilities [8] or evenbright polarization states [9, 10] a single photon detec-tor can not be used. In such scenarios, however, photonnumber resolving detectors, homodyne or heterodyne de-tectors are a better match, promising unconditionally se-cure key distribution.Free space QKD over an atmospheric channel was ∗ Electronic address: [email protected] first demonstrated in 1996 [11]. Since then, a numberof prepare-and-measure as well as entanglement basedschemes have been implemented in free space (for a re-view see [6]). The current world record in distance is144 km [12, 13] and satellite quantum communication isalready in preparation [14, 15]. All of these aforemen-tioned systems use single-photon detectors and thereforehave to employ spatial, spectral and/or temporal filteringin order to reduce background light. In our system, weuse an alternative approach: with the help of a bright lo-cal oscillator (LO), we perform homodyne measurementson weak coherent polarization states [10]. We focus onthe characterization of the quantum channel, which is a100 m free space link on the roof of our institute’s build-ing.In classical free space communication systems using ho-modyne detection (e.g. [16]), producing the LO locally atthe receiver is appropriate. In QKD, on the other hand,the requirements for detection efficiency are more strin-gent, as fragile quantum states are transmitted. Thuswe developed a protocol using the polarization degreeof freedom to multiplex signal and LO [9]. The LO isproduced by Alice and propagates in the same spatialchannel mode as the signal.In quantum mechanics, polarization is convenientlydescribed by the quantum Stokes operators, that arethe quantum counterpart of the classical Stokes param-eters [17]. The Stokes operators are introduced and de-fined for example in [18].In a homodyne detection of the Stokes parameters,the co-propagation of signal and LO leads to an intrinsi-cally excellent spatial interference between the two. Thistranslates to a high detection efficiency without any ad- a r X i v : . [ qu a n t - ph ] O c t ditional interference stabilization. For our free space sys-tem, there are also advantageous side effects of this co-propagation: firstly, the LO acts as a spatial filter, suchthat only those photons, that are spatially mode-matchedto it will result in a significant detector signal. Unlike insingle photon experiments there is no need for spatial fil-tering by pinholes or fibers. Secondly, the LO facilitatesspectral filtering, as the beat-note of signal and LO, in-terfering at a polarizing beam splitter (PBS), can be elec-tronically filtered at the detector. The detection band-width can thus be adjusted precisely and backgroundlight outside this range does not disturb the measure-ment. Finally, absolute phase fluctuations in the channelare auto-compensated, as they are identical for signal andLO.The theory for the propagation of classical lightthrough turbulent atmosphere including diverse phenom-ena such as beam wander or beam spreading has been in-vestigated in e.g. [19, 20, 21]. However, effects on quan-tum continuous variable states have only recently beenstudied in this context [22, 23, 24]. Influences of the at-mospheric channel may cause polarization and intensityexcess noise under certain conditions. Both might funda-mentally compromise the security of a QKD system andgenerally degrade transmitted nonclassical states. At-mospheric noise typically is of non-Gaussian character(e.g. on-off noise). Squeezed and entangled states thatwere degraded by this noise can be distilled with Gaus-sian operations [22, 23]. Intensity noise can easily stemfrom practical issues such as finite aperture size lead-ing to fluctuations of the detected intensity. Such effectshave thus to be studied and characterized in detail in or-der to determine if they could affect the quality of thequantum communication channel. In security analysis ofQKD systems, all excess noise is considered to originatefrom Eve’s interactions. In a worst case scenario, strongnoise effects would result in the fact that no key can beestablished. II. EXPERIMENTAL SETUPA. Quantum state measurements
The setup shown in figure 1 was used for our QKD-feasibility studies [10] and follows the principles of ourearlier laboratory work [9, 25]. We use a grating-stabilized CW diode laser, whose wavelength of 809 nmlies within an atmospheric transmission window. A lin-early polarized laser beam ( ˆ S in terms of Stokes opera-tors) is emitted by Alice and later serves as a LO in Bob’smeasurement. A modulator is used to generate the coher-ent signal states. (A magneto-optical-modulator (MOM)for example employs the Faraday effect to tilt the linearpolarization by small amounts.) The weak signal com-ponent of a mean photon number of typically less thanone photon per pulse is located in the same spatial modeas the LO, but is polarized orthogonally to it. After ex- FIG. 1: Experimental setup for our QKD feasibility stud-ies [10]: Alice’s laser emits a linearly polarized CW beamwhich later serves as a local oscillator (LO) for Bob’s mea-surements. In terms of Stokes operators, the local oscillator isˆ S -polarized. Alice’s modulator generates a weak signal thatBob then measures by an ˆ S Stokes detection. In-between,the beam is expanded and sent to a retro reflector at a dis-tance of 50 m. After reflection, Bob’s telescope again reducesthe beam diameter. PBS: polarizing beam splitter, HWP:half wave plate. panding the beam by a telescope, the signal/LO beam issent over the roof of our institute’s building and retro re-flected after 50 m. Bob reduces the beam diameter witha telescope and then performs a Stokes measurement ofthe ˆ S -operator to detect the signal states. B. Setups for Different Noise Measurements
Here we present measurements of the polarization, po-sition and intensity excess noise properties of the atmo-spheric channel.
1. Atmospheric polarization noise
In previous work [10, 26, 27] we investigated the po-larization excess noise introduced by the channel. Foran alphabet using two coherent polarization states wecompared the distributions of ˆ S -Stokes measurementsof the signal states before and after transmission throughthe channel. Additional polarization noise introduced bythe channel would broaden the measurement distribu-tion. The work in [26, 27] showed, that this is not thecase. Measurements of the RF frequency spectrum of un-modulated beams that were sent through the atmospherealso allow us to identify the frequency range above 10 kHzto be essentially noiseless [10].
2. Atmospheric intensity noise
Atmospheric intensity noise can be measured by directdetection of the beam. For calibration, we compare thenoise of a beam sent through the atmosphere with a beamsent over the optical table. The intensity noise is recordedby a spectrum analyzer. These measurements are sensi-tive to fluctuations of the laser’s intrinsic excess noise
FIG. 2: 2D-Beamprofiles under several conditions. Before be-ing sent over the roof, the beam is in a near TEM mode (up-per left picture). After transmission through the channel thebeam profiles are slightly distorted, but the intensity distri-butions along the two main beam axis are still approximatelyGaussian. Strong beam distortions are caused by opening ahatch over which the beam passes on its way to and backfrom the retro reflector. Plots thereof, recorded at differentinstances of time, are shown in the second row. All beamprofiles were recorded at an exposure time of 20 (cid:181) s. which we monitored accurately when recording the spec-tra. We use low noise detectors whose electronic noiseis significantly smaller than the shot noise, thus allowingus to measure at the quantum noise limit.
3. Atmospheric beam jitter
We used a beam profiling system to compare thechanges of the spatial beam profile caused by the atmo-spheric channel. For comparing both, the outgoing andthe returning beam were detected with the help of a CCDcamera (Metrolux ML3743). The pictures then were an-alyzed by the Metrolux BeamLux II software package.Figure 2 shows some typical spatial beam profiles underdifferent conditions. Sequences of pictures were taken atan exposure time of 20 (cid:181) s. FIG. 3: Intensity noise measured by a direct detection. Allcurves arise from an averaging over several measurements andare normalized to a quantum noise limited reference beam.The resolution bandwidth for the measurements consisting of401 points was 10 kHz, the video bandwidth 100 Hz. On theright hand side of the plot, the measurement accuracy of 5 %is shown in blue.
III. RESULTS AND DISCUSSION
In the following we will concentrate on atmosphericintensity noise and beam jitter (atmospheric polarizationnoise has been investigated in [10, 26, 27].
A. Atmospheric intensity noise
Measurements of the intensity noise were performedby detecting the amplified photocurrents of one photo-diode (Hamamatsu, S3399, active area 7 mm , diameter3 mm), and comparing the spectrum of an unmodulatedbeam transmitted over the optical channel with that of areference beam over the table. Constant attenuation wascompensated for by setting the optical power in front ofthe photodiode to the same value of 650 (cid:181) W in both cases.In figure 3, it can be seen that there is no atmosphericexcess noise measured for beams that are sent throughthe optical channel in good weather conditions (dry andsunny) as well as in light rain. This is valid for frequenciesabove the current QKD modulation frequency of 1 MHz.The measurement accuracy of 5% accomodates the factthat the beam moves on regions of the photodiode withslightly different sensitivities. Additionally, small inten-sity fluctuations of the laser are included. We can in-fer that the scattering effects of the atmosphere do notcause a measurable beam broadening or spatial beam jit-ter, and thus, the beam hits the photodiodes as well asit does when sent over the optical table.Intensity excess noise can occur if the collimated beamafter Bob’s telescope is detected without focussing it ontothe photodiodes. Then the spatial beam jitter caused bythe atmosphere exceeds the active area of the photodi-odes and thus leads to partial detection noise (red curvein figure 3). An estimation of this noise based on fluc-tuations of the beam centers will be given in the nextsection.
B. Atmospheric beam jitter
Figure 4 show the beam centers and standard devia-tions of sequences of beam profiles (typical spatial inten-sity distributions are shown in figure 2).Part a) in figure 4 shows the comparison of a beamthat was sent over the optical table with one sent overthe roof, both of which were focussed on the camera whenrecorded. As expected, the fluctuations of the beam cen-ters are much higher for atmospheric transmission, butstill small enough to be compensated by the aperture ofthe photodiodes. This is confirmed by the fact, that inthis case no intensity excess noise is shown in figure 3.One can estimate the relative quantum shot noise of thesestates by √ (cid:104) n (cid:105)(cid:104) n (cid:105) = 2 × − . The mean photon num-ber (cid:104) n (cid:105) of the 650 (cid:181) W-beams per measurement period isapproximated by dividing the total detected energy perperiod ( E total) by the energy of one photon of 809 nm( E photon): (cid:104) n (cid:105) = E total E photon = P opt VBW2 . × − Ws = 2 . × (1)with a video bandwidth (VBW) of the RF spectrum an-alyzer of 100 Hz.By a numerical evaluation we estimate the intensitynoise caused by the atmosphere for a focussed detection.The calculations are based on the measured beam cen-ter fluctuations and on values for beam diameters thatwe also gained from the spatial beam profiles. They re-fer to a certain frequency: 50 kHz, at which the camerameasurements were performed. In our calculation, we in-tegrate over the intensity distributions of beam profileswithin a region defined by the size of the photodiodes.The resulting intensity value then is normalized to thatof non-cropped beams. Aligning inaccuracies of 0.2 mmare also included in our calculations. We assume a Gaus-sian intensity distribution and a mean beam diameter of0.98 mm.Using the calculations explained above, we compare abeam whose center is shifted by 0.0134 mm, the meanstandard deviation of the beam center fluctuations (seefigure 4 a) ), to a centered beam. After normalizationto the intensity within the size of the photodiodes, theresult quotes a value for the relative intensity noise whichis around 7 × − . This value lies below the quantumshot noise estimated above, that marks the quantum me-chanical limitation of our measurement accuracy. Thus it is too small to be detected which is in agreement withfigure 3.As quoted in section III A and shown in figure 3, in-tensity noise can occur by a detection of the collimatedbeam directly after Bob’s telescope, without using a fo-cussing lens. This beam is broadened compared to a fo-cussed one and its beam center fluctuations are slightlyhigher, shown in figure 4 b). We perform the same eval-uation as for the focussed beam above, resulting in anintensity noise of 4 . × − at 50 kHz, which is abouttwice the estimated value for the relative quantum noise(2 × − ). This is in agreement with figure 3, showingthe intensity noise for an unfocussed beam to be about3 dB higher than shotnoise for the lowest measured fre-quencies at around 80 kHz. As the detection bandwidthwas limited we couldn’t perform the intensity noise mea-surements all the way down to 50 kHz. We expect aslight further increase of the noise for smaller frequencies.Thus, the estimation is in good over all agreement withthe measurements. We want to stress, that even thoughthis effect is small, it is still observable because of thelow-noise properties of the detectors having an electronicnoise level well below the quantum noise limit.The beam center fluctuations are even larger when ahatch (see figure 2) is open, over which the beam passeson its way to and from the retro reflector (see figure 4c)). In this particular case, atmospheric fluctuations aredramatically increased. This would cause further inten-sity noise when the beam is detected at the photodiode.Hence we are now working on an optimised detectionsystem to improve free space beam capture. The useof improved optical tapers can combat strong combinedspatial and angular fluctuations of the incident beam,better than a single lens could do [28]. We have exper-imental evidence for the non-Gaussian character of thenoise, which will be reported elsewhere. IV. CONCLUSION AND OUTLOOK
Within the framework of the first demonstration ofcontinuous variable quantum communication through areal atmospheric channel, we investigated different chan-nel noise properties. We precisely characterized atmo-spheric intensity fluctuations by quantum-noise limitedmeasurements. Our results indicate that in good weatherconditions and with an appropriate design of sending andreceiving optics, channel influences like polarization, in-tensity and position noise are sufficiently low to allow forquantum state transmission and QKD operation at day-light. For our 100 m link, there was no need for activebeam stabilization, as beam jitter effects caused by theatmosphere could be compensated by appropriate designof the passive optical components. For an extended linkof 1.6 km, on which we are working currently, active sta-bilization is probably necessary. Monitoring the brightLO can provide us with a control signal for active beamstabilization. Additionally, to synchronize Alice’s and
FIG. 4: ( x, y )-plots of the the beam centers and standard deviations of sequences of 650 beam profiles, recorded with anexposure time of 20 (cid:181) s. The mean values of the beam centers are shifted to (0,0) for each plot, the standard deviations areshown in colors corresponding to the particular beam centers. Plot a) shows the comparison of a beam that was sent over theoptical table with one that passed through the atmospheric channel, both being focussed on the camera while recorded. Asexpected, the fluctuations are much larger through the atmosphere. In the lower part, the difference between a focussed and anunfocussed ”atmospheric” beam is demonstrated (part b) ), corresponding to the intensity excess noise shown in red in figure 3.In part c), we compare this ”atmospheric” beam to one having passed directly over a hatch, whereby the temperature gradientbetween inside the building and outside caused strong atmospheric fluctuations (these measurements were taken during wintertime).
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Acknowledgments
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