Data capacity scaling of a distributed Rydberg atomic receiver array
J. Susanne Otto, Marisol K. Hunter, Niels Kjærgaard, Amita B. Deb
BBandwidth increase through distributed atomic receivers in a Rydberg vapour cell
J. Susanne Otto, ∗ Marisol K. Hunter, Niels Kjærgaard, and Amita B. Deb † Department of Physics, QSO-Centre for Quantum Science,and Dodd-Walls Centre, University of Otago, Dunedin, New Zealand (Dated: February 11, 2021)The data transfer capacity of a communication channel is limited by the Shannon-Hartley theoremand scales as log (1 + SNR) for a single channel with a given power signal-to-noise ratio (SNR).We implement an array of atom-optical receiver antennas in a single-input-multi-output (SIMO)configuration by using spatially distributed probe light beams. The data capacity of the distributedreceiver configuration is observed to scale as log (1+ N × SNR) for an array consisting of N receivers.Our result is independent on the modulation frequency, and we show that such enhancement of thebandwidth cannot be obtained by a single receiver with a similar level of combined optical powerdue to the saturation of the SNR. We investigate both theoretically and experimentally the originsof the single channel capacity limit for our implementation. I. INTRODUCTION
In recent years there has been a growing interest inatom-based techniques for the detection of microwave(MW) electric fields [1, 2], that allow for calibration freeSI-traceable measurements [3] achieving ultra-high sen-sitivity exceeding 55 nVcm − Hz − / [4]. Rydberg atomshave been identified as particularly suitable for such mea-surements of radio-frequency (RF) electric fields due totheir high polarizabilities and large microwave (MW)transition dipole moments [5]. Advances in Rydbergbased field measurements have been accompanied byapplications in atom-based communication technology.The fundamental working principles of analog and dig-ital communication, where a baseband signal is modu-lated onto an electromagnetic MW carrier wave, haverecently been demonstrated in a range of Rydberg-basedsystems. Examples include amplitude modulation (AM)[6–10], frequency modulation (FM) [11, 12] and phasedetection [13, 14], as well as multiple bands [11], multi-ple channels [15], and multiple species [10]. Common tothese methods is the broad carrier frequency range fromtens of MHz to several THz that can be covered by thenumerous Rydberg states of a single atomic species.Rydberg receivers generally rely on the phenomenonof electromagnetically induced transparency (EIT) [16]in a three-level system, where a coupling laser field ren-ders an otherwise opaque atomic medium transparent toa probe laser field due to quantum interference. If thethree-level atomic system is coupled to a fourth level viaan RF transition, the Autler-Townes (AT) effect [17] al-ters the transmission of a light field. For an RF electricfield, which is modulated with a signal measuring themodulation of the transmitted light allows to directly re-trieve the signal. In contrast to conventional receiversthat rely on band-specific electronic components, Ryd-berg based receivers benefit from a direct and real-time ∗ [email protected] † [email protected] read out of information, and physical reconfiguration isnot necessary when the carrier frequency is varied. Also,the received information is encoded in a light field, i.e.a laser beam, suitable for long-distance transport via afiber-link.One of the most important figures of merit for any com-munication system is the channel capacity C , which givesthe amount of information that can be passed through acommunication channel in a period of time without error.For a channel with a bandwidth (BW) its upper limit isdefined by the Shannon-Hartley theorem [18] as C = BW × log (1 + SNR) (bits/s) . (1)The crucial parameter for a communication system istherefore the power signal-to-noise ratio (SNR) at a givenbandwidth, which has to be maximised in order to attainthe maximal channel capacity. A common method for in-creasing the data transmission capability is to establishmultiple channels using, e.g., multiple identical receiverantennas while keeping the transmission power the same.This so called single-input-multi-output (SIMO) arrange-ment is formally equivalent to a single receiver with anincreased SNR, and therefore a higher data capacity isachieved. For N receiver antennas the data capacity in-creases with [19] C A ( N ) = BW × log (1 + N × SNR) . (2)In this paper we investigate the scalability of the datacapacity of atomic radio receivers. As opposed to con-ventional antennas, Rydberg receivers can work in theelectrically small regime [20]. Therefore it is possible, inprinciple, to improve the data capacity by increasing theatomic density within the receiver volume. In a vapourcell environment, this usually means that one needs toincrease the vapour pressure within the cell by heating itwhich introduces significantly larger Doppler width andcollisional broadening of the transition. Alternatively,the size of the receiver can be expanded, for example, byextending the optical path length – this has the conse-quence that the background optical depth, which doesnot contribute to the signal, rises, causing the detected a r X i v : . [ qu a n t - ph ] F e b Figure 1. (a) Experimental setup used in this work and de-scribed in the text. (b) Four-level energy diagram for ouratomic Rb system. Ω p and Ω c denote the probe and cou-pling Rabi frequencies. (c) Beam profiles of four probe beamsin the focal plane with frequency differences f − f = f − f = 3 MHz and f − f = 4 MHz. probe power to drop. While this can be counteractedby increasing the probe beam power to begin with, thedata capacity saturates beyond a certain probe beampower, as we demonstrate experimentally. Most impor-tantly, we show that an array of parallel and simultane-ous atomic receivers can significantly increase the datacapacity. Specifically, we employ up to four distributedreceiver volumes within a single vapour cell and we ob-serve a log (1 + N × SNR) scaling of the data capacity,where N is the number of closely spaced, distributed re-ceivers. II. EXPERIMENTAL SETUP
The heart of our experimental setup, see Fig. 1(a),is a 75 mm-long, 25 mm-diameter cylindrical vapour cellcontaining rubidium (Rb) atoms. The Rb atoms serveas receiver medium for an AM microwave signal emittedby a helical antenna. The AM microwave electric field isoptically detected in a ladder-type Rydberg-EIT scheme,where a strong coupling field generates a transparencywindow on resonance in presence of a weak probe field,see Fig. 1(b). This allows to transduce AM informationin the RF domain to optical information encoded as achange of the probe transmission.In our setup the coupling laser has a wavelength of480 nm and a power of 22 mW, and is focused to awaist of 60 µ m in the vapour cell. This corresponds toa Rayleigh length of ∼
25 mm and a Rabi frequencyof 2 π × P / ↔ D / . A probe light field at 780 nm passes through an acousto-optic modulator (AOM), which is driven by a multitonefrequency source. Up to four diffracted beams at fre-quencies f , f , f and f can be generated simultane-ously. The diffracted beams impinge on a lens whichgenerates parallel beams with ∼ µ m radii in its focalplane, see Fig. 1(c). The probe and coupling beams arecounter-propagating under an angle of ∼ ° to create spa-tially separated ( > . Rb D -line. With the AOMin front of the experimental cell, see Fig. 1(a), we ob-tain four beams with frequencies close to the transition5 S / ( F = 2) → P / ( F (cid:48) = 3). For the stabilisation ofthe tuneable coupling laser the EIT signal of a µ -metal-shielded Rb vapour cell is used, and the laser can bestabilized to different Rydberg levels nD / and nD / with n = 30 to 70. For the results presented in this pa-per we use the Rydberg level 52 D / .Our experimental vapour cell is kept at a tempera-ture of 85 ° C yielding a ground state atom density of ∼ cm − for a Rb sample. The two optical fieldsthat are passing through the cell are coupled to the Ryd-berg level 51 F / with a MW carrier field at 16 .
532 GHz.This field is generated by an analog signal generator thatfeeds a 10 dBm signal into a home-built helical end-fireantenna. In axial mode the antenna radiates microwavefields along the helical axis and the radiation is circu-larly polarized. The antenna is located 30 cm from thevapour cell. With increasing field strength of the MWfield, the amplitude of the EIT signal decreases while itswidth broadens and for MW electric fields >
10 dBm ourtransmission signal splits into two AT peaks.The MW carrier field is further amplitude modulatedwith a sinusoidal signal with frequencies between 100 kHzand 1 MHz. This results in a variation of the transmissionof the probe fields, which can be directly measured withthe combination of photodetector and spectrum analyserand permits to retrieve the initial AM signal.
III. RESULTSA. SNR and Bandwidth of a Single Receiver
Figure 2(a) shows the demodulated probe signal fora 200 kHz amplitude modulated carrier field, measuredwith the spectrum analyser for a resolution bandwidthof 3 kHz. The transmitted probe field is detected by -20 -10 0 10 20
Probe detunning/2 (MHz) T r an s m i ss i on (c) peak Figure 2. (a) Demodulated signal of the probe transmissionas a function of frequency measured with the spectrum anal-yser for three different probe powers. (b) Dependence of theSNR dB on the probe power for an AM frequency of 200 kHz.(c) Numerical data for the probe detuning versus probe trans-mission for two different probe Rabi frequencies (blue andblack), in the EIT regime (solid lines) and AT regime (dottedlines) for a resonant RF field (Ω RF = 2 π × the photodetector, and the spectrum analyser yields thepower spectral density for frequencies between 100 and350 kHz. The traces have distinct maxima at the modu-lation frequency, which are standing out against the noisefloor observed with the spectrum analyser. We define theSNR dB as the difference of the power spectral density at the modulation frequency to the noise floor on the spec-trum analyser at the respective frequency in absence ofthe modulation. As can be seen from Fig. 2(a), the sig-nal height and with it the SNR dB grow with increasingprobe power in the presented scenario (black to red).The development in SNR dB as a function of probepower is shown in more detail in Fig. 2(b). Initially, theSNR dB grows with increasing probe power, but saturatesat ∼ µ W, and slowly decreases for even higher probepowers. The growth of the SNR dB with probe power isrelated to an increase in peak contrast, as illustrated inFig. 2(c), which is the difference between the probe trans-missions at the EIT peak in presence and absence of thecarrier RF field. In the four level scheme in Fig. 1(b),the applied RF field suppresses the EIT transmission onresonance as shown as dotted lines in Fig. 2(c) for twodifferent probe Rabi frequencies (black and blue). Forstrong RF fields the EIT transmission on resonance issuppressed and can reach the baseline, which is given bythe absorption of the probe beam without RF and cou-pling field. The development of peak contrast with probepower ( ∝ Ω p ) can be qualitatively explained as follows:For larger Ω p , power broadening leads to an increase ofthe spectral width of the atomic absorption line, result-ing in an increased transmission of the probe beam anda rising baseline transmission. At the same time, for afixed Rabi frequency of the coupling field, Ω c , and weakΩ p the population of the meta-stable Rydberg state in-creases with Ω p , since the population ratio between theground and Rydberg state is determined by Ω p / Ω c . Ifless atoms are in the ground state the transmission ofthe probe field increases. Overall this results in a largerEIT peak [21], and the observed initial rise of the SNR dB in Fig. 2(b). For even larger Ω p the EIT peak trans-mission saturates [22] while the two-level baseline trans-mission continues to grow, see Fig. 2(c), explaining thesaturation and slow fall off of the SNR dB . Moreover, inour setup the noise floor on the spectrum analyser risesas quadratic function of the probe power for > µ W,see Fig. 7 in the Appendix A, and limits the achievableSNR dB . The dominant mode of noise in our experimentis introduced by an AOM which is far above that posedby photon shot noise (more discussed in Appendix A).In addition to the probe power, the SNR dB dependson the modulation frequency of the carrier RF field, asshown in Figure 3(a). The fall-off of the SNR dB withincreasing AM frequency sets the bandwidth of the re-ceiver system. We define the BW limit as the cutoff pointwith SNR dB = 10 dB. Figure 3(a) shows a measurementof the BW for a single probe beam and five differentprobe powers. For faster modulations the SNR dB de-creases and reaches its BW limit at 380 kHz for probepowers ≥ µ W. The slope of the curves is deter-mined by atomic parameters such as decoherence ratesset by Doppler broadening and transit times, which arethe same for all curves in Fig. 3(a). Increasing the SNR dB initially improves the cut-off frequency, but as the SNR dB saturates the BW saturates as well.
100 200 300 400 550 700
AM Frequency (kHz) S NR ( d B )
100 200 300 400 550 700
AM Frequency (kHz) S NR ( d B ) (a) (b) Figure 3. (a) SNR dB as a function of AM frequency of the carrier MW field for different power levels of a single probe beam.The plot shows average values of four probe beams with frequency offsets to the resonance of − . − . . . dB as a function of AM frequency for combinations of up tofour probe beams of 25 µ W with frequency offsets to the resonance frequency of f = − . f = − . f =1 . f =5 . f , f , f , f and average values and corresponding standarddeviations are presented. B. Multiple Atomic Receivers
To overcome the constraints in SNR dB and BW dis-cribed in Sec. III A we distribute the power of our singleprobe beam over multiple probe beams. In combinationwith a single coupling beam we obtain independent de-tection volumes within our vapour cell. We use AOMdriving frequencies with ∆ f = 3 − ∼ µ min the focal plane, see Fig. 1(c). By this means we obtaina single-input-multi-output (SIMO) configuration, as itis used in the context of smart antenna technology forimproved wireless communication performance. Addingmultiple antennas at both the transmitter and receiverhas proven to offer significant increases in data through-put [23]. Figure 3(b) shows the SNR dB versus AM fre-quency for one, two, three and four probe beams withindividual powers of 25 µ W at their probe frequencies f to f . The SNR dB drops towards higher AM frequency,similar to the scenario for a single receiver in Fig. 3(a),since the slope of the curves is determined by the atomicparameters of our experimental setup. Crucially, how-ever, the saturation of the SNR dB for a single beam canbe avoided. This allows us to exceed the BW limit ofthe single beam setup, as shown in Fig. 4. While for twobeams the BW limit approximately matches the scenarioof a single beam with twice the power, a clear improve-ment in BW appears for N ≥
3, where N is the numberof beams. The saturation in SNR dB for the single beam setup for probe powers > µ W can be avoided. Forthree and four beams the BW limit is extended to ∼
480 kHz and ∼
560 kHz respectively, exceeding the max-imum bandwidth of ∼
380 kHz of a single beam.Assuming that two receiver areas are independent andhave identical parameters such as Ω p , Ω c , beam frequen-cies, waists, and atomic densities, the signal amplitudeincreases by a factor of two. We note that doubling theoptical power that contributes to the signal increases theSNR dB by a factor of 4 (roughly 6 dB), as e.g. found inFig. 3(a) (black to blue line). This is because the opti-cal power P op hitting the photodetector translates intoa voltage V ∝ P op , and the spectrum analyser measuresthe corresponding electric power P el , for which P el ∝ V .In Fig. 4(a) we show the increase in BW for up tofour beams. As expected (see Appendix B) the BWscales logarithmically (black line). For the single receiverthe bandwidth saturates around ∼
380 kHz (blue shadedarea). In the context of Rydberg receivers the achievabledata capacity for N receiver volumes at a given ampli-tude modulation frequency f AM is given by Eq. (2) as C A ( N ) = f AM × log (1 + N × SNR N =1 ) . (3)Figure 4(b) presents the data capacity for three differ-ent modulation frequencies and up to four beams. Thedata capacity shows significant enhancement, followingthe predicted behaviour of Eq. (3) (dashed lines). More-over the data capacity depends on the amplitude modula- Figure 4. (a) 10 dB bandwidth limit of a single probe beam as function of the probe power (blue) and for multiple beams(red), where the probe power is distributed equally among N beams. The BW increases with logarithmically (black line), andsurpasses the BW limit of the single beam (shaded area). (b) Data capacity for three different AM frequencies and up to fourprobe beams. The dashed lines show the scaling of Eq. (3). (c) Data capacity versus amplitude modulation frequency for N beams. tion frequency, as presented in Fig. 4(c). For low AM fre-quencies the data capacity has a linear dependence on themodulation frequency. For faster modulations the datacapacity reaches a maximum, before decreasing due tothe decreasing SNR dB caused by a finite atom-switchingtime. The maximum data capacity moves towards highermodulation frequencies with the number of probe beams, N , as the SNR dB increases with N . C. Characterisation of the Distributed ReceiversSetup
1. Deviations of EIT peak heights for different receivervolumes
Unlike in an idealized scenario with identical dis-tributed receivers, in our experimental setup the re-ceiver volumes differ slightly due to the geometry of thesetup and since the probe power is distributed via anAOM. Therefore, we show average values of all four probebeam combinations in Figs. 3 and 4. Deviations betweenthe detected SNR dB for different probe frequencies arecaused by a change in the EIT condition. The main con-tribution comes from slightly different beam waists in theoverlap area, affecting the ratio Ω c / Ω p . To illustrate thechange of the EIT transmission for different probe fre-quencies, we present in Fig. 5 the EIT profiles for a scanof the coupling beam frequency. The AOM-offset of theprobe frequency with respect to the two-level resonancefrequency is denoted as ∆. For all frequencies we observetwo EIT peaks, since the EIT condition can be met forthe 52 D / and 52 D / Rydberg states. Relevant in thisinvestigation is the larger peak. For a fixed probe powerof 300 µ W the height of the EIT peak changes with ∆,since the position of the probe beam in the focal planemoves as ∼ ∆ × µ m / MHz. Considering the angle be-tween the counter-propagating probe and coupling beam, and a scan of ∆ = 40 MHz the overlap areas are separatedby up to 40 mm, which is approximately twice as largeas the Rayleigh lengths of the probe and coupling beam.The appearance of two maxima at ∆ = − . . f to f of Fig. 3(b) the EIT peak height differs by 10 %for a fixed probe power, which translates to deviations inthe SNR dB . This technical limitation can be mitigated,for example by adjusting the probe power of individualbeams. C oup li ng F r equen cy ( M H z ) ( M H z ) T r an s m i ss i on ( a r b . un i t s ) -20000.20.4 200.6 150 1000.8 50 01 -50 -100 = -10.5 MHz = 20.5 MHz Figure 5. Probe transmission for different AOM driving fre-quencies ∆ (EIT resonance at ∆ = 0) at a fixed probe powerof of 300 µ W. The coupling laser frequency is scanned and thefrequency axis calibrated using the known splittings betweenthe states 52 D / (large peaks) and 52 D / (smaller peaks onthe left side).
2. SNR dB for different probe beam frequencies With the objective to find optimised probe frequen-cies for our system, the coupling laser was locked tothe 5 P / ↔ D / transition and the probe frequencyswept over a 40 MHz range using the AOM. In differ-ence to the measurements in Fig. 5, we allow for changesin the probe power with AOM driving frequency, whichis caused by the dependence of the AOM efficiency onthe driving frequency. The SNR dB was determined as afunction of the AOM frequency (probe frequency) for anamplitude modulated MW field at 100 kHz. The resultsare presented in Fig. 6(a)-(c) as blue data points. In ourfrequency scan, we observe two minima (∆ = −
12 MHzand ∆ = 5 MHz), which can be associated with pointsfor which the EIT transmission does not change in pres-ence of the MW field. High SNR dB occurs for thelargest change in probe transmission due to the appliedMW field. For a symmetric AT profile, as depicted inFig. 6(d), we expect a symmetric scenario in Fig. 6(a)-(c) with three peaks with a vertical line symmetry at theresonance frequency ∆ = 0, and e.g. observed in [15]. Anumber of factors can lead to the observed asymmetry inFig. 6(a)-(c). First, with our MW electric field we intendto couple the Rydberg states 52 D / ↔ F / whichhas transition dipole moments up to 2260 a e. However,a second Rydberg state, 51 F / , is located only 1 . < −
10 MHz, as shown in Fig. 5, due to weaker overlapof coupling and probe beam. This explains the differencein height of the left and right peak in Fig. 6(a)-(c).
3. Effect of adding a second probe beam
In order to characterise the performance with multi-ple probe beams, we scanned the frequency of the firstprobe beam in presence of a second probe beam at threefixed frequencies -3 . . . dB of the two beams by combining thetwo probe signals on the photodetector. The obtainedvalues of SNR dB are presented in red in Fig. 6(a)-(c).One could naively expect the SNR dB to remain at leastat the value given by the fixed-frequency beam, depictedin Fig. 6(a)-(c) as horizontal black lines. Instead, fortwo probe beams we predominantly observe a drop inthe SNR dB if the two probe frequencies are in differentlyshaded frequency areas (grey and blue), and a rise fortwo frequencies in identically shaded areas. A descriptiveexplanation for this phenomenon is given in Fig. 6(d).While the grey and blue areas both show a significantchange of the EIT signal in presence of a MW electricfield, for grey areas the probe transmission increases inheight, whereas the signal drops in the blue area. By de-tecting both beams on the same photodetector the trans- Figure 6. SNR dB as function of the AOM frequency for a sin-gle beam (blue) and an additional beam (red) at (a) -3 . . . dB of the beam at frequencies (a)-(c) isrepresented as horizontal black line. A schematic representa-tion of the EIT spectrum with and without MW field for ascan of the coupling laser frequency is shown in (d). mission signals from two beams can be effectively “out ofphase”. If both components are equally strong, a zero oc-curs in the SNR dB . Otherwise, the stronger signal dom-inates the scenario. Hence, for our experimental setupit is crucial to choose frequencies within identically col-ored areas. All beams can then be detected with a simpledetection scheme, involving only a single photodetector.Alternatively, the probe transmission signal can be de-tected with individual photodetectors to allow for lesssensitivity in the choice of probe frequencies. For theSNR dB measurements of up to four distributed beams,we used a separation of 3 − dB levels, as well asspatially separated overlap areas.We note that the frequency deviations of the probebeams in or setup could be of potential benefit for anatom-based multi-input-multi-output (MIMO) configu-ration. These configurations use spatial multiplexing andallow for an even larger increase in data capacity pro-portional to N , where N is the number of transmitterand receive antennas. For the generation of multipleprobe beams with identical frequencies electromechani-cally driven mirrors or liquid crystal deflectors could beimplemented. IV. DISCUSSION
The fastest switching time in an atom-based receiveris given by the contrast of optical transmission and thedetection noise. Using purely classical light sources, thenoise floor is ultimately limited by the photon shot noise,while the transmission contrast is determined by the life-time of a single atom in a dark state, the number ofparticipating atoms in the EIT process, and the inputprobe beam power. The atom-switching time in our re-alisation is primarily limited by the EIT pumping rateΩ
EIT = Ω c /
2Γ [20], which describes the time atoms needto re-establish the EIT dark state when the MW field isturned off.For a given realisation with a particular atom-switching time, one expects that increasing the probepower or the number of the participating atoms withinthe receiver volume would lead to a larger contrast inoptical transmission, and therefore increase the SNR. Inthis paper we show that the SNR saturates beyond acertain probe power, but by spatially distributing theprobe power to address different atoms, one can increasethe SNR, which ultimately leads to an increase of thedata capacity. Our results demonstrate a scaling of thedata capacity with C SIMO = BW × log (1 + N × SNR)for N probe beams. The capacity advantage of the dis-tributed receiver stems from the reduction of the dephas-ing processes. These are caused by high atomic densitiesin the Rydberg state due to a high density of participat-ing atoms and due to high probe beam power.Our results particularly highlight the challenge of scal-ing up the data capacity of Rydberg atomic receiversby classical means, such as scaling up the probe vol-ume. To put this in perspective, in order to increasethe data capacity by an order of magnitude, one needsto implement more than thirty distributed receivers withthe receiver volume larger by the same ratio. Assuminga probe volume of a single receiver to be ∼ . , adistributed receiver with a volume of ∼ would leadto an increase of the data capacity by a factor of ∼ V. CONCLUSION AND OUTLOOK
We have considered a simple setup employing spa-tially distributed atomic receivers that allows to surpassthe SNR limit of an individual atom-based receiver, re-sulting in an improved channel capacity. The conceptof receiver arrays has proven beneficial in wireless com-munication systems for high-speed transmission and in-creased capacity. For example for a SIMO system, aspresented in this work, an increase in channel capacityto C SIMO = BW × log (1+ N × SNR) [19] for N receivingantennas has been derived. We experimentally confirmedthe same scaling for the channel capacity of our atomic-receiver system with N probe beams (receiver volumes).Moreover, we observed a significant increase in SNR dB ,and BW – the latter scaling logarithmically for our Ryd-berg atom-based RF-to-optical receiver. We showed thatsuch enhancement of the BW cannot be obtained by asingle receiver with a similar level of combined opticalpower due to the saturation of the power signal to noiseratio. Our approach benefits from the little additionalresource overhead needed to add multiple probe beams,if generated by the first order of an AOM and a multi-tone frequency source. While the results shown in thiswork were carried out with N ≤ × log (1 + N × SNR). This however requires thatthe spatial separation between the distributed atomic re-ceivers to be in the order of the RF wavelength [23].
ACKNOWLEDGMENTS
We acknowledge funding from the Marsden Fund ofNew Zealand (Contract No. UOO1729) and MBIE (Con-tract No. UOOX1915).
Appendix A: Noise assessment of the AOM
The key component for the generation of multipleprobe beams in our experimental setup is a continuouslyrunning AOM, see Fig. 1(a). This AOM is the mainsource for noise, degrading our SNR dB measurements.In general, two sources of noise can be induced by an N o i s e P o w e r ( m W ) -10 Spectrum AnalyzerDetectorWithout AOMWith AOM S NR ( d B ) measurementshot noise scenario (a)(b) Figure 7. (a) Measurement of the noise power on the spectrumanalyser over a frequency band of 50 to 350 kHz for a reso-lution bandwidth of 3 kHz and different probe powers. Thedata points show average values over the frequency range forthe first order of the AOM (red) and for a measurement with-out AOM (blue). The noise floor of the photodetector (blackline) lies ∼ . dB as function ofprobe power as shown for our setup in Fig. 2. If only shotnoise is present (black), the SNR dB saturates at ∼
32 dB.
AOM: phase noise from the AOM driver and RF am-plifier, and intensity noise. The intensity noise, whichoriginates from relative diffraction efficiency fluctuations(RDEF) [26], dominates in our setup. A measurement ofthe noise power on the spectrum analyser over a range of50 to 250 kHz with a resolution bandwidth of 3 kHz andfor different probe beam powers is shown in Fig. 7(a).The red data points represent a measurement in whichthe first order of the AOM ( ∼
65 % efficiency) was de-tected on the photodetector (Thorlabs PDA 100A-EC, bandwith up to 2 . > dB as depicted in Fig. 7(b) (blue points). Towardshigher probe powers the peak height of the AM signal,see Fig. 2(a), saturates while the noisefloor increasesquadratically with the probe power, resulting in a fall-off of the SNR dB . If we compensate for the increasingnoisefloor and only consider shot noise (black points)the SNR dB saturates at ∼
32 dB, and overall a higherSNR dB can be obtained. In order to reduce the inten-sity noise of the AOM the RF driving power of the AOMcan be set close to its saturation point. To further re-duce the noise, alternative options to generate multiplebeams can be employed, as discussed in Sec. III C 3. Fora photon-shot-noise-limited measurement an optical het-erodyne detection scheme can be employed, where a localoscillator beam is mixed with the transmitted probe ase.g. used in [20]. Appendix B: BW and data capacity for multiplereceivers
For a SIMO configuration with N identical receivervolumes (i.e. N probe beams) the signal-to-noise ratio indecibel is given by bySNR dB ( N ) = SNR dB ,N =1 + 20 × log ( N ) . (B1)This relates to an improvement of bandwidth with N asBW( N ) = BW N =1 + 20 /m × log ( N ) , (B2)where m is the slope defined by the falloff of the curvestowards higher AM frequencies in Figs. 3(a)-(b). Theslope, m , depends on atomic parameters, such as coher-ence rates and vary for different experimental setups.The achievable communication rate for a channel at agiven amplitude modulation frequency f AM is given bythe Shannon-Hartley theorem, Eq. (1), as C = f AM × log (1 + SNR). Equation (B1) can be used to derive thetheoretical data capacity for N receiver volumes C A ( N ) = f AM × log [1 + SNR( N )]= f AM × log [1 + 10 SNR dB ( N ) / ] B1 = f AM × log [1 + N × SNR N =1 ] . (B3) [1] J. Sedlacek, A. Schwettmann, H. K¨ubler, R. L¨ow,T. Pfau, and J. 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