Quantum interference of a single spin excitation with a macroscopic atomic ensemble
S. L. Christensen, J.-B. Béguin, E. Bookjans, H. L. Sørensen, J. H. Müller, J. Appel, E. S. Polzik
QQuantum interference of a single spin excitation with a macroscopic atomic ensemble
S. L. Christensen, J.-B. B´eguin, E. Bookjans, H. L. Sørensen, J. H. M¨uller, J. Appel, ∗ and E. S. Polzik ∗ QUANTOP, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
We report on the observation of quantum interference of a collective single spin excitation with aspin ensemble of N a ≈ atoms. Detection of a single photon scattered from the atoms createsthe single spin excitation, a Fock state embedded in the collective spin of the ensemble. Thestate of the atomic ensemble is then detected by a quantum nondemolition measurement of thecollective spin. A macroscopic difference of the order of √ N a in the marginal distribution of thecollective spin state arises from the interference between the single excited spin and N a atoms. Thesehybrid discrete-continuous manipulation and measurement procedures of collective spin states in anatomic ensemble pave the road towards generation of even more exotic ensemble states for quantuminformation processing, precision measurements, and communication. PACS numbers: 42.50.Dv, 42.50.Lc, 03.67.Bg
I. INTRODUCTION
The development of interfaces between quantum sys-tems plays a large role in present-day quantum infor-mation research. One of the most used interfaces isbased on the interaction between light and atomic en-sembles [1, 2]. Until now, predominantly, two differentapproaches based on either discrete or continuous vari-ables have been used. The discrete method is based oncollective single excitations, photon counting, and map-ping of the atomic state into a photonic state which is thencharacterized [2–6]. The continuous-variable schemes useatomic homodyne measurements which allow for deter-ministic protocols, such as quantum teleportation [1, 7],spin squeezing and atomic tomography [8–10], quantum-assisted metrology [11–13], and quantum memories [1, 14].A general feature of the continuous-variable approach is itshigh-efficiency state characterization and mode selectiv-ity. Hybrid discrete-continuous quantum state generationhas been demonstrated in pure photonic systems [15–19].Here, we report on a hybrid discrete-continuous protocolcombining a collective atomic excitation heralded by asingle-photon count with a continuous measurement ofthe atomic state directly in the ensemble. In combina-tion with quantum nondemolition (QND) measurement-induced squeezing [8], the discrete manipulation of theexcitation number allows for creation of Schr¨odinger’s catstates [15] within a quantum memory which are a val-ued resource for quantum repeater protocols [20]. Statescreated by this method can improve measurements be-yond the standard quantum limit [21]. The experimentpresented here unifies two main approaches to atom-lightquantum interfaces: first, a single excitation is generatedvia a Raman-type process (where a direct retrieval wouldresult in a single photon in the output mode) [5]; then, aFaraday-type (QND) memory readout [1] of the resultinginterference is performed. ∗ Corresponding Authors:[email protected]; [email protected]
II. THEORY
Our experiment is conducted on an atomic ensembleof pseudo-spin-1 / | Ψ (cid:105) = |↑↑ . . . ↑↑(cid:105) . (ii) A single spin is probabilisticallyflipped into the opposite state, without resolving whichatom was affected such that the excitation is distributedover the ensemble. The system state becomes | Ψ (cid:105) ≡ √ N a N a (cid:88) l =1 |↑↑ . . . ↑ ↓ (cid:122) (cid:125)(cid:124) (cid:123) l th atom ↑ . . . ↑↑(cid:105) . (1)(iii)) A π/ |↑(cid:105) → | + (cid:105) and |↓(cid:105) → |−(cid:105) with |±(cid:105) ≡ |↑(cid:105)±|↓(cid:105)√ . Depending on thepresence or absence of the spin flip, this leaves the systemin one of the two following states: | Ψ (cid:48) (cid:105) = | + + . . . + + (cid:105) , (2) | Ψ (cid:48) (cid:105) = 1 √ N a N a (cid:88) l =1 | + + . . . + − (cid:122) (cid:125)(cid:124) (cid:123) l th atom + . . . + + (cid:105) . (3)(iv) Atomic state analysis is performed by measuringthe population difference of atoms in the two spin states∆ N = N ↑ − N ↓ .An interesting and somewhat counterintuitive observa-tion is that the probability distribution of the measure-ment outcome is fundamentally altered whenever a singlespin-flip has taken place. The magnitude of the differencebetween the spin-flip and no-spin-flip distributions is com-parable to the atomic quantum projection noise ( ∼ √ N a )and is thus much bigger than an incoherent single-atomeffect. This enhancement is explained by quantum inter-ference between the single excited spin and the unaffectedatoms.The single-excitation state | Ψ (cid:48) (cid:105) has several interestingfeatures: in the limit of a large ensemble, N a (cid:29)
1, thestates | Ψ (cid:48) (cid:105) and | Ψ (cid:48) (cid:105) correspond to the atomic equivalent a r X i v : . [ qu a n t - ph ] A p r Probesµ-wave E xc i t a t i on S i ng l e pho t on (b)(a) (c) | ↑(cid:30) | ↓(cid:30)| e (cid:31) ∆ | ↑(cid:30) | ↓(cid:30)| e (cid:31) FIG. 1. (Color online) The simplified atomic level structureand the collective Bloch sphere at different stages of the exper-iment. (a) All atoms are prepared in the |↑(cid:105) state via opticalpumping. (b) Detection of a scattered anti-Stokes photon fol-lowing a weak excitation of the ensemble signals that a singleatom has been transferred to the |↓(cid:105) state. (c) A microwave π/ |↓(cid:105) to interferewith the remaining atoms in |↑(cid:105) by rotating all spins into theequatorial plane. This creates the collective state | Ψ (cid:48) (cid:105) , whichis characterized by a continuous-variable measurement. Theinset shows the probability density of ˆ J z for | Ψ (cid:48) (cid:105) [solid orange]and | Ψ (cid:48) (cid:105) [dashed blue]. of the vacuum and single-excitation states of a bosonicmode (Holstein-Primakoff approximation [23]). Unlikea single-photon state which is superposed with a stronglocal oscillator on a beam splitter to reveal its Wignerfunction [24], in the present case the atomic ensembleplays the role of the local oscillator and is inseparablefrom the single spin carrying the excitation. The state | Ψ (cid:48) (cid:105) is non-Gaussian with a negative Wigner functionstored within a quantum memory. As such it is potentiallyvaluable for quantum information applications [5, 25, 26].This negativity of the Wigner function leads to a non-Gaussian marginal distribution with an increased variancecompared to | Ψ (cid:48) (cid:105) [see inset in Fig. 1c [22]]. It is exactlythis increase that we will use to distinguish between thetwo states of interest. In our experiment various technicalimperfections (detector dark counts and so on.) limit thepurity of the | Ψ (cid:48) (cid:105) -state preparation. As shown in detaillater, this reduces the expected increase in the variance ofthe population difference. Due to the signal enhancementby interference, even for a low-purity state we are able todiscriminate the created state against | Ψ (cid:48) (cid:105) .We employ an ensemble of approximately 10 cesiumatoms; the pseudospin system is formed by two stablelevels in the ground-state hyperfine manifold, the clockstates |↑(cid:105) ≡ | F = 4 , m F = 0 (cid:105) and |↓(cid:105) ≡ | F = 3 , m F = 0 (cid:105) .Each atom l is described by pseudospin operators 2ˆ j ( l ) z = |↑(cid:105)(cid:104)↑| ( l ) − |↓(cid:105)(cid:104)↓| ( l ) , 2ˆ j ( l ) x = | + (cid:105)(cid:104) + | ( l ) − |−(cid:105)(cid:104)−| ( l ) , andˆ j ( l ) y = i [ˆ j ( l ) x , ˆ j ( l ) z ]. Introducing the collective operators ˆ J i = (cid:80) N a l =1 ˆ j ( l ) i , the ensemble state can be visualized on theBloch sphere [see Fig. 1 [27]]. For the characterizationof the interference effect, the observable of interest isˆ J z = ∆ N/ (b) ×4 C a l i b r a t i o n R e p u m p (a) M O T Time E x c i t a t i o n T r a p o ff R e m o v e a t o m s T r a n s f e r d i p o l e π / - p u l s e P r e p a r e | Ψ (cid:31) N a ∆ N FIG. 2. (Color online) (a) Experimental setup. The dipole-trapped atomic ensemble is overlapped with one arm of a MZI,using dichroic (DC) mirrors. The input mode of the MZI isused for the weak Raman excitation and for the dual-colorQND measurement of atoms. A single-photon counter module(SPCM) detects the heralding photon. To select a photonin the desired decay channel, polarization (via PBS) and fre-quency (via Fabry–P´erot cavities) filters are implemented. Theatomic state is characterized by a dispersive QND measure-ment using balanced homodyne detection. A beam splitter(BS) is used to calibrate the probe power. (b) Pulse sequence.
III. EXPERIMENT
To create the state | Ψ (cid:105) , we first load atoms in amagneto-optical trap (MOT), transfer them into a dipoletrap (formed by a P ≈ . |↓(cid:105) state using the D line [12]. With a microwave π pulse and a subsequentresonant F = 3 → F (cid:48) = 4 optical purifying pulse webring the atoms into the |↑(cid:105) state and remove any re-maining coherences between |↑(cid:105) and (cid:104)↓| [see Fig. 1a].To minimize the inhomogeneous broadening of the op-tical transitions, we briefly turn off the dipole trap andsubject the ensemble to a 2 . µ s off-resonant excitationpulse, detuned by ∆ = 5 . |↑(cid:105) ↔ | e (cid:105) = | F (cid:48) = 4 , m F (cid:48) = 1 (cid:105) transition, focused to awaist of 30 µ m and comprising n exct = 8 . × photons.By independently measuring the reduction of the mi-crowave π pulse contrast due to this excitation pulse, weinfer that 1 − η scatter = 23 % of the atoms scatter a photonfrom this excitation beam (see Appendix C: Scattering);with a probability p forward = 1 .
43 % the atoms forwardscatter a photon with an energy corresponding to a decayto the |↓(cid:105) state exactly into the detection spatial mode.The detection of a single |↑(cid:105) → F (cid:48) = 4 → |↓(cid:105) anti-Stokesphoton signals that a single atom has been transferredto the |↓(cid:105) state and thus heralds the preparation of the | Ψ (cid:105) state [see Fig. 1b [5]]. Using a microwave π/ |↑(cid:105) -state atoms [see Fig. 1c] and reestablish thedipole trap.The atoms are held in one arm of a Mach-Zehnderinterferometer (MZI) [see Fig. 2a], which allows us tomeasure ˆ J z ∝ ∆ N by dual-color QND tomography [12]with a precision much better than the projection noiseusing n probe = 1 . × photons in total [see Fig. 1c].We then repump all atoms into F = 4 and determine N a ,by again measuring the atomic induced phase shift [8].Depending on the detection of the heralding anti-Stokesphoton, the measurement outcomes are associated withˆ J z statistics of either the | Ψ (cid:48) (cid:105) or | Ψ (cid:48) (cid:105) states. To optimizethe measurement time we reuse the same MOT cloudfour times, allowing us to measure the atomic state forvarying atom numbers. Finally, the atoms are removedfrom the trap using resonant light, and calibration mea-surements are performed [see Fig. 2b]. To obtain therequired statistics the experiment is repeated more than ahundred thousand times. The atomic tomography methoddescribed above is an atomic analog of homodyne detec-tion of optical fields [24]: the strong local oscillator field isrepresented by the large number of atoms in the |↑(cid:105) state,and the quantum field is formed by the single |↓(cid:105) stateatom. The 50:50 beam splitter is realized by the π/ N .In order to enhance the probability of forward scatter-ing a photon in the desired channel |↑(cid:105) → F (cid:48) = 4 → |↓(cid:105) a bias magnetic field of B = 20 . z direction[see Fig. 2a] is applied [22]. Polarization and frequencyfiltering in the heralding photon path (see Fig. 2a) is usedto discriminate the unwanted decay channels originatingfrom F (cid:48) = 4. A polarizing beam splitter cube (PBS),attenuating π -polarized light by 1 / (7 × ), suppressesanti-Stokes photons leading to | F = 3 , , m F = ± (cid:105) . Twoconsecutive Fabry–P´erot cavities with a finesse of F = 300and a linewidth of δν c = 26 MHz filter out photons corre-sponding to decays into F = 4 states with a contrast of1 / (2 . × ). The decays into | F = 3 , m F = ± (cid:105) cannotbe filtered out and present a limitation on the purity ofthe state [22]. IV. ANALYSIS
The variable ˆ J z ∝ ∆ N is determined from the differ-ential phase shift ˜ φ i imprinted by the atoms onto twocollinear laser beams of different frequency in a MZI [12].The first step of the state analysis is to calibrate theoptical phase fluctuations to the atomic projection noise.Each measurement of ˜ φ i is referenced to an optimallyweighted average of 12 measurements on an ensemble in | Ψ (cid:48) (cid:105) in order to reduce the effect of slow drifts in probepowers. A small additional amount (9 %) of light shotnoise and atomic projection noise from these referencemeasurements causes a slight decrease of the detectionefficiency [see Fig. 3 and Appendix A: Technical fluctua-tions].A noise scaling analysis [8] confirms a predominantlinear scaling of the atomic noise with N a , which is char-acteristic for the atomic projection noise [see Fig. 3]. This ABA BA FIG. 3. (Color online) Variance of the measured optical phaseshift φ as a function of the atom number. Different noisecontributions are distinguished by a scaling analysis. Thedominating linear part (dashed line) corresponds to atomicquantum projection noise; the quadratic and constant con-tributions (A) originate from technical fluctuations and lightshot noise. Nine percent of the projection noise originatesfrom noise-canceling reference measurements; the remainingfraction (B) constitutes η noise = 50 % of the total noise. linear scaling with N a is analogous to linear noise scalingwith the local oscillator power in photonic homodynemeasurements. For larger N a , we observe classical noisewith its quadratic scaling. The main contribution tothis noise comes from frequency fluctuations of the ex-citation pulse which cause classical fluctuations in theatom number difference between the F = 3 and F = 4hyperfine manifolds. Additionally, in the bias magneticfield the |↑(cid:105) ↔ |↓(cid:105) transition frequency becomes sensitiveto magnetic fields (17 . J z probability distributions of states | Ψ (cid:48) (cid:105) and | Ψ (cid:48) (cid:105) . Here we include only the data with N a > × ,where the probability of detecting the heralding photon isthe highest. In order to compensate for slow drifts of thelight-atom coupling strength, we introduce a noise nor-malization procedure and divide each ∆ N measurementby the standard deviation of the neighboring M = 200measurement outcomes. This allows us to locally normal-ize the variance to unity for events where no heraldingphoton was detected. The results for the normalized vari-ances Z for the two states as a function of the number ofsamples are presented in Fig. 4. We findvar( Z no click ) = 1 . ± .
02 (4)var( Z click ) = 1 . ± . , (5)where the errors correspond to one standard deviation ofthe variance estimator (see Appendix B: State discrimi-nation). As expected from our normalization procedure,in the case of no heralding photon, we obtain the unityvariance. In the case of the presence of the heralding FIG. 4. (Color online) Cumulative statistics for the variance ofthe measurement outcomes of ∆ N for the two created states,showing an increased variance for heralding events. The samplevariance is plotted against the number of observations witha heralding photon (no heralding photon) as depicted on thetop (bottom) axis. photon, a statistically significant increase of 24 % in thevariance is observed.Heralding errors convert the pure target state into astatistical mixture described by a density operatorˆ ρ = p | Ψ (cid:48) (cid:105) (cid:104) Ψ (cid:48) | + (1 − p ) | Ψ (cid:48) (cid:105) (cid:104) Ψ (cid:48) | . (6)Here p is the classical probability that | Ψ (cid:48) (cid:105) is actuallyprepared when a photon is detected. For state ˆ ρ we wouldexpectvar(∆ N ) ˆ ρ = p (cid:104) Ψ (cid:48) | J z | Ψ (cid:48) (cid:105) + (1 − p ) (cid:104) Ψ (cid:48) | J z | Ψ (cid:48) (cid:105) = 3 pN a + (1 − p ) N a . (7)Heralding errors which reduce the purity of the stateinclude the detector dark counts with p dark = 0 . p click ,leakage of the excitation pulse through the filters with p exct = 0 . p click , and unfiltered photons originating fromthe decay into | F = 3 , m F = ± (cid:105) states with p decay =0 . p click . Here p click = 6 . × − is the observed photoncounting probability per excitation pulse. For the purityof the created state we find p state = 1 − p dark + p decay + p exct p click = 38 % . (8)With a stronger suppression of the false-positive eventsby better filtering cavities a state purity exceeding 70 %can be foreseen.The quantum efficiency of the atomic state detection isfinite due to several effects which add state-independentGaussian noise. It is well known [28] that when homodynequadrature measurements are normalized to a vacuumstate with added uncorrelated Gaussian noise, this effec-tively decreases the quantum efficiency η Q of the detection.This decrease can be modeled by assuming a vacuum ad-mixture of 1 − η Q followed by an ideal detection. In our experiment such additional noise that is uncorrelatedwith the quantum state of interest [red areas (dark gray)in Fig. 3] originates from the electronic detector noise,photon shot noise [22, 29], classical fluctuations in theatomic state initialization, and the noise from the 12reference measurements (Appendix A: Technical fluctua-tions). These noise sources lead to an effective detectionquantum efficiency of η noise = 50 % as indicated in Fig. 3.The non-perfect overlap between the excitation and thephoton-collection modes contributes η mm = 75 %. Spon-taneous emission of photons into modes which do notinterfere with the single excitation acts similar to theimperfect spatial overlap with the local oscillator in pho-tonic homodyning and leads to the factor η scatter = 77 %.Finally, the phase mismatch between the prepared spinwave and the detection mode caused by the inhomoge-neous ac-Stark shift induced by the excitation beam andthe refractive index of the atomic ensemble leads to a mi-nor correction of η inhom = 92 % (see Appendix D: Phasemismatch). The total efficiency of the state detectionis given by η Q = η noise η mm η inhom η scatter = 27 %. Theexpected variance of the created state can then be foundas var(∆ N ) ˆ ρ /N a = 3 p expect + (1 − p expect ) = 1 .
20, where p expect = p state η Q , in good agreement with the experi-mental value.The contribution of multiple excitations of the spin-wave mode which, in principle, can strongly affect theresults [30, 31] is negligible in our experiment. This is aresult of a relatively high probability to detect photonsscattered into the detection mode of p d = 18 % combinedwith a photon-number-resolving detector (the detectordead time of 50 ns is short compared to the 2 . µ s excita-tion pulse duration). A detailed calculation (Appendix E:Multiple spin-wave excitations) reveals that, on the con-dition of a heralding photon, the probability to find morethan one atomic excitation is p ( n > | × − which increases var( Z click ) by 2 %. The two-excitationcontribution amounts to less than 17 % of that of a coher-ent spin state with the same mean excitation number. V. CONCLUSION
In conclusion, we have implemented a hybrid discrete-continuous protocol where a collective single spin exci-tation heralded by the detection of a single photon ischaracterized by a direct measurement of a collectivecontinuous-variable atomic operator. Although, in gen-eral, an observed increase in the variance of the atomicoperator could be due to classical reasons, such as anadmixture of a thermal state, in our experiment theincrease is solely due to the detection of a scatteredsingle photon. Even stronger evidence of the success-ful generation of a single excitation state requires deter-mination of higher-order statistical moments, which inturn demands a higher purity of the produced quantumstate [24, 32, 33]. Steps towards this goal could includea stronger light-atom coupling achievable in ensemblestrapped around nanofibers [34, 35], atoms coupled to op-tical resonators [9, 36, 37], or photonic structures [38, 39].Furthermore, a better suppression of false photon countsand using atoms with a simpler level structure (e.g. Rb)could help. These improvements of the method shouldallow for observation of a negative Wigner function of amacroscopic atomic ensemble, certifying its non-classicalproperties [26]. Such a state is a building block for atomicSchr¨odinger’s cat states [20]; it can be used in precisionmeasurements [21, 40] and provides a non-Gaussian re-source for future quantum information processing [25].
ACKNOWLEDGMENTS
This work is funded by the Danish National ResearchFoundation, EU projects SIQS, MALICIA, ERC grantINTERFACE, and DARPA through the project QuASAR.We thank Emil Zeuthen and Anders S. Sørensen for helpfuldiscussions.
Appendix A: Technical fluctuations
Using the dual-color QND measurement we prepareand probe our ensemble, obtaining measurement out-comes ˜ φ i . To eliminate technical fluctuations, we subtractthe baseline of the empty interferometer. Further noisereduction is achieved by performing 12 reference mea-surements { ϕ ji } j ∈{− ,... − , ,... } on a | Ψ (cid:48) (cid:105) state, six eachimmediately before and after ˜ φ i is measured. Since thesemeasurements are performed on independently preparedatomic ensembles, all correlations between them are oftechnical nature. We therefore decorrelate ˜ φ i from itsreference measurements ϕ ji by subtracting the correlatednoise contributions: φ i = ˜ φ i − (cid:88) j = − j (cid:54) =0 w j ϕ ji . (A1)The 12 weight factors w j are chosen such that the samplevariance var( { φ i } ) is minimized. Since each referencemeasurement contains both the full (uncorrelated) atomicprojection and shot noise, this procedure not only reducestechnical fluctuations but also adds (cid:80) j w j = 0 .
09 units ofprojection and shot noise to each φ i measurement. Thisdecreases the state detection efficiency η noise , as explainedin Sec. IV. Analysis. The above choice of w j guaranteesan optimization of this trade-off. Appendix B: State discrimination
To compare the measurement statistics for the twocreated states | Ψ (cid:48) (cid:105) and | Ψ (cid:48) (cid:105) , we only consider data with N a > × . For these high-atom-number realizationswe obtain a high η noise , which directly leads to a large increase in the difference of variances, as explained inSec. IV. Analysis.As our data are acquired over a duration of 2 weeks,we observe slow, long-term changes in the variance ofthe measurements. These are caused mainly by drifts inthe relative optical power of the MOT beams, changes inthe background vapor pressure due to operation of thecesium dispensers, and accumulation of dust particles inthe shared optical path of the strong, focused dipole trap,excitation, and probe beams. We therefore perform alocal noise normalization to avoid long-term drifts in thevariance of our measurements.For each correlation-removed measurement outcome φ i we compute the sample variance of the M surroundingexperiments: Y i ≡ var (cid:0) { φ i − M/ , . . . , φ i + M/ } (cid:1) , (B1)and we use this to normalize the variance of φ i to thesurrounding data points: Z i ≡ φ i √ Y i . (B2)The Z i originating from a | Ψ (cid:48) (cid:105) measurement, by this con-struction, have an average variance of ≈ p click (cid:28) Y i contain almost entirely no-click events.Our final parameter of interest is the sample varianceof a set of Z i W L ≡ var ( { Z i } i ∈ L ) , (B3)both for the sets of indices L click = { i : click detected } and L no click = { i : no click } . In the following we focuson estimating the statistical uncertainty δW L on W L anddenote with L = | L | the number of samples used in thecalculation.Due to the finite number of points used in estimating Y i there is a statistical uncertainty on this estimator whichcarries over to Z i . Since all φ i from within the range[ i − M/ , . . . , i + M/
2] can be considered independentlyand identically distributed, we can give the uncertainty of Y i simply by the mean-square error (MSE) of the varianceestimator and find δY i = (cid:114) M − Y i . (B4)This allows us to find the variance of each of the Z i byTaylor expansion around (cid:104) Y i (cid:105) asvar( Z i ) = (cid:18) M + 1 (cid:19) var (cid:18) φ i (cid:104) Y i (cid:105) (cid:19) . (B5)One complication is that within a range of M neighbor-ing experiments the Z i are not statistically independentany longer due to our normalization procedure. As canbe seen from (B5), by ensuring that M (cid:29)
1, we can makethe contribution of correlated noise to each Z i negligiblysmall. If, additionally, we either choose L (cid:29) M or ensure R e p u m p M O T Time E x c i t a t i o n T r a p o ff T r a n s f e r d i p o l e π - p u l s e P r e p a r e | Ψ (cid:31) N a FIG. 5. (Color online) Experimental sequence in order toestimate η scatter . that the members of L are spaced much farther than M on average, W L is an unbiased estimator of the Z i variance, and its uncertainty is δW L = (cid:114) L − W L , (B6)which is simply the MSE on the variance estimator for L independent samples.We confirm all our error estimates experimentally bothby dividing our data set into subsets and evaluating thestandard deviation of the variance estimates and by thebootstrapping method (resampling). Appendix C: Scattering
To determine η scatter , the fraction of atoms that scattera photon from the excitation pulse, we perform a separatecalibration experiment [see Fig. 5]. First we prepare allatoms in the |↑(cid:105) state (see Sec. III. Experiment). Whilethe trap is switched off we send the excitation pulse fol-lowed by a microwave π pulse and an optional repumpingpulse, resonant to the F = 3 → F (cid:48) = 4 transition. Finallythe number of atoms N a in F = 4 is determined. Inthe presence of the repumping pulse, all atoms in thetrap are detected, whereas in the absence of the repump-ing pulse only atoms scattered into the | F = 4 , m F (cid:54) = 0 (cid:105) and | F = 3 , m F = 0 (cid:105) states are measured. Finally, usingthe relevant Clebsch-Gordan coefficients to determine thefraction of atoms scattering into each Zeeman state, wecan find the fraction of atoms that undergo a scatteringevent to be 1 − η scatter = 23 %. Appendix D: Phase mismatch
Inhomogeneous phase shifts can reduce the interferencevisibility η inhom = η phase η ac-Stark . Here we consider twoeffects.
1. Longitudinal phase profile
The first effect concerns the refractive index mismatchbetween the scattered single photon and the excitationbeam. In our one-dimensional model we describe this bya position dependent phase difference θ ( y ) = ∆ k y , where∆ k = k µ -wave + k exct − k photon is the wave-vector mismatchbetween microwave-, excitation- and heralding-photonfields. Since no atoms reside in |↓(cid:105) , the atomic phasemismatch emerges exclusively from the optical phase shiftof the excitation beam χ exct = θ ( L a ).For the sake of simplicity we assume a homogeneousatomic density distribution and average the phase overthe length of the atomic ensemble L a : η phase = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L a (cid:90) L a e − iθ ( y ) d y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sinc ( χ exct /
2) (D1)We can relate the excitation beam phase shift χ exct to the measured phase shift χ probe of the QND probeduring the atom number measurement. Let α denotethe resonant optical depth on a closed transition. Then,light traveling through a medium with optical depth α ,detuned by a frequency ∆ i with respect to a transitionwith relative strength ℘ i will experience an optical phaseshift, χ (∆ , α ) = α (cid:88) i ℘ i ∆ i ∆ i + (Γ / . (D2)When all our atoms are pumped into F (cid:48) = 4, withthe probe beam we measure an optical phase shift cor-responding to α = 31, from which, using the Clebsch-Gordan coefficients and the detuning corresponding toour excitation beam, we obtain χ exct = 42 ° , which gives η phase = 95 %.
2. Transversal phase profile
The off-resonant excitation pulse leads to an ac-Starkshift, both of the |↑(cid:105) state and the excited states. Onlythe spatially inhomogeneous shift of the |↑(cid:105) state affectsthe spin wave coherence, which reduces the interferencevisibility. Since the longitudinal extent of our atomicensemble is short compared to the Rayleigh length of thelight beams, we restrict our model to transversal effects,which can be evaluated as [1] η ac-Stark = (cid:82) (cid:12)(cid:12)(cid:82)(cid:82) (cid:37) ( x, z ) I ( x, z ) e − iω LS ( x,z ) t d x d z (cid:12)(cid:12) d tτ (cid:12)(cid:12)(cid:82)(cid:82) (cid:37) ( x, z ) I ( x, z ) d x d z (cid:12)(cid:12) (D3)where I ( x, z ) denotes the transverse Gaussian intensityprofile of the excitation beam, (cid:37) ( x, z ) is the atomic columndensity, ω LS ( x, y ) ∝ I ( x, z ) is the ac-Stark shift of the |↑(cid:105) state, and τ is the excitation pulse duration. Numericallyevaluating the above allows us to estimate the effect andwe find η ac-Stark = 97 %. Appendix E: Multiple spin-wave excitations
When the single photon counter reports a “click”, thiscan originate from dark counts, leakage of excitationphotons, or actual Stokes photons scattered from theatomic ensemble (either from the desired or from otherunwanted transitions). To investigate the influence ofmultiple-Stokes-photon events on our analysis, we calcu-late p ( n | n Stokes photons with an energy cor-responding to a decay to the state |↓(cid:105) exactly into thedetection spatial mode, on the condition of detectinga single click. Since the detector dead time of 50 ns ismuch smaller than the 2 . µ s excitation pulse length, weeffectively have a number-resolving photon detection. ByBayes’ rule we have p ( n | p ( n ) p (1click | n ) p (1click) , (E1)where p ( n ) is the probability to scatter n photons withan energy corresponding to a decay to the state |↓(cid:105) ex-actly into the detection spatial mode and p (1click) is theprobability to detect exactly one click. For a two-modesqueezed vacuum state the photon number statistics inthe individual modes is thermal [41]. Thus the probabilityto find n Stokes photons corresponding to a decay into |↓(cid:105) is given by p S ( n ) = (1 − p ) p n , (E2)where p is the probability to generate at least one of thedesired Stokes photons.Dark counts and leakage photons from the excitationpulse follow a Poisson distribution p DE ( n ) = p f n e − p f n ! , (E3)where p f is the mean number of such false-positive clicksin the absence of atoms. Stokes photons correspondingto decay into | F = 3 , m F = ± (cid:105) are not filtered out andtherefore also cause false positives. Their generation isdistributed as p S2 ( n ) = (1 − p ) p n , (E4)where p is the probability to scatter at least one photoncorresponding to a decay to | F = 3 , m F = ± (cid:105) .Finally, we introduce p d = 0 . × . × .
5, the proba-bility that a single Stokes photon in the detection mode causes an (additional) click which is given by the productof the transmission coefficients through the filter cavi-ties, through other optics, and by the detector quantumefficiency. The complementary probability is denoted˜ p d = 1 − p d .Then the probability for detecting n additional clicksdue to unwanted | F = 3 , m F = ± (cid:105) Stokes photons is p DS2 ( n ) = ∞ (cid:88) k = n p S2 ( k ) (cid:18) kk − n (cid:19) ˜ p d k − n p d n (E5)= (1 − p ) ( p p d ) n (1 − p ˜ p d ) n +1 . (E6)The probability to find no false-positive events is p F (0) = p DE (0) p DS2 (0), whereas the probability to findexactly one false positive is p F (1) = p DE (1) p DS2 (0) + p DE (0) p DS2 (1).With this, p (1click | n ), the probability to detect exactlyone click when the atoms scatter n photons correspondingto the desired |↓(cid:105) decay, is made up of two cases: (a)exactly one of the n photons makes it through the filteringoptical elements and causes a click in the detector whilesimultaneously no false-positive counts are detected, and(b) none of the n photons cause a detector click whileexactly one false positive count is detected: p (1click | n ) = n p d ˜ p d n − p F (0) + ˜ p d n p F (1) . (E7)Using the relation (cid:80) ∞ n =0 p ( n | p (1click) and obtain: p ( n | p d n p n (1 − ˜ p d p ) (cid:16) n p d ˜ p d + p f + p d p − ˜ p d p (cid:17) p f + p d p − ˜ p d p + p d p − ˜ p d p . (E8)Since we know p f = p dark + p exct from reference measure-ments without atoms and p = 0 . p from the ratio of thetransition strengths, we can deduce p = p forward = 0 . p ( n | n : p ( n = 0 | .
606 (E9) p ( n = 1 | .
385 (E10) p ( n = 2 | .
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