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Quantum-state transfer from an ion to a photon A. Stute,1,∗ B. Casabone,1,∗ B. Brandstätter,1 K. Friebe,1 T. E. Northup,1 and R.Blatt1,2 1Institut fu¨r Experimentalphysik, Universita¨t Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria 2O¨sterreichischen Akademie der Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria (Dated: December 11, 2013) A quantum network [1, 2] requires information transfer between distant quantum computers, whichwouldenabledistributedquantuminformationprocessing[3–5]andquantumcommunication [6,7]. Onemodelforsuchanetworkisbasedontheprobabilisticmeasurementoftwophotons,each entangled with a distant atom or atomic ensemble, where the atoms represent quantum computing nodes [8–12]. A second, deterministic model transfers information directly from a first atom onto a cavity photon, which carries it over an optical channel to a second atom [13]; a prototype with neutral atoms has recently been demonstrated [14]. In both cases, the central challenge is to find an efficient transfer process that preserves the coherence of the quantum state. Here, following 3 the second scheme, we map the quantum state of a single ion onto a single photon within an 1 optical cavity. Using an ion allows us to prepare the initial quantum state in a deterministic way 0 [15, 16], while the cavity enables high-efficiency photon generation [17–20]. The mapping process 2 is time-independent, allowing us to characterize the interplay between efficiency and fidelity. As n thetechniquesforcoherentmanipulationandstorageofmultipleionsatasinglequantumnodeare a well established [15, 16], this process offers a promising route toward networks between ion-based J quantum computers. 7 1 Intheoriginalproposalforquantum-statetransfer[13], drivetwosimultaneousRamantransitionsinwhichboth ] aphotonicqubitcomprisesthenumberstates |0(cid:105)and |1(cid:105). states |S(cid:105) and |S(cid:48)(cid:105) are coupled to the same final state h Such a qubit was subsequently employed for the cavity- |D(cid:105) via intermediate states |P(cid:105) and |P(cid:48)(cid:105) (Fig. 1b). One p - based mapping of a coherent state onto an atom [21]. arm of both Raman transitions is driven by a laser, the t n However, due to losses in a realistic optical path, it is second arm is mediated by the cavity field, and a sin- a advantageousinsteadtoencodethequbitwithinadegree gle photon is generated in the process [17–20]. If the u of freedom of a single photon. As a frequency qubit [22] initial state was |S(cid:105), this photon is in a horizontally po- q wouldbechallengingtorealizereversiblywithinacavity, larized state |H(cid:105); if it was |S(cid:48)(cid:105), a vertically polarized [ wechoosethepolarizationdegreeoffreedom. Thetarget photon |V(cid:105) is generated. As the polarization modes of 3 process then maps an electronic superposition of atomic the cavity are degenerate, entanglement of the polariza- v states |S(cid:105)and |S(cid:48)(cid:105)tothepolarizationstate |H(cid:105)and |V(cid:105) tionwiththefrequencydegreeoffreedomisavoided. We 0 of a photon, have recently used a similar Raman process to generate 9 4 ion–photon entanglement [24]. In contrast, here, by cou- 0 (cid:0)cosα|S(cid:105)+eiϕsinα|S(cid:48)(cid:105)(cid:1)⊗ |0(cid:105)−→ plingtwoinitialatomicstatestoonefinalstate,theion’s 1. |D(cid:105)⊗(cid:0)cosα|H(cid:105)+eiϕsinα|V(cid:105)(cid:1), (1) electronic state is transferred coherently to the photon, 0 and no information remains in the ion. The crux of this 3 preserving the superposition’s phase and amplitude, de- mapping problem is to maintain amplitude and phase 1 fined by ϕ and α; |D(cid:105) is a third atomic state. relationships during the transfer process. : v As an atomic qubit, we use two electronic states of a A magnetic field of 4.5 G lifts the degeneracy of elec- i single 40Ca+ ion in a linear Paul trap within an opti- tronic states |S(cid:105) and |S(cid:48)(cid:105), so that the two Raman tran- X cal cavity [23] (Fig. 1a). Any superposition state of the sitions have different resonance frequencies. We there- r a atomic qubit can be deterministically initialized via co- fore apply a phase-stable bichromatic driving field with herent laser manipulations [15, 16], where this initializa- detunings ∆ and ∆ from |P(cid:105) and |P(cid:48)(cid:105), respectively 1 2 tion is independent of the ion’s interaction with the cav- (Fig. 1b). If the difference frequency of the bichromatic ity field. Following optical pumping to the Zeeman state components is equal to the energy splitting of the qubit |S(cid:105)≡ |42S ,m =−1/2(cid:105), the atomic qubit is encoded states |S(cid:105) and |S(cid:48)(cid:105), both Raman transitions are driven 1/2 J inthestates |S(cid:105)and |S(cid:48)(cid:105)≡ |42S ,m =+1/2(cid:105)viatwo resonantly. The Rabi frequencies of the transitions are 1/2 J laserpulsesonthequadrupoletransitionthatcouplesthe determined not only by the field amplitudes Ω and 1 42S and 32D manifolds (Fig. 1b). The length and Ω but also by atomic transition probabilities and by 1/2 5/2 2 phaseofafirstpulseonthe |S(cid:105)↔ |D(cid:105)≡ |32D ,m = the cavity orientation with respect to the magnetic field 5/2 J +1/2(cid:105) transition set the amplitude and phase of the ini- [23]. In order to preserve the amplitudes of the initial tial state. The state is subsequently transferred back to state during mapping, we balance the Raman transition the S manifold via a π-pulse on the |D(cid:105) ↔ |S(cid:48)(cid:105) transi- probabilities to compensate for these factors by setting tion. Ω =2Ω . 1 2 To implement the state-mapping process of Eq. 1, we The mapping process is characterized via process to- 2 mography, in which the bichromatic Raman transition is applied to four orthogonal initial states of the atom: |S(cid:105), |S(cid:48)(cid:105), |S −S(cid:48)(cid:105), |S +iS(cid:48)(cid:105). For each input state, we measure the polarization state of the output photon via state tomography, using three orthogonal measurement settings [25] selected with two waveplates before a po- larizing beamsplitter (Fig. 1a). Avalanche photodiodes detect photons at both beamsplitter output ports. Process tomography extracts the process matrix χ, which parameterizes the map from an arbitrary input density matrix ρ to its corresponding output state in ρ in the basis of the Pauli operators σ ≡ out 0,1,2,3 (cid:80) {1,σ ,σ ,σ }: ρ = χ σ ρ σ . As the ideal map- x y z out i,j i in j i,j ping process preserves the qubit, the overlap χ with 0,0 the identity should be equal to one. We identify χ as 0,0 the process fidelity, which quantifies the success of the mapping. A maximum likelihood reconstruction [26] of χ is plotted in Fig. 2a for a 2 µs window of photons exiting the cavity. Here the matrix element χ indicates 0,0 a process fidelity of (92±2)%, well above the classical thresholdof1/2. Otherdiagonalelementsχ =(3±1)% 1,1 and χ =(4±2)% reveal a minor depolarization of the 3,3 quantum state. Another metric for quantum processes is the mean state fidelity, which evaluates the state fidelities (cid:104)ψi |ρi |ψi (cid:105) for a set of input states |ψi (cid:105), where ρi in out in in out represent the corresponding photon output states. The mean state fidelity can also be directly extracted from the process fidelity for an ideal unitary process [27]. For each of our four input states, state tomography of the output photon is shown in Fig. 2b, using the same pho- FIG. 1. Experimental configuration and mapping se- ton collection window as in Fig. 2a. The corresponding quence. a, A 40Ca+ ion is confined in a linear Paul trap state fidelities are (96±1)% for |S(cid:105), (94±2)% for |S(cid:48)(cid:105), (indicated schematically by two trap tips) and positioned at (97±2)%for |S−S(cid:48)(cid:105),and(95±2)%for |S+iS(cid:48)(cid:105),yield- theantinodeofanopticalcavity. Abichromatic393nmfield ing a mean of (96±1)%. This agrees with the value of drives a pair of Raman transitions, generating a single cav- (95±1)%extractedfromtheprocessfidelityandexceeds ity photon. The beam is linearly polarized in the zˆdirection the classical threshold [27] of 2/3. andpropagatesalongxˆ. Anexternalmagneticfieldischosen paralleltozˆ,orthogonaltothecavityaxisyˆ,inordertodrive We now consider the evolution over time of the pho- π transitions. The measurement basis of photons exiting the tonicoutputstatesρi generatedfromthesefouratomic out cavityissetbyhalf-andquarter-waveplates(L/2,L/4). Pho- input states. In Fig. 2c, we plot the temporal shape of tonsarethenseparatedbyapolarizingbeamsplitter(PBS)for the emitted photon in each of three measurement bases, detection on avalanche photodiodes (APD1, APD2). Tables a total of 12 cases. For each input state, there exists indicatethepolarizationofphotonsatAPD1andAPD2cor- one polarization measurement basis in which photons responding to three measurement bases. b, Two laser pulses would ideally impinge on only one detector. If the ion at 729 nm (1,2) prepare the ion in a superposition of levels S and S(cid:48). This superposition is subsequently mapped onto is prepared in the state |S(cid:105) and measured in the H/V the vertical (V) and horizontal (H) polarization of a cavity polarization basis, for example, the mapping scheme of photon. The 393 nm laser and the cavity field couple S and Fig.1bshouldonlyproducethephotonstate |H(cid:105). How- S(cid:48) to the metastable D level. The bichromatic laser field is ever, a few microseconds after the Raman driving field detuned from levels P and P(cid:48) by ∆ and ∆ (approximately 1 2 is switched on, we see that the photon state |V(cid:105) ap- 400 MHz) and has Rabi frequencies Ω and Ω . 1 2 pears and is generated with increasing probability over thenext55µs. Themechanismhereisoff-resonantexci- tationofthe42P manifoldanddecaytothepreviously 3/2 unpopulated state |S(cid:48)(cid:105), followed by a Raman transition generating the ‘wrong’ polarization. If the ion is prepared in |S(cid:48)(cid:105), the temporal photon shapes are inverted 3 a b c Re(ρ ) Im(ρ ) Ψi H V D A R L out out in 1 100 0.5 Process matrix χ 0 S 50 −0.5 0 n 1 1 100 s bi 0.5 µ 0 S−S´ 50 er 2 0.5 −0.5 0 s p nt 1 100 u o 0 0.5 S+iS´ 50 n c 0 0 oto 1 −0.5 0 Ph 2 3 0 1 2 3 1 100 0.5 S´ 50 0 0 −0.5 0 20 40 600 20 40 600 20 40 60 H H V H V V H V Photon detection time (µs) FIG. 2. Process and state fidelities of the ion–photon mapping. a, Absolute values of the process matrix χ reconstructed from cavity photons detected between 2 µs and 4 µs after the bichromatic field is switched on at time =0. b, State tomographyofphotonsinthesametimewindowforthefourinputstates |S(cid:105),|S−S(cid:48)(cid:105),|S+iS(cid:48)(cid:105),and |S(cid:48)(cid:105),showninrowsfrom toptobottom. c,EachstatetomographycorrespondstomeasurementsinthethreebasesH/V,D/A,andR/L(columns). For eachinputstate,thetemporalshapesofsinglephotonsareplottedinredforpolarizationsH,D,Randinblueforpolarizations V,A,L. In each row, in two of three columns, photons are equally distributed over both detectors, while in the third, photons are generated ideally with a single polarization. Master-equation simulations (red and blue lines) successfully reproduce the observed dynamics. The grey shaded area indicates the time window used for tomography. and symmetric, with the initial state |V(cid:105) followed by photon-detection window. The fidelity initially increases the gradual emergence of |H(cid:105). We have confirmed this becauseatshorttimes(<100ns), photonsareproduced process through master-equation simulations of the ion– primarily via the off-resonant rather than the resonant cavity system, also plotted in Fig. 2c and described in componentoftheRamanprocessandthusarenotinthe the Supplementary Information. target polarization state. This coherent effect, which we For the superposition input states |S −S(cid:48)(cid:105) and |S + haveinvestigatedthroughsimulations,isquicklydamped iS(cid:48)(cid:105), the mapping generates a photon with antidiago- due to the low amplitude of off-resonant Raman transi- √ nal polarization A = (H − V)/ 2 and right-circular tions. The cumulative process fidelity reaches a maxi- √ polarization R = (H + iV)/ 2, respectively. Thus, mumbetween2µsand4µsafterthebichromaticdriving photons impinge predominantly on one detector in the fieldisswitchedon,thetimeintervalusedtoanalyzethe diagonal(D)/antidiagonal(A) and right(R)/left(L)bases, data of Fig. 2a and b. The fidelity then slowly decreases √ √ whereD =(H+V)/ 2andL=(H−iV)/ 2(Fig.2c). as a function of time due to the increased likelihood of Here,asforstates |S(cid:105)and |S(cid:48)(cid:105),photonswiththe‘wrong’ off-resonant scattering. polarization are due to off-resonant scattering before the If all photons detected within 55 µs are taken into ac- mapping occurs. In this case, scattering destroys the count, the process efficiency exceeds 1%, while the pro- phase relationship between the S and S(cid:48) components. cess fidelity of (66 ± 1%) remains above the classical (Note that for eight of the cases in Fig. 2c, the measure- threshold of 1/2. This process efficiency includes losses mentbasisprojectsthephotonpolarizationontothetwo in the cavity mirrors, output path, and detectors. The detection paths with equal probability.) corresponding probability for state transfer within the The accumulation of scattering events over time sug- cavity is 14.7%. A longer detection time window would gests that the best mapping fidelities can be achieved allow transfer probabilities approaching one, but fideli- by taking into account only photons detected within a ties would fall below the classical threshold. Simulations certain time window. Such a window is used for the pre- thatincludetheeffectsofdetectordarkcounts,imperfect ceding analysis of process and state fidelities. For each state initialization, and magnetic-field fluctuations agree attempt to prepare and map the ion’s state, the proba- wellwiththedataofFig.3. Intheabsenceofthesethree bility to detect a photon within this window is 4·10−4, effects, simulations indicate that fidelities of 98% would whichweidentifyastheprocessefficiency. Thisefficiency be possible in our ion–cavity system. canbeincreasedattheexpenseoffidelitybyconsidering Theatomicsuperpositionof |S(cid:105)and |S(cid:48)(cid:105)experiencesa a broader time window. Fig. 3 shows both the cumu- 12.6 MHz Larmor precession, which corresponds to a ro- lative process fidelity and efficiency as a function of the tationofthestates’relativephaseatthisfrequency. One 4 1 a modified bichromatic scheme, a single ion-cavity sys- a y 0.9 tem can act as a deterministic source of photonic cluster elit 0.8 states [29], an essential resource for measurement-based d Data s fi 0.7 Model 1 quantum computation [30]. s ce 0.6 Model 2 We thank T. Monz and P. Schindler for assistance in Pro 0.5 Classical threshold tomography analysis and P. O. Schmidt for early contri- %) 0.4 butions to the experiment design. This work was sup- y ( 1 b ported by the Austrian Science Fund (FWF), the Euro- c en 0.5 pean Commission (AQUTE), the Institut fu¨r Quanten- ci Effi 0 information GmbH, and a Marie Curie International In- 0 10 20 30 40 50 coming Fellowship within the 7th European Framework Photon detection time window (µs) Program. FIG. 3. Time dependence of process fidelity and efficiency. a, Cumulative process fidelity and b, process efficiency are plotted as a function of the photon-detection time SUPPLEMENTARY INFORMATION window,whereerrorbarsrepresentones.d. (SeeSupplemen- tary Information.) A green dashed line indicates the sim- Coupling parameters ulated process fidelity for the same parameters as in Fig. 2. Tothismodel,wenowaddtheeffectsofdetectordarkcounts, imperfectstateinitialization,andmagnetic-fieldfluctuations, The coherent ion-cavity coupling rate is g = 2π × quantified in independent measurements, with the result in- 1.4 MHz. The cavity field decay rate is κ = 2π × dicated by a red line. A fit to the process efficiency is used 0.05 MHz, and the atomic polarization decay rate from to weight the effect of dark counts. The second model agrees the P state is γ = 2π×11.5 MHz. The effective cou- well with the data, while the first one represents achievable 3/2 values for this ion–cavity system. pling strength of the two Raman transitions i = 1,2 is givenbyΩeff =2π×gGiΩi,whereG isageometricfactor i 2∆i i thattakesintoaccountboththerelevantClebsch-Gordon coefficients and the projection of the vacuum-mode po- might expect that as a result, it would not be possible larization onto the atomic dipole moment [23]. The de- to bin data from photons generated from this superposi- tunings ∆ are approximately 400 MHz. We ensure tion across a range of arrival times as described above. 1,2 equal transition probabilities for the two Raman transi- However, because the frequency difference ∆ − ∆ of 1 2 tionsbysettingtheratiooftheRabifrequenciesΩ /Ω = the bichromatic Raman field matches the frequency dif- 1 2 G /G = 2, with absolute values Ω = 2π ×17.5 MHz ference between the two states, the Raman process gen- 2 1 1 and Ω =2π×8.75 MHz. erates a photon that preserves the initial states’ relative 2 phase. Asaresult,thephaseofthephotonsuperposition isindependentofdetectiontime. (SeeSupplementaryIn- formation.) This transfer scheme thus offers advantages Process tomography and measurement bases for any quantum system in which a magnetic field lifts the degeneracy of the states encoding a qubit. Inordertocharacterizethemappingprocess,wecarry Following the deterministic initialization of an atomic out process tomography. For this purpose, we carry out qubit within a cavity, we have shown the coherent map- state tomography of the photonic output state for four pingofitsquantumstateontoasinglephoton. Themap- orthogonal atomic input states [31]. Each state tomog- ping scheme achieves a high process fidelity, and by ac- raphy of a single photonic polarization qubit consists of ceptingcompromisesinfidelity,weincreasetheefficiency measurements in the three bases H/V, D/A, and R/L. of the process within the cavity up to 14%. The transfer Note that H and V correspond to the two cavity modes, measurementisprimarilylimitedbydetectordarkcounts where V is parallel to the zˆ axis of Fig. 1a in the main at 5.6 Hz, imperfect state initialization with a fidelity text and H to the xˆ axis. (In contrast, in Ref. [24], we of 99%, magnetic-field fluctuations corresponding to an defined H rather than V to be parallel to the magnetic atomic coherence time of 110 µs, and the finite strength field axis.) In each basis, we perform two measurements of the ion–cavity coupling in comparison to spontaneous ofequaldurationinwhichtheoutputpathstoavalanche decay rates. While a stronger coupling would improve photodiodes APD1 and APD2 are swapped by rotating the fidelity for a given efficiency, we note that the map- waveplates L/2 and L/4. Summing these measurements ping fidelity in our current intermediate-coupling regime allowsustocompensateforunequaldetectionefficiencies could also be improved by encoding the stationary qubit in the two paths [24]. across multiple ions [28]. A direct application of this The process and density matrices plotted in Fig. 2a bichromatic mapping scheme is state transfer between and b in the main text are reconstructed from the data two remote quantum nodes [13, 14]. Furthermore, via usingamaximumlikelihoodfit[26]. Thedataconsistsof 5 32,400single-photondetectionevents. Weextractfideli- (equation 1 of the main text). As both modes of the tiesandtheirstatisticaluncertaintiesvianon-parametric cavity are degenerate, the phase ϕ of the photonic state bootstrapping assuming a multinomial distribution [32]. remains constant after the transfer. Statistical uncertainties are stated as one standard devi- So far, we have neglected off-resonant Raman tran- ation. sitions, i.e., Ω coupling |S(cid:48),n(cid:105) to |P(cid:48),n(cid:105) and Ω cou- 1 2 pling |S,n(cid:105) to |P,n(cid:105). Taking these couplings into account,thetermsgeff intheHamiltonianareproportional i Time independence to Ωi +Ωjei(ωlj−ωli)t after transformation into the rotating frame and adiabatic elimination of |P,n(cid:105). Here, If the atomic qubit is comprised of nondegenerate the second term, oscillating at |ω −ω |, corresponds to li lj states, Larmor precession will change the qubit’s phase off-resonant Raman transitions in which a photon with over time. For a monochromatic mapping protocol in unwanted polarization is generated. These terms are which the photonic qubit is encoded solely in polariza- neglected in the rotating wave approximation because tion, the phase of the photonic qubit thus depends on |geff| (cid:28) |ω −ω |. These off-resonant coupling terms, i l1 l2 the time of photon generation [33]. In contrast, for the however, explain why the fidelity of the mapping pro- two Raman fields Ω1eiωl1t and Ω2eiωl2t at frequency ωl1 cess only reaches its maximum after about 3 µs (Fig. 3 and ωl2, the mapping pulse can be applied at any time of the main text). As confirmed by our simulations, the forthecorrectchoiceoffrequencydifferencebetweenthe off-resonant Raman processes generate photons with un- two Raman fields ωl1 −ωl2 = ∆ES,S(cid:48)/¯h, where ∆ES,S(cid:48) wantedpolarizationatthetimescaleof100nsafterturn- is the Zeeman splitting between the two qubit states S ingonthedrivelaserpulse. However, theprobabilityfor and S(cid:48). In this case, the atomic qubit is always mapped this process is very low due to the large detuning from to the same photonic qubit state, independent of photon Raman resonance, and the effect is quickly overcome by generation time. the much higher probability of generating photons with To show this, we define a model system consist- the desired polarization thereafter. ing of initial states |S,n(cid:105), |S(cid:48),n(cid:105), intermediate states |P,n(cid:105), |P(cid:48),n(cid:105) and target state |D,n(cid:105) with energies ES,S(cid:48),P,P(cid:48),D = h¯ωS,S(cid:48),P,P(cid:48),D. Here, n = 0,1 denotes the Simulations numberofphotonsineitherofthetwodegeneratecavity modes at energy ¯hωC. A similar model system was used Numerical simulations of the state-mapping process to explain the time independence of the bichromatic en- are based on the Quantum Optics and Computation tanglement protocol that we recently demonstrated [24]. Toolbox for MATLAB [34]. We formulate the master The |S,n(cid:105) ↔ |P,n(cid:105) transition is driven by the field equation for the 18-level 40Ca+ system interacting with Ω1eiωl1t with detuning ∆l1 = ωS −ωP −ωl1, while the two orthogonal modes of an optical cavity. We then nu- |S(cid:48),n(cid:105)↔ |P(cid:48),n(cid:105)transitionisdrivenbythefieldΩ2eiωl2t merically integrate the master equation to obtain the with detuning ∆l2 = ωS(cid:48) −ωP(cid:48) −ωl2. We choose a uni- system’s density matrix as a function of time. The sim- tarytransformationintoarotatingframethattakesinto ulation includes atomic and cavity decay and the laser account the atomic precession at frequency ωS − ωS(cid:48): linewidth. Relativemotionoftheionwithrespecttothe U = e−iωl1t|S(cid:105)(cid:104)S| e−iωl2t|S(cid:48)(cid:105)(cid:104)S(cid:48)|. After this transforma- cavity mode is taken into account by introducing an ef- tion and adiabatic elimination of the state |P,n(cid:105), the fectiveatom-cavitycouplingg smallerthang. This motion Hamiltonian reads motionresultsfromthe(presumablymechanical)oscilla- tion of the ion trap with respect to the cavity [23]. Fur- H/¯h=(ω +ω )|S(cid:105)(cid:104)S| +(ω +ω )|S(cid:48)(cid:105)(cid:104)S(cid:48)| S l1 S(cid:48) l2 thermore, small effects such as finite switching time of +ω |P(cid:48)(cid:105)(cid:104)P(cid:48)| +ω |D(cid:105)(cid:104)D| +ω |1(cid:105)(cid:104)1| P(cid:48) D C the laser, laser-amplitude noise and relative phase noise +(cid:0)geff|D,1(cid:105)(cid:104)S,0| +geff|D,1(cid:105)(cid:104)S(cid:48),0| +h.c.(cid:1), are neglected in the model. 1 2 (2) The simulations require us to specify the input parameters: magnetic field B, Raman-laser frequencies ω l1 wBohtehrecothuepleinngesrggyieffre=fer2Ωe∆in·lgciearisetthimee|-Pin,d0e(cid:105)psetnadteen(tω.PCh=oo0s)-. aqnudenωcile2s, pΩh1otaonnddeΩt2e,ctaiosnwpeallthaseffisycisetnemcy,paanrdamReatbeirsfrge-, ing the frequencies of the two fields to match the two κ, and γ. B and the laser frequencies are determined Raman conditions fromspectroscopyofthequadrupoletransitiontowithin ω +ω =ω +ω =ω +ω correspondstoafrequency 3kHz. Adetectionpathefficiencyof6.8%isusedtoscale S l1 D C S(cid:48) l2 difference |ω −ω |=|ω −ω |. The two states |S,0(cid:105) thesimulationresults,consistentwithpreviousmeasure- l1 l2 S S(cid:48) and |S(cid:48),0(cid:105) are degenerate in this frame, resulting in a ments [23]. Ω and Ω are determined experimentally 1 2 constant phase of the atomic state. As the couplings are via Stark-shift measurements with an uncertainty on also time-independent, the phase ϕ of the atomic state the order of 20%. However, the temporal shape of the doesnotchangeduringthetransfertothephotonicstate photons is highly dependent on Ω and Ω and on the 1 2 6 atom-cavity coupling g. In the simulation, we therefore [12] J. Hofmann, M. Krug, N. Ortegel, L. Grard, M. We- adjust Ω and Ω within the experimental uncertainty ber, W. Rosenfeld, and H. Weinfurter, Science 337, 72 1 2 range and find that the values Ω = 2π × 17.5 MHz (2012). 1 [13] J. I. Cirac, P. Zoller, H. J. Kimble, and H. 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