Enhanced Quantum Interface with Collective Ion-Cavity Coupling
B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt, T. E. Northup
EEnhanced Quantum Interface with Collective Ion-Cavity Coupling
B. Casabone, K. Friebe, B. Brandst¨atter, K. Sch¨uppert, R. Blatt,
1, 2 and T. E. Northup Institut f¨ur Experimentalphysik, Universit¨at Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria Institut f¨ur Quantenoptik und Quanteninformation,¨Osterreichische Akademie der Wissenschaften, Technikerstraße 21a, 6020 Innsbruck, Austria (Dated: August 19, 2018)We prepare a maximally entangled state of two ions and couple both ions to the mode of anoptical cavity. The phase of the entangled state determines the collective interaction of the ionswith the cavity mode, that is, whether the emission of a single photon into the cavity is suppressed orenhanced. By adjusting this phase, we tune the ion–cavity system from sub- to superradiance. Wethen encode a single qubit in the two-ion superradiant state and show that this encoding enhancesthe transfer of quantum information onto a photon.
Sub- and superradiance are fundamental effects inquantum optics arising in systems that are symmetricunder the interchange of any pair of particles [1–3]. Su-perradiance has been widely studied in many-atom sys-tems, in which effects such as a phase transition [4, 5] andnarrow-linewidth lasing [6] have recently been observed.For few-atom systems, each atom’s state and positioncan be precisely controlled, and thus collective emissioneffects such as Rydberg blockade [7] and the Lamb shift[8] can be tailored. In a pioneering experiment usingtwo trapped ions, variation of the ions’ separation al-lowed both sub- and superradiance to be observed, withthe excited-state lifetime extended or reduced by up to1.5% [9]. The contrast was limited because spontaneousemission from the ions was not indistinguishable, as theions’ separation was on the order of the wavelength ofthe emitted radiation. This limitation can be overcomeby observing preferential emission into a single mode,such as the mode defined by incident radiation [1] or byan optical cavity. In a cavity setting, indistinguishabilityis guaranteed when the emitters are equally coupled tothe mode, even if they are spatially separated. Subradi-ance corresponds to a suppressed interaction of the jointstate of the emitters with the cavity mode, while for thesuperradiant state, the interaction is enhanced.In the context of quantum networks [10, 11], super-radiance can improve a quantum interface when onelogical qubit is encoded across N physical qubits. Inthe DLCZ protocol for heralded remote entanglement,efficient retrieval of stored photons is based on super-radiance [12, 13]. Superradiance can also improve theperformance of a deterministic, cavity-based interface,which enables the direct transmission of quantum infor-mation between network nodes [14]. If a qubit is en-coded in the state √ N (cid:80) Ni | ↓ ... ↑ i ... ↓ N (cid:105) , the cou-pling rate to the cavity is enhanced from the single-qubitrate g to the effective rate g √ N , relaxing the techni-cal requirements for strong coupling between light andmatter [15]. This state corresponds to the first step inthe superradiant cascade described by Dicke [1]. In con-trast, subradiant states are antisymmetrized, resulting in suppressed emission. From a quantum-information per-spective, subradiant states are interesting because theyspan a decoherence-free subspace [16–18]. A subradiantstate of two superconducting qubits coupled to a cavityhas recently been prepared [19].Here, we generate collective states of two ions cou-pled to an optical cavity and use a state that maxi-mizes the coupling rate to improve ion–photon quantuminformation transfer. Our system is described by theTavis–Cummings Hamiltonian [21], the interaction termof which is H int = ¯ hg (cid:16) σ (1) − + e iζ σ (2) − (cid:17) a † + h. c. , (1)where σ ( j ) − is the lowering operator for the j th ion, ζ represents a relative phase [22], and a † is the creationoperator of a photon in the cavity mode. We preparea maximally entangled two-ion state and tune its emis-sion properties between sub- and superradiance, that is,between a dark state | Ψ sub (cid:105) and a state | Ψ super (cid:105) thatcouples with enhanced strength g √ Ca + separated by 5 . µ mare confined along the axis of a linear Paul trap andcoupled to an optical cavity in an intermediate couplingregime [22]. We position the ions so that g ≈ g , where g j represents the coupling strength of the j th ion to thecavity [26]. In a cavity-mediated Raman process, each ionprepared in a state from the 4 S / manifold produces asingle cavity photon [27]. The process is driven both by alaser at 393 nm detuned from the 4 S / − P / transi-tion and by the cavity, whose detuning from the 854 nm4 P / − D / transition satisfies a Raman resonancecondition [28]. Together, laser and cavity provide the in-teraction term of Eq. (1), in which the relative phase ζ between the ions’ coupling arises from the angle betweenthe Raman beam and the ion-trap axis [22]. Photonsleave the cavity preferentially through one mirror andare detected on photodiodes (Fig. 1a). a r X i v : . [ qu a n t - ph ] M a r pulse duration ( μ s) phase of π /2 rotation ( π ) popu l a ti on ( % ) p a r it y Raman beam addressing 729global 729 λ /2, λ /4platesPBS APD1APD2HVmirror mirror45°ab c20 1000 60 8040020406080100 0 0.5 1 1.5 2-1-0.500.51 FIG. 1. (a) Two Ca + ions in a linear Paul trap couplewith equal strength to the mode of a high-finesse optical cav-ity. A magnetic field orthogonal to the cavity axis defines thequantization axis. Quantum information stored in the ions ismanipulated using two 729 nm beams: the global beam cou-ples to both ions, while the addressing beam is focused ontoone ion. A 393 nm laser beam drives a cavity-mediated Ra-man transition, generating a single photon in the cavity. Atthe cavity output, two wave plates ( λ/ λ/
4) select the ba-sis in which photon polarization is analyzed. Two avalanchephotodiodes (APD1 and APD2) detect the horizontally (H)or vertically (V) polarized photons at the output of a polariz-ing beamsplitter (PBS). (b) Populations of the states | S (cid:105) | S (cid:105) (red diamonds), | D (cid:105) | D (cid:105) (blue circles), and | S (cid:105) | D (cid:105) or | D (cid:105) | S (cid:105) (green triangles) as a function of the Mølmer–Sørensen gateduration. After 55 µ s (dashed vertical line) a maximally en-tangled state is generated. Solid lines indicate the ideal timeevolution of the gate operation [20]. (c) Oscillations in theparity of the ion populations as a function of the phase of a π /2 pulse on the | S (cid:105) ↔ | D (cid:105) transition, following entangle-ment. The dashed vertical line at phase 1 . π corresponds to | Ψ + (cid:105) . Error bars represent projection noise. Entanglement between the ions is generated using a‘global’ 729 nm laser beam (Fig. 1a) that couples withequal strength to both ions on the 4 S / − D / quadrupole transition. The target state | Ψ + (cid:105) ≡ ( | S (cid:105) | D (cid:105) + | D (cid:105) | S (cid:105) ) / √ π/ | S (cid:105) ≡ | S / , m j = − / (cid:105) and | D (cid:105) ≡ | D / , m j = − / (cid:105) . In theMølmer–Sørensen gate, a bichromatic field that drivesblue and red motional sidebands generates a spin-dependent force, coupling the ion’s motion and internalstate [29]. Fig. 1b shows the evolution of the two-ion statepopulations during application of the gate. A maximallyentangled state | Φ (cid:105) = (cid:0) | S (cid:105) | S (cid:105) + i | D (cid:105) | D (cid:105) (cid:1) / √ µ s. Subsequently, a π/ | Φ (cid:105) to | Ψ + (cid:105) . A lower bound of 95(2)%on the state fidelity with respect to | Φ (cid:105) is determined a time t (µs)c D e t ec ti on p r ob a b ilit y p e r μ s ti m e b i n ( - ) r a ti o φ of the entangled state ( π )0.8 1.2 1 1.4 1.6b DSP D'
Raman laser
DSP D' cavity ion 1 entangled with ion 2
FIG. 2. (a) The two ions are prepared in either a separablestate | ψ (cid:105) or | ψ (cid:105) or an entangled state | Ψ( ϕ ) (cid:105) for variousvalues of ϕ . The global beam then drives a Raman transi-tion between | S (cid:105) and | D (cid:105) , generating a single cavity photonfor each ion in | S (cid:105) . Since | D (cid:48) (cid:105) is decoupled from the cavityinteraction, both | ψ (cid:105) and | ψ (cid:105) represent a single ion inter-acting with the cavity. (b) Ratio r ( ϕ ) of the probability todetect a photon for | Ψ( ϕ ) (cid:105) to that of | ψ (cid:105) as a function ofthe phase ϕ for the first 6 µ s of the Raman process. Thereference single-ion case is shown as a dashed horizontal line.(c) Temporal shape of the photon at the cavity output as afunction of detection time t , for the entangled states | Ψ super (cid:105) (circles) and | Ψ sub (cid:105) (diamonds) and the single-ion cases | ψ (cid:105) and | ψ (cid:105) (up and down triangles, respectively). The tempo-ral photon shapes are calculated by normalizing the detectedphoton counts per 1 µ s time bin by the number of photongeneration attempts. Data are shown until 20 µ s, the timescale for which enhancement and suppression are most promi-nent. Lines are simulations. The shaded area represents thetime window used in Fig. 2b. Error bars represent Poissonianstatistics and are mostly smaller than the plot symbols. by varying the phase of the π/ ϕ to theentangled state [31]: | Ψ( ϕ ) (cid:105) ≡ (cid:0) | S (cid:105) | D (cid:105) + e iϕ | D (cid:105) | S (cid:105) (cid:1) / √ . (2)By adjusting the length of the Stark-shift pulse, we shiftthis phase, which determines the effective coupling g eff of | Ψ( ϕ ) (cid:105) to the cavity mode under the action of H int .Specifically, the superradiant and subradiant states aregiven by | Ψ super (cid:105) ≡ | Ψ( ϕ = − ζ ) (cid:105) (3) | Ψ sub (cid:105) ≡ | Ψ( ϕ = − ζ + π ) (cid:105) . Note that if ζ were zero, | Ψ super (cid:105) and | Ψ sub (cid:105) would bethe Bell states | Ψ + (cid:105) and | Ψ − (cid:105) , respectively.The Raman process between | S (cid:105) and | D (cid:105) generatesa single cavity photon from | Ψ( ϕ ) (cid:105) , as only one ion isin | S (cid:105) . This photon has a temporal shape initially de-termined by g eff between the two-ion state and the cav-ity mode. For later times, the shape is determined bythe rates of both cavity decay and off-resonant scatter-ing. Varying g eff by changing the phase ϕ of | Ψ( ϕ ) (cid:105) thusmodifies the temporal shape, that is, the probability togenerate the photon early in the Raman process. Ideally,in the absence of scattering, the coupling of | Ψ sub (cid:105) to thecavity vanishes ( g eff = 0) so that no photon is generated.For | Ψ super (cid:105) , in contrast, the coupling is maximized suchthat g eff = g √
2. Thus, the probability to generateand detect a photon from | Ψ super (cid:105) early in the processis expected to be twice that of one ion. For time scalesmuch shorter than 1 /g , a photon generated in the cav-ity has not yet been reabsorbed, and therefore, cavityback-action does not play a role.We now determine this probability for a range ofphases ϕ . The experimental sequence starts with 1 ms ofDoppler cooling. The ions are then optically pumped to | S (cid:105) , followed by 1.3 ms of sideband cooling on the axialcenter-of-mass mode [32]. Next, global and addressing729 nm pulses generate the state | Ψ( ϕ ) (cid:105) . In the laststep, the cavity-mediated Raman transition is driven for55 µ s and photons are detected (Fig. 2a).In order to determine whether we achieve enhancementand suppression of the cavity coupling with respect to thesingle-ion rate g , we carry out a reference measurement.For this single-ion case, one of the two ions is hidden in astate | D (cid:48) (cid:105) ≡ | D / , m j = 3 / (cid:105) that is decoupled fromthe Raman process. Thus, the initial state is | ψ (cid:105) ≡| S (cid:105) | D (cid:48) (cid:105) or | ψ (cid:105) ≡ | D (cid:48) (cid:105) | S (cid:105) .For the states | Ψ( ϕ ) (cid:105) , we calculate η ( ϕ ), the proba-bility to detect a photon in the first 6 µ s of the Ramanprocess, an interval in which the effective coupling ratedetermines the initial slope. For the single-ion cases,we calculate η ψ , the average value of the photon de-tection probability for | ψ (cid:105) and | ψ (cid:105) in the same timewindow. Fig. 2b shows the ratio r ( ϕ ) = η ( ϕ ) /η ψ asthe phase ϕ is varied. For ϕ = 0 . π , the experimen-tally determined minimum, the ratio is 0.22(9): photongeneration is strongly suppressed. We therefore identify | Ψ( ϕ = 0 . π ) (cid:105) with | Ψ sub (cid:105) . As ϕ is increased, the ratioapproaches one, then enters the superradiant regime. Amaximum value of r ( ϕ ) is found for ϕ = 1 . π . For thecorresponding state, identified with | Ψ super (cid:105) , the proba-bility to detect a photon is 1.84(4), close to its maximum value of two, thus demonstrating strong enhancement inphoton generation.For these states | Ψ sub (cid:105) and | Ψ super (cid:105) , we now analyzethe temporal photon shapes at the detector (Fig. 2c).The temporal shapes corresponding to | ψ (cid:105) and | ψ (cid:105) areconsidered as a reference; from their overlap, we find thecoupling strengths of the two ions, g and g , to be within10% of one another. Photons generated from | Ψ super (cid:105) ex-hibit a steeper initial slope than the single-ion case, while | Ψ sub (cid:105) has a flatter slope. The photon shapes are consis-tent with enhanced and suppressed coupling to the cavityand are in good agreement with simulations. The simu-lations are based on numerical integration of the masterequation and include imperfect preparation of the ini-tial state, which together with off-resonant scattering ac-counts for the small but nonzero probability to generatephotons from | Ψ sub (cid:105) . For | Ψ super (cid:105) , these effects reducethe photon generation probability by about 10% for thefirst 6 µ s of the process [22].We now describe the implementation of a quantum in-terface that exploits the enhanced coupling of the su-perradiant state to the cavity [15]. The state | Ψ( ϕ ) (cid:105) asdefined in Eq. 2 contains two contributions: one fromthe ground state | S (cid:105) and the other from | D (cid:105) . We ex-tend this definition so that the ground-state componentcan be stored in either of two states, that is, in | S (cid:105) or in | S (cid:48) (cid:105) ≡ | S / , m j = +1 / (cid:105) . A logical qubit is encodedin these two states, and this qubit is mapped onto thepolarization state of a single cavity photon. To performthe mapping process, we use a phase-stable bichromaticRaman transition that coherently transfers | S (cid:105) to | D (cid:105) ,producing a horizontally polarized photon | H (cid:105) , and | S (cid:48) (cid:105) to | D (cid:105) , producing a vertically polarized photon | V (cid:105) [33](Fig. 3a). Defining a superposition state | α, β (cid:105) ≡ cos α | S (cid:105) + e iβ sin α | S (cid:48) (cid:105) , the mapping process can be represented by (cid:0) | α, β (cid:105) | D (cid:105) + e iϕ | D (cid:105) | α, β (cid:105) (cid:1) | (cid:105) / √ (cid:55)→ | D (cid:105) | D (cid:105) (cid:0) cos α | H (cid:105) + e iβ sin α | V (cid:105) (cid:1) , (4)where | (cid:105) stands for the cavity vacuum and the phase isset to ϕ = 1 . π , corresponding to | Ψ super (cid:105) .In order to characterize the mapping, we extract theprocess matrix χ , which describes the transformationfrom the input to the output density matrix: ρ out = (cid:80) i,j χ ij σ i ρ in σ j , where σ i ∈ { , σ x , σ y , σ z } are the Paulioperators [34]. Following Doppler cooling, optical pump-ing, and sideband cooling as above, the two ions areprepared in | Ψ super (cid:105) . Next, two global 729 nm pulsesprepare one of the four orthogonal input states | α, β (cid:105) ,with ( α, β ) ∈ { ( π/ , , (0 , , ( π/ , , ( π/ , π/ } . Fi-nally, the Raman transition is driven and the photon isdetected in one of three orthogonal polarization bases[35]. This set of measurements allows χ to be recon-structed via the maximum likelihood method. As the e ff i c i e n c y ( % ) bc p r o ce ss f i d e lit y ( % ) photon detection time window ( μ s)0 10 20 30 40 50 DS P S'D'H V DS P S'D'H V ion 1 entangled with ion 2 | χ ij | FIG. 3. (a) A bichromatic Raman transition maps a super-position of | S (cid:105) and | S (cid:48) (cid:105) onto a superposition of single-photonpolarization states | H (cid:105) and | V (cid:105) . The superposition is en-coded either in two entangled ions or in a single ion, with theother ion decoupled in | D (cid:48) (cid:105) . (b) Process fidelity for | Ψ super (cid:105) (filled blue circles) and | ψ (cid:105) (open black circles) as a functionof the photon detection time window. Lines are simulations(continuous line: two entangled ions; dashed line: single-ioncase). Inset: absolute value of the process matrix χ ij for | Ψ super (cid:105) reconstructed from photons detected between 2 and4 µ s, yielding the maximum process fidelity | χ | = 96 . | Ψ super (cid:105) (filled blue cir-cles) and | ψ (cid:105) (open black circles) as a function of the pho-ton detection time window. Error bars represent Poissonianstatistics and are smaller than the plot symbols. target mapping corresponds to the identity operation,the process fidelity is given by the matrix entry χ .For comparison, we carry out reference measurementsin which enhancement is not present, for which the ionsare prepared in | ψ (cid:105) . The mapping process is then givenby | α, β (cid:105) | D (cid:48) (cid:105) | (cid:105) (cid:55)→ | D (cid:105) | D (cid:48) (cid:105) (cid:0) cos α | H (cid:105) + e iβ sin α | V (cid:105) (cid:1) . (5)Fig. 3b shows the process fidelities χ for | Ψ super (cid:105) and | ψ (cid:105) as a function of the photon detection time win-dow. Not only is the fidelity of the superradiant casehigher for all data points, but also the improvement overthe single-ion case increases with the length of the detec-tion window. For a detection time window of 6 µ s, thefidelity is 93.3(3)% for | Ψ super (cid:105) and 90.9(5)% for | ψ (cid:105) ,indicating that in both cases the logical qubit is cor-rectly mapped onto photon polarization with very high probability. A maximum value of 96.0(3)% is found for | Ψ super (cid:105) for photons detected between 2 and 4 µ s. As thedetection window length is increased, χ decreases forboth cases because the probability for off-resonant exci-tation to the 4 P / -manifold increases with time. If suchan event happens during the Raman process, the initialstate | α, β (cid:105) is randomly projected onto | , (cid:105) = | S (cid:105) or | π/ , (cid:105) = | S (cid:48) (cid:105) , and the qubit is then mapped onto either | H (cid:105) or | V (cid:105) , regardless of the information in the initial su-perposition [33]. However, while the probability for scat-tering is the same for both states, photons are producedearlier from | Ψ super (cid:105) because of the enhanced effectivecoupling. Thus, the improvement in the fidelity stemsfrom an increased probability to generate a photon beforescattering occurs. After 55 µ s, we find χ = 73 . | Ψ super (cid:105) in comparison with 68.7(2)% for | ψ (cid:105) . Sim-ulations that take into account detector dark counts, im-perfect state initialization, different coupling strengthsof the ions to the cavity, and magnetic field fluctuationsare in good agreement with the data.We also investigate the cumulative process efficiency ε ( t ), defined as the probability to detect a photon be-fore time t (Fig. 3c). For t = 6 µ s, the process effi-ciency for | Ψ super (cid:105) is ε s ( t ) = 0 . | ψ (cid:105) ,it is ε ( t ) = 0 . ε s /ε of1.94(13). The ratio decreases monotonically with t , andby t = 55 µ s, it is 1.34(5). While the enhanced couplingmodifies the temporal shape of the photons early in theprocess, for longer times its effect on the cumulative pro-cess efficiency is small, such that the ratio is expected toapproach one. A single photon generated in the cavityis detected with an efficiency of 8(1)%, due to losses inthe cavity mirrors, optical path losses and the detectionefficiency of the avalanche photodiodes.The enhanced fidelity and efficiency of quantum statetransfer in the superradiant regime can be understood interms of a stronger effective ion–cavity coupling. Furtherimprovements are thus expected by encoding the logicalqubit across more physical qubits, as in a planar micro-fabricated trap [36]. Maximum enhancement would beachieved by encoding not just one but N/ N -ion state. The cooperative emissionrate would then be g (cid:113) N (cid:0) N + 1 (cid:1) , which scales with N for large N , as observed in atomic ensembles [4–6]. How-ever, it remains an open question how to transfer quan-tum information between such states and single photons,as required for a quantum transducer [15].Finally, we emphasize two advantages of ions as qubitsin these experiments: first, that the coupling strength ofeach ion to the cavity can be precisely controlled, andsecond, that a universal set of gate operations [37] allowspreparation of a range of states, from sub- to superra-diant. By tuning over this range, one could selectivelyturn off and on the coupling of logical qubits to the cav-ity. This technique would provide a versatile tool for ad-dressable read–write operations in a quantum register.We thank L. Lamata and F. Ong for helpful discus-sions and A. Stute for early contributions to the exper-iment design. We gratefully acknowledge support fromthe Austrian Science Fund (FWF): Project. Nos. F4003and F4019, the European Commission via the AtomicQUantum TEchnologies (AQUTE) Integrating Project,and the Institut f¨ur Quanteninformation GmbH. Whilepreparing this manuscript, we learned of related workwith two neutral atoms coupled to a cavity [38]. APPENDIXSystem parameters
Two Ca + ions are confined in a linear Paul trap andcoupled to an optical cavity. The cavity decay rate is κ = 2 π ×
50 kHz, and the atomic decay rate is γ = 2 π × . P to D and from P to S , where the manifolds are defined as P ≡ P / , D ≡ D / , and S ≡ S / . The couplingstrength of a single ion to the cavity mode on the P − D transition is g P D = 2 π × S − P transitions.The cavity parameters are described in further detail inRef. [39].The three-level system S - P - D can be mapped onto aneffective two-level system S - D if a Raman resonance con-dition is met, i.e., when both Raman beam and cavityresonance have the same detuning from P [28]. During acavity-mediated process, the electronic population of theion is coherently transferred from a state in S to a statein D , generating a cavity photon. For sufficiently large∆, negligible population is transferred to P . The ratesof the effective two-level system are g = ξ SD Ω g PD and γ eff = γ (cid:0) Ω2∆ (cid:1) . Here, ∆ ∼
400 MHz and ξ SD is a geo-metric factor that takes into account both the projectionof the vacuum-mode polarization onto the atomic dipolemoment and the Clebsch-Gordon coefficients of the S − P and D − P transitions [28].Ten individual Raman transitions between S and D canbe identified when all Zeeman sublevels are considered. Amagnetic field of B = 4 . S − D transitions.In the main text, two experiments are presented. Inthe first experiment, we examine the probability to gen-erate a photon as a function of the phase of the two-ion entangled state. To perform the experiment, a Ra-man beam with Rabi frequency Ω = 19 MHz drives the | S (cid:105) ≡ | S , m j = − / (cid:105) to | D (cid:105) ≡ | D , m j = − / (cid:105) transi- tion. For Ω = 19 MHz, the rates of the effective two-levelsystem are γ eff = 2 π × g eff = 2 π ×
18 kHz. Thecavity decay rate κ = 2 π ×
50 kHz is the fastest of thethree, placing the system in the bad cavity regime. Inthe second experiment, we use a superradiant state toenhance the performance of a cavity-based quantum in-terface. In this case, a bichromatic Raman beam withRabi frequencies 19 and 9 . | S (cid:105) to | D (cid:105) and | S (cid:48) (cid:105) ≡ | S , m j = 1 / (cid:105) to | D (cid:105) transitions. Thesetransitions do not have equal transition probabilities andadditionally, the orthogonally polarized photons coupledifferently to the cavity because of the orientation of thecavity with respect to the magnetic field [33]. By choos-ing the Rabi frequency for the | S (cid:48) (cid:105) to | D (cid:105) transition tohave twice the value of the | S (cid:105) to | D (cid:105) transition, thesedifferences are balanced and both transitions are drivenwith equal strength. In both experiments, the Rabi fre-quencies are first determined experimentally via Stark-shift measurements with an uncertainty on the order of10%. Next, in simulations of single-photon generation,we adjust the Rabi frequencies within the experimentaluncertainty range and find values for which the temporalphoton shapes have the best agreement with data. Relative Raman phase
In the first experiment, the part of the Hamiltonianthat describes the interactions of the Raman laser andthe cavity with the ion is H int = g P D (cid:0) σ (1) P D − σ (2) P D (cid:1) a † +Ω (cid:0) e iφ R σ (1) SP + e iφ R σ (2) SP (cid:1) + h.c. , (sm 1)where σ ( i ) P D ≡ | D (cid:105)(cid:104) P | , σ ( i ) SP ≡ | P (cid:105)(cid:104) S | , a † is the pho-ton creation operator, and φ R i is the optical phase ofthe Raman beam when interacting with the i th ion.Here, the rotating wave approximation has been usedand an appropriate transformation to the interaction pic-ture has been applied such that the Hamiltonian is time-independent. In this model, both ions are coupled tothe cavity with the same strength, and the minus signbetween the first and the second terms of Eq. (sm 1) ac-counts for the fact that in our cavity system the two ionsare located in adjacent antinodes [26].When the Raman resonance condition is met, Eq. (sm1) can be rewritten as Eq. (1), identifying ζ = ( φ R − φ R )and σ − = | D (cid:105)(cid:104) S | . The relative phase ζ is given by ζ =2 π d sin θ/λ , where d is the ions’ separation, θ ≈ ◦ is the angle between trap axis and Raman beam, and λ = 393 nm is the wavelength of the Raman beam. Initial state preparation
To generate | Ψ( φ ) (cid:105) = (cid:0) | S (cid:105) | D (cid:105) + e iφ | D (cid:105) | S (cid:105) (cid:1) / √ | Φ (cid:105) = (cid:0) | S (cid:105) | S (cid:105) + i | D (cid:105) | D (cid:105) (cid:1) / √ | S (cid:105) ↔ | D (cid:105) transition witha detuning δ . The ions are initialized in | S (cid:105) | S (cid:105) . Aftera time T = 1 /δ = 55 µ s, with a detuning δ = 18 . | Φ (cid:105) (seeFig. (1b)). For comparison, the coherence time for infor-mation stored in the S − D qubit is 475 µ s.We calculate the fidelity F Φ of the experimental statewith respect to | Φ (cid:105) in the following way [20]. After | Φ (cid:105) is created, we apply an ‘analysis’ π/ | S (cid:105) ↔ | D (cid:105) transition with a variable phase with respectto the previous entangling pulse. Subsequently, the par-ity operator P = p SS + p DD − p SD,DS is calculated fromfluorescence measurements of the ion populations, where p SS and p DD are the probabilities to find both ions in | S (cid:105) | S (cid:105) and | D (cid:105) | D (cid:105) , respectively, and p SD,DS is the prob-ability to find one ion in | S (cid:105) and the other in | D (cid:105) . Fig.(1c) shows the parity P as function of the phase of theanalysis pulse. If A is the amplitude of the parity os-cillation, then the fidelity F Φ is bound from above via F Φ ≥ A . From a fit to the data of Fig (2a), we calculatethat | Φ (cid:105) is created with a fidelity of at least 95(2)%.After the state | Φ (cid:105) is generated, a π/ | S (cid:105) ↔ | D (cid:105) transition rotates the state to ( | S (cid:105) | D (cid:105) + | D (cid:105) | S (cid:105) ) / √
2, identified in Fig. (1c). Finally, to convert( | S (cid:105) | D (cid:105) + | D (cid:105) | S (cid:105) ) / √ | Ψ( φ ) (cid:105) , we perform a single-ion rotation, introducing AC-Stark shifts to one ion us-ing the addressing beam [31]. The phase φ of | Ψ( φ ) (cid:105) isproportional to the duration τ of the Stark-shift pulse,where the proportionality constant depends on the Rabifrequency of the addressing beam, Ω AC , and the detun-ing of the laser from the | S (cid:105) ↔ | D (cid:105) transition, δ AC . Wechoose δ AC = 10 MHz and Ω AC = 8 . . µ s.The implementation of the Stark-shift gate is demon-strated via the generation of the state | S (cid:105) | D (cid:105) . Afteroptical pumping of both ions to | S (cid:105) | S (cid:105) , we apply a π/ | S (cid:105) ↔ | D (cid:105) transition using the globalbeam. Next, the Stark-shift gate is applied to one ion fora time τ . Subsequently, another global π/ | S (cid:105) ↔ | D (cid:105) transition is applied with the same phaseas the first π/ | S (cid:105) | S (cid:105) , | D (cid:105) | D (cid:105) and | S (cid:105) | D (cid:105) populations are extracted via fluorescence detec-tion. The results are shown in Fig (4) as a function of τ . After 2 . µ s, the ions are in a state with a fidelity of91(4)% with respect to | S (cid:105) | D (cid:105) .There are at least two other methods by which onecould tune the phase φ in the experiment. First, theangle of the Raman beam could be changed. Second, pulse duration (μs) popu l a ti on FIG. 4. Populations of the states | S (cid:105) | S (cid:105) (red diamonds), | D (cid:105) | D (cid:105) (blue circles) and | S (cid:105) | D (cid:105) (green triangles) as functionof the duration of the AC-Stark shift pulse. After 2 . µ s,state | S (cid:105) | D (cid:105) is generated with a fidelity of 91(4)%. Errorbars represent projection noise. the ion–ion separation could be changed by means of thevoltages that determine the trap potential. Both meth-ods would shift the relative phase seen by each ion. Ininitial experiments, we used the second method; however,when the ion–ion separation is adjusted to correspondto a desired phase, both ions must also remain equallyand near-maximally coupled to the cavity [26], and it isnot straightforward to satisfy both conditions simulta-neously. In practice, we found the Stark-shift gate de-scribed above to be the most precise and reproducibleapproach.To generate the single ion cases | ψ (cid:105) and | ψ (cid:105) , weuse the addressing beam. In this case, the frequencyof the addressing beam is set to drive the | S (cid:105) ↔ | D (cid:48) (cid:105) ≡| D , m j = 3 / (cid:105) transition on resonance. As the address-ing beam interacts with the second ion, a π -pulse trans-fers the state | S (cid:105) | S (cid:105) to | ψ (cid:105) = | S (cid:105) | D (cid:48) (cid:105) . To gener-ate | ψ (cid:105) = | D (cid:48) (cid:105) | S (cid:105) , we subsequently apply a π -rotationon the | S (cid:105) ↔ | D (cid:48) (cid:105) transition to both ions, such that | S (cid:105) | D (cid:48) (cid:105) is rotated to | D (cid:48) (cid:105) | S (cid:105) . The single-ion cases areprepared with a fidelity of 95(3)%. Two-ion crystal as a single–photon source
We have previously demonstrated that one ion in S produces a single photon when a Raman transition be-tween S and D is driven [27]. In the experiments pre-sented in the main text, we consider two ions in the entan-gled state | Ψ( φ ) (cid:105) , in which the probability to find one ionin | S (cid:105) is one. When a Raman transition is driven between | S (cid:105) and | D (cid:105) , the entangled state | Ψ( φ ) (cid:105) is transferred to | D (cid:105) | D (cid:105) and a single photon is expected. However, im-perfect preparation of | Ψ( φ ) (cid:105) leaves some population in | S (cid:105) | S (cid:105) , resulting in the generation of two photons.In order to estimate the number of two-photon detec-tion events, we consider detector dark counts and imper-fect preparation of the ions’ state. The following fourevents are relevant and contribute to two-photon detec-tions:1. State | S (cid:105) | S (cid:105) is generated; two photons are pro-duced and detected.2. State | S (cid:105) | S (cid:105) is generated; two photons are pro-duced, one is lost and the other is detected togetherwith a dark count.3. State | Ψ( φ ) (cid:105) is generated; one photon is producedand is detected together with a dark count.4. Two darks count are detected.State tomography reveals that in 3(2)% of attemptsto generate | Ψ( φ ) (cid:105) , the state | S (cid:105) | S (cid:105) is prepared instead.The probability to detect one photon during the 55 µ s du-ration of the Raman process is 5.4(3)%, which is mainlylimited by cavity absorption and detector efficiencies [28].Detector dark count rates are 3.2(1) s − and 3.8(1) s − for the two avalanche photodiodes. With these values,we expect one two-photon event in 8 . × attemptsto generate a single photon.To measure two-photon events, we generate (cid:0) | S (cid:105) | D (cid:105) + | D (cid:105) | S (cid:105) (cid:1) / √ (cid:0) | S (cid:48) (cid:105) | D (cid:105) + | D (cid:105) | S (cid:48) (cid:105) (cid:1) / √ g (2) (0)is calculated. After 223,106 attempts to generate pho-tons, 28 two–photon events were measured, and 27(3)two–photon events were expected from the considerationsabove. The observed number of two-photon detectionevents are thus consistent with single-photon generation. Process fidelity
Tomography of the state-mapping process consists ofstate tomography of the photonic output qubit for fourorthogonal input states. Measurements in the threebases of horizontal/vertical, diagonal/antidiagonal andright/left circular polarization constitute state tomogra-phy of the photonic qubit [23]. Each basis is measureda second time with the APDs swapped by rotating the λ/
2- and λ/ χ ij are reconstructed using amaximum-likelihood method. The process fidelity χ is given by the overlap of the reconstructed process ma-trix with the target process (i.e., the identity operation).Uncertainties in the process fidelities are given as onestandard deviation, derived from non-parametric boot-strapping assuming a multinomial distribution [24]. Simulations
Numerical simulations are based on the Quantum Op-tics and Computation Toolbox for MATLAB [25] via in-tegration of the master equation. We simulate two Ca + ions interacting with an optical cavity and a Ramanbeam. For each ion, we consider six levels: | S (cid:105) , | S (cid:48) (cid:105) , | D (cid:105) , | D (cid:48) (cid:105) , | P , m j = − / (cid:105) and | P , m j = 1 / (cid:105) . For theoptical cavity, we consider two orthogonal modes a and b with the Fock state basis truncated at 2 for each mode.Additional input parameters for the simulations arethe cavity parameters g, κ , and γ ; the magnetic field am-plitude B , the Rabi frequency Ω of the Raman laser, theRaman laser linewidth, and the output path losses. Thelaser linewidth, atomic decay, and cavity decay are intro-duced in the Lindblad form. The Raman laser linewidthis set to the measured value of 30 kHz.For the simulation of the first experiment, the initialdensity matrix ρ is assigned 5% of populations equallydistributed between | S (cid:105) | S (cid:105) and | D (cid:105) | D (cid:105) , and the coher-ence terms between | S (cid:105) | S (cid:105) and | D (cid:105) | D (cid:105) are set to zero,consistent with measurements. The rest of the popula-tion is distributed between | S (cid:105) | D (cid:105) and | D (cid:105) | S (cid:105) , preserv-ing the coherences such that ρ has an overlap of 95%with | Ψ( ϕ ) (cid:105) . In the case of the second experiment, thesuperposition state | α, β (cid:105) ≡ cos α | S (cid:105) + e iβ sin α | S (cid:48) (cid:105) , is introduced via an operator ˆ M that performs the map-ping | S (cid:105) → cos α | S (cid:105) + e iβ sin α | S (cid:48) (cid:105) for each ion. This operator ˆ M is applied to ρ .From the integration of the master equation up to atime t , we obtain the time-dependent density matrix ρ ( t ).The mean photon numbers of the cavity modes are cal-culated via the expectation values (cid:104) a † a ( t ) (cid:105) and (cid:104) b † b ( t ) (cid:105) .Contributions of the detector dark counts are added tothe mean photon number. Errors in the generation ofthe superposition state and magnetic field fluctuationsare introduced by scaling the off-diagonal terms of ρ ( t )by a factor of 0.96 and by the exponential e (2 t/τ ) re-spectively, where τ = 190 µ s is the coherence time of thequbit stored in | S (cid:105) and | S (cid:48) (cid:105) . Finally, the coupling of oneof the ions to the cavity mode is reduced to 90% of itmaximum value. This reduction is based on measureddrifts over the course of the experiment.Fig. (2c) shows the simulated and experimental tem-poral photon shapes as function of detection time. 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