PHYSICAL REVIEW X 7, 031007 (2017) Callum R. Murray and Thomas Pohl Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK 8000 Aarhus C, Denmark (Received 13 February 2017; revised manuscript received 21 June 2017; published 13 July 2017) Coupling photons to Rydberg excitations in a cold atomic gas yields unprecedentedly large optical nonlinearities at the level of individual light quanta. Here, the basic mechanism exploits the strong interactions between Rydberg atoms to block the formation of nearby dark-state polaritons. However, the dissipation associated with this mechanism ultimately limits the performance of many practical applications. In this work, we propose a new approach to strong photon interactions via a largely coherent mechanism at drastically suppressed photon losses. Rather than a polariton blockade, it is based on an interaction-induced conversion between distinct types of dark-state polaritons with different propagation characteristics. We outline a specific implementation of this approach and show that it permits us to turn a single photon into an effective mirror with a robust and continuously tunable reflection phase. We describe potential applications, including a detailed discussion of achievable operational fidelities. DOI: 10.1103/PhysRevX.7.031007 Subject Areas: Atomic and Molecular Physics, Optics, Quantum Physics I. INTRODUCTION can easily perturb and break the underlying EIT condition, thereby rendering light propagation highly nonlinear [48–50]. The notion that photons are devoid of mutual interactions Indeed, there have now been a number of experiments that in vacuum is well rooted in our elementary understanding demonstrated controllable photon-photon interactions of of light. Nevertheless, the ability to engineer such inter- unprecedentedstrength insuchsystems [37,48–59]. actions synthetically would hold profound implications Keytothis nonlinearityisthe destruction of EIT con- for both fundamental and applied science, and has since ditions that originates from an effective polariton blockade, ushered in a new era of research into nonlinear optics at the whereby multiple proximate photons are prevented from ultimate quantum level . Intense efforts have been simultaneously forming dark-state polaritons. As an imme- directed towards enhancing light-matter coupling through diate consequence, the emergent photon interactions inevi- tight mode confinement [2–14] in order to achieve local tably carry an intrinsic dissipative component. Nevertheless, nonlinearities by interfacing photons with a single quantum the nonlinear quantum optical response achieved in this emitter. A complementary strategy, which is rapidly gain- way can be utilized to facilitate a broad range of applications, ing momentum, exploits the collective coupling of light to such as imaging [60,61], all-optical switches and transistors particle ensembles with finite-range interactions [15–20] to [54–57,62], quantum gates [42,58], and single-photon establish large and nonlocal nonlinearities. sources [43,48] and subtractors . Yet, it turns out that Interfacing light with strongly interacting atomic Rydberg high-fidelity operations require conditions (e.g., high atomic ensembles [21–34] under conditions of electromagnetically densities) where the performance of such applications is induced transparency (EIT)  has emerged as a particularly ultimately eclipsed by additional decoherence effects promising way to implement this new type of mechanism [54,62–64]. [15,16,36–45]. EITin these systems is based on the formation As a solution to this outstanding issue, we describe here of Rydberg dark-state polaritons, which correspond to coher- a novel approach to quantum optical nonlinearities in a ent superposition states of light and matter that are immune to Rydberg-EIT medium without the polariton blockade. It dissipation [46,47]. While this polariton formation supports exploits the atomic interactions to modify EIT conditions, the lossless and form-stable propagation of single photons, rather than destroying them entirely. Generally, the devised the strong mutual interaction between two such polaritons strategy can thus be understood as a dark-state polariton switch, as opposed to the existing schemes based on the polariton blockade, Fig. 1(a). Consequently, this new Published by the American Physical Society under the terms of mechanism globally preserves EIT conditions such that the Creative Commons Attribution 4.0 International license. nonlinear dissipation is intrinsically suppressed, thereby Further distribution of this work must maintain attribution to alleviating the decoherence-related hindrances discussed in the author(s) and the published article’s title, journal citation, and DOI. Refs. [62,63]. 2160-3308=17=7(3)=031007(16) 031007-1 Published by the American Physical Society CALLUM R. MURRAY and THOMAS POHL PHYS. REV. X 7, 031007 (2017) the large induced level shift prevents the excitation of more than one Rydberg state within a so-called blockade radius. Since the van der Waals coefficient C ∼ n increases rapidly with the principal quantum number n of the chosen Rydberg state , the available interaction strengths can vastly exceed any other energy scale in the system, and the available blockade radii can become significant. In current approaches to nonlinear optics based on Rydberg EIT, this blockade phenomenon is used to break EIT conditions and realize an effective polariton blockade [15,16,36,37,39–43]. On the contrary, we consider here a situation where this level shift is used rather to establish a switching mechanism between different types of dark-state polar- itons. Consequently, the corresponding nonlinear optical response should thus be associated with minimal refraction and absorption, and only modify the dispersion relation that characterizes the photon propagation. As a specific exam- ple, we consider a situation in which the onset of inter- actions serves to cancel the linear dispersion of light and establish a locally quadratic dispersion, Fig. 1(b). This FIG. 1. Illustration of the basic principle of nonlinear polariton corresponds to a nonlinear switching between so-called switching. (a) A photon (target) propagates initially as one type of slow-light [46,47,66] and stationary-light [67–72] polar- dark-state polariton (type A, gray sphere), but is subsequently itons, both of which have been separately demonstrated in converted to another kind of dark-state polariton (type B, blue Refs. [73,74] and Refs. [75–77], respectively. sphere) upon interacting with a second (gate) polariton. (b) This induces a change in the dispersion relation that governs the EIT is an effect that uses destructive interference propagation of light and thereby mediates an effective photon between different excitation pathways to cancel the static interaction at greatly suppressed losses. optical response of a medium (which characterizes absorp- tion and refraction). In the simplest realization of slow-light EIT, this typically involves a single control field to induce We outline a specific implementation that can be realized transparency for a second weak probe field on two-photon with minimal extensions to current experiments [49,55,56] resonance with the transition to a stable excited atomic and is shown to yield a conditional coupling between two state. The transparency is associated with the formation of a distinct photonic modes. In particular, we show how this dark-state polariton that is composed of the probe photons can be used to establish a reflective nonlinearity, in which a and the stable atomic excitations (spin waves) [46,47]. single photon stored in a Rydberg spin-wave excitation acts Because of its spin-wave component, the dark-state polar- as an effective mirror, capable of reflecting photons with itons propagate at a greatly reduced speed according to the an arbitrary and continuously tunable reflection phase. one-dimensional propagation equation [46,47], The described realization of interaction-induced polariton switching can thus function as a single-photon router, ˆ ˆ ∂ Ψðz; tÞ¼ v∂ Ψðz; tÞ; ð1Þ t z facilitating a broad range of applications from quantum transistors to photonic gate operations. Finally, we discuss which describes the form-stable linear propagation of the the performance of such applications based on current slow-light polariton anihilation operator Ψðz; tÞ with a technology and in relation to previous blockade-based group velocity v that is typically much less than the approaches. vacuum speed of light c . Stationary light, on the other hand, can be realized when II. NONLINEAR POLARITON SWITCHING a pair of counterpropagating control fields is used to Rydberg dark-state polaritons acquire the properties of establish EIT for a pair of counterpropagating probe fields their constituents, inheriting kinetics from their photonic in a four-wave mixing configuration [67–72,75–77]. The admixture and interactions from their atomic Rydberg-state resulting polaritons are then composed of both probe field component. Typically, these interactions are of a van der modes, and become stationary when they contain an equal Waals type, causing a level shift VðzÞ¼ C =z of the admixture of the two counterpropagating fields, such that Rydberg state attached to one polariton when it interacts their linear dispersions cancel each other to yield a leading- with another at a distance z. The most dramatic conse- order quadratic dispersion. This effectively endows these quence of this effect is the Rydberg blockade , where polaritons with a kinetic energy and mass [69,71], akin to 031007-2 COHERENT PHOTON MANIPULATION IN INTERACTING … PHYS. REV. X 7, 031007 (2017) FIG. 2. (a) Schematics of the considered coupling scheme. Classical fields with indicated Rabi frequencies Ω and Ω establish EIT ˆ ˆ conditions for the counterpropagating photonic modes E and E . The Rydberg state jsi is subject to a spatially dependent level shift → ← VðzÞ upon interacting with the stored gate excitation as illustrated in (b). This shift modifies the underlying EIT conditions for E and E , rather than perturbing them. massive particles. To leading order in the polariton band- III. INTERACTION WITH A STORED SPIN WAVE width, the corresponding operator Φðz; tÞ describing the The proposed polariton-switching mechanism is best annihilation of a stationary-light polariton, thus, obeys the analyzed by considering the conceptually simplest type of one-dimensional evolution equation [69,71], photon-photon interaction, whereby a (gate) photon is first stored in the atomic ensemble [47,78–80] as a collective Rydberg spin-wave excitation. Subsequently, a second ˆ ˆ ∂ Φðz; tÞ¼ − ∂ Φðz; tÞ; ð2Þ t z (target) photon is sent through the medium and made to 2m interact with the stored gate excitation [see Fig. 2(b)]. This approach provides a well-controlled way to engineer two- where m is the effective mass acquired by Φðz; tÞ. photon interactions , and has been demonstrated in a In order to engineer a Rydberg-mediated switching number of recent experiments [54–58]. between the different types of polaritons described by The relevant atomic excitations are described by the Eqs. (1) and (2), we propose the level structure shown in ˆ ˆ continuous field operators  P ðz; tÞ, D ðz; tÞ, and Fig. 2(a). Here, EIT is achieved for two counterpropagating ⇄ † † ˆ ˆ ˆ S ðz; tÞ, which create an excitation in jp i, jdi, and jsi, light fields described by the field operators E and E , ⇄ → ← respectively, at position z. Moreover, we introduce the which create a probe photon in the right- and left-moving operator S ðz; tÞ that creates a stored gate excitation in an mode, respectively. As we see below, the precise nature of the resulting dark state is, however, controlled by the auxiliary Rydberg state js i that is not laser coupled during interaction-induced level shift of the Rydberg state jsi to ˆ ˆ the probe stage [42,54–58]. Along with E and E , all of → ← which the fields are coupled. Specifically, our proposed these field operators satisfy bosonic commutation rela- coupling scheme is shown to support slow-light EIT tions . ˆ ˆ conditions for E and E separately in the limit of weak In a rotating frame, the one-dimensional dynamics of → ← interactions, while it facilitates stationary-light EIT under this system are governed by the following (non-Hermitian) conditions of full Rydberg blockade. Hamiltonian: Z Z ∞ L † † † † ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ H ¼ −ic dz½E ðzÞ∂ E ðzÞ − E ðzÞ∂ E ðzÞ þ G dz½E ðzÞP ðzÞþ E ðzÞP ðzÞþ H:c: → z → ← z ← → → ← ← −∞ 0 Z Z L L † † † † iϕ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ − iγ dz½P ðzÞP ðzÞþ P ðzÞP ðzÞ þ Ω dz½D ðzÞP ðzÞþ e D ðzÞP ðzÞþ H:c: → → ← ← → ← 0 0 Z Z Z L L L † † † ˆ ˆ ˆ ˆ ˆ ˆ þ Ω dz½S ðzÞP ðzÞþ H:c:þ dzdxVðz − xÞS ðxÞS ðzÞSðzÞS ðxÞ; ð3Þ S ← g g 0 0 0 where L is the length of the medium. For simplicity, we assume that jp i and jp i decay with the scattering rate 2γ and that → ← pﬃﬃﬃﬃﬃ the probe photon modes couple to their respective transitions with G ≡ g ρ , where ρ is the homogeneous atomic density a a 031007-3 CALLUM R. MURRAY and THOMAS POHL PHYS. REV. X 7, 031007 (2017) and g is the single atom coupling strength. The state jdi is Ω ω ðkÞ ≈ c k þ O½k ; ð6Þ 2 2 coupled to jp i and jp i by classical control fields with → ← G þ Ω identical Rabi frequencies Ω ¼ Ω ¼ Ω, while we allow → ← for a relative phase difference ϕ between them. Finally, the state jp i is coupled by another classical field to the S 2 ω ðkÞ≈−c k þ O½k : ð7Þ Rydberg state jsi with a Rabi frequency Ω . The last term in ← S 2 2 G þ Ω Eq. (3) accounts for the spatially dependant level shift of the Rydberg state jsi due to its van der Waals interaction As expected, one finds linear dispersion relations with the stored gate excitation. The typical range over describing a form-stable propagation of the slow-light which this shift affects the probe photon propagation can polaritons with group velocities v ¼ðdω =dkÞ and be characterized by the blockade radius z according to → → v ¼ðdω =dkÞ, respectively. The two polaritons propa- Vðz Þ¼ Ω =γ . ← → 2 2 gate in opposite directions with v ¼ −ðΩ =Ω Þv under → ← the typical condition G ≫ Ω;Ω . This is further illustrated IV. POLARITON ANALYSIS in Fig. 3(a), where we plot the complete polariton spectrum Having established the basic idea and the specifics of the admitted in this noninteracting situation, indeed revealing considered setup, let us now discuss the characteristics of the emergence of two dark-state polariton branches at ˆ ˆ the dark-state polaritons involved in the underlying switch- k ¼ 0 corresponding to Ψ and Ψ . → ← ing protocol. The relevant dark-state polaritons can be Now we consider the polariton spectrum admitted well identified as the zero-energy eigenstate solutions of the within a blockade radius away from the stored spin wave, Hamiltonian Eq. (3) in the two limiting cases VðzÞ → 0 and i.e., under strong blockade conditions corresponding to VðzÞ → ∞, i.e., for vanishing interactions and in the limit VðzÞ → ∞. In this case, the shifted Rydberg state exposes a of a complete Rydberg-state blockade. modified effective level structure corresponding to a so- Focusing first on the noninteracting situation, diagonal- called dual-V coupling scheme [69,72], which can support izing the system Hamiltonian in the absence of photon stationary-light phenomena. For this system, one finds the dispersion yields two dark-state polaritons of the form emergence of only a single dark-state polariton Φ of the type in Eq. (2). Diagonalizing the underlying system Hamiltonian Eq. (3), again in the absence of photon iϕ ˆ ˆ ˆ ˆ Ψ ¼ ½ΩΩ E − GðΩ D − Ωe SÞ; ð4Þ → S → S ˆ kinetics, one finds Φ to be of the following form: iϕ ˆ ˆ ˆ ˆ Φ ¼ ½ΩðE þ e E Þ − GD; ð8Þ → ← ˆ ˆ ˆ N Ψ ¼ ½Ω E − GS; ð5Þ ← S ← pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 where N ¼ Ω þ G is the normalization factor. In pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 2 contrast to the noninteracting limit, a coherent coupling where N ¼ Ω Ω þ G ðΩ þ Ω Þ and N ¼ → ← S S pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 ˆ G þ Ω are the normalization factors required to obtain is now established between the two optical modes E ˆ ˆ and E . This is reflected in the photonic composition standard bosonic commutation relations for Ψ and Ψ . → ← These polaritons can be accredited to two separate slow- of Φ, which is composed of the symmetric superposition pﬃﬃﬃ ˆ ˆ iϕ ˆ light EIT schemes supported simultaneously by the level state of the optical fields, E ¼ð1= 2ÞðE þ e E Þ. þ → ← structure in Fig. 2(a): the five-level system formed by jgi, The corresponding dispersion relation ω ðkÞ that gov- jp i, jdi, jp i, and jsi (establishing EIT for E ), and the ˆ → ← → erns the propagation of Φ can be determined in a similar three-level system formed by jgi, jp i, and jsi (establish- ← fashion as before, and reads ing EIT for E ). We emphasize that the dark-state nature of ˆ ˆ Ψ and Ψ ensures that there is no coupling between the 2 → ← cΩ 2 3 ω ðkÞ≈−i2l k þ O½kð9Þ two underlying photonic modes. Hence, a right-moving ↔ abs 2 2 G þ 2Ω input photon will undergo low-loss and form-stable propa- gation through the medium, and so will a left-moving to lowest order in the photon momentum k, where l ¼ photon. abs cγ=G is the resonant two-level absorption length. Indeed, The dispersion relations, ω ðkÞ and ω ðkÞ, governing → ← this propagation dynamics are readily obtained from the obtained dispersion is quadratic in k, such that Φ a momentum space formulation of Eq. (3). To leading behaves as a stationary-light polariton. Figure 3(b) shows order in the photon momentum k (and the ratio Ω=Ω ), the complete polariton spectrum for VðzÞ → ∞ and illus- one finds trates the above discussion of the dark-state polariton. 031007-4 COHERENT PHOTON MANIPULATION IN INTERACTING … PHYS. REV. X 7, 031007 (2017) (a) (b) FIG. 3. (a) Real part of the polariton spectrum in the absence of interactions, indicating the emergence of two slow-light dark-state polaritons. These are separately governed by the linear dispersion relations ω ðkÞ and ω ðkÞ as given by Eqs. (6) and (7), respectively. → ← Here, Ω =G ¼ 1, Ω=G ¼ 0.5, and γ=Ω ¼ 1. (b) Imaginary part of the polariton spectrum for strong interactions, i.e., under conditions of a complete Rydberg blockade. In this case, one finds a single stationary-light dark-state polariton described by a quadratic dispersion relation ω ðkÞ, Eq. (9). Here, G=Ω ¼ 1 and γ=Ω ¼ 1. For each bright-state branch there are two solutions with identical Im½ωðkÞ. The ˆ ˆ blue to red color coding indicates the relative fraction of E and E comprising the underlying state of each polariton branch, while the → ← gray scale indicates the overall atomic fraction. V. PHOTON PROPAGATION of a single susceptibility χ ðzÞ ≡ χ ðz; 0Þ¼ −χ ðz; 0Þ¼ 0 → ← −χðz; 0Þ, given by In order to develop an intuitive physical picture of the target photon dynamics, we first model the stored gate χ ðzÞ¼ : ð12Þ excitation as a spatially localized Rydberg impurity, and ðz=z Þ þ 2i generalize this analysis to the consideration of a collective spin-wave state in Sec. VI. First, we transform into the Here, we define 2d ¼ 2z =l as the medium’s optical b b abs Schrödinger picture. Introducing jψðtÞi as the general time- depth per blockade radius. χ ðzÞ basically characterizes an dependent wave function of the system, we define the two- effective potential through which the stored spin wave can † † ˆ ˆ body amplitudes E ðz; x; tÞ¼h0jE ðz; tÞS ðx; tÞjψi and → → g affect the target photon propagation dynamics. In particu- † † ˆ ˆ E ðz; x; tÞ¼h0jE ðz; tÞS ðx; tÞjψi corresponding to a lar, one finds that χ ðzÞ → 0 outside the blockade radius ← ← g stored gate excitation at position x and a target photon of the stored excitation, jzj >z , consistent with the slow- at position z in the right- and left-moving mode, respectively. light EIT conditions supported in this region and the Denoting the temporal Fourier transform of E ðz; x; tÞ and → associated decoupling of the photonic modes. However, ~ ~ χ ðzÞ approaches −id =2 within the blockade volume, E ðz; x; tÞ by E ðz; x; ωÞ and E ðz; x; ωÞ, the photon 0 b ← → ← reflecting the fact that a coupling between these modes dynamics can be formulated in terms of a matrix equation is established, which gives rise to stationary-light EIT of the form conditions. i∂ Eðz; x; ωÞ¼ Mðz − x; ωÞEðz; x; ωÞ; ð10Þ Considering a target photon incident on the medium from the left at z ¼ 0, its transmission and reflection can ~ ~ then be characterized by the following relations: where Eðz; x; ωÞ¼fE ðz; x; ωÞ;E ðz; x; ωÞg and the → ← propagation matrix is given by ~ ~ E ðL; x; ωÞ¼ T ðω;xÞE ð0;x; ωÞ; ð13Þ → n → iϕ χ ðz; ωÞ χðz; ωÞe Mðz; ωÞ¼ : ð11Þ ~ ~ −iϕ E ð0;x; ωÞ¼ R ðω;xÞE ð0;x; ωÞ; ð14Þ −χðz; ωÞe χ ðz; ωÞ ← n → The susceptibilities χ ðz; ωÞ and χ ðz; ωÞ characterize the where T ðω;xÞ and R ðω;xÞ are the transmission and → ← n n ~ ~ reflection coefficients of the medium containing n ∈ ½0; 1 propagation of E and E , respectively, while χðz; ωÞ → ← stored gate excitations at position x. describes the coupling between the two modes. A derivation Let us first consider the situation in which the gate of Eq. (10) is outlined in Appendix A, along with the explicit excitation is absent. In this case, the target photon will expressions for the susceptibilities. initially generate the slow-light polariton described by Ψ In the continuous wave (cw) limit (ω → 0) the propa- gation matrix in Eq. (11) can be parametrized in terms at the entrance of the medium. As we describe above, the 031007-5 CALLUM R. MURRAY and THOMAS POHL PHYS. REV. X 7, 031007 (2017) formed polariton will then traverse the medium with a vanishing coupling to the counterpropagating mode and experience full transmission under perfect EIT conditions. The actual mechanism underlying this decoupling of E and E can be traced back to quantum interference effects involving the dressed states of the laser-driven Rydberg transition. Specifically, the resonant coupling of jp i and jsi via the classical field Ω [see Fig. 2(a)] establishes a pair of light-shifted states, jp ijsi jf i¼ pﬃﬃﬃ ; ð15Þ FIG. 4. Transmission coefficient in the absence of a stored gate excitation for various indicated values of Ω =Ω. The total optical which are shifted in energy by Ω , respectively. It is the depth is 2d ¼ 50, while γ=Ω ¼ 0.5 and G=Ω ¼ 0.1. destructive interference between competing excitation pathways involving these states that ultimately decouples demands a large level splitting of the dressed states jf i, the two modes of the target photon. and this is given by Ω . We note that this decoupling is exact on EIT resonance To verify this picture, we plot the solution for T ðωÞ in (ω ¼ 0) for any finite value of Ω . However, this is not Fig. 4 for various ratios of Ω =Ω, and indeed find that true for a finite bandwidth of the target photon. In this the transmission spectrum converges to that of the jgi ↔ case, a nonvanishing coupling is established between the jp i ↔ jdi Λ system as Ω increases. Importantly, Fig. 4 → S off-resonant frequency components of E ðz; x; ωÞ and demonstrates that near optimal transmission is already E ðz; x; ωÞ. Such bandwidth limitations exist for any reached for remarkably small ratios of Ω =Ω, which are realistic EIT setting, but can be minimized through a well within current experimental capabilities. proper choice of parameters. To establish these conditions, Let us now consider the propagation dynamics in the we first expand jT ðωÞj to lowest order in ω as presence of a stored gate excitation. As before, upon jT ðωÞj ≈ 1 − ðω=Δω Þ . Δω then corresponds to the 0 0 0 entering the medium, the target photon propagates accord- characteristic width of the transmission resonance, defining ing to the linear dispersion relation ω ðkÞ in the form the range of frequencies over which the target photon is of a slow-light polariton Ψ . Upon entering the blockade transparent, and is given explicitly as volume established by the stored excitation, however, the target photon is subject to stationary-light EIT conditions, 2 4 4 −1=2 Ω Ω d Ω as described in the preceding sections. In this case, a Δω ¼ Γ 1 þ 2 þ 2 þ ; ð16Þ 2 4 4 Ω Ω Ω ˆ ˆ S S S coherent coupling between the E and E modes is → ← established and the symmetric photonic state E is gen- where 2d ¼ 2d L=z is the total optical depth of the b b pﬃﬃﬃ erated. Because of the lack of a linear dispersion, the medium and Γ ¼ Ω =γ d. photon can now only traverse the blockade region through a According to Eq. (16), the transparency width tends slow dispersion following ω ðkÞ¼ −ik =2m. Upon cross- to a maximum value of Δω → Γ when the conditions ing the blockade region the photon can reestablish the slow- 2 2 2 4 4 2Ω =Ω ≪ 1 and d Ω =2Ω ≪ 1 are satisfied. The con- S S light polariton Ψ and be transmitted through the medium. 2 4 4 dition d Ω =2Ω ≪ 1 accounts for a weak oscillatory However, there is typically a larger amplitude for the behavior in jT ðωÞj. However, the effects of this only photon to diffuse into the counterpropagating direction become observable at significant values of d, and can and exit the medium as a Ψ polariton, corresponding to be safely neglected for our purposes. To understand the reflection by the stored excitation. This reflection bias is 2 2 second condition, 2Ω =Ω ≪ 1, first notice that in the limit essentially due to a boundary effect; the photon is more 2 2 Ω =Ω → ∞ the Rydberg-state admixture of the Ψ likely to diffuse the much shorter distance in the counter- polariton vanishes, and the effective coupling scheme propagating direction upon entering the interaction volume establishing EIT for E reduces to a Λ system formed → than diffuse the full 2z length in the forward direction. The from jgi ↔ jp i ↔ jdi [see Fig. 2(a)]. In this ideal limit, → schematics of this overall process are depicted in Fig. 5(a). the optical response of the medium is then set solely by this To determine the relative importance of reflection and effective EIT scheme with an effective transparency width transmission, we can solve the propagation dynamics of pﬃﬃﬃ of Ω =γ d , which is consistent with the limiting value the target photon exactly in the cw limit, for which the of Δω . Physically, though, Ω is required to be large since propagation matrix M in Eq. (11) is parametrized solely 0 S the effective decoupling of the photonic modes ultimately by χ ðzÞ, Eq. (12). With the boundary conditions 031007-6 COHERENT PHOTON MANIPULATION IN INTERACTING … PHYS. REV. X 7, 031007 (2017) FIG. 5. (a) Illustration of target photon propagation in the presence of a stored gate excitation, where R ðωÞ and T ðωÞ are the 1 1 reflection and transmission coefficients, respectively. (b) The resonant values of R ðω ¼ 0Þ and T ðω ¼ 0Þ are shown as a function of d 1 1 b ˆ ˆ in red and blue, respectively. The inset shows the spatial dependence of the photonic amplitudes E ðzÞ and E ðzÞ within the medium for → ← d ¼ 5, where the gray shaded region indicates the extent of the blockade volume established by the stored gate excitation. (c)–(e) The transmission and reflection spectra T ðωÞ and R ðωÞ are shown in blue and red, respectively, for various indicated values of d , where 1 1 b the stored excitation is located at the center of the medium. The total optical depth is 2d ¼ 50, and the remaining parameters are γ=Ω ¼ 0.5, Ω=G ¼ 0.1, Ω =G ¼ 0.5. The transmission coefficient T ðωÞ in the absence of interactions is indicated by the gray shaded S 0 region for reference. ~ ~ ~ traversing the entire blockade region is strongly suppressed E ðz ¼ 0;x; 0Þ¼ E and E ðz ¼ L; x; 0Þ¼ 0, the solu- → 0 ← such that photons predominantly exit in the opposite tions for the photonic fields can be readily obtained as direction right at the incident boundary. Additionally, Eq. (20) shows that one can imprint an νðz; xÞ ~ ~ E ðz; x; 0Þ¼ 1 − E ; ð17Þ → 0 arbitrary phase onto the reflected photon. This phase, ϕ,is 1 þ νðL; xÞ the relative phase difference between the classical control fields that establish stationary-light conditions within the νðL; xÞ − νðz; xÞ −iϕ ~ ~ E ðz; x; 0Þ¼ e E ; ð18Þ ← 0 interaction volume (see Fig. 2) and, therefore, can be well 1 þ νðL; xÞ controlled and tuned in a continuous manner. For large d ,the z 0 0 phase is a pure result of the reflection physics and does not where we introduce νðz; xÞ¼ i χ ðz − xÞdz . Assuming depend on any system parameter other than the relative phase that the gate excitation is stored farther than z away from between the two classical control fields. Our setup can thus the medium boundaries, the problem effectively becomes function as a photonic quantum router with an arbitrary and independent of the spin-wave position x. One can continuously tunable reflection phase. This robust condi- ∞ 6 then write νðL; xÞ ≈ d ν, with ν ¼ i dz=ðz þ 2iÞ¼ −∞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ tional phase presents an important distinguishing feature ðπ 1 þ i=3Þ ≈ 1.1 þ 0.3i, and obtain [from Eqs. (13) compared to previous Rydberg-EIT-based protocols [42,58]. and (14)] the following simple expressions for the trans- Figure 5(c)–5(e) shows numerically obtained spectra mission and reflection coefficients: T ðωÞ and R ðωÞ for various values of d . These are 1 1 b compared to the transmission spectrum T ðωÞ in the absence of the stored spin wave. As d increases, one T ðω ¼ 0Þ¼ ; ð19Þ 1 b 1 þ νd finds that the reflection spectrum develops an asymmetry. This asymmetry emerges since the Rydberg level shift VðzÞ νd −iϕ R ðω ¼ 0Þ¼ e : ð20Þ induces a nonsymmetric optical response, where the 1 þ νd positive ðω > 0Þ and negative ðω < 0Þ frequency compo- nents are affected differently. However, the effects of this We plot T ð0Þ and R ð0Þ in Fig. 5(b), along with the spatial 1 1 can be minimized by choosing a sufficiently narrow ~ ~ solutions of E ðz; 0Þ and E ðz; 0Þ within the medium. → ← bandwidth of the target photon, as we discuss in Sec. VIII. The reflection amplitude increases with d , and even- tually dominates the transmission beyond d ≈ 1. VI. COHERENCE PROPERTIES Physically, this can be understood from the effective mass m ∝ 1=l  of the stationary-light polariton and the Another distinguishing feature of the described polariton- abs length 2z of the blockade region. Since both increase switching mechanism is that it maintains EIT conditions and with d ¼ z =l , reflection dominates at large d , where therefore operates at inherently suppressed photon losses. In b b abs b 031007-7 CALLUM R. MURRAY and THOMAS POHL PHYS. REV. X 7, 031007 (2017) 2 2 Fig. 6(a), we show the loss A ¼ 1 − jT ð0Þj − jR ð0Þj as a by dissipative interactions with the incident target photon 1 1 function of d . Remarkably, absorption decreases with [57,62,81], as we discuss below. To this end, we consider the dynamics of the spin- increasing d , even though the target photon is resonantly R R L L coupled to the medium. This, in turn, permits us to work wave density matrix ρ ˆðtÞ≔ dx dyρðx;y;tÞS ðx;tÞj0i 0 0 under conditions of strong light-matter coupling, and stands h0jS ðy;tÞ, where the complex elements ρðx; y; tÞ indicate in marked contrast to conventional Rydberg-EIT schemes the spatial coherence between spin-wave components at where resonant photon coupling implies large interaction- positions x and y. Using the theoretical framework devel- induced losses, A ≈ 1 − exp½−4d [42,54–57,62] [see oped in Ref. , the final spin-wave state after the Fig. 6(a)], and one requires large single-photon detunings interaction with the target photon can be obtained exactly for coherent nonlinear operations [15,16,42,50]. in the cw limit and is given by Absorption in the current situation originates from the fact that the target photon does not switch fully adiabati- 1 1 cally between the slow- and stationary-light polariton ρðx; yÞ¼ 1 þ id 1 þ νðL; xÞ 1 þ ν ðL; yÞ solutions. Specifically, this means that the target photon 6 6 partially populates the bright-state polariton branches L ðz − xÞ − ðz − yÞ × dz ρ ðx; yÞ; depicted in Fig. 3. However, the associated energy cost 0 6 6 ½ðz − xÞ þ 2i½ðz − yÞ − 2i 2 −1 of ∼G =γ ¼ cl ensures that this population is suppressed abs ð21Þ with increasing d and leads to the observed decrease of the loss coefficient A. Thus far we have focused on the dynamics of the target where ρ ðx; yÞ denotes the initial (pure) state of the stored photon, where it is sufficient to consider a localized gate gate excitation. A detailed derivation of Eq. (21) is outlined excitation at a given position in the medium. Storage of the in Appendix B. gate photon, however, generates a spatially delocalized While the spin-wave density ρðx; xÞ remains unaffected collective spin-wave excitation, and the preservation of its , target photon scattering results in partial decoherence, coherent nature is essential for subsequent photon retrieval. i.e., a reduction of the off-diagonal elements of ρðx; yÞ. Respectively, the retrieval efficiency is typically diminished Assuming that the length of the medium is significantly (a) (c) (d) (e) (f) (b (b (b (b (b (((b) b)))) (b) (h) (g) (i) (j) FIG. 6. (a) Loss coefficient A as a function of d (solid line) compared with the conventional polariton blockade mechanism (dashed line). (b) Initial density matrix, ρ ðx; yÞ¼ sinðxπ=LÞ sinðyπ=LÞ, of the stored spin wave for L ¼ 5z . (c)–(f) Final density matrix of the 0 b stored spin wave corresponding to the present coherent polariton-switching mechanism for various indicated values of d . Panels (g)–(j) show the final density matrix for the dissipative polariton blockade, which is shown to cause much stronger decoherence. 031007-8 COHERENT PHOTON MANIPULATION IN INTERACTING … PHYS. REV. X 7, 031007 (2017) longer than the extent of the spin-wave state, one can show blockade, the switching fidelity is given by the nonlinear that ρðx;yÞ≈ð1−AÞρ ðx;yÞ at large distances jx − yj ≫ z . loss coefficient ∼1 − exp½−4d . 0 b b Indeed, this shows explicitly that spin-wave decoherence However, we emphasize that this purely dissipative non- is directly related to the nonlinear photon losses and, linearity fundamentally prevents the realization of a quantum therefore, can be greatly suppressed by increasing d in switch, since the underlying photon scattering completely the present approach. decoheres any quantum superposition state of the gate Figure 6 illustrates this difference between the present excitation. On the contrary, this does not affect the proposed and conventional Rydberg-EIT schemes for an explicit routing approach, which does not rely on photon scattering. initial spin wave with ρ ðx; yÞ¼ sinðxπ=LÞ sinðyπ=LÞ [see The quantum switching fidelity for the present scheme is Fig. 6(b)]. In Figs. 6(c)–6(f), we show the final spin-wave given by the coherent reflection coefficient according to ðquantumÞ state according to Eq. (21) for various values of d .As 2 F ¼jR ð0Þj , with R ð0Þ given by Eq. (20). 1 1 switch expected, the suppressed photon losses, shown by the solid The ability to retrieve the stored gate photon presents line in Fig. 6(a), result in very little spin-wave decoherence, another essential performance aspect relevant for optical as reflected in a marginal deformation of ρðx; yÞ, which transistor operation or applications for nondestructive reduces with increasing d . In Figs. 6(g)–6(j), we plot photon detection [54–57]. This factor is critically deter- the final spin-wave state corresponding to conventional mined by dissipative scattering of target photons, as we Rydberg-EIT conditions [57,62,81]. In this case, the discuss above. To account for the finite retrieval efficiency, virtually complete scattering of the target field, as indicated we adopt the strategy of Ref.  and optimize the gate by the dashed line in Fig. 6(a), causes strong decoherence storage and retrieval efficiency η in the presence of such that turns the initial spin wave into a near-classical decoherence processes by shaping the spatial profile of the distribution of the stored excitation. stored spin-wave mode. We consider the target field in the cw limit, and describe the induced spin-wave decoherence by VII. APPLICATIONS Eq. (21). Spin-wave decoherence for conventional Rydberg EIT is calculated according to Ref. . The overall The demonstrated conditional reflection of the target transistor fidelity is then given by the product of η with field realizes a quantum nonlinear photon router [11,82] in the corresponding switching fidelity we discuss above. which a single gate photon can be used to control or redirect In Fig. 7(a), we compare the different switch fidelities for the flow of target photons between the two optical modes the proposed photon router to the conventional approach ˆ ˆ E and E . This capability facilitates a broad range of → ← based on nonlinear photon scattering. For classical oper- functionalities. ations, the fidelities are nearly identical since the apparent The ability to modify the transmissive properties gain in the storage and retrieval efficiency due to reduced of the medium via gate storage has immediate practical spin-wave decoherence is compensated by the higher applications in the context of optical switching switch fidelity in the dissipative case. More importantly, [9,13,83–89]. Classical switching requires only a gate- however, the proposed routing mechanism also functions photon-induced blocking of the target field transmission as a quantum transistor, which otherwise is fundamentally (via either dissipative scattering or coherent reflection), for impossible  for conventional Rydberg EIT. This is ðclassicalÞ which we can define a fidelity F ¼ 1 − jT ð0Þj , demonstrated by the red lines in Fig. 7(a). switch where T ð0Þ is given by Eq. (19). In previous Rydberg-EIT As indicated by Eq. (20), the present approach permits us schemes [42,54–57,62] relying on a dissipative polariton to imprint any relative phase between the applied control (a) (b) Classical Quantum FIG. 7. (a) Overall switch fidelity, taking into account both the target field switching as well as the gate photon storage and retrieval efficiencies. The gray lines show the fidelity of a classical switch realized for a dissipative polariton blockade nonlinearity (dashed) and the coherent polariton switching nonlinearity (solid). The red lines show the corresponding fidelities for a quantum transistor. (b) Fidelity of a two-photon phase gate realized with the proposed photon router (solid line) compared with the corresponding fidelity of a π-phase gate based on the dispersive phase shift obtained from the polariton blockade mechanism (dashed line). The black dot indicates the critical value of d ∼ 6 below which a dispersive phase shift of π is fundamentally impossible with the polariton blockade mechanism under EIT conditions. 031007-9 CALLUM R. MURRAY and THOMAS POHL PHYS. REV. X 7, 031007 (2017) fields onto the reflected target photon. In this way, the respect to E can be straightforwardly compensated with router can perform robust two-photon phase gate opera- standard techniques or by sending the reflected light tions. The imprinted phase is largely independent of d ,in b through an identical medium without the Rydberg-state contrast to previous schemes [42,58] based on conventional ˆ ˆ coupling. This permits us to reconvert E into E with an ← → Rydberg EIT. There, one requires a large single-photon 2 2 efficiency ∼d =ð1 þ dÞ that is much larger than that of the detuning Δ to suppress the absorptive contribution to the photon router. nonlinear response, along with a large value of d in order In Fig. 8, we show the reflection efficiency, to account for the reduced single atom coupling and achieve a significant phase shift. In Fig. 7(b), we compare the resulting fidelity for a π-phase gate to the gate fidelity R ¼ dωR ðωÞjE ðωÞj ; ð22Þ 1 0 −∞ of the present scheme, F ¼jR ð0Þj [see Eq. (20)]. gate 1 Again, one finds a significant gain by the proposed routing for realistic experimental parameters of the described mechanism for experimentally relevant parameters of coupling scheme and different pulse lengths of the incom- moderate d . In Appendix C, we present a more detailed ing target photon. Its spectrum E ðωÞ is normalized discussion of the classical switch and phase gate based on ∞ 2 the conventional polariton blockade mechanism. according to dωjE ðωÞj ¼ 1. In calculating R,we −∞ also account for a finite linewidth of the Rydberg state γ by including an additional non-Hermitian term VIII. EXPERIMENTAL CONSIDERATIONS S ˆ ˆ −iγ dzS ðz; tÞSðz; tÞ in the Hamiltonian of Eq. (3). Finally, we discuss a concrete physical implementation We take a value of γ =2π ¼ 300 kHz, which was used of the described physics, using Rb atoms as the most to describe recent experiments , accounting for both relevant example for current experiments [37,49,55,56]. the natural linewidth of the Rydberg state and additional Slow-light polaritons involving Rydberg states are being dephasing and decoherence processes. In spite of these employed in a growing number of experiments [15,16], and additional effects, and the frequency asymmetry of R ðωÞ stationary-light polaritons as emerging from the double-Λ (see Fig. 5), the fidelity can approach the ideal limit coupling scheme of Fig. 2(a) have also been demonstrated jR ðω ¼ 0Þj for realistic pulse lengths used in current experimentally [76,90,91]. Following this approach, the experiments [55,56,58]. atomic cloud may be initialized in the jgi≡j5S ;F ¼ 1; 1=2 Interestingly, a finite Rydberg decay rate does not m ¼ 0i ground state. One can then choose the low-lying directly lead to enhanced losses, as in the conventional excited states jp i and jp i from the j5P ;F ¼ 2i → ← 3=2 setting, but instead causes enhanced reflection. This is hyperfine manifold. In this case, choosing the E and because such decay no longer allows for a symmetric − þ E fields to have σ and σ polarizations, respectively, splitting of the dressed states jf i [cf. Eq. (15)] that is ˆ ˆ E will drive the jgi → jp i ≡ j5P ;F ¼ 2;m ¼ −1i responsible for decoupling E and E in the limit of → → 3=2 F → ← vanishing interactions. Stationary-light EIT is, however, transition, while E will drive the jgi→ jp i≡j5P ; ← ← 3=2 F ¼ 2;m ¼þ1i transition. Stationary-light conditions are established by coupling these excited states back to the jdi ≡ j5S ;F ¼ 2;m ¼ 0i hyperfine ground state by 1=2 F the counterpropagating classical fields with Rabi frequen- iϕ cies Ω ¼ Ω and Ω ¼ Ωe , as indicated in Fig. 2(a).We → ← note that using m ¼ 0 ground states for jgi and jdi ensures that the classical Rabi frequencies have equal magnitude, ˆ ˆ along with the coupling strengths of E and E . Finally, → ← taking an nS Rydberg state, σ -polarized light can be 1=2 used to drive the jp i→ jsi¼jnS ;J ¼ 1=2;m ¼ 1=2i ← 1=2 J FIG. 8. Reflection efficiency, according to Eq. (22), for differ- transition. ent indicated durations of a Gaussian incoming target pulse E . Because of the degeneracy of the involved hyperfine The total optical depth is fixed at 2d ¼ 50, γ=2π ¼ 3.05 MHz, and fine-structure manifolds, the optical fields would and the classical Rabi frequencies are Ω=2π ¼ 5 MHz and resonantly drive a number of additional transitions. Ω =2π ¼ 20 MHz. The solid lines include a finite Rydberg-state These additional couplings can, however, be suppressed linewidth γ =2π ¼ 300 kHz . The gate excitation is located by applying a magnetic field along the light propagation at the center of the medium and generates an interaction region pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ axis. The resulting Zeeman shifts can be made sufficiently 6 with z ¼ C γ=Ω ∼ 8.7 μm, as obtained for the 100S state b 6 1=2 strong to isolate the desired transitions upon adjusting the 87 of Rb atoms . The dashed line indicates the ideal limit field frequencies accordingly to maintain a resonant of a vanishing photon bandwidth and γ ¼ 0, corresponding coupling. The resulting slight frequency shifts of E with to Eq. (20). 031007-10 COHERENT PHOTON MANIPULATION IN INTERACTING … PHYS. REV. X 7, 031007 (2017) still supported for a finite γ , such that when the broadening Rydberg blockade of dark-state polaritons [15,16,36,42]. of the Rydberg state becomes large, the entire medium thus This in turn permits us to achieve a strong optical response behaves like a stationary-light medium and is associated and high operational fidelities already at moderate values of with large reflection, rather than large absorption. However, d that are well within the domain of accessible atomic for typical Rydberg decay rates, this unwanted reflection is densities, where additional decoherence processes  can negligible. be kept at a minimum. The Rydberg-state loss and decoherence also limits the The results of this work thus indicate that the exploration target photon transmission in the absence of a stored gate of nonlinearity mechanisms beyond the conventional excitation. For conventional Rydberg EIT, this is respon- Rydberg blockade holds new perspectives for all-optical quantum computing with Rydberg systems, such as the sible for enhanced scattering, and the amplitude trans- recent combination of EIT and dipolar Rydberg-state mission decreases exponentially with the total optical depth exchange interactions . d according to jT ð0Þj ≈ exp½−dðγ =Γ Þ, where Γ ¼ 0 S EIT EIT While we focus in this work on the interaction of a single Ω =γ and γ ≪ Γ is assumed. However, for the present S EIT propagating photon with a single stored gate excitation, the system, the transmission decreases polynomially as interaction of multiple freely propagating photons also jT ð0Þj ≈ 1=½1 þ dðγ =Γ Þ, and therefore yields a sub- 0 S EIT holds interesting perspectives. The described type of non- stantial improvement for large γ . For the parameters of linearity could be used to coherently filter out highly Fig. 8 this gives jT ð0Þj ≈ 0.95, and thus does not limit the nonclassical states of light from a classical light source, overall routing fidelity. such as single-photon states [43,49] or strongly correlated Finally, considering the finite spatial extent of both the trains of photons . As compared to the dissipative target and the gate photon, one needs to account for another nonlinearity based on the polariton blockade considered in effect, namely, entanglement between the target field and previous work [43,49,63], the present coherent nonlinear the gate excitation emerging from the Rydberg-Rydberg reflection mechanism might, for example, require signifi- atom interaction. More specifically, the reflection time of cantly lower optical densities for generating spatially the target photon becomes correlated with the spatial ordered photons  and, thereby, make such exotic states position of the gate excitation in the atom cloud. While of light accessible with present experimental capabilities. this presents another decoherence mechanism for the Moreover, the broken left-right symmetry of the under- collective gate excitation, its actual effect is greatly sup- lying coupling scheme in Fig. 2(a) implies that the optical pressed as long as the target pulse length exceeds the response of the medium acquires a strong dependence on the characteristic difference of possible reflection times. The propagation direction of the light. This effect can be inves- maximum time difference can be estimated as τ ≈ tigated in multiphoton transmission measurements, where, 2 2 dðγ=Ω þ γ=Ω Þ , which corresponds to the time delay for example, a head-on collision of two photons would between photons reflected at z ¼ 0 and z ¼ L, respectively. generate a pair of copropagating photons with a strong bias Using the same parameters for d, γ, Ω, and Ω as used in in one direction. We emphasize that this symmetry breaking is Fig. 8, we find that τ ∼ 0.5 μs. This is an order of magnitude a nonlinear effect, which is in contrast to the chiral linear shorter than the largest pulse length in Fig. 8 and typical response currently being explored in nanoscale waveguides pulse lengths used in current experiments [55,58]. [94–97]. The availability of such unusual types of photon interactions combined with the freely tunable reflection phase IX. SUMMARY AND CONCLUSIONS suggests intriguing perspectives for the collective engineer- ing of nonclassical multiphoton states or the exploration of In summary, we devise a new approach to engineering exotic many-body physics with photons. effective photon interactions via particle interactions in an EIT medium. The basic principle is based on a modification ACKNOWLEDGMENTS rather than a breaking of EIT conditions to achieve a nonlinear alteration of light propagation under low-loss We thank C. S. Adams, M. Baghery, H. Gorniaczyk, M. conditions. We present a specific implementation using Gullans, S. Hofferberth, D. Paredes-Barato, and E. Zeuthen laser-driven Rydberg-atom ensembles which realizes an for valuable discussions. This work was funded by the effective photon-photon interaction that is reflective in EU through the FET Grant No. 512862 (HAIRS), and the character and highly coherent. H2020-FETPROACT-2014 Grant No. 640378 (RySQ), by We demonstrate that in this way a single photon acts like the DFG through the SPP 1929, and by the DNRF. a mirror with a robust and continuously tunable reflection phase and discuss a number of applications entailed by APPENDIX A: PHOTON PROPAGATION such a quantum nonlinear photon router. Here, the EQUATIONS enhanced coherence properties of the developed polar- iton-switching mechanism offer a significant performance Here, we outline the derivation of the propagation gain compared to existing approaches based on the equations in Eq. (10) and provide explicit expressions 031007-11 CALLUM R. MURRAY and THOMAS POHL PHYS. REV. X 7, 031007 (2017) for the susceptibility functions appearing in Eq. (11). ∂ ρðx; y; tÞ¼ i dz½Vðz − xÞ − Vðz − yÞ Starting from the Heisenberg equations derived from the Hamiltonian in Eq. (3), one obtains equations of motion × S ðz; x; tÞSðz; y; tÞ: ðB1Þ for all relevant two-body amplitudes describing a stored gate excitation at position x and a target excitation at Sðz; x; tÞ is again the two-body probability amplitude to position z. Using the shorthand notation E ≡ E ðz; x; tÞ, → → have an jsi Rydberg excitation at position z and a stored E ≡ E ðz; x; tÞ, etc. for convenience, these coupled ← ← gate excitation at position x. Imposing the initial condition equations of motion can be written as ρðx; y; 0Þ¼ ρ ðx; yÞ, the general solution to Eq. (B1) is given by ∂ E ¼ −c∂ E − iGP ; ðA1Þ t → z → → ∂ E ¼ c∂ E − iGP ; ðA2Þ ρðx; y; tÞ¼ 1 þ i dz½Vðz − xÞ − Vðz − yÞ t ← z ← ← ∂ P ¼ −iGE − iΩD − γP ; ðA3Þ t → → → × dτS ðz; x; τÞSðz; y; τÞ ρ ðx; yÞ: ðB2Þ −∞ −iϕ ∂ P ¼ −iGE − iΩe D − iΩ S − γP ; ðA4Þ t ← ← S ← Solving the target field dynamics governed by iϕ ∂ D ¼ −iΩðP þ e P Þ; ðA5Þ Eqs. (A1)–(A6) in the cw limit, one can obtain a solution t → ← for the Rydberg spin-wave amplitude Sðz; xÞ in terms of the photonic amplitudes: ∂ S ¼ −iΩ P − iVðz − xÞS: ðA6Þ t S ← GΩ E ðz; xÞ − E ðz; xÞ Transforming to frequency space, we then obtain a series of S ← → R Sðz; xÞ¼ : ðB3Þ −1=2 ∞ −iωt 2 γ Ω =γ þ 2iVðz − xÞ equations for E ðz;x;ωÞ¼ð2πÞ dte E ðz;x;tÞ, → → S −∞ −1=2 ∞ −iωt E ðz; x; ωÞ¼ð2πÞ dte E ðz; x; tÞ etc. Upon ← ← −∞ Using the solutions for E ðz; xÞ and E ðz; xÞ according → ← ~ ~ solving for P ðz; x; ωÞ and P ðz; x; ωÞ in terms of the → ← to Eqs. (17) and (18), the target spin-wave amplitude can photonic amplitudes, and inserting these into Eqs. (A1) and be written as (A2), one immediately arrives at a closed system of ~ ~ equations for E ðz; x; ωÞ and E ðz; x; ωÞ as given by → ← GΩ 1 1 Eqs. (10) and (11). The explicit expressions for Sðz; xÞ¼ − E ; ðB4Þ γ Ω =γ þ 2iVðz − xÞ 1 þ νðL; xÞ χ ðz; ωÞ, χ ðz; ωÞ, and χðz; ωÞ appearing in Eq. (11) → ← are given by where the amplitude E describes the incoming target ∞ 2 photon, with the normalization dtjE j ¼ 1=c. S −∞ ðξ − Þ ω G ω−VðzÞ Inserting this result into Eq. (B2) and carrying out the χ ðz; ωÞ¼ − þ ; ðA7Þ Ω 4 c c S time integration, one arrives at the result given in Eq. (20), ξðξ − Þ − ω−VðzÞ ω where distances have been rescaled by z , defined accord- 2 ing to Vðz Þ¼ Ω =γ. ω G ξ χ ðz; ωÞ¼ − ; ðA8Þ Let us finally consider Eq. (21) in the limit jx − yj ≫ z . ← 2 c c S Ω ξðξ − Þ − In this case, the spatial integral has two separate contri- ω−VðzÞ butions around z ∼ x and z ∼ y, such that the solution for 2 2 G Ω 1 ρðx; yÞ can be approximated by χðz; ωÞ¼ ; ðA9Þ c ω S Ω ξðξ − Þ − ω−VðzÞ νðL;xÞþ ν ðL;yÞ ρðx;yÞ≈ 1− ρ ðx;yÞ: ðB5Þ ½1 þ νðL;xÞ½1 þ ν ðL;yÞ where we introduce ξ ¼ ω þ iγ − Ω =ω. Assuming further that the medium length L greatly exceeds the spatial extent of the gate spin wave, we can APPENDIX B: SPIN-WAVE use νðL; xÞ¼ νðL; yÞ¼ d ν and obtain the expression DECOHERENCE DYNAMICS Here, we outline the solution to the scattering-induced 2d Re½ν ρðx; yÞ ≈ 1 − ρ ðx; yÞðB6Þ decoherence dynamics of the stored gate spin wave, as 0 j1 þ d νj given by Eq. (21). To this end, we begin with the equation of motion for the density matrix elements, ρðx; y; tÞ, of the ¼ð1 − AÞρ ðx; yÞ; ðB7Þ stored spin wave, which can be expressed as 031007-12 COHERENT PHOTON MANIPULATION IN INTERACTING … PHYS. REV. X 7, 031007 (2017) discussed in the main text, where A is the loss coefficient For large d , the gate infidelity is thus given by introduced in Sec. VI. 1 − F ≈ 5π=ð2d Þ. This is the same scaling as for gate b 1 − jR j ≈ 1.69=d obtained from Eq. (20), which, how- 1 b ever, features a significantly smaller prefactor. 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