Theory of Interacting Cavity Rydberg Polaritons
TTheory of Interacting Cavity Rydberg Polaritons
Alexandros Georgakopoulos , Ariel Sommer , and Jonathan Simon Department of Physics and James Franck Institute, University of Chicago, Chicago, IL and Department of Physics, Lehigh University, Bethlehem, PA (Dated: May 21, 2018)Photonic materials are an emerging platform to explore quantum matter [1, 2] and quantum dy-namics [3]. The development of Rydberg electromagnetically induced transparency [4, 5] provideda clear route to strong interactions between individual optical photons. In conjunction with carefullydesigned optical resonators, it is now possible to achieve extraordinary control of the properties ofindividual photons, introducing tunable gauge fields [6] whilst imbuing the photons with mass andembedding them on curved spatial manifolds [7]. Building on work formalizing Rydberg-mediatedinteractions between propagating photons [8, 9], we develop a theory of interacting Rydberg po-laritons in multimode optical resonators, where the strong interactions are married with tunablesingle-particle properties to build and probe exotic matter. In the presence of strong coupling be-tween the resonator field and a Rydberg-dressed atomic ensemble, a quasiparticle called the “cavityRydberg polariton” emerges. We investigate its properties, finding that it inherits both the fastdynamics of its photonic constituents and the strong interactions of its atomic constituents. Wedevelop tools to properly renormalize the interactions when polaritons approach each other, andinvestigate the impact of atomic motion on the coherence of multi-mode polaritons, showing thatmost channels for atom-polariton cross-thermalization are strongly suppressed. Finally, we proposeto harness the repeated diffraction and refocusing of the optical resonator to realize interactionswhich are local in momentum space. This work points the way to efficient modeling of polaritonicquantum materials in properly renormalized strongly interacting effective theories, thereby enablingexperimental studies of photonic fractional quantum Hall fluids and crystals [2, 10, 11], plus photonicquantum information processors and repeaters [12–14].
A. INTRODUCTION
Current efforts to produce and explore the properties ofsynthetic quantum materials take numerous forms, fromultracold atoms [15] to superconducting circuits [16–18] and electronic heterostructures [19, 20] and super-lattices [21]. Cold atom techniques allow for precisecontrol through lattice tuning [22] and Feshbach res-onances [23, 24]. Superconducting quantum circuitspresent an opportunity to create materials from stronglyinteracting microwave photons, as they exhibit excellentcoherence [25], strong interactions [26], and have re-cently been shown to be compatible with low disorderlattices [27], low loss lattice gauge fields [28], and inter-action & dissipation driven phase transitions [29].In parallel, there is now growing interest in creatingmaterials from optical photons. Non-interacting photonshave been Bose-condensed in a resonator using a dye asa thermalization medium [30]; photons have been madeto interact weakly and subsequently Bose condense bycoupling them to interacting excitons [20]. To explorestrongly interacting photonic materials, it has previouslybeen proposed to marry Rydberg electromagnetically in-duced transparency (EIT) tools developed to induce free-space photons to interact [3, 8, 31, 32] with multimodeoptical resonators [2] to control the properties of indi-vidual photons [7], thereby introducing a real mass for2D photons, and effective magnetic fields [6], in conjunc-tion with Rydberg mediated interactions. It was recentlyexperimentally demonstrated that individual cavity pho-tons do indeed hybridized with Rydberg excitations toform “cavity Rydberg polaritons,” quasiparticles [33– 35] that collide with one another with high probability[36].Formal modeling of these complex systems is incom-plete. The properties of interacting free-space Rydbergpolaritons have been explored in the dispersive regime[9], as well as the resonant regime for Van der Waals[37, 38] and dipolar interactions [39]. Effective models ofstrongly interacting two-level cavity polaritons have beendeveloped [40], along with blockade “bubble” approxi-mations that qualitatively reflect the physics of three-level polaritons [41], but to date no effective theories ofthree-level cavity Rydberg polaritons exist which quanti-tatively reproduce the observed strong interactions, as aconsequence of the intricate renormalization of the two-polariton wavefunction once the polaritons overlap inspace.In this paper, we first show that cavity Rydberg po-laritons at large separations are described by the Hamil-tonian H pol ∼ cos θ d H phot + sin θ d H int , where H phot is the Hamiltonian describing the bare cavity-photon dy-namics, determined through the resonator geometry; and H int is the Hamiltonian describing the Rydberg-Rydberginteractions. The polaritons thus inherit properties fromboth photonic and atomic constituents, with the propor-tion of each contribution determined by the dark staterotation angle θ d [42], providing an interaction tuningknob akin to an atomic Feshbach resonance [23]. Inthe remainder of the paper we examine the limitations ofthis model, providing quantitative refinements to variousaspects of it.In section B we begin with the Floquet Hamiltonian fornon-interacting resonator photons [7] and formally couple a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y these photons to an ensemble of Rydberg-dressed three-level atoms residing in a waist of the resonator [2]. Insection C we explore the physics of an individual photonin the resonator, discovering one long lived dark polari-ton (with renormalized mass relative to the bare pho-ton) and two short-lived bright polaritons. In section Dwe generalize to the case of two dark polaritons in theresonator, derive the form of the low-energy polariton-polariton interaction potential, investigate scattering intobright-polariton manifolds as well as the regime of valid-ity of the two polariton picture in the face of interactionsand loss, focusing in section E on collisional loss of po-laritons by dark → bright scattering. Once the interactionenergy becomes larger than the dark/bright splitting thesimple polaritonic picture breaks down, so in section Fwe explore the maximally challenging case of two darkpolaritons in single mode optical resonator, developing aproperly renormalized effective theory of interacting po-laritons (with first principles calculable parameters) thatwe benchmark against a complete (and numerically ex-pensive) microscopic theory. We find excellent agreementin experimentally relevant parameter regimes, pointingthe way to a fully renormalized effective field theory ofmultimode cavity Rydberg polaritons. In section G wedemonstrate that a properly situated Rydberg-dressedatomic ensemble produces interactions between polari-tons that are local in momentum-space. In section Hwe relax the assumption of stationary atoms and investi-gate the effect of atomic motion on polariton coherencein both a single- and multi- mode regimes. Finally, insection I we conclude with a discussion of applications ofcavity Rydberg polaritons to quantum information pro-cessing and strongly-correlated matter.Table of Results Equation PagePolariton Projected Hamiltonian D.2 5Renormalized Theory F.1, F.2, F.3 8Role of Atomic Motion H.1 9Doppler Decoherence H.3 10Crossmode Thermalization H.4 10 B. COUPLING THE PHOTONS TO ANATOMIC ENSEMBLE
Here we explore how the coupling to an ensemble ofthree-level atoms impacts the physics of non-interacting2D resonator photons. We find the emergence of longlived “dark” polaritons, with dynamics similar to thoseof a resonator photon but renormalized mass and har-monic trapping. Off-resonant, nonadiabatic couplings to“bright” polaritons limit the lifetime of the dark polari-tons. We operate in the limit that the light-matter cou-pling energy scale is much larger than the energy scaleof the photonic dynamics within the resonator, makingthe polaritonic quasiparticles a nearly “good” basis for describing the physics, with corrections that we derive.The second quantized Hamiltonian for photons withina single longitudinal manifold of a resonator is givenby [7]: H phot = (cid:90) d x a † ( x ) h phot a ( x ) , where a † ( x ) creates a photon at transverse location x and h phot is the single particle Hamiltonian for a pho-ton within the resonator, typically given by h phot ( x ) = Π m phot + m phot ω trap | x | − i κ . Here x runs over theplane transverse to the resonator axis, κ parametrizes the(mode independent) resonator loss, and Π ≡ i (cid:126) ∇ − e A is the mechanical momentum; the parameters of thisphotonic “Floquet” Hamiltonian are determined by res-onator geometry: mirror locations and curvatures, plusthe twist of the resonator out of a single plane [6, 7].We now insert a Rydberg-dressed atomic ensemble intothe resonator as a tool to mediate interactions betweenthe photons. To this end, the lower (S → P) transition ofthis ensemble is coupled to the quantized resonator field,while the upper (P → Rydberg) transition is coupled toa strong coherent field (see Fig. 1a). Before exploringthe resulting photon-photon interactions, we must firstunderstand how the Rydberg-dressed atoms impact thelinear dynamics of individual photons. The light-mattercoupling induced by the introduction of the atomic en-semble takes the form (in the frame rotating with theresonator- and Rydberg-dressing fields): H at = (cid:90) d x { φ † r ( x, z ) φ r ( x, z ) (cid:16) δ − i γ r (cid:17) + φ † e ( x, z ) φ e ( x, z ) (cid:16) δ e − i γ e (cid:17) (B.1)+ φ † r ( x, z ) φ e ( x, z ) Ω2 ( x, z ) + h . c .φ † e ( x, z ) a ( x ) G ( x, z )2 + h . c . } . Here φ † r ( x, z ) and φ † e ( x, z ) are the bosonic creation op-erators for atomic excitations at 3D location (x,z) in theRydberg and excited (P) states respectively, from the“vacuum” of ground state atoms. γ r and γ e are theFWHM of the Rydberg and excited states respectively. δ e is the detuning of an untrapped, zero-transverse-momentum resonator photon from the atomic line and δ is the two-photon detuning from EIT resonance. Ω( x, z )is the laser induced Rabi coupling between excited (P)and Rydberg states, while G ( x, z ) is the vacuum-Rabicoupling strength between a resonator photon localizedat transverse location x and a collective atomic excita-tion localized at longitudinal location z , and thereforemust reflect the atom density. It does not reflect thetransverse spatial structure of any particular resonatormode, as a † ( x ) creates a transversely localized photon. (a) (b) FIG. 1.
Resonator Rydberg polariton three-level sys-tem and setup schematic. (a)
The atomic ground state | g (cid:105) is coupled to the excited state | e (cid:105) through the cavity mode,with detuning δ e and collective coupling strength G . The con-trol beam then excites the atoms to the Rydberg state | r (cid:105) ,with Rabi frequency Ω and two-photon detuning δ .The hori-zontal displacement of the energy levels indicates a change incircular polarization. (b) The four mirror resonator supportsrunning wave modes and two waists, allowing for both realand momentum space interactions. The primary (real space)atomic ensemble sits in the lower waist, where it mediatesphoton-photon interactions by admixing in a Rydberg stateusing the blue control field.
Indeed it may be written as G ( x, z ) ≈ d ge (cid:113) L res ρ ( x ) (cid:126) ω ge (cid:15) ,where L is the length of the atomic ensemble along theresonator axis, L res is the length of the resonator itself, d ge is the dipole moment of the atomic transition coupledto the optical resonator, ω ge is the angular frequency ofthis transition, and ρ ( x ) is the number density of atomsat location x .Here and throughout, we incorporate losses throughnon-Hermitian Hamiltonians rather than through Lind-bladian master equations. This allows us to identify theimaginary part of one- and two- particle eigenstates withparticle decay rates ([43] III.B.2). C. POLARITON BASIS
The atomic density distribution exists in three dimen-sions, while the manifold of nearly-degenerate resonatormodes to which the atoms couple is two-dimensional(here we assume G (cid:28) cL res , ensuring that the atomscouple to only a single longitudinal manifold of theresonator). In order to develop a formalism of two-dimensional polaritons, we define longitudinally delocal-ized, transversely localized collective atomic excitationoperators (for k laser and k cav the wave-vector magnitudesof the coupling-laser and resonator fields, respectively),normalized to ensure a mode-independent bosonic com-mutation relation. To this end we choose the minimalcase of an atomic ensemble of uniform density, and auniform coupling-field that propagates counter to the res- onator field: φ † e ( x ) ≡ (cid:114) L (cid:90) L − L d z φ † e ( x, z ) e − ik cav z φ † r ( x ) ≡ (cid:114) L (cid:90) L − L d z φ † r ( x, z ) e i ( k laser − k cav ) z . We can now rewrite the atomic Hamiltonian as: H at = (cid:90) d x [ W † ( x )][ h at ][ W ( x )]where [ W † ] ≡ ( a † ( x ) φ † e ( x ) φ † r ( x ))and [ h at ] = G G δ e − i γ e Ω2 δ − i γ r . Here G ≈ d ge (cid:113) LL res ρ (cid:126) ω ge (cid:15) , where L is the length of theatomic ensemble along the resonator axis. Note that [ h at ]has no position dependence. In the basis where [ h at ] isdiagonal, the resulting creation operators are the gen-erators of three varieties of polaritons: one dark (withlittle to no excited state participation, depending on κ and γ r ), sandwiched between two bright (with large ex-cited state participation). We are primarily concernedwith the long-lived and strongly interacting dark polari-tons, but will include off-resonant couplings to the brightpolaritons to accurately model dark-polariton lifetime.We diagonalize [ h at ] = (cid:80) µ m ˜ µ m (cid:15) m , where m is anelement of [d,b − ,b + ], meaning [dark, lower bright, upperbright], (cid:15) m is the energy of (ket) µ m and (bra) ˜ µ m andthe µ m , ˜ µ l satisfy the generalized orthogonality condition(due to the non-hermicity of h at ): ˜ µ l (cid:5) µ m = δ lm . Notethat because of the non-hermicity of h at , ˜ µ m (cid:54) = µ † m .The resulting polariton creation and annihilation op-erators are thus: χ † j ( x ) = [ W † ] · µ j ,χ j ( x ) = ˜ µ j · [ W ] . For notational convenience only have we named thepolariton creation/annihilation operators “ χ † j ( x )” and“ χ j ( x )”; these operators are not precisely Hermitian con-jugates of one another, but are instead defined to preservethe bosonic commutation relations: [ χ j ( x ) , χ † k ( x (cid:48) )] = δ jk δ ( x − x (cid:48) ). We may now write H at as: H at = (cid:88) m (cid:90) d x χ † m ( x ) χ m ( x ) (cid:15) m . The last step in writing H tot in the polariton basis is todecompose a † ( x ) , a ( x ) into polariton field operators: a † ( x ) = (cid:88) j c j χ † j ( x ) ,a ( x ) = (cid:88) j ˜ c j χ j ( x ) , where the c j , ˜ c j are elements of the inverse µ, ˜ µ matrices: c j ≡ µ − cav,j , ˜ c j ≡ ˜ µ − cav,j and the index “cav” denotes thephotonic slot of µ − , ˜ µ − . H tot can now be written as: H tot = (cid:88) ij (cid:90) d x c i ˜ c j χ † i ( x ) h ijtot ( x ) χ j ( x ) , where h ijtot ( x ) ≡ δ ij (cid:15) i + h phot ( x ).We will operate in the limit that the difference of theeigenvalues of h at (the “dark”-“bright” spitting) is muchlarger than the spectrum of h phot , making the [ d, b + , b − ]basis that diagonalizes the atomic Hamiltonian a near-diagonal basis for the multimode system. This is equiv-alent to the statement that a particle oscillating in thetrap is more accurately described as a polariton ratherthan a photon if the light-matter coupling is much greaterthan the transverse optical mode spacing. To first orderin h phot /(cid:15) , the Hamiltonian projected into the dark po-lariton manifold is: H poltot = (cid:90) d x c d ˜ c d χ † d ( x )( (cid:15) d + h phot ( x )) χ d ( x ) . For γ r = δ = 0: (cid:15) d = 0 and c d ˜ c d = Ω Ω + G = cos θ d ,where θ d is the dark state rotation angle [42]. We thenhave: H poltot = cos θ d (cid:90) d x χ † d ( x ) (cid:20) Π m phot + 12 m phot ω trap | x | − iκ (cid:21) χ d ( x ) . (C.1)Thus, we see that to lowest order in ratio of the pho-tonic dynamics energy scale to the atomic coupling en-ergy scale ω trap Ω + G , the atoms simply slow down all pho-tonic dynamics and loss by a factor of cos θ d . The dom-inant correction to this story is a second-order resonator( h phot ) -induced dark → bright coupling, producing an ef-fective Hamiltonian [43] (in the above limits): δH tot ≈ (cid:88) j ∈ [ b − ,b + ] (cid:90) d x H phot χ † j ( x ) χ j ( x ) H phot (cid:15) j − (cid:15) d For a dark-polariton in an eigenstate with energy E = Ω G +Ω δ (cid:28) (cid:112) Ω + g ,where δ is the detuning ofthe corresponding bare photon eigenstate from EIT res-onance, the correction is largely imaginary (assuming forsimplicity that δ e = 0 , g, Ω (cid:29) γ e ): δ Γ pol ≡ − i (cid:104) δH tot (cid:105) ≈ Ω + G G Ω + G δ Ω + G γ e = 2 tan θ d E Ω + G γ e (C.2)Thus we see that loss is maximized (at fixed G + Ω )for a polariton which is equal parts photon and Rydbergexcitation, yielding δ Γ pol ≈ E Ω + G γ e , a worst-case ap-proximation we will employ for simplicity going forward.Additional contributions to polariton loss arise frominhomogeneous broadening of the Rydberg manifold, e.g.atomic motion and electric field gradients (to which Ry-dberg atoms are susceptible due to their large DC po-larizability [13]). Such processes generate a ladder ofcouplings from the collective Rydberg state with thesymmetry of the resonator mode into modes orthogo-nal to it (which therefore bright). We explore how (ran-dom) atomic motion induces polariton decoherence in SecH; for inhomogeneous E-fields, the coupling rate to thebright manifold is γ b ≈ αEδE , where α is the DC polar-izability of the Rydberg state, E is the DC electric fieldat the atomic sample, and δE is the field-variation acrossit. The resultant broadening of the dark manifold is then(in the limit Ω (cid:29) γ e ) [35]: δ Γ pol ∼ γ e γ b Ω . (C.3)It is instructive to compare this result with the lossinduced by detuning a resonator mode out of the EITwindow (Eqn. C.2). Both channels are quadratically sup-pressed, but the suppression factor is different, emphasiz-ing the distinction between the underlying physical pro-cesses: detuning from the EIT window couples to brightpolariton manifolds that live in the resonator and arethus suppressed by both the light-matter-coupling- andcontrol- fields, while inhomogeneous broadening couplesto non-resonator bright polariton manifolds that “see”only the control- field. D. POLARITON-POLARITON INTERACTIONS
The interaction between Rydberg atoms will result inan interaction between polaritons, much as in the 1Dfree-space situation [3, 4, 38, 44]. In the limit that theinteraction length-scale is comparable to the mode-waistof the resonator, there can be a substantial renormaliza-tion of the collective atomic excitation which we inves-tigate in Sec F. For now, we assume sufficiently weakinteractions that photonic- and collective-atomic- com-ponents of the polariton wave-function share the samespatial structure; note that these interactions can stilldominate over kinetic and potential energies, as well asparticle decay rates, so this need not be a “weakly in-teracting” polaritonic gas in the traditional (mean-field)sense.The bare 3D interaction between two Rydberg atomstakes the form V ( x − x (cid:48) ) = c r ( θ ) C | x − x (cid:48) | [40], where c r ( θ ) is theangular dependence of the interaction. S-Rydberg atomshave radially symmetric wavefunctions and so c r ( θ ) ≈ c r ≡
1. In the second quantized picture, for a thin atomiccloud (thickness T (cid:28) d , where d is the cavity analogof the blockade radius [8]) the 2D-projected interactiontakes the form (with ˜ V ( x ) ≈ T × V ( x ; z = 0)): H int = 12 (cid:90) (cid:90) d x d x (cid:48) φ † r ( x ) φ † r ( x (cid:48) ) ˜ V ( x − x (cid:48) ) φ r ( x (cid:48) ) φ r ( x ) .φ † r ( x ) may be written in the polariton basis in analogy tothe way a † ( x ) was written in the polariton basis in thepreceding section: H int = 12 (cid:88) ijkl ∈ [ d,b − ,b + ] d i d j ˜ d k ˜ d l (cid:90) (cid:90) d x d x (cid:48) χ † i ( x ) χ † j ( x (cid:48) ) ˜ V ( x − x (cid:48) ) χ k ( x (cid:48) ) χ l ( x ) . (D.1)Here d i , ˜ d j are matrix elements of the inverse µ, ˜ µ matri-ces: d j ≡ µ − ryd,j , ˜ d j ≡ ˜ µ − ryd,j and the index “ryd” denotesthe Rydberg slot of µ − , ˜ µ − . In the absence of Rydbergloss and 2-photon detuning, d d = sin θ d .If the interaction energy ˜ V is small compared to thesplitting between dark- and bright- polariton branches(Sec. E explores the couplings and loss that violatethis condition), the diagonal elements of H int domi-nate, yielding a lowest-order polariton-projected effectiveHamiltonian: ˆ P d ( H tot + H int ) ˆ P d =cos θ d (cid:16)(cid:90) d xχ † d [ Π m p + 12 m p ω t | x | − iκ χ d (cid:17) + 12 sin θ d (cid:18)(cid:90) (cid:90) d x d x (cid:48) ˆ n d ( x )ˆ n d ( x (cid:48) ) ˜ V ( x, x (cid:48) ) (cid:19) , (D.2)where ˆ P d is dark-polariton projection operator, and wehave defined the dark polariton number density operatorˆ n d ( x ) = χ † d ( x ) χ d ( x ).By tuning the dark-state rotation angle θ d (via atomicdensity and control-field intensity) it is possible to movefrom a weakly interacting gas of “nearly-photonic” po-laritons (for θ d ≈
0) to a strongly interacting gas of“nearly-Rydberg” polaritons (for θ d = π ) and explore the correlations which then develop [2]. In this latterlimit it is likely that the interactions ˜ V become com-parable to the dark-bright splitting, so the interactionpotential must be renormalized as explored in section F. E. INTERACTION DRIVEN POLARITON LOSS
The polariton-projected field theory of the preced-ing sections neglects loss from collisional coupling tobright manifolds. We investigate processes of this sortby exploring the minimal model of two δ -interactingdark polaritons in the TEM mode of a resonator, with˜ V ( x, x (cid:48) ) = U eff δ ( x − x (cid:48) ), where U eff is a phenomenolog-ical interaction strength. Such an interaction couples thetwo dark polariton state | dd (cid:105) to a final two-polariton state | f (cid:105) with Rabi frequency Ω dd → f = (cid:104) f | ˜ V | dd (cid:105) . We considertwo final states: (1) one each upper and lower bright po-laritons ( | b + b − (cid:105) ), which is energetically degenerate with | dd (cid:105) , and (2) dark- and (upper/lower) bright- polariton( | db (cid:105) ), which is off-resonant, but sufficiently spectrallybroad that its enhanced matrix element makes it impor-tant. (a) (b) FIG. 2.
Interaction driven loss. (a)
In the rotating frame,the dark polariton | d (cid:105) has zero energy. The upper and lowerbright polariton branches ( | b + (cid:105) and | b − (cid:105) respectively) are sep-arated in energy from the dark polariton by the coupling andcontrol lasers, giving a detuning of ± √ G +Ω . (b) Two darkpolaritons that experience a contact interaction U eff can be-come a pair of bright polaritons | b + b − (cid:105) (upper diagram) ora dark and (upper- or lower- branch) bright polariton | db ± (cid:105) (lower diagram). Dark polaritons are depicted by a straightblack line while the wavy gray lines are bright polaritons.The form of the additional dark polariton loss introduced bythese scattering events are determined by how “resonant”the processes are: the ∼ energy conserving | dd (cid:105) → | b + b − (cid:105) process introduces a loss of Γ dd → bb = sin θ d U eff γ e , while the | dd (cid:105) → | db ± (cid:105) coupling is “off-resonant” and so its additionalloss Γ dd → db = sin θ d sin θ d γ e G U eff is suppressed by thelight-matter coupling field. In a frame rotating with the cavity and control fields( δ e = δ = 0), the dark polaritons have zero energyand the upper/lower bright polariton branches are en-ergetically shifted by ± √ G +Ω (Fig. 2). The process | dd (cid:105) → | b + b − (cid:105) process is thus energy conserving, withcollisional Rabi coupling given by:Ω dd → bb = (cid:104) b + b − | ˜ V | dd (cid:105) = √ θ d θ d U eff = 12 √ θ d U eff . Because this process is resonant, it induces a lossΓ dd → bb = Ω bb Γ bb , where Γ bb = 2 γ e is the intrinsic loss ofthe bright polariton branches which are predominantlycomposed of the lossy P-state. Plugging in, the loss rateis: Γ dd → bb = sin θ d U eff γ e . (E.1)This loss depends heavily on the interaction strength be-tween the Rydberg atoms and the dark state rotationangle. Making the particles more Rydberg-like createsstronger interactions and increases the loss rate; simi-larly, using a higher principle quantum number increases U eff , further enhancing the loss.We now investigate the second scattering process, | dd (cid:105) → | db (cid:105) (upper or lower bright polariton). Follow-ing the same procedure, the collisional Rabi frequencyis: Ω dd → db = (cid:104) db | ˜ V | dd (cid:105) = √ θ d θ d U eff . Since this is an off-resonant process (final and initial stateare detuned by ∆ = √ G +Ω ), the resulting loss rate isΓ dd → db = Ω db ∆ Γ db , with Γ db = γ e (we ignore resonator-and Rydberg- loss, as bright-state loss is dominated byp-state loss in alkali metal atom Rydberg cQED exper-iments [36]). Combining these effects/approximationsyields: Γ dd → db = sin θ d θ d U eff G γ e . (E.2)Once again, stronger interactions and a more Rydberg-like character for the polaritons increases the loss fromthis process but due to the off-resonant nature of thisprocess there is a quadratic suppression from the light-matter coupling which separates the bright- and dark-polaritons in energy.We can compare the two loss processes:Γ dd → bb Γ dd → db = sin θ d θ d G γ e . (E.3) In the limit G (cid:29) Ω , γ e ( θ d → π ⇒ sin θ d → , sin θ d → | dd (cid:105) → | db (cid:105) loss channel will dominate, while inthe opposite limit, interaction-driven loss is dominatedby the ∼ energy conserving process. F. EFFECTIVE THEORY FOR INTERACTINGPOLARITONS
The simple polariton-projected interacting theory in-troduced in section D is an accurate description only forpolaritons whose interaction energy is less than the EITlinewidth [35] for all pairs of atoms comprising the polari-tons. As two polaritons approach one another and theirwavepackets begin to spatially overlap, some terms in theinteraction energy diverge; full numerics (see SI of [36]for details of the approach) reveal that the two-polaritonwavefunction is renormalized to suppress such overlap,at the cost of additional (finite) interaction energy, andloss. We now explore the extreme case of this physics:a single-mode optical resonator that is moderately-to-strongly blockaded, to develop a low-dimensional effec-tive model in the basis of near-symmetric collective statesthat the full numerics.The “brute force” numerical approach that we havepreviously employed accounts for the three-level struc-ture of each atom in the atomic-ensemble, and the in-teractions between atoms. It accurately reproduces ob-served correlations [36] at the expense of a Hilbert spacewhich grows as N m , where N is the number of atomsin the atomic ensemble and m the number of polaritonsin the system; including multiple resonator transversemodes to allow for motional dynamics of the polaritonsrapidly becomes computationally intractable. The prob-lem of an extremely large Hilbert space is exacerbatedsince many-body physics [2, 45] demands both multipleresonator modes and significantly more than two exci-tations, which makes numerically computing the behav-ior of the system completely untenable without a coarse-grained effective theory.To the extent that “polaritons” are well-defined collec-tive excitations whose atomic spatial structure reflectsthe cavity mode functions, it should be possible to de-velop an effective theory whose Hilbert space size is in-dependent of the atom number, making explorations ofmultimode/many-body physics tractable. In this sec-tion we demonstrate, for a single optical resonator mode,an approach to handle the suppression of short-rangedouble-excitation of the Rydberg-manifold, arriving at acoarse-grained effective theory including both dark andbright polaritons whose parameters may be calculatedfrom first principles.Consider two excitations, either atomic or photonic, inan atomic ensemble coupled to a single-mode resonator.In the absence of interactions, we can explicitly write outthe collective states that couple to the resonator field: | CC (cid:105) , | CE (cid:105) , | CR (cid:105) , | EE (cid:105) , | ER (cid:105) , | RR (cid:105) . These states rep-resent two excitations as photons in the resonator, one (a) (b) (c) (d) (e) FIG. 3.
Comparing the effective field theory to full numerics.
Effective theory renormalized interactions and couplings(blue line) and full numerical model (orange line) are compared by looking at the temporal intensity autocorrelation function g ( τ = 0), which characterizes the strength of interactions in the system. Common parameters are (unless varied in the plot ormentioned otherwise): G = 2 π × π × . δ e = 0 MHz, C = 2 π ×
56 THz µ m (corresponding approximatelyto the interaction strength of the 100S Rydberg state), N atom = 700, γ e = 2 π × κ = 2 π × γ r = 2 π × .
15 MHz, δ c = 0 MHz, w c = 14 µ m. (a) g ( τ = 0) vs probe laser detuning. Scanning the probe laser frequency affects both the detuningto the P-state and the two-photon detuning to the Rydberg state. (b) g ( τ = 0) vs P-state detuning δ e . The detuning of theP-state δ e is varied while keeping the two photon detuning δ zero. (c) g ( τ = 0) vs light-matter coupling field strength G , g ( τ = 0) vs probe detuning δ l (inset). The light-matter coupling strength per atom g is varied from 2 π × .
07 MHz to 2 π × (d) g ( τ = 0) vs control field strength Ω. There is good agreement betweenthe full numerics and the renormalized effective theory except for a small region at about Ω = 2 π × − (e) g ( τ = 0) vsRydberg-Rydberg interaction coefficient C . C is varied from 2 π × MHz µ m to 2 π × MHz µ m , which correspondsto Rydberg states in the range n ∼ π × photon and one p-state excited atom, one photon andone Rydberg atom, two excited p-state atoms, one p-state and one Rydberg atoms and two Rydberg atomsrespectively. The Hamiltonian is closed in this basis, andtakes the form [46]: H = | CC (cid:105) | CE (cid:105) | CR (cid:105) | EE (cid:105) | ER (cid:105) | RR (cid:105)− iκ G √ G √ − i κ +˜ γ e G √ Ω2 − i κ + γ r G G √ − i ˜ γ e Ω √
00 0 G √ − i ˜ γ e + γ r √ Ω √ − iγ r where ˜ γ e ≡ γ e + 2 iδ e is a complex linewidth incorpo-rating the P-state detuning. The above basis and corre-sponding Hamiltonian no longer accurately describe thephysics once the Rydberg-Rydberg interactions becomecomparable to the dark-bright splitting: under such con-ditions, the | RR (cid:105) is renormalized due to Zeno suppressionof excitation of Rydberg-atom-pairs at small separation.We posit that the model can be “fixed” by consideringcoupling to a new collective “two-Rydberg” state | (cid:103) RR (cid:105) ,where the tilde signifies that the relative two-Rydbergamplitudes are renormalized by interaction; furthermore,the coupling from | ER (cid:105) to | (cid:103) RR (cid:105) will no longer be Ω √ .To ascertain the form of the state | (cid:103) RR (cid:105) , we will ex-amine the equations of motion in the frequency domain under the non-Hermitian Hamiltonian in the bare-atomicbasis within the two- excitation manifold. We work ina frame that rotates with an energy 2Ω p , convenient for performing scattering experiments of pairs of photons in-jected by a probe at energy Ω p . We assume that whilethe state | RR (cid:105) is renormalized by the interactions, thestate | ER (cid:105) is not , and reflects the non-interacting polari-tonic wave-functions of the preceding sections; this is thecentral assumption of this section, and is validated bynumerics. Corrections to | ER (cid:105) would enlarge the Hilbertspace and may be included as higher-order terms in theeffective theory.Following notation from [36] SI L, the equation ofmotion for the amplitude of two Rydberg excitationsin atoms α and β , C αβRR , is given by i p C αβRR = i (cid:16) U αβRR + 2 δ + iγ r (cid:17) C αβRR + i Ω (cid:16) C αβER + C αβRE (cid:17) , where wehave assumed that the control field Ω is uniform acrossthe atomic ensemble. Here C αβER and C αβRE are the am-plitudes to have P- and Rydberg- excitations in atoms α, β and β, α , respectively, and satisfy C αβER = C βαRE .The assumption that | ER (cid:105) is not renormalized by theinteractions is equivalent to C αβER = g α g β (cid:80) ν | g ν | . Plug-ging this expression into the equation of motion for C αβRR yields the un-normalized two-Rydberg state amplitude: C αβRR = Ω gαgβ (cid:80) ν | gν | ( U αβRR +2˜ δ r ) , where we defined the complex detun-ing ˜ δ r = δ + i γ r − Ω p . We can now write the normalizedcollective state | (cid:103) RR (cid:105) , its effective interaction energy ˜ U and effective coupling ˜Ω √ to | ER (cid:105) as: | (cid:103) RR (cid:105) = (cid:80) αβ g α g β U αβRR +2˜ δ r | R α R β (cid:105) (cid:113)(cid:80) µν | g µ g ν U µνRR +2˜ δ r | , ˜ U = (cid:104) (cid:103) RR | U | (cid:103) RR (cid:105) = (cid:80) αβ | g α g β U αβRR +2˜ δ r | U αβRR (cid:80) µν | g µ g ν U µνRR +2˜ δ r | , ˜Ω √ (cid:104) (cid:103) RR | (cid:88) j Ω2 (cid:0) σ jer + σ jre (cid:1) | ER (cid:105) = √ Ω (cid:80) αβ | g α g β | U αβRR +2˜ δ r (cid:113)(cid:80) µν | g µ g ν U µνRR +2˜ δ r | (cid:113)(cid:80) µν | g µ g ν | . (F.1)(F.2)(F.3)In the extreme limit of strong interactions acrossall space U αβRR (cid:29) ˜ δ r : ˜ U = (cid:80) αβ | gαgβ | UαβRR (cid:80) µν | gµgν | ( UµνRR ) = C w c (cid:82) d ˜ A d ˜ A (cid:48) e − r r (cid:48) ˜ d (cid:82) d ˜ A d ˜ A (cid:48) e − r r (cid:48) ˜ d ≈ C w c , where w c is the modewaist; the pre-factor makes this interaction substantiallyweaker than one might na¨ıvely anticipate- the interac-tion predominantly arises from particles separated by ∼ . w c , and not w c .We next benchmark the validity of this effective the-ory against a full microscopic numerical model [36]. Asa figure of merit we have chosen the temporal intensityautocorrelation function g ( τ ), which compares the rateat which pairs of photons escape the resonator with sepa-ration in time of τ to what would be expected for uncor-related photons escaping at the same average rate; in anexperiment where our only access to the Rydberg physicsis through photons leaking from the resonator, g ( τ )characterizes the strength of polariton interactions, with g ( τ = 0) (cid:28) g (0) vs. probe detuning δ l (Fig. 3a), g (0) vs. p-state detuning δ e (Fig. 3b), light-matter coupling strength G (Fig. 3c), Rydberg controlfield strength Ω (Fig. 3d) and van der Waals interactioncoefficient C (Fig. 3e) between brute-force numerics ofmany individual three-level atoms and the effective the-ory developed above. It is apparent that our approachlargely agrees with the full numerical model up to “noise”arising from randomness in the atom locations. We ex-pect that residual deviations can be parameterized as cor-rections to ˜Ω and ˜ U due to coupling to bright-polaritonmanifolds, and a slight enlargement of the Hilbert spaceto incorporate the additional states coupled to. G. MOMENTUM-SPACE INTERACTIONS
A resonator which exhibits manifolds of nearly-degenerate modes may be understood as a self-imaging cavity: a localized spot living within such a manifold isre-focused onto itself after a full transit around the cav-ity. In-between, the localized spot undergoes diffraction,equivalent to the time-of-flight expansion of a free atomicgas [47]. Indeed, what the optics community calls a“fourier plane” is what a cold-atom experimentalist calls“momentum space”: the momentum of the photon in theinitial (“reference” or “image”) plane has been mappedonto its position in the “fourier plane” [48]. Accordingly,it should be possible to realize interactions which are lo-cal in momentum-space by placing a Rydberg-dressedatomic-ensemble that mediates these interactions in afourier- or nearly-fourier- plane of the optical resonator.We explore this idea formally by extending the cav-ity Floquet Hamiltonian engineering tools of our priorwork [7] to the interacting regime. A thin gas of Rydberg-dressed atoms placed in a plane separated from the“reference”/“image” plane by a ray-propagation matrix M = ( a bc d ) produces interactions of the form: H int = 12 sin θ d (cid:90) (cid:90) d x d x (cid:48) χ † d ( x ) χ † d ( x (cid:48) )˜ V (cid:18) a ( x − x (cid:48) ) − b (cid:126) k (ˆ p − ˆ p (cid:48) ) (cid:19) χ d ( x (cid:48) ) χ d ( x ) . (G.1)For the simple case of a delta-interacting gas of atomsplaced in such an intermediate plane (a distance z from the reference plane), we employ this result totransform an expression where the dark polariton cre-ation/destruction operators and interaction potential arewritten in the intermediate plane to one where the inter-action is transformed and all operators are written in the“reference” plane: H int = 12 sin θ d (cid:90) (cid:90) d x d x (cid:48) χ † d ( x ; z ) χ † d ( x (cid:48) ; z ) δ ( x − x (cid:48) ) χ d ( x (cid:48) ; z ) χ d ( x ; z )= 12 sin θ d (cid:90) (cid:90) (cid:90) d x d δ d∆ χ † d ( x + δ ) χ † d ( x − δ ) e i k z ( δ − ∆ ) χ d ( x + ∆) χ d ( x − ∆) . The resulting polariton interaction is no longer purely lo-cal in real-space, and indeed can “instantaneously” trans-port polaritons through space.The most extreme example of such an interaction oc-curs if the mediating gas is placed in a fourier plane ofthe system, a = 0 , b = fH int = 12 sin θ d (cid:90) (cid:90) d x d x (cid:48) χ † d ( x ) χ † d ( x (cid:48) )˜ V (cid:18) − f (cid:126) k (ˆ p − ˆ p (cid:48) ) (cid:19) χ d ( x (cid:48) ) χ d ( x ) , an interaction that is local in momentum-space. H. IMPACT OF ATOMIC MOTION
In this section we investigate the effects of atomicmotion on the coherence properties of individual Ryd-berg polaritons in both single- and multi-mode regimes.We relax the assumption, employed to this point in themanuscript, that the atoms remain spatially fixed, andinstead allow them to move ballistically through space.The impact of this motion upon the P-state is ignoredbecause the P-state linewidth of an alkali-metal atom( ∼ π × µ K temperatures ( ∼ π × θ d .We incorporate atomic motion into the Hamiltonianin the bare-atom basis by allowing each atom to have atime-dependent coupling-phase to the probe and controlfields resulting from its time-varying position: H = ω c ˆ a † ˆ a | c (cid:105)(cid:104) c | + (cid:88) j ω e | e (cid:105)(cid:104) e | + ω r | r (cid:105)(cid:104) r | + (cid:88) j (cid:110)(cid:104) G j ( x j + v j t ) | e (cid:105)(cid:104) c | + Ω j ( x j + v j t ) | r (cid:105)(cid:104) e | (cid:105) + h . c . (cid:111) , where x j and v j are the positions and velocities of theatoms, drawn from a normal distribution reflecting thesample r.m.s. size and temperature. The effect of atomicmotion, then, is to mix the collective states that coupleto the resonator modes with those that, in the absence ofatomic motion, do not couple to it. To see this formally,we write the Hamiltonian in the basis of the instanta-neous collective eigenstates, resulting in a Hamiltonianof the form: H = (cid:88) m E m ( t ) | m ( t ) (cid:105)(cid:104) m ( t ) | + | ˙ r ( t ) (cid:105)(cid:104) r ( t ) | , (H.1)where | m ( t ) (cid:105) , E m ( t ) are the instantaneous polaritoniceigenstates and their corresponding energies, and | ˙ r ( t ) (cid:105)(cid:104) r ( t ) | is an extra term introduced by this time-dependent change of basis, capturing the effects of atomicmotion in the instantaneous collective Rydberg state | r ( t ) (cid:105) . In what follows, we examine the form of this finalterm for the particular case of twisted resonators whichproduce a Landau level for light [6, 45], so the mode func-tions are Laguerre-Gauss Ψ l ( z ≡ x + iy ) = (cid:113) l +1 πl ! z l e −| z | ,with angular momentum L = l (cid:126) . For a polariton ina mode with angular momentum l (cid:126) , | r ( t ) (cid:105) = | r l ( t ) (cid:105) ,with | r l ( t ) (cid:105) = (cid:80) j e i(cid:126)k · ( (cid:126)xj + (cid:126)vjt ) Ψ l (cid:16) (cid:126)xj + (cid:126)vjtwc (cid:17) α l ( t ) | j (cid:105) , where | j (cid:105) is the state where all atoms are in the ground state ex-cept for the j th which is in the Rydberg state, and α l ( t ) = (cid:114)(cid:80) j | Ψ l (cid:16) x j + v j tw c (cid:17) | is the normalization factorfor mode l . The time derivative ddt | r l ( t ) (cid:105) is: | ˙ r l ( t ) (cid:105) = (cid:80) j (cid:126)k · (cid:126)v j e i(cid:126)k · (cid:126)r Ψ l (cid:16) (cid:126)rw c (cid:17) α l ( t ) | j (cid:105) + | r l ( t ) (cid:105) (cid:18) − ˙ α l ( t ) α l ( t ) (cid:19) (H.2)+ (cid:80) j e i(cid:126)k · (cid:126)r w c (cid:16) (cid:126)v j · (cid:126) ∇ (cid:126)r (cid:17) Ψ l (cid:16) (cid:126)rw c (cid:17) α l ( t ) | j (cid:105) , where the index | j (cid:105) runs over all atoms in the sam-ple, (cid:126)r = (cid:126)x + (cid:126)vt , w c is the resonator mode waist, k isthe wavevector defined by the relative orientation of thecavity- and control- fields and (cid:126) ∇ (cid:126)r refers to the gradientwith respect to (cid:126)r . These terms of H in the instanta-neous eigen-basis have three effects: mixing polaritons inmodes of different angular momenta, coupling to brightpolariton manifolds orthogonal to the resonator field, andrandom shifts of the energy of the mode in which the po-lariton resides. We now investigate the extent to whicheach of the terms above induce each of these effects. De-fine: | T (cid:105) = (cid:80) j (cid:126)k · (cid:126)v j e i(cid:126)k · (cid:126)r j Ψ l (cid:16) (cid:126)r j w c (cid:17) α l ( t ) | j (cid:105) . Even for a maximally degenerate concentric cavity, mostcollective Rydberg states that one can generate (for ex-ample through atomic motion, above) are orthogonal toall resonator modes, because their spatial form along thecavity axis does not match the cavity field (equivalently,their longitudinal momentum is not that of a cavity pho-ton). As a consequence, most of the dynamics generatedby coupling to | T (cid:105) , | T (cid:105) , and | T (cid:105) consists of coupling tobright polariton manifolds with no corresponding dark(resonator-like) mode. We bound these effects by assum-ing, at zeroth order, that all of each coupling is to thesebright manifolds. The strength of this coupling is thusthe normalization of the corresponding | T i (cid:105) : (cid:104) T | T (cid:105) = (cid:80) j k v j | Ψ l | α l , (cid:104) T | T (cid:105) vt =0 = ( kv th ) (cid:80) j | Ψ l | α l = ( kv th ) . We can now write | T (cid:105) = kv th | T (cid:105) kv th = kv th | ˜ T (cid:105) , where | ˜ T i (cid:105) is the normalized state-vector corresponding to state | T i (cid:105) .This corresponds to a Rabi-coupling of strength ∼ kv th to a bright polaritonic state which is detuned by Ω, and0a resulting dark → bright loss rate of: δ Γ T = ( kv th ) (cid:0) Ω2 (cid:1) γ e kv th ) Ω γ e . (H.3)A small fraction of | T (cid:105) overlaps with other degener-ate resonator modes, corresponding to an atomic-motion-induced polaritonic motional diffusion: (cid:104) r m ( t ) | T (cid:105) = (cid:80) p kv p Ψ ∗ m (cid:16) r p w c (cid:17) Ψ l (cid:16) r p w c (cid:17)(cid:114)(cid:80) µ | Ψ l (cid:16) r µ w c (cid:17) | (cid:80) ν | Ψ m (cid:16) r ν w c (cid:17) | . The expected value of this term is zero since the averageatomic velocity is zero: (cid:104) v p (cid:105) t =0 = 0. The r.m.s. coupling,however, is non-zero: (cid:113) (cid:104)|(cid:104) r m | T (cid:105)| (cid:105) v.p. = kv th √ N C (cid:48) l → m , (H.4)where N is the number of atoms in mode l =0 and C (cid:48) l → m = C l → m √ √ lm = (cid:114) − l − m ( l + m )! l ! m ! √ lm is a generalized theDoppler coupling matrix element between modes l and m , incorporating the fact that higher angular momentummodes contain more atoms, and thus provide a smootheratom distribution. We can expand this matrix elementfor large l ≈ m yielding C l → m ≈ e − ( l − m ) / l √ πl ) / , in-dicating diffusion only into nearly-adjacent modes.Last, the r.m.s. energy shift (inhomogeneous broaden-ing) of the collective Rydberg state induced by this termis given by: (cid:113) (cid:104)|(cid:104) r l | T (cid:105)| (cid:105) v.p. = kv th √ N (cid:115) − l (2 l )! l ( l !) . (H.5)The second term of eq. H.2 is: | T (cid:105) = | r l (cid:105) w c (cid:80) n (cid:126)v n · (cid:126) ∇ (cid:126)r | Ψ l | α l = | r l (cid:105) (cid:18) − ˙ α l α l (cid:19) . Again, we examine how this term couples a dark polari-ton to the lossy manifold of bright polaritons: (cid:112) (cid:104) T | T (cid:105) = v th w c Γ( l + ) l ! ≈ v th (cid:113) l + w c . (H.6)This broadening comes from the time-dependent probefield coupling that the atoms experience as they movewithin the mode; it is much smaller than kv th . Fromthe functional form of | T (cid:105) we can also see that it does not couple modes of different angular momenta (cid:16) (cid:104) r m | T (cid:105) = δ l,m (cid:16) − ˙ α l α l (cid:17)(cid:17) .The third term, similar to the first, produces both abroadening and a shift in the dark polariton energy. Wecan see that this term couples to the bright collectivemanifold with matrix element: (cid:112) (cid:104) T | T (cid:105) = v th w c √ l + 1 . (H.7)We can similarly evaluate how this term couples to otherstates in the dark collective-state manifold: (cid:104)|(cid:104) r m | T (cid:105)| (cid:105) vp = v th w c N ( C (cid:48)(cid:48) l → m ) , (H.8)where C (cid:48)(cid:48) l → m = (cid:113) − l − m ( l +9 l + m +2 lm + m )Γ( l + m ) l ! m ! √ lm is thecoupling element between modes of the resonator thatcaptures mode spatial overlaps, coupling induced byatomic motion and increasing mode area. This cross-thermalization coupling element converges for large l ≈ m to C (cid:48)(cid:48) l → m ≈ (cid:113) √ π l − e − ( l − m − ) / l . The l dependencein both the broadening and energy shift terms arises fromthe more rapid phase accrual of higher angular momen-tum modes.The total broadening of a mode with angular momen-tum l is thus Γ Dopplerl ≈ ( kv th ) Ω γ e (cid:16) l +1( kw c ) (cid:17) ; ther.m.s. Rabi-coupling of mode l to mode m is | Ω lm | ≈ ( π ) kv th √ N l e − δl / l (cid:113) √ l ( kw c ) , where 2 δl ≡ l − m ,2 l ≡ l + m . Noting that kw c ≈
100 for typical ex-periments [36], approximating further yields: Γ
Dopplerl ≈ ( kv th ) Ω γ e , | Ω lm | ≈ ( π ) kv th √ N l e − δl / l . In fact, sum-ming over all final states, the net coupling out of mode l is | Ω l | ≈ (16 π ) / kv th √ N , independent of l .To summarize, atomic motion results in homogeneousand inhomogeneous broadening of the dark polaritons,along with state diffusion. All effects arise from recoil-induced differential motion of the Rydberg-excited atom,expressed as ˙ r l ( t ): the first couples (eqn’s H.3, H.6, H.7)the system to modes that decay into free space due totheir spatial symmetry, while the second term (eqn’s H.4,H.8) quantifies thermalization of atomic degrees of free-dom into the polaritonic degrees of freedom as a resultof atomic motion. The former effect is suppressed by thedetuning of the uncoupled (and therefore bright) modesfrom the dark manifold, while the latter effect is sup-pressed because atomic motion is random, so it is onlythe shot-noise in the motion of the ensemble comprisingthe polariton that leads to polaritonic mode coupling.Doppler decoherence fundamentally arises from therelative motion of the atoms comprising the matter-component of a polariton relative to the field comprisingthe photonic component; as a consequence, the Dopplerdecoherence is sensitive to the canonical momentum of1Doppler Decoherence SummaryBroadening Cross thermalization T ( kv th ) Ω γ e ( π ) kv th √ N l e − δl / l T l +1) ( v th /w c ) Ω γ e T l + 1) ( v th /w c ) Ω γ e ( π ) v th w c √ N l (9 l ) e − ( δl − ) / l TABLE I. Atomic motion induced homoge-neous/inhomogeneous broadening, as well as r.m.s.diffusion/cross-thermalization matrix elements. Forbroadening terms, the angular momentum of the stateunder consideration is l ; for cross-thermalization we considernearby angular momentum states with mean l and sepa-ration 2 δl . Computed broadening and cross-thermalizationterms are upper bounds on the respective processes. Notethat kw c ≈
100 for typical experiments [36], so kv th termsdominate strongly over v th /w c terms until l becomes large. the optical field, not its mechanical momentum [7]. Thisdistinction is particularly important in cavities whosenear-degenerate manifolds represent a particle in a mag-netic field, because although the Landau level is trans-lationally invariant in a fundamental sense, the choiceof gauge arising from resonator twist means that polari-tons further from the resonator axis are more susceptibleto Doppler decoherence, apparent in the l -dependence ofthe loss terms above. I. OUTLOOK
In this paper we have presented a field theory of in-teracting cavity polaritons in the strongly interacting regime, including a formal treatment of interaction andatomic-motion-induced loss channels, and the develop-ment of a renormalized single-mode theory. We alsodemonstrate that by varying the location of one or moreRydberg-dressed atomic ensembles within the resonator,the interactions can be tuned continuously from local inposition-space to local in momentum-space.The renormalized single-mode theory suggests thatit should be possible to develop a renormalized cavity-Rydberg-polariton field-theory, analogous to its free-space counterpart [9], and in conjunction with re-cently demonstrated cavity Rydberg polariton Keldysh-techniques [49], we are now in a position to accuratelymodel the physics of cavity polariton crystals and Laugh-lin puddles [2], plus quantitative analysis of photonic QIPand quantum repeater protocols [14, 50].
ACKNOWLEDGEMENTS
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In Sections B- E, we treat photons as objects that can occupy a completely arbitrary spatial mode in the 2Dplane transverse to the resonator axis. In practice, resonator geometry can impose additional symmetries on theallowed photon wave-functions, and limit the permissible degree of photon localization. In what follows we explorethe wave-function constraints imposed by various resonator configurations.It is convenient to begin by considering a general non-degenerate spherical mirror Fabry-perot resonator [51], whosemodes are enumerated with three indices l, m, n ; the first index l , is the longitudinal mode number, while m, n indexthe transverse mode:The most extreme case is degeneracy of all the transverse modes of a resonator, ω lmn = ω l , achieved in planar andconcentric cavities [51] (note that we have re-indexed the longitudinal modes of the concentric resonator). Planarresonators are constructed with flat mirrors (radius of curvature → ∞ ) very close together, while concentric cavitiesconsist of two mirrors separated by the sum of their radii of curvature. In both configurations the space of allowedphoton wave-functions is not constrained by any symmetries, and arbitrarily small spots can be created at any pointin 2D space (see Figure 4b).The next most extreme case is families of degenerate modes that still require two indices to enumerate, but withrestrictions on the indices. An example of this is the confocal resonator: two mirrors with radii of curvature R placeda distance R apart. Such a resonator exhibits ω lmn = ω l,mod ( m + n, . The constraint that m + n is either even or oddimposes a reflection symmetry across the origin: the photon may be arbitrarily well localized in space at any location,but must simultaneously exist at this mirror-image location (see Figure 4c).The next case is degenerate families that may be indexed with only a single parameter, which may themselves befurther broken down into two sub-categories: (1) families in which the index takes on only a finite number of values;and (2) families in which the index takes on a countably infinite number of values. A spherical mirror Fabry-Perot fallsinto the first category; ω lmn = ω l,m + n ; another example is an astigmatic resonator whose length is tuned to enforcedegeneracy such as ω lmn = ω l,m +2 n [35] (see Figure 4d). The latter category could be achieved in an astigmaticresonator tuned to confocality on only one axis: ω lmn = ω l,m,mod ( n, , or in a non-planar resonator by imposing twistwhich is a rational fraction of 2 π , as in [6]: ω lmn = ω l,m,mod ( n, (see Figure 4e). The reduced degeneracy stronglyconstrains the wavefunctions which may be represented by the family, to the point that the physical interpretationof these families is often quite unclear. Indeed, the twisted resonator explored in [6] exhibits three-fold rotationalsymmetry, and quantum geometry, meaning that has a minimum spot size; this system may be understood as aLandau level on the surface of a cone, quite an exotic manifold indeed. (a) (b) (c) (d) (e) FIG. 4.
Resonator degeneracies and their effects on photon localization and spot size. (a)
Decomposition of anoff-center Gaussian into 1D HG modes. A Gaussian at any arbitrary location can be written as the sum of 1D HG modes, bytaking the spatial overlap of the Gaussian with each mode. The overlaps are shown in the plot, with even modes represented asblue dots and odd modes as red dots. (b)
Localization of a photon in a planar or concentric cavity. When all resonator modesare degenerate, we can create a photon at any location inside the resonator and of any spot size. This can be understood as ageneralized version of the decomposition in (a), but in general it is done in a two dimensional parameter space. The red lines inthe plot signify x = 0 , y = 0, showing than an off-center spot has been created. (c) In a confocal resonator, we have degeneracywithin the even and odd mode manifolds only. As such, a decomposition of any spot would include either the even (blue) orodd modes (red dots) only. Any spot size at any location can still be created but now we pick up reflections through the origin,resulting in a mirrored spot. (d)
Superposition of TEM and TEM . This is an example of a finite one parameter family.The resonator mirrors and mirror positions are picked in such a way as to make these two modes energetically degenerate. Theresult is interference between the two modes, resulting in a new mode shape [35]. (e) Threefold symmetry in a resonator. In acountably infinite single parameter family of degenerate modes where every third LG mode is degenerate (LG l ,LG l +3 ,LG l +6 ,...),a localized spot can be created at any point but must satisfy a threefold rotational symmetry, resulting in three copies, each ofwhich cannot be smaller than the LG0