Multimode-Floquet-Polariton Superradiance
MMultimode-Floquet-Polariton Superradiance
Christian H. Johansen, ∗ Johannes Lang, † and Francesco Piazza ‡ Max-Planck-Institut f¨ur Physik komplexer Systeme, 01187 Dresden, Germany
We consider an ensemble of ultracold bosonic atoms within a near-planar cavity, driven by afar detuned laser whose phase is modulated at a frequency comparable to the transverse cavitymode spacing. We show that a strong, dispersive atom-photon coupling can be reached for manytransverse cavity modes at once. The resulting Floquet polaritons involve a superposition of a set ofcavity modes with a density excitation of the atomic cloud. We show that their mutual interactionslead to distinct avoided crossings between the polaritons. Increasing the laser drive intensity, a low-lying multimode Floquet polariton softens and eventually becomes undamped, corresponding to thetransition to a superradiant, self-organized phase. We demonstrate the stability of the stationarystate for a broad range of parameters, including modulation frequencies larger than the separationbetween transverse cavity modes.
I. INTRODUCTION
The implementation of strong interactions betweenphotons mediated by a medium [1] is essential forquantum-information processing [2, 3], and at the sametime allows for the exploration of the quantum many-body physics of light [4, 5]. Photons inherit interactionsfrom the material through the formation of polaritons[6], hybrid quasiparticles corresponding to an excitationpresent both in the electromagnetic field and in the ma-terial.The creation of interacting polaritons having access toa macroscopic number of modes is essential for the studyof thermodynamic phases of photons and complex typesof order [4, 5, 7–15]. The strong-coupling between matterand light, required to implement photon interactions, canbe realized by reducing the electromagnetic-mode volumeusing optical cavities or evanescent fields [1]. The formercan also be combined with the use of Rydberg atoms tofurther enhance interactions [16]. This task obviously be-comes more challenging if it needs to be achieved for awhole set of electromagnetic modes, which are in generalseparated in frequency. One solution is to shape the cav-ity geometry such that a set of quasidegenerate modesis formed [4, 17–19]. Another option is to use a set ofpropagating modes as in cavity arrays [14] or photoniccrystals [20].An alternative route towards interacting multimodepolaritons has been recently demonstrated experimen-tally via Floquet engineering of Rydberg levels in a non-degenerate optical cavity [21]. Starting from a givenatomic transition resonantly coupled to a single cavitymode, the periodic amplitude modulation of the laserdressing effectively splits the transition into multiple lev-els separated by the modulation frequency, one of thembecoming resonant with a second cavity mode. The re-sulting two-mode polaritons strongly interact via the Ry- ∗ [email protected] † [email protected] ‡ [email protected] dberg component inherited from the atomic levels. Thisdemonstrated the potential to Floquet engineer interact-ing multimode polaritons for the exploration of the many-body physics of light.In this work, adopting similar ideas, we theoreticallystudy a first example of a phase transition to an orderedphase emerging for multimode Floquet polaritons. Dif-ferently from the setup used in [21], the realization ofpolaritons and their interactions is achived here via thedispersive coupling between the motional degrees of free-dom of an ultracold Bose gas and the transverse elec-tromagnetic (TEM) modes of a near-planar Fabry-Perotresonator [22, 23], with mode-spacing in the GHz range.Similarly to the case of plasmon-polaritons in electronicmatter [24], here the polaritons mix a cavity-photon witha density excitation in the gas of neutral atoms.If performed at a frequency close to the TEM mode-spacing, a periodic phase modulation of the laser canbring a large number of modes into resonance. This isachieved without heating the atoms, since ultracold den-sity excitations are in the kHz range, and the internaldynamics can be at hundreds of GHz for a correspond-ingly detuned laser. The mutual interactions betweenmultimode Floquet polaritons induce avoided crossings,controlled not only by the effective light-matter couplingand detunings, but also by the phase-modulation am-plitude. As a consequence, a low-lying multimode Flo-quet polariton can be red-shifted to zero frequency andsubsequently become undamped, corresponding to an in-stability towards a multimode superradiant phase withmacroscopic occupation of the polariton.The observation of the avoided crossings between inter-acting Floquet polaritons requires sub-recoil resolutionachievable in long cavities [23]. Using ultracold bosonscoupled to such cavities, single-mode superradiance hasalready been observed [25], and Floquet modulation hasalso been studied, but at frequencies far below the trans-verse mode spacing [26, 27].Multimode superradiance has been recently observedusing ultracold bosons coupled to degenerate confocalresonators [28–31], and is expected to give access tobeyond-mean-field effects due to the increased localityof the light-matter coupling [32, 33]. On the other hand, a r X i v : . [ qu a n t - ph ] N ov Floquet superradiance has so far only been studied in thesingle-cavity-mode case. Here we show that one can enterthe deep multimode regime even in near-planar cavities.
II. THEORETICAL DESCRIPTIONA. Model
We consider a cloud of bosonic atoms trapped inside anear-planar optical cavity and transversally pumped bya laser, as depicted in Fig. 1. This laser is phase mod-ulated (PM) with a frequency comparable to the energydifference between the TEM modes of the cavity. Thepump laser is described as a classical field E ( t, x ) = 2 λ η p ( x ) cos ( ω p t + f ( t )) , (1)with ω p being the carrier frequency of the pump and η p its spatial profile. The PM function, f ( t ), is assumed tobe periodic f ( t + T ) = f ( t ) and real. Because of theperiodicity one can represent e − if ( t ) as a discrete Fourierseries e − if ( t ) = ∞ (cid:88) α = −∞ c α e − iα Ω t , (2)where Ω = πT . In experiments the PM is generated us-ing an electro-optic modulator with a limited bandwidth.We therefore consider a series with finite support and de-fine α M as the index where the sum can be truncated, c | α | >α M ∼
0. The carrier frequency of the pump is reddetuned from the atomic transition ω e , in such a waythat the detuning between atoms and pump ∆ a , satisfies∆ a = ω e − ω p (cid:28) ω e + ω p . For instance, for Rb atoms,the S / → P / transition has a frequency ω e ≈
378 THz[34]. In that case ∆ a ∼ GHz will easily satisfy theinequality. Similarly, the PM frequency is chosen to bemuch smaller than ∆ a , such that ∆ a + α M Ω ∼ ∆ a . Dueto this condition it is well justified to apply the rotatingwave approximation [35].Since ∆ a is large compared to the inverse lifetime ofthe excited state, the occupation of the latter remainssmall and saturation effects are negligible. Therefore,the ground and excited states of the atoms can be repre-sented as two independent bosonic fields. The resultingHamiltonian, in the frame rotating with the carrier fre-quency, in units where (cid:126) = 1, reads H = (cid:90) d r (cid:26) ψ † (cid:18) − ∇ m + V g ( r ) (cid:19) ψ + φ † ∆ a φ + φ † (cid:32)(cid:88) i g i η i ( r ) a i + λη p ( r )e − if ( t ) (cid:33) ψ + H.c.+ (cid:88) i ∆ i a † i a i (cid:41) . (3)Here ψ ( φ ) is the bosonic annihilation field operatorfor the ground (excited)-state of the atoms with mass FIG. 1: The physical setup considered, consists of a linearcavity with a waist of w . Inside the cavity ultracold bosonicatoms are confined within a cigar-shaped harmonic trap,with a transverse harmonic oscillator length of L H . Eachatom is modelled as a two-level system with excited-stateenergy ω e . The atoms are transversally pumped with a laserof frequency ω p and pump strength λ . The laser is sentthrough an electro-optical modulator which introduces aperiodic phase-modulation with period 1 / Ω and depth B m .Photons are lost from the cavity mirrors with rate κ , makingthe system driven-dissipative. m . The space-time dependence of field operators hasbeen suppressed for brevity. As ∆ a is much larger thanboth kinetic and trapping energy of the excited state,these terms have been neglected. η j ( r ) is the spatialmode function for the j ’th cavity mode that couples withstrength g j to the atoms. The j ’th transverse cavitymode has detuning ∆ j = ω j − ω p and is annihilatedby the bosonic operator a j . We consider the case wherehigher-order cavity modes have an approximate linear en-ergy spacing ∆ j ≈ ∆ + jω T . Choosing a PM frequencyΩ that is comparable to ω T means that the detuningbetween higher-order modes and the atomic transitioncan be compensated by coupling to the higher frequencypump sidebands, generated by the PM. As the PM isperiodic, the sidebands are equidistant in frequency, sep-arated by Ω.All modes couple to all sidebands but the coupling isinversely proportional to the detuning. Because Ω is largethe only relevant modes are the near-resonant ones∆ i ± α Ω ∼ ∆ . (4)The cavity is an intrinsically lossy system, which com-pensates the energy input from the continuous pumping.This loss is included by coupling the cavity to a contin-uum of electromagnetic modes playing the role of a bath[36]. At optical frequencies the electromagnetic bath is (a) ω + α Ω ω + α Ω ω ω + β Ω ω + β Ω α Ω β Ω (b) − ω + α Ω α Ω ω + β Ω ω + β Ω ω β Ω α Ω FIG. 2: The scattering processes that determine the polarizability of a BEC driven by a PM laser. In these Feynmandiagrams, the straight (wavy) lines represent propagation of a bare atom (photon) labeled by its frequency. Themomentum/mode indices have been suppressed for brevity. Green lines represent atoms in the electronic ground state andblue dashed ones atoms in the excited state. The purple wavy line represent cavity photons. In addition, there are externalsources, represented by lines ending in a cross. Those can be photons from a given sideband of the phase-modulated laser (seediscussion in section II A), depicted as a black wavy line, or atoms from the BEC, depicted as solid green lines. Panel a)shows the normal process, whereas panel b) shows the anomalous process where the laser acts as a false vacuum. effectively at zero temperature and is therefore well de-scribed as a Markovian loss for the photons. This meansthat in the von Neumann equation dρdt = − i (cid:126) [ H, ρ ] +
Dρ , (5)the unitary Hamiltonian term is supplemented with aLindblad dissipator [36] Dρ = − (cid:88) j κ j (cid:18) (cid:110) a † j a j , ρ (cid:111) − a j ρa † j (cid:19) . (6)This Markovian modeling of the environment is also validwhen the PM is included, as long as the environmentspectral function appears flat over the energy range ω p ± α M Ω.As usual, the atom number is assumed to be con-stant over the duration of the experiment. By cou-pling the atoms to the cavity the total energy, on theother hand, is not conserved. In the regime where thecavity-mode detunings are comparable to the recoil en-ergy (cid:15) r = Q / (2 m ), this leads to a non-thermal stateof the bosonic atoms [37]. In the following, we will ne-glect this effect by assuming that the time τ rel requiredto reach that state is longer than the experimental time,which is realistic for sufficiently large detunings and trapsizes, since τ rel scales inversely with these quantities [37].We will therefore assume the atoms to be initially cooleddown to an ultracold temperature T much smaller thanthe recoil energy, so that thermal excitations are of littleconcern and we can model the atoms as a perfect BEC. B. Polarisability of the Floquet-driven bosonic gas
Collective polaritonic excitations correspond to thenormal modes of the electromagnetic field including po-larization effects, that is, the modification of the propa-gation of light due to the medium. In our case, the latteris a gas of ultracold bosonic atoms, whose internal elec-tronic transition is driven by a PM laser, as described in section II A. The assumption that the bosons are initiallyin a perfect BEC at zero temperature corresponds to allatoms being in the electronic ground state and also inthe lowest motional state.The effect of this driven, polarizable medium on thecavity photons is depicted in Fig. 2, where we illustratethe relevant scattering processes. In Fig. 2 (a), a cavityphoton excites an atom out of the BEC, and takes it toan electronically and motionally excited state (see secondline in Eq. (3)). The atom then emits a photon intothe laser and returns back to its electronic ground stateand at a momentum which does not belong to the BEC(otherwise the net effect of the whole process would notcontribute to the polarizability). The whole sequence isthen repeated backwards, leading to the emission of aphoton back into the cavity.The motional scales of cold atoms are in the kHzrange while the considered PM frequencies are compa-rable to the TEM mode spacing, which is of the or-der of GHz. The scattering process is therefore signif-icantly suppressed if the incoming cavity photon excitesthe atomic ground state to energies which are large com-pared to the recoil energy, unless excess is compensatedby the laser. With a single laser frequency this is impos-sible, which leads back to a single-mode scenario. How-ever, via PM the laser sidebands can be brought closeto resonance with the high-energy modes. In the for-malism this can be efficiently accounted for by splittingthe energy ( ω (cid:48) ) into Floquet sectors ω (cid:48) = ω ± α Ω, where ω ∈ {− Ω / , Ω / } is the quasienergy. In the diagramsof Fig. 2, this means that if a photon at energy ω + α Ωexcites the atoms, then the laser has to remove an energy α Ω for the process to be non-negligible.The same scattering process can clearly also take placewith the role of cavity and laser being exchanged, whichcorresponds to the anomalous process [38, 39] depictedin Fig. 2 (b), where the laser plays the role of a falsevacuum.The crucial point is that, in all these processes, due tothe periodic PM of the driving laser, the initial and thefinal cavity photon can have different frequencies withoutthe atomic ground state propagating at a high energy andthus far off-shell. Due to the absorption and emission oflaser photons from different sidebands the polarizabilityof such a Floquet-driven BEC is thus not diagonal infrequency space as it couples different Floquet sectors ofthe cavity field differing by multiples of Ω.Under the condition, stated in section II A, that thehighest sideband of the laser is still far detuned from theelectronic transition, one can adiabatically eliminate theelectronic excited state from the diagrams in Fig. 2. Thisleads to the following dynamical polarizability: χ i,j ; α,β ( ω ) = Λ ¯ c α c β Π Ri,j ( ω ) , (7)which has been expressed as a matrix in the TEM-modebasis, with indices i, j , and in the Floquet basis, denotedby α, β . Here we also introduced a single parameter, theeffective light-matter coupling strength Λ = λg √ N / ∆ a ,which increases linearly with the pump strength. Thiscan be captured in a single parameter because the ener-getic difference between the transverse modes is on theorder of GHz while the total energy in the electric fieldis hundreds of THz [40]. This justifies approximating allmodes to have a similar coupling to the atoms ( g i = g ).Again due to the large energy difference between theatomic ground-state motion and the PM frequencyΩ thedensity response, Π R in Eq. (7), is diagonal in frequency,and takes the form of a bosonic analog of the Lindhardfunction [41], which specified for a perfect BEC in a trap[38] readsΠ Ri,j ( ω ) = (cid:88) n − (cid:15) r + (cid:15) n )( ω + i + ) − ( (cid:15) r + (cid:15) n ) × (cid:104) ψ | η j η p | ψ n (cid:105)(cid:104) ψ n | η i η p | ψ (cid:105) , (8)where (cid:15) n is the energy of the atomic transverse modes andwe have assumed that the wave functions of the cavitymodes, as well as those of the atomic eigenstates, arereal.In summary, we see that in this regime the non-diagonal frequency structure of the polarizability, cou-pling two different Floquet sectors α, β , is simply encodedin the product of two Fourier coefficients c α c β of the pe-riodic modulation of the laser phase. Each coefficientquantifies how much of the laser intensity goes into thecorresponding sideband i.e. how strong that sidebandcouples to the electronic transition. According to theprocess shown in Fig. 2, these weights clearly have toenter the polarizability. The remaining part of the polar-izability is then given by the retarded density response ofthe medium, which only depends on the low energy mo-tional degree of freedom of the atoms in their electronicground state.The physical content of the density-response functionis quite transparent: it features the matrix element of thetransition from the trap ground state | ψ (cid:105) to an excitedstate | ψ n (cid:105) , and back to the ground state. The transi-tion out of (back to) the ground state is induced by thespatially-varying optical potential η i η p ( η j η p ), created bythe interference between the laser and the cavity field of the i ’th ( j ’th) TEM mode. As it is clear from inspectionof the matrix elements, for a generic choice of the atomtrap, the density response, and in turn the polarizability,need not be diagonal in the cavity-mode basis.In summary, the Floquet-driven bosonic medium canchange both the frequency (by multiples of the modu-lation frequency Ω) and the TEM mode of an incomingphoton. Whether and how this happens depends on thespecific choice of the PM of the laser, the cavity geometryand the trapping potential for the atoms, as we discussnext.
1. Symmetric trap and harmonic phase modulation
We consider the radially symmetric configuration of acigar-shaped harmonic trap in the center of a near-planarcavity. Assuming a long cigar-shaped zero-temperatureBEC the longitudinal wave function is well localized inmomentum space with a negligible width ∆ k (cid:28) Q . Uponabsorption of a photon the atoms therefore scatter intoa state with longitudinal momentum Q and recoil energy (cid:15) r .Assuming that the laser wavelength, λ p , is muchsmaller than the transverse diameter of the BEC L H ,means that the pump oscillates many times over thesize of the cloud. The transverse spatial modes of theatoms then satisfies a Mathieu equation with an addi-tional quadratic term. The momentum kicks associatedwith the laser oscillation leads to far off-shell processeswhich can be neglected, unless the kicks during one scat-tering process, Fig. 2, cancels out. As the cavity modefunctions are considered larger than the atom cloud thefast oscillations of the Mathieu equation can be neglectedand the radially symmetric trap determines the BECshape. Following these considerations, we can simplifythe pump mode function as a constant, η p = 1. Whenthis approximation is not valid, the pump breaks the ra-dial symmetry of the trap and introduces periodic mod-ulations along the laser axis, which requires the full nu-merical solution of the modified Mathieu equation.For a small λ p , the transverse modes which we seekto couple are Laguerre-Gauss (LG) modes, which for thecavity read [42]LG jp ( r, Θ) = e − r / w + ip Θ (cid:113) j !( p + j )! (cid:16) rw (cid:17) | p | L | p | j (cid:16) r w (cid:17) w , (9)with L | p | j ( x ) being the associated Laguerre polynomialof order j . The ground-state atoms have similar modefunctions but with w replaced by L H .For atom clouds that are radially symmetric aroundthe cavity axis, angular momentum conservation impliesthat for each overlap the angular index must be con-served. The BEC is in the ground state with zero angularmomentum and therefore only radial cavity modes withthe same absolute angular momentum are coupled. Hav- B m (mod. depth) − . − . . . . . . . J ( B m ) J ( B m ) J ( B m ) J ( B m ) J ( B m ) J ( B m ) FIG. 3: A plot of the first six phase-modulation coefficientsas a function of the modulation amplitude. The coefficientsare given by the Bessel functions of the first kind. ing the carrier frequency being only slightly detuned fromthe zeroth TEM mode means that only cavity modes withp=0 are relevant and thus η j ( x ) = w LG j ( r ) cos ( Qz ),rendering all mode functions and eigenstates in the den-sity response function real. The remaining radial overlaphas a closed form solution shown in appendix A. The sim-plest case is when the BEC is significantly narrower thanthe cavity waist. In this case all overlaps are negligibleexcept the one with the atomic state in the zeroth radialmode and a longitudinal momentum given by the recoilmomentum. In this case, the density response takes asimple form Π Ri,j ( ω ) = − (cid:15) r ( ω + i + ) − (cid:15) r , (10)which is the same as the single mode result [38].For non-zero modulation depth it is necessary toparametrize the PM, which can be any real function thatis periodic in time. A simple yet flexible choice is a har-monic modulation: f ( t ) = B m sin(Ω t ) , with the amplitude B m being a real number. From theJacobi-Anger expansion [43], the discrete Fourier coeffi-cients are known to be Bessel functions of the first kinde if ( t ) = (cid:80) α c α e iα Ω t , with c α = J α ( B m ). As shown inFig. 3, for zero modulation depth, the zeroth-order coef-ficient is the only non-zero component, which is exactlyequivalent to having no PM. Because of the orthonormalnature of the coefficients, the weight in the zeroth-ordercomponent is distributed as the modulation is increased.As discussed in section II B, since the coefficients directlydetermine how strongly different modes are coupled by the medium, one can tune the amount of multimodal-ity to a large extent by simply changing the modulationdepth.The modulation frequency Ω is equally important, as itdetermines the effective detunings of the different cavitymodes. If the modes are exactly linearly spaced one canmake the system energetically degenerate by choosing Ωequal to the mode spacing. If one wants to energeti-cally suppress either higher or lower order modes, onecan choose a frequency that is either smaller or greaterthan the mode spacing. Thus, even though the speci-fied harmonic modulation has only two parameters, it isnevertheless highly tunable. III. MULTIMODE FLOQUET POLARITONS
Having determined the polarizability of the Floquet-driven BEC, we can now investigate how this modifies thepropagation of cavity photons and leads to the formationof polaritons. As already mentioned above, polaritons arethe normal modes of the electromagnetic field inside themedium. As such, they appear as poles of the Green’sfunction of the electromagnetic field. In the present case,this Green’s function reads D Ri,j ; α,β ( ω ) = (cid:18) P Ri,j ; α,β ( ω ) + χ i,j ; α,β ( ω ) χ i,j ; α,β ( ω ) χ i,j ; α,β ( ω ) P Ai,j ; α,β ( − ω ) + χ i,j ; α,β ( ω ) (cid:19) − , (11)where the positive- and negative-frequency componentsof the electromagnetic field have been separated. Theresulting Nambu matrix depends on the inverse Green’sfunction of the bare cavity P R/Ai,j,α,β ( ω ) = δ i,j δ α,β ( ω − ∆ j − α Ω ± iκ j ) , where R/A indicate the causality of the Green’s functionbeing retarded or advanced. This combination of Green’sfunction appears here because both positive and negativefrequencies are present.Due to the polarization function in Eq. (11), theGreen’s function of the electromagnetic field is non-diagonal in both frequency and LG-mode space, suchthat its computation, in general, becomes rather cumber-some. However, as long as the polarizability decays on anenergy scale much smaller than Ω ∼ ω T , the calculationsallow for simplifications. In this case the cavity modesthat actually contribute to the electromagnetic Green’sfunction are the ones that are near-resonant modulo amultiple of the modulation frequency: ∆ i + α Ω ∼ ∆ .This effectively couples the sideband α with the modeindex i , largely reducing the number of elements of thematrix in Eq. (11).In order to visualize the poles of the electromagneticGreen’s function, we will use the so-called spectral func- . . . / Λ c ) . . . . . . . ω / (cid:15) r LG (a) 0 . . . / Λ c ) LG (b) 0204060 A ( ω ) FIG. 4: Distribution of the spectral weight in the LG -component of the cavity field. a) when no phase-modulation isapplied to the laser, B m = 0 and b) when B m = 0 .
9, Ω = ( ω T + 0 . (cid:15) r . The remaining parameters are ∆ = 0 . (cid:15) r , w /Q = 200, L H = 10 − w and κ = 0 . (cid:15) r with the parameters described in section II B 1. The horizontal dashed lines in b)represent the effective detunings of higher order modes, while the decreasing dashed line represents the position of thespectral line from a). The three red crosses mark the values used in Fig. 5. tion of the cavity A ij ; αβ = i (cid:16) D Rij ; αβ − (cid:2) D Rji ; βα (cid:3) † (cid:17) . (12)This function has peaks in correspondence to the realpart of the poles (the polariton frequency), with a widthset by the imaginary part (the polariton damping or in-verse lifetime). The features observed in the spectralfunction can be measured by pump-probe or transmis-sion experiments. A. No phase-modulation
To put our results into perspective and highlight ef-fects of the coupling between many cavity modes, wefirst review some features of the single-mode calculation[38, 44]. The spectral function for the unmodulated cav-ity is shown in Fig. 4 (a). When the cavity couplesweakly to the atoms the cavity spectrum is dominatedby the free Lorentzian peak centered at the detuning ∆ with a width determined by the cavity loss κ . As thepump strength is increased the atoms start to hybridizewith the cavity. This leads to a second peak in the cav-ity spectral function initially at the recoil frequency, cor-responding to the characteristic energy of the xdensityexcitation in the atoms. This signature is initially ex-tremely narrow but broadens as the pump strength isincreased. In addition, the repulsion between the hy-bridized modes is also continuously increasing, therebymoving away from the bare resonances. This repulsionbetween the cavity and atomic peaks signals appreciable hybridization i.e. mixed atom-photon character of thepolaritons.By increasing the coupling Λ, the low-lying polaritonis pushed to lower and lower energies until its frequencyreaches zero. At this point, its damping is still finite andthe peak in the spectral function is no longer Lorentzian[37, 38]. By further increasing the coupling to a criticalvalue Λ c , also the polariton damping vanishes and thenormal phase becomes unstable. In the current parame-ter regime where only the lowest-order atomic transverse-mode is relevant, the critical coupling strength Λ c can befound analytically [45]Λ c = κ + ∆ (cid:15) r . (13)If the cavity loss is significantly smaller than the detun-ing, then Λ c is approximately linearly dependent on ∆ . B. Including phase-modulation
Considering the same system, but turning on the phasemodulation, more cavity modes can be brought into play.How strongly these couple to the atoms is determinedby the modulation depth according to Eq. (7). As seenin section II B 1, for small atom clouds the density re-sponse becomes independent of the mode index, that isthe spatial structure of the LG j cavity modes plays norole in determining how strongly they couple. Therefore,assuming that they have similar detunings (modulo thePM frequency Ω), the cavity mode admixture is fully de-termined by the PM of the laser. This allows one to LG LG LG LG Cavity mode0 . . . . . . O v e r l a p p p p r/w . . . I / I p p p FIG. 5: The overlap of different LG modes with the totalcavity field at the three points marked by red crosses inFig. 4 (b): p = (0 . , . p = (0 . , . p = (0 . , . create polaritons with a photonic part consisting of a su-perposition of several cavity modes.When multiple cavity modes are available, the photonGreen’s function is a matrix in the cavity-mode basis (seeEq. (11)). As argued above, for the parameters consid-ered here, the matrix structure in the Floquet basis canbe suppressed since its index is fixed by the cavity mode,i.e. the Green’s function in Eq. (11) is proportional to δ iα δ jβ . The spectral function is thus also a matrix in thecavity-mode basis. In order to illustrate the effect of thePM, we show the diagonal entry of the spectral functioncorresponding to the LG mode in Fig. 4 (b). This al-lows a direct comparison with the unmodulated case ofFig. 4 (a). Using a PM frequency of the form Ω = ω T + (cid:15) ,the effective detunings of the higher-order modes become∆ i = ∆ − (cid:15). (14)Choosing the sign of (cid:15) allows one to switch between caseswhere higher-order modes are effectively lower or higherin frequency than the zeroth mode.For the spectral function plotted in Fig. 4 (b), (cid:15) ischosen to be a small positive number, which causes thehigher order modes to be energetically preferred. With aweak modulation depth ( B m = 0 .
9) the only new rel-evant modes are LG and LG , which are given byEq. (9). The magnitude of the pump-sidebands, gen-erated by the PM, is given by the corresponding Fouriercoefficient squared. The effective light-matter coupling strength between each mode and the atoms is thereforegiven by the Fourier coefficient of the nearest sideband(Fig. 3). The relevance of each mode is determined byits atom-coupling strength and the magnitude of the ef-fective detuning. For the specific case of the figure, thismeans that the LG j mode couples only to the j ’th side-band, which has a magnitude of J j ( B m ) . Coupling to allthe other sidebands leads to far detuned processes withvery small magnitudes.A first clear signature of the multimode nature is theappearance of more peaks in the spectral function. Look-ing at Fig. 3 at B m = 0 . starts having a non-negligible couplingto the atoms. Because this mode is (cid:15) closer to its side-band than LG is to the carrier frequency, LG becomesmore favourable for the system and one observes a clearavoided crossing near the frequency of the bare LG -mode. This repeats when the LG starts being relevantand a second avoided crossing is observed at ∆ .These avoided crossings signal the strong hybridizationbetween cavity modes and the emergence of multimodeFloquet polaritons. This is confirmed by Fig. 5, wherethe overlap of the LG modes with the total cavity fieldis shown for different points on the polariton branchesof Fig. 4 (b). This also explicitly demonstrates that theLG can be neglected as presumed based on its coeffi-cient in Fig. 3. In the inset of Fig. 5 the resulting modeprofile of the total cavity field is shown. Already for thefew modes involved here, it is clearly seen that the PMleads to a significant decrease of the waist of the cavityfield, which is bounded from below by the central waistfor the highest involved TEM mode. The observed re-duction of the waist of the cavity field directly implies areduction of the range of the cavity-mediated interactions[29]. More prominent effects can obviously be achievedby increasing the PM amplitude, but also by displacingthe atom cloud from the center of the cavity. The oscil-lating nature of higher-order transverse modes can thenlead to interference effects that further decrease the in-teraction range.Changing the modulation depth B m gives the freedomto choose how strongly the different modes couple to theatoms. The parameters B m and (cid:15) are completely inde-pendent and therefore allow one to tune between a widerange of different multimode scenarios. In Fig. 4 (b)we showed how choosing (cid:15) > (cid:15) < / Λ c = 0 . B m = 0).The LG mode’s spectral function shows the familiar . . . . . . . ω / (cid:15) r LG LG B m . . . . . . . ω / (cid:15) r LG B m LG A jj ( ω ) FIG. 6: Distribution of the spectral weight as a function of the modulation amplitude for different components of the cavityfield, corresponding to the first four radial transverse modes LG pl with p ∈ { , , , } . The PM frequency is chosen such thatthe higher order modes have a large effective detuning ( (cid:15) = − . (cid:15) R ) and Λ = 0 . c ( B m ) , with the renormalized couplingstrength defined in Eq. (15). The zeroth mode has detuning ∆ = 0 .
6, while the remaining parameters are as in Fig. 4. Thered lines are poles of the real part of the cavity propagator in Eq. (11) and therefore indicate the polariton frequencies. two peak structure discussed in Fig. 4 (a), whereas thehigher-order modes have no atom peak in the spectrumbut only their bare Lorentzian lineshape at the respectivedetunings. As B m is increased the other modes start tocouple through the atoms at the price of decreasing theatom coupling of the LG mode. This is clearly seenby the fact that the LG line starts showing up in A .Near B m ≈ .
3, the coefficient c vanishes, see Fig. 3.This means that the LG mode no longer couples tothe atoms, and both the atom peak and what will be-come the superradiant peak vanish from A . At thatmodulation depth, A is simply the free Lorentzian lineshape at ∆ . Meanwhile, both LG and LG couplerelatively strongly to the atoms, and therefore effectivelyto each other, which introduces multiple peaks in the corresponding components of the spectral function. Fur-ther increasing B m reintroduces the LG mode in thepolariton peaks.This shows that one can tune B m such that the polari-tons have contributions from many modes. In particu-lar, the polariton becoming unstable at the superradianttransition will then be a linear combination of all modesLG j with j < α M , with weights essentially given bythe histograms in Fig. 5. As the higher modes have alarger detuning than the zeroth mode, it is necessary toincrease the pump power in order to keep the ratio Λ / Λ c fixed. This shows up in the spectral functions, where theatomic peak is pushed to higher energies as B m is in-creased. It is interesting to note that, even though thehigher-energy modes are red detuned, one can choose a B m . . . . . Λ c / ∆ observablerenormalised obsv. detunedrenorm. detuned FIG. 7: Critical coupling strength Λ c as a function of themodulation amplitude. The two different colors representΩ = ω T (black) and an effective detuning between the modesΩ = ω T + 0 . modulation depth which results in anticrossings, espe-cially prominent at around B m = 4. This demonstratesthat non-trivial multimode effects can be seen using both (cid:15) < (cid:15) > IV. MULTIMODE SUPERRADIANCE
Having multiple cavity modes available affects severalfeatures of the transition to the superradiant phase. Thefirst clear effect is that the hybridization between multi-ple cavity modes can give rise to an increase in Λ c com-pared to the unmodulated system, as seen in Fig. 6. Thebehaviour of Λ c is plotted as a function of B m in Fig. 7.To isolate the effect, consider first the case where Ω = ω T .In this case all modes in the system have the same effec-tive detuning. The value of Λ c one would observe inan experiment is the black dashed line increasing non-monotonically as the PM is increased. This rise in thecritical coupling strength is due to the PM being sym-metric around the carrier frequency. The higher-ordermodes see only the closest laser sideband, but the PMalso induces sidebands at a lower energy than the carrierfrequency. For these negative frequency sidebands thecavity has no stable modes as they correspond to veryhigh transverse quantum numbers with a lower longitu-dinal quantum number. These modes are lossy due totheir large transverse size and are generally not stable.The power in the lower frequency peaks is therefore effec-tively lost. This means that the total effective pumping power of the cavity is decreased by the weight in the neg-ative frequency coefficients. This effect can be taken intoaccount by renormalising the couplingΛ → Λ (cid:80) ≤ α<α M c α = Λ( B m ) . (15)We always use the explicit B m dependence to denote therenormalized coupling strength. Employing this renor-malization one sees that Λ c ( B m ) stays constant throughall modulation depths. When (cid:15) >
0, as in Fig. 4 (b),the higher-order modes have a smaller detuning, and oneagain sees that Λ c is increasing with B m . This mightseem counter-intuitive, as the higher modes have smallereffective detunings and one would therefore expect Λ c todecrease with B m , which is indeed observed when usingthe renormalized coupling strength.Another important modification to single-mode super-radiance is that with a positive (cid:15) , higher-order modeshave a lower frequency than the zeroth mode. For a rangeof values of (cid:15), B m and ∆ , the situation arises wheresome higher-order modes are effectively red-detuned fromtheir nearest sideband. In this case the magnitudes oftheir detunings are still small, compared to ω T , but nowhave negative signs. For a single-mode system, this leadsto an instability at a much smaller Λ c than for a simi-larly blue-detuned cavity mode. The critical coupling ismuch smaller because the bare atomic part of the systemhas a vanishingly small loss and once the cavity is reddetuned the atomic part of the system is easily renderedunstable, even at very weak interaction strengths. Thistype of “atomic instability” happens at finite frequency,that is, the atomic polariton becoming unstable still hasa finite energy. This instability is thus of a very differentnature than the superradiant transition happening at Λ c ,via a zero frequency excitation.The results presented so far assume that the systemremains stable up to Λ c , which means that we have tochoose the parameters such that the finite-frequency in-stability is avoided. Fig. 8 (a) shows the nature of theleading instability as a function of the PM parameters.For (cid:15) < .
15 the transition always happens to a station-ary, superradiant phase. This is explained by the factthat we have considered a system with 5 available modesand with ∆ = 0 .
6. This means that the smallest de-tuning for this region is ∆ = ∆ − (cid:15) ≥
0, which doesnot allow for a finite-frequency instability. Clearly, theconstraint on the number of cavity modes imposed by thephysical setup plays an important role for the existence ofthe finite frequency instability. When moving to largervalues of (cid:15) it can be seen that, due to the effective in-teractions between multimode polaritons, the boundarybetween zero- and finite-frequency instabilities is highlynon-linear. For the parameters considered in the presentwork, there is thus a large region where the PM can beused to generate multimode Floquet polaritons withoutencountering a finite-frequency instability.0 . . . . (cid:15)/(cid:15) r B m (a) ω p = 0 ω p = 0 0 . . . . (cid:15)/(cid:15) r ω p = 0 ω p = 0 FIG. 8: Plot of the dominant instability of the system as a function of the PM parameters. In panel (a) we have chosen thesame parameters as in Fig. 6 but with κ = 0 . (cid:15) r . In (b), we employed the parameters from the Hamburg experiment ofreference [23]: κ = 0 . (cid:15) r , w /Q = 244 , L H = w /
10, as well as a cavity detuning ∆ = 4 . κ . In the blue region the systemexperiences a transition to a superradiant phase at the finite critical coupling strength Λ c . The bright color indicatesparameters where a finite-frequency instability takes place at Λ < Λ c . V. EXPERIMENTAL OBSERVABILITY
The scheme proposed here can be realized in severalstate-of-the-art laboratories. The main feature we dis-cussed is the presence of anticrossings, which is a clearsignature of the formation of the multimode Floquet po-laritons. A prerequisite for achieving appreciable hy-bridization between different cavity modes, is that tworesonances in the spectral function have to be broughtclose to each other. As all cavity modes have a positivedetuning, the corresponding peaks move towards zero fre-quency when coupling to the atoms. In order to generatean (avoided) crossing, we therefore need to make a modeof higher energy move to zero faster than a lower-energymode, while they both couple to the atoms. From theprevious discussions it is clear that one way of achievingthis is by having (cid:15) >
0, such that there is a higher-ordermode that has a smaller detuning. This was shown inFig. 4 (b). By introducing a shallow PM, the zerothmode will couple strongly to the atoms. This will pushit faster towards zero than the higher-order mode whichcouples weaker to the atoms. As the zeroth mode comesclose to the higher-order cavity mode, their repulsion willbe determined by how strongly they effectively interactthrough the atoms. Similarly, one could use (cid:15) < ∼ V F SR − ω T .This is easily possible for the small free spectral range V F SR of the long cavity in the Hamburg experiment, asit would require Ω ∼ . . . . / Λ c ) . . . . . . . ω / (cid:15) r LG (a) 0 . . . . ω /(cid:15) r A ( ω )0 . . . . . FIG. 9: Spectral function of LG mode for B m = 0 . (cid:15) = 2 κ . b) Cut at the white dashed line shown in a). All otherparameters are taken directly from the experimental setup used in Hamburg [23] given in Fig.8 (b). range of current fiber-based electro-optical modulators.In this experiment κ/(cid:15) R ≈ .
2, allowing for the subrecoilresolution necessary to see the described avoided cross-ings. We suggest to filter the cavity output for the LG mode and measure the distribution of the spectral weight.In Fig. 8 (b), we have seen that there is a large range ofparameters available to adjust the PM while preservinga zero frequency superradiant instability at a finite cou-pling strength. Choosing PM parameters that allow ananticrossing for the experimental parameters of [23], ourprediction is shown in Fig. 9. We see that an anticrossingis visible in the spectral weight of the LG component ofthe cavity field. In addition, clear multimodality shouldbe observable by decomposing the cavity output light af-ter entering the superradiant phase, as is typically donefor degenerate cavities [28]. Differently from the avoidedcrossings, a multimodal cavity output should be visibleeven when the loss rate is much larger than the recoilenergy. VI. CONCLUSION
We have shown that the periodic phase modulation ofthe driving laser can generate a large dispersive couplingbetween an ultracold atomic cloud and many modes ofa non-degenerate cavity. This leads to the formation ofmultimode Floquet polaritons. Their mutual interactionsmediated by the atoms are visible as avoided crossingsand ultimately lead to a phase transition to a multimodesuperradiant state. This scenario should be experimen-tally testable in state-of-the-art platforms.In this investigation, we have focused on the energeticeffect of the PM and simplified the degrees of freedom by considering small atom clouds at the center of the cavity.Having seen that multimode Floquet polaritons can begenerated, a very interesting aspect is to explore regimeswhere the cloud has significantly different overlaps withthe different resonant modes. This allows one to establishan added competing effect that can further enrich themultimode correlations. Moreover, the multimode natureof the polaritons and their mutual interactions might giverise to a richer scenario for finite-frequency instabilities.We defer these studies to future work.
ACKNOWLEDGEMENT
We thank Alexander Baumg¨artner, Tobias Donner,Davide Dreon, and Tilman Esslinger for useful discus-sions.
Appendix A: Mode overlaps
To calculate the density response, Eq. (8), one has tocompute the overlaps between atom cloud, cavity andlaser. In general these overlaps will have to be solved nu-merical but a closed form solution can be found when theatomic states are well described as eigenstates of the ra-dially symmetric, harmonic trap. Furthermore, the laserform-factor η p also has to be approximated as a constant.With a shallow longitudinal trap and if the atoms are in azero-temperature BEC then the longitudinal part of theatom eigenstate is tightly localized in momentum space.This means that the longitudinal part of the overlap leadsto the atoms scattering into a state with momentum Q ,set by the cavity geometry. The transverse part of the2overlap can be computed both for centered and and non-centered atom clouds. For the more general case whenradial symmetry is broken, one finds that the integralover three Hermite polynomials leads to three finite sums with a shared combinatoric pre-factor. However, in theradially symmetric case the result simplifies significantlywith the integral to be solved given by (cid:104) ψ | η jα | ψ nβ (cid:105) = 1 πL H (cid:115) j ! n !( j + | α | )!( n + | β | )! (cid:90) π d θ (cid:90) ∞ d r r exp (cid:2) iθ ( α − β ) − r (cid:0) L − H + w − (cid:1)(cid:3) × (cid:18) rL H (cid:19) | β | (cid:18) rw (cid:19) | α | L | α | j (cid:18) r w (cid:19) L | β | n (cid:18) r L H (cid:19) , (A1)where the mode indices have been split into a radial modeindex (roman letter) and a angular index (greek letter).The non-BEC scattering state in Eq. (8) leads to a de-coupling of cavity modes with different angular index, through the δ α,β from the angular integral. Starting inangular momentum zero we therefore condsider only thestates with zero angular momentum. The remaining ra-dial integral then has a closed form solution given by (cid:104) ψ | η j | ψ n (cid:105) = (cid:104) ψ n | η j | ψ (cid:105) = Γ( j + n (cid:48) + 1)2 n (cid:48) n (cid:48) ! j ! (cid:0) δ − (cid:1) j (cid:0) δ + (cid:1) j + n (cid:48) +1 2 F (cid:20) − n (cid:48) , − j ; − n (cid:48) − j ; − δ + δ − (cid:21) , (A2)where δ = w /L H is the relative size of the cavity waistcompared to the transverse harmonic trapping length, F is the Gauss hypergeometric function and Γ theGamma function. Clearly these overlaps are fully de-termined by the cavity waist and radially symmetricharmonic trap strength which sets L H . The simplest case is when the BEC is significantly narrower thanthe cavity waist. In this limit δ (cid:29) δ →∞ (cid:104) ψ | η j | ψ n (cid:105) = δ n . In all presented calcula-tions the result from Eq. (A2) has been used and thenumber of atomic states have been truncated only afterconvergence has been achieved. [1] Darrick E. Chang, Vladan Vuleti´c, and Mikhail D. Lukin.Quantum nonlinear optics —photon by photon. NaturePhotonics, 8(9):685–694, 2014.[2] H. J. Kimble. The quantum internet. Nature,453(7198):1023–1030, 2008.[3] Mark Saffman, Thad G Walker, and Klaus Mølmer.Quantum information with Rydberg atoms. Reviews ofmodern physics, 82(3):2313, 2010.[4] Iacopo Carusotto and Cristiano Ciuti. Quantum fluidsof light. Reviews of Modern Physics, 85(1):299, 2013.[5] Changsuk Noh and Dimitris G Angelakis. Quantum sim-ulations and many-body physics with light. Reports onProgress in Physics, 80(1):016401, Nov 2016.[6] Michael Fleischhauer, Atac Imamoglu, and Jonathan P.Marangos. Electromagnetically induced transparency:Optics in coherent media. Rev. Mod. Phys., 77:633–673,Jul 2005.[7] Michael J Hartmann, Fernando GSL Brandao, and Mar-tin B Plenio. Strongly interacting polaritons in coupledarrays of cavities. Nature Physics, 2(12):849–855, 2006.[8] Alexey V Gorshkov, Rejish Nath, and Thomas Pohl. Dis-sipative many-body quantum optics in Rydberg media.Physical review letters, 110(15):153601, 2013. [9] James S Douglas, Hessam Habibian, C-L Hung, Alexey VGorshkov, H Jeff Kimble, and Darrick E Chang. Quan-tum many-body models with cold atoms coupled to pho-tonic crystals. Nature Photonics, 9(5):326–331, 2015.[10] James S Douglas, Tommaso Caneva, and Darrick EChang. Photon molecules in atomic gases trappednear photonic crystal waveguides. Physical Review X,6(3):031017, 2016.[11] Stefan Ostermann, Francesco Piazza, and Helmut Ritsch.Spontaneous crystallization of light and ultracold atoms.Physical Review X, 6(2):021026, 2016.[12] Emil Zeuthen, Michael J. Gullans, Mohammad F.Maghrebi, and Alexey V. Gorshkov. Correlated Pho-ton Dynamics in Dissipative Rydberg Media. Phys. Rev.Lett., 119:043602, Jul 2017.[13] Kieran A Fraser and Francesco Piazza. Topologicalsoliton-polaritons in 1D systems of light and fermionicmatter. Communications Physics, 2(1):1–7, 2019.[14] Ruichao Ma, Brendan Saxberg, Clai Owens, Nelson Le-ung, Yao Lu, Jonathan Simon, and David I Schus-ter. A dissipatively stabilized Mott insulator of photons.Nature, 566(7742):51–57, 2019.[15] Johannes Lang, Darrick Chang, and Francesco Piazza.3