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Superradiance in Inverted Multi-level Atomic Clouds R.T. Sutherland1∗ and F. Robicheaux 1,2† 1Department of Physics and Astronomy, Purdue University, West Lafayette IN, 47907 USA and 2Purdue Quantum Center, Purdue University, West Lafayette, Indiana 47907, USA (Dated: February 15, 2017) This workexaminessuperradiancein initially invertedcloudsofmulti-level atoms. Wedevelopa setofequationsthatcanapproximatelycalculate thetemporalevolutionofN coupledatoms. This allowsustosimulatecloudscontaininghundredsofmulti-levelatomswhileeschewingtheassumption and/or approximation of symmetric dipole-dipole interactions. This treatment is used to explore the effects that dephasing caused by elastic dipole-dipole interactions, and competition between multiple transitions have on superradiance. Both of these mechanisms place strong parametrical restrictions on a given transition’s ability to superradiate. These results are likely important to recent experiments that probe superradiance in Rydbergatoms. PACSnumbers: 42.50.Nn,42.50.Ct,32.70.Jz,37.10.Jk 7 1 0 I. INTRODUCTION 2 b Thefactthatcoherentradiationcandramaticallyalter e physicalphenomena[1]hasresurfacedatthe forefrontof F physics. Studies have shown that superradiance must 4 be considered when studying atomic [2–22], biological 1 [23,24],andcondensedmattersystems[25–27]. Uponits recent revival, superradiance has led to conflicting ideas ] in the Rydberg atom community. At first glance, tran- h sitions between high-lying Rydberg states seem like per- p - fectcandidatesforsuperradiance[12,13,19]. Becauseof nt their long wavelengths,Rydberg transitions allowexper- a imentaliststoreachthe limitfirstdescribedbyDicke[1], u whereatomsareplacedinavolumewitharadiusthatis FIG.1. Thispaperstudiessuperradiantcascadesincloudsof q very small relative to the wavelength. Later theoretical (a) two, (b) three, and (c) four -level atoms. (a) Represents [ works [19, 28, 29], however, argued that putting a cloud atwo-levelsystemcoupledviadipole-dipoleinteractions. (b) 2 ofatomsinthisregimewouldresultinlargedipole-dipole Represents a three-level system, where dipole-dipole interac- v interactions that quickly dephase the transition and de- tions for the a transition are considered, but are not consid- 9 stroy superradiance. This implies that transitions be- ered for the g transition, due to the relatively small value of 1 tweenhigh-lyingRydberglevelsshouldnot superradiate. λg. (c) Represents a four-level system, where an additional 7 interactingtransition,b,hasbeenaddedtothesystemrepre- Toexacerbatethe confusion,conflictingexperimentalre- 3 sented by (b). sults have been reported, with some groups claiming to 0 . haveeitherdirectlyorindirectlyobservedRydbergatom 1 superradiance [12–14, 30], and one claiming to have ob- 0 served no superradiance at all [31]. In order to unravel 7 where Γ is the decay rate of the transition e → α in α 1 these experimental and theoretical differences, new de- an isolated atom (since the excited level is the same for : velopments are needed. every transition (see Fig. 1), the lower energy level, α, v The physics of superradiance in a vacuum is governed i is used as the label of a given transition). Resultantly, X by the set of dipole-dipole interactions between every this part of the dipole-dipole interaction is completely r atom pair. These interactions can be traced back to two symmetric for sufficiently dense clouds, i.e. the Dicke a distincttypesofphotonexchanges,realandvirtual. The limit [1]. Not included in Dicke’s work are the interac- exchangeof realphotons (see Eq. (5)) contributes to the tions that result from exchanges of virtual photons (see collective decay of the atomic ensemble. These inelastic Eq. (3)). These exchanges cause shifts between the en- dipole-dipole interactions, result in either superradiance ergy levels of the system, often referred to as the collec- or subradiance. As the value of |~r −~r | → 0, where i j tive/cooperative Lamb shifts [32]. These couplings will |~r −~r | is the spatial distance between atoms i and j, i j be referred to as elastic dipole-dipole interactions from the value of every inelastic coupling approaches Γ /2, α here on. In a vacuum, the magnitude of the elastic in- teractionbetweentwoatomsdiverges∝1/(k |~r −~r |)3, α i j where k ≡2π/λ . For dense atomic clouds, this results α α ∗ [email protected] in large and random energy shifts between levels that † [email protected] dephasethe system,quelling superradiance. This willbe 2 referred to as elastic dephasing in the rest of this paper. multiple transitions can superradiate at once. This sec- A rigorous numerical treatment of this effect requires tion indicates that the previously proposed mechanism an implementation of the superradiance master equa- for superradiance in Rydberg atoms [13], where states tion[29],orequivalentlythequantumMonte-Carlowave- tend to superradiate via transitions with the largest val- functionalgorithm[33]. Unfortunately,bothofthesecal- uesofλα,is likelynotthe dominantmechanisminmany culations grow exponentially with the number of atoms Rydbergatomsystems. Hereitis arguedthatonlytran- being simulated, N. So far, this has limited numerical sitions with λα lying within a certain range of λαN1/3 calculations to systems such that N ∼ 10 or less. While superradiate. This might lead to an explanation of the recent treatments have been developed that permit sim- current experimental disagreements [12–14, 31]. ulations involvinglargenumbers ofhighly-excitedatoms [13, 34, 35], so far they rely on symmetries that result fromeithertheassumptionorapproximationofsymmet- II. NUMERICAL TREATMENT ric dipole-dipole interactions. These approximations are notvalidinthisworkbecausetheyignoreelasticdephas- A. Master Equation Evaluation ing. Because of this, we derive and implement a numerical Thetimedependenceofthereduceddensitymatrix,ρˆ, approach that scales ∝ N4, rather than exponentially. is given by the master equation [36] Thisenablesthesimulationofinitially invertedcloudsof hundreds of multi-level atoms. Note that a system con- dρˆ i taining initially inverted atoms is qualitatively different =− [H ,ρˆ]+L ρˆ . (1) dt ~ ed than a system where a particular transition is triggered, (cid:0) (cid:1) such as that in [14]. Unlike previous methods, our ap- Here H represents the elastic dipole-dipole interaction ed proach can fully incorporate the inhomogeneous dipole- defined by the Hermitian Hamiltonian: dipole interactions present in the cloud. This is done by solving the set of differential equations that describe the expectation values of the operators: bα−bα+, where Hed = ~fiαjbαi+bαj−, (2) bα−(+) represents the lowering(raising) opieratjor for the iX6=j,α i e→αtransitionoftheith atom. Theresultingequations where bα+ ≡|e ihα |, bα− ≡|α ihe |, and i i i i i i are then truncated by factorizing the higher-ordercorre- 3Γ sinξα cosξα lation operators. Since a full analysis of superradiating fα = α 1−3cos2φα ij + ij ij 4 ij ξα2 ξα3 Rydberg atoms requiresan understanding of the compe- n(cid:16) (cid:17)(cid:16) ij ij (cid:17) titionbetweenmanypotentiallysuperradianttransitions, cosξα −sin2φα ij . (3) the derivationassumesaninitially excitedstate thatcan ij ξα ij o decay into an arbitrary number of lower energy states. The equations are then implemented in order to study L(ρˆ) is the Lindblad superoperator given by: the rich physics that occurs in clouds of two, three, and 1 1 four -level atoms (see Fig. 1). L(ρˆ)= Γα bα−ρˆbα+− bα+bα−ρˆ− ρˆbα+bα− , (4) ij j i 2 i j 2 i j This paper is organized in the following way: first in iX,j,α (cid:16) (cid:17) Sec. II, the equations that describe a system containing where Γα is the inelastic dipole-dipole interaction of N multi-level atoms are derived. In Sec. III, this sys- ij atoms i and j for the e→α transition, tem of equations is used to simulate a cloud of two-level atoms. Here, it is demonstrated how superradiance is limited by elastic dephasing in high density systems and Γα = 3Γα 1−3cos2φα cosξiαj − sinξiαj bydiffractioninlowdensitysystems. Theresultofthisis ij 2 ij ξα2 ξα3 n(cid:16) (cid:17)(cid:16) ij ij (cid:17) thatforacloudwithagivenN anddensity,N,thereisa sinξα particularvalue ofλg, suchthatthe coherentemissionis +sin2φαij ξαij . (5) at a maximum. In Sec. IV, it is shown that when a sys- ij o temhasmultiple decaychannels,superradiancedevelops In this notation, φα is the angle between the αth dipole in a very different manner than when there is only one. ij momentandtherelativepositionofatomsiandj,~r −~r . HerewemimicanelementaryRydbergsystembyinclud- i j ξα = k |~r −~r |, where k is the wavenumber of the ingonetransitiontoahigh-lyingstate,a,withavalueof ij α i j α e→α transition, 2π/λ . λ such that atoms couple via dipole-dipole interactions α a The fact that the system of interest starts in state significantly. On top of this, we include one transition |eee...ei, and has no driving term, leads to a massive to the ground state, g, with a very small λ such that g truncation of the Liouville-space. If one expresses ρˆ in dipole-dipole interactions are negligible (see Fig. 1(b)). the form: This section shows that the presence of an alternate decay path strongly diminishes the buildup of superradi- ρˆ= c |mihn|, (6) mn ance. Finally,Sec. Vshowsthephysicsthatresultswhen Xm,n 3 it can be shown straightforwardly that the operators in where α 6= β. These factorizations are choosen with Eq. (1) will only connect to elements of ρˆsuch that |mi two ‘rules’ in mind. First, the quartic operators must and hn| contain the same number of atoms in each level. be grouped such that both the raising and lowering op- This constitutes only a small fraction of ρˆ. For a system erators act on the same transition, in order to insure a ofN two-levelatoms,thefullLiouville-spacecontains4N closed system of equations. When this is satisfied, the matrix elements. However,only quartic operator representing the population of state α for atom m (i.e. bα−bα+) is factored out. This is done m m N N 2 2N so that the factorized terms are smaller when little de- = (7) cayhas occured,makingourapproximationsaccurateat (cid:18)k(cid:19) (cid:18)N (cid:19) kX=0 early times. Implementing these approximations results in the following closed system of equations: of these elements are actually non-zero. Incorporating this into our numerical algorithm, enables the exact cal- d hbα−bα+i=−hbα−bα+i Γ dt n m n m β culation of Eq. (1) up to 10 atoms (see Fig. (2)). Xβ + gα∗hbα−bα+i 1−hbα−bα+i− hbβ−bβ+i jm n j m m m m B. Approximate Evaluation of Operators j6=Xm,n n Xβ o + gα hbα−bα+i 1−hbα−bα+i− hbβ−bβ+i nj j m n n n n The problems addressedin this paper require simulat- j6=Xm,n n Xβ o ing hundreds of atoms, while avoiding the usual mean- − gβ hbα−bα+ihbβ−bβ+i field approximations that ultimately ignore elastic de- nj n m j n phasing [13, 34, 35]. This is accomplished by solving βX6=αj6=Xm,n the differentialequationsthatdescribetheprobabilityof − gβ∗hbα−bα+ihbβ−bβ+i atom i being in state α, or in operator form: hbαi−bαi+i, βX6=αj6=Xm,n jm n m m j as well as the quadratic correlation functions for atoms i and j, defined as hbα−bα+i. The change in the expec- +2Re gnαm (1− hbβn−bβn+i)(1− hbβm−bβm+i) i j tation value of an operator, Ω, with time is determined (cid:8) (cid:9) Xβ Xβ by: −gα hbα−bα+i(1− hbβ−bβ+i) nm n n m m Xβ d dρˆ −gα∗hbα−bα+i(1− hbβ−bβ+i). (11) hΩi=Tr Ω . (8) nm m m n n dt n dto Xβ Using Eq. (1) and Eq. (8), one can derive dhbα−bα+i Thefactorizedtermsareinitiallynegligible. Therefore, dt i i at early times the results from Eq. (11) agree quanti- (see appendix). This gives: tatively with the results from Eq. (1). At later times, d the two equations agree qualitatively. This is shown hbα−bα+i=Γ 1− hbβ−bβ+i dt i i α i i in Fig. 2(a), where the photon emission rate per atom, (cid:16) Xβ (cid:17) γ′, versus time is obtained by solving both Eq. (1) and +2 Re gαhbα−bα+i , (9) Eq. (11) for clouds of 10 atoms. Fig. 2(b) shows the ij j i Xj6=i (cid:16) (cid:17) probability of excitation: wifhiαjer+e gΓiαjαij/is2)t.heTcohmespeleexqduiaptoiolen-sdiaproeledienpteernadcetniotno(ngiαjth≡e NNe ≡ N1 Xi hbαi+bαi−i (12) quadratic correlationfunctions, which may be solved for versus time for 10 atom clouds, using several values for inthesamemanner(seeappendix). Thisyieldsasystem density, N. The figure not only shows how Eq. (11) is of equations that is not closed, because solving for the qualitatively accurate at later times, but that it scales quadratic correlation functions results in equations that correctly with N. Fig. 4 also shows this by demonstrat- depend on the quartic correlation functions. To avoid ingthatthephotonemissionmaximaversusN calculated this, the system of equations is truncated by factorizing with both methods, closely match for 10 atom clouds. thequarticcorrelationfunctionsinthefollowingmanner: Sec. IIIA further illustrates this fact for clouds where elastic dephasing is neglected (see Fig. 3). Figure 2(c) hbα−bα+bβ−bβ+i≃hbα−bα+ihbβ−bβ+i shows that Eq. (11) also scales correctly with N. Here n j m m n j m m Eq. (11) is compared with Eq. (1) for the pure Dicke hbβm−bβj+bαn−bαm+i≃hbβm−bβj+ihbαn−bαm+i model (i.e. λ3gN → ∞, Im(giαj) = 0, and Hed = 0) hbα−bα+bβ−bβ+i≃hbα−bα+ihbβ−bβ+i [1]. Using this model and clouds such that N = 10,40, m m n n m m n n and 160, the results from Eq. (11) scale with N in the hbα−bα+bα−bα+i≃hbα−bα+ihbα−bα+i n j m m n j m m same way as the results from Eq. (1). Since the calcula- hbα−bα+bα−bα+i≃hbα−bα+ihbα−bα+i, (10) tionsofthisworkareintendedtoprobesuperradiancein m m n n m m n n 4 One may also note some differences in the calculations shown in Fig. 2(c), at longer times. For example, in the simulation of a cloud containing 40 atoms, the value of N /N approaches approximately 0.022 at later times, e rather than 0. This illustrates the fact that the results of Eq. (11) are only exact when a small amount of population has decayed via an interacting transition. For simplicity, all of the states considered here are as- sumed to be M = 0. Unless specified otherwise, all J calculations average over 15360/N randomly generated frozen atomic clouds. Each cloud is given a Gaussian density distribution: N −r2 N(r)= exp , (13) σ3(2π)3/2 (cid:16)2σ2(cid:17) with an average density, N, determined by N = N/(4πσ2)3/2. III. TWO-LEVEL ATOMS The Dicke model describes two-level atoms (see Fig. 1(a)) radiating from a volume that is small relative to the transition’s wavelength, λ . In this limit, all the g atomsradiatefromeffectivelythe sameposition,making their emissionin alldirections coherent. This dissipative coherencecausestheincreaseinphotonemissionrateas- sociated with superradiance [1]. In reality, the system is morecomplexintwoimportantways: experimentallyre- alizable clouds canbe muchlargerthan λ , andphysical g clouds undergo elastic dephasing. A. Finite Size Effects FIG. 2. Comparison of the time dependence resulting from Eq. (1) (solid line) and Eq. (11) (dashed line) for clouds of Large and dilute clouds (compared to the Dicke limit) two-level atoms. (a) Photon emission rate per atom, γ′ ≡ can also superradiate [37, 38]. While atoms in such γan/Nd,λ3foNr c≃lo3u7d.s(wb)heErexcNitat=ion10p,roabnadbdileitnys,itNies/:Nλ,3gfNor c=lou12d5s cloudsare usuallyseparatedby distances largerthanλg, g e the emission of successive photons in a particular direc- whereN =10,andλ3N ≃1000,125,and37. Notethatinthe figure,thevaluesofλg3N areinverslyproportionaltotheslope tion, kˆg, projects the cloud onto a quantum state with g a diffraction maxima, and therefore coherent radiation, of N . (c) N /N for N = 10, 40, and 160 two-level atoms e e usingtheDickemodel,i.e. N →∞andHed =0. Resultsare in kˆg. This causes superradiance [1, 33, 37]. Ignoring shown for the full calculation obtained using Eq. (1), as well elastic dephasing and invoking a semiclassical approx- as the approximate Eq. (11). Note that the two calculations imation, the time-dependence of a given cloud can be scale with N in the same manner. shown to be [37]: N˙ =−Γ N +µ(k σ)N N , (14) e g e g e g regimesthat areunreachableusing Eq.(1), the factthat n o the output of Eq. (11) scales correctly with both N and wherek ≡2π/λ ,N istheexpectationvalueofnumber g g g λ3N indicates the validity of the calculations presented of atoms in the ground state (N = N −N ), Γ is the g g e g below. single-atom decay rate of the g transition, and µ(k σ) g One may note that the results of Fig. 2(b) appear to is a shape parameter, which for a cloud of xˆ polarized agree significantly better than those of Fig. 2(a). This two-level atoms is given by: is because the photonemissionrates,shownin Fig.2(a), correspondtothenegativederivativesofthecalculations µ(k σ)= 3 dΩ 1−(kˆ·xˆ)2 eikg(kˆ−kˆg)·(~rm−~rn), in Fig. 2(b). These small differences in slope, produce g 8πN2 Z kn omX6=n very small differences in N over the time frame shown. (15) e 5 where kˆ is the direction of the diffraction maxima of g the radiation. This equation shows that µ(k σ) is de- g termined by the diffraction pattern of the cloud’s emission[37]. Solvingthis non-lineardifferentialequationfor thetime dependenceofthecloud’sphotonemissionrate, γ(t), yields: 2 NΓg 1+Nµ(kgσ) eΓgt(1+Nµ(kgσ)) γ(t)= (cid:16) (cid:17) , (16) 2 Nµ(kgσ)+eΓgt(1+Nµ(kgσ)) (cid:16) (cid:17) which reaches a radiative maximum after a time delay, t , equal to: d ln Nµ(k σ) g t = . (17) d Γ 1(cid:0)+Nµ(k σ(cid:1)) g g (cid:0) (cid:1) FIG.3. Themaximumphotonemission rateperatom,γ′ , This equation may be applied to the present system, a max given by both Eq. (11) and Eq. (16) for a Gaussian cloud of spherically symmetric Gaussiancloud, by convertingthe 160atomswhenelasticdephasingisneglected. Notethatthe discrete sums in Eq. (15) to integrals, assuming kˆg = plots agree well, indicating the validity of both equations in zˆ. Performing the integrals over both kˆ and the atomic theabsence of dephasing. positions yields: 3 µ(k σ)= 1−2k2σ2+4k4σ4 is seen for both calculations in Fig. 3, when the value g 32kg6σ6n g g of γ′ increases towards a constant, as 1/(λgN1/3) de- max −e−4kg2σ2 1+2k2σ2+4k4σ4 . (18) creases towards 0. The quantitative agreement shown in g g the two simulations is a good indication of the useful- (cid:0) (cid:1)o In the limit, k σ → ∞, µ(k σ) → 3 . This agrees ness of Eq. (16) when elastic interactions are neglected. g g 8kg2σ2 However,this is often not valid. withrecentresultsthatshowthe collectiveenhancement to the decay rate of a large, singly-excited, and dilute atomic cloud is 3(N−1)Γg [4]. Conversely, in the small 8k2σ2 B. Dephasing Due to Elastic Dipole-dipole g cloud (k σ → 0) limit, µ(k σ) → 1, which reduces the Interactions g g more general Eq. (14) to the equation derived by Dicke [1]. TheseminalworkofDickeignorestheoff-resonant,vir- Eq. (16) is a reasonable approximation when dephas- tual photon exchanges that lead to elastic dipole-dipole ingduetoelasticinteractionscanbeignored. Thisstate- interactions. In dense clouds, such as the ones described mentismadeapparentbycomparingEq.(16)toEq.(11) by Dicke, elastic interactions cause large and random whengmαn →Γαmn/2,thusartificiallykeepingonlythein- energy shifts and can lead the system to dephase on elastic part of the each dipole-dipole interaction. Since timescales much shorter than its collective decay rate one of the key features of aninitially invertedsuperradi- [28, 29]. This has been described semi-classically[28], as ating cloud is an increase in photon emission, the max- wellasnumericallyforsmallvaluesofN (i.e. N=3−10) imum photon emission rate, γ(td), is used to quantify [33, 39]. In this section, Eq. (11) is used to simulate the ‘amount’ of superradiance in a givencloud. Figure 3 cloudscontainingupto640initiallyinvertedatoms. This compares the two calculations by showing the maximum enables the exploration of the fundamental limits that photonemissionrateperatom,γ′ ≡γ(td)/(NΓg),ver- elastic dipole-dipole interactions have on superradiance. max sus 1/(λ N1/3) when N = 160. As the value of λ3N For a given atomic cloud, the calculations presented in g g increases, γ′ increases as well. This increase of γ′ this section place strong restrictions on the parameters max max withN maybeunderstoodbyconsideringthediffraction thatcanleadtosignificantsuperradiantbuildup. Aswill patternofacloudthathasemittedaseriesofphotonsin be shown in Sec. V, this has important implications on a particular direction, kˆ. As the value of λ3N increases, the Rydberg atom problem. g the size of the atomic cloud decreases. The result of this Therelevantquantitywhendeterminingthedephasing is that µ(k σ) increases due to the broadening of the rateofaspecificatomiccloudisλ3N. InFig. 4,Eq.(11) g g diffractionmaximacenteredatkˆ. Asthesizeofthecloud issolvedinordertodemonstratehowγ′ foraparticu- max decreases,photonscanradiatecoherentlyintomoresolid lar transition depends on N and 1/(N1/3λ ). In Fig. 4, g angles, until finally the Dicke limit is reached and pho- γ′ is shown for clouds such that N = 10, 20, 40, 80, max tons are coherent in all directions (µ(k σ) → 1). This and160. Thereareseveralimportanteffectsvisiblehere. g 6 First, increasingN indilute clouds causes anincreasein γ′ . Thisisbecauseindilutecloudselasticdephasingis max slow relative to the collective decay of the cloud. There- fore little dephasing occurs within t . Thus decreasing d the cloud size simply increases the directional coherence (the value of µ(k σ)) discussed in Sec. IIIA. As one in- g creases λ3N, however, the dephasing rate due to elastic g interactions (Eq. (3)) grows linearly, while the radiative enhancement due to inelastic interactions (Eq. (5)) approaches a constant value (Γ /2). Therefore, in dense g clouds the system dephases significantly before t , and d the value of γ′ begins to diminish ∝ λ3N. This is max g seen in Fig. 4, where γ′ increases as 1/(λ N1/3) de- g max creases for dilute clouds, followed by a rapid decrease as 1/(λ N1/3) →0. Due to computational limitations, the g largestvalue of λ3N inFig. 4is 37037,however,the val- g ues ofγ′ do seemto approach0 for increasingly dense FIG. 4. Eq. (11) is used to calculate the maximum pho- max clouds. This is in contrast to Fig. 3, where there is no ton emission rate divided by the number of atoms, N, times elastic dephasing. Here, as clouds condense the value of the single atom decay rate, Γ , (γ′ ≡ γ(t )/(NΓ )) ver- g max d g γ′ increases to a constant with approximately 6 times sus 1/(λ N1/3). This is done for clouds such that N = thmeaxmaximum values of γ′ seen in Fig. 4. In order 10, 20, 4g0, 80, 160. As 1/(λ N1/3) decreases, at first γ′ max g max to demonstrate the accuracy of our approximate simula- increases, due to the broadening of the diffraction maxima. tions, Fig. 4 compares the photon emission maxima for However, for every value of N there is a certain λ3N where g 10 atom clouds using both Eq. (1) and Eq. (11). Note thelargeelasticdipole-dipoleinteractionsbegintodominate, that for a given density, there is a narrow range of λg ashnodwtshethvaatluteheofvaγlm′uaexosftλar3tNs tsoudchectrheaastesruappeirdrlayd.iaTnhcee iinssaett where superradiance maximizes. g a maximum , λ3N , increases linearly with Nµ(k σ) due g max g tothecollectiveenhancementtothedecayrate. Notethatin Figure 4 shows that for clouds with more atoms, the themainfigure,thesolidblackdotsrepresentthecalculations value of λ3gN with the largestvalue of γm′ ax, λ3gNmax, in- for 10 atom clouds using Eq. (1). creases. This is because the collective decay rate of a cloud increases ∝ Nµ(k σ) while the dephasing rate of g a dense cloud increases ∝ λ3gN. Assuming λ3gNmax oc- C. Parallel with Classically Radiating Dipoles curs when the two rates are equal, up to some constant, λ3N should increase linearly with Nµ(k σ). This in- g max g There is a notable parallel between the physics of a crease of N with Nµ(k σ) is shown in the inset of max g cloud of two-level atoms and that of a cloud of coupled Fig. (4). harmonicoscillators. Inmatrixform,thesetofequations that describes coupled harmonic oscillators is given by: In the single excitation regime, it has been demonstrated that large elastic interactions produce negligible ~a˙ =−G~a, (19) effects when clouds are sufficiently dilute. This enables relatively straightforwardanalytic treatments that agree where each element of the vector ~a represents the com- well with the exact numerical results [4, 11, 20, 40]. For plex amplitude of a specific oscillator. The matrix el- more dense clouds, however, the numerical results begin ements of the complex symmetric matrix, G, are given to deviate from the analytic ones [4, 11]. Figure 5 shows by: that clouds of initially inverted atoms are similar in this respect. As seen in Fig. 5(a), for initially inverted and Γg dilute clouds, Eq. (16) gives similar results to the full Gmn ≡ m2n +ifmgn(1−δmn). (20) calculation of Eq. (11). For lower values of λ3N, the g numerical calculation of γ′ grows with N in a similar The eigenvalues of G often have important physical sig- max manner to Eq. (16). For more dense clouds, the results nificance [5, 9, 41–43], where the real part of an eigen- are very different. When comparing Fig 5(b) and 5(c), value corresponds to half of that eigenmode’s decay rate we see that the presence of largeelastic dipole-dipole in- and the imaginary part corresponds to its energy shift, teractions significantly decreases the rate at which γ′ often called the collective/cooperative Lamb shift. This max grows with N. Counterintuitively, in clouds where elas- section will be concerned with each eigenmode’s decay ticdephasingisimportant,suchasinFig.5(c),theslope rate, Γ , defined by the equation: j of γ′ increases with N. This is because the larger the max lveaslsuetimofeNtoµ(dkegpσh)a,stehbeesfoooreneirttddecoacycsurssig,nailfilocwanintgly.a cloud G~aj = Γj +iǫj ~aj. (21) 2 (cid:16) (cid:17) 7 FIG. 6. The average value of the decay rate of the most superradiant eigenmode, hΓ i, in clouds such that N = 10,20,40,80, and 160. Themainxset shows the value of λ3N g such that hΓ i is at a maximum, λ3N . max g max as the clouds become condensed. Also, as seen in the inset ofFig. 6, for everycloud with more than10 atoms, the value of λ3N is equivalent to its corresponding g max value in the inset of Fig. 4. Similar to a cascade of initially inverted atoms, there is competition between the FIG.5. Forcloudswith agiven density,N, thisfigureshows increasinglysymmetric inelastic interactions (Eq.(5)) in the maximum photon emission rate per atom, γ′ , versus max G, and the highly disorderedandincreasingly largeelas- the numberof atoms, N. (a) For dilute clouds, where elastic ticdipole-dipoleinteractions(Eq.(3)). Notethattheval- dipole-dipole interactions may be ignored, the analytic re- ues of hΓ i in Fig. 6 do not approach0 as 1/(λ N1/3) sultsofEq.(16)aresimilartothefullnumericaltreatmentof max g Eq.(11). HereacloudwithN =30.5/λ3 isshown. Notethat approaches 0. This is likely due to the fact that the g forbothcalculations,theslopeofγ′ versusN decreasesat eigenmodes in highly dense clouds become localizedover max largerN. Thisisduetothediffractivecoherenceasdescribed several atoms [44]. in thetext. Formore denseclouds where N =15625/λ3, (b) g shows the analytic results given by Eq. (16), and (c) shows the full numerical results obtained by solving Eq. (11). Here IV. THREE-LEVEL ATOMS: MULTIPLE the presence of large elastic dipole-dipole interactions in (c) DECAY CHANNELS causes thetwo calculations to notably deviate. Two-levelsystems,inmanyrespects,arethe best pos- In the originalDicke model, when a clouddecays,it cas- siblescenarioformaximizingtheeffectsofsuperradiance. cades through a set of states with a specific ‘coopera- Realistically,superradianceexperimentstypicallyinvolve tion number’. For an initially inverted cloud of atoms, populating an excited state that can decay via multi- the cloud cascades throughthe most superradiantstates ple transitions. These transitions then ‘compete’ for the (cooperationnumber = N/2)until it reachesthe ground buildup in cooperativity that results in superradiance. state[1]. Sincethesystemsinvestigatedinthispaperare As an example, a Rydberg-like system (see Fig. 1(b)), of initially inverted, we imagine a physical situation where anexited statethat candecayinto a high-lyingRydberg the atomic cloud in question has decayedinto the ‘most’ state, a, as well as a low-lying ground state, g, is con- superradiant singly excited eigenmode, as a parallel to sidered. For typical Rydberg systems λa ≫ λg. Since the results of Fig. 4. Rydbergexperimentsareusuallyconductedforvaluesof In Fig. 6, the maximum value of Γ , hΓ i, aver- N such that λ3N ≪ 1, setting Γg = 0 and fg = 0 is j max g ij ij aged over 1.6×105/N runs, is shown for clouds where valid. Thisisnot thecasefortransitiona,whereinmany N = 10,20,40,80, and 160. More configurations are av- experimentalsetupsλ3N ≫1. Despitethis potentialfor a eraged here than in the rest of the paper, since only one cooperative behavior, the presence of an alternate decay data point from each run is kept. Fig. 6 shows a pattern routesignificantlydampensthebuildupofsuperradiance that is remarkably similar to Fig. 4. At low densities, in the a transition. hΓ i increases with λ3N followed by a sharp decrease Thequalitativephysicsstudiedherecanbegleanedby max g 8 examining the two linearly independent equations that above physics, the qualitative features of Fig. 7 can be correspond to Eq. (14) for the three-level system de- replicatedusingthisequation;however,elasticdephasing scribed here: dampens the results by a factor of approximately 2. We N˙ =−(Γ +Γ )N −Γ µ(k σ)N N also note that the larger the value of Γg/Γa is, the more e g a e a a e a accurate Eq. (11) becomes. This is because the terms N˙a =ΓaNe+Γaµ(kaσ)NeNa, (22) thatareapproximatelyfactorizedinEq. (11)aresmaller when less population decays via channel a. where k ≡ 2π/λ . Note that N = N +N +N and a a g a e 0 = N˙ +N˙ +N˙ . Since transition a can potentially As far as the superradiant enhancement to transition g a e a is concerned, the process of atoms decaying via an al- superradiate, one may imagine that if Nµ(k σ)Γ >Γ , a a g ternate decay route is similar to removing atoms from then the a transition should dominate the g transition. the system. Resultantly,the temporaldependenceofthe However,thisisoftennotthecasesincethesuperradiant enhancement to N˙ is proportionalto the number of de- transition a for larger values of Γg/Γa is like that of a a two-level system with less atoms. This can be seen in cays via transition a (N ). In Rydberg systems, Γ /Γ a a g is a small value. Therefore, until Γ µ(k σ)N ∼ Γ , N˙ the inset of Fig. 7. Note that similar to clouds of two- a a a g a level atoms (see Eq. (17)), t increases for larger values will be smaller than N˙ . Often the entire system will d g of Γ /Γ . g a decay before this occurs, preventing any significant superradiant enhancement to the transition a. V. FOUR-LEVEL ATOMS: COMPETING SUPERRADIANT CHANNELS In Rydberg atom experiments, there are multiple po- tentiallysuperradianttransitionsthatcompetewitheach other. To properly explore this, an additional transition to an upper-lying Rydberg state, b, is added to the system of the previous section (see Fig. 1(c)). This allows simulations of more realistic Rydberg systems. Here, the development of superradiance in a particular transition not only competes with the decay to state g, as described in the previous section, but also with another superradiating channel. For an isolated atom, the com- FIG. 7. Eq. (11) is used to calculate the maximum pho- petitionbetweenmultipletransitionscanbesummarized, ton emission rate per atom for transition a divided by Γ , γ′ , versus Γ /Γ . This is shown for clouds such thaat straightforwardly,bythebranchingratiosofthosetransi- (a)max g a tions. These branching ratios are determined by the sin- N = 1000/λ3 and N = 10,20,40, and 80. For every value a gle atom decay rates of the system. When an ensemble of N shown, γ′ decreases as therelative decay rate into (a)max of atoms radiates coherently, however, the competition state g increases. The inset shows the temporal dependence of γ′ /Γ . Note that when Γ /Γ increases, t increases. between transitions is more complex. As will be shown, (a) a g a d when considering the competition between superradiating transitions in a cloud, there are two important pa- The superradiant enhancement to the decay rate is il- rameters: the single atom decay rates, and the relative sluhsotwrasteγd′ throuvgehrsvuasluΓego/fΓγa′ fcoarlcturalantseidtiounsian,gγ(′Ea)q..F(i1g1.)7. denIfsietliaesstifcordeeapchhastirnagnsiistinoeng(lei.cet.edλ,3αtNhe).superradiant en- (a)max Here the presence of an alternate decay channel quells hancement to a transition is proportional to the num- superradiantbehavior. Thisisseeninthefactthatwhen ber of atoms that have decayed via that transition (see the value of Γg/Γa is increased,a decrease in γ′ fol- Eq. (14)). It follows that transitions with larger decay (a)max lows. As shown in Fig. 7, for clouds with larger values rates will experience more coherent enhancement, sim- of N, the system is progressively more resistant to de- ply because they decay more. In Sec. III, it was demon- cay into state g. This is because clouds consisting of stratedthatforagivenvalueofN1/3 thereisaverynar- more atoms must radiate more photons before all atoms rowrangeinλ suchthattransitionαcandevelopstrong α reach the ground state. This provides more opportu- superradiantcharacter. Thisisbecausethe superradiant nities for the system to decay into state a. When the enhancement to transition α is limited by diffraction if system has decayed sufficiently into state a, such that λ N1/3 is too small and by elastic dephasing if λ N1/3 α α N µ(k σ)Γ ≫Γ , the presence of transition g becomes istoolarge. Resultantly,agivenRydbergstatewillhave a a a g unimportant. This means that alternate decay paths only a small number of transitions with the potential are less relevant for clouds with more atoms. However, to develop significant superradiant behavior. This dif- in many Rydberg-atom schemes Γ /Γ ∼ 50, making fersfromprevioustreatmentsignoringelasticdephasing, g a this mechanism likely to be at least quantitatively im- since they argue that transitions with the largest values portant. Since the much simpler Eq. (22) includes the of λ always show the most superradiant enhancement α 9 [13, 37]. in part because Γ > Γ . Also, since λ3N is not large a b a enough for elastic dephasing to be important, the fact that µ(k σ) > µ(k σ) causes the superradiant develop- a b ment to favor transition a. Therefore, for more dilute clouds, as N is increased, the system tends to decay more into state a. This indicates that at this density, for clouds with N ∼ 104, such as those of recent experiments [12, 13, 31], the a transition is likely to dominate. Conversely, Fig. 8(b) shows that for more condensed clouds, N ≃ 37037/λ3 ≃ 1098.3/λ3, elastic dephasing a b can be fast enough to significantly diminish superradiance in one transition, while still allowing superradiance to build in another. Here we can see that even though Γ > Γ and µ(k σ) > µ(k σ), at longer times N /N a b a b a b decreaseswithN. Earlierintheevolution,N /N inFig. a b 8(b)growsinasimilarmannerasFig.8(a)bothbecause Γ > Γ and because µ(k σ) > µ(k σ). However, the a b a b elasticdephasingrateissignificantlylargerfortransition a than for transition b. Resultantly, later in the cloud’s evolution, elastic dephasing causes the buildup of super- radianceintransitionatodiminish,whichinturnallows FIG.8. AnatomiccloudisplacedinaRydbergstate,e,that the collective behavior to favor transition b. subsequentlydecaysintothreestates: a,b,andg. Thevalues Itisprobablethatforcloudswithenoughatoms,tran- of Γ and λ for each transition are given in the text. Only α α sition a will likely dominate again. This is because, as transitionstostatesaandbexperiencedipole-dipoleinterac- is shown in the inset of Fig. 4, the value of λ3N in- tions. Forbothofthefiguresabove,thecompetitionbetween a max creaseswithN. Ifagivencloudcontainedenoughatoms thetwotransitions’superradiantbehaviorisillustratedbythe suchthatλ3N ≃37037,thentransitionawouldlikely ratio of thepopulation in states a and b, Na/Nb. This is cal- a max re-emerge as the dominant superradiating transition at culatedusingEq.(11). (a)N /N versustimeforcloudssuch a b thatN =1000/λ3. Formorediluteclouds,thedephasingrate all times. Traces of this are visible in Fig. 8(b), where a forbothtransitionsisrelativelylow,causingtheenhancement for clouds with larger values of N, Na/Nb increases for to the a transition’s decay rate to grow with N faster than slightly longer periods of time before elastic dephasing the b transition’s enhancement. (b) Na/Nb versus time for decimates it. If a cloud reaches the point where the en- N ≃ 37037/λ3a. Initially, the coherent enhancement to the tire system decays before elastic dipole-dipole interac- a transition’s decay rate is stronger than the b transition’s tions can significantly dephase transition a, then it will enhancement. However, the elastic dipole-dipole interactions likely dominate the cascade. aremuchlarger fortheatransition thanfor theb transition. Lastly, one must consider the decay to state g, in or- Thiscausestheatransitiontoquicklydephase,whiletheco- der to fully understand what is occurring in Fig. 8. For herentenhancementtothebtransition’sdecayratecontinues reasons argued in Sec. IV, the fact that Γ /Γ ≃21 and to build with N. g a Γ /Γ ≃ 43 suggests that the decay via the g transition g b strongly suppresses the cooperative behavior in Fig. 8. These effects are made apparent in Fig. 8. So as to This is illustrated concretely in Fig. 9, where it is shown obtain experimentally relevantresults, Eq.(11) is solved that for even the most superradiant cloud in Fig. 8, less using values of Γ and λ calculated for specific transi- α α tions ofRb. For the calculations ofFig. 8, Γ =169s−1 than 15% of the population decays to either of the two a andλ =1.134×10−3mcorrespondingtothe26p→26s Rydbergstates. ThisshowsthatforthevaluesofN that a transition, while Γ = 80.8 s−1 and λ = 3.51×10−4 m are currently computationally feasible, the superradiant b b behavior of a system is dominated by the decay to the corresponding to the 26p → 25s transition. Lastly, the ground state. As was argued in Sec IV, for clouds with numbers for transition g correspond to the 26p → 5s transition,where Γ =3.5×103 s−1 and λ is negligibly significantlymoreatomsthiseffectwillbelessimportant. g g small. In Fig. 8, the value of N /N versus time allows one a b to observe the temporal behavior of the two transitions’ VI. CONCLUSION collective decay rates. Initially, N /N =Γ /Γ because a b a b superradiantbehaviorhasnotdevelopedyet. Asthesys- Thisworkconsistedofthedevelopmentandimplemen- tem begins to decay however, this ratio tends towards tationofarobustsetofdifferentialequationsthatcanbe the transition experiencing the most collective enhance- used to study superradiance in ensembles of initially in- ment. For example, Fig. 8(a) shows that for lower den- vertedatoms. This setof equationsincorporatedthe de- sity clouds, N = 1000/λ3 ≃ 29.65/λ3, superradiant be- phasing due to elastic dipole-dipole interactions present a b havior develops the most in transition a. This occurs in dense atomic clouds, and reduced the calculation so 10 studied. Further,itwasarguedthatwhichtransitionde- velops the strongest superradiant behavior depends on the values of N, Γ and λ for the system. α α The parametrical dependence of a transition’s potential to superradiate,described in this work, may provide a starting point for an explanation of why some experiments observe superradiance and some do not. For example,theobservationsin[12]wereconductedforclouds where N ∼ 107 cm−3 while in [31] they were conducted at N ∼109 cm−3. If elastic dephasing were not present, this two orders of magnitude difference in N would not have a tremendous effect, since the high lying transitions from an initial Rydberg state are all in the Dicke regime. When elastic dephasing is considered, however, therangeofλ N1/3thatcansuperradiateissignificantly α narrowed, meaning that the specific value of N for an experimentis extremely important. For transitions from high-lying Rydberg states to nearby Rydberg states, the value ofΓ decreasessharplywiththe differencein prin- α ciplequantumnumber,duetothesmallerdipolemoment forthosetransitions. However,Γ fortransitionstolow- α FIG. 9. The number of atoms in a cloud that have de- lying states can be two orders of magnitude larger than cayed into either of the two Rydberg states compared to the the Rydberg transitions, due to the fact that Γ ∝ ω3 α α number of atoms that have decayed into the ground state, [45]. Therefore it is possible that for very dense clouds, (Na+Nb)/Ng. Thisisshownforcloudswithvariousnumbers theRydbergtransitionswiththe largestvaluesofΓ are ofatoms,N,suchthat(a)λ3N =1000and(b)λ3N =37037. α a a prevented from superradiating by elastic dephasing (see Notethatforeverycloudshown,onlyaverysmallfractionof Sec. III), while decay rates to lower Rydberg states are the population actually decays into the Rydberg states. For prevented by the competition between those transitions thereasonsarguedinSec.IV,thisdiminishesthecooperative and the transitions to the ground state, (see Sec. IV). behavior of the cloud. This is calculated using Eq. (11) For more dilute clouds, however, transitions to nearby Rydberg states will undergo significantly less elastic dephasing, and will therefore be much more likely to superradiate. Thiscouldbe relevanttoelucidatingthe dis- that 100s of multi-level atoms could be simulated. Note crepancies currently present in experiments [12–14, 31]. that unlike some recent experiments [14], the superradi- The diversity of the transitions available in Rydberg ant cascades studied here were not triggered. ‘Rydberg- atoms implies that they have tremendous potential for like’systemswerestudiedsothattwofundamentallydif- studying superradiance. Nevertheless, quantitatively ferent effects could be examined. Sec. III focused on predicting the superradiant decay in a Rydberg cloud dephasing due to the elastic part of the dipole-dipole in- is a daunting task. Certain sophisticated approaches to teraction. Herewefoundthatbecausetheelasticdephas- doing this have been conducted [13]; however, these ap- ing rate of a given transition is ∝ λ3N, where N is the proachesdonotincorporateelasticdephasing. Thiswork α averagedensityforagivencloud,thereisanarrowrange fillsthatgap. However,thereisclearlyaneedforfurther of λ such that superradiance can develop significantly. experimental and theoretical developments that provide α Further in Sec. IV, it was shown that the presence of an the community with much needed insights into Rydberg alternatedecaychannelalsosuppressesthebuildupofsu- atoms, and superradiance in general. perradiance. In Sec. V, both of these mechanisms were This material is based upon work supported by the incorporated into one four-level system. 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