Hyperradiance from collective behavior of coherently driven atoms
aa r X i v : . [ qu a n t - ph ] J un Hyperradiance from collective behavior of coherently driven atoms
M.-O. Pleinert,
1, 2, 3
J. von Zanthier,
1, 3 and G. S. Agarwal
2, 4 Institut f¨ur Optik, Information und Photonik, Friedrich-Alexander-Universit¨atErlangen-N¨urnberg (FAU), 91058 Erlangen, Germany Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA Erlangen Graduate School in Advanced Optical Technologies (SAOT),Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg (FAU), 91052 Erlangen, Germany Institute for Quantum Science and Engineering and Department of Biological and Agricultural Engineering,Texas A&M University, College Station, Texas 77843, USA
The collective behavior of ensembles of atoms has been studied in-depth since the seminal paperof Dicke [R. H. Dicke, Phys. Rev. , 99 (1954)], where he demonstrated that a group of emittersin collective states is able to radiate with increased intensity and modified decay rates in particulardirections, a phenomenon which he called superradiance. Here, we show that the fundamentalsetup of two atoms coupled to a single-mode cavity can be distinctly exceeding the free-spacesuperradiant behavior, a phenomenon which we call hyperradiance. The effect is accompanied bystrong quantum fluctuations and surprisingly arises for atoms radiating out-of-phase, an alleged non-ideal condition, where one expects subradiance. We are able to explain the onset of hyperradiance ina transparent way by a photon cascade taking place among manifolds of Dicke states with differentphoton numbers under particular out-of-phase coupling conditions. The theoretical results can berealized with current technology and thus should stimulate future experiments. I. INTRODUCTION
Arguably one of the most enigmatic phenomena in thehistory of quantum optics is the discovery of superra-diance by Dicke [1–26]. A key requirement in Dicke’swork is the initial preparation of an ensemble of two-level atoms in a special class of collective states, so-calledsymmetric Dicke states. The startling gist is that eventhough atoms in these states have no dipole moment,they radiate with an intensity which is enhanced by afactor of N compared to N independent atoms. Thepreparation of such states has been a challenge since. Thefirst experiments on superradiance were done in atomicvapors [6–8], where it was assumed that the fully excitedsystem in the course of temporal evolution would at sometime be found in a Dicke state leading to the emission ofsuperradiant light [2, 3, 6]. The same assumption led tothe recent observation of subradiance [9].Many of the mysteries behind this effect have begunto unfold only recently. In the seventies, it was realizedthat the superradiant emission results from the strongquantum correlations among the atoms being preparedin symmetric Dicke states [5, 6]. Only lately, it becameclear that the N -fold radiative enhancement can be ex-plained by the multiparticle entanglement of the Dickestates. For example, studying superradiance in a chainof Dicke-entangled atoms on a lattice enables one to iden-tify that multiple interfering quantum paths lead to thecollective subradiant and superradiant behavior [14, 27].The strong entanglement of the states can already beinferred from the simpler two atom case and holds for al-most all Dicke states of a multi-atom-system. With thecurrent advances in quantum information science, we in-deed understand the great difficulties in precisely prepar-ing such highly entangled multiparticle states. There areproposals for the generation of whole classes of Dicke states using projective measurements [10, 11, 28], yetthese schemes have very low success probability. Deter-ministic entanglement has been produced with about adozen qubits in the form of W -states [12, 13], but we arestill far away from the realization of W -states for an ar-bitrary number of qubits. Calculations show that thesestates, which can be considered to be the analog to singleexcitation Dicke states with appropriate phase factors,produce also enhancement by a factor of N [14–16].In comparably simpler systems, yet with a higher num-ber of excitations, one can also study the quantum sta-tistical aspects of the collective emission [25]. However,down to the present day the generation and measurementof higher excited multi-particle entangled Dicke statesare challenging, so mostly systems with no more thanone excitation have been realized. In these systems, thedynamics is still quite complex, yet superradiance andalso subradiance can be fruitfully explored [14–22]. Ex-periments on superradiance with single photon excitedDicke states were also reported for nuclear transitions[23, 24]. A recent work discusses preparation of a singlephoton subradiant state and its radiation characteristicsfor atoms in free space [20]. Even applications of super-radiance are beginning to appear. Lately, a laser witha frequency linewidth less than that of a single-particledecoherence linewidth was realized [29] by using morethan one million intracavity atoms and operating in asteady-state superradiant regime [30, 31].Despite these advances, one is still faced with the diffi-culties in the optical domain which arise from the infinitenumber of modes in free space and interatomic effectslike the dipole-dipole interaction [4, 5]. It is thus evi-dent that one needs to work with systems which havefewer degrees of freedoms and where a precise prepara-tion of entangled states is possible. This brings us towork with single-mode cavities [26, 32] with few atoms. FIG. 1. Basic properties of the system. (a) Sketch of the system consisting of two atoms (A) that are coupled to a single-modecavity (C) and driven by a coherent laser (L) with Rabi frequency η . Intracavity photons can leak through the mirrors bycavity decay ( κ ) and be registered by a detector (D). Another possible dissipative process is spontaneous emission ( γ ) by theatoms. The inset shows a magnified section of the arrangements of the atoms: One atom (depicted left) is fixed at an anti-nodeof the cavity field, while the other atom (right) can be scanned along the cavity axis causing a relative phase shift φ z betweenthe radiation of the atoms. (b,c) Energy levels and transitions of the system for (b) in-phase and (c) out-of-phase radiation ofthe atoms. In the case of two atoms, the state space consists of manifolds of four Dicke states with different intracavity photonnumbers: For a fixed cavity state | n i , the unentangled two atom ground and two atom excited state | gg i and | ee i , respectively,as well as the maximally entangled symmetric and anti-symmetric Dicke states |±i . For clarity, we neither draw the transitionsdue to cavity decay nor the detuning. (d) Energy levels and transitions of the corresponding system containing a single atom. The current technological progress in atom trapping andthe availability of well characterized single-mode cavitiesis making this ideal situation becoming more and morea reality. Several experiments in the last two years havebeen reported using such well-characterized systems [33–35]. The experiments consist of two coherently drivenatoms [33, 35] or entangled ions [34] coupled to a single-mode cavity as depicted in Fig. 1(a). This setup enablesone to study collective behavior as a function of variousatomic and cavity parameters, e.g., the precise locationof the atoms.In spite of this recent progress on superradiant andsubradiant behavior [14–22] and the surge of new classesof experiments [29, 33–35], there is yet no report ofatomic light emission beyond that of superradiance. Inthis paper, we demonstrate that a two-atom system cou-pled to a single-mode cavity is capable of radiating up toseveral orders of magnitude higher than a correspondingsystem consisting of two uncorrelated atoms, thereby ex-ceeding the free-space superradiant emission by far. Wecall this effect hyperradiance. Surprisingly, hyperradi-ance occurs in a regime which one usually considers tobe non-ideal, namely when the two atoms radiate out-of-phase. Such nonideal conditions are rather expectedto suppress superradiance and thus one is not inclined toimagine in this regime an emission burst exceeding theone of superradiance.Although the study that we present is in the contextof atomic systems, the results should be applicable to other types of two-level systems like ions [26, 34, 36], su-perconducting qubits [37–39] and quantum dots [40–44].We thus expect that our findings stimulate a multitudeof new experiments in various domains of physics.
II. METHODSA. System
The investigated system follows the experiments of[33, 35] consisting of two atoms (A) coupled to a single-mode cavity (C) as shown in Fig. 1(a). A laser (L) ori-ented perpendicular to the cavity axis coherently drivesthe atoms. In this paper, we fix one atom at an antin-ode of the cavity field, while we vary the position of theother atom along the cavity axis inducing a relative phaseshift between the radiation of the atoms. The atomswithin the cavity are modeled as two-level systems withtransition frequency ω A , driven by a laser field at fre-quency ω L , and couple to a single-mode of the cavitywith frequency ω C = 2 πc/λ C . The i th atom is charac-terized by spin-half operators S + i = | e i i h g | i , S − i = ( S + i ) † and S zi = ( | e i i h e | i − | g i i h g | i ) /
2. Bosonic annihilationand creation operator a and a † describe the intracavitymode. The dynamical behavior of the entire system canbe treated in a master equation approach [45] and is gov-erned by ddt ρ = − i ~ [ H + H I + H L , ρ ] + L γ ρ + L κ ρ , (1)where ρ is the density operator of the atom-cavity sys-tem. In the interaction frame rotating at the laser fre-quency, atoms and cavity are described by H = ~ ∆( S z + S z ) + ~ δa † a . Here, ∆ = ω A − ω L is the atom-laserdetuning and δ = ω C − ω L the cavity-laser detuning.The Tavis-Cummings interaction term of atom-cavitycoupling is given by H I = ~ P i =1 , g i (cid:0) S + i a + S − i a † (cid:1) and can be obtained by utilizing the dipole approxima-tion and applying the rotating wave approximation [46].The term g i = g cos(2 πz i /λ C ) describes the position-dependent coupling strength between cavity and i thatom. The interatomic distance ∆ z induces a phase shift φ z = 2 π ∆ z/λ C between the radiation emitted by thetwo atoms. Since φ z can be chosen (mod 2 π ), sepa-rations of the atoms much larger than the cavity wave-length λ C can be achieved in order to avoid direct atom-atom interactions as in [47]. Observe that in our setup at φ z = π/ π/ H L = ~ η P i =1 , (cid:0) S + i + S − i (cid:1) . Hereby, itis assumed that the pumping laser with Rabi frequency η propagates perpendicular to the cavity axis. Neglectingpossible interatomic displacements in y -direction leads toa homogeneous driving of the atoms. Varying pump ratesdue to spatial variation of the laser phase could be ab-sorbed into effective coupling constants of the atoms [34].For fixed atomic transition dipole moment, η indicatesthe strength of the coherent pump. Spontaneous emis-sion of the atoms at rate γ is taken into account by theterm L γ ρ = γ/ P i =1 , (cid:0) S − i ρS + i − S + i S − i ρ − ρS + i S − i (cid:1) ,whereas cavity decay at rate κ is considered by the Liou-villian L κ ρ = κ/ (cid:0) aρa † − a † aρ − ρa † a (cid:1) . In this letter,we neglect marginal dephasing effects, which, for exam-ple, become relevant in the case of quantum dots.In order to work out the dynamical behavior of theatom-cavity system, we have to solve Eq. (1), whichdepends on many parameters. Whereas η , δ , and ∆ canbe easily varied, g , κ , and γ are intrinsic properties anddepend on the design of the cavity and the atomic systemused. The specific dynamics very much depends on thecavity coupling and the cavity Q -factor. Thus to keep ourdiscussion fairly general it becomes necessary to solve themaster equation quite universally so that the behaviorin different regimes can be studied. We thus resort tonumerical techniques based on QuTiP [48]. We ensuredthe numerical convergence of our results by consideringdifferent cutoffs of the photonic Hilbert space. B. Transitions
To clarify the dynamical behavior of the system, wemake use of the collective basis states | gg i , | ee i and |±i todescribe the atoms. The symmetric and anti-symmetric Dicke state |±i = D †± | gg i = ( | eg i ± | ge i ) / √ D †± = ( S +1 ± S +2 ) / √ D †± [49], yields a clearpicture of the occurring transitions as can be seen inFig. 1(b) and (c). The pumping term is then givenby H L = ~ √ η ( D † + + D + ) and gives rise to the tran-sitions | gg, n i η → | + , n i η → | ee, n i with n being the num-ber of photons in the cavity mode. Hence, only sym-metric Dicke state | + i and doubly excited state | ee i arepumped. The interaction term, on the other hand, cou-ples the cavity to | + i or |−i depending on the inter-atomic phase φ z . It reads H I = H + + H − with H ± = ~ g ± ( φ z )( aD †± + a † D ± ) and g ± ( φ z ) = g (1 ± cos( φ z )) / √ g − ( φ z = 0) = 0 and the anti-symmetric Dicke state |−i is uncoupled from the dynamics. Possible atom-cavity interactions are then via the states | ee, n i g + ←→| + , n + 1 i g + ←→ | gg, n + 2 i , see also Fig. 1(b).For atoms radiating out of phase, however, g + ( φ z = π ) = 0 and the cavity only couples via |−i , i.e. | ee, n i g − ←→ |− , n + 1 i g − ←→ | gg, n + 2 i . Note that al-though only the symmetric Dicke state | + i is pumpedby the applied coherent field, the photon number in thecavity is non-zero for an out of phase radiation of theatoms due to higher-order processes, which can pop-ulate the state |−i . These are direct cavity coupling | ee, n i g − → |− , n + 1 i and spontaneous emission | ee, n i γ →|± , n i γ → | gg, n i , see also Fig. 1(c). Note that the lat-ter process, of course, takes place for φ z = 0 as well as φ z = π .For a phase in between, both couplings are presentas g − ( φ z ) and g + ( φ z ) will be nonzero. For the sake ofcompleteness, we list the transitions due to cavity de-cay which read | ., n i κ → | ., n − i and are possible for allvalues of φ z . C. Radiance witness R In the considered setup, it is natural to measure theemitted radiation at an external detector (D) placedalong the cavity axis, see Fig. 1(a). As the pumpingbeam (L) is perpendicular to the cavity axis and thuswill not contribute photons along the cavity axis, theregistered mean photon number at the detector (D) isproportional to the corresponding intracavity quantity.By performing a reference simulation of a single atom lo-cated at an antinode of the cavity field, we are thus ableto quantify the radiant character of the two-atom systemas a function of the correlations of the two atoms by useof a radiance witness R := h a † a i − h a † a i h a † a i , (2) FIG. 2. Radiance witness R for different regimes as a function of the interatomic phase φ z and pumping rate η . The colorencodes six different regimes of radiation, i.e., extremely subradiant (black), subradiant (blue), uncorrelated (light blue),enhanced (yellow), superradiant (orange), and hyperradiant (red). Dotted, dashed, and solid curve in the figures indicate themean photon numbers h a † a i = 0 . , . ,
1, respectively. (a) 3D plot and 2D surface map of the predominant hyperradiant areafor γ = κ , g = 10 κ and no detuning. Here, the superradiant and uncorrelated scattering area are very small and can hardly beseen. (b,c) Results for bad and intermediate cavity with γ = κ , no detuning and (b) g = 0 . κ , (c) g = κ . (d,e) Influence of thedetuning on hyperradiance with γ = κ , g = 10 κ and (d) δ = ∆ = κ , (e) δ = ∆ = 10 κ . involving the intracavity bosonic operators a and a † .Here, h a † a i i is the steady-state mean photon numberwith i = 1 , h a † a i results from the comparison of thecoupled two-atom system to the system of two uncorre-lated atoms, while the denominator in Eq. (2) yields anormalization of R .The witness R is composed of experimental observ-ables, i.e., number of photons, which can be measuredas in [33]. A possible detection strategy for R is, forinstance, scanning the second atom from φ z = π/ φ z = θ simulating the transition from effectively oneatom coupled to the cavity to two atoms radiating in-phase ( θ = 0) or out-of-phase ( θ = π ) into the cavitymode. By evaluating the experimental data according toEq. (2), the radiance witness R can be obtained. R = 0 reveals an uncorrelated scattering, where thescattering of the two atom-cavity system is simply thesum of two independent atoms in the cavity. A value of R different from zero thus indicates correlations betweenthe atoms. Negative or positive values of R signal a sup-pressed or enhanced radiation of the two atom-cavity sys-tem, respectively. R = 1, in particular, implies that theradiation scales with the square of the number of atoms ∝ N , which is called superradiance with respect to thefree-space scenario [1]. Atoms confined to a cavity, how- ever, feel a back-action of the cavity field which modifiestheir collective radiative behavior, allowing for a remark-ably new possibility R >
1. In fact, we found regimeswith h a † a i > h a † a i yielding R greater than 24. Inorder to emphasize this phenomenon, we call the domainof R > h S + S − i , on the otherhand, can be used to obtain the sideway radiation of theatoms. Note that in the bad cavity regime, R reduces tothe definition in terms of atomic operators like in [29–31] due to adiabatic elimination, while in good cavitieswith g > γ , the emission of photons into the cavity modedominates over spontaneous emission in side-modes. R thus constitutes a very natural witness for the setup ofFig. 1(a). D. Semiclassical treatment
Several phenomena of fundamental atom-light interac-tion can be fully analyzed within a semiclassical frame-work, even atoms coupled to a cavity in the weak atomicexcitation limit. In a semiclassical approximation, onedecouples the dynamics of atoms and cavity, i.e., h aS z i ≈h a i h S z i and assumes a vanishing atomic excitation lead-ing to h S zi i ≈ − /
2. In steady state, one is able to deducean analytical result for h a i , which is proportional to theintracavity field. In terms of the parameters of the sys-tem, it reads h a i = ηg N G g (cid:0) γ + i ∆ (cid:1) (cid:0) κ + iδ (cid:1) − N H , (3)where N is the number of atoms inside the cavity. Wefurther introduced the two collective coupling parame-ters, H = N − P Ni =1 cos [2 πz i /λ C ] along the cavity and G = N − P Ni =1 cos[2 πz i /λ C ] for the incident beam [50],which involve the position-dependent atom-cavity cou-plings g i . In the investigated two-atom system, these canbe written as a function of the interatomic phase only: H ( φ z ) = [1 + cos ( φ z )] / G ( φ z ) = [1 + cos( φ z )] / G ( φ z ).For an out-of-phase configuration, it holds G ( φ z = π ) = 0and thus semiclassical treatment predicts a vanishing in-tracavity field. III. RESULTS AND DISCUSSION
In what follows, we study the radiance witness R ofEq. (2) for the setup of Fig. 1(a) in a very broad regimeof parameters. In Fig. 2, for example, we plot R as afunction of the interatomic phase φ z and the pumpingrate η for weak and strong values with respect to theatomic spontaneous emission rate γ . In all figures weset γ = κ , where κ is the cavity decay rate, while theother parameters are varied from figure to figure. Wecategorize the value range of R into six different classes,which are depicted in unified colors: extremely subradi-ant ( R < − .
5, black), subradiant ( − . < R <
0, blue),uncorrelated ( R = 0, light blue), enhanced (0 < R < R = 1, orange) and hyperradi-ant (1 < R , red) scattering. For the color scheme seealso the color palette of Fig. 2. Dotted, dashed, andsolid curve in the figures indicate a mean photon number h a † a i = 0 . , . ,
1, respectively.In good cavities and for atoms radiating out of phasethe system can exhibit the phenomenon of hyperradi-ance, see Fig. 2(a). The radiation can exceed the oneof two atoms emitting in phase with otherwise iden-tical parameters distinctly, thereby also surpassing thefree-space limit R = 1. This is the synergy of twoeffects: Higher-order processes can populate the dou-bly excited atomic Dicke state | ee i , see Fig. 1(b) and(c). In the case of φ z = π , this leads to the emis-sion of single photons into the cavity via the transition | ee, n i γ → |− , n i g − → | gg, n + 1 i or even photon pairs via | ee, n i g − → |− , n + 1 i g − → | gg, n + 2 i producing superradi- − π/ π π/ πR φ z − π/ π π/ π ↑ FIG. 3. Results for different cavities. Vertical cuts of theradiance witness at η ≈ . κ as a function of the interatomicphase φ z for different types of cavities: blue (dashed) for abad cavity corresponding to Fig. 2(b); green (dotdashed) foran intermediate cavity corresponding to Fig. 2(c); and black(bold) with highlighted hyperradiant area ( R >
1) for a goodcavity corresponding to Fig. 2(a). ant or even hyperradiant light. For φ z = 0, however,cavity backaction prevents the excitation of the atoms.This is due to vacuum Rabi splittings [51–55] of the in-tracavity field, which for a driving laser on resonanceleads to a suppressed excitation of the atoms. The lat-ter can also be interpreted as a destructive quantumpath interference [49] between the laser-induced excita-tion | gg, n i η → | + , n i and the cavity-induced excitation | gg, n + 1 i g + → | + , n i , see Fig. 1(b), resulting in subradi-ant light. The interpretation holds true for uncorrelatedatoms, where the interfering terms can be seen in Fig.1(d) and read | g, n i η → | e, n i and | g, n + 1 i g → | e, n i , re-spectively, which when superimposed yield little excita-tion of the atom. For two atoms radiating out of phase,however, this back reaction is suppressed as the cavitycouples to the anti-symmetric Dicke state |−i and thusthe pathway | gg, n + 1 i to | + , n i is not allowed. As aresult, we observe hyperradiance.Note that in contrast to the coherent light emittedby a laser, the hyperradiant light is (super-)bunched (asrevealed by a second-order correlation function at zerotime g (2) (0) >
1) due to the emission of photon pairsin the out-of-phase configuration (see Fig. 1(c)). More-over, commonly lasing is observed when atoms radiatein phase [56]. Opposed to that, in the investigated sys-tem the atoms radiate out of phase in the hyperradiantregime.In intermediate and bad cavities with g . κ the ra-diation of two atoms out of phase is, however, highlysuppressed, see Fig. 2(b) and (c). At φ z = π , the cavitycouples to the anti-symmetric Dicke state |−i , which isoften also called the dark state [34]. When the atomsare driven well below saturation, the coherent laser onlypumps the symmetric Dicke state | + i (bright state). Asa consequence, the cavity mode is almost empty due todestructive interference of the radiation emitted from thetwo atoms [49]. The radiant character is extremely sub-radiant R < − . h a † a i < . h a † a i . By contrast,two atoms emitting in phase into an intermediate cav-ity change their radiant character at higher pumping η . . . g/κ η/κ . a . b FIG. 4. Comparison of in-phase and out-of-phase radiation.Both figures constitute a plot of R as a function of pumpingrate η and atom-cavity coupling g , where g = 0 . κ → κ reflects the transition from bad to good cavities. Results areshown for γ = κ , no detuning and (a) atoms radiating inphase ( φ z = 0), (b) atoms radiating out of phase ( φ z = π ).For the clarification of the color code as well as dotted, dashedand solid line, see Fig. 2. Here, even at low pumping rates photons can be emittedinto the cavity mode via the laser-pumped state | + i . InFig. 2(c), for instance, for η . κ the pumping strengthis not sufficient to pump both atoms leading to a sub-radiant behavior. At higher η , both emitters at firstscatter uncorrelated, while then higher order processesvia | ee i reinforce the atom-cavity coupling leading to anenhanced radiation. If η gets too high, the already men-tioned destructive quantum path interference takes place.This also occurs at high pumping rates in bad cavities,see Fig. 2(b), while at lower η the in-phase radiationis mainly enhanced. In the limit of an extremely badcavity, corresponding to a free-space setup, superradiantscattering is recovered ( R →
1) for an in-phase configu-ration.The detuning can change the radiant behavior drasti-cally, see Fig. 2(d) and (e). By comparing to the unde-tuned results of Fig. 2(a), we can infer that small detun-ing of the order of δ = ∆ = κ (d) weakens the hyperradi-ant behavior while in systems with stronger detuning ofthe order of δ = ∆ = 10 κ (e), no hyperradiance can beobserved and the light is predominantly subradiant foratoms radiating out-of-phase.In the experimental realization by Reimann et al. [33],the authors measure the intensity of the system in aregime where the radiation is suppressed independent ofthe interatomic phase. Using the parameters of [33], weobserve a transition of the witness from R = − .
37 incase of φ z = 0 to R = − .
00 in case of φ z = π . Thus,the system becomes extremely subradiant, as the atomstend to radiate out-of-phase. In fact, one could guessthat atoms radiating out-of-phase scatter predominantlysubradiantly, as observed in all previously mentioned ex-periments [33–35]. This is the case in bad and interme-diate cavities (see dashed and dot-dashed curve in Fig.3), or at high detuning. Yet, when studying the behav-
01 0 π/ π π/ π | h a i | / h a † a i φ z
01 0 π/ π π/ π FIG. 5. Comparison of classical and quantum mechanicaltreatment. The ratio |h a i| / h a † a i comparing the classicalintensity of the intracavity field with the quantum-mechanicalmean photon number is shown as a function of the interatomicphase φ z for g = 10 κ , γ = κ and η = 0 . κ . ior in a good cavity and zero detuning, we find that thenumber of photons within the system can become muchlarger than in the corresponding setup with uncorrelatedatoms. The bold line of Fig. 3 displays this tendency of R for η ≈ . κ . Here, the transition of atoms radiatingin phase to atoms radiating out of phase is accompaniedby the transition from (extreme) subradiance to hyperra-diance. In order to observe hyperradiance, the previousexperiments [33–35] would need to adapt to the parame-ters of Figs. 2(a) and 4(b).A brief comparison of atoms located at anti-nodes ofa cavity can be seen in Fig. 4. Here, the radiation oftwo atoms radiating in-phase ( φ z = 0) and two atomsradiating out-of-phase ( φ z = π ) is compared over a widerange of coupling constants g : 0 . κ → κ reflectingthe transition from a bad to a good cavity. For an in-phase radiation of the atoms, see Fig. 4(a), the radiantcharacter hardly depends on the pumping rate as long as η . κ but is determined by g : for g & . κ ( g . . κ ),the radiation is subradiant (enhanced) and at g ≈ . κ uncorrelated. Note that g/κ → R →
1. In Fig. 4(b), we comparethese findings to atoms radiating out of phase in the sameparameter range. While for atoms radiating in phase,the transition from bad to good cavities goes along withthe transfer from superradiance or enhanced radiation tosubradiance, the situation is reversed for atoms radiatingout of phase. Here they radiate subradiantly in bad cav-ities, whereas their radiation in good cavities can exceedthe superradiant limit distinctly, finally ending up in hy-perradiance, which can be explained via quantum pathinterference.Interestingly, the classical treatment of the discussedsetup predicts an intracavity field that vanishes in thecase of an out-of-phase configuration, see Eq. (3) with G ( φ z = π ) = 0. One can quantify the deviation from theclassical approach by considering the ratio |h a i| / h a † a i ,which compares the classical intensity of the intracavityfield with the quantum-mechanical mean photon num-ber. A deviation from unity reveals quantum featuresdisplayed by the system. For the investigated two atom-cavity system, |h a i| / h a † a i equals one for an in-phaseconfiguration, but tends to zero as φ z → π , see Fig.5. A value below one of the ratio displayed in Fig. 5corresponds to the quantum theory predicting a higherintensity than the classical approach. Therefore, theoccurrence of hyperradiance even in the low pumpingregime η ≈ . κ can only be explained in a full quantum-mechanical treatment revealing the true quantum originof the phenomenon hyperradiance. IV. CONCLUSIONS
In conclusion, we have shown a new phenomenon in thecollective behavior of coherently driven atoms, which wecall hyperradiance. In this regime, the radiation of twoatoms in a single-mode cavity coherently driven by an ex-ternal laser can exceed the free-space superradiant behav-ior considerably. Hyperradiance occurs in good cavitiesand, surprisingly, for atoms radiating out of phase. Theeffect cannot be explained in a (semi-)classical treatmentrevealing a true quantum origin. Moreover, by modify- ing merely the interatomic phase, crossovers from sub-radiance to hyperradiance can be observed. Our resultsshould stimulate various new experiments examining thepossibility for the observation of hyperradiance in thisfundamental system, consisting of a cavity coupled toany kind of two-level systems like atoms, ions, supercon-ducting qubits or quantum dots.
ACKNOWLEDGMENTS
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