Abstract

The collective behavior of ensembles of atoms has been studied in-depth since the seminal paper of Dicke [Phys. Rev. 93, 99 (1954) [CrossRef]  ], where he demonstrated that a group of emitters in collective states is able to radiate with increased intensity and modified decay rates in particular directions, a phenomenon that he called superradiance. Here, we show that the fundamental setup of two atoms coupled to a single-mode cavity can distinctly exceed the free-space superradiant behavior, a phenomenon we call hyperradiance. The effect is accompanied by strong 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 in a transparent way by a photon cascade taking place among manifolds of Dicke states with different photon numbers under particular out-of-phase coupling conditions. These theoretical results can be realized with current technology and should thus stimulate future experiments.

© 2017 Optical Society of America

1. INTRODUCTION

Arguably one of the most enigmatic phenomena in the history of quantum optics is the discovery of superradiance by Dicke [126]. A key requirement in Dicke’s work is the initial preparation of an ensemble of two-level atoms in a special class of collective states, so-called symmetric Dicke states. The startling gist is that even though atoms in these states have no dipole moment, they radiate with an intensity that is enhanced by a factor of N compared to N independent atoms. The preparation of such states has been a challenge since. The first experiments on superradiance were done in atomic vapors [68], where it was assumed that the fully excited system in the course of temporal evolution would at some time be found in a Dicke state leading to the emission of superradiant light [2,3,6]. The same assumption led to the recent observation of subradiance [9].

Many of the mysteries behind this effect have begun to unfold only recently. In the seventies, it was realized that superradiant emission results from the strong quantum correlations among the atoms being prepared in symmetric Dicke states [5,6]. Only lately, it became clear that the N-fold radiative enhancement can be explained by the multiparticle entanglement of the Dicke states. For example, studying superradiance in a chain of Dicke-entangled atoms on a lattice enables one to determine that multiple interfering quantum paths lead to the collective subradiant and superradiant behavior [14,27]. The strong entanglement of the states can already be inferred from the simpler two-atom case and holds for almost all Dicke states of a multi-atom-system. With current advances in quantum information science, we indeed understand the great difficulties in precisely preparing such highly entangled multiparticle states. There are proposals for the generation of whole classes of Dicke states using projective measurements [10,11,28], yet these schemes have very low success probabilities. Deterministic entanglement has been produced with about a dozen qubits in the form of W-states [12,13], but we are still far away from the realization of W-states for an arbitrary number of qubits. Calculations show that these states, which can be considered to be the analog to single excitation Dicke states with appropriate phase factors, also produce enhancement by a factor of N [1416].

In comparably simpler systems, yet with a higher number of excitations, one can also study the quantum statistical aspects of the collective emission [25]. However, down to the present day the generation and measurement of higher excited multiparticle entangled Dicke states are challenging, so mostly systems with no more than one excitation have been realized. In these systems, the dynamics are still quite complex, yet superradiance and also subradiance can be fruitfully explored [1422]. Experiments of superradiance with single photon excited Dicke states were also reported for nuclear transitions [23,24]. A recent work discusses preparation of a single photon subradiant state and its radiation characteristics for atoms in free space [20]. Even applications of superradiance are beginning to appear. Lately, a laser with a frequency linewidth less than that of a single-particle decoherence linewidth was realized [29] by using more than one million intracavity atoms and operating in a steady-state superradiant regime [30,31].

Despite these advances, one is still faced with difficulties in the optical domain that arise from the infinite number of modes in free space and interatomic effects like dipole-dipole interactions [4,5]. It is thus evident that one needs to work with systems that have fewer degrees of freedoms and where a precise preparation of entangled states is possible. This brings us to work with single-mode cavities [26,32] with few atoms. The current technological progress in atom trapping and the availability of well-characterized single-mode cavities is making this ideal situation become a reality. Several experiments in the last two years have been reported using such well-characterized systems [3335], which consist of two coherently driven atoms [33,35] or entangled ions [34] coupled to a single-mode cavity as depicted in Fig. 1(a). This setup enables one to study collective behavior as a function of various atomic and cavity parameters, e.g., the precise location of the atoms.

 

Fig. 1. Basic properties of the system. (a) Sketch of the system consisting of two atoms (A) that are coupled to a single-mode cavity (C) and driven by a coherent laser (L) with Rabi frequency η. Intracavity photons can leak through the mirrors by cavity decay (κ) and are registered by a detector (D). Another possible dissipative process is spontaneous emission (γ) by the atoms. The inset shows a magnified section of the arrangements of the atoms: one atom (depicted left) is fixed at an anti-node of the cavity field, while the other atom (right) can be scanned along the cavity axis causing a relative phase shift ϕz between the radiation of the atoms. (b) and (c) Energy levels and transitions of the system for (b) in-phase and (c) out-of-phase radiation of the atoms. In the case of two atoms, the state space consists of manifolds of four Dicke states with different intracavity photon numbers. For a fixed cavity state |n, these are unentangled two atom ground and two atom excited states |gg and |ee, respectively, as well as the maximally entangled symmetric and anti-symmetric Dicke states |±. For clarity, we draw neither the transitions due to cavity decay nor the detuning. (d) Energy levels and transitions of the corresponding system containing a single atom.

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In spite of this recent progress on superradiant and subradiant behavior [1422] and the surge of new classes of experiments [29,3335], there is yet no report of atomic light emission beyond that of superradiance. In this paper, we demonstrate that a two-atom system coupled to a single-mode cavity is capable of radiating up to several orders of magnitude higher than a corresponding system consisting of two uncorrelated atoms, thereby exceeding the free-space superradiant emission by far. We call this effect hyperradiance. Surprisingly, hyperradiance occurs in a regime that one usually considers to be non-ideal, namely when the two atoms radiate out-of-phase. Such non-ideal conditions are rather expected to suppress superradiance and thus one is not inclined to imagine in this regime an emission burst exceeding that of superradiance.

Although the study that we present is in the context of atomic systems, the results should be applicable to other types of two-level systems like ions [26,34,36], superconducting qubits [3739], and quantum dots [4044]. We thus expect that our findings stimulate a multitude of new experiments in various domains of physics.

2. METHODS

A. 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) oriented perpendicular to the cavity axis coherently drives the atoms. In this paper, we fix one atom at an anti-node of the cavity field while we vary the position of the other atom along the cavity axis inducing a relative phase shift between the radiation of the atoms. The atoms within the cavity are modeled as two-level systems with transition frequency ωA, driven by a laser field at frequency ωL, and coupled to a single-mode of the cavity with frequency ωC=2πc/λC. The ith atom is characterized by the spin-half operators Si+=|eig|i, Si=(Si+), and Siz=(|eie|i|gig|i)/2. Bosonic annihilation and creation operator a and a describe the intracavity mode. The dynamic behavior of the entire system can be treated in a master equation approach [45] and is governed by

ddtρ=i[H0+HI+HL,ρ]+Lγρ+Lκρ,
where ρ is the density operator of the atom-cavity system. In the interaction frame rotating at the laser frequency, atoms and cavity are described by H0=Δ(S1z+S2z)+δaa. Here, Δ=ωAωL is the atom-laser detuning and δ=ωCωL is the cavity-laser detuning. The Tavis–Cummings interaction term of atom-cavity coupling is given by HI=i=1,2gi(Si+a+Sia) and can be obtained by utilizing the dipole approximation and applying the rotating wave approximation [46]. The term gi=gcos(2πzi/λC) describes the position-dependent coupling strength between the cavity and ith atom. The interatomic distance Δz induces a phase shift ϕz=2πΔz/λC between the radiation emitted by the two atoms. Since ϕz can be chosen (mod2π), separations of the atoms much larger than the cavity wavelength λC can be achieved to avoid direct atom-atom interactions as in Ref. [47]. Observe that in our setup at ϕz=π/2(3π/2), only one atom is coupled to the cavity. The coherent pumping of the atoms is characterized by the Hamiltonian HL=ηi=1,2(Si++Si). Hereby, it is assumed that the pumping laser with Rabi frequency η propagates perpendicular to the cavity axis. Neglecting possible interatomic displacements in the y-direction leads to a homogeneous driving of the atoms. Varying pump rates due to spatial variation of the laser phase could be absorbed into effective coupling constants of the atoms [34]. For fixed atomic transition dipole moments, η indicates the strength of the coherent pump. Spontaneous emission of the atoms at rate γ is taken into account by the term Lγρ=γ/2i=1,2(2SiρSi+Si+SiρρSi+Si), whereas cavity decay at rate κ is considered by the Liouvillian Lκρ=κ/2(2aρaaaρρaa). In this paper, we neglect marginal dephasing effects, which, for example, become relevant in the case of quantum dots.

In order to work out the dynamical behavior of the atom-cavity system, we have to solve Eq. (1), which depends on many parameters. Whereas η, δ, and Δ can be easily varied, g, κ, and γ are intrinsic properties that depend on the design of the cavity and the atomic system used. The specific dynamics very much depend on the cavity coupling and the cavity Q-factor. Thus to keep our discussion fairly general, it becomes necessary to solve the master equation quite universally so that the behavior in different regimes can be studied. We thus resort to numerical techniques based on QuTiP [48]. We ensured the numerical convergence of our results by considering different cutoffs of the photonic Hilbert space.

B. Transitions

To clarify the dynamical behavior of the system, we make use of the collective basis states |gg, |ee, and |± to describe the atoms. The symmetric and anti-symmetric Dicke states |±=D±|gg=(|eg±|ge)/2 are created by the collective Dicke operators D±=(S1+±S2+)/2 [45]. Rewriting interaction and pumping Hamiltonian in terms of the collective operators D± [49] yields a clear picture of the occurring transitions, as can be seen in Figs. 1(b) and 1(c). The pumping term is then given by HL=2η(D++D+) and gives rise to the transitions |gg,nη|+,nη|ee,n with n being the number of photons in the cavity mode. Hence, only the symmetric Dicke state |+ and doubly excited state |ee are pumped. The interaction term, on the other hand, couples the cavity to |+ or | depending on the interatomic phase ϕz. It reads HI=H++H with H±=g±(ϕz)(aD±+aD±) and g±(ϕz)=g(1±cos(ϕz))/2.

In case of an in-phase radiation of the atoms, apparently g(ϕz=0)=0 and the anti-symmetric Dicke state | is uncoupled from the dynamics. Possible atom-cavity interactions are then via the states |ee,ng+|+,n+1g+|gg,n+2, see also Fig. 1(b).

For atoms radiating out-of-phase, however, g+(ϕz=π)=0 and the cavity only couples via |, i.e., |ee,ng|,n+1g|gg,n+2. Note that although only the symmetric Dicke state |+ is pumped by the applied coherent field, the photon number in the cavity is non-zero for an out-of-phase radiation of the atoms due to higher-order processes, which can populate the state |. These are direct cavity coupling |ee,ng|,n+1 and spontaneous emission |ee,nγ|±,nγ|gg,n, see also Fig. 1(c). Note that the latter process, of course, takes place for ϕz=0 as well as ϕz=π.

For a phase in between, both couplings are present as g(ϕz), and g+(ϕz) will be non-zero. For the sake of completeness, we list the transitions due to cavity decay, which read |.,nκ|.,n1 and are possible for all values of ϕz.

C. Radiance Witness R

In the considered setup, it is natural to measure the emitted radiation at an external detector (D) placed along the cavity axis, see Fig. 1(a). As the pumping beam (L) is perpendicular to the cavity axis and thus will not contribute photons along the cavity axis, the registered mean photon number at the detector (D) is proportional to the corresponding intracavity quantity. By performing a reference simulation of a single atom located at an anti-node of the cavity field, we are thus able to quantify the radiant character of the two-atom system as a function of the correlations of the two atoms by use of a radiance witness:

Raa22aa12aa1,
involving the intracavity bosonic operators a and a. Here, aai is the steady-state mean photon number with i=1,2 atoms in the cavity. The factor 2 arising in front of aa1 results from the comparison of the coupled two-atom system to the system of two uncorrelated atoms, while the denominator in Eq. (2) yields a normalization of R.

The witness R is composed of experimental observables, i.e., number of photons, which can be measured as in Ref. [33]. A possible detection strategy for R is, for instance, scanning the second atom from ϕz=π/2 to ϕz=θ, simulating the transition from effectively one atom coupled to the cavity to two atoms radiating in-phase (θ=0) or out-of-phase (θ=π) into the cavity mode. By evaluating the experimental data according to Eq. (2), the radiance witness R can be obtained.

R=0 reveals an uncorrelated scattering, where the scattering of the two-atom cavity system is simply the sum of two independent atoms in the cavity. A value of R different from zero thus indicates correlations between the atoms. Negative or positive values of R signal a suppressed or enhanced radiation of the two-atom cavity system, respectively. R=1, in particular, implies that the radiation scales with the square of the number of atoms N2, which is called superradiance with respect to the free-space scenario [1]. Atoms confined to a cavity, however, feel a backaction of the cavity field, which modifies their collective radiative behavior, allowing for a remarkable new possibility R>1. In fact, we found regimes with aa2>50aa1 yielding R greater than 24. In order to emphasize this phenomenon, we call the domain of R>1 hyperradiant.

The atomic correlation quantity S+S, on the other hand, can be used to obtain the sideway radiation of the atoms. Note that in the bad cavity regime, R reduces to the definition in terms of atomic operators like in Ref. [2931] due to adiabatic elimination, while in good cavities with g>γ, the emission of photons into the cavity mode dominates over spontaneous emission in side-modes. R thus constitutes a very natural witness for the setup of Fig. 1(a).

D. Semiclassical Treatment

Several phenomena of fundamental atom-light interactions can be fully analyzed within a semiclassical framework, even atoms coupled to a cavity in the weak atomic excitation limit. In a semiclassical approximation, one decouples the dynamics of atoms and cavity, i.e., aSzaSz and assumes a vanishing atomic excitation leading to Siz1/2. In steady state, one is able to deduce an analytical result for a, which is proportional to the intracavity field. In terms of the parameters of the system, it reads

a=ηgNG1g2(γ2+iΔ)(κ2+iδ)NH,
where N is the number of atoms inside the cavity. We further introduced the two collective coupling parameters, H=N1i=1Ncos2[2πzi/λC] along the cavity and G=N1i=1Ncos[2πzi/λC] for the incident beam [50], which involve the position-dependent atom-cavity couplings gi. In the investigated two-atom system, these can be written as a function of the interatomic phase only: H(ϕz)=[1+cos2(ϕz)]/2 and G(ϕz)=[1+cos(ϕz)]/2. Equation (3) can also be derived in a classical framework, where the atoms are treated as radiating dipoles that couple to a non-quantized standing-wave optical resonator [50]. Hereby, one exploits the condition that the intracavity field needs to match itself after a round trip to be sustained by the resonator. Observe that the (semi-)classical intracavity field is proportional to G(ϕz). For an out-of-phase configuration, it holds G(ϕz=π)=0 and thus semiclassical treatment predicts a vanishing intracavity field.

3. RESULTS AND DISCUSSION

We study the radiance witness R of Eq. (2) for the setup of Fig. 1(a) in a very broad regime of parameters. In Fig. 2, for example, we plot R as a function of the interatomic phase ϕz and the pumping rate η for weak and strong values with respect to the atomic spontaneous emission rate γ. In all figures we set γ=κ, where κ is the cavity decay rate, while the other parameters are varied from figure to figure. We categorize the value range of R into six different classes, which are depicted in unified colors: extremely subradiant (R<0.5, black), subradiant (0.5<R<0, blue), uncorrelated (R=0, light blue), enhanced (0<R<1, yellow), superradiant (R=1, orange) and hyperradiant (1<R, red) scattering. For the color scheme see also the color palette of Fig. 2. Dotted, dashed, and solid curves in the figures indicate a mean photon number aa2=0.01,0.1,1, respectively.

 

Fig. 2. Radiance witness R for different regimes as a function of the interatomic phase ϕz and pumping rate η. The color encodes 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 curves in the figures indicate the mean photon numbers aa2=0.01,0.1,1, respectively. (a) 3D plot and 2D surface map of the predominant hyperradiant area for γ=κ, g=10κ, and no detuning. Here, the superradiant and uncorrelated scattering area are very small and can hardly be seen. (b) and (c) Results for bad and intermediate cavity with γ=κ, no detuning, and (b) g=0.1κ or (c) g=κ. (d) and (e) Influence of the detuning on hyperradiance with γ=κ, g=10κ, and (d) δ=Δ=κ or (e) δ=Δ=10κ.

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In good cavities and for atoms radiating out-of-phase, the system can exhibit the phenomenon of hyperradiance, see Fig. 2(a). The radiation can distinctly exceed that of two atoms emitting in-phase with otherwise identical parameters, thereby also surpassing the free-space limit R=1. This is the synergy of two effects: higher-order processes can populate the doubly excited atomic Dicke state |ee, see Figs. 1(b) and 1(c). In the case of ϕz=π, this leads to the emission of single photons into the cavity via the transition |ee,nγ|,ng|gg,n+1 or even photon pairs via |ee,ng|,n+1g|gg,n+2 producing superradiant or even hyperradiant light. For ϕz=0, however, cavity backaction prevents the excitation of the atoms. This is due to vacuum Rabi splitting [5155] of the intracavity field, which for a driving laser on resonance leads to a suppressed excitation of the atoms. The latter can also be interpreted as destructive quantum path interference [49] between the laser-induced excitation |gg,nη|+,n and the cavity-induced excitation |gg,n+1g+|+,n, see Fig. 1(b), resulting in subradiant light. The interpretation holds true for uncorrelated atoms, where the interfering terms can be seen in Fig. 1(d) and read |g,nη|e,n and |g,n+1g|e,n, respectively, which when superimposed yield little excitation of the atom. For two atoms radiating out-of-phase, however, this back reaction is suppressed as the cavity couples to the anti-symmetric Dicke state | and thus the pathway |gg,n+1 to |+,n is not allowed. As a result, we observe hyperradiance.

Note that in contrast to the coherent light emitted by a laser, the hyperradiant light is (super-)bunched (as revealed by a second-order correlation function at zero time g(2)(0)>1) due to the emission of photon pairs in the out-of-phase configuration [see Fig. 1(c)]. Moreover, lasing is commonly observed when atoms radiate in-phase [56]. Opposed to that, in the investigated system, the atoms radiate out-of-phase in the hyperradiant regime.

In intermediate and bad cavities with gκ, the radiation of two atoms out-of-phase is, however, highly suppressed, see Figs. 2(b) and 2(c). At ϕz=π, the cavity couples to the anti-symmetric Dicke state |, which is often also called the dark state [34]. When the atoms are driven well below saturation, the coherent laser only pumps the symmetric Dicke state |+ (bright state). As a consequence, the cavity mode is almost empty due to destructive interference of the radiation emitted from the two atoms [49]. The radiant character is extremely subradiant R<0.9 or aa2<0.2aa1. By contrast, two atoms emitting in-phase into an intermediate cavity change their radiant character at higher pumping η. Here, even at low pumping rates, photons can be emitted into the cavity mode via the laser-pumped state |+. In Fig. 2(c), for instance, for ηκ the pumping strength is not sufficient to pump both atoms, leading to subradiant behavior. At higher η, both emitters at first scatter uncorrelated, while then higher-order processes via |ee reinforce the atom-cavity coupling, leading to an enhanced radiation. If η gets too high, the already mentioned 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 radiation is mainly enhanced. In the limit of an extremely bad cavity, corresponding to a free-space setup, superradiant scattering is recovered (R1) for an in-phase configuration.

The detuning can change the radiant behavior drastically, see Figs. 2(d) and 2(e). By comparing to the undetuned results of Fig. 2(a), we can infer that small detuning of the order of δ=Δ=κ (d) weakens the hyperradiant behavior, while in systems with stronger detuning of the order of δ=Δ=10κ (e), no hyperradiance can be observed and the light is predominantly subradiant for atoms radiating out-of-phase.

In the experimental realization by Reimann et al. [33], the authors measure the intensity of the system in a regime where the radiation is suppressed independent of the interatomic phase. Using the parameters of [33], we observe a transition of the witness from R=0.37 in the case of ϕz=0 to R=1.00 in the case of ϕz=π. Thus, the system becomes extremely subradiant, as the atoms tend to radiate out-of-phase. In fact, one could guess that atoms radiating out-of-phase scatter predominantly subradiantly, as observed in all previously mentioned experiments [3335]. This is the case in bad and intermediate cavities (see dashed and dot-dashed curve in Fig. 3), or at high detuning. Yet, when studying the behavior in a good cavity with zero detuning, we find that the number of photons within the system can become much larger than in the corresponding setup with uncorrelated atoms. The bold line of Fig. 3 displays this tendency of R for η0.5κ. Here, the transition of atoms radiating in-phase to atoms radiating out-of-phase is accompanied by the transition from (extreme) subradiance to hyperradiance. To observe hyperradiance, the previous experiments [3335] would need to adapt to the parameters of Figs. 2(a) and 4(b).

 

Fig. 3. Results for different cavities. Vertical cuts of the radiance witness at η0.5κ as a function of the interatomic phase ϕz for different types of cavities: blue (dashed) for a bad cavity corresponding to Fig. 2(b); green (dot-dashed) for an intermediate cavity corresponding to Fig. 2(c); and black (bold) with highlighted hyperradiant area (R>1) for a good cavity corresponding to Fig. 2(a).

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Fig. 4. Comparison of in-phase and out-of-phase radiation. Both figures constitute a plot of R as a function of pumping rate η and atom-cavity coupling g, where g=0.1κ10κ reflects the transition from bad to good cavities. Results are shown for γ=κ with no detuning and (a) atoms radiating in-phase (ϕz=0) or (b) atoms radiating out-of-phase (ϕz=π). For clarification of the color code as well as dotted, dashed, and solid lines, see Fig. 2.

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A brief comparison of atoms located at anti-nodes of a cavity can be seen in Fig. 4. Here, the radiation of two atoms radiating in-phase (ϕz=0) and two atoms radiating out-of-phase (ϕz=π) is compared over a wide range of coupling constants g:0.1κ10κ, reflecting the transition from a bad to a good cavity. For in-phase radiation of the atoms, see Fig. 4(a), the radiant character hardly depends on the pumping rate as long as ηκ, but it is determined by g: for g0.5κ (g0.5κ), the radiation is subradiant (enhanced) and at g0.5κ uncorrelated. Note that g/κ0 reflects a free-space setting for which our calculations show that superradiant scattering is recovered, R1. In Fig. 4(b), we compare these findings to atoms radiating out-of-phase in the same parameter range. While for atoms radiating in-phase, the transition from bad to good cavities goes along with the transfer from superradiance or enhanced radiation to subradiance, the situation is reversed for atoms radiating out-of-phase. Here they radiate subradiantly in bad cavities, whereas their radiation in good cavities can distinctly exceed the superradiant limit, finally ending up in hyperradiance, which can be explained via quantum path interference.

Interestingly, the classical treatment of the discussed setup predicts an intracavity field that vanishes in the case of an out-of-phase configuration, see Eq. (3) with G(ϕz=π)=0. One can quantify the deviation from the classical approach by considering the ratio |a|2/aa, which compares the classical intensity of the intracavity field with the quantum-mechanical mean photon number. A deviation from unity reveals quantum features displayed by the system. For the investigated two-atom cavity system, |a|2/aa equals one for an in-phase configuration, but tends to zero as ϕzπ, see Fig. 5. A value below one of the ratio displayed in Fig. 5 corresponds to the quantum theory predicting a higher intensity than the classical approach. Therefore, the occurrence of hyperradiance, even in the low pumping regime η0.1κ, can only be explained in a full quantum-mechanical treatment revealing the true quantum origin of the phenomenon hyperradiance.

 

Fig. 5. Comparison of classical and quantum-mechanical treatment. The ratio |a|2/aa comparing the classical intensity of the intracavity field with the quantum-mechanical mean photon number is shown as a function of the interatomic phase ϕz for g=10κ, γ=κ, and η=0.1κ.

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4. CONCLUSIONS

In conclusion, we have shown a new phenomenon in the collective behavior of coherently driven atoms, which we call hyperradiance. In this regime, the radiation of two atoms in a single-mode cavity coherently driven by an external laser can considerably exceed the free-space superradiant behavior. Hyperradiance occurs in good cavities and, surprisingly, for atoms radiating out-of-phase. The effect cannot be explained in a (semi-)classical treatment revealing a true quantum origin. Moreover, by modifying merely the interatomic phase, crossovers from subradiance to hyperradiance can be observed. Our results should stimulate various new experiments examining the possibility for the observation of hyperradiance in this fundamental system, consisting of a cavity coupled to any kind of two-level system, like atoms, ions, superconducting qubits, or quantum dots.

Funding

Erlangen Graduate School of Advanced Optical Technologies (SAOT).

Acknowledgment

M.-O.P. gratefully acknowledges the hospitality at the Oklahoma State University. The authors gratefully acknowledge funding by the Erlangen Graduate School in Advanced Optical Technologies (SAOT) by the German Research Foundation (DFG) in the framework of the German excellence initiative. Some of the computing for this project was performed at the OSU High Performance Computing Center at Oklahoma State University supported in part through the National Science Foundation grant OCI-1126330.

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17. A. A. Svidzinsky, L. Yuan, and M. O. Scully, “Quantum amplification by superradiant emission of radiation,” Phys. Rev. X 3, 041001 (2013). [CrossRef]  

18. T. Bienaimé, R. Bachelard, N. Piovella, and R. Kaiser, “Cooperativity in light scattering by cold atoms,” Fortschr. Phys. 61, 377–392 (2013). [CrossRef]  

19. W. Feng, Y. Li, and S.-Y. Zhu, “Effect of atomic distribution on cooperative spontaneous emission,” Phys. Rev. A 89, 013816 (2014). [CrossRef]  

20. M. O. Scully, “Single photon subradiance: quantum control of spontaneous emission and ultrafast readout,” Phys. Rev. Lett. 115, 243602 (2015). [CrossRef]  

21. P. Longo, C. H. Keitel, and J. Evers, “Tailoring superradiance to design artificial quantum systems,” Sci. Rep. 6, 23628 (2016). [CrossRef]  

22. A. A. Svidzinsky, F. Li, H. Li, X. Zhang, C. H. R. Ooi, and M. O. Scully, “Single-photon superradiance and radiation trapping by atomic shells,” Phys. Rev. A 93, 043830 (2016). [CrossRef]  

23. R. Röhlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Rüffer, “Collective lamb shift in single-photon superradiance,” Science 328, 1248–1251 (2010). [CrossRef]  

24. R. Röhlsberger, “Cooperative emission from nuclei: the collective lamb shift and electromagnetically induced transparency,” Fortschr. Phys. 61, 360–376 (2013). [CrossRef]  

25. D. Bhatti, J. von Zanthier, and G. S. Agarwal, “Superbunching and nonclassicality as new hallmarks of superradiance,” Sci. Rep. 5, 17335 (2015). [CrossRef]  

26. R. G. DeVoe and R. G. Brewer, “Observation of superradiant and subradiant spontaneous emission of two trapped ions,” Phys. Rev. Lett. 76, 2049–2052 (1996). [CrossRef]  

27. G. Nienhuis and F. Schuller, “Spontaneous emission and light scattering by atomic lattice models,” J. Phys. B 20, 23–36 (1987). [CrossRef]  

28. T. Bastin, C. Thiel, J. von Zanthier, L. Lamata, E. Solano, and G. S. Agarwal, “Operational determination of multiqubit entanglement classes via tuning of local operations,” Phys. Rev. Lett. 102, 053601 (2009). [CrossRef]  

29. J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012). [CrossRef]  

30. D. Meiser and M. J. Holland, “Steady-state superradiance with alkaline-earth-metal atoms,” Phys. Rev. A 81, 033847 (2010). [CrossRef]  

31. D. Meiser and M. J. Holland, “Intensity fluctuations in steady-state superradiance,” Phys. Rev. A 81, 063827 (2010). [CrossRef]  

32. R. G. DeVoe and R. G. Brewer, “Laser-frequency division and stabilization,” Phys. Rev. A 30, 2827–2829 (1984). [CrossRef]  

33. R. Reimann, W. Alt, T. Kampschulte, T. Macha, L. Ratschbacher, N. Thau, S. Yoon, and D. Meschede, “Cavity-modified collective Rayleigh scattering of two atoms,” Phys. Rev. Lett. 114, 023601 (2015). [CrossRef]  

34. B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt, and T. E. Northup, “Enhanced quantum interface with collective ion-cavity coupling,” Phys. Rev. Lett. 114, 023602 (2015). [CrossRef]  

35. A. Neuzner, M. Körber, O. Morin, S. Ritter, and G. Rempe, “Interference and dynamics of light from a distance-controlled atom pair in an optical cavity,” Nat. Photonics 10, 303–306 (2016). [CrossRef]  

36. A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstatter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012). [CrossRef]  

37. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004). [CrossRef]  

38. J. M. Fink, R. Bianchetti, M. Baur, M. Göppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Dressed collective qubit states and the Tavis-Cummings model in circuit QED,” Phys. Rev. Lett. 103, 083601 (2009). [CrossRef]  

39. J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014). [CrossRef]  

40. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gulde, S. Falt, E. L. Hu, and A. Imamoglu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007). [CrossRef]  

41. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Coherent generation of non-classical light on a chip via photon-induced tunneling and blockade,” Nat. Phys. 4, 859–863 (2008). [CrossRef]  

42. J. Miguel-Sánchez, A. Reinhard, E. Togan, T. Volz, A. Imamoglu, B. Besga, J. Reichel, and J. Estève, “Cavity quantum electrodynamics with charge-controlled quantum dots coupled to a fiber Fabry-Perot cavity,” New J. Phys. 15, 045002 (2013). [CrossRef]  

43. A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014). [CrossRef]  

44. H. A. M. Leymann, A. Foerster, F. Jahnke, J. Wiersig, and C. Gies, “Sub- and superradiance in nanolasers,” Phys. Rev. Appl. 4, 044018 (2015). [CrossRef]  

45. G. S. Agarwal, Quantum Optics (Cambridge University, 2012).

46. M. Tavis and F. W. Cummings, “Exact solution for an n-molecule-radiation-field Hamiltonian,” Phys. Rev. 170, 379–384 (1968). [CrossRef]  

47. E. V. Goldstein and P. Meystre, “Dipole-dipole interaction in optical cavities,” Phys. Rev. A 56, 5135–5146 (1997). [CrossRef]  

48. J. Johansson, P. Nation, and F. Nori, “Qutip: an open-source python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 183, 1760–1772 (2012). [CrossRef]  

49. S. Fernández-Vidal, S. Zippilli, and G. Morigi, “Nonlinear optics with two trapped atoms,” Phys. Rev. A 76, 053829 (2007). [CrossRef]  

50. H. Tanji-Suzuki, I. D. Leroux, M. H. Schleier-Smith, M. Cetina, A. T. Grier, J. Simon, and V. Vuletić, “Chapter 4—interaction between atomic ensembles and optical resonators: classical description,” in Advances in Atomic, Molecular, and Optical Physics (Academic, 2011), Vol. 60, pp. 201–237.

51. G. S. Agarwal, “Vacuum-field Rabi splittings in microwave absorption by Rydberg atoms in a cavity,” Phys. Rev. Lett. 53, 1732–1734 (1984). [CrossRef]  

52. Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990). [CrossRef]  

53. R. J. Thompson, G. Rempe, and H. J. Kimble, “Observation of normal-mode splitting for an atom in an optical cavity,” Phys. Rev. Lett. 68, 1132–1135 (1992). [CrossRef]  

54. Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014). [CrossRef]  

55. L. V. Abdurakhimov, Y. M. Bunkov, and D. Konstantinov, “Normal-mode splitting in the coupled system of hybridized nuclear magnons and microwave photons,” Phys. Rev. Lett. 114, 226402 (2015). [CrossRef]  

56. D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a milliHertz-linewidth laser,” Phys. Rev. Lett. 102, 163601 (2009). [CrossRef]  

References

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  19. W. Feng, Y. Li, and S.-Y. Zhu, “Effect of atomic distribution on cooperative spontaneous emission,” Phys. Rev. A 89, 013816 (2014).
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  20. M. O. Scully, “Single photon subradiance: quantum control of spontaneous emission and ultrafast readout,” Phys. Rev. Lett. 115, 243602 (2015).
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  23. R. Röhlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Rüffer, “Collective lamb shift in single-photon superradiance,” Science 328, 1248–1251 (2010).
    [Crossref]
  24. R. Röhlsberger, “Cooperative emission from nuclei: the collective lamb shift and electromagnetically induced transparency,” Fortschr. Phys. 61, 360–376 (2013).
    [Crossref]
  25. D. Bhatti, J. von Zanthier, and G. S. Agarwal, “Superbunching and nonclassicality as new hallmarks of superradiance,” Sci. Rep. 5, 17335 (2015).
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  26. R. G. DeVoe and R. G. Brewer, “Observation of superradiant and subradiant spontaneous emission of two trapped ions,” Phys. Rev. Lett. 76, 2049–2052 (1996).
    [Crossref]
  27. G. Nienhuis and F. Schuller, “Spontaneous emission and light scattering by atomic lattice models,” J. Phys. B 20, 23–36 (1987).
    [Crossref]
  28. T. Bastin, C. Thiel, J. von Zanthier, L. Lamata, E. Solano, and G. S. Agarwal, “Operational determination of multiqubit entanglement classes via tuning of local operations,” Phys. Rev. Lett. 102, 053601 (2009).
    [Crossref]
  29. J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
    [Crossref]
  30. D. Meiser and M. J. Holland, “Steady-state superradiance with alkaline-earth-metal atoms,” Phys. Rev. A 81, 033847 (2010).
    [Crossref]
  31. D. Meiser and M. J. Holland, “Intensity fluctuations in steady-state superradiance,” Phys. Rev. A 81, 063827 (2010).
    [Crossref]
  32. R. G. DeVoe and R. G. Brewer, “Laser-frequency division and stabilization,” Phys. Rev. A 30, 2827–2829 (1984).
    [Crossref]
  33. R. Reimann, W. Alt, T. Kampschulte, T. Macha, L. Ratschbacher, N. Thau, S. Yoon, and D. Meschede, “Cavity-modified collective Rayleigh scattering of two atoms,” Phys. Rev. Lett. 114, 023601 (2015).
    [Crossref]
  34. B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt, and T. E. Northup, “Enhanced quantum interface with collective ion-cavity coupling,” Phys. Rev. Lett. 114, 023602 (2015).
    [Crossref]
  35. A. Neuzner, M. Körber, O. Morin, S. Ritter, and G. Rempe, “Interference and dynamics of light from a distance-controlled atom pair in an optical cavity,” Nat. Photonics 10, 303–306 (2016).
    [Crossref]
  36. A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstatter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012).
    [Crossref]
  37. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
    [Crossref]
  38. J. M. Fink, R. Bianchetti, M. Baur, M. Göppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Dressed collective qubit states and the Tavis-Cummings model in circuit QED,” Phys. Rev. Lett. 103, 083601 (2009).
    [Crossref]
  39. J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
    [Crossref]
  40. K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gulde, S. Falt, E. L. Hu, and A. Imamoglu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
    [Crossref]
  41. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Coherent generation of non-classical light on a chip via photon-induced tunneling and blockade,” Nat. Phys. 4, 859–863 (2008).
    [Crossref]
  42. J. Miguel-Sánchez, A. Reinhard, E. Togan, T. Volz, A. Imamoglu, B. Besga, J. Reichel, and J. Estève, “Cavity quantum electrodynamics with charge-controlled quantum dots coupled to a fiber Fabry-Perot cavity,” New J. Phys. 15, 045002 (2013).
    [Crossref]
  43. A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
    [Crossref]
  44. H. A. M. Leymann, A. Foerster, F. Jahnke, J. Wiersig, and C. Gies, “Sub- and superradiance in nanolasers,” Phys. Rev. Appl. 4, 044018 (2015).
    [Crossref]
  45. G. S. Agarwal, Quantum Optics (Cambridge University, 2012).
  46. M. Tavis and F. W. Cummings, “Exact solution for an n-molecule-radiation-field Hamiltonian,” Phys. Rev. 170, 379–384 (1968).
    [Crossref]
  47. E. V. Goldstein and P. Meystre, “Dipole-dipole interaction in optical cavities,” Phys. Rev. A 56, 5135–5146 (1997).
    [Crossref]
  48. J. Johansson, P. Nation, and F. Nori, “Qutip: an open-source python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 183, 1760–1772 (2012).
    [Crossref]
  49. S. Fernández-Vidal, S. Zippilli, and G. Morigi, “Nonlinear optics with two trapped atoms,” Phys. Rev. A 76, 053829 (2007).
    [Crossref]
  50. H. Tanji-Suzuki, I. D. Leroux, M. H. Schleier-Smith, M. Cetina, A. T. Grier, J. Simon, and V. Vuletić, “Chapter 4—interaction between atomic ensembles and optical resonators: classical description,” in Advances in Atomic, Molecular, and Optical Physics (Academic, 2011), Vol. 60, pp. 201–237.
  51. G. S. Agarwal, “Vacuum-field Rabi splittings in microwave absorption by Rydberg atoms in a cavity,” Phys. Rev. Lett. 53, 1732–1734 (1984).
    [Crossref]
  52. Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
    [Crossref]
  53. R. J. Thompson, G. Rempe, and H. J. Kimble, “Observation of normal-mode splitting for an atom in an optical cavity,” Phys. Rev. Lett. 68, 1132–1135 (1992).
    [Crossref]
  54. Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014).
    [Crossref]
  55. L. V. Abdurakhimov, Y. M. Bunkov, and D. Konstantinov, “Normal-mode splitting in the coupled system of hybridized nuclear magnons and microwave photons,” Phys. Rev. Lett. 114, 226402 (2015).
    [Crossref]
  56. D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a milliHertz-linewidth laser,” Phys. Rev. Lett. 102, 163601 (2009).
    [Crossref]

2016 (4)

A. A. Svidzinsky, F. Li, H. Li, X. Zhang, C. H. R. Ooi, and M. O. Scully, “Single-photon superradiance and radiation trapping by atomic shells,” Phys. Rev. A 93, 043830 (2016).
[Crossref]

W. Guerin, M. O. Araújo, and R. Kaiser, “Subradiance in a large cloud of cold atoms,” Phys. Rev. Lett. 116, 083601 (2016).
[Crossref]

P. Longo, C. H. Keitel, and J. Evers, “Tailoring superradiance to design artificial quantum systems,” Sci. Rep. 6, 23628 (2016).
[Crossref]

A. Neuzner, M. Körber, O. Morin, S. Ritter, and G. Rempe, “Interference and dynamics of light from a distance-controlled atom pair in an optical cavity,” Nat. Photonics 10, 303–306 (2016).
[Crossref]

2015 (6)

H. A. M. Leymann, A. Foerster, F. Jahnke, J. Wiersig, and C. Gies, “Sub- and superradiance in nanolasers,” Phys. Rev. Appl. 4, 044018 (2015).
[Crossref]

D. Bhatti, J. von Zanthier, and G. S. Agarwal, “Superbunching and nonclassicality as new hallmarks of superradiance,” Sci. Rep. 5, 17335 (2015).
[Crossref]

L. V. Abdurakhimov, Y. M. Bunkov, and D. Konstantinov, “Normal-mode splitting in the coupled system of hybridized nuclear magnons and microwave photons,” Phys. Rev. Lett. 114, 226402 (2015).
[Crossref]

B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt, and T. E. Northup, “Enhanced quantum interface with collective ion-cavity coupling,” Phys. Rev. Lett. 114, 023602 (2015).
[Crossref]

M. O. Scully, “Single photon subradiance: quantum control of spontaneous emission and ultrafast readout,” Phys. Rev. Lett. 115, 243602 (2015).
[Crossref]

R. Reimann, W. Alt, T. Kampschulte, T. Macha, L. Ratschbacher, N. Thau, S. Yoon, and D. Meschede, “Cavity-modified collective Rayleigh scattering of two atoms,” Phys. Rev. Lett. 114, 023601 (2015).
[Crossref]

2014 (4)

W. Feng, Y. Li, and S.-Y. Zhu, “Effect of atomic distribution on cooperative spontaneous emission,” Phys. Rev. A 89, 013816 (2014).
[Crossref]

A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
[Crossref]

Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, “Hybridizing ferromagnetic magnons and microwave photons in the quantum limit,” Phys. Rev. Lett. 113, 083603 (2014).
[Crossref]

J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
[Crossref]

2013 (4)

T. Bienaimé, R. Bachelard, N. Piovella, and R. Kaiser, “Cooperativity in light scattering by cold atoms,” Fortschr. Phys. 61, 377–392 (2013).
[Crossref]

A. A. Svidzinsky, L. Yuan, and M. O. Scully, “Quantum amplification by superradiant emission of radiation,” Phys. Rev. X 3, 041001 (2013).
[Crossref]

J. Miguel-Sánchez, A. Reinhard, E. Togan, T. Volz, A. Imamoglu, B. Besga, J. Reichel, and J. Estève, “Cavity quantum electrodynamics with charge-controlled quantum dots coupled to a fiber Fabry-Perot cavity,” New J. Phys. 15, 045002 (2013).
[Crossref]

R. Röhlsberger, “Cooperative emission from nuclei: the collective lamb shift and electromagnetically induced transparency,” Fortschr. Phys. 61, 360–376 (2013).
[Crossref]

2012 (3)

J. Johansson, P. Nation, and F. Nori, “Qutip: an open-source python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 183, 1760–1772 (2012).
[Crossref]

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[Crossref]

A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstatter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012).
[Crossref]

2011 (2)

R. Wiegner, J. von Zanthier, and G. S. Agarwal, “Quantum-interference-initiated superradiant and subradiant emission from entangled atoms,” Phys. Rev. A 84, 023805 (2011).
[Crossref]

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref]

2010 (3)

R. Röhlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Rüffer, “Collective lamb shift in single-photon superradiance,” Science 328, 1248–1251 (2010).
[Crossref]

D. Meiser and M. J. Holland, “Steady-state superradiance with alkaline-earth-metal atoms,” Phys. Rev. A 81, 033847 (2010).
[Crossref]

D. Meiser and M. J. Holland, “Intensity fluctuations in steady-state superradiance,” Phys. Rev. A 81, 063827 (2010).
[Crossref]

2009 (6)

J. M. Fink, R. Bianchetti, M. Baur, M. Göppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Dressed collective qubit states and the Tavis-Cummings model in circuit QED,” Phys. Rev. Lett. 103, 083601 (2009).
[Crossref]

M. O. Scully, “Collective lamb shift in single photon Dicke superradiance,” Phys. Rev. Lett. 102, 143601 (2009).
[Crossref]

M. O. Scully and A. A. Svidzinsky, “The super of superradiance,” Science 325, 1510–1511 (2009).
[Crossref]

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a milliHertz-linewidth laser,” Phys. Rev. Lett. 102, 163601 (2009).
[Crossref]

T. Bastin, C. Thiel, J. von Zanthier, L. Lamata, E. Solano, and G. S. Agarwal, “Operational determination of multiqubit entanglement classes via tuning of local operations,” Phys. Rev. Lett. 102, 053601 (2009).
[Crossref]

A. Maser, U. Schilling, T. Bastin, E. Solano, C. Thiel, and J. von Zanthier, “Generation of total angular momentum eigenstates in remote qubits,” Phys. Rev. A 79, 033833 (2009).
[Crossref]

2008 (2)

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Coherent generation of non-classical light on a chip via photon-induced tunneling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature 453, 1008–1015 (2008).
[Crossref]

2007 (3)

C. Thiel, J. von Zanthier, T. Bastin, E. Solano, and G. S. Agarwal, “Generation of symmetric Dicke states of remote qubits with linear optics,” Phys. Rev. Lett. 99, 193602 (2007).
[Crossref]

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gulde, S. Falt, E. L. Hu, and A. Imamoglu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

S. Fernández-Vidal, S. Zippilli, and G. Morigi, “Nonlinear optics with two trapped atoms,” Phys. Rev. A 76, 053829 (2007).
[Crossref]

2004 (1)

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref]

1997 (1)

E. V. Goldstein and P. Meystre, “Dipole-dipole interaction in optical cavities,” Phys. Rev. A 56, 5135–5146 (1997).
[Crossref]

1996 (1)

R. G. DeVoe and R. G. Brewer, “Observation of superradiant and subradiant spontaneous emission of two trapped ions,” Phys. Rev. Lett. 76, 2049–2052 (1996).
[Crossref]

1992 (1)

R. J. Thompson, G. Rempe, and H. J. Kimble, “Observation of normal-mode splitting for an atom in an optical cavity,” Phys. Rev. Lett. 68, 1132–1135 (1992).
[Crossref]

1990 (1)

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
[Crossref]

1987 (1)

G. Nienhuis and F. Schuller, “Spontaneous emission and light scattering by atomic lattice models,” J. Phys. B 20, 23–36 (1987).
[Crossref]

1984 (2)

R. G. DeVoe and R. G. Brewer, “Laser-frequency division and stabilization,” Phys. Rev. A 30, 2827–2829 (1984).
[Crossref]

G. S. Agarwal, “Vacuum-field Rabi splittings in microwave absorption by Rydberg atoms in a cavity,” Phys. Rev. Lett. 53, 1732–1734 (1984).
[Crossref]

1982 (1)

M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
[Crossref]

1977 (1)

Q. H. F. Vrehen, H. M. J. Hikspoors, and H. M. Gibbs, “Quantum beats in superfluorescence in atomic cesium,” Phys. Rev. Lett. 38, 764–767 (1977).
[Crossref]

1973 (2)

R. Friedberg, S. Hartmann, and J. Manassah, “Frequency shifts in emission and absorption by resonant systems of two-level atoms,” Phys. Rep. 7, 101–179 (1973).
[Crossref]

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[Crossref]

1971 (2)

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302–313 (1971).
[Crossref]

N. E. Rehler and J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971).
[Crossref]

1968 (1)

M. Tavis and F. W. Cummings, “Exact solution for an n-molecule-radiation-field Hamiltonian,” Phys. Rev. 170, 379–384 (1968).
[Crossref]

1954 (1)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[Crossref]

Abdumalikov, A. A.

J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
[Crossref]

Abdurakhimov, L. V.

L. V. Abdurakhimov, Y. M. Bunkov, and D. Konstantinov, “Normal-mode splitting in the coupled system of hybridized nuclear magnons and microwave photons,” Phys. Rev. Lett. 114, 226402 (2015).
[Crossref]

Agarwal, G. S.

D. Bhatti, J. von Zanthier, and G. S. Agarwal, “Superbunching and nonclassicality as new hallmarks of superradiance,” Sci. Rep. 5, 17335 (2015).
[Crossref]

R. Wiegner, J. von Zanthier, and G. S. Agarwal, “Quantum-interference-initiated superradiant and subradiant emission from entangled atoms,” Phys. Rev. A 84, 023805 (2011).
[Crossref]

T. Bastin, C. Thiel, J. von Zanthier, L. Lamata, E. Solano, and G. S. Agarwal, “Operational determination of multiqubit entanglement classes via tuning of local operations,” Phys. Rev. Lett. 102, 053601 (2009).
[Crossref]

C. Thiel, J. von Zanthier, T. Bastin, E. Solano, and G. S. Agarwal, “Generation of symmetric Dicke states of remote qubits with linear optics,” Phys. Rev. Lett. 99, 193602 (2007).
[Crossref]

G. S. Agarwal, “Vacuum-field Rabi splittings in microwave absorption by Rydberg atoms in a cavity,” Phys. Rev. Lett. 53, 1732–1734 (1984).
[Crossref]

G. S. Agarwal, Quantum Optics (Cambridge University, 2012).

G. S. Agarwal, Springer Tracts in Modern Physics: Quantum Optics (Springer, 1974), p. 55.

Alt, W.

R. Reimann, W. Alt, T. Kampschulte, T. Macha, L. Ratschbacher, N. Thau, S. Yoon, and D. Meschede, “Cavity-modified collective Rayleigh scattering of two atoms,” Phys. Rev. Lett. 114, 023601 (2015).
[Crossref]

Araújo, M. O.

W. Guerin, M. O. Araújo, and R. Kaiser, “Subradiance in a large cloud of cold atoms,” Phys. Rev. Lett. 116, 083601 (2016).
[Crossref]

Atature, M.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gulde, S. Falt, E. L. Hu, and A. Imamoglu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

Bachelard, R.

T. Bienaimé, R. Bachelard, N. Piovella, and R. Kaiser, “Cooperativity in light scattering by cold atoms,” Fortschr. Phys. 61, 377–392 (2013).
[Crossref]

Badolato, A.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gulde, S. Falt, E. L. Hu, and A. Imamoglu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

Bajcsy, M.

A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
[Crossref]

Barreiro, J. T.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref]

Bastin, T.

A. Maser, U. Schilling, T. Bastin, E. Solano, C. Thiel, and J. von Zanthier, “Generation of total angular momentum eigenstates in remote qubits,” Phys. Rev. A 79, 033833 (2009).
[Crossref]

T. Bastin, C. Thiel, J. von Zanthier, L. Lamata, E. Solano, and G. S. Agarwal, “Operational determination of multiqubit entanglement classes via tuning of local operations,” Phys. Rev. Lett. 102, 053601 (2009).
[Crossref]

C. Thiel, J. von Zanthier, T. Bastin, E. Solano, and G. S. Agarwal, “Generation of symmetric Dicke states of remote qubits with linear optics,” Phys. Rev. Lett. 99, 193602 (2007).
[Crossref]

Baur, M.

J. M. Fink, R. Bianchetti, M. Baur, M. Göppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Dressed collective qubit states and the Tavis-Cummings model in circuit QED,” Phys. Rev. Lett. 103, 083601 (2009).
[Crossref]

Besga, B.

J. Miguel-Sánchez, A. Reinhard, E. Togan, T. Volz, A. Imamoglu, B. Besga, J. Reichel, and J. Estève, “Cavity quantum electrodynamics with charge-controlled quantum dots coupled to a fiber Fabry-Perot cavity,” New J. Phys. 15, 045002 (2013).
[Crossref]

Bhatti, D.

D. Bhatti, J. von Zanthier, and G. S. Agarwal, “Superbunching and nonclassicality as new hallmarks of superradiance,” Sci. Rep. 5, 17335 (2015).
[Crossref]

Bianchetti, R.

J. M. Fink, R. Bianchetti, M. Baur, M. Göppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Dressed collective qubit states and the Tavis-Cummings model in circuit QED,” Phys. Rev. Lett. 103, 083601 (2009).
[Crossref]

Bienaimé, T.

T. Bienaimé, R. Bachelard, N. Piovella, and R. Kaiser, “Cooperativity in light scattering by cold atoms,” Fortschr. Phys. 61, 377–392 (2013).
[Crossref]

Blais, A.

J. M. Fink, R. Bianchetti, M. Baur, M. Göppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Dressed collective qubit states and the Tavis-Cummings model in circuit QED,” Phys. Rev. Lett. 103, 083601 (2009).
[Crossref]

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref]

Blatt, R.

B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt, and T. E. Northup, “Enhanced quantum interface with collective ion-cavity coupling,” Phys. Rev. Lett. 114, 023602 (2015).
[Crossref]

A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstatter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012).
[Crossref]

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref]

R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature 453, 1008–1015 (2008).
[Crossref]

Bohnet, J. G.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[Crossref]

Bonifacio, R.

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302–313 (1971).
[Crossref]

Brandstatter, B.

A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstatter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012).
[Crossref]

Brandstätter, B.

B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt, and T. E. Northup, “Enhanced quantum interface with collective ion-cavity coupling,” Phys. Rev. Lett. 114, 023602 (2015).
[Crossref]

Brewer, R. G.

R. G. DeVoe and R. G. Brewer, “Observation of superradiant and subradiant spontaneous emission of two trapped ions,” Phys. Rev. Lett. 76, 2049–2052 (1996).
[Crossref]

R. G. DeVoe and R. G. Brewer, “Laser-frequency division and stabilization,” Phys. Rev. A 30, 2827–2829 (1984).
[Crossref]

Buckley, S.

A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
[Crossref]

Bunkov, Y. M.

L. V. Abdurakhimov, Y. M. Bunkov, and D. Konstantinov, “Normal-mode splitting in the coupled system of hybridized nuclear magnons and microwave photons,” Phys. Rev. Lett. 114, 226402 (2015).
[Crossref]

Carlson, D. R.

D. Meiser, J. Ye, D. R. Carlson, and M. J. Holland, “Prospects for a milliHertz-linewidth laser,” Phys. Rev. Lett. 102, 163601 (2009).
[Crossref]

Carmichael, H. J.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
[Crossref]

Casabone, B.

B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt, and T. E. Northup, “Enhanced quantum interface with collective ion-cavity coupling,” Phys. Rev. Lett. 114, 023602 (2015).
[Crossref]

A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstatter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012).
[Crossref]

Cetina, M.

H. Tanji-Suzuki, I. D. Leroux, M. H. Schleier-Smith, M. Cetina, A. T. Grier, J. Simon, and V. Vuletić, “Chapter 4—interaction between atomic ensembles and optical resonators: classical description,” in Advances in Atomic, Molecular, and Optical Physics (Academic, 2011), Vol. 60, pp. 201–237.

Chen, Z.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012).
[Crossref]

Chwalla, M.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref]

Coish, W. A.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
[Crossref]

Couet, S.

R. Röhlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Rüffer, “Collective lamb shift in single-photon superradiance,” Science 328, 1248–1251 (2010).
[Crossref]

Cummings, F. W.

M. Tavis and F. W. Cummings, “Exact solution for an n-molecule-radiation-field Hamiltonian,” Phys. Rev. 170, 379–384 (1968).
[Crossref]

DeVoe, R. G.

R. G. DeVoe and R. G. Brewer, “Observation of superradiant and subradiant spontaneous emission of two trapped ions,” Phys. Rev. Lett. 76, 2049–2052 (1996).
[Crossref]

R. G. DeVoe and R. G. Brewer, “Laser-frequency division and stabilization,” Phys. Rev. A 30, 2827–2829 (1984).
[Crossref]

Dicke, R. H.

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954).
[Crossref]

Eberly, J. H.

N. E. Rehler and J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971).
[Crossref]

Eichler, C.

J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
[Crossref]

Englund, D.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Coherent generation of non-classical light on a chip via photon-induced tunneling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Estève, J.

J. Miguel-Sánchez, A. Reinhard, E. Togan, T. Volz, A. Imamoglu, B. Besga, J. Reichel, and J. Estève, “Cavity quantum electrodynamics with charge-controlled quantum dots coupled to a fiber Fabry-Perot cavity,” New J. Phys. 15, 045002 (2013).
[Crossref]

Evers, J.

P. Longo, C. H. Keitel, and J. Evers, “Tailoring superradiance to design artificial quantum systems,” Sci. Rep. 6, 23628 (2016).
[Crossref]

Falt, S.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gulde, S. Falt, E. L. Hu, and A. Imamoglu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

Faraon, A.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Coherent generation of non-classical light on a chip via photon-induced tunneling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Feld, M. S.

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973).
[Crossref]

Feng, W.

W. Feng, Y. Li, and S.-Y. Zhu, “Effect of atomic distribution on cooperative spontaneous emission,” Phys. Rev. A 89, 013816 (2014).
[Crossref]

Fernández-Vidal, S.

S. Fernández-Vidal, S. Zippilli, and G. Morigi, “Nonlinear optics with two trapped atoms,” Phys. Rev. A 76, 053829 (2007).
[Crossref]

Filipp, S.

J. M. Fink, R. Bianchetti, M. Baur, M. Göppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Dressed collective qubit states and the Tavis-Cummings model in circuit QED,” Phys. Rev. Lett. 103, 083601 (2009).
[Crossref]

Fink, J. M.

J. M. Fink, R. Bianchetti, M. Baur, M. Göppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Dressed collective qubit states and the Tavis-Cummings model in circuit QED,” Phys. Rev. Lett. 103, 083601 (2009).
[Crossref]

Fischer, K.

A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90, 023846 (2014).
[Crossref]

Foerster, A.

H. A. M. Leymann, A. Foerster, F. Jahnke, J. Wiersig, and C. Gies, “Sub- and superradiance in nanolasers,” Phys. Rev. Appl. 4, 044018 (2015).
[Crossref]

Friebe, K.

B. Casabone, K. Friebe, B. Brandstätter, K. Schüppert, R. Blatt, and T. E. Northup, “Enhanced quantum interface with collective ion-cavity coupling,” Phys. Rev. Lett. 114, 023602 (2015).
[Crossref]

Friedberg, R.

R. Friedberg, S. Hartmann, and J. Manassah, “Frequency shifts in emission and absorption by resonant systems of two-level atoms,” Phys. Rep. 7, 101–179 (1973).
[Crossref]

Frunzio, L.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref]

Fushman, I.

A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Coherent generation of non-classical light on a chip via photon-induced tunneling and blockade,” Nat. Phys. 4, 859–863 (2008).
[Crossref]

Gauthier, D. J.

Y. Zhu, D. J. Gauthier, S. E. Morin, Q. Wu, H. J. Carmichael, and T. W. Mossberg, “Vacuum Rabi splitting as a feature of linear-dispersion theory: analysis and experimental observations,” Phys. Rev. Lett. 64, 2499–2502 (1990).
[Crossref]

Gerace, D.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gulde, S. Falt, E. L. Hu, and A. Imamoglu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

Gibbs, H. M.

Q. H. F. Vrehen, H. M. J. Hikspoors, and H. M. Gibbs, “Quantum beats in superfluorescence in atomic cesium,” Phys. Rev. Lett. 38, 764–767 (1977).
[Crossref]

Gies, C.

H. A. M. Leymann, A. Foerster, F. Jahnke, J. Wiersig, and C. Gies, “Sub- and superradiance in nanolasers,” Phys. Rev. Appl. 4, 044018 (2015).
[Crossref]

Girvin, S. M.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431, 162–167 (2004).
[Crossref]

Goldstein, E. V.

E. V. Goldstein and P. Meystre, “Dipole-dipole interaction in optical cavities,” Phys. Rev. A 56, 5135–5146 (1997).
[Crossref]

Göppl, M.

J. M. Fink, R. Bianchetti, M. Baur, M. Göppl, L. Steffen, S. Filipp, P. J. Leek, A. Blais, and A. Wallraff, “Dressed collective qubit states and the Tavis-Cummings model in circuit QED,” Phys. Rev. Lett. 103, 083601 (2009).
[Crossref]

Grier, A. T.

H. Tanji-Suzuki, I. D. Leroux, M. H. Schleier-Smith, M. Cetina, A. T. Grier, J. Simon, and V. Vuletić, “Chapter 4—interaction between atomic ensembles and optical resonators: classical description,” in Advances in Atomic, Molecular, and Optical Physics (Academic, 2011), Vol. 60, pp. 201–237.

Gross, M.

M. Gross and S. Haroche, “Superradiance: an essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982).
[Crossref]

Guerin, W.

W. Guerin, M. O. Araújo, and R. Kaiser, “Subradiance in a large cloud of cold atoms,” Phys. Rev. Lett. 116, 083601 (2016).
[Crossref]

Gulde, S.

K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atature, S. Gulde, S. Falt, E. L. Hu, and A. Imamoglu, “Quantum nature of a strongly coupled single quantum dot-cavity system,” Nature 445, 896–899 (2007).
[Crossref]

Haake, F.

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302–313 (1971).
[Crossref]

Hänsel, W.

T. Monz, P. Schindler, J. T. Barreiro, M. Chwalla, D. Nigg, W. A. Coish, M. Harlander, W. Hänsel, M. Hennrich, and R. Blatt, “14-qubit entanglement: creation and coherence,” Phys. Rev. Lett. 106, 130506 (2011).
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A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vuckovic, “Coherent generation of non-classical light on a chip via photon-induced tunneling and blockade,” Nat. Phys. 4, 859–863 (2008).
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A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstatter, T. E. Northup, and R. Blatt, “Tunable ion-photon entanglement in an optical cavity,” Nature 485, 482–485 (2012).
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A. A. Svidzinsky, F. Li, H. Li, X. Zhang, C. H. R. Ooi, and M. O. Scully, “Single-photon superradiance and radiation trapping by atomic shells,” Phys. Rev. A 93, 043830 (2016).
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T. Bastin, C. Thiel, J. von Zanthier, L. Lamata, E. Solano, and G. S. Agarwal, “Operational determination of multiqubit entanglement classes via tuning of local operations,” Phys. Rev. Lett. 102, 053601 (2009).
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Phys. Rev. X (1)

A. A. Svidzinsky, L. Yuan, and M. O. Scully, “Quantum amplification by superradiant emission of radiation,” Phys. Rev. X 3, 041001 (2013).
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Sci. Rep. (2)

D. Bhatti, J. von Zanthier, and G. S. Agarwal, “Superbunching and nonclassicality as new hallmarks of superradiance,” Sci. Rep. 5, 17335 (2015).
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H. Tanji-Suzuki, I. D. Leroux, M. H. Schleier-Smith, M. Cetina, A. T. Grier, J. Simon, and V. Vuletić, “Chapter 4—interaction between atomic ensembles and optical resonators: classical description,” in Advances in Atomic, Molecular, and Optical Physics (Academic, 2011), Vol. 60, pp. 201–237.

G. S. Agarwal, Quantum Optics (Cambridge University, 2012).

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Figures (5)

Fig. 1.
Fig. 1. Basic properties of the system. (a) Sketch of the system consisting of two atoms (A) that are coupled to a single-mode cavity (C) and driven by a coherent laser (L) with Rabi frequency η. Intracavity photons can leak through the mirrors by cavity decay (κ) and are registered by a detector (D). Another possible dissipative process is spontaneous emission (γ) by the atoms. The inset shows a magnified section of the arrangements of the atoms: one atom (depicted left) is fixed at an anti-node of the cavity field, while the other atom (right) can be scanned along the cavity axis causing a relative phase shift ϕz between the radiation of the atoms. (b) and (c) Energy levels and transitions of the system for (b) in-phase and (c) out-of-phase radiation of the atoms. In the case of two atoms, the state space consists of manifolds of four Dicke states with different intracavity photon numbers. For a fixed cavity state |n, these are unentangled two atom ground and two atom excited states |gg and |ee, respectively, as well as the maximally entangled symmetric and anti-symmetric Dicke states |±. For clarity, we draw neither the transitions due to cavity decay nor the detuning. (d) Energy levels and transitions of the corresponding system containing a single atom.
Fig. 2.
Fig. 2. Radiance witness R for different regimes as a function of the interatomic phase ϕz and pumping rate η. The color encodes 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 curves in the figures indicate the mean photon numbers aa2=0.01,0.1,1, respectively. (a) 3D plot and 2D surface map of the predominant hyperradiant area for γ=κ, g=10κ, and no detuning. Here, the superradiant and uncorrelated scattering area are very small and can hardly be seen. (b) and (c) Results for bad and intermediate cavity with γ=κ, no detuning, and (b) g=0.1κ or (c) g=κ. (d) and (e) Influence of the detuning on hyperradiance with γ=κ, g=10κ, and (d) δ=Δ=κ or (e) δ=Δ=10κ.
Fig. 3.
Fig. 3. Results for different cavities. Vertical cuts of the radiance witness at η0.5κ as a function of the interatomic phase ϕz for different types of cavities: blue (dashed) for a bad cavity corresponding to Fig. 2(b); green (dot-dashed) for an intermediate cavity corresponding to Fig. 2(c); and black (bold) with highlighted hyperradiant area (R>1) for a good cavity corresponding to Fig. 2(a).
Fig. 4.
Fig. 4. Comparison of in-phase and out-of-phase radiation. Both figures constitute a plot of R as a function of pumping rate η and atom-cavity coupling g, where g=0.1κ10κ reflects the transition from bad to good cavities. Results are shown for γ=κ with no detuning and (a) atoms radiating in-phase (ϕz=0) or (b) atoms radiating out-of-phase (ϕz=π). For clarification of the color code as well as dotted, dashed, and solid lines, see Fig. 2.
Fig. 5.
Fig. 5. Comparison of classical and quantum-mechanical treatment. The ratio |a|2/aa comparing the classical intensity of the intracavity field with the quantum-mechanical mean photon number is shown as a function of the interatomic phase ϕz for g=10κ, γ=κ, and η=0.1κ.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

ddtρ=i[H0+HI+HL,ρ]+Lγρ+Lκρ,
Raa22aa12aa1,
a=ηgNG1g2(γ2+iΔ)(κ2+iδ)NH,

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