Superbunching and Nonclassicality as new Hallmarks of Superradiance

Superradiance, i.e., spontaneous emission of coherent radiation by an ensemble of identical two-level atoms in collective states introduced by Dicke in 1954, is one of the enigmatic problems of quantum optics. The startling gist is that even though the atoms have no dipole moment they radiate with increased intensity in particular directions. Following the advances in our understanding of superradiant emission by atoms in entangled $W$ states we examine the quantum statistical properties of superradiance. Such investigations require the system to have at least two excitations as one needs to explore the photon-photon correlations of the radiation emitted by such states. We present specifically results for the spatially resolved photon-photon correlations of systems prepared in doubly excited $W$ states and give conditions when the atomic system emits nonclassial light. Equally, we derive the conditions for the occurrence of bunching and even of superbunching, a rare phenomenon otherwise known only from nonclassical states of light like the squeezed vacuum. We finally investigate the photon-photon cross correlations of the spontaneously scattered light and highlight the nonclassicalty of such correlations.

predicted that if an ensemble of two-level atoms is prepared in a collective state where half of the atoms are in the excited state and half of the atoms are in the ground state the spontaneous emission is proportional to the square of the number of atoms as if the particles would radiate coherently in phase like synchronized antennas 2 . To analyze the phenomenon Dicke introduced the concept of collective spins where N two level atoms are described by the collective spin eigenstates / , N M 2 , with M running from = − / , …, + / M N N 2 2 in steps of unity. Among these states the state / , N 2 0 radiates with an intensity N 2 times as strong as that of a single atom. The origin of superradiance is difficult to see since all states / , N M 2 exhibit no macroscopic dipole moment whereas such a dipole moment is commonly assumed to be required for a radiation rate proportional to N 2 . The reason is that the Dicke states display strong quantum entanglement. The entangled character of the states is particularly apparent for the case of two two-level atoms where the individual atomic states are labeled by e l and | 〉 g l , = , l 1 2, for the excited and ground state, respectively. In this case the Dicke state | , 〉= / (| , 〉 + | , 〉) e g g e 1 0 1 2 1 2 1 2 , also known as the Bell state or the EPR state, is clearly maximally entangled. For three atoms one of the Dicke states is denoted by |3/2, −1/2�, which in current language would be the W-state / (| , , 〉 + | , , 〉 + | , , 〉) e g g g e g g g e 1 3 1 2 3   1  2  3  1  2  3 3 . The single excited generalized W-state, where only one atom is excited and − N 1 atoms are in the ground state, is also known to be fully entangled and plays a particularly important role for single-photon superradiance [4][5][6][7][8] . In fact, it has been recognized that most of the important aspects of superradiance 1,[9][10][11][12] can be studied by examining samples in single excited generalized W-states 4,5,7,8,[13][14][15][16][17][18] as the emission from these states possesses all the features of superradiance that originally were calculated for samples with an arbitrary number of excitations 5,15 . The spatial features of one-photon superradiance have been extensively studied for example from the Scientific RepoRts | 5:17335 | DOI: 10.1038/srep17335 perspective of timed Dicke states 4,5,8 and also the spectral and temporal aspects have been investigated in a large variety of systems 7,14,[18][19][20][21][22] . Note that a number of recent works [23][24][25][26][27][28][29] have also discussed how single excited generalized W-states for a small number of atoms can be prepared in the laboratory.
The single excited generalized W-state does however not allow one to study the quantum statistical properties of superradiance. In order to explore these aspects the system must emit at least two photons. Only then one has access to the photon-photon correlations which display amongst others the particular quantum characteristics of the spontaneously scattered radiation [30][31][32] . To this end it is required to investigate what we will term two-photon superradiance from generalized W-states with ≥ n 2 e excitations. In the present paper we show that in two-photon superradiance the emitted radiation can exhibit both bunched as well as nonclassical and antibunched light depending on the angle of observation, i.e., the position of the detectors collecting the scattered photons, and on the particular W-state, i.e., the number of atoms N and the number of excitations n e , considered. In particular, in certain cases it is also possible to observe the phenomenon of superbunching, i.e., photon-photon correlations larger than those maximally measurable for classical light sources. In all the cases the mean intensity displays the familiar features of superradiance produced by the corresponding W-state. While we derive our results for two-photon superradiance for arbitrary generalized W-states we focus in this paper on systems in doubly excited W-states; the outcomes for arbitrary W-states with more than two excitations are presented in the Supplementary Information section.
Note that bunching in the radiation of generalized W-states can be explained semi-classically. However, the phenomenon of superbunching as well as the emission of nonclassical light, first demonstrated in 1977 31 , can only be understood in a quantum mechanical description 31,33 . The latter is a feature arising from the light's particle nature where photon fluctuations become smaller than for coherent light. Demonstrating nonclassicality in the light of arbitrary W-states thus directly leads to a manifestation of the particular quantum mechanical characteristics of these superradiant states. We finally discuss also the spatial cross correlations of photons in two-photon superradiance. Here, likewise, superbunching and nonclassicality can be observed.
We note that recent investigations for a mesoscopic number of atoms in a cavity have already reported the observation of superbunching 34,35 . In this paper we bring out for a simple model system in free space the reasons for the appearance of this phenomenon, a curio which does not commonly occur, the squeezed vacuum being one of the rare examples 36 . The theoretical predictions could be verified experimentally, e.g., using ions localized in a linear rf-trap or atoms trapped in an optical lattice.

Results
To focus on the key aspects of two-photon superradiance we consider a linear system of N equidistantly aligned identical emitters, e.g., atoms or ions with upper state e l and ground state | 〉 g l , = ,…, l N 1 , trapped in a linear arrangement [37][38][39][40] at positions R l with spacing λ  d such that the dipole-dipole coupling between the particles can be neglected (see Fig. 1). This configuration -without any loss of generality -simplifies the calculations and leads to intuitive results which can be easily compared to an array of unentangled atoms emitting coherently in phase like a regular collection of synchronized antennae. The atoms are assumed to be prepared initially in a generalized W-state with n e excitations, i.e., in } . In what follows we study the second order correlation functions at equal times of the light emitted by the atoms in the described W-state. To this end two detectors are placed at positions r 1 and r 2 in the far field each measuring a single photon coincidentally, i.e., within a small time window much smaller than the lifetime of the upper state. To simplify the calculations we suppose that the emitters and the detectors are in one plane and that the atomic dipole moments of the transition | 〉 → | 〉 e g l l are oriented perpendicular to this plane (see Fig. 1). Due to the far field condition and therefore the inability to identify the individual photon sources, the electric field operator at r j takes the form 41 ( l is the atomic lowering operator for atom l, and ϕ θ φ the relative optical phase accumulated by a photon emitted by source l and recorded by detector j with respect to a photon emitted at the origin. Hereby, θ j and φ j denote the polar angle and the azimuth angle of the j-th detector, respectively. Note that the field operators have been chosen dimensionless as all dimension defining prefactors cancel out in the normalized correlation functions. The first and second order spatial correlation functions at equal times are defined as 30 1 is proportional to the mean intensity of the emitted radiation, i.e., To compare the photon statistics of various systems radiating with different intensities we further introduce the normalized second order correlation function 30 ( , the first order correlation function in the configuration of Fig. 1 has been calculated 15 to Note that this function is even with respect to δ 1 if adding an angle π due to the symmetry of the setup (see Fig. 1).
In order to access whether the system displays bunching for this configuration we have to search for values δ δ ( , )> 1 1 attains its maximal value (see Fig. 2). To investigate this outcome quantitatively we have to study the case of even and odd N separately since χ π ( ) and χ π ( ) 2 yield different results in these two cases. Note that in the regime  kd 1 of widely spaced atoms we are investigating, it is simple to obtain δ π = i or δ π = 2 i , = , i 1 2, as these values can be achieved already with observation angles θ  1 (see Fig. 1). For an even number N of atoms one obtains the following two identities χ π ( )=0, χ π ( ) = − 2 1 , from which we deduce π π ( , ) = ( − )/ even as a function of N has in principal no upper limit as it increases ∼N 2 for  N 1. This means that we can produce principally unlimited values of superbunching if we add more and more atoms in the ground state to the system 34 .
In case of odd N the above identities read χ π ( ) = ± /N 1 , χ π ( ) = +   To determine the possibilities for superbunching in cross correlations of the scattered photons we look for the maximum of Eq. (4). This is attained at δ =  , the ratio of which decreases with increasing N and converges to / 1 2 for  N 1.

Discussion
In conclusion we investigated for a prototype ensemble of N identical non-interacting two-level atoms prepared in collective superradiant generalized W-states with n e excitations the particular quantum statistical properties of the emitted radiation. Such investigations require the collective system to have at least two excitations as we explore the photon-photon correlations of the scattered light. We derived conditions for which the atomic system emits bunched and even superbunched light, as well as nonclassial and antibunched radiation. Here, superbunching refers to values of the normalized second order correlation function δ δ ( , )> , where Ω is the Rabi frequency (in units of the spontaneous decay rate γ). The effect is even stronger if the two atoms are subject to a strong dipole-dipole interaction as in dipole blockade systems; here π π δ ( , ) ∼ /Ω = ( ) g N 2 2 0 2 4 where δ 0 is the level shift of the doubly excited state (again in units of γ) 45 . In the last part of the paper we finally investigated the spatial cross correlations in two-photon superradiance, i.e., the second order correlation functions δ δ ( , )  , corresponding in this case to superbunching and antibunching of the cross correlations of the scattered photons.