Superradiance of non-Dicke states

In 1954, Dicke predicted that a system of quantum emitters confined to a subwavelength volume would produce a superradiant burst. For such a burst to occur, the emitters must be in the special Dicke state with zero dipole moment. We show that a superradiant burst may also arise for non-Dicke initial states with nonzero dipole moment. Both for Dicke and non-Dicke initial states, superradiance arises due to a decrease in the dispersion of the quantum phase of the emitter state. For non-Dicke states, the quantum phase is related to the phase of long-period envelopes which modulate the oscillations of the dipole moments. A decrease in dispersion of the quantum phase causes a decrease in the dispersion of envelope phases that results in constructive interference of the envelopes and the superradiant burst.


INTRODUCTION
Superradiance (SR) is a sharp enhancement of the spontaneous radiation rate of an ensemble of N independent emitters (two-level atoms) compared to the radiation rate of a single emitter, 0 γ . This phenomenon was predicted by Dicke [1] for a subwavelength collection of N quantum emitters that are coupled by their own radiation field. Various aspects of this phenomenon are reviewed in Refs. [2][3][4][5][6][7][8][9].
Dicke assumed that all emitters are indistinguishable and their wave function is symmetric with respect to permutations of any two emitters. In a general form, the Dicke state of N twolevel atoms, n of which are excited, has the form where P denotes all possible permutations. As an initial state, Dicke considered a state in which all N emitters are excited [3,4]. The dipole moment of such a system is equal to zero. Dicke took into account only one channel of the system's evolution in which at each time step only one of the emitters relaxes to the ground state and the system proceeds to another pure Dicke state Thus, when half of the emitters are excited, the radiation rate depends quadratically on the number of emitters, while initially, this dependence is linear. This increase in the radiation rate of two-level atoms is characteristic to SR. Dicke showed that the peak in the radiation intensity is reached in a time ~log / N N , while the duration of the SR burst is smaller than the radiation time of a single emitter by a factor of 1 / N . SR may arise for any Dicke state with n N ≤ excited atoms. For example, an SR state can be a state with a single excited atom, which is symmetric with respect to all possible permutations [10][11][12]. However, SR depends strongly on the initial state. When the initial state is antisymmetric with respect to atomic permutations, instead of SR, radiation is suppressed. It becomes even smaller than radiation of independent emitters. This phenomenon is called subradiance [4].  [4]. These mixed states still have a zero dipole moment. Second, SR is not the sole prerogative of quantum systems. It can also occur in an ensemble of nonlinear classical oscillators, which surely has a nonzero dipole moment [13,14]. In such a system, SR results from the constructive interference of long-period envelopes of rapidly oscillating dipoles [13]. For this reason, it is interesting to investigate whether SR may occur from quantum states with nonzero dipole moments, which are not Dicke states. It is worth noting that there are several phenomena discussed in the literature which one way or another are similar to SR. These are superfluorescence [15,16], superluminescence [17,18], and amplified spontaneous emission (ASE) [19][20][21]. Some of the conditions required for the observation of these phenomena are the same as for SR. We want to emphasize that we consider a subwavelength system of quantum emitters that are initially excited non-coherently. The focus of our study is the effect of the initial dipole moment of the system dynamics.
In this paper, we study the possibility of SR in an ensemble of two-level atoms in the general case, in which the system is not initially in a Dicke state. We show that for a quantum system, there is a unified mechanism for SR for both Dicke states with zero dipole moment and non-Dicke states for which the total dipole moment is not zero. We introduce a phase operator for a quantum state and show that this mechanism is related to a decrease in the dispersion of the state phase. The SR burst occurs when the dispersion reaches its minimum value. The expectation value of the initial dipole moment only affects the time delay. The greater the expected value, the smaller the time delay. We also show that nonlinearity is essential for SR to arise.

THE DICKE MODEL OF SUPPERRADIANCE
Let us first briefly consider the Dicke theory (see for details Refs. [3,4] are transition operators from excited e and ground g states, respectively, and Ê is the 2 2 × identity matrix. The corresponding Hamiltonian of the Jaynes-Cummings type for the interaction between free-space field modes and two-level atoms in the rotating wave approximation is: where ˆk a + and ˆk a are creation and annihilation operators of a photon in a mode with the frequency k ω , ω is the transition frequency of two-level atoms, k Ω is the interaction constant between photons and atoms,ˆˆj  Using the Heisenberg approach and the integral of motion, ( ) ( ) The Markovian approximation allows for the elimination of the field variables ˆk a + and ˆk a [3,4].
At this stage, the rate 0 γ of the spontaneous radiation into free-space modes is introduced. This parameter sets the characteristic time-scale 1 Solving this equation Dicke obtained the time dependence of the inversion: where delay t is determined from the initial condition ˆ( 0) The radiation intensity has a form of a burst From Eq. (6) one can see that the intensity maximum occurs at delay t t = . Thus, delay t has a meaning of the delay time of the SR burst. Note, that at the initial moment, the system inversion This expression is identical to expression (5) obtained by solving Dicke equation (3).
If the initial number of excited emitters n in the Dicke state is smaller than N, then the transition time to the , / 2 N N state decreases because of the decrease in the number of terms in the sum in Eq. (7). As the number of excited emitters approaches N/2, the delay time tends to zero. This is confirmed by our computer simulations shown in Fig. 1 and is in agreement with the results of Refs. [10][11][12] in which the case 1 n = has been considered. (the black line).
The Dicke system can also be described in a different way without using the Dicke assumptions. If we restrict ourselves to the case of 0 Nγ ω << , then the dynamics of the system can be described by the density matrix governed by the Lindblad master equation [4,22,23]: where [ ] , denotes a commutator of the respective operators. Using the interaction representation, , we consider smooth oscillations (envelopes). Switching from single-particle operators, ˆj σ and ˆj σ + , to collective operators, Ĵ − and Ĵ + , we obtain the master equation in the form [4]: Computer simulation shows (see also Ref. [4]) that the solutions of Eqs. (3) and (9)

SUPERRADIANCE OF NON-DICKE STATES
In a number of works, the Dicke approach is refined [24][25][26][27][28][29][30][31]. In these papers, it was assumed that dipole moments for both pure and mixed states are zero. In this section, by using Eq. (9) we study a possibility of SR from non-Dicke states with a nonzero dipole moment of each atom.

A. Phase operator for a two-level atom
When quantum emitters are excited by pulse pumping, the probability that a two-level atom is in a ground state is nonzero, regardless of the pump power. In general, even within the framework of pure states, the wave function of the final state of a two-level atom is a superposition,  To characterize such a system, we can use the phase of the dipole moment. This is not convenient, though, because we cannot treat in such a manner the Dicke states as their dipole moments are equal to zero. Instead, following Refs. [32][33][34] For a two-level atom, 2 M = , we have For an ensemble of N atoms, in the 2 N -dimensional space, the phase operator ˆi Φ of an i-th atom is defined as a direct product of the operator of the phase of the i-th atom, Eq. (12), and unit operators of the other atoms: The consideration of the time evolution of the expected value of the operator (13) sheds light on the origin of SR. Obviously, an average value of the operator of the phase difference of any two atoms of a system in the Dike state with n excited atoms is equal to zero ˆ, cos cos , 0 However, the dispersion of this operator is not zero As follows from Eq. (15), ij D , has a minimum value for / 2 n N = . This is the moment at which SR occurs. Thus, the time of an SR burst can be identified as the moment when the quantum system reaches the phase synchronism, i.e., when the dispersion, ij D , is minimal.

B. Mixed states
Below, to emphasize the difference of our approach from the Dicke model, we consider mixed, non-Dicke states with a nonzero dipole moment as an initial state. The phase operator (11)-(13) can also be applied to such states. To do this, we represent the initial density matrix of non-Dicke states as the direct product of density matrices of individual atoms The initial density matrix of the i-th atom can be represented as, where i k , i α , and i ϕ are assumed to be real numbers [35]. Note that the average value of the complex dipole moment ˆi σ coincides with the expression obtained for a pure state, Eq. (10): Thus, the values i α and i ϕ have the same physical meanings as in Eq. (10).
The phase operator (11)-(13) can be also applied to non-Dicke states with a nonzero dipole moment. The dynamics of the dispersion of the phase difference is studied by computer simulation of master equation (9). The dimension of the whole system is 2 2 N . In our computer simulation, 8 N ≤ , i.e., the order of the system of equations is 16 2 1 65,535 − = . The results are displayed in Fig. 3 where the radiation intensity, defined by Dicke as the time derivative of , as well as the dispersion ij D are shown. We can see that in this case, similar to the Dicke case, the SR burst and the minimum of the dispersion happen at the same time.
Equation (18) shows that that phase of a dipole moment is uniquely related to the state phase.
Note that the difference of cosines of phases of dipole moments is equal to average values of the difference of operators of cosines of phases for states of any two atoms are ˆĉ os cos cos cos Equation (  Let us now consider the dynamics of phases of dipole moments. Our computer simulation shows that at the moment 0 t , which is near delay t , emitter phases become close to each other as shown in Fig. 4. This time coincides with the time at which dispersions of the dipole phase, reach their minimum (see Fig. 5). The duration of the radiation burst is close to the prediction of the Dicke model.
Numerical simulations shows that approaching delay t both dispersions, ij D and ∆ , decrease.
They reach their minimum near the SR burst, i.e. near the time delay t (see Fig. 5). This is in qualitative agreement with Eq. (19). Similar to classical dipoles [13], the phase convergence shown in Fig. 4 indicates constructive interference in envelopes of the fast dipole oscillations. Note that the time delay t is smaller than the time of the phase convergence 0 t . This happens because the maximum of the total dipole moment is defined by two processes. First, the convergence of phases of emitters of dipole moments leads to constructive interference and to increase in the total dipole moment. Second, during time evolution the system radiates that results in a decrease in the dipole moments. The interplay of these processes leads the SR burst to occur before the phase convergence at time moment delay t where the individual dipole moments are still large enough to form a large total moment.
Thus, we may conclude that SR from non-Dicke states arises due to the phase convergence of emitter dipole moments.

C. Role of dipole moment
If a classical system of emitters initially oscillates in phase, then similar to SR, the radiation intensity is proportional to the square of the number of particles. However, the delay time of such a system is zero [36][37][38]. In order to have a delay time, the emitters must have a phase spread [13] which also results in a decrease in the initial dipole moment. This is also true for a quantum system: an increase in the average initial dipole moment leads to a decrease in the delay time.
The limiting case of the zero initial dipole moment corresponds to the Dicke model and produces the maximum delay time.  Fig. 1). This is confirmed by the results of a numerical experiment shown in Fig. 6. We can conclude, that there is a close connection between SR in systems of nonlinear classical emitters and two-level atoms. Both systems have attraction points for emitter phases and the delay times decrease when both classical and quantum dipole moments decreases. Since nonlinearity of the classical system is critical for SR, below we show that the same is true for the quantum system.

D. Role of nonlinearity
Due to the effect of saturation of the population inversion, a two-level emitter is a nonlinear system [39,40]. As we show, this nonlinearity causes the phase conversion of emitters and a decrease in the dispersion of the phase of the emitter state. A many-level system, e.g., a harmonic oscillator, does not superradiate. This is similar to a system of classical oscillators which does not superradiate for a random uniform distribution of phases.
Let us consider whether an SR burst can arise in a system of identical quantum linear harmonic oscillators. We assume that as a system of two-level atoms, oscillators are in a subwavelength volume, and they interact with modes of the electromagnetic field of the free space. This interaction has a form The dynamics of the density matrix is described by the Lindblad equation where ˆˆi i A a = ∑ is the operator of the collective dipole moment of oscillators, ˆi a is the annihilation operator of the i-th oscillator. This equation is derived in a similar way as master equation (9) for a system of two-level atoms.
Equation (21) allows one to obtain dynamics equations for a dipole moment of each oscillator: Let us estimate each average at the right-hand side of Eq. (22). The first one determines an attenuation of a harmonic oscillator in vacuum: a a a a a a a γ γ Combining Eqs. (23) and (24) Thus, the final system of equation that describes the dynamics of harmonic oscillators is closed with respect to variables ˆk a . This is a linear system of differential equations for oscillator amplitudes ˆk a . Since in system (25), all oscillator velocities are the same, any solution of this system attenuates with exactly this velocity. Therefore, there is no SR burst in this system.

CONCLUSION
We show that there is an analogy between SR in quantum and nonlinear classical systems [13,41,42]. This analogy can be recognized by considering SR from non-Dicke states. In both systems, at the moment of the phase convergence, all dipole moments of the emitters are in phase resulting in an SR burst. The convergence of emitter phases for a system of nonlinear classical emitters arises due to the formation of an attraction point for the phase evolution of the dipole moments of emitters [13]. The existence of an attraction point is a consequence of the nonlinear nature of the process. Our numerical simulations show that an attraction point of phases exists for a system of quantum emitters as well (see Fig. 4). This is likely caused by a nonlinear response of two-level atoms on the electromagnetic field due to the effect of saturation [43].
The behavior of the delay time in a quantum system is also similar to that in a classical system. If the dipole moment of a quantum system initially has its maximum value, then the delay time is zero. In a nonlinear classical system, a system has maximum dipole moment when all emitters are initially in phase. In this case, SR starts without any time delay.
Nonlinearity plays a critical role for SR in both classical and quantum systems. As shown in Ref. [14], in a linear classical system, SR does not occurs. Nonlinearity of a quantum system of two-level atoms is due to their saturation at excitation [39,40]. A system of linear quantum oscillators, which has no saturation, does not superradiate.
To conclude, we have studied the dynamics of quantum emitters interacting via their radiation field. In contrast to the Dicke model, in which all emitters are assumed to be in a state with zero dipole moment, the new SR regime arises in a more realistic system in which the initial state may have a nonzero dipole moment. We demonstrate that the Dicke state is not necessary for SR. Since the Dicke state can be realized only in a limited number of physical systems we expect that our study will stimulate the search for SR which we have shown may be observed in simpler and more realistic systems.
A.A.L would like to acknowledge support from the NSF under Grant No. DMR-1312707.