## Abstract

Energy and lifetime of collective optical excitations in regular arrays of atoms and molecules are significantly influenced by dipole-dipole interaction. While the dynamics of closely positioned atoms can be approximated well by the Dicke superradiance model, the situation of finite regular configurations is hard to access analytically. Most treatments use an exciton based description limited to the lowest excitation manifold. We present a general approach studying the complete decay cascade of a finite regular array of atoms from the fully inverted to the ground state. We explicitly calculate all energy shifts and decay rates for two generic cases of a three-atom linear chain and an equilateral triangle. In numerical calculations we show that despite fairly weak dipole-dipole interactions, collective vacuum coupling allows for superradiant emission as well as subradiant states in larger arrays through multi-particle interference. This induces extra dephasing and modified decay as important limitations for Ramsey experiments in lattice atomic clock setups as well as for the gain and frequency stability of superradiant lasers.

© 2012 Optical Society of America

## 1. Introduction

The spontaneous decay of an excited atom arises due to its coupling to vacuum fluctuations of the electromagnetic field as first proposed by Dirac and later analyzed by Wigner and Weis-skopf. Dicke [1] showed that the vacuum coupling of several identical atoms at almost the same position leads to correlations among the atoms and as a result to collective superradiant spontaneous emission. In general, the decay rate of low energy collective excitations grows linearly with the particle number *N*. The resulting collective decay exhibits a delayed intensity maximum proportional to *N*^{2} as a significant deviation from the exponential decay of individual atoms [2, 3] and is called ’*superradiance*’. The phenomenon has been observed in a large number of experiments in gases and solids [2, 4] and recently also for ultracold quantum gases [5, 6].

The basic phenomenon of superradiance was widely studied theoretically already decades ago using a variety of analytical approximation methods [7, 8] with a particularly comprehensive and detailed review by Haroche [3]. These treatments are mostly based on spatially well-confined samples neglecting finite distance dipole-dipole interactions or other near-field effects as Van der Waals shifts by collisions (see [3] for the limits). While in such small-sample configurations all atoms are exposed to the same environment (vacuum fluctuations) and virtually indistinguishable, in extended systems of atoms or molecules, as e.g. in optical lattices, this approach has to be refined to account for finite interaction and correlation lengths. In systems with finite resonant dipole-dipole interactions along with a lattice symmetry, the lowest energy eigenstates are given by coherent collective electronic excitations in the material, called ’excitons’ [9]. Due to their wave-like periodic structure excitons can feature superradiance as well [10]. Interestingly, it has been noted only very recently, that also subradiant exciton states can appear in regular optical lattices [11]. In previous work, we studied the energies and lifetimes of such excitons for ultracold atoms in 1D and 2D optical lattices [11, 12] and in more general configurations. In most cases it was sufficient to limit ourselves to electrostatic dipole-dipole interactions and consider coupling among nearest-neighbour sites only. Such collective states were also considered by other authors, e.g. [13].

In contrast to these treatments, in the present paper we will investigate the full model for the collective decay process of a few atoms involving multiple excitations up to the fully inverted state. We refrain from any limiting size and distance approximations and in particular do not restrict ourselves to the single-excitation manifold. While the underlying equations for the dynamics are well-established, exact analytic treatments of the full decay problem for more than two particles are hardly possible and apart from some special cases, we have to rely on numerical solutions. Besides exhibiting the underlying basic physical mechanisms of decay channels, correlation buildup and entanglement, our study aims at direct implications for atomic clock configurations based on magic wavelength lattices [14], optical storage of qubits in atomic ensembles and ultrastable superradiant lasers [15–17]. Here, super- and subradiance can play a decisive limiting or helpful role, as e.g. the Ramsey signal crucially depends on the remaining excited state population at the time of the second pulse.

To some extent important physical effects can be seen already in the simplest configuration of two two-level atoms with identical transition wavelength *λ*_{0} as a function of the interatomic distance *r*[18]. As each particle can decay, the doubly excited state exhibits twice the single atom damping rate to a singly excited state. Depending on the distance *r* the excitation distributes differently among the two single-excitation atomic eigenstates, which due to dipole-dipole interaction are given by a symmetric and an anti-symmetric superposition of the two dipoles. For small distances, i.e. *r*/*λ*_{0} ≪ 1, the symmetric state becomes populated dominantly and is superradiant, while the anti-symmetric one is almost dark. At large distances, *r*/*λ*_{0} ≫ 1, the damping rate of both states tends to the one of a single atom leading to quite different effective dynamics. Here, everything can be calculated explicitly in an analytical fashion.

For more particles the situation becomes much more difficult to solve immediately, since already for the first step we get an increasing number of intermediate states with the total physical Hilbert space growing as 2* ^{N}*. We will show that analytical results can still be obtained for some special configurations, like a regular triangle, while most calculations need to be performed numerically.

The paper is organized as follows: in section 2 we describe the model and exhibit the dependence of the interaction terms on its geometry. In section 3 we review general properties and present an analytical solution for three atoms positioned in an equilateral triangle and compare it to the numerical solution for three atoms in a chain. Finally we study superradiance in larger systems and its implications on physical applications numerically.

## 2. Model

Let us consider *N* identical two-level systems held in a regular spaced configuration e.g. in a far detuned optical lattice. We describe the spontaneous decay process by common dipole coupling of the atoms to the free space radiation modes in vacuum state. Upon rotating wave and Markov approximation one ends up with a standard Lindblad type master equation including dipole-dipole interaction [19]. Explicitly, the time-evolution of the density operator is governed by

*i*-th atom with the atomic transition energy given by

*h̄ω*

_{0}, and Ω

*denotes the resonant dipole-dipole energy transfer between the atoms*

_{ij}*i*and

*j*. The collective damping is accounted for by the Liouvillian

*being generalized spontaneous emission rates arising from the coupling of the atomic transition dipoles through the vacuum field [21].*

_{ij}Note, that collective coupling and decay matrices [Ω* _{ij}*] and [Γ

*] possess non-diagonal elements, which have to be calculated as a function of the system’s geometry and its relative angle*

_{ij}*θ*to the atomic dipoles

*μ*. In many cases due to the finite correlation length of vacuum fluctuations these non-diagonal parts can be safely neglected. Here, we assume the same linear dipole moments and orientation for all particles (

_{i}*μ*=

_{i}*μ*) and

**r**

*=*

_{ij}**r**

*−*

_{i}**r**

*denotes the vector connecting the atom’s positions. Fortunately, the damping Liouvillian is bilinear in the dipole moment operators, so that the total interaction is composed of pairwise terms depending on the relative coordinates only. Thus, for identical atoms we have [18]*

_{j}*k*

_{0}=

*ω*

_{0}/

*c*= 2

*π*/

*λ*

_{0}and

*ξ*=

*k*

_{0}

*r*.

_{ij}It is noteworthy to point out that *F* (*ξ* → 0) = 2/3, *G*(*ξ*) diverges for *ξ* → 0 and *F* (*ξ* → ∞) = *G*(*ξ* → ∞) = 0. As a reminder and for later reference the two (scaled) functions for *θ* = *π*/2 are shown in Fig. 1. For this work we will be concerned with lattice constants (atomic distances) large enough to keep the effect of the divergence small only. For the collective states of two atoms it is possible to find distances *d* at which there is either no energy shift, *G*(*k*_{0}*d*) = 0, or no modified spontaneous emission, *F*(*k*_{0}*d*) = 0. Due to the non-periodicity of *F* and *G* this cannot be achieved for more than two atoms in a periodic arrangement. Similarly one can expect the most significant effects to occur at distances where either *F* or *G* has an extremal value, which also cannot be fulfilled for all atoms in a regular array.

As a first step in investigating the decay properties of the collective states of the system, we will consider the energy eigenstates including the dipole-dipole couplings Ω* _{ij}*. In this basis the Hamiltonian can be rewritten in diagonal form

*ω*depend on the geometry. When we represent ℒ

_{k}_{cd}in this same basis, we see that only in very special cases the Liouvillian gets diagonal as well. In these special cases spontaneous decays occur between these same eigenstates only, which allows for a simple analytical treatment of the entire system. In the general case, however,

*Ĥ*and ℒ

*have different eigenstates. Hence, spontaneous decay processes will lead to superpositions of energy eigenstates inducing oscillatory dynamics and we need to resort to a numerical analysis. An approach based on the damping basis leads to equivalent phenomena as it will not diagonalize the Hamiltonian. Despite these problems, a diagonalization of the |Γ*

_{cd}*] and [Ω*

_{ij}*] matrices can be at least performed numerically, even for hundreds of atoms. Important properties of energy shifts and decay rates appearing in the system can be obtained from their eigenvalues without the need to solve the full dynamics in the excessively large corresponding Hilbert space. To get some intuitive insight intro their connection to the full system dynamics we will study this relation closely by investigating special fully solvable simple examples.*

_{ij}## 3. Collective system dynamics and examples

#### 3.1. General results

At first we will exhibit some general features of the dynamics. Interestingly, independent of the geometry we find that the totally inverted state |*e*〉 with all *N* atoms in the excited state will always decay with a collective emission rate of Γ* _{e}* = ∑

*Γ =*

_{i}*N*Γ. The state |

*e*〉 is also a simultaneous eigenstate of the Hamiltonian with energy

*E*=

_{e}*Nh̄ω*

_{0}without any interaction shifts and the Liouvillian with the decay

*N*and the rate Γ

*is split among many possible paths. The resulting state manifold is closely related to the exciton states containing only a single excitation, but with the role of ground and excited state reversed. This manifold now includes super- and subradiant states and in the successive steps even more channels become available until half of the energy is dissipated.*

_{e}As a second general result we present the collective decay rate of the single-excitation symmetric state of *N* identical atoms,

*a*. This decay rate can be derived analytically as shown in [22] and gives

*θ*= 0 and

*θ*=

*π*/2 as a function of the number of atoms constituting the chain. The chain’s length is

*L*= (

*N*− 1)

*a*. Let us remark here that the scattering intensity in a particular directional mode can still grow for large

*N*but the solid angle of this mode shrinks so that the effect on the total decay rate decreases and the effective lifetime will saturate. This is good news for atomic lattice clocks, which in this case will not suffer too much from superradiant decay.

Throughout this work we will mainly concentrate on ^{87}*Sr* as a specific example. To trap these atoms one usually uses a ’magic wavelength’ optical lattice, which refers to the specific wavelength to minimize or even eliminate the differential light shift in the ^{87}*Sr*: ^{1}*S*_{0} →^{3}*P*_{0} clock transition. This wavelength turns out to be *λ _{m}* = 813.5 ± 0.9nm [23]. The optical lattice will confine the atoms at a distance of

*λ*/2 [24] which given in units of the transition wavelength [25] will be

_{m}*T*

_{1}-time for readout.

#### 3.2. Three particle regular arrays

In this section we will investigate two different regular geometric arrangements for *N* = 3. We compare a linear chain, where we go beyond the single excitation and nearest-neighbour coupling limits, discussed in [11], to an equilateral triangle, which has the advantage of being fully analytically treatable. Let us point out, that for two atoms, e.g. [1], the particular relative arrangement is irrelevant, and therefore the system can always be handled analytically.

### 3.2.1. Linear Chain

First, we consider a linear chain of lattice constant *a*, where the angle between the atomic dipoles and the direction of the chain is given by *θ* (see Fig. 3).

The collective states that arise from the dipole-dipole interaction are listed in Table 1, where the values that have been chosen for the numerical treatment are *ω*_{0} = 10^{14} Hz, Γ = 1Hz and thus with *a* = *λ*_{0}/2 and *θ* = *π*/2 we obtain Ω_{12} = Ω_{23} = 0.21Hz and Ω_{13} = −0.12Hz, as well as Γ_{12} = Γ_{23} = −0.15Hz and Γ_{13} = 0.04Hz. The energy shifts Δ of the collective states are independent from *ω*_{0} and can be expressed in terms of the collective parameters as −Ω_{13} and
$\left({\mathrm{\Omega}}_{13}\pm \sqrt{8{\mathrm{\Omega}}_{12}^{2}+{\mathrm{\Omega}}_{13}^{2}}\right)/2$.

With this we can now study the system’s decay properties for arbitrary initial preparations. Figure 4 (right) depicts the decay from the |2* _{z}*〉 state, which involves all three single excitation states |1

*〉, |1*

_{x}*〉 and |1*

_{y}*〉 and finally populates the ground state |*

_{z}*g*〉. The initial state decays exponentially and ’feeds’ the intermediate states whose populations (per feeding state) over time obey

*A*is the amplitude,

*ν*denotes the feeding rate responsible for increasing the population and

*γ*is the state’s decay rate. In this manner we have studied the system’s behaviour for arbitrary initial preparations, where our results are summarized in Table 2. Here, the diagonal entries refer to the states’ decay rates, while the off-diagonal ones describe the feeding rate from an upper to a lower state. A scheme visualizing the various decay channels is given in Fig. 4 (left).

Let us point out that the decay rates of the single-excitation states correspond exactly to the eigenvalues of the matrix [Γ* _{ij}*], that can be built up from the Γ

*, since*

_{ij}*σ*([Γ

*]) = {1.23, 0.96, 0.81}, while the decay rates of the doubly excited states are larger by exactly one Γ.*

_{ij}### 3.2.2. Equilateral triangle

Now, we consider an arrangement of the atoms in an equilateral triangle of length *a* with the atomic dipoles drawing a right angle to the plane of the triangle (see Fig. 3). Due to the fact that in this particular configuration Ω* _{ij}* = Ω and Γ

*=*

_{ij}*γ*for all

*i*≠

*j*the coefficient matrices [Ω

*] and [Γ*

_{ij}*] assume the same structure. As a consequence, the Hamiltonian as well as the Liouvillian are diagonal in the same basis, which allows for an analytical discussion of the system.*

_{ij}Again, we diagonalize the Hamiltonian, where the diagonal states with their energy are put down in Table 3. In this setup, since all mutual couplings have the same value Ω, the states |*a*〉 and |*b*〉 are degenerate, as they experience the same energy shift. Notice the fairly close correspondence to the states that appear in the chain (Table 1). Moreover, the existence of two symmetric states |*s*^{1}〉 and |*s*^{2}〉 shall be pointed out, which is a consequence of the uniform mutual coupling as well.

Figure 5 shows the decay from the fully inverted state |*e*〉 for *a* = *λ*_{0}/5, corresponding to *γ* = 0.71Γ. Notice, that the majority of the population decays via the symmetric channels. For a negative *γ* the symmetric states |*s ^{i}*〉 (yellow and green) feature a diminished decay rate, while the states |

*a*〉 and |

^{i}*b*〉, which employ the same behaviour (black and grey) become superradiant. As above, we show the decay and feeding rates in Table 4.

^{i}The decay scheme for this situation looks quite similar to the one of the chain (Fig. 4), except that for one and two excitations there is only one state with a positive energy shift, while the other two states are degenerate and shifted downwards.

In Fig. 6 (left) we compare the decay process to the ground state for a magic wavelength distance (*a* = *λ _{m}*/2, dashed line) and close positioning of the atoms (

*a*=

*λ*

_{0}/5, solid line). The decay of the fully inverted state is not affected by the system’s geometry and will therefore show the same exponential decay for any configuration.

For *γ* = Γ the symmetric decay channel |*e*〉 → |*s*^{2}〉 → |*s*^{1}〉 → |*g*〉 decouples from the two channels |*a*^{2}〉 → |*a*^{1}〉 and |*b*^{2}〉 → |*b*^{1}〉, where the latter two will not decay to the ground state, yielding two dark states in the single-excitation manifold. For *γ* = 0 the system behaves as independent two-level subsystems and no distinction in terms of emission rates can be made. Again, we observe, that the decay rates for the single-excitation states coincide with the eigenvalues of the [Γ* _{ij}*]-matrix, which are

*σ*([Γ

*]) = {Γ + 2*

_{ij}*γ*, Γ −

*γ*}.

## 4. Superradiance in larger extended arrays

One characteristic feature of Dicke superradiance is the pulsed emission with an increase of the energy emission as a function of time in an initial pulse buildup phase [17]. We will now study this phenomenon in our finite spaced arrays and look at the system’s energy emission given by *W*(*t*) = −*∂ _{t}* 〈

*Ĥ*〉

_{ρ̂(t)}. For very close atoms in the Dicke limit where decay occurs only via the symmetric states, the maximum occurs exactly when half of the energy is lost and is given by

*W*

_{max}(

*N*)/Γ =

*N*(

*N*+ 2)/4. This is strongly modified when other decay channels get mixed due to finite atom-atom distance, even for two atoms only [26]. Surprisingly, as shown in the following numerical solutions of the master equation, one obtains an effectively much smaller maximum of the energy emission,

*W*

_{max}, for finite lattice constants. In Fig. 7 the maximum emission intensity relative to the initial decay rate of the fully inverted state is depicted as a function of the number of atoms in the chain for different lattice constants. The distance

*d*is the first root of

_{f}*F*, namely

*F*(

*k*

_{0}

*d*) = 0, and

_{f}*d*is the first root of

_{g}*G*, analogously. We note that the closer the atoms are positioned in the chain the more obvious the superradiant nature of the system becomes, but even for

*aλ*

_{0}/10 we are far from the values of the Dicke case. Even a small contribution of slowly decaying states has a a large influence due to their long lifetime. On the other hand, even for larger lattice constants, i.e.

*a*>

*λ*

_{0}/2, the emission per atom increases with the number of atoms in contrast to independent decay.

#### 4.1. Ramsey signal

As the decay of excitation is directly accompanied by loss of atomic coherence, enhanced decay rates influence the spectroscopic properties of the collective system. As a practical example we consider two-pulse Ramsey spectroscopy, where the first *π*/2-pulse prepares a product state of half-exited atoms, which potentially exhibit strong superradiance. Using the two generic three-atom configurations discussed above, we now study the maximum possible Ramsey signal contrast, which emerges if we start with all atoms in the ground state, apply a resonant *π*/2-pulse (’Hadamard’-gate) with the same phase to each atom, then leave the systems to its free dynamics, and after a time *t* apply a second *π*/2-pulse (once in-phase and once with a phase shift o *π*), again to each atom with the same phase and look at the difference of these two signals. Figure 6(right) shows the survival probability of the fully inverted state |*e*〉 as a function of the time *t* in between the two pulses for independent atoms (black), close positioning in a triangle (red), where dispersive dephasing via Ω* _{ij}* occurs, a chain of lattice constant

*a*=

*λ*

_{0}/4 (green), where we observe a superradiant decay via Γ

*, and the magic wavelength chain (blue), which is clearly subradiant.*

_{ij}## 5. Conclusions and outlook

We have shown that despite the system size being much larger than a wavelength, collective effects in the decay and energy shifts of atoms in regular optical lattices will lead to important changes in the system dynamics. In conjunction with the appearance of fast decay via super-radiant states, one usually also finds subradiant channels and states, where the population can be trapped and which feature different energy shifts. In general, we see that superradiance can persist in spatially distributed arrays to a surprising extent, but it will be accompanied by subra-diant states, so that we get a large spread in the behaviour of individual trajectories. In contrast, for average quantities the changes get less and less significant.

The discussion in the present paper, even though presented in the language of ultracold atoms in optical lattices, can be adopted for any set-up of ordered active materials, e.g. an array of semiconductor quantum dots, a chain of colour centers in solids, or a cluster of organic molecules. Collective states in these structures can play a key role in the physical implementation of quantum information processing, and their lifetimes are critical in this context. Furthermore, these phenomena can be relevant in the context of cooling molecules by superradiant emissions [27].

## Acknowledgments

We thank Sebastian Krämer for his help with the numerics and acknowledge support by DARPA through the QUASAR project.

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