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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 28 — Dec. 31, 2012
  • pp: 29634–29645
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Cascaded collective decay in regular arrays of cold trapped atoms

Laurin Ostermann, Hashem Zoubi, and Helmut Ritsch  »View Author Affiliations


Optics Express, Vol. 20, Issue 28, pp. 29634-29645 (2012)
http://dx.doi.org/10.1364/OE.20.029634


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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 OSA

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

1. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954). URL http://link.aps.org/doi/10.1103/PhysRev.93.99. [CrossRef]

] 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 N2 as a significant deviation from the exponential decay of individual atoms [2

2. J. MacGillivray and M. Feld, “Theory of superradiance in an extended, optically thick medium,” Rhys. Rv. A 14(3), 1169–1189 (1976).

, 3

3. M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. 93(5), 301–396 (1982). URL http://www.sciencedirect.com/science/article/pii/0370157382901028. [CrossRef]

] and is called ’superradiance’. The phenomenon has been observed in a large number of experiments in gases and solids [2

2. J. MacGillivray and M. Feld, “Theory of superradiance in an extended, optically thick medium,” Rhys. Rv. A 14(3), 1169–1189 (1976).

, 4

4. N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke Superradiance in optically pumped HFgGas,” Phys. Rev. Lett. 30, 309–312 (1973). URL http://link.aps.org/doi/10.1103/PhysRevLett.30.309. [CrossRef]

] and recently also for ultracold quantum gases [5

5. S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, and W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science 285(5427), 571–574 (1999). [CrossRef] [PubMed]

, 6

6. M. Moore and P. Meystre, “Theory of superradiant scattering of laser light from Bose-Einstein condensates,” Phys. Rev. Lett. 83(25), 5202–5205 (1999). [CrossRef]

].

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

14. M. Takamoto, F. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435(7040), 321–324 (2005). [CrossRef] [PubMed]

], optical storage of qubits in atomic ensembles and ultrastable superradiant lasers [15

15. L. Casperson, “Spectral narrowing in double-pass superradiant lasers,” Optics Communications 8(1), 85–87 (1973). [CrossRef]

17

17. K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82(3), 033810 (2010). [CrossRef]

]. 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 λ0 as a function of the interatomic distance r[18

18. Z. Ficek and R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372(5), 369–443 (2002). [CrossRef]

]. 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/λ0 ≪ 1, the symmetric state becomes populated dominantly and is superradiant, while the anti-symmetric one is almost dark. At large distances, r/λ0 ≫ 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 2N. 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

19. R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2(3), 883–888 (1970). [CrossRef]

]. Explicitly, the time-evolution of the density operator is governed by
ρ^t=ih¯[H^,ρ^]cd[ρ^],
(1)
with the Hamiltonian
H^=ih¯ω0Si+Si+ijh¯ΩijSi+Sj.
(2)
The above master equation could also be formulated using a non-Hermitian Hamiltonian, as e.g. in [20

20. J. Guo and J. Cooper, “Cooling and resonance fluorescence of two atoms in a one-dimensional optical molasses,” Phys. Rev. A 51(4), 3128–3135 (1995). [CrossRef] [PubMed]

], which is equivalent to our formulation. Here, Si+ and Si are the rising and lowering operators for the atomic dipole of the i-th atom with the atomic transition energy given by h̄ω0, and Ωij denotes the resonant dipole-dipole energy transfer between the atoms i and j. The collective damping is accounted for by the Liouvillian
cd[ρ^]=12i,jΓij(Si+Sjρ^+ρ^Si+Sj2Siρ^Sj+)
(3)
with Γij being generalized spontaneous emission rates arising from the coupling of the atomic transition dipoles through the vacuum field [21

21. Z. Ficek, R. Tanaś, and S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Physica A 146(3), 452–482 (1987). [CrossRef]

].

Note, that collective coupling and decay matrices [Ωij] and [Γij] possess non-diagonal elements, which have to be calculated as a function of the system’s geometry and its relative angle θ to the atomic dipoles μi. 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 (μi = μ) and rij = rirj 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

18. Z. Ficek and R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372(5), 369–443 (2002). [CrossRef]

]
Γij=3Γ2F(k0rij)andΩij=3Γ4G(k0rij)
(4)
with Γ the single-atom linewidth, k0 = ω0/c = 2π/λ0 and
F(ξ)=(1cos2θ)sinξξ+(13cos2θ)(cosξξ2sinξξ3),G(ξ)=(1cos2θ)cosξξ+(13cos2θ)(sinξξ2+cosξξ3),
(5)
where ξ = k0rij.

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(k0d) = 0, or no modified spontaneous emission, F(k0d) = 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.

Fig. 1 Collective spontaneous emission Γij/Γ and resonant dipole-dipole coupling Ωij/Γ for θ = π/2 as a function of inter-atomic distance in units of the resonant wavelength λ0

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 Ωij. In this basis the Hamiltonian can be rewritten in diagonal form
H^=kh¯ωkSk+Sk,
(6)
where the energies ωk depend on the geometry. When we represent ℒcd 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 ℒcd 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 |Γij] and [Ωij] 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.

3. Collective system dynamics and examples

3.1. General results

As a second general result we present the collective decay rate of the single-excitation symmetric state of N identical atoms,
|s=1Ni=1N|g1eign,
(8)
which corresponds to the zero-momentum exciton in a chain of lattice constant a. This decay rate can be derived analytically as shown in [22

22. H. Zoubi, “Collective light emission of a finite size atomic chain,” EPL 100(24002) (2012). [CrossRef]

] and gives
Γs(N)=Γ[1+2n=1N1(1nN)F(k0an)].
(9)
We see that the spontaneous decay rate is collectively enhanced by pairwise interactions and can grow linearly with particle number for small chains as predicted by the Dicke model [3

3. M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. 93(5), 301–396 (1982). URL http://www.sciencedirect.com/science/article/pii/0370157382901028. [CrossRef]

]. For longer chains the long-range contributions become small and the enhancement saturates with chain length. In Fig. 2 the damping rate of the symmetric state for different lattice constants is shown for 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.

Fig. 2 Symmetric exciton state decay rate Γs(N)/Γ as function of atom number for θ = 0 (left) and θ = π/2 (right)

Throughout this work we will mainly concentrate on 87Sr 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 87Sr: 1S03P0 clock transition. This wavelength turns out to be λm = 813.5 ± 0.9nm [23

23. M. Takamoto and H. Katori, “Spectroscopy of the 1S0-3P0 Clock Transition of 87Sr in an optical lattice,” Phys. Rev. Lett. 91(22), 223001 (2003). [CrossRef] [PubMed]

]. The optical lattice will confine the atoms at a distance of λm/2 [24

24. I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Physics 1(1), 23–30 (2005). [CrossRef]

] which given in units of the transition wavelength [25

25. G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, and S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008). [CrossRef]

] will be
λm2λ00.5824.
(10)
As a matter of fact we will see that at the magic wavelength even subradiance can appear for long lattices, which should prolong the available T1-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

11. H. Zoubi and H. Ritsch, “Lifetime and emission characteristics of collective electronic excitations in two-dimensional optical lattices,” Phys. Rev. A 83(6), 063831 (2011). [CrossRef]

], to an equilateral triangle, which has the advantage of being fully analytically treatable. Let us point out, that for two atoms, e.g. [1

1. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954). URL http://link.aps.org/doi/10.1103/PhysRev.93.99. [CrossRef]

], 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).

Fig. 3 Linear chain and triangular array of three atoms with lattice constant a and angle θ

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 ω0 = 1014 Hz, Γ = 1Hz and thus with a = λ0/2 and θ = π/2 we obtain Ω12 = Ω23 = 0.21Hz and Ω13 = −0.12Hz, as well as Γ12 = Γ23 = −0.15Hz and Γ13 = 0.04Hz. The energy shifts Δ of the collective states are independent from ω0 and can be expressed in terms of the collective parameters as −Ω13 and (Ω13±8Ω122+Ω132)/2.

Table 1. Collective states (non-normalised) and energy shifts Δ [Hz] for lattice constants a = λ0/2 and a = λm/2. β1 ≈ 1.71, β2 ≈ 1.18 (for a = λ0/2)

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With this we can now study the system’s decay properties for arbitrary initial preparations. Figure 4 (right) depicts the decay from the |2z〉 state, which involves all three single excitation states |1x〉, |1y〉 and |1z〉 and finally populates the ground state |g〉. The initial state decays exponentially and ’feeds’ the intermediate states whose populations (per feeding state) over time obey
ρinterm(t)=A[1exp(νt)]]exp(γt),
(11)
where 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).

Fig. 4 Decay scheme (left) and cascade dynamics from |2z〉 state via the single-excitation states |1z〉, |1x〉 and |1y〉 to the ground state for a chain with spacing a = λ0/2

Table 2. [Hz]. Decay (diagonal entries) and feeding rates for the collective states in a chain of lattice constant a = λ0/2

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Let us point out that the decay rates of the single-excitation states correspond exactly to the eigenvalues of the matrix [Γij], that can be built up from the Γij, since σ([Γij]) = {1.23, 0.96, 0.81}, while the decay rates of the doubly excited states are larger by exactly one Γ.

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 Ωij = Ω and Γij = γ for all ij the coefficient matrices [Ωij] and [Γij] 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.

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 |s1〉 and |s2〉 shall be pointed out, which is a consequence of the uniform mutual coupling as well.

Table 3. Collective states (non-normalised) in the equilateral triangle with Ω = 3ΓG(k0a)/4

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Figure 5 shows the decay from the fully inverted state |e〉 for a = λ0/5, corresponding to γ = 0.71Γ. Notice, that the majority of the population decays via the symmetric channels. For a negative γ the symmetric states |si〉 (yellow and green) feature a diminished decay rate, while the states |ai〉 and |bi〉, which employ the same behaviour (black and grey) become superradiant. As above, we show the decay and feeding rates in Table 4.

Fig. 5 Decay from the fully inverted state via |s2〉, |a2〉 (and |b2〉, which due to the degeneracy shows the sam behaviour) and the single-excitation states |s1〉 and |a1〉 (and |b1〉) to the ground state for γ = 0.71Γ (a = λ0/5)

Table 4. Decay rates (diagonal entries) and feeding rates for the equilateral triangle where we have a uniform collective spontaneous emission rate γ = Γij

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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 = λm/2, dashed line) and close positioning of the atoms (a = λ0/5, 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.

Fig. 6 Collective decay from fully excited state to the ground state in the equilateral triangle for a = λ0/5 and a = λm/2 (left) and its influence on the maximal Ramsey signal contrast for independent atoms, atoms closely positioned in a triangle as well as in a chain (green) and at magic wavelength-distance (right)

For γ = Γ the symmetric decay channel |e〉 → |s2〉 → |s1〉 → |g〉 decouples from the two channels |a2〉 → |a1〉 and |b2〉 → |b1〉, 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 [Γij]-matrix, which are σ([Γij]) = {Γ + 2γ, 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

17. K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82(3), 033810 (2010). [CrossRef]

]. We will now study this phenomenon in our finite spaced arrays and look at the system’s energy emission given by W(t) = −tĤρ̂(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 Wmax(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

26. P. Horak, K. Gheri, and H. Ritsch, “Quantum dynamics of a single-atom cascade laser,” Phys. Rev. A 51(4), 3257–3266 (1995). [CrossRef] [PubMed]

]. Surprisingly, as shown in the following numerical solutions of the master equation, one obtains an effectively much smaller maximum of the energy emission, Wmax, 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 df is the first root of F, namely F(k0df) = 0, and dg is the first root of 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 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 > λ0/2, the emission per atom increases with the number of atoms in contrast to independent decay.

Fig. 7 Maximum of the energy emission as a function of the number of atoms in the chain for small (left) and large (right) lattice constants a. The distances df and dg refer to the first root of the functions F and G, respectively.

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 Ωij occurs, a chain of lattice constant a = λ0/4 (green), where we observe a superradiant decay via Γij, and the magic wavelength chain (blue), which is clearly subradiant.

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

27. G. Lin and S. Yelin, “Superradiance in spin-j particles: effects of multiple levels,” Phys. Rev. A 85(3), 033831 (2012). [CrossRef]

].

Acknowledgments

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

References and links

1.

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954). URL http://link.aps.org/doi/10.1103/PhysRev.93.99. [CrossRef]

2.

J. MacGillivray and M. Feld, “Theory of superradiance in an extended, optically thick medium,” Rhys. Rv. A 14(3), 1169–1189 (1976).

3.

M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. 93(5), 301–396 (1982). URL http://www.sciencedirect.com/science/article/pii/0370157382901028. [CrossRef]

4.

N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke Superradiance in optically pumped HFgGas,” Phys. Rev. Lett. 30, 309–312 (1973). URL http://link.aps.org/doi/10.1103/PhysRevLett.30.309. [CrossRef]

5.

S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, and W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science 285(5427), 571–574 (1999). [CrossRef] [PubMed]

6.

M. Moore and P. Meystre, “Theory of superradiant scattering of laser light from Bose-Einstein condensates,” Phys. Rev. Lett. 83(25), 5202–5205 (1999). [CrossRef]

7.

R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4(1), 302–313 (1971). [CrossRef]

8.

N. E. Rehler and J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971). URL http://link.aps.org/doi/10.1103/PhysRevA.3.1735. [CrossRef]

9.

S. Davydov, Theory of Molecular Excitons (Plenum PressNew York, 1971).

10.

V. Agranovich, Excitations in Organic Solids (Oxford University Press, UK, 2009).

11.

H. Zoubi and H. Ritsch, “Lifetime and emission characteristics of collective electronic excitations in two-dimensional optical lattices,” Phys. Rev. A 83(6), 063831 (2011). [CrossRef]

12.

H. Zoubi and H. Ritsch, “Metastability and directional emission characteristics of excitons in 1D optical lattices,” Europhys. Lett. 90, 23001 (2010). [CrossRef]

13.

I. Carusotto, M. Antezza, F. Bariani, S. De Liberato, and C. Ciuti, “Optical properties of atomic Mott insulators: from slow light to dynamical Casimir effects,” Phys. Rev. A 77(6), 063621 (2008). [CrossRef]

14.

M. Takamoto, F. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature 435(7040), 321–324 (2005). [CrossRef] [PubMed]

15.

L. Casperson, “Spectral narrowing in double-pass superradiant lasers,” Optics Communications 8(1), 85–87 (1973). [CrossRef]

16.

J. Bohnet, Z. Chen, J. Weiner, D. Meiser, M. Holland, and J. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484(7392), 78–81 (2012). [CrossRef] [PubMed]

17.

K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82(3), 033810 (2010). [CrossRef]

18.

Z. Ficek and R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372(5), 369–443 (2002). [CrossRef]

19.

R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2(3), 883–888 (1970). [CrossRef]

20.

J. Guo and J. Cooper, “Cooling and resonance fluorescence of two atoms in a one-dimensional optical molasses,” Phys. Rev. A 51(4), 3128–3135 (1995). [CrossRef] [PubMed]

21.

Z. Ficek, R. Tanaś, and S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Physica A 146(3), 452–482 (1987). [CrossRef]

22.

H. Zoubi, “Collective light emission of a finite size atomic chain,” EPL 100(24002) (2012). [CrossRef]

23.

M. Takamoto and H. Katori, “Spectroscopy of the 1S0-3P0 Clock Transition of 87Sr in an optical lattice,” Phys. Rev. Lett. 91(22), 223001 (2003). [CrossRef] [PubMed]

24.

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Physics 1(1), 23–30 (2005). [CrossRef]

25.

G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, and S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008). [CrossRef]

26.

P. Horak, K. Gheri, and H. Ritsch, “Quantum dynamics of a single-atom cascade laser,” Phys. Rev. A 51(4), 3257–3266 (1995). [CrossRef] [PubMed]

27.

G. Lin and S. Yelin, “Superradiance in spin-j particles: effects of multiple levels,” Phys. Rev. A 85(3), 033831 (2012). [CrossRef]

OCIS Codes
(020.7010) Atomic and molecular physics : Laser trapping
(140.6630) Lasers and laser optics : Superradiance, superfluorescence
(270.0270) Quantum optics : Quantum optics

ToC Category:
Atomic and Molecular Physics

History
Original Manuscript: October 4, 2012
Revised Manuscript: November 28, 2012
Manuscript Accepted: November 28, 2012
Published: December 20, 2012

Citation
Laurin Ostermann, Hashem Zoubi, and Helmut Ritsch, "Cascaded collective decay in regular arrays of cold trapped atoms," Opt. Express 20, 29634-29645 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-28-29634


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References

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  2. J. MacGillivray and M. Feld, “Theory of superradiance in an extended, optically thick medium,” Rhys. Rv. A14(3), 1169–1189 (1976).
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  8. N. E. Rehler and J. H. Eberly, “Superradiance,” Phys. Rev. A3, 1735–1751 (1971). URL http://link.aps.org/doi/10.1103/PhysRevA.3.1735 . [CrossRef]
  9. S. Davydov, Theory of Molecular Excitons (Plenum PressNew York, 1971).
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  11. H. Zoubi and H. Ritsch, “Lifetime and emission characteristics of collective electronic excitations in two-dimensional optical lattices,” Phys. Rev. A83(6), 063831 (2011). [CrossRef]
  12. H. Zoubi and H. Ritsch, “Metastability and directional emission characteristics of excitons in 1D optical lattices,” Europhys. Lett.90, 23001 (2010). [CrossRef]
  13. I. Carusotto, M. Antezza, F. Bariani, S. De Liberato, and C. Ciuti, “Optical properties of atomic Mott insulators: from slow light to dynamical Casimir effects,” Phys. Rev. A77(6), 063621 (2008). [CrossRef]
  14. M. Takamoto, F. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature435(7040), 321–324 (2005). [CrossRef] [PubMed]
  15. L. Casperson, “Spectral narrowing in double-pass superradiant lasers,” Optics Communications8(1), 85–87 (1973). [CrossRef]
  16. J. Bohnet, Z. Chen, J. Weiner, D. Meiser, M. Holland, and J. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature484(7392), 78–81 (2012). [CrossRef] [PubMed]
  17. K. Henschel, J. Majer, J. Schmiedmayer, and H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A82(3), 033810 (2010). [CrossRef]
  18. Z. Ficek and R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep.372(5), 369–443 (2002). [CrossRef]
  19. R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A2(3), 883–888 (1970). [CrossRef]
  20. J. Guo and J. Cooper, “Cooling and resonance fluorescence of two atoms in a one-dimensional optical molasses,” Phys. Rev. A51(4), 3128–3135 (1995). [CrossRef] [PubMed]
  21. Z. Ficek, R. Tanaś, and S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Physica A146(3), 452–482 (1987). [CrossRef]
  22. H. Zoubi, “Collective light emission of a finite size atomic chain,” EPL100(24002) (2012). [CrossRef]
  23. M. Takamoto and H. Katori, “Spectroscopy of the 1S0-3P0 Clock Transition of 87Sr in an optical lattice,” Phys. Rev. Lett.91(22), 223001 (2003). [CrossRef] [PubMed]
  24. I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Physics1(1), 23–30 (2005). [CrossRef]
  25. G. Campbell, A. Ludlow, S. Blatt, J. Thomsen, M. Martin, M. de Miranda, T. Zelevinsky, M. Boyd, J. Ye, S. Diddams, T. Heavner, T. Parker, and S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia45, 539–548 (2008). [CrossRef]
  26. P. Horak, K. Gheri, and H. Ritsch, “Quantum dynamics of a single-atom cascade laser,” Phys. Rev. A51(4), 3257–3266 (1995). [CrossRef] [PubMed]
  27. G. Lin and S. Yelin, “Superradiance in spin-j particles: effects of multiple levels,” Phys. Rev. A85(3), 033831 (2012). [CrossRef]

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