## Cascaded collective decay in regular arrays of cold trapped atoms |

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

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]

*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

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]

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

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]

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]

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]

*λ*

_{0}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]

*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.

*. 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.*

^{N}## 2. Model

*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]

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]

*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 with Γ

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

_{ij}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]

*] and [Γ*

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

*k*

_{0}=

*ω*

_{0}/

*c*= 2

*π*/

*λ*

_{0}and where

*ξ*=

*k*

_{0}

*r*.

_{ij}*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.

*. In this basis the Hamiltonian can be rewritten in diagonal form where the energies*

_{ij}*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

*N*identical atoms, 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]

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]

*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.

^{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*= 813.5 ± 0.9nm [23

_{m}23. M. Takamoto and H. Katori, “Spectroscopy of the ^{1}*S*_{0}-3*P*_{0} Clock Transition of 87*Sr* in an optical lattice,” Phys. Rev. Lett. **91**(22), 223001 (2003). [CrossRef] [PubMed]

*λ*/2 [24

_{m}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 ^{87}Sr optical clock transition,” Metrologia **45**, 539–548 (2008). [CrossRef]

*T*

_{1}-time for readout.

### 3.2. Three particle regular arrays

*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]

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]

#### 3.2.1. Linear Chain

*a*, where the angle between the atomic dipoles and the direction of the chain is given by

*θ*(see Fig. 3).

*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

*〉 state, which involves all three single excitation states |1*

_{z}*〉, |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 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).

*], that can be built up from the Γ*

_{ij}*, 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

*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

*= Ω and Γ*

_{ij}*=*

_{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}*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.

*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*〉 (yellow and green) feature a diminished decay rate, while the states |

^{i}*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}*a*=

*λ*/2, dashed line) and close positioning of the atoms (

_{m}*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.

*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

*]-matrix, which are*

_{ij}*σ*([Γ

*]) = {Γ + 2*

_{ij}*γ*, Γ −

*γ*}.

## 4. Superradiance in larger extended arrays

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]

*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

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]

*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

*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

*occurs, a chain of lattice constant*

_{ij}*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

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

## Acknowledgments

## References and links

1. | R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. |

2. | J. MacGillivray and M. Feld, “Theory of superradiance in an extended, optically thick medium,” Rhys. Rv. A |

3. | M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. |

4. | N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke Superradiance in optically pumped HFgGas,” Phys. Rev. Lett. |

5. | S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, and W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science |

6. | M. Moore and P. Meystre, “Theory of superradiant scattering of laser light from Bose-Einstein condensates,” Phys. Rev. Lett. |

7. | R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A |

8. | N. E. Rehler and J. H. Eberly, “Superradiance,” Phys. Rev. A |

9. | S. Davydov, |

10. | V. Agranovich, |

11. | H. Zoubi and H. Ritsch, “Lifetime and emission characteristics of collective electronic excitations in two-dimensional optical lattices,” Phys. Rev. A |

12. | H. Zoubi and H. Ritsch, “Metastability and directional emission characteristics of excitons in 1D optical lattices,” Europhys. Lett. |

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 |

14. | M. Takamoto, F. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature |

15. | L. Casperson, “Spectral narrowing in double-pass superradiant lasers,” Optics Communications |

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 |

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 |

18. | Z. Ficek and R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. |

19. | R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A |

20. | J. Guo and J. Cooper, “Cooling and resonance fluorescence of two atoms in a one-dimensional optical molasses,” Phys. Rev. A |

21. | Z. Ficek, R. Tanaś, and S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Physica A |

22. | H. Zoubi, “Collective light emission of a finite size atomic chain,” EPL |

23. | M. Takamoto and H. Katori, “Spectroscopy of the |

24. | I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Physics |

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 |

26. | P. Horak, K. Gheri, and H. Ritsch, “Quantum dynamics of a single-atom cascade laser,” Phys. Rev. A |

27. | G. Lin and S. Yelin, “Superradiance in spin-j particles: effects of multiple levels,” Phys. Rev. A |

**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

- 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]
- J. MacGillivray and M. Feld, “Theory of superradiance in an extended, optically thick medium,” Rhys. Rv. A14(3), 1169–1189 (1976).
- 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]
- 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]
- S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, and W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science285(5427), 571–574 (1999). [CrossRef] [PubMed]
- 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]
- R. Bonifacio, P. Schwendimann, and F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A4(1), 302–313 (1971). [CrossRef]
- 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]
- S. Davydov, Theory of Molecular Excitons (Plenum PressNew York, 1971).
- V. Agranovich, Excitations in Organic Solids (Oxford University Press, UK, 2009).
- 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]
- H. Zoubi and H. Ritsch, “Metastability and directional emission characteristics of excitons in 1D optical lattices,” Europhys. Lett.90, 23001 (2010). [CrossRef]
- 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]
- M. Takamoto, F. Hong, R. Higashi, and H. Katori, “An optical lattice clock,” Nature435(7040), 321–324 (2005). [CrossRef] [PubMed]
- L. Casperson, “Spectral narrowing in double-pass superradiant lasers,” Optics Communications8(1), 85–87 (1973). [CrossRef]
- 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]
- 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]
- Z. Ficek and R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep.372(5), 369–443 (2002). [CrossRef]
- R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A2(3), 883–888 (1970). [CrossRef]
- 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]
- 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]
- H. Zoubi, “Collective light emission of a finite size atomic chain,” EPL100(24002) (2012). [CrossRef]
- 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]
- I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Physics1(1), 23–30 (2005). [CrossRef]
- 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]
- P. Horak, K. Gheri, and H. Ritsch, “Quantum dynamics of a single-atom cascade laser,” Phys. Rev. A51(4), 3257–3266 (1995). [CrossRef] [PubMed]
- 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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