## A superradiant clock laser on a magic wavelength optical lattice |

Optics Express, Vol. 22, Issue 11, pp. 13269-13279 (2014)

http://dx.doi.org/10.1364/OE.22.013269

Acrobat PDF (1902 KB)

### Abstract

An ideal superradiant laser on an optical clock transition of noninteracting cold atoms is predicted to exhibit an extreme frequency stability and accuracy far below mHz-linewidth. In any concrete setup sufficiently many atoms have to be confined and pumped within a finite cavity mode volume. Using a magic wavelength lattice minimizes light shifts and allows for almost uniform coupling to the cavity mode. Nevertheless, the atoms are subject to dipole-dipole interaction and collective spontaneous decay which compromises the ultimate frequency stability. In the high density limit the Dicke superradiant linewidth enhancement will broaden the laser line and nearest neighbor couplings will induce shifts and fluctuations of the laser frequency. We estimate the magnitude and scaling of these effects by direct numerical simulations of few atom systems for different geometries and densities. For Strontium in a regularly filled magic wavelength configuration atomic interactions induce small laser frequency shifts only and collective spontaneous emission weakly broadens the laser. These interactions generally enhance the laser sensitivity to cavity length fluctuations but for optimally chosen operating conditions can lead to an improved synchronization of the atomic dipoles.

© 2014 Optical Society of America

## 1. Introduction

1. T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics **6**, 687–692 (2012). [CrossRef]

*Q*and the linewidth of long lived atomic clock states. At this point, further technological improvements seem extremely challenging. Therefore, it has been suggested recently [2

2. D. Meiser, J. Ye, D. Carlson, and M. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. **102**, 163601 (2009). [CrossRef] [PubMed]

3. 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**, 78–81 (2012). [CrossRef] [PubMed]

4. J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, and J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A **88**, 013826 (2013). [CrossRef]

5. V. Vuletic, “Atomic physics: An almost lightless laser,” Nature **484**, 43–44 (2012). [CrossRef] [PubMed]

7. 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**, 033810 (2010). [CrossRef]

8. F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, and S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. **71**, 995–998 (1993). [CrossRef] [PubMed]

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

*N*and squeezed output light was predicted. Their superb accuracy in the regime of a cavity linewidth much larger than the atomic linewidth were highlighted just recently [3

3. 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**, 78–81 (2012). [CrossRef] [PubMed]

10. M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, and J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science **331**, 1043–1046 (2011). [CrossRef] [PubMed]

11. M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. **93**, 301–396 (1982). [CrossRef]

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

13. N. E. Rehler and J. H. Eberly, “Superradiance,” Phys. Rev. A **3**, 1735–1751 (1971). [CrossRef]

*N*[14

14. H. Zoubi, “Collective light emission of a finite-size atomic chain,” Europhys. Lett. **100**, 24002 (2012). [CrossRef]

*N*

^{2}as a significant deviation from the exponential decay of individual atoms [11

11. M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. **93**, 301–396 (1982). [CrossRef]

15. J. MacGillivray and M. Feld, “Theory of superradiance in an extended, optically thick medium,” Phys. Rev. A **14**, 1169–1189 (1976). [CrossRef]

16. L. Ostermann, H. Zoubi, and H. Ritsch, “Cascaded collective decay in regular arrays of cold trapped atoms,” Opt. Express **20**, 29634–29645 (2012). [CrossRef]

15. J. MacGillivray and M. Feld, “Theory of superradiance in an extended, optically thick medium,” Phys. Rev. A **14**, 1169–1189 (1976). [CrossRef]

17. N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. **30**, 309–312 (1973). [CrossRef]

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

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

8. F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, and S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. **71**, 995–998 (1993). [CrossRef] [PubMed]

2. D. Meiser, J. Ye, D. Carlson, and M. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. **102**, 163601 (2009). [CrossRef] [PubMed]

*) gets prohibitively large. Interesting results can still be obtained for special finite configurations, which should exhibit the qualitative consequences of dipole-dipole coupling quite well. Besides demonstrating the underlying basic physical mechanisms, our study aims at direct implications for the laser linewidth of a magic wavelength lattice laser in the superradiant regime [3*

^{N}3. 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**, 78–81 (2012). [CrossRef] [PubMed]

7. 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**, 033810 (2010). [CrossRef]

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

21. L. Casperson, “Spectral narrowing in double-pass superradiant lasers,” Opt. Commun. **8**, 85–87 (1973). [CrossRef]

## 2. Model

*N*identical two-level atoms held in a regular spaced configuration, e.g. in a far detuned optical trap, each of them symmetrically coupled to a single mode of a high

*Q*optical resonator. Due to the inherent exposure of the atoms to the vacuum bath the ensemble is affected by coherent dipole-dipole energy exchange processes and also by collective spontaneous emission [22

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

*N*-atom density matrix is governed by (

*h̄*= 1) with the Hamiltonian where

*i*-th atom with the transition energy

*ω*

_{0}, the operators

*a*

^{†}and

*a*correspond to the creation and annihilation of a photon with the frequency

*ω*in the cavity mode, Ω

_{c}*denotes the resonant dipole-dipole energy transfer between the atoms*

_{ij}*i*and

*j*, and represents the Jaynes-Cummings type interaction between the individual atomic transition dipoles and the cavity mode with

*g*being the coupling that emerges if a constant mode function is assumed. This approximation is justified in the situation where the atomic ensemble is aligned transversely to the propagation direction of the cavity mode or its dimensions are much smaller than the length of the resonator.

*arising from the coupling of the atomic transition dipoles through the vacuum field [23*

_{ij}23. Z. Ficek, R. Tanaś, and S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Physica A: Stat. Mech. Appl. **146**, 452–482 (1987). [CrossRef]

*R*quantifying the pumping rate and cavity loss with the rate

*κ*is described by

*] and [Γ*

_{ij}*] possess non-diagonal elements, which have to be calculated as a function of the system’s geometry [16*

_{ij}16. L. Ostermann, H. Zoubi, and H. Ritsch, “Cascaded collective decay in regular arrays of cold trapped atoms,” Opt. Express **20**, 29634–29645 (2012). [CrossRef]

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

*k*

_{0}=

*ω*

_{0}/

*c*= 2

*π/λ*

_{0}and where

*ξ*=

*k*

_{0}

*r*. Here,

_{ij}*r*denotes the relative distance between the atoms

_{ij}*i*and

*j*and

*θ*is the angle the transition dipole draws with the vector connecting the two atoms.

*ρ*, which can be calculated as the kernel of the Liouvillian, i.e. solving

_{S}*ℒ*[

*ρ*] = 0. Now, the annihilation operator

_{S}*a*is applied and we let this state evolve. After a time

*τ*has elapsed, we apply the creation operator

*a*

^{†}and Fourier-transform the trace of this aggregate, as the Fourier transformation of the expectation value of the field correlation function equates to the spectrum of the intra cavity and output light field.

## 3. Superradiant laser dynamics with confined ensembles

### 3.1. General properties of superradiant lasing

*= Γ for the collective case as discussed in [8*

_{ij}8. F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, and S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. **71**, 995–998 (1993). [CrossRef] [PubMed]

*= Γ*

_{ij}*δ*for independent decay as studied in [2

_{ij}2. D. Meiser, J. Ye, D. Carlson, and M. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. **102**, 163601 (2009). [CrossRef] [PubMed]

*N*+ 1 states of a spin-

*N*/2 system. The effective pumping of the atoms can also be described as an independent or collective, which results in analogous expressions as those describe the respective decay processes, see 4. Here we refrain from including dipole-dipole induced excitonic shifts of the energy levels. This assumption can be justified for a completely homogeneous atomic density [11

11. M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. **93**, 301–396 (1982). [CrossRef]

*R*and the single atom decay rate Γ for the three cases of collective pump and collective decay, individual pump and collective decay and independent pump and independent decay for

*N*= 4 atoms. We see that the maximum photon number is not so different for the three cases and appears at small spontaneous decay rates. For fully collective pump and collective spontaneous decay Fig. 2(a) superradiant emission into free space limits the optimal operation regime to a lower pump intensity, though.

### 3.2. The superradiant lattice laser

14. H. Zoubi, “Collective light emission of a finite-size atomic chain,” Europhys. Lett. **100**, 24002 (2012). [CrossRef]

26. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. **93**, 99–110 (1954). [CrossRef]

#### 3.2.1. A square lattice of four atoms

*g*

^{(2)}(0) correlation function for a fixed cavity loss

*κ*while tuning the pumping rate

*R*and the individual atom decay rate Γ for a four atom laser in a square lattice. The chosen lattice constant is half of the magic wavelength for Strontium,

*λ*/(2

_{magic}*λ*

_{0}) ≈ 0.58 [20

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

27. 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]

*R/κ*= 2.2, which is equal to the result from above for individual pumping and collective decay as depicted in Fig. 2(b).

*σ*operator is illustrated, where the black line represents the crossover to population inversion. On the right-hand side of the line the atom population is inverted, corresponding to the lasing case. Figure 4(c) presents the

_{z}*g*

^{2}(0) function, where the white line highlights a value of

*g*

^{2}(0) = 1, indicating a perfectly coherent light field. The area where

*g*

^{2}(0) < 1 could be referred to as an anti-bunching regime.

#### 3.2.2. Comparison of different geometrical configurations

*d*=

*λ*

_{0}/10 and a fixed atomic decay rate of Γ/

*κ*= 0.2.

*d*with a fixed spontaneous emission rate of Γ/

*κ*= 0.2. As one might have expected, Fig. 6 demonstrates a much more pronounced effect when varying the distance as opposed to changing the geometry.

## 4. Laser stability and frequency shifts for different atomic distances

### 4.1. Laser linewidth and frequency shift

7. 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**, 033810 (2010). [CrossRef]

*γ*and the energy shift

_{L}*δ*for different interatomic distances and geometrical configurations as a function of the pumping rate

*R*. For these calculations we used the same parameters as above and we don’t include a detuning between the atoms and the cavity mode (Δ =

*ω*−

_{c}*ω*

_{0}= 0).

*R/κ*≈ 1.9, which corresponds to an operation at the maximally achievable photon number, as shown in Fig. 4.

*d*<

*λ*

_{0}/2, only. Even with just four atoms it is possible to achieve a linewidth significantly below the resonator’s linewidth. The predicted frequency shift with respect to the bare atom frequency (as depicted in Fig. 7) remains very small for larger interatomic distances and reaches a maximum value when the laser is operated at

*R/κ*≈ 3, close to the maximum photon number. This could certainly be an observable phenomenon, but it is not detrimental for the operation of such a laser. Obviously, for a realistic setup we assume much too high a value for the atom-mode coupling

*g*, which however seems justified as one of our individual atoms could represents 10

^{3}to 10

^{4}atoms in an experiment.

### 4.2. Laser sensitivity to cavity length fluctuations

28. B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10^{−18} level,” Nature **506**, 71–75 (2014). [CrossRef] [PubMed]

*δ*=

_{a}*ω*

_{0}−

*ω*) as seen in Fig. 10 depending on the average atomic distance. As shown in Fig. 9(a) for closely positioned atoms the interaction evokes a significant blue shift of the cavity frequency, generating the maximum photon number with respect to the clock transition. For atoms in a magic wavelength lattice Fig. 9(b) this shift is much smaller and close to the interaction-free case. The detuning sensitivity of the laser output spectrum in these two cases is depicted in Fig. 10.

_{L}## 5. Conclusions and outlook

## Acknowledgments

## References and links

1. | T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, and J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics |

2. | D. Meiser, J. Ye, D. Carlson, and M. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. |

3. | 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 |

4. | J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, and J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A |

5. | V. Vuletic, “Atomic physics: An almost lightless laser,” Nature |

6. | M. Xu, D. Tieri, and M. Holland, “Simulating Open Quantum Systems using the Simple Lie Group SU (4),” arXiv preprint arXiv:1302.6284 (2013). |

7. | 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 |

8. | F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, and S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. |

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

10. | M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, and J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science |

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

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

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

14. | H. Zoubi, “Collective light emission of a finite-size atomic chain,” Europhys. Lett. |

15. | J. MacGillivray and M. Feld, “Theory of superradiance in an extended, optically thick medium,” Phys. Rev. A |

16. | L. Ostermann, H. Zoubi, and H. Ritsch, “Cascaded collective decay in regular arrays of cold trapped atoms,” Opt. Express |

17. | N. Skribanowitz, I. P. Herman, J. C. MacGillivray, and M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. |

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

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

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

21. | L. Casperson, “Spectral narrowing in double-pass superradiant lasers,” Opt. Commun. |

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

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

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

25. | P. Meystre and M. Sargent, |

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

27. | 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 |

28. | B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, and J. Ye, “An optical lattice clock with accuracy and stability at the 10 |

**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:**

Quantum Optics

**History**

Original Manuscript: February 18, 2014

Revised Manuscript: April 1, 2014

Manuscript Accepted: April 2, 2014

Published: May 27, 2014

**Citation**

Thomas Maier, Sebastian Kraemer, Laurin Ostermann, and Helmut Ritsch, "A superradiant clock laser on a magic wavelength optical lattice," Opt. Express **22**, 13269-13279 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13269

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### References

- T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. Martin, L. Chen, J. Ye, “A sub-40-mHz-linewidth laser based on a silicon single-crystal optical cavity,” Nat. Photonics 6, 687–692 (2012). [CrossRef]
- D. Meiser, J. Ye, D. Carlson, M. Holland, “Prospects for a millihertz-linewidth laser,” Phys. Rev. Lett. 102, 163601 (2009). [CrossRef] [PubMed]
- J. Bohnet, Z. Chen, J. Weiner, D. Meiser, M. Holland, J. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature 484, 78–81 (2012). [CrossRef] [PubMed]
- J. G. Bohnet, Z. Chen, J. M. Weiner, K. C. Cox, J. K. Thompson, “Active and passive sensing of collective atomic coherence in a superradiant laser,” Phys. Rev. A 88, 013826 (2013). [CrossRef]
- V. Vuletic, “Atomic physics: An almost lightless laser,” Nature 484, 43–44 (2012). [CrossRef] [PubMed]
- M. Xu, D. Tieri, M. Holland, “Simulating Open Quantum Systems using the Simple Lie Group SU (4),” arXiv preprint arXiv:1302.6284 (2013).
- K. Henschel, J. Majer, J. Schmiedmayer, H. Ritsch, “Cavity QED with an ultracold ensemble on a chip: Prospects for strong magnetic coupling at finite temperatures,” Phys. Rev. A 82, 033810 (2010). [CrossRef]
- F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, S. Reynaud, “Superradiant laser,” Phys. Rev. Lett. 71, 995–998 (1993). [CrossRef] [PubMed]
- P. Horak, K. Gheri, H. Ritsch, “Quantum dynamics of a single-atom cascade laser,” Phys. Rev. A 51, 3257–3266 (1995). [CrossRef] [PubMed]
- M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science 331, 1043–1046 (2011). [CrossRef] [PubMed]
- M. Gross, S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep. 93, 301–396 (1982). [CrossRef]
- R. Bonifacio, P. Schwendimann, F. Haake, “Quantum statistical theory of superradiance. I,” Phys. Rev. A 4, 302 (1971). [CrossRef]
- N. E. Rehler, J. H. Eberly, “Superradiance,” Phys. Rev. A 3, 1735–1751 (1971). [CrossRef]
- H. Zoubi, “Collective light emission of a finite-size atomic chain,” Europhys. Lett. 100, 24002 (2012). [CrossRef]
- J. MacGillivray, M. Feld, “Theory of superradiance in an extended, optically thick medium,” Phys. Rev. A 14, 1169–1189 (1976). [CrossRef]
- L. Ostermann, H. Zoubi, H. Ritsch, “Cascaded collective decay in regular arrays of cold trapped atoms,” Opt. Express 20, 29634–29645 (2012). [CrossRef]
- N. Skribanowitz, I. P. Herman, J. C. MacGillivray, M. S. Feld, “Observation of Dicke superradiance in optically pumped HF gas,” Phys. Rev. Lett. 30, 309–312 (1973). [CrossRef]
- S. Inouye, A. Chikkatur, D. Stamper-Kurn, J. Stenger, D. Pritchard, W. Ketterle, “Superradiant Rayleigh scattering from a Bose-Einstein condensate,” Science 285, 571–574 (1999). [CrossRef] [PubMed]
- M. Moore, P. Meystre, “Theory of superradiant scattering of laser light from Bose-Einstein condensates,” Phys. Rev. Lett. 83, 5202–5205 (1999). [CrossRef]
- M. Takamoto, F. Hong, R. Higashi, H. Katori, “An optical lattice clock,” Nature 435, 321–324 (2005). [CrossRef] [PubMed]
- L. Casperson, “Spectral narrowing in double-pass superradiant lasers,” Opt. Commun. 8, 85–87 (1973). [CrossRef]
- R. Lehmberg, “Radiation from an N-atom system. I. General formalism,” Phys. Rev. A 2, 883–888 (1970). [CrossRef]
- Z. Ficek, R. Tanaś, S. Kielich, “Quantum beats and superradiant effects in the spontaneous emission from two nonidentical atoms,” Physica A: Stat. Mech. Appl. 146, 452–482 (1987). [CrossRef]
- Z. Ficek, R. Tanaś, “Entangled states and collective nonclassical effects in two-atom systems,” Phys. Rep. 372, 369–443 (2002). [CrossRef]
- P. Meystre, M. Sargent, Elements of Quantum Optics (Springer-Verlag, 1990).
- R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev. 93, 99–110 (1954). [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, S. Jefferts, “The absolute frequency of the 87Sr optical clock transition,” Metrologia 45, 539–548 (2008). [CrossRef]
- B. Bloom, T. Nicholson, J. Williams, S. Campbell, M. Bishof, X. Zhang, W. Zhang, S. Bromley, J. Ye, “An optical lattice clock with accuracy and stability at the 10−18 level,” Nature 506, 71–75 (2014). [CrossRef] [PubMed]

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