## A classical simulation of nonlinear Jaynes–Cummings and Rabi models in photonic lattices |

Optics Express, Vol. 21, Issue 10, pp. 12888-12898 (2013)

http://dx.doi.org/10.1364/OE.21.012888

Acrobat PDF (1369 KB)

### Abstract

The interaction of a two-level atom with a single-mode quantized field is one of the simplest models in quantum optics. Under the rotating wave approximation, it is known as the Jaynes-Cummings model and without it as the Rabi model. Real-world realizations of the Jaynes-Cummings model include cavity, ion trap and circuit quantum electrodynamics. The Rabi model can be realized in circuit quantum electrodynamics. As soon as nonlinear couplings are introduced, feasible experimental realizations in quantum systems are drastically reduced. We propose a set of two photonic lattices that classically simulates the interaction of a single two-level system with a quantized field under field nonlinearities and nonlinear couplings as long as the quantum optics model conserves parity. We describe how to reconstruct the mean value of quantum optics measurements, such as photon number and atomic energy excitation, from the intensity and from the field, such as von Neumann entropy and fidelity, at the output of the photonic lattices. We discuss how typical initial states involving coherent or displaced Fock fields can be engineered from recently discussed Glauber-Fock lattices. As an example, the Buck-Sukumar model, where the coupling depends on the intensity of the field, is classically simulated for separable and entangled initial states.

© 2013 OSA

## 1. Introduction

19. S. Longhi, “Jaynes-Cummings photonic superlattices,” Opt. Lett. **36**, 3407–3409 (2011) [CrossRef] [PubMed] .

20. A. Crespi, S. Longhi, and R. Osellame, “Photonic realization of the quantum Rabi model,” Phys. Rev. Lett. **108**, 163601 (2012) [CrossRef] [PubMed] .

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

22. E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE **51**, 89–109 (1963) [CrossRef] .

23. J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneuous collapse and revival in a simple quantum model,” Phys. Rev. Lett. **44**, 1323–1326 (1980) [CrossRef] .

24. B. Buck and C. V. Sukumar, “Exactly soluble model of atom–phonon coupling showing periodic decay and revival,” Phys. Lett. **81**, 132–135 (1981) [CrossRef] .

25. E. A. Kochetov, “Exactly solvable non-linear generalisations of the Jaynes-Cummings model,” J. Phys. A: Math. Gen. **20**, 2433–2442 (1987) [CrossRef] .

*Ŝ*with

_{i}*i*=

*z*, +, − that obey an

*su*(2) algebra, [

*Ŝ*,

_{z}*Ŝ*

_{±}] = ±

*Ŝ*

_{±}and [

*Ŝ*

_{+},

*Ŝ*

_{−}] = 2

*Ŝ*, and the inter-level energy is given by

_{z}*ω*

_{0}. The field is described by the creation (annihilation) operators,

*â*

^{†}(

*â*), and the frequency

*ω*. The parameter

_{f}*λ*is a coupling constant and the function

*f*(

*â*

^{†}

*â*) is a real well-behaved function of the number operator. Peculiar phenomena has been found among the years with specific realizations of the Kochetov model; e.g. for just two levels [26

26. R. R. Schlicher, “Jaynes-Cummings model with atomic motion,” Opt. Commun. **70**, 97–102 (1989) [CrossRef] .

32. B. M. Rodríguez-Lara, A. Z. Cárdenas, F. Soto-Eguibar, and H. M. Moya-Cessa, “A photonic crystal realization of a phase driven two-level atom,” Opt. Commun. **292**, 87–91 (2013) [CrossRef] .

33. A. Joshi, “Nonlinear dynamical evolution of the driven two-photon Jaynes-Cummings model,” Phys. Rev. A **62**, 043812 (2000) [CrossRef] .

34. M. Abdel-Aty, S. Furuichi, and A.-S. F. Obada, “Entanglement degree of a nonlinear multiphoton Jaynes-Cummings model,” J. Opt. B: Quantum Semiclass. Opt. **4**, 37–43 (2002) [CrossRef] .

36. O. de los Santos-Sanchez and J. Récamier, “The f-deformed Jaynes-Cummings model and its nonlinear coherent states,” J. Phys. B: At. Mol. Opt. Phys. **45**, 015502 (2012) [CrossRef] .

37. M. H. Naderi, M. Soltanolkotabi, and R. Roknizadeh, “A theoretical scheme for generation of nonlinear coherent states in a micromaser under intensity-dependent Jaynes-Cummings model,” Eur. Phys. J. D **32**, 397–408 (2005) [CrossRef] .

## 2. The quantum optics model

*h*(

*n̂*) and

*f*(

*n̂*) in terms of the number operator,

*n̂*=

*â*

^{†}

*â*, and where the operators

*σ̂*with

_{i}*i*=

*z*, +, − are Pauli matrices. We have split the nonlinear Rabi coupling

*σ̂*

_{−}+

*σ̂*

_{+}) into Jaynes-Cummings coupling described by the coupling parameter

*g*

_{−}and counter-rotating terms described by the parameter

*g*

_{+}for reasons that will become apparent in the next section. As mentioned before, this general Hamiltonian is not physical realizable with current experimental setups. For example, in order to implement (2) in cavity- or circuit-QED the function

*h*(

*n̂*) must be linear to account for the free-field energy or quadratic at most to describe the effect of a Kerr medium,

*h*(

*n̂*) =

*ωn̂*+

*κn̂*

^{2}, and the coupling function

41. M. D. Crisp, “Ed Jaynes’ steak dinner problem II,” in *Physics and Probability, Essays in Honor of Edwin T. Jaynes*, W. T. Grandy Jr. and P. W. Milonni, eds. (Cambridge University Press, 1993) [CrossRef] .

42. R. L. de Matos Filho and W. Vogel, “Even and Odd Coherent States of the Motion of a Trapped Ion,” Phys. Rev. Lett. **76**, 608–611 (1996) [CrossRef] [PubMed] .

44. H. Moya-Cessa, F. Soto-Eguibar, J. M. Vargas-Martínez, R. Juárez-Amaro, and A. Zúñiga Segundo, “Ion-laser interactions: The most complete solution,” Physics Reports **513**, 229–261 (2012) [CrossRef] .

45. T. Esslinger, “Fermi-Hubbard Physics with atoms in an optical lattice,” Annu. Rev. Condens. Matter Phys. **1**, 129–152 (2010) [CrossRef] .

*Ĥ*, Π̂] = 0. Conservation of parity allows us to define two orthogonal parity bases, with

46. E. A. Tur, “Jaynes-Cummings model: Solutions without rotating wave approximation,” Opt. Spectrosc. **89**, 574–588 (2000) [CrossRef] .

48. J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep strong coupling regime of the Jaynes-Cummings model,” Phys. Rev. Lett. **105**, 263603 (2010) [CrossRef] .

*g*

_{+}= 0, then the model reduces to a nonlinear JC model that also conserves the total number of excitations defined as

*N̂*=

*n̂*+

*σ̂*/2; i.e. [

_{z}*Ĥ*,

*N̂*] = 0. It is simple to obtain the time evolution for this case by using a method that makes use of Susskind-Glogower operators [49

49. B. M. Rodríguez-Lara and H. M. Moya-Cessa, “Exact solution of generalized Dicke models via Susskind-Glogower operators,” J. Phys. A: Math. Theor. **46**, 095301 (2013) [CrossRef] .

*g*

_{+}= 0 becomes with the elements of the similarity transformation given by and the dispersion relation, Photon transport is then given by the time evolution operator, In other words, by using the analogy between transport of single-photon states and propagation of classical field, it is very simple to calculate the propagation through the equivalent nonlinear JC photonic lattice via quantum optics methods.

## 3. Classical simulation in arrays of coupled waveguides

39. B. M. Rodríguez-Lara, “Exact dynamics of finite Glauber-Fock photonic lattices,” Phys. Rev. A **84**, 053845 (2011) [CrossRef] .

*i∂*

_{t}ℰ⃗^{(±)}=

*H*

^{(±)}

*ℰ⃗*

^{(±)}leading to

*ℰ⃗*

^{(±)}(

*t*) =

*e*

^{−}

^{iH}^{(±)t}

*ℰ⃗*

^{(±)}(0), as long as the functions

*d*

^{(±)}(

*j*) and

*f*(

*j*) are time independent. In short, the dispersion relation of the truncated parity optical lattices of size

*N*are given by the roots of the characteristic polynomial

*g*

_{+}= 0, its truncation depends heavily on the initial state of the field thanks to the fact that the Hamiltonian conserves the number of excitations. In the nonlinear Rabi case the truncation depends heavily on both the initial state and the value of the coupling parameter

*g*

_{+}, it increases rapidly with the value of

*g*

_{+}. For experimental realizations this just means that the lattice must be large enough to keep the propagated classical field far from the last segment of waveguides.

51. S. Longhi, “Photonic analog of zitterbewegung in binary waveguide arrays,” Opt. Lett. **35**, 235–237 (2010) [CrossRef] [PubMed] .

53. B. M. Rodríguez-Lara and H. Moya-Cessa, “Photon transport in binary photonic lattices,” Phys. Scr. **87**, 038116 (2013) [CrossRef] .

*n*even the input at the positive and negative parity lattices are

*ϕ*given by the phase difference between

*c*and

_{e}*c*and the equivalent for

_{g}*n*odd. For coherent states, say the simplest case, This state is simply obtained by impinging a field on the 0th waveguide of a Glauber-Fock lattice [39

39. B. M. Rodríguez-Lara, “Exact dynamics of finite Glauber-Fock photonic lattices,” Phys. Rev. A **84**, 053845 (2011) [CrossRef] .

54. R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. **107**, 103601 (2011) [CrossRef] [PubMed] .

*α*

_{±}〉 = |

*α*〉±|−

*α*〉. The Glauber-Fock lattice can be used to simulate input corresponding to both coherent and displaced number states.

## 4. An example: The Buck-Sukumar model

*g*= 0.1

*ω*; i.e. we use a photonic lattice described by the differential sets (14–17) with parameter values

_{f}*ω*

_{0}=

*ω*,

_{f}*g*

_{−}= 0.1

*ω*, and

_{f}*g*

_{+}= 0. The numerical propagation considers a photonic lattice composed by three hundred coupled waveguides. We consider the initial state |

*ψ*(0)〉 = |

*α*

_{+},

*g*〉 with parameter values

*α*= 5 in Fig. 1. This is a separable state with positive parity; i.e. just the positive parity photonic lattice is needed to classically simulate its evolution. The intensity of the light field is shown in Fig. 1(a). The time evolution of the mean value for the photon number, atomic excitation energy and von Neumann entropy are shown in Figs. 1(b)–1(d). Figure 1(e) shows the time evolution of the fidelity, note how the evolution of this separable initial state returns periodically to its original state. In Fig. 2, we show the classical simulation of an entangled initial state |

*ψ*(0)〉 = |

*α*

_{+},

*g*〉 + |

*α*

_{−},

*e*〉 with identical parameter values as those in Fig. 1. Again, the state has positive parity and just the positive parity photonic lattice is needed to simulate the quantum system. The evolution of this entangled initial state also returns to its original state as witnessed by the fidelity in Fig. 2(e).

*ω*

_{0}=

*ω*and

_{f}*g*

_{−}=

*g*

_{+}= 2

*ω*. Figure 3 shows the numerical results for the propagation of a light field that simulates the initial state |

_{f}*ψ*(0)〉 = |0,

*e*〉 which has negative parity and corresponds to impinging the first waveguide of the negative parity photonic lattice. Again, the intensity of the light field is shown in Fig. 3(a). The time evolution of the mean value for the photon number, atomic excitation energy and von Neumann entropy are shown in Figs. 3(b)–3(d). The numerical propagation considers a photonic lattice of size two thousand and the probability of finding light at the last waveguide has a maximum value of 7 × 10

^{−4}within the parameter range considered here. Note that this is a thought experiment at the time because current technology can produce a couple hundred coupled waveguides at most.

## 5. Conclusion

## Acknowledgments

## References and links

1. | E. C. Kemble, |

2. | R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. |

3. | N. J. Cerf, C. Adami, and P. G. Kwiat, “Optical simulation of quantum logic,” Phys. Rev. A |

4. | S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical Zeno effect,” Phys. Rev. Lett. |

5. | H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett. |

6. | G. Della Valle, M. Ornigotti, T. T. Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett. |

7. | Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett. |

8. | F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tunnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett. |

9. | S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev. |

10. | Y. Lahini, Y. Bromberg, D. N. Christodoulides, and Y. Silberberg, “Quantum correlations in two-particle Anderson localization,” Phys. Rev. Lett. |

11. | F. Dreisow, M. Heinrich, R. Keil, A. Tunnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic zitterbewegung in photonic lattices,” Phys. Rev. Lett. |

12. | S. Longhi, “Klein tunneling in binary photonic superlattices,” Phys. Rev. B |

13. | S. Longhi, “Photonic Bloch oscillations of correlated particles,” Opt. Lett. |

14. | S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B |

15. | S. Longhi, “Many-body dynamic localization of strongly correlated electrons in ac-driven hubbard lattices,” J. Phys.: Condens. Matter |

16. | S. Longhi and G. Della Valle, “Photonic realization of |

17. | I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation an localization in modulated photonic lattices and waveguides,” Phys. Rep |

18. | A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B.M. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A |

19. | S. Longhi, “Jaynes-Cummings photonic superlattices,” Opt. Lett. |

20. | A. Crespi, S. Longhi, and R. Osellame, “Photonic realization of the quantum Rabi model,” Phys. Rev. Lett. |

21. | R. H. Dicke, “Coherence in spontaneaous radiation processes,” Phys. Rev. |

22. | E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE |

23. | J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneuous collapse and revival in a simple quantum model,” Phys. Rev. Lett. |

24. | B. Buck and C. V. Sukumar, “Exactly soluble model of atom–phonon coupling showing periodic decay and revival,” Phys. Lett. |

25. | E. A. Kochetov, “Exactly solvable non-linear generalisations of the Jaynes-Cummings model,” J. Phys. A: Math. Gen. |

26. | R. R. Schlicher, “Jaynes-Cummings model with atomic motion,” Opt. Commun. |

27. | S. J. D. Phoenix and P. L. Knight, “Periodicity, phase, and entropy in models of two-photon resonance,” J. Opt. Soc. Am. B |

28. | G. Benivegna, A. Messina, and A. Napoli, “Canonical dressing in nonlinear Jaynes-Cummings models,” Phys. Lett. A |

29. | W. Vogel and R. L. de Matos Filho, “Nonlinear Jaynes-Cummings dynamics of a trapped ion,” Phys. Rev. A |

30. | R. L. de Matos Filho and W. Vogel, “Nonlinear coherent states,” Phys. Rev. A |

31. | X. Yang, Y. Wu, and Y. Li, “Unified and standarized procedure to solve various nonlinear Jaynes-Cummings models,” Phys. Rev. A |

32. | B. M. Rodríguez-Lara, A. Z. Cárdenas, F. Soto-Eguibar, and H. M. Moya-Cessa, “A photonic crystal realization of a phase driven two-level atom,” Opt. Commun. |

33. | A. Joshi, “Nonlinear dynamical evolution of the driven two-photon Jaynes-Cummings model,” Phys. Rev. A |

34. | M. Abdel-Aty, S. Furuichi, and A.-S. F. Obada, “Entanglement degree of a nonlinear multiphoton Jaynes-Cummings model,” J. Opt. B: Quantum Semiclass. Opt. |

35. | S. Cordero and J. Récamier, “Selective transition and complete revivals of a single two-level atom in the Jaynes-Cummings Hamiltonian with an additional Kerr medium,” J. Phys. B: At. Mol. Opt. Phys. |

36. | O. de los Santos-Sanchez and J. Récamier, “The f-deformed Jaynes-Cummings model and its nonlinear coherent states,” J. Phys. B: At. Mol. Opt. Phys. |

37. | M. H. Naderi, M. Soltanolkotabi, and R. Roknizadeh, “A theoretical scheme for generation of nonlinear coherent states in a micromaser under intensity-dependent Jaynes-Cummings model,” Eur. Phys. J. D |

38. | A. Perez-Leija, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Glauber-Fock photonic lattices,” Opt. Lett. |

39. | B. M. Rodríguez-Lara, “Exact dynamics of finite Glauber-Fock photonic lattices,” Phys. Rev. A |

40. | A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the correlation and anticorrelation behavior of path-entangled photons in Glauber-Fock lattices,” Phys. Rev. A |

41. | M. D. Crisp, “Ed Jaynes’ steak dinner problem II,” in |

42. | R. L. de Matos Filho and W. Vogel, “Even and Odd Coherent States of the Motion of a Trapped Ion,” Phys. Rev. Lett. |

43. | H. Moya-Cessa and P. Tombesi, “Filtering number states of the vibrational motion of an ion,” Phys. Rev. A |

44. | H. Moya-Cessa, F. Soto-Eguibar, J. M. Vargas-Martínez, R. Juárez-Amaro, and A. Zúñiga Segundo, “Ion-laser interactions: The most complete solution,” Physics Reports |

45. | T. Esslinger, “Fermi-Hubbard Physics with atoms in an optical lattice,” Annu. Rev. Condens. Matter Phys. |

46. | E. A. Tur, “Jaynes-Cummings model: Solutions without rotating wave approximation,” Opt. Spectrosc. |

47. | E. A. Tur, “Energy spectrum of the Hamiltonian of the Jaynes-Cummings model without rotating-wave approximation,” Opt. Spectrosc. |

48. | J. Casanova, G. Romero, I. Lizuain, J. J. García-Ripoll, and E. Solano, “Deep strong coupling regime of the Jaynes-Cummings model,” Phys. Rev. Lett. |

49. | B. M. Rodríguez-Lara and H. M. Moya-Cessa, “Exact solution of generalized Dicke models via Susskind-Glogower operators,” J. Phys. A: Math. Theor. |

50. | J. W. Chan, T. Huser, S. Risbud, and D. M. Krol, “Structural changes in fused silica after exposure to focused femtosecond laser pulses,” Opt. Lett. |

51. | S. Longhi, “Photonic analog of zitterbewegung in binary waveguide arrays,” Opt. Lett. |

52. | C. Thompson, G. Vemuri, and G. S. Agarwal, “Quantum physics inspired optical effects in tight-binding lattices: Phase-controlled photonic transport,” Phys. Rev. B |

53. | B. M. Rodríguez-Lara and H. Moya-Cessa, “Photon transport in binary photonic lattices,” Phys. Scr. |

54. | R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical analogue of displaced Fock states and quantum correlations in Glauber-Fock photonic lattices,” Phys. Rev. Lett. |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.5580) Quantum optics : Quantum electrodynamics

(350.5500) Other areas of optics : Propagation

(310.2785) Thin films : Guided wave applications

(230.4555) Optical devices : Coupled resonators

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: February 14, 2013

Revised Manuscript: March 23, 2013

Manuscript Accepted: April 29, 2013

Published: May 17, 2013

**Citation**

B. M. Rodríguez-Lara, Francisco Soto-Eguibar, Alejandro Zárate Cárdenas, and H. M. Moya-Cessa, "A classical simulation of nonlinear Jaynes–Cummings and Rabi models in photonic lattices," Opt. Express **21**, 12888-12898 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12888

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### References

- E. C. Kemble, The Fundamental Principles of Quantum Mechanics with Elementary Applications (Dover, 1958).
- R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys.28, 361–374 (1998). [CrossRef]
- N. J. Cerf, C. Adami, and P. G. Kwiat, “Optical simulation of quantum logic,” Phys. Rev. A57, R1477–R1480 (1998). [CrossRef]
- S. Longhi, “Nonexponential decay via tunneling in tight-binding lattices and the optical Zeno effect,” Phys. Rev. Lett.97, 110402 (2006). [CrossRef] [PubMed]
- H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, “Realization of quantum walks with negligible decoherence in waveguide lattices,” Phys. Rev. Lett.100, 170506 (2008). [CrossRef] [PubMed]
- G. Della Valle, M. Ornigotti, T. T. Fernandez, P. Laporta, S. Longhi, A. Coppa, and V. Foglietti, “Adiabatic light transfer via dressed states in optical waveguide arrays,” Appl. Phys. Lett.92, 011106 (2008). [CrossRef]
- Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silberberg, “Quantum and classical correlations in waveguide lattices,” Phys. Rev. Lett.102, 253904 (2009). [CrossRef] [PubMed]
- F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tunnermann, and S. Longhi, “Bloch-Zener oscillations in binary superlattices,” Phys. Rev. Lett.102, 076802 (2009). [CrossRef] [PubMed]
- S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photon. Rev.3, 243–261 (2009). [CrossRef]
- Y. Lahini, Y. Bromberg, D. N. Christodoulides, and Y. Silberberg, “Quantum correlations in two-particle Anderson localization,” Phys. Rev. Lett.105, 163905 (2010). [CrossRef]
- F. Dreisow, M. Heinrich, R. Keil, A. Tunnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic zitterbewegung in photonic lattices,” Phys. Rev. Lett.105, 143902 (2010). [CrossRef]
- S. Longhi, “Klein tunneling in binary photonic superlattices,” Phys. Rev. B81, 075102 (2010). [CrossRef]
- S. Longhi, “Photonic Bloch oscillations of correlated particles,” Opt. Lett.36, 3248–3250 (2011). [CrossRef] [PubMed]
- S. Longhi, “Classical simulation of relativistic quantum mechanics in periodic optical structures,” Appl. Phys. B104, 453–468 (2011). [CrossRef]
- S. Longhi, “Many-body dynamic localization of strongly correlated electrons in ac-driven hubbard lattices,” J. Phys.: Condens. Matter24, 435601 (2012). [CrossRef]
- S. Longhi and G. Della Valle, “Photonic realization of 𝔓 𝔗-symmetric quantum field theories,” Phys. Rev. A85, 012112 (2012). [CrossRef]
- I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation an localization in modulated photonic lattices and waveguides,” Phys. Rep518, 1–79 (2012). [CrossRef]
- A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L.-C. Kwek, B.M. Rodríguez-Lara, A. Szameit, and D. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A87, 012309 (2013). [CrossRef]
- S. Longhi, “Jaynes-Cummings photonic superlattices,” Opt. Lett.36, 3407–3409 (2011). [CrossRef] [PubMed]
- A. Crespi, S. Longhi, and R. Osellame, “Photonic realization of the quantum Rabi model,” Phys. Rev. Lett.108, 163601 (2012). [CrossRef] [PubMed]
- R. H. Dicke, “Coherence in spontaneaous radiation processes,” Phys. Rev.93, 99–110 (1954). [CrossRef]
- E. T. Jaynes and F. W. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE51, 89–109 (1963). [CrossRef]
- J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, “Periodic spontaneuous collapse and revival in a simple quantum model,” Phys. Rev. Lett.44, 1323–1326 (1980). [CrossRef]
- B. Buck and C. V. Sukumar, “Exactly soluble model of atom–phonon coupling showing periodic decay and revival,” Phys. Lett.81, 132–135 (1981). [CrossRef]
- E. A. Kochetov, “Exactly solvable non-linear generalisations of the Jaynes-Cummings model,” J. Phys. A: Math. Gen.20, 2433–2442 (1987). [CrossRef]
- R. R. Schlicher, “Jaynes-Cummings model with atomic motion,” Opt. Commun.70, 97–102 (1989). [CrossRef]
- S. J. D. Phoenix and P. L. Knight, “Periodicity, phase, and entropy in models of two-photon resonance,” J. Opt. Soc. Am. B7, 116–124 (1990). [CrossRef]
- G. Benivegna, A. Messina, and A. Napoli, “Canonical dressing in nonlinear Jaynes-Cummings models,” Phys. Lett. A194, 353–357 (1994). [CrossRef]
- W. Vogel and R. L. de Matos Filho, “Nonlinear Jaynes-Cummings dynamics of a trapped ion,” Phys. Rev. A52, 4214–4217 (1995). [CrossRef] [PubMed]
- R. L. de Matos Filho and W. Vogel, “Nonlinear coherent states,” Phys. Rev. A54, 4560–4563 (1996). [CrossRef] [PubMed]
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