## Photonic analogue of Josephson effect in a dual-species optical-lattice cavity |

Optics Express, Vol. 18, Issue 14, pp. 14586-14597 (2010)

http://dx.doi.org/10.1364/OE.18.014586

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### Abstract

We extend the idea of quantum phase transitions of light in the photonic Bose-Hubbard model with interactions to two atomic species by a self-consistent mean field theory. The excitation of two-level atoms interacting with a coherent photon field is analyzed with a finite temperature dependence of the order parameters. Four ground states of the system are found, including an isolated Mott-insulator phase and three different superfluid phases. Like two weakly coupled superconductors, our proposed dual-species lattice system shows a photonic analogue of Josephson effect. i.e., the crossovers between two superfluid states. The dynamics of the proposed two species model provides a promising quantum simulator for possible quantum information processes.

© 2010 Optical Society of America

## 1. Introduction

4. M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B **40**, 546–570 (1989). [CrossRef]

5. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. **2**, 856–862 (2006). [CrossRef]

8. M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Effective Spin Systems in Coupled Microcavities,” Phys. Rev. Lett. **99**, 160501 (2007). [CrossRef] [PubMed]

9. M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “A polaritonic two-component Bose-Hubbard model,” N. J. Phys. **10**, 033011 (2008). [CrossRef]

10. D. Rossini and R. Fazio, “Mott-Insulating and Glassy Phases of Polaritons in 1D Arrays of Coupled Cavities,” Phys. Rev. Lett. **99**, 186401 (2007). [CrossRef] [PubMed]

13. J. Quach, M. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A **80**, 063838 (2009). [CrossRef]

14. P. Wurtz, T. Langen, T. Gericke, A. Koglbauer, and H. Ott, “Experimental Demonstration of Single-Site Addressability in a Two-Dimensional Optical Lattice,” Phys. Rev. Lett. **103**, 080404 (2009). [CrossRef] [PubMed]

15. E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, “Phase diagram of two-component bosons on an optical lattice,” N. J. Phys. **5**, 11301–11319 (2003). [CrossRef]

16. T. Mishra, B. K. Sahoo, and R. V. Pai, “Phase-separated charge-density-wave phase in the two-species extended Bose-Hubbard model,” Phys. Rev. A **78**, 013632 (2008). [CrossRef]

## 2. Model Hamiltonian

*A*and

*B*, as illustrated in Fig. 1. The array is made from high-Q electromagnetic cavities. Each cavity contains a single TLA of the type

*A*or

*B*, which is spaced at intervals. The atoms are assumed to interact strongly with photons in the photon-blockade regime so that a coherent composite polaritonic (photon and atom) system is formed in the presence of radiation fields [5

5. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. **2**, 856–862 (2006). [CrossRef]

11. S.-C. Lei and R.-K. Lee, “Quantum phase transitions of light in the Dicke-Bose-Hubbard model,” Phys. Rev. A **77**, 033827 (2008). [CrossRef]

17. 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]

*i*-th unit cell (two cavities numbered as 2

*j*and 2

*j*+ 1 for

*A-*and

*B*-type atoms, respectively),

*ε*

_{A(B),j}is the transition energy for the TLA in

*j*-th cavity and

*ω*is the radiation field frequency which we have assumed to be the same at all cavities. The atom-photon coupling,

*g*

_{A(B),j}, is assumed to be real here.

*â*

^{†}

_{j}and

*â*

_{j}are the raising and lowering operators for photons, respectively. The excitation of the A (or B)-type TLA from the low energy state to the high energy state is described by the operator

σ ^

^{†}

_{A(B),j}, while the operator

σ ^

^{−}

_{A(B),j}describes the reverse process. By including the hopping process for photons, the total Hamiltonian describing our proposed system for a configuration of 2

*N*cavities is

*κ*is the hopping matrix element and

_{i,j}*i,j*are the cavity indices. We also introduce a chemical potential term

*μ*for photons which control the overall strength of the photon field, and

*n̂*=

_{i}*â*

^{†}

_{i}

*â*is the photon number operator. For simplicity, in this configuration, each cavity has four nearest inter-species neighbors (different atomic type) with the hopping coefficient

_{i}*κ*=

_{A↔B}*κ*≡

_{B↔A}*κ*, and four next-nearest intra-species neighbors (the same atomic type) with the hopping coefficient

*κ*=

_{A↔A}*κ*≡

_{B↔B}*κ*′.

*â*

^{†}

σ ^

^{±}

_{A(B)}. It is convenient to introduce two superfluid order parameters

*ψ*

_{A(B)}for photons and two TLA order parameters

*J*

_{A(B)}for the atomic species

*A*and

*B*, respectively, where

5. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. **2**, 856–862 (2006). [CrossRef]

*ψ*

_{A(B)}≠ 0, and in the MI phase otherwise,

*ψ*

_{A(B)}= 0. The parameter

*J*

_{A(B)}can be viewed as an order parameter for the atomic coherent states. The physical meaning for

*J*

_{A(B)}as a coherent state order parameter for the atoms lies on the coherent superposition of the ground and excited state of the atoms.With these order parameters, the possibility to have a QPT at low temperature would be demonstrated in the right parameter regimes. In the following we assume that both

*ψ*

_{A(B)}and

*J*

_{A(B)}are real numbers and spatially independent, i.e., there is no spontaneous symmetry-breakings in the translation and rotation for the infinite system considered here. The assumption that

*ψ*

_{A(B)}and

*J*

_{A(B)}are real numbers will be justified later by the self-consistent mean-field solution.

### 2.1. Mean-field solution for H, with the atomic operators only

*H*and

^{a}*H*, where only the atomic and photonic operators are considered separately. For one atom (

^{p}*A*or

*B*) inside each cavity, the mean-field Hamiltonian for atomic operators is

*J*

_{A(B)}at zero temperature can be derived as

*β*= 1/

*k*and we set the Boltzmann constant

_{B}T*k*= 1 for the sake of convenience. For a finite temperature,

_{B}*T*, the TLA order parameter

*J*

_{A(B)}becomes,

*J*

_{A(B)}is a real number and consistent with our initial assumption. From Eq. (16), it can be clearly seen that only at zero temperature, the TLA order parameter

*J*

_{A(B)}has a maximum value. The atomic order parameter

*J*

_{A(B)}is a linear function of the photonic order parameter

*ψ*

_{A(B)}, with the coefficient defined by the coupling strength

*g*

_{A(B)}and the atomic eigen-energy

*E*

_{A(B)}. Whenever the atom-photon coupling strength

*g*

_{A(B)}is zero, the atomic order parameter

*J*

_{A(B)}goes to zero as there is no atom-photon interaction. The larger the coupling strength, the easier to have a non-zero atomic order parameter.

### 2.2. Mean-field solution for H, with the photonic operators only

*â*

_{A(B),k}, the combined photonic mean-field Hamiltonian of two sites in a unit cell becomes

*k⃗*=

*k*+

_{x}x̂*k*.

_{y}ŷ*H*in Eq. (21) can be diagonalized in the following form

^{p}*â*and

_{sym,k}*â*as,

_{asym,k}_{sym(asym)}(

*k⃗*) ≥ 0. Equivalently, we have the condition (

*ω*−

*μ*) − 4

*κ*′ > 4

*κ*required for photon fields. The assumption that

*J*

_{A(B)}is spatially independent implies that the atoms couple only to the

*k⃗*= 0 photon mode. In other words, we only search for the spatially homogeneous solutions in our mean-field theory. From the coupling of photon field to atoms, one may imply that the

*k⃗*= 0 mode of photon develops a ground state, with the expectation values

*â*

_{sym(asym)}〉 is a linear function of the atomic order parameter

*J*

_{A(B)}, with the coefficient defined by the coupling strength

*g*

_{A(B)}and the atomic eigen-energy Ω(

*k⃗*= 0). Whenever the atom-photon coupling strength

*g*

_{A(B)}is zero, the photonic order parameter goes to zero as there is no atom-photon interactions.

*ψ*

_{A(B)}and the atomic coherent state order parameter for the TLA

*J*

_{A(B)}. Based on these results, the proposed two-site Hamiltonian will be used to study the QPT of light analytically and numerically in the following.

## 3. Analysis of the mean field equations

*F*can be written as

_{s}*F*=

_{s}*F*+

_{a}*F*−

_{p}*E*, where

_{m}*F*,

_{a}*F*, and

_{p}*E*are the free energy densities associated with the atomic-only Hamiltonian in Eq. (8), photonic-only Hamiltonian in Eq. (24), and the correction term for the double counting in our mean-field decomposition in Eq. (3), respectively, i.e.,

_{m}*V*is the volume of the system.

### 3.1. Solutions for the special case of identical atoms

*ψ*

_{A(B)}, in the following matrix form,

*A*and

*B*are identical. For this case

*ψ*=

_{A}*ψ*=

_{B}*ψ*,

*E*=

_{A}*E*=

_{B}*E*,

*ε*=

_{A}*ε*=

_{B}*ε*, and

*g*=

_{A}*g*=

_{B}*g*, we can derive the photonic superfluid order parameter explicitly,

*ψ*≠ 0, to exist at zero temperature one can find

*g*, our composite system is in the Mott-insulation phase

*ψ*= 0 or in the superfluid phase

*ψ*≠ 0 depending the coupling strength is smaller or larger than the critical value,

**2**, 856–862 (2006). [CrossRef]

7. D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A **76**, 031805 (2007). [CrossRef]

11. S.-C. Lei and R.-K. Lee, “Quantum phase transitions of light in the Dicke-Bose-Hubbard model,” Phys. Rev. A **77**, 033827 (2008). [CrossRef]

13. J. Quach, M. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A **80**, 063838 (2009). [CrossRef]

*κ*and the chemical potential

*μ*, in our approach we absorb these two effects into the parameter Ω

_{sym}(

*k⃗*= 0) in Eq. (27), i.e.,

*κ*′ =

*κ*, as expected, when the hopping term becomes larger, then Ω

_{sym}becomes smaller, and our system approaches the superfluid state. On the other hand, when the hopping coefficient is smaller, our system would be in the Mott-insulator state.

*T*given by

_{c}### 3.2. The phase diagram for non-identical atoms

*A*and

*B*are different, we consider the limit case without inter-atomic species hopping effects, i.e.,

*κ*= 0 (or Ω

^{−1}

_{−}= 0). In this case, at

*T*= 0 the atoms

*A*and

*B*couple separately to the radiation fields. Now each of the two species has a MI to SF phase transition at

*g*,

_{A}*g*), as the regions defined by the solid lines shown in Fig. 2. These four phases correspond to (1) both of the two-species atoms are in the MI state,

_{B}*B*-type atoms are in the SF state; and (4) all the atoms are in the SF state, i.e., the co-existence SF-AB state for

*κ*≠ 0, the four phase states mentioned above should be modified. To give a qualitative analysis on the possible four phase states, we assume Ω

^{−1}

_{−}to be small and perform a perturbative expansion on the limit case Ω

^{−1}

_{−}= 0 for the solutions of the mean-field Hamiltonian in Eq. (37) at zero temperature

*T*= 0. By expanding the superfluid order parameter

*ψ*

_{A(B)}=

*ψ*

^{(0)}

_{A(B)}+

*δψ*

_{A(B)}, to the zero-th order one can again obtain,

^{−1}

_{−},

*ψ*

^{(0)}

_{A}=

*ψ*

^{(0)}

_{B}= 0, both of the perturbed superfluid order parameters

*δψ*and

_{A}*δψ*are zero as expected. The MI phase of the system is not modified by the perturbation in Ω

_{B}^{−1}

_{−}. On the contrary, the properties of SF states are strongly modified. From the second line in Eq. (44), one can easily find that

*δψ*

_{A(B)}is non-zero as long as

*ψ*

^{(0)}

_{B(A)}≠ 0, which is independent of whether

*ψ*

^{(0)}

_{A(B)}is zero or not. In such a scenario, one of the atomic species in the superfluid phase can induce a non-zero superfluid order parameter on the other species of atoms, which is originally in the MI state. Unlike the case of only one atomic species, in our system both species of atoms develop nonzero order parameters with the superfluidity driven by the other type of atomic species.

18. B. D. Josephson, “The discovery of tunnelling supercurrents,” Rev. Mod. Phys. **46**, 251–254 (1974). [CrossRef]

*et al.*[19

19. D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. **5**, 281–284 (2009). [CrossRef]

*ε*in our system, that is the new input light resonates strongly with the B-type atoms, modifies sequentially the superfluidity of A-type atoms depending on the original state of the system. For an original SF-A phase, the superfluidity of A-type atoms is destroyed due to the decoupling from B-type atoms; while an original co-existence SF-AB phase is driven into a SF-B phase.

_{B}## 4. Numerical results and discussions

*g*and

_{A}*g*by fixing the value of transition energy

_{B}*ε*

_{A(B)}and photon dispersion Ω

_{+(−)}. Figure 2 demonstrates the phase diagram of our system in the parameter plane

*g*and

_{A}*g*at zero temperature. As conjectured by the perturbative analysis, we have the phase diagram for the QPT of light in our dual-species configuration, where the original well-defined boundaries for SF-A/B to SF-AB phases for

_{B}*κ*= 0 is now turned into crossovers.

*ψ*and

_{A}*ψ*as a function of the temperature,

_{B}*T*, with different values of

*g*and

_{A}*g*, respectively. As shown in Fig. 3(a), for the parameters

_{B}*g*= 0.1 and

_{A}*g*= 2.0, the system is in the SF-B phase at zero temperature. A clear finite temperature insulator to superfluid transition is found at the critical temperature

_{B}*T*≈ 5. Notice that

_{c}*ψ*>

_{B}*ψ*but the differences between

_{A}*ψ*and

_{A}*ψ*remains small throughout the whole temperature range in the SF-B phase, indicating the importance of Josephson coupling effect. On the other hand, for the co-existence SF-AB state at zero temperature, the two curves for

_{B}*ψ*and

_{A}*ψ*stay close to each other for the whole temperature range. The overall magnitudes of

_{B}*ψ*and

_{A}*ψ*are larger by a ratio of factor 2 when compared with the case with

_{B}*g*= 0.1 and

_{A}*g*= 2.0.

_{B}*ψ*

_{A(B)}in order to carefully examine the effect of Josephson-like coupling. All the parameters are kept the same to those studied before except for the hopping constants changed to

*κ*= 0.35 and

*κ*′ = 0.175. Comparing to the cases in Fig. 3, we find that the magnitudes of

*ψ*and

_{A}*ψ*become both smaller due to the reduced Josepshon coupling effect, and the difference between

_{B}*ψ*and

_{A}*ψ*becomes larger. Notice that the values of Ω

_{B}_{sym}and Ω

_{asym}are increased by decreasing

*κ*′ and

*κ*and the system is driven towards the MI phase. In fact we find that

*ψ*=

_{A}*ψ*= 0 and the system is already in the MI regime for

_{B}*κ*= 0.3 and

*κ*′ = 0.2 with our chosen set of parameters.

## 5. Conclusion

## Acknowledgement

## References and links

1. | S. Sachdev, |

2. | M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature |

3. | D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. |

4. | M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B |

5. | A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. |

6. | M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. |

7. | D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A |

8. | M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Effective Spin Systems in Coupled Microcavities,” Phys. Rev. Lett. |

9. | M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “A polaritonic two-component Bose-Hubbard model,” N. J. Phys. |

10. | D. Rossini and R. Fazio, “Mott-Insulating and Glassy Phases of Polaritons in 1D Arrays of Coupled Cavities,” Phys. Rev. Lett. |

11. | S.-C. Lei and R.-K. Lee, “Quantum phase transitions of light in the Dicke-Bose-Hubbard model,” Phys. Rev. A |

12. | T. Giamarchi, C. Regg, and O. Tchernyshyov, “Bose-Einstein condensation in magnetic insulators,” Nat. Phys. |

13. | J. Quach, M. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A |

14. | P. Wurtz, T. Langen, T. Gericke, A. Koglbauer, and H. Ott, “Experimental Demonstration of Single-Site Addressability in a Two-Dimensional Optical Lattice,” Phys. Rev. Lett. |

15. | E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, “Phase diagram of two-component bosons on an optical lattice,” N. J. Phys. |

16. | T. Mishra, B. K. Sahoo, and R. V. Pai, “Phase-separated charge-density-wave phase in the two-species extended Bose-Hubbard model,” Phys. Rev. A |

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

18. | B. D. Josephson, “The discovery of tunnelling supercurrents,” Rev. Mod. Phys. |

19. | D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. |

**OCIS Codes**

(020.5580) Atomic and molecular physics : Quantum electrodynamics

(270.0270) Quantum optics : Quantum optics

(270.1670) Quantum optics : Coherent optical effects

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: April 14, 2010

Revised Manuscript: June 6, 2010

Manuscript Accepted: June 18, 2010

Published: June 23, 2010

**Citation**

Soi-Chan Lei, Tai-Kai Ng, and Ray-Kuang Lee, "Photonic analogue of Josephson effect in a dual-species optical-lattice cavity," Opt. Express **18**, 14586-14597 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14586

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### References

- S. Sachdev, Quantum Phase Transitions, (Cambridge University Press, 1999).
- M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, “Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms,” Nature 415, 39–44 (2002). [CrossRef] [PubMed]
- D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, “Cold Bosonic Atoms in Optical Lattices,” Phys. Rev. Lett. 81, 3108–3111 (1998). [CrossRef]
- M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson localization and the superfluid-insulator transition,” Phys. Rev. B 40, 546–570 (1989). [CrossRef]
- A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–862 (2006). [CrossRef]
- M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006). [CrossRef]
- D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007). [CrossRef]
- M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “Effective Spin Systems in Coupled Microcavities,” Phys. Rev. Lett. 99, 160501 (2007). [CrossRef] [PubMed]
- M. J. Hartmann, F. G. S. L. Brandao, and M. B. Plenio, “A polaritonic two-component Bose-Hubbard model,” N. J. Phys. 10, 033011 (2008). [CrossRef]
- D. Rossini, and R. Fazio, “Mott-Insulating and Glassy Phases of Polaritons in 1D Arrays of Coupled Cavities,” Phys. Rev. Lett. 99, 186401 (2007). [CrossRef] [PubMed]
- S.-C. Lei, and R.-K. Lee, “Quantum phase transitions of light in the Dicke-Bose-Hubbard model,” Phys. Rev. A 77, 033827 (2008). [CrossRef]
- T. Giamarchi, C. Rügg, and O. Tchernyshyov, “Bose-Einstein condensation in magnetic insulators,” Nat. Phys. 4, 198–204 (2008). [CrossRef]
- J. Quach, M. Makin, C.-H. Su, A. D. Greentree, and L. C. L. Hollenberg, “Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems,” Phys. Rev. A 80, 063838 (2009). [CrossRef]
- P. Wurtz, T. Langen, T. Gericke, A. Koglbauer, and H. Ott, “Experimental Demonstration of Single-Site Addressability in a Two-Dimensional Optical Lattice,” Phys. Rev. Lett. 103, 080404 (2009). [CrossRef] [PubMed]
- E. Altman, W. Hofstetter, E. Demler, and M. D. Lukin, “Phase diagram of two-component bosons on an optical lattice,” N. J. Phys. 5, 11301–11319 (2003). [CrossRef]
- T. Mishra, B. K. Sahoo, and R. V. Pai, “Phase-separated charge-density-wave phase in the two-species extended Bose-Hubbard model,” Phys. Rev. A 78, 013632 (2008). [CrossRef]
- 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]
- B. D. Josephson, “The discovery of tunneling supercurrents,” Rev. Mod. Phys. 46, 251–254 (1974). [CrossRef]
- D. Gerace, H. E. Tureci, A. Imamoglu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009). [CrossRef]

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