Physics:Jaynes–Cummings–Hubbard model

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Short description: Model in quantum optics
Tunnelling of photons between coupled cavities. The [math]\displaystyle{ \kappa }[/math] is the tunnelling rate of photons.
Illustration of the Jaynes–Cummings model. In the circle, photon emission and absorption are shown.

The Jaynes–Cummings–Hubbard (JCH) model is a many-body quantum system modeling the quantum phase transition of light. As the name suggests, the Jaynes–Cummings–Hubbard model is a variant on the Jaynes–Cummings model; a one-dimensional JCH model consists of a chain of N coupled single-mode cavities, each with a two-level atom. Unlike in the competing Bose–Hubbard model, Jaynes–Cummings–Hubbard dynamics depend on photonic and atomic degrees of freedom and hence require strong-coupling theory for treatment.[1] One method for realizing an experimental model of the system uses circularly-linked superconducting qubits.[2]

History

The JCH model was originally proposed in June 2006 in the context of Mott transitions for strongly interacting photons in coupled cavity arrays.[3] A different interaction scheme was synchronically suggested, wherein four level atoms interacted with external fields, leading to polaritons with strongly correlated dynamics.[4]

Properties

Using mean-field theory to predict the phase diagram of the JCH model, the JCH model should exhibit Mott insulator and superfluid phases.[5]

Hamiltonian

The Hamiltonian of the JCH model is ([math]\displaystyle{ \hbar=1 }[/math]):

[math]\displaystyle{ H = \sum_{n=1}^{N}\omega_c a_{n}^{\dagger}a_{n} +\sum_{n=1}^{N}\omega_a \sigma_n^+\sigma_n^- + \kappa \sum_{n=1}^{N} \left(a_{n+1}^{\dagger}a_{n}+a_{n}^{\dagger}a_{n+1}\right) + \eta \sum_{n=1}^{N} \left(a_{n}\sigma_{n}^{+} + a_{n}^{\dagger}\sigma_{n}^{-}\right) }[/math]

where [math]\displaystyle{ \sigma_{n}^{\pm} }[/math] are Pauli operators for the two-level atom at the n-th cavity. The [math]\displaystyle{ \kappa }[/math] is the tunneling rate between neighboring cavities, and [math]\displaystyle{ \eta }[/math] is the vacuum Rabi frequency which characterizes to the photon-atom interaction strength. The cavity frequency is [math]\displaystyle{ \omega_c }[/math] and atomic transition frequency is [math]\displaystyle{ \omega_a }[/math]. The cavities are treated as periodic, so that the cavity labelled by n = N+1 corresponds to the cavity n = 1.[3] Note that the model exhibits quantum tunneling; this process is similar to the Josephson effect.[6][7]

Defining the photonic and atomic excitation number operators as [math]\displaystyle{ \hat{N}_c \equiv \sum_{n=1}^{N}a_n^{\dagger}a_n }[/math] and [math]\displaystyle{ \hat{N}_a \equiv \sum_{n=1}^{N} \sigma_{n}^{+}\sigma_{n}^{-} }[/math], the total number of excitations is a conserved quantity, i.e., [math]\displaystyle{ \lbrack H,\hat{N}_c+\hat{N}_a\rbrack=0 }[/math].[citation needed]

Two-polariton bound states

The JCH Hamiltonian supports two-polariton bound states when the photon-atom interaction is sufficiently strong. In particular, the two polaritons associated with the bound states exhibit a strong correlation such that they stay close to each other in position space.[8] This process is similar to the formation of a bound pair of repulsive bosonic atoms in an optical lattice.[9][10][11]

Further reading

  • D. F. Walls and G. J. Milburn (1995), Quantum Optics, Springer-Verlag.

References

  1. ↑ Schmidt, S.; Blatter, G. (Aug 2009). "Strong Coupling Theory for the Jaynes-Cummings-Hubbard Model". Phys. Rev. Lett. 103 (8): 086403. doi:10.1103/PhysRevLett.103.086403. PMID 19792743. Bibcode: 2009PhRvL.103h6403S. http://link.aps.org/doi/10.1103/PhysRevLett.103.086403. 
  2. ↑ A. Nunnenkamp; Jens Koch; S. M. Girvin (2011). "Synthetic gauge fields and homodyne transmission in Jaynes-Cummings lattices". New Journal of Physics 13 (9): 095008. doi:10.1088/1367-2630/13/9/095008. Bibcode: 2011NJPh...13i5008N. 
  3. ↑ 3.0 3.1 D. G. Angelakis; M. F. Santos; S. Bose (2007). "Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays". Physical Review A 76 (3): 1805(R). doi:10.1103/physreva.76.031805. Bibcode: 2007PhRvA..76c1805A. 
  4. ↑ M. J. Hartmann, F. G. S. L. Brandão and M. B. Plenio (2006). "Strongly interacting polaritons in coupled arrays of cavities". Nature Physics 2 (12): 849–855. doi:10.1038/nphys462. Bibcode: 2006NatPh...2..849H. 
  5. ↑ A. D. Greentree; C. Tahan; J. H. Cole; L. C. L. Hollenberg (2006). "Quantum phase transitions of light". Nature Physics 2 (12): 856–861. doi:10.1038/nphys466. Bibcode: 2006NatPh...2..856G. 
  6. ↑ B. W. Petley (1971). An Introduction to the Josephson Effects. London: Mills and Boon. 
  7. ↑ Antonio Barone; Gianfranco Paternó (1982). Physics and Applications of the Josephson Effect. New York City: John Wiley & Sons. 
  8. ↑ Max T. C. Wong; C. K. Law (May 2011). "Two-polariton bound states in the Jaynes-Cummings-Hubbard model". Phys. Rev. A (American Physical Society) 83 (5): 055802. doi:10.1103/PhysRevA.83.055802. Bibcode: 2011PhRvA..83e5802W. http://link.aps.org/doi/10.1103/PhysRevA.83.055802. 
  9. ↑ K. Winkler; G. Thalhammer; F. Lang; R. Grimm; J. H. Denschlag; A. J. Daley; A. Kantian; H. P. Buchler et al. (2006). "Repulsively bound atom pairs in an optical lattice". Nature 441 (7095): 853–856. doi:10.1038/nature04918. PMID 16778884. Bibcode: 2006Natur.441..853W. 
  10. ↑ Javanainen, Juha and Odong, Otim and Sanders, Jerome C. (Apr 2010). "Dimer of two bosons in a one-dimensional optical lattice". Phys. Rev. A 81 (4): 043609. doi:10.1103/PhysRevA.81.043609. Bibcode: 2010PhRvA..81d3609J. http://link.aps.org/doi/10.1103/PhysRevA.81.043609. 
  11. ↑ M. Valiente; D. Petrosyan (2008). "Two-particle states in the Hubbard model". J. Phys. B: At. Mol. Opt. Phys. 41 (16): 161002. doi:10.1088/0953-4075/41/16/161002. Bibcode: 2008JPhB...41p1002V. 



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