# Jaynes–Cummings–Hubbard model

(Redirected from Jaynes-Cummings-Hubbard model)

The Jaynes-Cummings-Hubbard (JCH) model is a many-body quantum system modeling the quantum phase transition of light. As the name suggests, the Jayne-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, Jayne-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]

Tunnelling of photons between coupled cavities. The ${\displaystyle \kappa }$ is the tunnelling rate of photons.
Illustration of the Jaynes-Cummings model. In the circle, photon emission and absorption are shown.

## 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 (${\displaystyle \hbar =1}$):

${\displaystyle H=\sum _{n=1}^{N}\omega _{c}a_{n}^{\dagger }a_{n}+\sum _{n=1}^{N}\omega _{a}\sigma _{n}^{+}\sigma _{n}^{-}+\kappa \sum _{n=1}^{N}\left(a_{n+1}^{\dagger }a_{n}+a_{n}^{\dagger }a_{n+1}\right)+\eta \sum _{n=1}^{N}\left(a_{n}\sigma _{n}^{+}+a_{n}^{\dagger }\sigma _{n}^{-}\right)}$

where ${\displaystyle \sigma _{n}^{\pm }}$ are Pauli operators for the two-level atom at the n-th cavity. The ${\displaystyle \kappa }$ is the tunneling rate between neighboring cavities, and ${\displaystyle \eta }$ is the vacuum Rabi frequency which characterizes to the photon-atom interaction strength. The cavity frequency is ${\displaystyle \omega _{c}}$ and atomic transition frequency is ${\displaystyle \omega _{a}}$. 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 is process is similar to the Josephson effect.[6][7]

Defining the photonic and atomic excitation number operators as ${\displaystyle {\hat {N}}_{c}\equiv \sum _{n=1}^{N}a_{n}^{\dagger }a_{n}}$ and ${\displaystyle {\hat {N}}_{a}\equiv \sum _{n=1}^{N}\sigma _{n}^{+}\sigma _{n}^{-}}$, the total number of excitations a conserved quantity, i.e., ${\displaystyle \lbrack H,{\hat {N}}_{c}+{\hat {N}}_{a}\rbrack =0}$.[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]

• 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. American Physical Society. 103 (8): 086403. Bibcode:2009PhRvL.103h6403S. arXiv:. 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: 095008. Bibcode:2011NJPh...13i5008N. arXiv:. doi:10.1088/1367-2630/13/9/095008.
3. ^ a b 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 (03): 1805(R). Bibcode:2007PhRvA..76c1805A. arXiv:. doi:10.1103/physreva.76.031805.
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: 849. Bibcode:2006NatPh...2..849H. arXiv:. doi:10.1038/nphys462.
5. ^ A. D. Greentree; C. Tahan; J. H. Cole; L. C. L. Hollenberg (2006). "Quantum phase transitions of light". Nature Physics. 2: 856. Bibcode:2006NatPh...2..856G. arXiv:. doi:10.1038/nphys466.
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: Wiley.
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. Bibcode:2011PhRvA..83e5802W. arXiv:. 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; P. Zoller (2006). "Repulsively bound atom pairs in an optical lattice". Nature. 441: 853. Bibcode:2006Natur.441..853W. PMID 16778884. arXiv:. doi:10.1038/nature04918.
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. American Physical Society. 81 (4): 043609. Bibcode:2010PhRvA..81d3609J. arXiv:. 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: 161002. Bibcode:2008JPhB...41p1002V. doi:10.1088/0953-4075/41/16/161002.