Jaynes–Cummings–Hubbard model

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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. arXiv:. Bibcode:2009PhRvL.103h6403S. 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. arXiv:. Bibcode:2011NJPh...13i5008N. 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). arXiv:. Bibcode:2007PhRvA..76c1805A. 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. arXiv:. Bibcode:2006NatPh...2..849H. 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. arXiv:. Bibcode:2006NatPh...2..856G. 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. arXiv:. Bibcode:2011PhRvA..83e5802W. 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. arXiv:. Bibcode:2006Natur.441..853W. 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. arXiv:. Bibcode:2010PhRvA..81d3609J. 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.