Jaynes–Cummings–Hubbard model

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.

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 combination of Hubbard-type models with Jaynes-Cummings (atom-photon) interactions near the photon blockade ^[3][4]regime originally appeared in three, roughly simultaneous papers in 2006.^[5][6][7]

All three papers explored systems of interacting atom-cavity systems, and shared much of the essential underlying physics. Nevertheless, the term Jaynes–Cummings–Hubbard was not coined until 2008.^[8]

Properties

Using mean-field theory to predict the phase diagram of the JCH model, the JCH model should exhibit Mott insulator and superfluid phases.^[9]

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.^[5] Note that the model exhibits quantum tunneling; this process is similar to the Josephson effect.^[10][11]

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 is 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.^[12] This process is similar to the formation of a bound pair of repulsive bosonic atoms in an optical lattice.^[13][14][15]

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. arXiv:0905.3344. Bibcode:2009PhRvL.103h6403S. doi:10.1103/PhysRevLett.103.086403. PMID 19792743. S2CID 32092406.
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. arXiv:1105.1817. Bibcode:2011NJPh...13i5008N. doi:10.1088/1367-2630/13/9/095008. S2CID 118557639.
3. Imamoḡlu, A.; Schmidt, H.; Woods, G.; Deutsch, M. (25 August 1997). "Strongly Interacting Photons in a Nonlinear Cavity". Physical Review Letters. 79 (8): 1467–1470. doi:10.1103/PhysRevLett.79.1467.
4. Birnbaum, K. M.; Boca, A.; Miller, R.; Boozer, A. D.; Northup, T. E.; Kimble, H. J. (July 2005). "Photon blockade in an optical cavity with one trapped atom". Nature. 436 (7047): 87–90. doi:10.1038/nature03804. PMID 16001065.
5. 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). arXiv:quant-ph/0606159. Bibcode:2007PhRvA..76c1805A. doi:10.1103/physreva.76.031805. S2CID 44490741.
6. 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. arXiv:quant-ph/0606097. Bibcode:2006NatPh...2..849H. doi:10.1038/nphys462. S2CID 9122839.
7. A. D. Greentree; C. Tahan; J. H. Cole; L. C. L. Hollenberg (2006). "Quantum phase transitions of light". Nature Physics. 2 (12): 856–861. arXiv:cond-mat/0609050. Bibcode:2006NatPh...2..856G. doi:10.1038/nphys466. S2CID 118903056.
8. Makin, M. I.; Cole, Jared H.; Tahan, Charles; Hollenberg, Lloyd C. L.; Greentree, Andrew D. (21 May 2008). "Quantum phase transitions in photonic cavities with two-level systems". Physical Review A. 77 (5): 053819. arXiv:0710.5748. doi:10.1103/PhysRevA.77.053819.
9. A. D. Greentree; C. Tahan; J. H. Cole; L. C. L. Hollenberg (2006). "Quantum phase transitions of light". Nature Physics. 2 (12): 856–861. arXiv:cond-mat/0609050. Bibcode:2006NatPh...2..856G. doi:10.1038/nphys466. S2CID 118903056.
10. B. W. Petley (1971). An Introduction to the Josephson Effects. London: Mills and Boon.
11. Antonio Barone; Gianfranco Paternó (1982). Physics and Applications of the Josephson Effect. New York: Wiley.
12. Max T. C. Wong; C. K. Law (May 2011). "Two-polariton bound states in the Jaynes-Cummings-Hubbard model". Phys. Rev. A. 83 (5). American Physical Society: 055802. arXiv:1101.1366. Bibcode:2011PhRvA..83e5802W. doi:10.1103/PhysRevA.83.055802. S2CID 119200554.
13. 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 (7095): 853–856. arXiv:cond-mat/0605196. Bibcode:2006Natur.441..853W. doi:10.1038/nature04918. PMID 16778884. S2CID 2214243.
14. 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. arXiv:1004.5118. Bibcode:2010PhRvA..81d3609J. doi:10.1103/PhysRevA.81.043609. S2CID 55445588.
15. M. Valiente; D. Petrosyan (2008). "Two-particle states in the Hubbard model". J. Phys. B: At. Mol. Opt. Phys. 41 (16): 161002. arXiv:0805.1812. Bibcode:2008JPhB...41p1002V. doi:10.1088/0953-4075/41/16/161002. S2CID 115168045.