Jaynes–Cummings–Hubbard model explained

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(

\hbar =1 ):H = \sum ^{N} _{n =1} \omega _{c} a ^{\dagger} _{n} a _{n} + \sum ^{N} _{n =1} \omega _{a} \sigma ^{-} _{n} + \kappa \sum ^{N} _{n =1} \left(a ^{\dagger} _{n +1} a _{n} + a ^{\dagger} _{n} a _{n+1} \right) + η \sum ^{N} _{n =1} \left(a _{n} \sigma ^{+} _{n} + a ^{\dagger} _{n} \sigma ^{-} _{n} \right) where \sigma ^{\pm} _{n} are Pauli operators for the two-level atom at then-th cavity. The \kappa is the tunneling rate between neighboring cavities, and η is the vacuum Rabi frequency which characterizes to the photon-atom interaction strength. The cavity frequency is \omega _{c} and atomic transition frequency is \omega _{a} . The cavities are treated as periodic, so that the cavity labelled by n = N+1 corresponds to the cavity n = 1. 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

\hat{N} _{c} \equiv \sum ^{N} _{n =1} a ^{\dagger} _{n} a _{n} and \hat{N} _{a} \equiv \sum ^{N} _{n =1} \sigma ^{+} _{n} \sigma ^{-} _{n} , the total number of excitations is a conserved quantity,i.e., \lbrack H,\hat{N} _{c+\hat{N}} _{a\rbrack=0} .

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.

Notes and References

1. Strong Coupling Theory for the Jaynes-Cummings-Hubbard Model. Schmidt, S. . Blatter, G.. Phys. Rev. Lett.. 103. 8. 086403. Aug 2009. 10.1103/PhysRevLett.103.086403. 19792743 . 0905.3344 . 2009PhRvL.103h6403S . 32092406 .
2. A. Nunnenkamp . Jens Koch . S. M. Girvin . Synthetic gauge fields and homodyne transmission in Jaynes-Cummings lattices. New Journal of Physics. 2011. 13. 9 . 095008. 10.1088/1367-2630/13/9/095008. 1105.1817 . 2011NJPh...13i5008N . 118557639 .
3. Imamoḡlu . A. . Schmidt . H. . Woods . G. . Deutsch . M. . Strongly Interacting Photons in a Nonlinear Cavity . Physical Review Letters . 25 August 1997 . 79 . 8 . 1467–1470 . 10.1103/PhysRevLett.79.1467 .
4. Birnbaum . K. M. . Boca . A. . Miller . R. . Boozer . A. D. . Northup . T. E. . Kimble . H. J. . Photon blockade in an optical cavity with one trapped atom . Nature . July 2005 . 436 . 7047 . 87–90 . 10.1038/nature03804 . 16001065 .
5. D. G. Angelakis . M. F. Santos . S. Bose . Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays. Physical Review A. 2007. 76. 3. 1805(R). 10.1103/physreva.76.031805. quant-ph/0606159 . 2007PhRvA..76c1805A . 44490741 .
6. M. J. Hartmann, F. G. S. L. Brandão and M. B. Plenio. Strongly interacting polaritons in coupled arrays of cavities. Nature Physics. 2006. 2. 12. 849–855. 10.1038/nphys462. quant-ph/0606097 . 2006NatPh...2..849H . 9122839.
7. A. D. Greentree . C. Tahan . J. H. Cole . L. C. L. Hollenberg . Quantum phase transitions of light. Nature Physics. 2006. 2. 12 . 856–861. 10.1038/nphys466. cond-mat/0609050 . 2006NatPh...2..856G . 118903056 .
8. Makin . M. I. . Cole . Jared H. . Tahan . Charles . Hollenberg . Lloyd C. L. . Greentree . Andrew D. . Quantum phase transitions in photonic cavities with two-level systems . Physical Review A . 21 May 2008 . 77 . 5 . 053819 . 10.1103/PhysRevA.77.053819 . 0710.5748 .
9. A. D. Greentree . C. Tahan . J. H. Cole . L. C. L. Hollenberg . Quantum phase transitions of light. Nature Physics. 2006. 2. 12 . 856–861. 10.1038/nphys466. cond-mat/0609050 . 2006NatPh...2..856G . 118903056 .
10. Book: B. W. Petley. An Introduction to the Josephson Effects. Mills and Boon. London. 1971.
11. Book: Antonio Barone . Gianfranco Paternó . Physics and Applications of the Josephson Effect. Wiley. New York. 1982.
12. Two-polariton bound states in the Jaynes-Cummings-Hubbard model. Max T. C. Wong . C. K. Law . Phys. Rev. A. 83. 5. 055802. May 2011. 10.1103/PhysRevA.83.055802. American Physical Society. 1101.1366 . 2011PhRvA..83e5802W . 119200554 .
13. K. Winkler . G. Thalhammer . F. Lang . R. Grimm . J. H. Denschlag . A. J. Daley . A. Kantian . H. P. Buchler . P. Zoller . Repulsively bound atom pairs in an optical lattice. Nature. 2006. 441. 7095 . 853–856. 10.1038/nature04918. cond-mat/0605196 . 2006Natur.441..853W . 16778884. 2214243 .
14. Dimer of two bosons in a one-dimensional optical lattice. Javanainen, Juha and Odong, Otim and Sanders, Jerome C.. Phys. Rev. A. 81. 4. 043609. Apr 2010. 10.1103/PhysRevA.81.043609. 1004.5118 . 2010PhRvA..81d3609J . 55445588.
15. M. Valiente . D. Petrosyan . Two-particle states in the Hubbard model. J. Phys. B: At. Mol. Opt. Phys.. 2008. 41. 16 . 161002. 10.1088/0953-4075/41/16/161002. 2008JPhB...41p1002V . 0805.1812. 115168045 .