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The '''Jaynes-Cummings-Hubbard (JCH) model''' is a combination of the [[Jaynes–Cummings model]] and the coupled cavities. The one-dimensional JCH model consists of a chain of ''N''-coupled single-mode cavities and each cavity contains a two-level [[atom]] as illustrated in the figures. This model was originally proposed in June 2006 in the context of Mott transitions for strongly interacting photons in coupled cavity arrays in Ref.<ref>
{{cite journal
| author = D. G. Angelakis, M. F. Santos and S. Bose
| title = Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays
| journal = [[Physical Review A]]
| year = 2007
| volume = 76
| number = 03
| pages = 1805(R)
}}
</ref> A different [[interaction]] scheme has been suggested at the same time, where four level atoms were interacting with external fields and strongly correlated dynamics of [[polaritons]] were studied.<ref>
{{cite journal
| author = M. J. Hartmann, F. G. S. L. Brandão and M. B. Plenio
| title = Strongly interacting polaritons in coupled arrays of cavities
| journal = [[Nature Physics]]
| year = 2006
| volume = 2
| pages = 849
}}
</ref> In Ref.<ref>
{{cite journal
| author = A. D. Greentree, C. Tahan, J. H. Cole and L. C. L. Hollenberg
| title = Quantum phase transitions of light
| journal = [[Nature Physics]]
| year = 2006
| volume = 2
| pages = 856
}}
</ref>
the phase diagram of the JCH using mean field theory has been calculated in which the [[Mott insulator]] phase and [[superfluid]] phase are identified.

The [[Quantum tunneling|tunnelling]] effect comes from the junction between cavities which is an analogy of the [[Josephson effect]].<ref>
{{cite book
| author = B. W. Petley
| title = An Introduction to the Josephson Effects
| publisher = [[Mills and Boon]]
| address = [[London]]
| year = 1971
}}
</ref><ref>
{{cite book
| author = Antonio Barone and Gianfranco Paternó
| title = Physics and Applications of the Josephson Effect
| publisher = [[John Wiley & Sons|Wiley]]
| address = [[New York]]
| year = 1982
}}
</ref> The model can be made using [[circuit QED]] with [[superconducting qubits]]. More information can be found in Ref.<ref>
{{cite journal
| author = A. Nunnenkamp, Jens Koch and S. M. Girvin
| title = Synthetic gauge fields and homodyne transmission in Jaynes-Cummings lattices
| journal = [[New Journal of Physics]]
| year = 2011
| volume = 13
| pages = 095008
}}
</ref>

[[File:Jaynes-Cummings-Hubbard model.png|thumb|[[Quantum tunneling|Tunnelling]] of photons between coupled cavities. The <math>\kappa</math> is the tunnelling rate of photons.|600px]]

[[File:Jaynes-Cummings model.png|thumb|Illustration of the [[Jaynes-Cummings model]]. In the circle, [[photon]] [[emission (electromagnetic radiation)|emission]] and [[absorption (electromagnetic radiation)|absorption]] are shown.]]

==Basic description==
The investigation of [[quantum electrodynamics]] (QED) in coupled-cavity systems provides insight about the behavior of strongly interacting [[photons]] and [[atoms]].
With the capability of tunable coupling and measurement of individual cavity fields, coupled-cavity QED could serve as a useful tool to address the control of quantum [[many-body]] phenomena <ref>
{{cite journal
| author = M. J. Hartmann, F. G. S. L. Brandão and M. B. Plenio
| title = Quantum many-body phenomena in coupled cavity arrays
| journal = [[Laser Photonics Review]]
| year = 2008
| volume = 2
| number = 6
| pages = 527
}}

</ref><ref>
{{cite journal
| author = A. Tomadin and R. Fazio
| title = Many-body phenomena in QED-cavity arrays
| journal = [[J. Opt. Soc. Am. B]]
| year = 2010
| volume = 27
| number = 6
| pages = A130
}}
</ref> as well as the transmission and storage of [[quantum information]].<ref>
{{cite book
| editor = Hoi-Kwong Lo, Sandu Popescu and Tim Spiller
| title = Introduction to Quantum Computation and Information
| publisher = [[World Scientific]],
| address = [[Singapore]],
| year = 1998
}}
</ref> In particular, the JCH model corresponds to a fundamental configuration exhibiting the [[quantum phase transition]] of [[light]]. In the original version of this model in Ref.[1], single two-level atoms are embedded in each cavity and the [[dipole]] interaction leads to dynamics involving photonic and atomic [[degrees of freedom (physics and chemistry)|degrees of freedom]], which is in contrast to the widely studied [[Bose-Hubbard model]]. More recent treatment using strong coupling theory can be found at Ref.
<ref>
{{cite journal
| title = Strong Coupling Theory for the Jaynes-Cummings-Hubbard Model
| author = Schmidt, S. and Blatter, G.
| journal = [[Phys. Rev. Lett.]]
| volume = 103
| issue = 8
| pages = 086403
| numpages = 4
|date=Aug 2009
| doi = 10.1103/PhysRevLett.103.086403
| url = http://link.aps.org/doi/10.1103/PhysRevLett.103.086403
| publisher = [[American Physical Society]]
}}
</ref>

==Formulation==

===Hamiltonian===
The Hamiltonian of the model are laid out in Ref. [1] is given by
(<math>\hbar=1</math>):
:<math>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>\sigma_{n}^{\pm}</math> are [[Pauli]] operators for the two-level atom at the
''n''-th cavity. The <math>\kappa</math> is the tunnelling rate between neighboring cavities, and <math>\eta</math> is the [[vacuum Rabi frequency]] which characterizes to the [[photon]]-atom interaction strength. The cavity [[frequency]] is <math>\omega_c</math> and atomic transition frequency is <math>\omega_a</math>. We assume the periodic [[boundary condition]] such that the cavity labelled by ''n'' = ''N''+1 corresponds to the cavity ''n'' = 1.

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

==Two-polariton bound states==
The [[eigenstates]] of the JCH Hamiltonian in the two-excitation subspace for the ''N''-cavity system are examined. The research focus is placed on the existence of [[bound states]] as well as their features. It is interesting to note that two repulsive [[bosonic]] [[atoms]] can form a bound pair in an [[optical lattice]].<ref>
{{cite journal
| author = K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. H. Denschlag, A. J. Daley, A. Kantian, H. P. Buchler and P. Zoller
| title = Repulsively bound atom pairs in an optical lattice
| journal = [[Nature]]
| year = 2006
| volume = 441
| pages = 853
}}
</ref><ref>
{{cite journal
| title = Dimer of two bosons in a one-dimensional optical lattice
| author = Javanainen, Juha and Odong, Otim and Sanders, Jerome C.
| journal = [[Phys. Rev. A]]
| volume = 81
| issue = 4
| pages = 043609
| numpages = 12
|date=Apr 2010
| doi = 10.1103/PhysRevA.81.043609
| url = http://link.aps.org/doi/10.1103/PhysRevA.81.043609
| publisher = [[American Physical Society]]
}}
</ref><ref>
{{cite journal
| author = M. Valiente and D. Petrosyan
| title = Two-particle states in the Hubbard model
| journal = [[J. Phys. B]]: At. Mol. Opt. Phys.
| year = 2008
| volume = 41
| pages = 161002
}}
</ref> The JCH Hamiltonian also 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]]. The results discussed has been published in Ref.<ref>
{{cite journal
| title = Two-polariton bound states in the Jaynes-Cummings-Hubbard model
| author = Max T. C. Wong and C. K. Law
| journal = [[Phys. Rev. A]]
| volume = 83
| issue = 5
| pages = 055802
| numpages = 4
|date=May 2011
| doi = 10.1103/PhysRevA.83.055802
| url = http://link.aps.org/doi/10.1103/PhysRevA.83.055802
| publisher = [[American Physical Society]]
}}
</ref> The [[analytic solution]] of the [[eigenvalues]] and [[eigenvectors]] in strong coupling regime is also given. The [[time evolution]] of such system is also studied for the cases of different [[initial conditions]].

==Further reading==
* D. F. Walls and G. J. Milburn (1995), ''Quantum Optics'', Springer-Verlag.

==See also==
* [[Bose-Hubbard model]]
* [[Jaynes-Cummings model]]
* [[Quantum phase transition]]

==References==
{{Reflist|2}}

{{DEFAULTSORT:Jaynes-Cummings-Hubbard model}}
[[Category:Quantum optics]]

Rule of mixtures - Revision history

en>Bibcode Bot: Adding 0 arxiv eprint(s), 1 bibcode(s) and 0 doi(s). Did it miss something? Report bugs, errors, and suggestions at User talk:Bibcode Bot