# Jaynes–Cummings–Hubbard model

(Redirected from Jaynes-Cummings-Hubbard model)

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.[1] 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.[2] In Ref.[3] 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 tunnelling effect comes from the junction between cavities which is an analogy of the Josephson effect.[4][5] The model can be made using circuit QED with superconducting qubits. More information can be found in Ref.[6]

Tunnelling of photons between coupled cavities. The $\kappa$ is the tunnelling rate of photons.
Illustration of the Jaynes-Cummings model. In the circle, photon emission and 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 [7][8] as well as the transmission and storage of quantum information.[9] 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, which is in contrast to the widely studied Bose-Hubbard model. More recent treatment using strong coupling theory can be found at Ref. [10]

## Formulation

### Hamiltonian

The Hamiltonian of the model are laid out in Ref. [1] is given by ($\hbar=1$):

$H = \sum_{n=1}^{N}\omega_c a_{n}^{\dagger}a_{n} +\sum_{n=1}^{N}\omega_a \sigma_n^+\sigma_n^- + \kappa \sum_{n=1}^{N} \left(a_{n+1}^{\dagger}a_{n}+a_{n}^{\dagger}a_{n+1}\right) + \eta \sum_{n=1}^{N} \left(a_{n}\sigma_{n}^{+} + a_{n}^{\dagger}\sigma_{n}^{-}\right)$

where $\sigma_{n}^{\pm}$ are Pauli operators for the two-level atom at the n-th cavity. The $\kappa$ is the tunnelling rate between neighboring cavities, and $\eta$ is the vacuum Rabi frequency which characterizes to the photon-atom interaction strength. The cavity frequency is $\omega_c$ and atomic transition frequency is $\omega_a$. 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 $\hat{N}_c \equiv \sum_{n=1}^{N}a_n^{\dagger}a_n$ and $\hat{N}_a \equiv \sum_{n=1}^{N} \sigma_{n}^{+}\sigma_{n}^{-}$, it is easy to check that the total number of excitations is still a conserved quantity, i.e., $\lbrack H,\hat{N}_c+\hat{N}_a\rbrack=0$.

## 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.[11][12][13] 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.[14] 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.

• D. F. Walls and G. J. Milburn (1995), Quantum Optics, Springer-Verlag.

## References

1. ^ D. G. Angelakis, M. F. Santos and S. Bose (2007). "Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays". Physical Review A 76 (03): 1805(R). arXiv:quant-ph/0606159. Bibcode:2007PhRvA..76c1805A. doi:10.1103/physreva.76.031805.
2. ^ 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:quant-ph/0606097. Bibcode:2006NatPh...2..849H. doi:10.1038/nphys462.
3. ^ A. D. Greentree, C. Tahan, J. H. Cole and L. C. L. Hollenberg (2006). "Quantum phase transitions of light". Nature Physics 2: 856. arXiv:cond-mat/0609050. Bibcode:2006NatPh...2..856G. doi:10.1038/nphys466.
4. ^ B. W. Petley (1971). An Introduction to the Josephson Effects. London: Mills and Boon.
5. ^ Antonio Barone and Gianfranco Paternó (1982). Physics and Applications of the Josephson Effect. New York: Wiley.
6. ^ A. Nunnenkamp, Jens Koch and S. M. Girvin (2011). "Synthetic gauge fields and homodyne transmission in Jaynes-Cummings lattices". New Journal of Physics 13: 095008. arXiv:1105.1817. Bibcode:2011NJPh...13i5008N. doi:10.1088/1367-2630/13/9/095008.
7. ^ M. J. Hartmann, F. G. S. L. Brandão and M. B. Plenio (2008). "Quantum many-body phenomena in coupled cavity arrays". Laser Photonics Review 2 (6): 527. doi:10.1002/lpor.200810046.
8. ^ A. Tomadin and R. Fazio (2010). "Many-body phenomena in QED-cavity arrays". J. Opt. Soc. Am. B 27 (6): A130. arXiv:1005.0137. Bibcode:2010JOSAB..27..130T. doi:10.1364/josab.27.00a130.
9. ^ Hoi-Kwong Lo, Sandu Popescu and Tim Spiller, ed. (1998). Introduction to Quantum Computation and Information. Singapore,: World Scientific,.
10. ^ Schmidt, S. and Blatter, G. (Aug 2009). "Strong Coupling Theory for the Jaynes-Cummings-Hubbard Model". Phys. Rev. Lett. (American Physical Society) 103 (8): 086403. arXiv:0905.3344. Bibcode:2009PhRvL.103h6403S. doi:10.1103/PhysRevLett.103.086403.
11. ^ K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. H. Denschlag, A. J. Daley, A. Kantian, H. P. Buchler and P. Zoller (2006). "Repulsively bound atom pairs in an optical lattice". Nature 441: 853. arXiv:cond-mat/0605196. Bibcode:2006Natur.441..853W. doi:10.1038/nature04918.
12. ^ 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:1004.5118. Bibcode:2010PhRvA..81d3609J. doi:10.1103/PhysRevA.81.043609.
13. ^ M. Valiente and 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.
14. ^ Max T. C. Wong and 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:1101.1366. Bibcode:2011PhRvA..83e5802W. doi:10.1103/PhysRevA.83.055802.