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Cavity array of Quantum Electrodynamic systems have received much
interests due to its usability as a tunable laboratory made quantum
simulator to study many-body physics. For instance, the lattice model
of Jaynes-Cummings cavities has been shown to mimic the quantum phase
diagram of the Bose-Hubbard model [1]. In this talk, we shall discuss
our recent work on the lattice model of Rabi quantum cavities [2]. For
the Rabi lattice problem, we have rigorously shown that, in the strong
atom-photon coupling limit, it behaves as the quantum Ising model. It
is well known that, there are two zero energy Majorana modes at the
two ends of a quantum Ising chain (in the ordered phase) with open
boundaries. Thus, we propose to look for the Majorana edge modes in
the one-dimensional Rabi lattice. We have proposed a useful relation
between the end-to-end dipole correlation and polarization to identify
these edge modes. We support our findings also with numerical
calculations using DMRG (density matrix renormalization group) method.
We also checked the stability of these edge modes against uniform as
well as local longitudinal fields. To see the effects of frustration
on the edge modes, we studied the quantum Ising model with the nearest
($J_1$) and the next-nearest ($J_2$) neighbor interactions, which is
the second part of my talk [3].
The quantum phase diagram of the $J_1 - J_2$ quantum Ising chain has
two ordered phases, one is simple ferromagnetic phase (or N\'eel phase
depending on the sign of nearest neighbor interaction) and the other
is the double-staggered antiferromagnetic phase, separated by a
quantum paramagnetic and a critical quantum paramagnetic phase. By
doing systematic DMRG and cluster-mean-field diagonalization
calculations, we identify the Majorana-like edge modes in these
ordered phases by calculating the end-to-end spin-spin correlations on
open chains.
References:
[1] Quantum phase transition of light,
A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, Nature
Physics, 2:856, 2006;
Strongly interacting polaritons in coupled arrays of cavities,
M. J. Hartmann, F. G. S L. Brandao, and M. B. Plenio. Nature Physics,
2:849, 2006
[2] Quantum Ising dynamics and Majorana-like edge modes in the Rabi
lattice model,
B. Kumar and Somenath Jalal, Phys. Rev. A {\bf 88}, 011802 (R) (2013)
[3] Edge modes in a frustrated quantum Ising chain,
Somenath Jalal and B. Kumar, (submitted to Phys. Rev. B), arxiv:1407.0201 |