LWP: Dicke Quantum Spin Gas of Atoms and Photons

by P. Strack and S. Sachdev
Physical Review Letters 107, 277202 (2011)

A few years ago, Nagy and collaborators proposed to use Dicke model to describe atoms in a quantized cavity driven by a classical field in one-dimension [Eur. Phys. J. D, 2008, 48, 127 - 137]. Later Baumann and collaborators experimentally demonstrated a checkerboard transition in the two-dimensional center of mass motion  of a condensate in a cavity and an optical lattice which may be described by Dicke's model [Nature, 2010, 464, 1301 - 1306]; see also Nagy and collaborators [Phys. Rev. Lett., 2010, 104, 130401]. The experiments of Baumann and collaborators realized a supersolid atomic phase with long range interactions mediated by photons. Theoretically, Dicke's model in the thermodynamic limit (the number of two-level atoms, a.k.a. qubits, is infinitely large)  looking just at the atoms delivers a phase transition in the ground state into a ferromagnet-like structure.

Here, Strack and Sachdev study what happens in Dicke model when multiple electromagnetic modes interact with the atomic ensemble without the rotating-wave-approximation instead of just the single-mode. They propose to integrate out the photonic degrees of freedom in a path integral representation to obtain a  Hamiltonian similar to the Ising model in a transverse field, where long range interactions depend inversely on the imaginary frequencies of the qubits in the path integral. They mention that such a condensed matter model is similar to the Hopfield model and allows for Mattis ground states which critical properties should be similar to those of a single-mode Dicke model but focus on the case where the long range interaction distribution is Gaussian. This, in the infinitely large number of qubits limit, allows for an extra ground state quantum spin-glass phase appart from the paramagnet and the ferromagnet known in the single-mode Dicke model. They also show that it is also possible to calculate the photon correlation function. But most importantly, discuss that the paramagnet to ferromagnet phase transition is to be considered classical while the transition to a quantum spin-glass is a genuine phase transition from the radio-frequency spectral response function of the qubits (there is a spectral weight going to zero for a continuum of frequencies in the latter case).

I really liked the paper, it is very interesting how a well-known model still delivers new ways of studying condensed matter phenomena in quantum optics setups. I'm still trying to work my way with the formalism they used to integrate out the photon fields but the good thing is that they present the procedure in the last two pages of the manuscript and it is being very helpful.

Note: I'm sorry for the one-day delay, yesterday was a public holiday and I was lazy enough to stay at home.