Photon Berry phases, Instantons, Schrodinger Cats with oscillating parities and crossover from $ U(1) $ to $ Z_2 $ limit in cavity QED systems

TL;DR

This paper investigates the $U(1)/Z_2$ Dicke model at finite $N$, revealing energy level regimes, quantum tunneling phenomena, and Berry phase effects leading to Schrödinger cat states, with results applicable to quantum optics experiments.

Contribution

It introduces a combined $1/J$ expansion and exact diagonalization approach to analyze the $U(1)/Z_2$ Dicke model, uncovering new quantum tunneling processes and Schrödinger cat oscillations.

Findings

Identification of three energy regimes: normal, $U(1)$, and QT.

Discovery of Schrödinger cat states with oscillating parities.

Mapping of energy level evolution from $U(1)$ to QT regime.

Abstract

In this work, we study the $ U(1)/Z_2 $ Dicke model at a finite $ N $ by using the $ 1/J $ expansion and exact diagonization. This model includes the four standard quantum optics model as its various special limits. The $ 1/J $ expansions is complementary to the strong coupling expansion used by the authors in arXiv:1512.08581 to study the same model in its dual $ Z_2/U(1) $ representation. We identify 3 regimes of the system's energy levels: the normal, $ U(1) $ and quantum tunneling (QT) regime. The system's energy levels are grouped into doublets which consist of scattering states and Schrodinger Cats with even ( e ) and odd ( o ) parities in the $ U(1) $ and quantum tunneling (QT) regime respectively. In the QT regime, by the WKB method, we find the emergencies of bound states one by one as the interaction strength increases, then investigate a new class of quantum tunneling processes through the instantons between the two bound states in the compact photon phase. It is the Berry phase interference effects in the instanton tunneling event which leads to Schrodinger Cats oscillating with even and odd parities in both ground and higher energy bound states. We map out the energy level evolution from the $ U(1) $ to the QT regime and also discuss some duality relations between the energy levels in the two regimes. We also compute the photon correlation functions, squeezing spectrum, number correlation functions in both regimes which can be measured by various experimental techniques. The combinations of the results achieved here by $ 1/J $ expansion and those in arXiv:1512.08581 by strong coupling method lead to rather complete understandings of the $ U(1)/Z_2 $ Dicke model at a finite $ N $ and any anisotropy parameter $ \beta $.