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

Abstract

The four standard quantum optics models such as Rabi, Dicke, Jaynes-Cummings ( JC ) and Tavis-Cummings (TC) model were proposed by the old generation of great physicists many decades ago. Despite their relative simple forms and many previous theoretical works, their solutions at a finite N , especially inside the superradiant regime, remain unknown. In this work, we address this outstanding problem by using the 1/J expansion and exact diagonization to study the U (1)/Z2 Dicke model at a finite N . 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 Z2/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)/Z2 Dicke model at a finite N and any anisotropy parameter β. Experimental realizations and detections are presented. Connections with past works and future perspectives are also discussed.I.

Comments on "Controlling Discrete and Continuous Symmetries in Superradiant Phase Transitions with Circuit QED Systems "

“…Then all the results on Goldstone and Higss modes shown in Fig. 1-5 at and near the U (1) limit in [2] and many other novel phenomena demonstrated in [4] at generic β can be detected for a few N = 2 ∼ 9 qubits.…”

“…In a recent preprint [4], using Eqn.12, we investigated the quantum fluctuations at order 1/N at any 0 ≤ g ′ /g = β ≤ 1 and full interaction strength.…”