Goldstone and Higgs modes of photons inside a cavity

Abstract

Goldstone and Higgs modes have been detected in various condensed matter, cold atom and particle physics experiments. Here, we demonstrate that the two modes can also be observed in optical systems with only a few (artificial) atoms inside a cavity. We establish this connection by studying the U(1)/Z2 Dicke model where N qubits (atoms) coupled to a single photon mode. We determine the Goldstone and Higgs modes inside the super-radiant phase and their corresponding spectral weights by performing both 1/J = 2/N expansion and exact diagonalization (ED) study at a finite N. We find nearly perfect agreements between the results achieved by the two approaches when N gets down even to N = 2. The quantum finite size effects at a few qubits make the two modes quite robust against an effectively small counterrotating wave term. We present a few schemes to reduce the critical coupling strength, so the two modes can be observed in several current available experimental systems by just conventional optical measurements.

Parity oscillations and photon correlation functions in theZ2−U(1)Dicke model at a finite number of atoms or qubits

“…In general, in such a ultra-strong coupling regime, the system is described well by Eq.1 dubbed as the U (1)/Z 2 Dicke model [8,26,27] which includes the four standard quantum optics model as its various special limits. Here, we study the U (1)/Z 2 Dicke model Eq.1 at any finite N and any ratio between the RW and the CRW term 0 ≤ g ′ /g = β ≤ 1 by the strong coupling expansion [28] and exact diagonization (ED) [8,12,29]. We show that in the super-radiant phase, the system's energy levels are grouped into doublets each of which consists of two Schrodinger Cat states with even and odd parity.…”

“…Strong coupling expansion-In the strong coupling limit, it is more convenient to rewrite the U (1)/Z 2 Dicke model [8] in its dual Z 2 /U (1) presentation:…”

“…Here, we address this outstanding problem. It is convenient to classify the four well known quantum optics models from a simple symmetry point of view: the TC and Dicke model as the U (1) and Z 2 Dicke model [7][8][9] respectively, while JC and Rabi model are just as the N = 1 version of the two.Due to recent tremendous advances in technologies, ultra-strong couplings in cavity QED systems were achieved in at least two experimental systems (1) a BEC atoms inside an ultrahigh-finesse optical cavity [16][17][18][19][20] and (2) superconducting qubits inside a microwave circuit cavity [21][22][23][24][25]. In general, in such a ultra-strong coupling regime, the system is described well by Eq.1 dubbed as the U (1)/Z 2 Dicke model [8,26,27] which includes the four standard quantum optics model as its various special limits.…”

“…Here, we address this outstanding problem. It is convenient to classify the four well known quantum optics models from a simple symmetry point of view: the TC and Dicke model as the U (1) and Z 2 Dicke model [7][8][9] respectively, while JC and Rabi model are just as the N = 1 version of the two.…”

Collective Modes in the Cooperative Jahn–Teller Model: Path Integral Approach

“…A prominent example is the recent experimental demonstration of the quantum phase transition from Mott insulator state to superfluid state of spin-boson excitations in a system of trapped ions [2]. It was shown that with the spontaneous breaking of continuous U(1) symmetry at the quantum phase transition in such models, including for example Jaynes-Cummings-Hubbard system [3] and Dickelike system [4,5], two-types of collective excitations emerge. One is the the gapless Goldstone mode, and the other is the gapped amplitude mode corresponding to phase and density fluctuations.…”