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

Abstract

In this work, by using the strong coupling expansion and exact diagonization (ED), we study the Z2/U (1) Dicke model with independent rotating wave (RW) coupling g and counter-rotating wave (CRW) coupling g ′ at a finite N . This model includes the four standard quantum optics model: Rabi, Dicke, Jaynes-Cummings ( JC ) and Tavis-Cummings (TC) model as its various special limits. We show that in the super-radiant phase, the system's energy levels are grouped into doublets with even and odd parity. Any anisotropy β = g/g ′ = 1 leads to the oscillation of parities in both the ground and excited doublets as the atom-photon coupling strength increases. The oscillations will be pushed to the infinite coupling strength in the isotropic Z2 limit β = 1. We find nearly perfect agreements between the strong coupling expansion and the ED in the super-radiant regime when β is not too small. We also compute the photon correlation functions, squeezing spectrum, number correlation functions which can be measured by various standard optical techniques.Introduction: There are several well known quantum optics models to study atom-photon interactions [1,2]. In the Rabi model[3], a single mode photon interacts with a two level atom with equal rotating wave (RW) and counter rotating wave (CRW) strength. When the coupling strength is well below the transition frequency, the CRW term is effectively much smaller than that of RW, so it was dropped in the Jaynes-Cummings ( JC ) model [4]. Then the Rabi and JC model were extended to an assembly of N two level atoms to the Dicke model [5] and the Tavis-Cummings (TC) model [6] respectively. Despite their relative simple forms and many previous theoretical works [10][11][12][13][14][15], their solutions at a finite N , especially inside the superradiant regime, remain unknown. 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 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 lev-

Classification of the quantum chaos in colored Sachdev-Ye-Kitaev models

“…The only exception is that in the two color case with N (mod 4) = 2, as shown in Appendix A-a, both Q 1 and Q 2 are imaginary, so it is in GUE. As shown in [37][38][39][40][41] on Dicke model, K is the only relevant anti-unitary operator which commutes the Dicke Hamiltonian, so K 2 = 1 only leads to GOE for the Dicke model.…”

“…Despite the lack of quenched disorders, the quantum chaos in tensor models seems much more difficult to analyze by either OTOC or RMT [33]. The OTOC and RMT [37] may also be used to demonstrate the quantum chaos in a clean quantum optics model called Dicke model which describes the N qubits interacting with a single photon mode with both rotating wave and counter-rotating wave interacting term [38][39][40][41].…”

Slow-Goldstone mode generated by order from quantum disorder and its experimental detection

“…The Goldstone theorem states [1][2][3] that if a system has a continuous global symmetry G which is spontaneously broken by its ground state to a smaller symmetry group H, then 1) the symmetry broken ground state must support gapless modes, 2) the number of which is just the difference of the number of generators of G and H. These gapless modes with linear dispersions are called Goldstone modes. The Goldstone theorem has had tremendous applications in essentially all the branches of physics such as string theory/quantum gravity [4,5], especially the gapless reparametrization mode in the Sachdev-Ye-Kitaev (SYK) models leading to its maximal chaos tying that of a quantum black hole [6][7][8][9], particle physics [10], condensed matter systems [11,12], cold-atoms/quantum optics [13][14][15] and quantum information science [16]. The number of Goldstone modes can be counted just from the symmetry breaking analysis.…”

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

“…In 21 , by the 1/J expansion, we focused on the U (1)/Z 2 Dicke model Eqn.1 near the U (1) limit ( namely, with a small anisotropy parameter g ′ /g = β ≪ 1 and not too far from the critical strength g c ) at any finite N . In a very recent preprint 39 , by the strong coupling expansion and the ED, the authors studied the U (1)/Z 2 Dicke model in its dual presentation starting from from the Z 2 limit β = 1. Here, by the 1/J expansion and ED, we will study the U (1)/Z 2 Dicke model Eqn.1 starting from from the U (1) limit β = 0 which is complementary to the strong coupling expansion in 39 .…”

“…In a very recent preprint 39 , by the strong coupling expansion and the ED, the authors studied the U (1)/Z 2 Dicke model in its dual presentation starting from from the Z 2 limit β = 1. Here, by the 1/J expansion and ED, we will study the U (1)/Z 2 Dicke model Eqn.1 starting from from the U (1) limit β = 0 which is complementary to the strong coupling expansion in 39 . The combinations of both approaches will lead to rather complete understandings of the U (1)/Z 2 Dicke model Eqn.1 in the full range of 0 < β < 1.…”