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Physical Review A

Hidden continuous symmetry and Nambu-Goldstone mode in a two-mode Dicke model

Jingtao Fan, Zhiwei Yang, Yuanwei Zhang, Jie Ma, Gang Chen, Suotang Jia

Published Feb 10, 2014• 49 citations
Publisher

Abstract

In this paper we investigate the ground- and excited-state properties of a two-mode Dicke model, which is realized in an ensemble of many four-level atoms interacting simultaneously with two quantized cavity fields and two pairs of Raman lasers. We reveal rich phase diagrams, including a first-order quantum phase transition from an electric superradiant phase to a magnetic superradiant phase. In particular, when the two collective atom-photon coupling strengths are equal, we find a hidden continuous $\text{U}(1)$ symmetry for the two-mode Dicke model. If these identical coupling strengths are sufficiently strong, a quantum phase called the electromagnetic superradiant phase is predicted. In such a phase, the predicted continuous $\text{U}(1)$ symmetry breaks down and the gapless excitation spectrum (the Nambu-Goldstone mode), which could be measured experimentally by combining Bragg spectroscopy and cavity-enhanced Bragg scattering, is thus achieved. Our results pave the way to explore the physics for a model with continuous symmetry in quantum optics without using the rotating-wave approximation.

DOI
10.1103/physreva.89.023812

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In this paper we investigate the ground- and excited-state properties of a two-mode Dicke model, which is realized in an ensemble of many four-level atoms interacting simultaneously with two quantized cavity fields and two pairs of Raman lasers. We reveal rich phase diagrams, including a first-order quantum phase transition from an electric superradiant phase to a magnetic superradiant phase. In particular, when the two collective atom-photon coupling strengths are equal, we find a hidden continuous U(1)\text{U}(1)U(1) symmetry for the two-mode Dicke model. If these identical coupling strengths are sufficiently strong, a quantum phase called the electromagnetic superradiant phase is predicted. In such a phase, the predicted continuous U(1)\text{U}(1)U(1) symmetry breaks down and the gapless excitation spectrum (the Nambu-Goldstone mode), which could be measured experimentally by combining Bragg spectroscopy and cavity-enhanced Bragg scattering, is thus achieved. Our results pave the way to explore the physics for a model with continuous symmetry in quantum optics without using the rotating-wave approximation.
We investigate the two-mode quantum Rabi model (QRM) describing the interaction between a two-level atom and a two-mode cavity field. The quantum phase transitions are found when the ratio η of transition frequency of atom to frequency of cavity field approaches infinity. We apply the Schrieffer–Wolff (SW) transformation to derive the low-energy effective Hamiltonian of the two-mode QRM, thus yielding the critical point and rich phase diagram of quantum phase transitions. The phase diagram consists of four regions: a normal phase, an electric superradiant phase, a magnetic superradiant phase and an electromagnetic superradiant phase. The quantum phase transition between the normal phase and the electric (magnetic) superradiant phase is of second order and associates with the breaking of the discrete Z 2 symmetry. On the other hand, the phase transition between the electric superradiant phase and the magnetic superradiant phase is of first order and relates to the breaking of the continuous U(1) symmetry. Several important physical quantities, for example the excitation energy and average photon number in the four phases, are derived. We find that the excitation spectra exhibit the Nambu–Goldstone mode. We calculate analytically the higher-order correction and finite-frequency exponents of relevant quantities. To confirm the validity of the low-energy effective Hamiltonians analytically derived by us, the finite-frequency scaling relation of the averaged photon numbers is calculated by numerically diagonalizing the two-mode quantum Rabi Hamiltonian.
Abstract We investigate the ground-state properties and quantum phase transitions of an ensemble consisting of NNN four-level atoms within an optical cavity coupled to the single cavity mode and external laser fields. The system is described by an extended imbalanced Dicke model, in which the co- and counterrotating coupling terms are allowed to have different coupling strengths. Some novel polarized phases characterized by the phase differences between the cavity field or the atomic spin excitation and the Raman laser are found analytically. Meanwhile, the full phase diagram and quantum phase transitions are also revealed. Two sign-dependent coexisting phases have been identified, where the normal phase persists as a metastable state alongside the superradiant ground state. Finally, the breaking or restoration of the intrinsic symmetry in this system is addressed. It is found that besides the continuous U(1)U(1)U(1) and discrete Z2\mathbb{Z}_2Z2​ symmetries, the system also exhibits two reflection symmetries σv\sigma_vσv​s, a central symmetry C2C_2C2​ in the abstract position-momentum representation, and a discrete reflection parity-time (PT\mathcal{PT}PT) symmetry, a parameter exchange symmetry Tex\mathcal{T}_\mathrm{ex}Tex​ in the parameters space. These additional symmetries are governed by two Coxeter groups.
Abstract We carefully investigate the comprehensive impact of atom-cavity interaction and artificial magnetic fields on quantum phase transitions of anti-Jaynes-Cummings triangle model in the infinite frequency limit. We discover that ground states of the optical field can be a gapped normal phase (NP) or three kinds of gapless superradiant phases with infinite degeneracy. When the atom-cavity coupling is weak, the optical field is in a NP, which is a vacuum with no photons. Otherwise, it will stay at one of the superradiant phases: a normal superradiant phase without photon currents and another two chiral superradiant phases with opposite photon currents. The former only breaks the continuous U (1) symmetry and its gapless excitations are normal Goldstone modes. Nevertheless, the latter, mainly induced by an external synthetic gauge field, break both the continuous U (1) symmetry and chiral symmetry, thereby corresponding gapless excitations are chiral Goldstone modes. In addition, we also propose a detecting scheme to distinguish these superradiant phases.
We explore theoretically the physics of a collection of two-level systems coupled to a single-mode bosonic field in the non-standard configuration where each (artificial) atom is coupled to both field quadratures of the boson mode. We determine the rich phase diagram showing 'superradiant' phases with different symmetries. We demonstrate that it is possible to pass from a discrete, parity-like Z2\mathbb{Z}_2Z2​ symmetry to a continuous U(1) symmetry even in the ultrastrong coupling regime where the rotating wave approximation for the interaction between field and two-level systems is no longer applicable. By applying this general paradigm, we propose a scheme for the experimental implementation of such continuous U(1) symmetry in circuit QED systems, with the appearance of photonic Goldstone and amplitude modes above a critical point.
The Dicke model (DM) serves as a paradigm for understanding collective light-matter interactions. We introduce the chiral Dicke model, a generalization where an atomic ensemble couples to a two-mode cavity via chiral interactions. Unlike the standard DM, the chiral DM is endowed with an inherent continuous U(1)U(1)U(1) symmetry associated with angular momentum conservation. The ground-state phase diagram and the associated quantum phase transitions are charted out, revealing a U(1)U(1)U(1)-broken superradiant phase that spans a broad parameter space. We demonstrate that the spectrum of quantum fluctuations is highly tunable in both the symmetric and broken phases. Strikingly, our calculations reveal that the system exhibits `multiversality', where distinct universality classes govern the transition between the same two phases. In particular, along a special line in parameter space, the dynamical critical exponent for the normal-superradiant phase transition changes from zν=1zν=1zν=1 to zν=1/2zν=1/2zν=1/2. Our work establishes the chiral Dicke model as a powerful platform to realize novel quantum phases and multiversal critical phenomena in light-matter coupled systems.
We investigate the physics of a gas of ultracold atoms coupled to three single-mode optical cavities and transversely pumped with a laser. Recent work has demonstrated that, for two cavities, the Z2{\mathbb{Z}}_{2}Z2​ symmetries of each cavity can be combined into a global U(1) symmetry [J. L\'eonard et al., Nature (London) 543, 87 (2017)]. Here we show that when adding an extra cavity mode, the low-energy description of this system can additionally exhibit an SO(3) rotational symmetry which can be spontaneously broken. This leads to a superradiant phase transition in all the cavities simultaneously, and the appearance of Goldstone and amplitude modes in the excitation spectrum. We determine the phase diagram of the system, which shows the emergence and breaking of the continuous symmetries and displays first- and second-order phase transitions. We also obtain the excitation spectrum for each phase and discuss the atomic self-organized structures that emerge in the different superradiant phases. We argue that coupling the atoms equally to nnn different modes will in general generate a global SO(n)\text{SO}(n)SO(n) symmetry if the mode frequencies can be tuned to the same value.