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arXiv:1310.3780

Controlling discrete and continuous symmetries in 'superradiant' phase transitions

Alexandre Baksic, Cristiano Ciuti

Published Oct 14, 2013• 0 citations
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Abstract

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 $\mathbb{Z}_2$ 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coupling regime where the rotating wave approximation for the interaction between field and two-level systems is no longer applicable.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.

DOI
10.48550/arxiv.1310.3780

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

Yu Yi-Xiang, Jinwu Ye, W. M. Liu +1 more

Recently, the authors of the commented PRL presented the $ N=\infty $ solution of the $ U(1)/Z_2 $ Dicke model studied by us previously. Here we point out that (1) The authors missed an important transformation relating the two parameter regimes, so their separate discussions on the two regimes is redundant. (2) Both $ N=\infty $ classical limit and $ 1/N $ quantum fluctuations have been achieved in two of our previously published papers. It is the $ 1/N $ quantum fluctuations which lead to the non-trivial new quantum phenomena. In view of only a few $ N=2\sim 9 $ qubits inside a circuit QED microwave cavity, they can be tested in near future experiments. (3) Several possible experimental implementations of the $ U(1)/Z_2 $ Dicke model have been proposed before and recently experimentally realized.

arXiv:1506.06382•Jun 21, 2015•1 citation
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.
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 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.
We explore theoretically the physics of a collection of two-level systems coupled to a single-mode bosonic field in the nonstandard configuration where each (artificial) atom is coupled to both field quadratures of the boson mode. We show that such an unusual coupling scheme can be implemented in circuit QED systems, where artificial Josephson atoms are coupled both capacitively and inductively to a superconducting resonator. We demonstrate that it is possible to pass from a discrete, paritylike Z(2) symmetry to a continuous U(1) with the appearance of photonic Goldstone and amplitude modes above a critical point even in the ultrastrong coupling regime (where the rotating wave approximation for the interaction between field and two-level systems is no longer applicable). We determine the rich phase diagram showing "superradiant" phases with different symmetries and phase boundaries of both first and second order.
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.
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 study ultrastrong-coupling quantum-phase-transition phenomena in a few-qubit system. In the one-qubit case, three second-order transitions occur and the Goldstone mode emerges under the condition of ultrastrong-coupling strength. Moreover, a first-order phase transition occurs between two different superradiant phases. In the two-qubit case, a two-qubit Hamiltonian with qubit-qubit interactions is analyzed fully quantum mechanically. We show that the quantum phase transition is inhibited even in the ultrastrong-coupling regime in this model. In addition, in the three-qubit model, the superradiant quantum phase transition is retrieved in the ultrastrong-coupling regime. Furthermore, the NNN-qubit model with U(1)U(1)U(1) symmetry is studied and we find that the superradiant phase transition is inhibited or restored with the qubit-number parity.
Recently, the authors of the commented PRL presented the $ N=\infty $ solution of the $ U(1)/Z_2 $ Dicke model studied by us previously. Here we point out that (1) The authors missed an important transformation relating the two parameter regimes, so their separate discussions on the two regimes is redundant. (2) Both $ N=\infty $ classical limit and $ 1/N $ quantum fluctuations have been achieved in two of our previously published papers. It is the $ 1/N $ quantum fluctuations which lead to the non-trivial new quantum phenomena. In view of only a few $ N=2\sim 9 $ qubits inside a circuit QED microwave cavity, they can be tested in near future experiments. (3) Several possible experimental implementations of the $ U(1)/Z_2 $ Dicke model have been proposed before and recently experimentally realized.