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arXiv:1805.04392 · Physical Review A

Emergence of continuous rotational symmetries in ultracold atoms coupled to optical cavities

E. I. Rodríguez Chiacchio, Andreas Nunnenkamp

Published Aug 15, 2018• 13 citations
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Abstract

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 ${\mathbb{Z}}_{2}$ 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 $n$ different modes will in general generate a global $\text{SO}(n)$ symmetry if the mode frequencies can be tuned to the same value.

DOI
10.1103/physreva.98.023617

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Ultracold atomic gases loaded into optical cavities constitute one of the most versatile platforms for the study of out-of-equilibrium dynamics and quantum simulation of many- body systems. In this thesis, we explore these two possibilities, motivated by current state-of-the-art atom-cavity experiments, uncovering novel phenomena and opening up new theoretical and experimental avenues into the intriguing world of quantum many- body physics. First, we study a gas of ultracold atoms coupled to three optical resonators. We show that the inherent Z2 symmetries, associated with each atom-cavity coupling, can be combined into a global SO(3) rotational symmetry which can be spontaneously broken. 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 argue that by coupling the atoms equally to n different modes, it will in general be possible to generate a global SO(n) symmetry. Second, we investigate the dynamics of a spinor BEC inside an optical cavity. We focus on a two-component Dicke model with complex light-matter couplings, accounting for photon losses. We compute the steady-state phase diagram and find dynamical in- stabilities in the form of limit cycles, heralded by the presence of exceptional points and level attraction. We show that the instabilities are induced by dissipative processes that generate nonreciprocal couplings between the two collective spins. Lastly, we explore the dynamics of a BEC inside an optical cavity in the presence of an optical lattice. We derive an effective master equation by adiabatically eliminating the cavity field. In the bad-cavity regime, we find an infinite-temperature steady state, but with relaxation dynamics that can be highly nontrivial. For small hopping, the interplay between dissipation and strong interactions leads to an algebraic relaxation of the system, whereas for large hopping, the approach to the steady state is exponential. In the good- cavity regime, we show that the system allows for optical pumping between momentum modes, enabling cavity cooling.
Atoms in transversely pumped optical cavities ``self-organize'' by forming a density wave and emitting superradiantly into the cavity mode(s). For a single-mode cavity, the properties of this self-organization transition are well characterized both theoretically and experimentally. Here, we explore the self-organization of a Bose-Einstein condensate in the presence of two cavity modes---a system that recently was realized experimentally [L\'eonard et al., Nature (London) 543, 87 (2017)]. We argue that this system can exhibit a ``vestigially ordered'' phase in which neither cavity mode exhibits superradiance but the cavity modes are mutually phase locked by the atoms. We argue that this vestigially ordered phase should generically be present in multimode cavity geometries.
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.
The self-organization of a Bose-Einstein condensate (BEC) in a transversely pumped optical cavity is a process akin to crystallization: when pumped by a laser of sufficient intensity, the coupled matter and light fields evolve, spontaneously, into a spatially modulated pattern, or crystal, whose lattice structure is dictated by the geometry of the cavity.In cavities having multiple degenerate modes, the quasicontinuum of possible lattice arrangements, and the continuous symmetry breaking associated with the adoption of a particular lattice arrangement, give rise to phenomena such as phonons, defects, and frustration, which have hitherto been unexplored in ultracold atomic settings involving neutral atoms.The present work develops a nonequilibrium field-theoretic approach to explore the self-organization of a BEC in a pumped, lossy optical cavity.We find that the transition is well described, in the regime of primary interest, by an effective equilibrium theory.At nonzero temperatures, the self-organization occurs via a fluctuation-driven first-order phase transition of the Brazovskii class; this transition persists to zero temperature and crosses over into a quantum phase transition.We make further use of our field-theoretic description to investigate the role of nonequilibrium fluctuations in the self-organization transition, as well as to explore the nucleation of ordered-phase droplets, the nature and energetics of topological defects, supersolidity in the ordered phase, and the possibility of frustration controlled by the cavity geometry.In addition, we discuss the range of experimental parameters for which we expect the phenomena described here to be observable, along with possible schemes for detecting ordering and fluctuations via either atomic correlations or the correlations of the light emitted from the cavity.
We investigate harmonically trapped, laser-pumped bosons with infinite-range interactions induced by a dissipative high-finesse red-detuned optical cavity with numerical and analytical methods. We obtain multiple cavity and atomic observables as well as the full phase diagram of the system using the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X) approach. Besides the transition from an unorganized normal phase to a superradiant phase where the atoms self-organize, we focus on an in-depth investigation of the self-organized superfluid to self-organized Mott-insulator phase transition in the superradiant phase as a function of the cavity-atom coupling. The numerical results are substantiated by an analytical study of an effective Bose-Hubbard model. We numerically analyze cavity fluctuations and emergent strong correlations between atoms in the many-body state across the Mott transition via the atomic density distributions and Glauber correlation functions. Unexpectedly, the weak harmonic trap leads to features like a lattice switching between the two symmetry-broken Z2{\mathbb{Z}}_{2}Z2​ configurations of the untrapped system and a reentrance of superfluidity in the Mott-insulating phase. Our analytical considerations quantitatively explain the numerically observed correlation features.