Higher symmetry breaking and nonreciprocity in a driven-dissipative Dicke model

Higher symmetries in interacting many-body systems often give rise to new phases and unexpected dynamical behavior. Here, we theoretically investigate a variant of the Dicke model with higher-order discrete symmetry, resulting from complex-valued coupling coefficients between quantum emitters and a bosonic mode. We propose a driven-dissipative realization of this model focusing on optomechanical response of a driven atom tweezer array comprised of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>n</a:mi> </a:math> subensembles and placed within an optical cavity, with the phase of the driving field advancing stepwise between subensembles. Examining stationary points and their dynamical stability, we identify a phase diagram for <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>n</b:mi> <b:mo>≥</b:mo> <b:mn>3</b:mn> </b:mrow> </b:math> with three distinctive features: a <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msub> <c:mi mathvariant="double-struck">Z</c:mi> <c:mi>n</c:mi> </c:msub> </c:math> ( <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msub> <e:mi mathvariant="double-struck">Z</e:mi> <e:mrow> <e:mn>2</e:mn> <e:mi>n</e:mi> </e:mrow> </e:msub> </e:math> ) symmetry-breaking superradiant phase for even (odd) <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>n</g:mi> </g:math> , a normal unbroken-symmetry phase that is dynamically unstable due to nonreciprocal forces between emitters, and a first-order phase transition separating these phases. This <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mi>n</h:mi> </h:math> -phase Dicke model may be equivalently realized in a variety of optomechanical or optomagnonic settings, where it can serve as a test bed for studying high-order symmetry breaking and nonreciprocal interactions in open systems.