Abstract
The coupled nonlinear dynamics of ultracold quantum matter and electromagnetic field modes in an optical resonator exhibits a wealth of intriguing collective phenomena. Here we study a Λ-type, three-component Bose–Einstein condensate coupled to four dynamical running-wave modes of a ring cavity, where only two of the modes are externally pumped. However, the unpumped modes play a crucial role in the dynamics of the system due to coherent backscattering of photons. On a mean- field level we identify three fundamentally different steady-state phases with distinct characteristics in the density and spatial spin textures: a combined density and spin-wave, a continuous spin spiral with a homogeneous density, and a spin spiral with a modulated density. The spin-spiral states, which are topological, are intimately related to cavity-induced spin–orbit coupling emerging beyond a critical pump power. The topologically trivial density-wave–spin-wave state has the characteristics of a supersolid with two broken continuous symmetries. The transitions between different phases are either simultaneously topological and first-order, or second-order. The proposed setup allows the simulation of intriguing many-body quantum phenomena by solely tuning the pump amplitudes and frequencies, with the cavity output fields serving as a built-in nondestructive observation tool.
Highlights
The experimental progresses in reaching the quantum degeneracy limit in atomic gases paved the way for the realization of quantum many-body phenomena in these highly tunable systems [1, 2]
Almost all experimental works and most theoretical studies of coupled atom-cavity systems in the past were limited to systems where either the atomic internal states [29,30,31,32] or the atomic external degrees of freedom [33,34,35,36,37,38,39,40,41,42,43,44] are taken into account
We theoretically studied an effective two-component BEC inside a ring cavity, which possesses two pairs of nearly resonant running-wave modes with orthogonal polarizations
Summary
The experimental progresses in reaching the quantum degeneracy limit in atomic gases paved the way for the realization of quantum many-body phenomena in these highly tunable systems [1, 2]. The cavity-induced spin-orbit coupling for the atoms only emerges above a critical pump power, which in turn gives rise to novel nonequilibrium quantum phases and quantum phase transitions of various natures in our system It is this dynamical population of the unpumped cavity modes and its nontrivial interplay with the other degrees of freedom which marks a sharp contrast to the free-space spin-orbit coupling [12,13,14, 58, 59] as well as all other previous cavity-based spin-orbit coupling schemes [45,46,47, 50]. If the relative atomic detunings with respect to the pump lasers ∆↓(↑) := ωa(b) − [ωe − ω↓(↑)] are large compared to the two-photon detuning δ := ∆↑ − ∆↓ and the atom-photon couplings {G↓, G↑}, the atomic excited state reaches a steady-state on a short time scale and its dynamics can be adiabatically eliminated This results in a set of six coupled effective Heisenberg equations for the atomic pseudospin and photonic field operators i. In the strong pumping limit the pumped cavity fields {a+, ˆb−} behave as classical fields, the unpumped modes {a−, ˆb+} still retain their quantum nature and behave as dynamical fields
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