Driven-dissipative dynamics of atomic ensembles in a resonant cavity: Nonequilibrium phase diagram and periodically modulated superradiance

Abstract

We study the dynamics of two ensembles of atoms (or equivalently, atomic clocks) coupled to a bad cavity and pumped incoherently by a Raman laser. Our main result is the nonequilibrium phase diagram for this experimental setup in terms of two parameters -detuning between the clocks and the repump rate. There are three main phases -trivial steady state (Phase I), where all atoms are maximally pumped, nontrivial steady state corresponding to monochromatic superradiance (Phase II), and amplitude-modulated superradiance (Phase III). Phases I and II are fixed points of the mean-field dynamics, while in most of Phase III stable attractors are limit cycles. Equations of motion possess an axial symmetry and a Z2 symmetry with respect to the interchange of the two clocks. Either one or both of these symmetries are spontaneously broken in various phases. The trivial steady state loses stability via a supercritical Hopf bifurcation bringing about a Z2-symmetric limit cycle. The nontrivial steady state goes through a subcritical Hopf bifurcation responsible for coexistence of monochromatic and amplitude-modulated superradiance. Using Floquet analysis, we show that the Z2-symmetric limit cycle eventually becomes unstable and gives rise to two Z2-asymmetric limit cycles via a supercritical pitchfork bifurcation. Each of the above attractors has its own unique fingerprint in the power spectrum of the light radiated from the cavity. In particular, limit cycles in Phase III emit frequency combs -series of equidistant peaks, where the symmetry of the frequency comb reflects the symmetry of the underlying limit cycle. For typical experimental parameters, the spacing between the peaks is several orders of magnitude smaller than the monochromatic superradiance frequency, making the lasing frequency highly tunable.

Driven-dissipative dynamics of atomic ensembles in a resonant cavity: Quasiperiodic route to chaos and chaotic synchronization

“…(8), all the energies (including δ and W ) are written in the units of N Γ c , and time is written in the units of (N Γ c ) −1 . For a typical 87 Sr based experimental setup, N Γ c ≈ 1.4kHz [9,28]. The above mean-field equations (8) have the following two symmetries.…”

“…To obtain this line we employ the Floquet analysis. In our previous work [9] we used a complementary numerical method to obtain the same line. First, we regroup the spin components in the following way -(1) the symmetric ones on the Z 2 -symmetric manifold (15)…”

“…In this paper, we analyze the signatures of chaos in the light radiated by two atomic ensembles resonantly coupled to a "bad" (fast leaking) cavity and incoherently pumped by a transverse laser [7][8][9][10]. Each ensemble contains a large number (N ≈ 10 6 ) of 87 Rb or 87 Sr atoms.…”

“…1 shows the full nonequilibrium phase diagram of the system, which we have already introduced in Ref. 9, and Fig. 2 shows the cartoon of the experimental setup.…”

“…To get around this issue, we analyze the projected 2D Poincaré sections, henceforth simply referred to as the Poincaré sections, for s A in different phases (see Figs. 8,9,16,and 23 Table 1: Properties of power spectra of radiated light for three distinct amplitude-modulated superradiances. In the second, third and fourth columns we list whether the spectra are discrete or continuous, whether they possess a reflection symmetry about the f = 0 axis, and if they have the most prominent peak at f = 0, respectively.…”