Self-pulsing in driven-dissipative photonic Bose-Hubbard dimers

We report on the observation of nonlinear dynamics in driven-dissipative Bose-Hubbard dimers using a pair of coupled fiber ring resonators coherently driven by a single laser beam. We comprehensively explore the optical switching arising when scanning the detuning of the undriven cavity and demonstrate the key role played by the driven cavity detuning in the nonlinear regime. By driving the photonic dimer out of equilibrium, we observe the occurrence of robust self-switching oscillations.

Figure 1

(a) Schematic of an asymmetrically excited photonic dimer: The two resonators (L, R) are coupled (with normalized coupling strength $C$) and excited by an input beam coupled to the left resonator. (b) Detuning diagram, by analogy with the energy-level diagram of diatomic molecules. The coupling induces a resonance splitting giving rise to bondinglike (B) and antibondinglike (AB) modes. The resonance splitting is $2{\mathrm{\Delta }}^{\prime }=\sqrt{4C^2+{({\mathrm{\Delta }}_L-{\mathrm{\Delta }}_R)}^2}$, with ${\mathrm{\Delta }}_{L,R}$ being the detunings from the driving laser. (c) and (d) Real part of the intracavity field amplitudes as a function of the detunings for $C=5$ [computed from Eq. (1) in the linear regime]. The fields are in phase (B mode) and out of phase (AB mode) for ${\mathrm{\Delta }}_{L,R}>0$ and ${\mathrm{\Delta }}_{L,R}<0$, respectively.

Figure 2

(a) Phase diagram in the (${\mathrm{\Delta }}_R,S_L$) parameter space for ${\mathrm{\Delta }}_L=1.75$. The saddle-node bifurcations ${\mathrm{SN}}_j$ are plotted as green lines, and $c_{ij}$ mark cusp bifurcations. The dimer is bistable in the shaded green areas. (b), (d), and (f) Bifurcation diagrams showing the normalized intracavity power, in the driven (blue) and undriven (red) resonators as a function of ${\mathrm{\Delta }}_R$, for $S_L=0.2$ (b), 1.5 (d), and 2.8 (f) [see also dashed lines in (a)]. Solid (dashed) lines represent stable (unstable) solutions of Eq. (1). (c), (e), and (g) Corresponding experimental forward and backward scans. The arrows show the direction of the scans.

Figure 3

Same as in Fig. 2, but for ${\mathrm{\Delta }}_L=3.1$. (a) Phase diagram. The Hopf bifurcations ${\mathrm{HB}}_{1,2}$ (red lines) mark the boundaries of the self-pulsing region, and the Shilnikov bifurcation SB (black line) marks the boundaries of the area where there is no self-pulsing. $n_{41}$, necking bifurcation; $c_{ij}$, cusp bifurcation. The star indicates the parameters of the experiment reported in Fig. 4. (b), (d), (f), and (h) Normalized intracavity powers (thin lines) as a function of ${\mathrm{\Delta }}_R$ at a driving $S_L$ value of 0.2, 2.0, 2.1, and 2.8, respectively. The thick lines between the bifurcations ${\mathrm{HB}}_{1,2}$ show the maximum and minimum oscillation amplitudes. Note that in (h), ${\mathrm{HB}}_2$ (not shown for clarity) is located at ${\mathrm{SN}}_4$. (c), (e), (g), and (i) Corresponding forward and backward experimental scans. See main text for further explanation.

Figure 4

(a) Normalized power in the left (blue) and right (red) resonators, recorded over more than 2000 lifetimes ($10000$ round trips) and demonstrating periodic oscillations. The parameters are $S_L=2.8$, ${\mathrm{\Delta }}_L=3.1$, and ${\mathrm{\Delta }}_R=2.3$. (b) Zoom over $\approx 12$ periods and (c) corresponding trajectory in the $\left(\right|{\psi }_L{|}^2,\left|{\psi }_R{|}^2\right)$-phase plane showing a limit cycle. (d) Sustained oscillations and (e) limit cycle obtained by numerical integration of Eq. (1). (f) Spectrum of the hybrid modes ($\mathcal{F}$ denotes the discrete Fourier transform operator) corresponding to the self-pulsing shown in (d) and (e).