Mesoscopic Limit Cycles in Coupled Nanolasers

Two coupled nanolasers exhibit a mode switching transition, theoretically described by mode beating limit cycle oscillations. Their decay rate is vanishingly small in the thermodynamic limit, i.e., when the spontaneous emission noise tends to zero. We provide experimental statistical evidence of mesoscopic limit cycles ($\sim {10}^3$ intracavity photons). Specifically, we show that the order parameter quantifying the limit cycle amplitude can be reconstructed from the mode intensity statistics. We observe a maximum of the averaged amplitude at the mode switching, accounting for limit cycle oscillations. We finally relate this maximum to a dip of mode cross-correlations, reaching a minimum of $g_{ij}^{(2)}=2/3$, which we show to be a mesoscopic limit. Coupled nanolasers are thus an appealing test bed for the investigation of spontaneous breaking of time translation symmetry in the presence of strong quantum fluctuations.

Figure 1

Numerical simulations of mode beating limit cycles in two strongly coupled nanolasers. (a) Schematics of two evanescently coupled laser nanocavities, with coupling strength $K$. Bottom frame: SEM image of the fabricated indium phosphide photonic crystal cavities; right: the two eigenmodes ($B$: bonding, $A$: antibonding) are split in energy. (b) Bloch sphere showing noiseless limit cycles close to a switching point [$F_{a_{1,2}}(t)=0$ in Eq. (1)]; blue trajectories correspond to orbits increasing pumping ($P/P_0$, blue labels); other parameters [Eqs. (1) and (2)] are $K=12\kappa$, $\gamma =0.05\kappa$, $\alpha =7$, ${\gamma }_{\parallel }=2.2\mathrm{GHz}$, ${\gamma }_{\mathrm{tot}}=5\mathrm{GHz}$, $\kappa =140.84\mathrm{GHz}$, $n_0={10}^{18}{\mathrm{cm}}^{-3}\times V_a$ with $V_a=0.016\times {10}^{-12}{\mathrm{cm}}^3$. (c),(d) Ensemble average of the intensity in the cavity 1 for $P/P_0=6.015$; the average is taken over 100 different noise realizations and the same initial conditions (see text). Insets: enlargement of a few oscillation periods (left) and histogram of the limit cycle phase $\phi$ (right, color map; average and standard deviation in black and red lines, respectively). (e),(f) Mean value and fluctuations of the order parameter as a function of the pump power $P$ normalized to the transparency pump $P_0$. (c),(e) $\beta =1.7\times {10}^{-5}$ and $V_a=16\times {10}^{-12}{\mathrm{cm}}^{-3}$. (d),(f) $\beta =0.017$ and $V_a=0.016\times {10}^{-12}{\mathrm{cm}}^3$. Other parameters are $F_P=1.03$ and $B=3\times {10}^{10}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$; $P_{\mathrm{th},B}/P_0=1.4496$.

Figure 2

Experimental time traces as the pump power is ramped up. (a) Intensity traces for $B$ (blue) and $A$ (red) modes measured with two APD detectors. Pump ramp duration = 6 ns. Thick lines: average corresponding to ${10}^4$ time traces. (b) Second order correlations (left axis): $g_{BB}^{(2)}(0)$ (blue), $g_{AA}^{(2)}(0)$ (red) and $g_{BA}^{(2)}(0)$ (green); moments of the mode population imbalance [$⟨x⟩$ (black) and $(\mathrm{\Delta }x{)}^2$ (grey)] computed using the full statistics (right axis). (c) Two first moments of the order parameter $\mathcal{A}$: mean value $⟨\mathcal{A}⟩$ (blue, left axis) and variance $(\mathrm{\Delta }\mathcal{A}{)}^2$ (yellow, right axis).

Figure 3

Experimental time-resolved cross-correlation around a mode switching transition. (a) Time traces of $B$ (blue) and $A$ (red) modes (thick lines: average over 100 plateaus). The switching is induced by a slow thermal drift. (b) Time dependent correlations $g_{BA}^{(2)}(t,\tau )$; the regions before, near, and after the switching point are framed in blue, red, and yellow, respectively. (c) Average of $g_{BA}^{(2)}(t,\tau )$ over $t$ within the boxes of (b). (d) Blowup of the normalized cross-correlation for positive $\tau$.

Figure 4

Minimum of cross-correlations (green) and maximum of the squared mean limit cycle amplitude (blue squares) between eigenmodes of a laser dimer for increasing system size ${\beta }^{-1}$. Green line: guide to the eye. Right column: zero time delay intensity correlations $g_{ij}^{(2)}$ [$(i,j)=(B,B)$: blue, $(A,A)$: red, and $(B,A)$: green lines] as a function of the pump around the mode switching point, showing a double dip in the macroscopic regime (a) and a single dip in the mesoscopic regime (b). Red symbol: experimental result.