Probing a Dissipative Phase Transition via Dynamical Optical Hysteresis

We experimentally explore the dynamical optical hysteresis of a semiconductor microcavity as a function of the sweep time. The hysteresis area exhibits a double power law decay due to the influence of fluctuations, which trigger switching between metastable states. Upon increasing the average photon number and approaching the thermodynamic limit, the double power law evolves into a single power law. This algebraic behavior characterizes a dissipative phase transition. Our findings are in good agreement with theoretical predictions for a single mode resonator influenced by quantum fluctuations, and the present experimental approach is promising for exploring critical phenomena in photonic lattices.

Figure 1

(a) Sketch of a microcavity with mode frequency ${\omega }_0$, loss rate $\gamma$, photon-photon interactions of strength $U$, driven by an electromagnetic field of intensity $I$ and frequency $\omega$. (b) Normalized photon density in a semiconductor microcavity under weak driving. Experimental data points for the sample studied in Figs. 2, 3, and 4 are fitted with a Lorentzian line shape. The shaded area indicates the mean-field bistable regime $\mathrm{\Delta }\equiv \omega -{\omega }_0>\sqrt{3}\gamma /2$ for $U>0$. (c) Mean-field (black curves) and quantum (gray curves) solutions for a cavity with $U=0.0075$ $\gamma$ probed at $\mathrm{\Delta }=\gamma$. In the mean-field solution, the solid and dashed curves are stable and unstable states, respectively. (d) Tunneling time ${\tau }_{\text{tunn}}$ between the two mean-field states. Data points are residence time measurements [35].

Figure 2

(a) Experimental setup: $\lambda /2$, MO, PD, and $\mathrm{EOM}+\mathrm{pol}$, stand for half-wave plate, microscope objective, photodiode, and electro-optic modulator with a polarizer, respectively. The green (purple) traces in the waveform generator and in the oscilloscope are measurements of the incident (transmitted) signals. The hysteresis cycles in the oscilloscope are the transmitted versus the incident signal, overlaid for various scanning times. The colored and black lines in (b)–(e) represent the transmission when the power is ramped down and up, respectively. (b) and (c) show single shot (thin lines) and averages over 1000 realizations (thick lines) of dynamic hysteresis. The scanning time is $t_s=0.11\mathrm{ms}$ in (b), and $t_s=0.43\mathrm{ms}$ in (c). (d) and (e) show dynamic hysteresis averaged over 1500 realizations for values of $t_s$ indicated between the two panels. The dashed line in (d) is the mean-field calculation corresponding to the experiment.

Figure 3

(a) Measured average hysteresis area $A_{\mathrm{av}}$ as a function of $t_s/P_s$, with $P_s$ the scanned power range. Different colors correspond to different values of $\mathrm{\Delta }/\gamma$. The gray and blue lines are power law fits. The exponents fitted in the regime influenced by fluctuations (blue lines) are shown with $2\sigma$ confidence intervals. (b) Calculations of the nonadiabatic range $\delta I$ of the driving intensity using the scaling analysis described in the text. All power laws in (b) have the same exponents retrieved from the fits in (a). The inset in (b) shows the system reaction time ${\tau }_R$ in gray, and the sweep time scale ${\tau }_S$ in black, for $U=0.0075\gamma$, $\mathrm{\Delta }=1.01\gamma$, and $t_s/P_s={10}^{-6}\mathrm{s}/\mu \mathrm{W}$. The dashed lines indicate the nonadiabatic range $\delta I$. The conversion of the theoretical intensity units to the experimental power units is described in the Supplemental Material [35].

Figure 4

(a) Measured average hysteresis area $A_{\mathrm{av}}$ for cavities with different $U/\gamma$ (decreasing from cavity 1 to cavity 3) and approximately equal $\mathrm{\Delta }/\gamma$. Cavity 2 is the same one studied in Figs. 1, 2, 3. (b) Calculations of $\delta I$ as explained in Fig. 3. For the highest and lowest curves $\mathrm{\Delta }/\gamma =1.15\pm 0.1$, while for the middle curve $\mathrm{\Delta }/\gamma =1.13\pm 0.1$, in experiments and calculations. In (a), data for cavity 3 was divided by 80. In (b), data for the smallest $U/\gamma$ was divided by 4, and data for the largest $U/\gamma$ was multiplied by 2. These multiplications (for improving visibility) only shift the curves vertically and do not change the exponent.