Pump-Power-Driven Mode Switching in a Microcavity Device and Its Relation to Bose-Einstein Condensation

We investigate the switching of the coherent emission mode of a bimodal microcavity device, occurring when the pump power is varied. We compare experimental data to theoretical results and identify the underlying mechanism based on the competition between the effective gain, on the one hand, and the intermode kinetics, on the other. When the pumping is ramped up, above a threshold, the mode with the largest effective gain starts to emit coherent light, corresponding to lasing. In contrast, in the limit of strong pumping, it is the intermode kinetics that determines which mode acquires a large occupation and shows coherent emission. We point out that this latter mechanism is akin to the equilibrium Bose-Einstein condensation of massive bosons. Thus, the mode switching in our microcavity device can be viewed as a minimal instance of Bose-Einstein condensation of photons. Moreover, we show that the switching from one cavity mode to the other always occurs via an intermediate phase where both modes are emitting coherent light and that it is associated with both superthermal intensity fluctuations and strong anticorrelations between both modes.

Popular Summary

All quantum particles can be classified as either fermions or bosons, which cannot or can occupy the same state, respectively. Under suitable conditions, even a macroscopically large share of bosons can occupy the same state. Two famous examples are lasers, which emit photons predominately from one state, and Bose-Einstein condensates, an exotic phase of matter that occurs when massive bosons such as helium atoms are cooled to near absolute zero. In this paper, we show a connection between these two phenomena, in a photonic system behaving either like a laser or a Bose-Einstein condensate.

Using experimental and theoretical analyses, we investigate how a bimodal microresonator—a tiny laser with two orthogonal polarized resonator modes—responds as its pump power is increased. We find that the microresonator undergoes a transition from lasing to an exotic regime corresponding to the Bose-Einstein condensation of photons. For weaker (but still sufficiently strong) pumping, the majority of the photons are found to occupy the mode favored by the largest gain-to-loss ratio, and this mode emits coherent radiation. For strong pumping, it is the (photon-number conserving) intermode kinetics alone that dictates which of the modes is occupied by the majority of the photons and emits coherently.

Our results not only reveal the mechanism responsible for mode switching in a bimodal photonic resonator but also the minimal requirements for Bose-Einstein condensation of photons. Some of our observations could also be relevant to more efficient fluorescence microscopy, as well as the design of optical memories.

Figure 1

Illustration of the relevant processes in our bimodal system. The intermode kinetics between the modes $h$ and $l$ is determined by the rates $R_{i\to j}$. The modes lose their photons with rates ${\ell }_i$. This loss is compensated by new excitation from the quantum dots (QDs) with gain rates $g_i$. The quantum dots are pumped with rate $P$, and they lose their excitations spontaneously with rate ${\tau }^{-1}$.

Figure 2

Measured and calculated microcavity input-output characteristics (high-effective-gain mode in red, low-effective-gain mode in gray). The injection current $I$ is proportional to the pump rate $P$. Here, circles (connected to guide the eye), solid curves, and dashed curves refer to experimentally measured data, exact Monte Carlo results, and asymptotic mean-field theory, respectively. Panel (a) depicts the mean occupations of the modes and the number of excited carriers (light green) in arbitrary units. The colored bars at the bottom of this panel mark the phases determined by the asymptotic theory (see Sec. 4c). Panel (b) shows the autocorrelation $g_{ii}^{(2)}$ for both modes in comparison with the value 2 expected for thermal emission (dotted black line). Panel (c) depicts the cross-correlation $g_{hl}^{(2)}$ in comparison to the value 1 for correlated emission (dotted black line). For the theoretical curves, we use $s=0$, $G=0.77$, as well as (in units of the spontaneous loss rate ${\tau }^{-1}$) $g_h=1.6\times {10}^{-3}$, $g_l=2.1\times {10}^{-3}$, ${\ell }_h=2.2\times {10}^{-2}$, ${\ell }_l=3.8\times {10}^{-2}$, $R_{l\to h}=1.7\times {10}^{-4}$, and $A_{h\to l}=8.5\times {10}^{-6}$ (see Appendix pp1).

Figure 3

Phase diagram with phase A for no selected mode, phase B when mode $h$ is selected, phase C when both modes are selected, and phase D when mode $l$ is selected. Plotted versus the injection current $I$ that is proportional to pump rate $P$ and the ratio of the effective-gain rates $G=g_l{\ell }_h/g_h{\ell }_l$ with fixed $g_h{\ell }_l/{\ell }_h$. Except for $g_l$, all parameters are the same as in Fig. 2. The dashed line indicates the value of $g_l$ used in Fig. 2. While for $G<1$ all phases are present when increasing the pump rate $P$, for $G>1$ the mode $l$, which is favored by the direct mode coupling, actually becomes the high-effective-gain mode; thus, no mode switching occurs.

Figure 4

Two-mode photon statistics obtained from the reduced master equation (see Appendix pp2) with single-mode statistics for mode $l$ at the bottom (black) and mode $h$ at the right (red) axis, for injection currents $I=12.5\mu \mathrm{A}$, $27.4\mu \mathrm{A}$, $41.7\mu \mathrm{A}$, $60.0\mu \mathrm{A}$ (phases A, B, C, D). These pump rates are also indicated by arrows in Fig. 2. The labels $⟨{\widehat{n}}_h⟩$ and $⟨{\widehat{n}}_l⟩$ mark the position of the expectation value for each mode. For a better comparison, the single-mode statistics are depicted up to 0.025% by solid lines, while the remaining part is shown by dashed lines. Note that the plotted range of photon numbers increases from left to right.

Figure 5

Photon statistics of mode $l$ (${\rho }^{n_l}$) for pump rates below ($P_{BC}-ϵ$, left panel) and above ($P_{BC}+ϵ$, right panel) with $ϵ=0.016\mu \mathrm{A}$. Only in the right panel the photon statistics exhibits a second local maximum. The system parameters are the same as in Fig. 2.

Figure 6

Two-mode photon statistics obtained from the reduced master equation for the same parameters as in Fig. 4, B and C, but with spontaneous transitions fully included between the modes. In contrast to the case with suppressed spontaneous transitions, the statistics are no longer attracted to the axis. The color bar is the same as in Fig. 4. For a better comparison, the single-mode statistics are depicted up to 0.015% by solid lines, while the remaining part is shown by dashed lines.

Figure 7

Comparison of $⟨n_i⟩$ and $g_{ij}^{(2)}$ obtained from the numerical, exact Monte Carlo simulation of Eq. (2) (solid lines) with the ones obtained from the direct solution of the reduced density matrix Eq. (b1) (dashed lines). Note that the pump in the reduced density matrix is scaled to compensate for $\beta \ne 1$.

Figure 8

Calculated input-output characteristics and fluctuation for both modes in the presence of spontaneous transitions between the modes, $s=1$. The fluctuations are restricted to values $g_{ii}^{(2)}\le 2$. All parameters are chosen as in Fig. 2.