Figure 1

Transition rates in and out of state ($n$, $s$). Positive signs indicate flow into the state and negative signs flow out of the state. $-G_{n}s=(a\Gamma cn/V{n}_r)s$ is the stimulated emission rate in the system at photon energy $\hslash \omega$, $-As=(a\Gamma c{n}_0/V{n}_r)s$ is the stimulated absorption rate, $n_0$ is the transparency carrier number, $c$ is the speed of light in vacuum, $\Gamma$ is the overlap of the optical field intensity with the gain medium, $a$ is the gain slope coefficient, and $n_r$ is the refractive index of the active volume $V$. The total optical loss rate from the Fabry-Perot cavity is $\kappa s=\frac{c}{n_r}\mathbf{(}{\alpha }_i+\frac{1}{2L_c}\ln (\frac{1}{r_1{r}_2})\mathbf{)}s$, where $r_1=r_2$ is the mirror reflectivity, ${\alpha }_i$ is the internal loss, and $L_{\mathrm{c}}$ is the cavity length. $I$ is the injection (pump) current and $e$ is the electron charge.Reuse & Permissions

Figure 2

Steady-state characteristics. (a) Calculated mean photon number as a function of current showing that M.E. predict suppression of lasing threshold relative to mean-field R.E. calculations. Suppression in lasing is due to quantum fluctuations. (b) Calculated mean electron number as a function of current. M.E. show carrier depinning due to quantum fluctuations. (c) Fano-factor $F={\sigma }_s^2/⟨s⟩$ as a function of current, $I$. (d) Electron-photon correlation and product of means versus current. (e) Probability of photons for different currents. (f) Probability of electrons for different currents. Parameters: $V=(0.1\mu \mathrm{m}\times 0.1\mu \mathrm{m}\times 10\mathrm{nm}$), $\Gamma =0.25$, $a=2.5\times {10}^{-18}{\mathrm{cm}}^2{\mathrm{s}}^{-1}$, $B^{\prime }={10}^{-10}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$, $n_0={10}^{18}{\mathrm{cm}}^{-3}$, ${\alpha }_i=1{\mathrm{cm}}^{-1}$, $n_r=4$, $r=1–{10}^{-6}$, $\beta ={10}^{-4}$.Reuse & Permissions

Figure 3

Transient behavior of mean electron number and photon number for a step change in current from $I(t<0)=0$ to $I(t\ge 0)=100\mathrm{\text{electron}}/\mathrm{ns}$ ($=16\mathrm{nA}$). (a) Mean photon number vs time. (b) Mean electron number vs time. The dot at the last time point denotes the mean calculated from the probability distribution obtained by the steady-state technique. (c) $P_{n,s}$ calculated at time $t=1.25\mathrm{ns}$ and indicated by arrow labeled 1 in (a) and (b). (d) $P_{n,s}$ calculated at time $t=5\mathrm{ns}$ and indicated by arrow labeled 2 in (a) and (b). Parameters as in Fig. 2 but with $V=(0.1\mu \mathrm{m}\times 10\mathrm{nm}\times 10\mathrm{nm})$ and $\beta ={10}^{-1}$.Reuse & Permissions

Figure 4

Time evolution of electrons and photons calculated by the random walk method. (a) Current, $I=9.6\mathrm{nA}$. (b) $I=48\mathrm{nA}$. (c) $I=72\mathrm{nA}$. (d) $I=192\mathrm{nA}$. The inset shows discrete step changes in photon number with time. Parameters as in Fig. 2.Reuse & Permissions

Figure 5

Comparison of steady-state laser characteristics for 3 cavities. (a) Normalized mean photon number and Fano factor versus current. (b) Normalized mean electron number versus current. (i) Represents a large cavity, (ii) represents an intermediate cavity, (iii) represents a small cavity (Fig. 2). The large cavity matches the rate equation data closely. The rate equation data for the other two cases are multiplied by constant factors (normalization constants) to overlap the large cavity data. The current values are divided by the respective threshold currents predicted by rate equations ($I_{\mathrm{th}\mathrm{\text{−}}\mathrm{i}}=112\mu \mathrm{A}$, $I_{\mathrm{th}\mathrm{\text{−}}\mathrm{ii}}=320\mathrm{nA}$, $I_{\mathrm{th}\mathrm{\text{−}}\mathrm{iii}}=12.8\mathrm{nA}$). Normalization constants, $N$ are Mean photon number, $N(s{)}_{\mathrm{i}}=1$, $N(s{)}_{\mathrm{ii}}=350$, $N(s{)}_{\mathrm{iii}}=800$. Mean electron number, $N(n{)}_{\mathrm{i}}=1$, $N(n{)}_{\mathrm{ii}}=607$, $N(n{)}_{\mathrm{iii}}=21594$. Fano factor, $N(F{)}_{\mathrm{i}}=10$, $N(F{)}_{\mathrm{ii}}=10$, $N(F{)}_{\mathrm{iii}}=100$. Parameters: $V_{\mathrm{i}}=(5\mu \mathrm{m}\times 1\mu \mathrm{m}\times 1\mu \mathrm{m})$, $V_{\mathrm{ii}}=(5\mu \mathrm{m}\times 0.1\mu \mathrm{m}\times 10\mathrm{nm})$, $V_{\mathrm{iii}}=(0.1\mu \mathrm{m}\times 0.1\mu \mathrm{m}\times 10\mathrm{nm})$, ${\Gamma }_{\mathrm{i},\mathrm{iii}}=0.25$, ${\Gamma }_{\mathrm{ii}}=0.05$, $a_{\mathrm{i},\mathrm{ii}}=2.5\times {10}^{-16}{\mathrm{cm}}^2{\mathrm{s}}^{-1}$, $a_{\mathrm{iii}}=2.5\times {10}^{-18}{\mathrm{cm}}^2{\mathrm{s}}^{-1}$, $B^{\prime }={10}^{-10}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$, $n_0={10}^{18}{\mathrm{cm}}^{-3}$, ${\alpha }_i=10{\mathrm{cm}}^{-1}$ for (i),(ii), ${\alpha }_i=1{\mathrm{cm}}^{-1}$ for (iii), $n_r=4$, $r_{\mathrm{i},\mathrm{ii}}=0.999$, $r_{\mathrm{iii}}=1–{10}^{-6}$ and $\beta ={10}^{-4}$.Reuse & Permissions