Two-photon lasing by a single quantum dot in a high-$Q$ microcavity

We investigate theoretically two-photon processes in a microcavity containing one quantum dot in the strong-coupling regime. The cavity mode can be tuned to resonantly drive the two-photon transition between the ground and the biexciton states while the exciton states are far-off resonance due to the biexciton binding energy. We study the steady state of the quantum dot and cavity field in presence of a continuous incoherent pumping. We identify the regime where the system acts as two-photon emitter and discuss the feasibility and performance of realistic single quantum-dot devices for two-photon lasing.

Figure 1

(Color online) (a) A high-$Q$ microcavity coupled to a single quantum dot can be experimentally realized, for instance, by embedding the quantum dot in a photonic crystal (Ref. 24). In (b) the ground $\left(|G⟩\right)$, excitons $\left(|E_1⟩,|E_2⟩\right)$ , and biexciton $\left(|B⟩\right)$ energy levels of the quantum dot are displayed, the arrows indicate the coupling due to the resonator mode, the horizontal dashed line shows the frequency of the cavity-mode photon (see Sec. 2 for details on the parameters). The cavity mode is resonantly coupled with the biexciton-ground-state transition via two-photon processes. Due to the biexciton-binding-energy shift, the exciton states are coupled far-off resonance. In this regime we explore the performance of the system as a two-photon laser.

Figure 2

(Color online) Energy levels of the QD for different cavity-mode frequencies, when the biexciton is bound $\left(\chi >0\right)$. In (I) the cavity mode is resonant with the transition $|G⟩\leftrightarrow |E⟩$, namely, $\delta =-\chi \left({\omega }_E={\omega }_a\right)$. In (II) the $|G⟩\leftrightarrow |B⟩$ transition couples resonantly with two photons of the cavity mode, $\delta \simeq 0\left(2{\omega }_a\simeq {\omega }_B\right)$. In (III) the cavity mode is resonant with the exciton-biexciton transition $|E⟩\leftrightarrow |B⟩$ ($\delta =\chi$, corresponding to ${\omega }_a={\omega }_B-{\omega }_E$).

Figure 3

(Color online) Populations of the QD levels $p_{\xi }\left(t\right)$ as a function of time, in units of $1/g$, for the initial state $|G,2⟩$, obtained by evaluating numerically the equation $p_{\xi }\left(t\right)={\sum }_n{\left|⟨\xi ,n|{\text{e}}^{-iHt/\hslash }|G,2⟩\right|}^2$. The curves correspond to $p_B$ (solid blue line), $p_E=p_{E_1}+p_{E_2}$ (dotted purple), and $p_G$ (dashed brown). In (a) $\delta =-20g$, (b) $\delta =-2g$, and (c) $\delta =-{\delta }_{2P}$ while $\chi =20g$ in all three cases. See also Fig. 2.

Figure 4

(Color online) Amplitude of oscillations of the population of the QD states, $V_{\xi }=\text{max}\left\{p_{\xi }\left(t\right)\right\}-\text{min}\left\{p_{\xi }\left(t\right)\right\}$ as a function of $\delta$, in units of $g$, with $\chi =20g$ and initial state $|B,0⟩$. The curves correspond to $V_B$ (solid blue line), $V_E$ (dotted purple), and $V_G$ (dashed brown).

Figure 5

(Color online) Steady-state populations of the QD energy levels as a function of the cavity detuning $\delta$ (in units of $g$) for $\kappa =0.1g$, ${\Gamma }_d=0$, $\gamma =0.05g$, $\chi =20g$ and pump rate (a) $P=0.16g$, (b) $P=0.5g$, (c) $P=0.8g$, (d) $P=1.8g$. The curves correspond to the occupation of the states $|B⟩$ (solid line), $|E_j⟩$ (dotted line), and $|G⟩$ (dashed line). Two-photon resonance is at $\delta =-{\delta }_{2P}\sim 0$. At $\delta =20g$ transition $|B⟩\leftrightarrow |E_j⟩$ couples resonantly with the cavity mode.

Figure 6

(Color online) (a) Average photon number $⟨\widehat{n}⟩$, (b) Fano factor $F$, and (c) $g^{\left(2\right)}\left(0\right)$ as a function of the cavity detuning $\delta$ (in units of $g$) for $\kappa =0.1g$, ${\Gamma }_d=0$, $\gamma =0.05g$, $\chi =20g$ and pump strengths $P=0.16g$ (dashed lines), $P=0.5g$ (solid lines), $P=0.8g$ (dotted lines), $P=1.8g$ (dotted-dashed lines). The horizontal dotted lines indicate (a) the value $F=1$, (b) the values $g^{\left(2\right)}\left(0\right)=1$ (Poisson distribution) and (c) $g^{\left(2\right)}\left(0\right)=2$ (thermal distribution).

Figure 7

(Color online) Same as in Fig. 6 but for a fixed value of the pump rate, $P=0.5g$, and for different values of the radiative decay rate: $\gamma =0.05g$ (solid line), $\gamma =0.1g$ (dot-dashed line), and $\gamma =0.5g$ (dashed line).

Figure 8

(Color online) (a) Average photon number, (b) Fano factor, and (c) $g^2\left(0\right)$ as a function of the pumping rate $P$ (in units of $g$) for $\delta =0$, ${\Gamma }_d=0$, $\gamma =0.05g$, $\chi =20g$, and different values of the cavity decay rate: $\kappa =0.2g$ (dashed line), $\kappa =0.1g$ (solid line), and $\kappa =0.05g$ (dot-dashed line).

Figure 9

(Color online) (a) Cavity-field average photon number $⟨\widehat{n}⟩$ at steady state as a function of the dephasing rate ${\Gamma }_d$ in units of $g$. The solid line corresponds to $\delta =0$ (two-photon resonance) and the dashed line to $\delta =\chi$ (one-photon resonance). (b) Average photon number, (c) Fano factor, and (d) $g^{\left(2\right)}\left(0\right)$ as a function of the two-photon detuning $\delta$ in units of $g$. The dot-dashed line corresponds to ${\Gamma }_d=0$, the solid line to ${\Gamma }_d=0.1g$, and the dashed line to ${\Gamma }_d=0.5g$. The other parameters are $P=0.5g$, $\gamma =0.05g$, and $\kappa =0.1g$.

Figure 10

(a) Spectrum of the cavity-field intensity at the output for different values of the pump rate $P$ with $\kappa =10g$, $\delta =0$, $\chi =2000g$, and $\gamma =5g$. The curves refer to $P=g$ (dotted line), $P=5g$ (dotted-dashed line), $P=50g$ (dashed line), and $P=100g$ (solid line). The plot in (b) shows the curves in the interval of frequency close to $\omega -{\omega }_a=0$ and plotted in linear scales. The thick gray curves in (a) and (b) indicate the (Lorentzian) emission line of the cavity in absence of the coupling with the quantum dot. The plot (c) shows the linewidth at half height of the peaks in (b) as a function of $P$. The horizontal dashed line indicates the value of $\kappa$.

Figure 11

Same as Fig. 10 but with $\kappa =0.1g$, $\delta =0$, $\chi =20g$, $\gamma =0.05g$ and $P=0.01g$ (dotted line), $P=0.05g$ (dotted-dashed line), $P=0.5g$ (dashed line), $P=1.8g$ (solid line).

Figure 12

(a) Time evolution of the variance $\left[\Delta X\left(\varphi \right)=⟨X^2⟩-{⟨X⟩}^2\right]$ of two orthogonal quadrature of the field, defined as $X=a{\text{e}}^{-i\varphi }+a^{†}{\text{e}}^{i\varphi }$ (the horizontal dotted line indicates the shot-noise limit). The solid (dashed) lines correspond to the quadrature $\Delta X\left(\varphi \right){\mid }_{\text{min}}\left[\Delta X\left(\varphi +\pi /2\right){\mid }_{\text{max}}\right]$ with minimum (maximum) value at the given time. The corresponding phase $\varphi$ of the quadrature with minimum variance is plotted in (b). The field is initially prepared in a coherent state with average photon number $⟨\widehat{n}⟩=4$. The black lines are found for $P=0.01g$, $\gamma =0.01g$, and $\kappa =0.01$ whereas the gray lines are found for larger rates of incoherent processes: $P=0.1g$, $\gamma =0.05g$, and $\kappa =0.02$. The other parameters are $\chi =50g$ and $\delta =-{\delta }_{2P}$.

Figure 13

Sketch of the processes considered in the rate equation (18). The horizontal lines indicate the energy levels of the cavity mode, the vertical arrows indicate the flow of probability in and out of the energy level at $n$ photons with the corresponding rate.

Figure 14

(Color online) Average cavity-photon number $⟨\widehat{n}⟩$, Fano factor $F$, and $g^2\left(0\right)$, evaluated numerically from master equation (11) (solid lines) and analytically using rate equation (18) (dashed lines) as a function of the pumping rate $P$ (in units of $g$). The parameters are $\chi =50g$, $\delta ={\delta }_{2P}$, ${\Gamma }_d=0$, $\gamma =4g$, and $\kappa =0.02g$. When the pump is sufficiently small, $P<\gamma$, such that the number of intracavity photon does not saturate the biexciton-ground-state transition, the rate equation provides a good description of the dynamics at all times. In this regime the system behaves as a laser below threshold and the cavity field approaches a thermal state.

Figure 15

Cavity emission spectrum. The solid-thick-gray lines correspond to the situation in which all four transitions have equal coupling strength, namely, $\alpha =\beta =1$ and $g_{G1}=g_{1B}\equiv g$ while the dashed-thick-gray lines give the cavity emission spectrum in absence of coupling with the quantum dot. The solid black lines are found for (a) $\alpha =\beta =0$, $g_{G1}=g_{1B}=\sqrt{2}g$, $\delta =-0.2g$, and $ϵ=0$; (b) $\alpha =\beta =1$, $g_{G1}=1.5g$, $g_{1B}=0.66g$, $\delta =-3g$, and $ϵ=0$; (c) $\alpha =\beta =0$, $g_{G1}=2g$, $g_{1B}=g$, $\delta =-2.7g$, and $ϵ=0$; (d) $\alpha =\beta =1$, $g_{G1}=2g$, $g_{1B}=0.5g$, $\delta =-4.2g$, and $ϵ=0$. The crosses are found by setting $ϵ=g$. The other parameters are $\chi =20g$, $\gamma =0.05g$, ${\Gamma }_d=0$, $P=1.8g$, and $k=0.1g$. The effective two-photon coupling is the same for all the curves, ${\tilde{g}}_{\text{eff}}=-g^2/\chi$. In each plot it is explicitly indicated the steady-state average photon number corresponding to each curve. The insets depict the coupling scheme for each set of parameters.