Nondegenerate two-photon lasing in a single quantum dot

We propose a two-mode, two-photon microlaser using a single semiconductor quantum dot (QD) grown inside a two-mode microcavity. We explore both incoherent and coherent pumping at low temperatures to achieve suitable conditions for the lasing. The two-mode, two-photon-stimulated emission is strongly suppressed when exciton-phonon interactions occur, but the single-photon-stimulated emission is enhanced in this case. In coherently pumped QDs, one can achieve large two-mode, two-photon lasing, while the single-photon lasing is almost absent here. We further extend our study for the generation of a two-mode entangled state using two-photon resonant pumping.

Figure 1

The schematic energy level diagram for the quantum dot (QD)-cavity model. Left solid blue arrows indicate incoherent (coherent) pumping from $|g\rangle \to |u\rangle$, with pumping rates ${\eta }_1\left({\mathrm{\Omega }}_1\right)$ and ${\eta }_2\left({\mathrm{\Omega }}_2\right)$. Right dashed green and dashed dot red arrows indicate cavity coupling with $|y\rangle \to |g\rangle$ and $|u\rangle \to |y\rangle$, with coupling constants $g_1$ and $g_2$ and frequencies ${\omega }_1$ and ${\omega }_2$. Here, ${\mathrm{\Delta }}_{xx}$ and ${\delta }_x$ stand for the biexciton binding energy and fine structure splitting energy between two exciton states, respectively. The effective detunings for the first and second cavity modes are ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ respectively.

Figure 2

The steady-state populations in quantum dot (QD) states $|u\rangle$ (blue short dash), $|x\rangle$ (red solid), $|y\rangle$ (black short dash dot), and $|g\rangle$ (green long dash) as a function of ${\mathrm{\Delta }}_1$ for phonon bath temperature T = 5 K. Other parameters are cavity leakage ${\kappa }_1={\kappa }_2=0.1g$, cavity field couplings $g_1=g_2=g$, spontaneous decay rates ${\gamma }_1={\gamma }_2=0.01g$, pure dephasing rate ${\gamma }_d=0.01g$, biexciton binding energy ${\mathrm{\Delta }}_{xx}=15g$, hyperfine energy gap between excitons ${\delta }_x=-g$, detuning of second cavity mode ${\mathrm{\Delta }}_2=-5g$, and ${\eta }_1={\eta }_2=0.5g$.

Figure 3

(a) The average photon number of first and second modes $\langle n_1\rangle$ (blue solid) and $\langle n_2\rangle$ (red dash); (b) Fano factors $F_1$ (green solid) and $F_2$ (orange dash); and (c) second-order photon correlation function with zero time delay $g_1^{\left(2\right)}\left(0\right)$ (cyan solid), $g_2^{\left(2\right)}\left(0\right)$ (pink dash), and $g_{12}^{\left(2\right)}\left(0\right)$(brown dash-dot) varying with first cavity mode detuning ${\mathrm{\Delta }}_1$ for temperature T = 5 K. All other parameters are the same as Fig. 2.

Figure 4

The steady-state populations in quantum dot (QD) states $|u\rangle$ (blue short dash), $|x\rangle$ (red solid), $|y\rangle$ (black short dash-dot), and $|g\rangle$ (green long dash) vs incoherent pumping rate $\eta$ for phonon bath temperature T = 5 K. All other parameters are the same as Fig. 2 except ${\mathrm{\Delta }}_1=5g$ and ${\eta }_1={\eta }_2=\eta$.

Figure 5

(a) Average photon number for first and second modes $\langle n_1\rangle$ (blue solid) and $\langle n_2\rangle$ (red dash); (b)Fano factors $F_1$ (green solid) and $F_2$ (orange dash); and (c) second-order correlation function with zero time delay $g_1^{\left(2\right)}\left(0\right)$ (cyan solid), $g_2^{\left(2\right)}\left(0\right)$ (pink dash), and $g_{12}^{\left(2\right)}\left(0\right)$ (brown dash-dot) as a function of $\eta$ for temperature T = 5 K. All other parameters are the same as Fig. 2 except ${\mathrm{\Delta }}_1=5g$ and ${\eta }_1={\eta }_2=\eta$.

Figure 6

First-mode single-photon emission rate (SPE1, red solid), second-mode single-photon emission rate (SPE2, green dash), and two-photon stimulated emission rate (TPE, blue dash dot) dependency on incoherent pumping rate using ${\mathrm{\Delta }}_1=5g_1$, ${\eta }_1={\eta }_2=\eta$ for (a) T = 5 K and (b) T = 20 K with all other parameters the same as Fig. 2.

Figure 7

The steady-state populations in the various quantum dot energy states $|u\rangle$ (blue short dash), $|x\rangle$ (red solid), $|y\rangle$ (black short dash-dot), and $|g\rangle$ (green long dash) plotted against first-cavity-mode detuning ${\mathrm{\Delta }}_1$ for phonon bath temperature T = 5 K. Other parameters are the same as Fig. 2 except ${\mathrm{\Delta }}_p=0$ and ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=2g$.

Figure 8

The strong field coupled with the $|g\rangle \leftrightarrow |x\rangle$ transition produces dressed states $|\pm \rangle$ with frequency separation $2{\mathrm{\Omega }}_1$. These dressed states facilitate phonon-assisted pumping to the biexciton state, indicated by black dotted lines. Subsequently, two different values of ${\mathrm{\Delta }}_1$ can be obtained, satisfying the two-photon resonance condition, as illustrated by red and blue dotted lines.

Figure 9

The variation of (a) average photon number $\langle n_1\rangle$ (blue solid) and $\langle n_2\rangle$ (red dash); (b) Fano factors $F_1$ (green solid) and $F_2$ (orange dash); and (c) second-order correlation $g_1^{\left(2\right)}\left(0\right)$ (cyan solid), $g_2^{\left(2\right)}\left(0\right)$ (pink dash), and $g_{12}^{\left(2\right)}\left(0\right)$ (brown dash dot) with respect to ${\mathrm{\Delta }}_1$ at a phonon bath temperature T = 5 K. All other parameters are the same as Fig. 7.

Figure 10

The figure illustrates the single-photon emission rate of the first (SPE1, red solid) and second (SPE2, green dash) modes, as well as the two-mode, two-photon stimulated emission rate (TPE, blue dash-dot) vs the first-mode detuning ${\mathrm{\Delta }}_1$ by considering ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=2g_1$, for (a) T = 5 K and (b) T = 20 K with all other parameters the same as Fig. 7.

Figure 11

The two-photon resonantly pumped steady-state populations variation with ${\mathrm{\Delta }}_1$ in quantum dot energy states $|u\rangle$ (blue short dash), $|x\rangle$ (red solid), $|y\rangle$ (black short dash-dot), and $|g\rangle$ (green long dash) for the same parameters as Fig. 7 except ${\mathrm{\Delta }}_p=7g$.

Figure 12

The graph displays the variation of (a) average photon numbers $\langle n_1\rangle$ (blue solid) and $\langle n_2\rangle$ (red dash); (b) Fano factors $F_1$ (green solid) and $F_2$ (orange dash); and (c) second-order correlations $g_1^{\left(2\right)}\left(0\right)$ (cyan solid), $g_2^{\left(2\right)}\left(0\right)$ (pink dash), $g_{12}^{\left(2\right)}\left(0\right)$ (brown dash-dot) as a function of ${\mathrm{\Delta }}_1$ for the same parameters as Fig. 9 except ${\mathrm{\Delta }}_p=7g$.

Figure 13

The total variance of an Einstein-Podolsky-Rosen (EPR)-like variable pair plotted with first-mode, single-photon detuning ${\mathrm{\Delta }}_1$ for different bath temperatures. The black dotted line indicates the margin for entanglement phenomena according to the Duan-Giedke-Cirac-Zoller (DGCZ) criterion. All the parameters are the same as Fig. 11 except ${\varphi }_1={\varphi }_2=-0.5$, with coherent pumping Rabi frequencies (a) ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=0.5g$ and (b) ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=2g$.