Phonon-Assisted Gain in a Semiconductor Double Quantum Dot Maser

We develop a microscopic model for the recently demonstrated double-quantum-dot maser. In characterizing the gain of this device we find that, in addition to the direct stimulated emission of photons, there is a large contribution from the simultaneous emission of a photon and a phonon, i.e., the phonon sideband. We show that this phonon-assisted gain typically dominates the overall gain, which leads to masing. Recent experimental data are well fit with our model.

Figure 1

(a) Schematic of the combined DQD, cavity, and phonon system. (b) DQD energy spectrum versus detuning $\epsilon$ for $t_c=16.4\mu \mathrm{eV}$, the direct photon emission process, and the phonon sideband for ${\omega }_c/2\pi =8\mathrm{GHz}$. (c) Transport cycle for the DQD maser: a finite source drain bias leads to current flow via single electron tunneling. The interdot charge transition is accompanied by direct photon emission into the cavity (near zero detuning) or a second order process involving emission of a photon and a phonon (large detuning). Electrons tunnel onto(off of) the left(right) dots at rate ${\mathrm{\Gamma }}_{\ell (r)}$.

Figure 2

(a) Population inversion $⟨{\sigma }_z⟩$, direct photon rate $R$, and phonon-assisted emission ${\gamma }_e$ and absorption ${\gamma }_a$ rates as a function of $\epsilon$ with $t_c=16.4\mu \mathrm{eV}$, $g_c/2\pi =90\mathrm{MHz}$, ${\omega }_c/2\pi =8\mathrm{GHz}$, $J(2t_c/\hslash )=5\mathrm{GHz}$, ${\mathrm{\Gamma }}_{\ell }/2\pi ={\mathrm{\Gamma }}_r/2\pi =4\mathrm{GHz}$, and $T_{\mathrm{eff}}=0$. (b) ${\gamma }_e/\kappa$ plotted as a function $\epsilon$ and ${\omega }_c/2\pi$. The dashed line corresponds to ${\omega }_c/2\pi =8\mathrm{GHz}$ used in Ref. [15].

Figure 3

(a) Circles: experimentally measured gain in one of the DQDs with ${\omega }_c/2\pi =8\mathrm{GHz}$, $\kappa /2\pi =2.6\mathrm{MHz}$, $t_c=50(10)\mu \mathrm{eV}$, ${\mathrm{\Gamma }}_{\ell }/2\pi ={\mathrm{\Gamma }}_r/2\pi =17(2)\mathrm{GHz}$, $T_{\mathrm{eff}}=3(1)\mathrm{K}$, and $g_c/2\pi =100(20)\mathrm{MHz}$. Black curve: fit to the theory with $J(2t_c/\hslash )=2.4(2)\mathrm{GHz}$ a free parameter and only including contributions from the lowest phonon branch. The blue curve includes contributions to $J(\omega )$ from the second phonon mode and substrate phonons [31]. Dashed curve: gain neglecting the phonon-assisted contributions. (b) Gain rate function $g(u_0)$ plotted versus $\epsilon$ and ${\omega }_c/2\pi$ with parameters as in Fig. 2. The second peak near $\epsilon =150\mu \mathrm{eV}$ arises from the second phonon mode in the nanowire.