Bistable Photon Emission from a Solid-State Single-Atom Laser

We predict a bistability in the photon emission from a solid-state single-atom laser comprising a microwave cavity coupled to a voltage-biased double quantum dot. To demonstrate that the single-atom laser is bistable, we evaluate the photon emission statistics and show that the distribution takes the shape of a tilted ellipse. The switching rates of the bistability can be extracted from the electrical current and the shot noise in the quantum dots. This provides a means to control the photon emission statistics by modulating the electronic transport in the quantum dots. Our prediction is robust against moderate electronic decoherence and dephasing and is important for current efforts to realize single-atom lasers with gate-defined quantum dots as the gain medium.

Figure 1

Solid-state single-atom laser. (a) Schematic of a DQD coupled to a microwave cavity forming a hybrid circuit-QED structure. (b) The coupling between the cavity (with frequency ${\omega }_c$ and decay rate $\kappa$) and the DQD (with tunnel coupling $\mathrm{\Omega }$ and dealignment $ϵ$) is denoted as $g$. Single electrons tunnel in and out of the DQD at rates ${\mathrm{\Gamma }}_L$ and ${\mathrm{\Gamma }}_R$, respectively. The electronic dephasing rate is $\gamma$. (c) Wigner functions for the cavity photons in the lasing (${\mathrm{\Gamma }}_R=0.55{\omega }_c/2\pi$), intermediate (${\mathrm{\Gamma }}_R=0.84{\omega }_c/2\pi$), and thermal states (${\mathrm{\Gamma }}_R=1.25{\omega }_c/2\pi$). The other parameters are $ϵ=\sqrt{(\hslash {\omega }_c{)}^2-(2\mathrm{\Omega }{)}^2}$, $\mathrm{\Omega }=0.1\hslash {\omega }_c$, ${\mathrm{\Gamma }}_L=10{\omega }_c$, $g=0.05\hslash {\omega }_c$, $\kappa =0.013{\omega }_c/2\pi$, and $\gamma =0.05{\omega }_c/2\pi$.

Figure 2

Electrical current, noise, and switching rates. (a) Average electrical current $⟨I⟩$ and the Fano factor $F=S/e⟨I⟩$ as functions of the coupling to the right lead. The other parameters are chosen as in Fig. 1. Results without the cavity are shown with dashed lines [37]. (b) Switching rates of the bistability extracted from the current and noise using Eq. (6). The circles indicate the values of ${\mathrm{\Gamma }}_R$ for which the cavity Wigner functions are shown in Fig. 1 and the photon emission statistics are shown in Fig. 3.

Figure 3

Photon emission statistics. The solid lines show exact results for the distribution $\ln [P(Jt,t)]/t$ of the photon current $J$ in the lasing state, the intermediate regime, and the thermal state corresponding to the values of ${\mathrm{\Gamma }}_R$ indicated in Fig. 2. Together with the exact results we show ellipses whose forms are determined by the switching rates in Fig. 2. The ellipses are delimited by the average emission rates in the lasing $J_{\mathbb{L}}$ and thermal $J_{\mathbb{T}}$ states (the dashed lines). The fact that the distributions have the shape of a tilted ellipse demonstrates that the single-atom laser is bistable.

Figure 4

Electronic skewness. The green line shows the skewness $⟨(\widehat{I}-⟨\widehat{I}⟩{)}^3⟩$. The yellow line is the skewness for a bistable system, $⟨(\widehat{I}-⟨\widehat{I}⟩{)}^3⟩=6(I_{\mathbb{T}}-I_{\mathbb{L}}{)}^3{\mathrm{\Gamma }}_{\mathbb{T}\leftarrow \mathbb{L}}{\mathrm{\Gamma }}_{\mathbb{L}\leftarrow \mathbb{T}}({\mathrm{\Gamma }}_{\mathbb{L}\leftarrow \mathbb{T}}-{\mathrm{\Gamma }}_{\mathbb{T}\leftarrow \mathbb{L}})/({\mathrm{\Gamma }}_{\mathbb{L}\leftarrow \mathbb{T}}+{\mathrm{\Gamma }}_{\mathbb{T}\leftarrow \mathbb{L}}{)}^5$ [44]. The agreement between the curves constitutes additional evidence for the bistability based on electrical measurements only.