Proposal for an Optical Laser Producing Light at Half the Josephson Frequency

We describe a superconducting device capable of producing laser light in the visible range at half of the Josephson generation frequency with the optical phase of the light locked to the superconducting phase difference. It consists of two single-level quantum dots embedded in a $p\mathrm{\text{−}}n$ semiconducting heterostructure and surrounded by a cavity supporting a resonant optical mode. We study decoherence and spontaneous switching in the device.

Figure 1

(a) The Josephson LED: electron (above) and hole (below) QD levels are close to the chemical potentials ${\mu }_{e,h}$ of the SC leads which differ by an energy $eV$. Charge transfer is possible only either through electron-hole recombination with the emission of a red photon at $\frac{1}{2}{\omega }_J$ or through a Cooper pair transfer with the emission of a blue photon at ${\omega }_J$. (b) The HJL is a Josephson LED embedded in an optical cavity with a resonance frequency ${\omega }_0\approx eV/\hslash$, i.e., close to the red emission frequency. The separately colored regions in between the depleted areas represent the two QDs.

Figure 2

SSRs in the even states for QD parameters given in the text. The five eigenstates are labeled with numbers. (a) Lasing thresholds $G_c$ for three eigenstates. At the line with the number $m$, the energy gain at $\lambda =0$ for the state $|m⟩$ exactly equals the energy loss. (b) Number of photons $n$ for eigenstate $|2⟩$ versus the coupling constant $G$ for $\omega /\Gamma =-0.1$ [dotted line in (a)] above the lasing threshold at $G=G_c\approx 0.5\sqrt{\hslash \Gamma E}$. Plots (c) and (d) illustrate the radiation amplitudes ${\lambda }_s^m$ (marked with circles) for SSRs corresponding to the different eigenstates of the QD. The parameter choice is given by the cross in (a) where only the states $|2⟩$ and $|3⟩$ are lasing. (c) Nonzero ${\lambda }_s^m$ come in pairs with opposite sign. Changing the SC phase difference will rotate the ${\lambda }_s^m$ with respect to the origin of the plot. (d) Eigenenergies versus $\lambda$ (at $\mathrm{Im}\lambda =0$). The ${\lambda }_s^m$ are different for each $|m⟩$.

Figure 3

Sketch of the radiation intensity $|\lambda {|}^2$ evolving in time. (a) Switching events not altering the parity of the QDs occur at the time scale $\simeq {\Gamma }_{\mathrm{SW}}^{-1}$. After switching, the radiation amplitude attains its new stationary value at a time scale $\simeq {\Gamma }^{-1}\ll {\Gamma }_{\mathrm{SW}}^{-1}$ during which the QD remains in the same eigenstate. (b) Switching events altering the parity occur at a longer time scale $\simeq {\Gamma }_{e\mathrm{\text{−}}o}^{-1}\gg {\Gamma }_{\mathrm{SW}}^{-1}$. These can change between the dark and lasing states.