Lasing and transport in a quantum-dot resonator circuit

We study a double quantum-dot system coherently coupled to an electromagnetic resonator. A current through the dot system can create a population inversion in the dot levels and, within a narrow resonance window, a lasing state in the resonator. The lasing state correlates with the transport properties. On one hand, this allows probing the lasing state via a current measurement. On the other hand, the resulting narrow current peak allows resolving small differences in the dot properties (e.g., a small difference in the Zeeman splittings of the two dots). For realistic situations relaxation processes have pronounced consequences. Remarkably, they may even enhance the resolution between different spin states by releasing a trapped population in the off-resonant spin channel.

Figure 1

Illustration of a double quantum-dot-resonator circuit. The dot is placed at a maximum of the electric field of the transmission line to maximize the dipole interaction with the resonator. For gate-defined quantum dots most parameters can be tuned by the applied voltages.

Figure 2

Tunneling sequence in the double dot system. The chemical potentials ${\mu }_{\mathrm{L}}(1,0)$ and ${\mu }_{\mathrm{R}}(0,1)$ of the two dots are assumed to be arranged as indicated.

Figure 3

Average photon number $⟨n⟩$, compared with the analytical result ${⟨n⟩}_{\mathrm{a}}$ in Eq. (9), Fano factor $F$, decay rate $\lambda$, and transport current $I$ as functions of the detuning with $\Gamma ={10}^{-3}{\omega }_{\mathrm{r}}$ and $t=0.3\hslash {\omega }_{\mathrm{r}}$. Throughout this paper we choose the bare coupling strength $g_0={10}^{-3}{\omega }_{\mathrm{r}}$.

Figure 4

Average photon number and Fano factor versus the bare energy difference $\epsilon$ and the incoherent tunneling rate $\Gamma$ for interdot tunneling strength $t=0.1\hslash {\omega }_{\mathrm{r}}$ and vanishing relaxation and decoherence rates in the dot system.

Figure 5

Current through the double-dot system as a function of the bare energy difference $\epsilon$. The incoherent tunneling rate is $\Gamma ={10}^{-3}{\omega }_{\mathrm{r}}$, while the relaxation and decoherence are set to zero. We choose the interdot tunneling strength $t=0.3\hslash {\omega }_{\mathrm{r}}$, creating a nearly maximized population inversion ${\tau }_0\simeq 1$ as well as a strong enough coupling strength $g\simeq 3\times {10}^{-4}{\omega }_{\mathrm{r}}$ to optimally satisfy the requirement (7). The resonance with the oscillator occurs around $\epsilon \simeq 0.95\hslash {\omega }_{\mathrm{r}}$, leading to the sharp peak in the current.

Figure 6

Average photon number $⟨n⟩$, Fano factor $F$, decay rate $\lambda$, and current $I$ through the double dot as functions of the detuning for different relaxation rates but vanishing decoherence in the left column, and for different decoherence rates but vanishing relaxation in the right column. Other parameters are chosen to be the same in Fig. 5.

Figure 7

Eigenenergies for the spin-up channel, $E_{\mathrm{e}\uparrow }$ and $E_{\mathrm{g}\uparrow }$ (red solid lines), and spin-down channel, $E_{\mathrm{e}\downarrow }$ and $E_{\mathrm{g}\downarrow }$ (blue dotted lines), in the presence of different Zeeman splittings in each dot. The two spin channels become resonant with the radiation field at separate values of the bare energy difference (denoted as ${\epsilon }_{\uparrow ,\downarrow }$), as indicated by the double-headed arrows. For large $\epsilon \gg t$, the eigenstates of both channels approach the basis states $|\downarrow ,0⟩$, $|0,\downarrow ⟩$, $|\uparrow ,0⟩$, and $|0,\uparrow ⟩$, as shown in the figure.

Figure 8

Pumping with spin-split states induced by a bias voltage across the double dots. The eigenstates in both spin channels are connected to the common state $|0,0⟩$ with transition rates depending on the mixing angles ${\theta }_{\uparrow /\downarrow }$. For an effective pumping in both channels, the transitions from the ground state $|g_{\uparrow /\downarrow }⟩$ via $|0,0⟩$ to the corresponding excited state $|e_{\uparrow /\downarrow }⟩$, denoted by solid arrows, are stronger than those flowing backward (dotted arrows). The energy of state $|0,0⟩$ is drawn at a position for a clear illustration.

Figure 9

Average photon number and current in the absence of both relaxation and decoherence. The two spin-resolved resonances ${\epsilon }_{\uparrow ,\downarrow }$ are marked by the green solid lines, which distance gives the difference in Zeeman splittings $h_{-}$. In the discussion with spin-split states, we choose the interdot tunneling strength $t=0.5\hslash {\omega }_{\mathrm{r}}$ and incoherent tunneling rate $\Gamma =0.008{\omega }_{\mathrm{r}}$.

Figure 10

Total population in each spin channel ${\rho }_{\uparrow /\downarrow }^{\mathit{tot}}$ as a function of bare energy difference $\epsilon$. Results obtained without relaxation are indicated by dotted lines while those with a relaxation rate ${\Gamma }_{\downarrow }={10}^{-3}{\omega }_{\mathrm{r}}$ by solid lines. Here we choose $h_{-}=0.025\hslash {\omega }_{\mathrm{r}}$.

Figure 11

Average photon number and transport current as functions of the bare energy difference $\epsilon$ with $h_{-}=0.025\hslash {\omega }_{\mathrm{r}}$ for different relaxation rates.

Figure 12

Average photon number and current in the presence of both relaxation and decoherence with rates ${\Gamma }_{\downarrow }={\Gamma }_{\varphi }^{*}=3\times {10}^{-3}{\omega }_{\mathrm{r}}$.