Quantum photovoltaic effect in double quantum dots

We analyze the photovoltaic current through a double quantum dot system coupled to a high-quality driven microwave resonator. The conversion of photons in the resonator to electronic excitations produces a current flow even at zero bias across the leads of the double quantum dot system. We demonstrate that due to the quantum nature of the electromagnetic field in the resonator, the photovoltaic current exhibits a double peak dependence on the frequency $\omega$ of an external microwave source. The distance between the peaks is determined by the strength of interaction between photons in the resonator and electrons in the double quantum dot. The double peak structure disappears as strengths of relaxation processes increases, recovering a simple classical condition for maximal current when the microwave frequency is equal to the resonator frequency.

Figure 1

(a) An illustration of a DQD and a transmission line resonator coupled to an external microwave source $\mu$w. (b) A schematic view of the DQD. Electrons are confined to the left (L) and right (R) dots by barrier gates BL, BM, and BR that also control electron tunneling rates between the source, S, and the left dot, the left and right dots, and the right dot and the drain, D, respectively. Electrostatic energies of two quantum dots are defined by the plunger gates, PL and PR, and the PL gate is also connected to an antinode of the resonator, see, e.g., Refs. 9 and 7. (c) Electronic states of the DQD are presented in both the eigenstate basis (solid lines) and the left-right basis (dashed lines). Tunneling from the excited state, $|e\rangle$, to the left/right lead, with rate ${\Gamma }_{e,L/R}$ and from the left/right lead to the ground state, $|g\rangle$, with rate ${\Gamma }_{L/R,g}$ are illustrated by arrows.

Figure 2

The average number of photons in the resonator and the photovoltaic current as functions of level bias $\varepsilon$ for $\mathcal{T}/2\pi =1$ GHz, $F=50\mu {\text{s}}^{-1}$ and ${\omega }_0/2\pi =8\text{GHz}$. The current is generated near the resonant condition when $\varepsilon =\pm \sqrt{\hslash {}^2{\omega }_0^2-4{\mathcal{T}}^2}$ (vertical lines). The three curves represent different dephasing rates ${\gamma }_{\varphi }$ of the DQD.

Figure 3

The average number of photons in the resonator and the photovoltaic current as a function of the frequency $\omega$ of the microwave drive for $\mathcal{T}/2\pi =1$ GHz, $F=50\mu {\text{s}}^{-1}$ and $\Omega ={\omega }_0=2\pi \times 8\text{GHz}$, $g/2\pi =48.5\text{MHz}$. For ${\gamma }_{\varphi }=0$, both average number of photons $\overline{N}$ and the photovoltaic current show local minima at $\omega ={\omega }_0$ and local maxima near $\omega =(E_{1,\pm }-E_0)/\hslash$, shown by vertical lines. As the dephasing rate ${\gamma }_{\varphi }$ increases, the double peaks merge to a single peak at $\omega ={\omega }_0$.

Figure 4

The average number of photons in the resonator and the photovoltaic current as a function of the frequency $\omega$ of the microwave drive for detuned DQD and resonator system with $\Omega /2\pi =8.1\text{GHz}$, ${\omega }_0/2\pi =8.0\text{GHz}$, the intradot tunneling $\mathcal{T}/2\pi =1$ GHz, and the drive amplitude $F=50$ MHz. The photon average number has a peak at $\omega =(E_{1,-}-E_0)/\hslash$, Eq. (10), while the photovoltaic current exhibits a double peak feature at $\omega =(E_{1,\pm }-E_0)/\hslash$ (vertical lines).

Figure 5

(a) The average number of photons in the resonator and the photovoltaic current as a function of the frequency $\omega$ of the microwave drive for $\Omega ={\omega }_0=2\pi \times 8\text{GHz}$, the intradot tunneling $\mathcal{T}/2\pi =1$ GHz, and the drive amplitude $F/2\pi =30$ MHz and extremely low tunneling rates to the leads and the energy relaxation rate, ${\Gamma }_l={\Gamma }_r=\gamma =2\pi \times 100\text{kHz}$. The photon average number and the photocurrent have several peaks at ${\omega }_{n,\pm }=(E_{n,\pm }-E_0)/n\hslash$ with $n=1,2,3$ [these frequencies, calculated from Eq. (10) are shown by vertical lines]. (b) The histogram presents the probabilities $P_n$ to have $n$ photons in the resonator steady state at drive frequency ${\omega }_1$ (dark bars) and ${\omega }_2$ (grey bars). (c) The diagram is a schematic picture for the Jaynes-Cummings energy levels showing single and two-photon excitations.

Figure 6

Dependence of the average photon number and the current on microwave frequency $\omega$ for several values of the drive amplitude $F/2\pi =5,10,15$ MHz, at zero dephasing ${\gamma }_{\varphi }=0$ and other parameters the same as in Fig. 3. As the amplitude of the drive increases, the two peaks merge together to a single peak at $\omega ={\omega }_0$.