Resonance fluorescence in driven quantum dots: Electron and photon correlations

We study the counting statistics for electrons and photons being emitted from a driven two-level quantum dot. Our technique allows us to calculate their mutual correlations as well. We study different transport configurations by tuning the chemical potential of one of the leads to find that the electronic and photonic fluctuations can be externally manipulated by tuning the ac and transport parameters. We also propose special configurations where electron-photon correlation is maximal meaning that spontaneous emission of photons with a well-defined energy is regulated by single electron tunneling. Interesting features are also obtained for energy-dependent tunneling.

Figure 1

When the chemical potential in the right lead $\mu$ is above the energy of both levels, the electron remains in the quantum dot and photons are spontaneously emitted analogously as photons in resonance fluorescent atoms.

Figure 2

$\mu \gtrsim {\epsilon }_2$: dependence of the photonic current, Fano factor, and skewness with the detuning for different field intensities in the regime where no levels are in the transport window: ${\epsilon }_{1,2}<\mu$. ${\Gamma }_L={\Gamma }_R=\Gamma =1$, $\gamma =0.1$, and $x\approx 0$. Since electronic transport is canceled in this regime, the photonic statistics are equivalent to the resonance fluorescence problem. Sub-Poissonian photonic statistics are only found near resonance. It must be noted here, however, that the validity of these results, obtained within the rotating wave approximation, is guaranteed only for ${\Delta }_{\omega }\approx 0$.

Figure 3

$\mu \gtrsim {\epsilon }_2$: dependence of the photonic current, Fano factor, and skewness and the electron-photon correlation with (a) the field intensity, $\Omega$, for different photon emission rates and (b) the photon emission rate, $\gamma$, for different field intensities in the regime where no levels are in the transport window: ${\epsilon }_{1,2}<\mu$. ${\Gamma }_L={\Gamma }_R=\Gamma =1$ and $x\approx 0$. $F_p$ and ${\eta }_p$ show a pronounced minimum in their dependence with the field intensity typical for resonance fluorescence. In the nondriven case, $I_p=0$, $F_p={\eta }_p=1$, and $r=0$.

Figure 4

Dynamical channel blockade configuration, where the electronic transport is strongly suppressed through the lower level, though there is a small probability introduced by thermal smearing of the Fermi surface in the right lead. Again, the chemical potential of the left lead is considered infinite.

Figure 5

Dynamical channel blockade: dependence of $I_e$, $F_e$, and ${\eta }_e$ with (a) the field intensity, $\Omega$, for different photon emission rates and (b) the photon emission rate, $\gamma$, for different field intensities in the dynamical channel blockade regime: ${\epsilon }_1<\mu <{\epsilon }_2$. $\mu <{\epsilon }_{1,2}$. ${\Gamma }_L={\Gamma }_R=\Gamma =1$, $\tilde{\Omega }=\Omega /\Gamma$, $\tilde{\gamma }=\gamma /\Gamma$, and $x\approx 0$. As discussed in the text, the super-Poissonian electronic Fano factor, typical for dynamical channel blockade, is turned sub-Poissonian by the ac intensity.

Figure 6

Dynamical channel blockade: dependence of the photonic current, Fano factor, and skewness and the electron-photon correlation coefficient with (a) the field intensity for different photon emission rates and (b) the photon emission rate, $\gamma$, for different field intensities in the dynamical channel blockade regime: ${\epsilon }_1<\mu <{\epsilon }_2$. $\mu <{\epsilon }_{1,2}$. ${\Gamma }_L={\Gamma }_R=\Gamma =1$ and $x\approx 0$.

Figure 7

System configuration discussed in Sec. 5, with the two levels in the transport window, ${\epsilon }_1,{\epsilon }_2>\mu$.

Figure 8

${\epsilon }_1,{\epsilon }_2>\mu$: dependence of the photonic current, Fano factor, and skewness and the electron-photon correlation coefficient on (a) the field intensity, $\Omega$, for different photon emission rates and (b) the photon emission rate, $\gamma$, for different field intensities for $\mu <{\epsilon }_{1,2}$. ${\Gamma }_L={\Gamma }_R=\Gamma =1$. The electronic statistics (not shown) is sub-Poissonian and not affected by the ac field or by photonic relaxation. The electron-photon correlation coefficient is positive if $\gamma >\Gamma$.

Figure 9

Diagrams of the selective tunneling configuration, where each level is coupled to a different lead. Up: the particular system considered here. Bottom: a possible physical realization by considering a triple quantum dot where the lateral ones are strongly coupled to the leads so they behave as zero-dimensional leads.

Figure 10

Selective tunneling: electronic current, Fano factor, and skewness as a function of (a) the driving intensity for different photon emission rates and (b) the photon emission rate for different field intensities. ${\Gamma }_L={\Gamma }_R=\Gamma =1$.

Figure 11

Selective tunneling: electronic current, Fano factor, skewness, and electron-photon correlation as functions of the frequency detuning for different field intensities. ${\Gamma }_L=1$, ${\Gamma }_R=0.001$, and $\gamma =0.001$. In the insets the corresponding photonic values are shown. When the photon emission rate is much smaller than the tunneling rates and ${\Gamma }_L>{\Gamma }_R$, the system behaves as a coherently coupled double quantum dot, showing a sub-Poissonian minimum in the Fano factor which is between two super-Poissonian peaks.

Figure 12

Selective tunneling: photonic current, Fano factor, skewness, and electron-photon correlation as functions of the frequency detuning for different field intensities. ${\Gamma }_L=\gamma =1$ and ${\Gamma }_R=0.001$. The insets show the electronic correspondents. The double peak in the electronic Fano factor seen in Fig. 11 is washed out by a larger photon emission rate.

Figure 13

Selective tunneling: photonic current, Fano factor, and skewness as a function of (a) the driving intensity for different photon emission rates and (b) the photon emission rate for different field intensities. ${\Gamma }_L={\Gamma }_R=\Gamma =1$. Inset: electron-photon correlation coefficient as a function of the field intensity for different couplings to the left contact, ${\Gamma }_L$, with ${\Gamma }_R=\gamma =1$. It is possible to tune the sign of the electron-photon correlation by means of the tunneling coupling asymmetry.

Figure 14

Schematic diagram of the proposed setup for a level-dependent tunneling configuration where ${\Gamma }_1⪡{\Gamma }_2$.

Figure 15

Level-dependent tunneling: electronic current, Fano factor, and skewness as a function of (a) the field intensity for different photon emission rates and (b) the photon emission rate for different field intensities for the case ${\Gamma }_1={10}^{-5}$ and ${\Gamma }_2=1$.

Figure 16

Level-dependent tunneling: photonic current, Fano factor, skewness, and electron-photon correlation coefficient as a function of (a) the field intensity for different photon emission rates and (b) the photon emission rate for different field intensities for the case ${\Gamma }_1={10}^{-5}$ and ${\Gamma }_2=1$.

Figure 17

Schematic diagram of the proposed setup for a level-dependent tunneling configuration where ${\Gamma }_2⪡{\Gamma }_1$.

Figure 18

Level-dependent tunneling: electronic current, Fano factor, and skewness as a function of (a) the field intensity for different photon emission rates and (b) the photon emission rate for different field intensities for the case ${\Gamma }_1=1$ and ${\Gamma }_2={10}^{-5}$.

Figure 19

Level-dependent tunneling: Photonic current, Fano factor, skewness, and electron-photon correlation coefficient as a function of (a) the field intensity for different photon emission rates and (b) the photon emission rate for different field intensities for the case ${\Gamma }_1=1$ and ${\Gamma }_2={10}^{-5}$.