Resonance Fluorescence in Transport through Quantum Dots: Noise Properties

We study a two-level quantum dot embedded in a phonon bath and irradiated by a time-dependent ac field, and develop a method that allows us to extract simultaneously the full counting statistics of the electronic tunneling and relaxation (by phononic emission) events as well as their correlation. We find that the quantum noise of both the transmitted electrons and the emitted phonons can be controlled by the manipulation of external parameters such as the driving field intensity or the bias voltage.

Figure 1

A two-level QD coupled to two electronic leads. Electrons entering the QD from the left oscillate between both levels with Rabi frequency $\Omega$, can relax with rate $\gamma$, or tunnel out if its energy is greater than $\mu$.

Figure 2

Electron Fano factor as a function of (a) $\tilde{\gamma }=\gamma /\Gamma$ for $\tilde{\Omega }=\Omega /\Gamma =0$ ($\Gamma ={\Gamma }_{L/R}$) and (b) $\tilde{\Omega }$ for $\tilde{\gamma }=0.1$ for different chemical potentials: $\mu <{\epsilon }_1$ (solid), $\mu ={\epsilon }_1$ (dashed), ${\epsilon }_1<\mu <{\epsilon }_2$ (dash-dotted), $\mu ={\epsilon }_2$ (dash-dot-dotted) and $\mu \gtrsim {\epsilon }_2$ (dotted). Note that, in the nondriven case (a), the phonon bath does not affect the sub or super-Poissonian character of the electron noise which can be manipulated introducing the resonant driving field (b). Insets: (a) $F_e$ as a function of $\mu$ for $\Omega =0$ and different phonon emission rates: $\tilde{\gamma }=0.01$ (solid line), $\tilde{\gamma }=0.1$ (dashed), $\tilde{\gamma }=1$ (dash-dotted line) and $\tilde{\gamma }=10$ (dotted line) and (b) for $\tilde{\gamma }=0.1$ and different driving intensities: $\tilde{\Omega }=0$ (solid line), $\tilde{\Omega }=0.15$ (dashed line), $\tilde{\Omega }=0.3$ (dash-dotted line), $\tilde{\Omega }={\tilde{\Omega }}_P=0.492$ (dash-dot-dotted line), $\tilde{\Omega }=1$ (dotted line). Parameters (holding for all figures, in meV): ${\epsilon }_1=0.5$, ${\epsilon }_2=0.8$, ${\kappa }_{B}T={\beta }^{-1}=2\times {10}^{-3}$.

Figure 3

$F_{\mathrm{ph}}$ as a function of $\tilde{\Omega }$ for $\tilde{\gamma }=1$ and different chemical potentials: $\mu <{\epsilon }_1$ (solid line), ${\epsilon }_1<\mu <{\epsilon }_2$ (dashed line) and $\mu >{\epsilon }_2$ (dotted line). In the large-bias case, the noise always increases with $\tilde{\Omega }$ and becomes super-Poissonian (in this concrete configuration) for $\tilde{\Omega }>4.046$. In the case $\mu >{\epsilon }_2$, there is a pronounced minimum at $\tilde{\Omega }=1/\sqrt{2}$ typical of the RF. The case ${\epsilon }_1<\mu <{\epsilon }_2$ follows an intermediate behavior showing the RF-like minimum and the super-Poissonian noise (typical of the large-bias case) for $\tilde{\Omega }>5\sqrt{2}$ (see text). Inset: $F_{\mathrm{ph}}$ as a function of $\mu$ for $\tilde{\gamma }=10$ and different Rabi frequencies: $\tilde{\Omega }=0.1$ (solid line), $\tilde{\Omega }=2$ (dashed line), $\tilde{\Omega }=6$ (dash-dotted line), $\tilde{\Omega }=10$ (dotted line). For high $\tilde{\Omega }$, $\mu$ changes the character of the noise.

Figure 4

Color plot showing the different regions where one can find $F_e$, $F_{\mathrm{ph}}\ge 1$ (white, appearing only for $\tilde{\Omega }=0$), $F_e<1$, $F_{\mathrm{ph}}\ge 1$ (light gray), $F_e\ge 1$, $F_{\mathrm{ph}}<1$ (dark gray), and $F_e$, $F_{\mathrm{ph}}<1$ (black) by tuning $\mu$ and $\tilde{\Omega }$, for $\tilde{\gamma }=1$.