Resonance Fluorescence in Transport through Quantum Dots ...

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PRL 98, 146805 (2007) PHYSICAL REVIEW LETTERS week ending 6 APRIL 2007 z 0 Pm;n>0c mn s e 1 m s ph 1 n , we obtain G t;s e ;s ph g s e ;s ph e z 0t and the central moments hn e ph i 2 e ph @g 1; 1 @s e ph c 10 01 t; (3a) @ 2 g 1; 1 @s 2 e ph @g 1; 1 @s e ph 2 c 10 01 2c 20 02 t; (3b) or the electron-phonon correlation (not discussed here), given by hn e n ph i hn e ihn ph i @ 2 g 1; 1 @s e @s ph c 11 t: (4) Higher moments can be straightforwardly obtained by this formalism. In the large time asymptotic limit, all the information is included in the coefficents c mn . Then, the Fano factor is F e ph 1 2c 20 02 =c 10 01 so the sign of the second term in the right-hand side defines the sub- (F1) Poissonian character of the noise. We describe our system by the Hamiltonian ^H t i" i ^d y i P ^d i 2 e i!t ^d y ^d 2 1 H:c: P Q! Q â y Q â P ^d Q Q Q y ^d P 2 1 â Q H:c: k " k ^c y k ^c k P k iV c y k ^d i H:c: , where â Q , ^c k , and ^d i are annihilation operators of phonons and electrons in the leads and in the QD, respectively. Writing the density matrix as a vector, 00; 11; 12; 21; 22 T , the equation of motion of the GF (2) is described, in the Born-Markov approximation, by the matrix 0 2 L 1 2 R s e 1 R 0 0 s 1 e 2 R L se 1 1 R 1 R i 2 i 2 s ph 1 2 R M s e ;s ph 0 i 2 2 i ! 0 i 2 B @ C ; (5) 1 2 R 0 i 2 0 2 i ! i 2 A L se 1 2 R 0 i 2 i 2 2 R where 2 j " 2 " 1 j 2 is the spontaneous phonon emission rate, 2 jV j 2 is the tunneling rate through the contact , is the Rabi frequency, which is proportional to the intensity of the ac field, and i f " i 1 e " i 1 and i 1 i weight the tunneling of electrons between the right lead (with a chemical potential ) and the state i in the QD. The Fermi energy of the left lead is considered infinite, so no electrons can tunnel from the QD to the left lead. All the parameters in these equations, except the sample-depending coupling to the phonon bath, can be externally manipulated. In the limit L R ! 0, we recover the pure RF case for the statistics of the emitted phonons [20], F ph i 0 1 2 2 3 2 4 2 ! 2 2 2 4 2 ! 2 ; (6) yielding the famous sub-Poissonian noise result at resonance ( ! 0). In the following, we will restrict ourselves to the resonant case. Electron noise.—By tuning , we control the tunneling of electrons from the two levels. Let us first consider the nondriven case ( 0). If
PRL 98, 146805 (2007) Fe 2.5 2.25 2 1.75 1.5 1.25 1 0.75 Fe 1.8 2 1.6 1.4 1.2 a PHYSICAL REVIEW LETTERS week ending 6 APRIL 2007 0.8 1 0.6 0.4 0.5 0.6 µ 0.7 0.8 Fe 2.5 2.25 2 1.75 1.5 1.25 1 0.75 Fe b 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.5 0.6 µ 0.7 0.8 0 2 4 6 8 10 12 14 γ 0 1 2 3 4 5 FIG. 2. Electron Fano factor as a function of (a) ~ = for ~ = 0 ( L=R) and (b) ~ for ~ 0:1 for different chemical potentials: " 2 ) cf. Fig. 3. It can be easily seen that the noise p suppression for this last case is largest when = 2 . Thus, in the more interesting intermediate regime, " 1 < " 2 (dotted line). In the large-bias case, the noise always increases with ~ and becomes super-Poissonian (in this concrete configuration) for ~ > 4:046. In the case >" 2 , there is a pronounced minimum at ~ p 1= 2 typical of the RF. The case " 1 <
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