Resonance Fluorescence in Transport through Quantum Dots ...

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 ^a y Q ^a P ^d Q Q Q y ^d P
2 1 ^a Q H:c: k " k ^c y k ^c k
P
k iV c y k
^d i H:c: , where ^a 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