DETUNING-DEPENDENT NARROWING OF MOLLOW TRIPLET LINES OF DRIVEN QUANTUM DOTS
A. P. Saiko"*, R. Fedarukb**, S. A. Markevicha
a Scientific-Practical Materials Research Centre, National Academy of Sciences of Belarus
220072, Minsk, Belarus
bInstitute of Physics, University of Szczecin 70-451, Szczecin, Poland
Received August 27, 2013
We study the two-time correlation function and the resonance fluorescence spectrum of a semiconductor quantum dot excited by a strong off-resonant laser pulse. The obtained analytic expressions exhibit a specific detuning-dependent damping of Rabi oscillations of the dressed quantum dot as well as a detuning-dependent width of Mollow triplet lines. In the absence of pure dephasing, the central peak of the triplet is broadened upon increasing detuning, but the blue- and red-side peaks are narrowed. We demonstrate that the pure dephasing processes can invert these dependences. A crossover between the regimes of detuning-dependent narrowing and broadening of the side and central peaks is identified. The predicted effects are consistent with recent experimental results and numerical calculations.
DOI: 10.7868/S0044451014040168
Emission properties of driven semiconductor quantum dots (QDs) have attracted much interest recently due to their potential applications in the fields of photonics and quantum information technology fl 3]. Semiconductor QDs provide an atomlike light matter interaction demonstrating typical quantum dynamical features of isolated natural atoms. In particular, artificial atoms such as QDs excited by a strong resonant continuous-wave optical field exhibit a resonance fluorescence spectrum containing three peaks, known as the Mollow triplet [4]. Under pulsed resonant excitation in time-resolved experiments, QDs undergo the Rabi oscillations [5,6]. In the dressed-atom approach [7], these effects are understood as resulting from the quantum transitions in the total coupled system of the atom and driving photons. In contrast with natural atoms, QDs interact with their solid-state environment in a more complicated manner. There is a variety of loss mechanisms for quantum dots [5]. The damping of the Rabi oscillations as well as the width of the Mollow triplet lines can be used for identifying these mechanisms. One of the main consequences of the solid-state character
* E-mail: saiko'flifttp.bas-net .by
E-mail: fedaruk'fflwmf. univ.szczecin.pl
of QDs is a specific dephasing caused by coupling to acoustic phonons. A well-known signature of phonon coupling is revealed in an excitation-induced dephasing with a rate proportional to the square of the effective Rabi frequency. Experimental evidence of the excitation-induced dephasing effects was recently observed as oscillation damping in pulsed photocurrent measurements on a resonantly driven QD [5]. The effect of the excitation-induced dephasing has also been observed as the Mollow triplet sideband broadening under resonant continuous-wave excitation of a single QD in a microcavity [1]. In that paper, the phenomenon of the spectral Mollow sideband narrowing depending on the laser-excitation detuning from the bare emitter resonance was also demonstrated, but that effect had to be left open for further theoretical analysis. Previous numerical studies using the polaron master equation approach with cavity coupling did not reveal the narrowing effect [8]. Recently, it has been shown that a crossover between the regimes of detuning-dependent sideband narrowing and broadening can be qualitatively understood from a theoretical model based on the polaron ME without inclusion of the QD cavity-coupling [3]. Numerical calculations presented in [9] have also shown that for off-resonant driving, narrowing in the spectral sideband width can occur under cer-
tain conditions. In [10], it was shown that the docay rate of the Rabi oscillations in QDs can decrease as the detuning increases.
In this paper, we analytically calculate the two-time correlation function and resonance fluorescence spectrum of a semiconductor QD excited by an off-resonant laser pulse taking the exciton phonon interaction into account. Our analytic results allow us to give a clear physical treatment of the detuning-dependent narrowing and broadening of the Mollow triplet lines. We show that pure dephasing processes radically influence the character of these phenomena and can cause a crossover between the detuning-dependent narrowing and broadening of the triplet peaks.
We model the QD as a two-level system with an energy splitting u>o between the ground |1) and excited |2) states (we set h = 1). The QD is driven by a coherent laser field of a frequency u>l and is coupled to a phonon bath. The master equation for the density matrix p of the QD [3] in the Markovian approximation can be written as
where
= [H,p] + i\p,
n
H = As' + - (.S'-
il)
(2)
vice versa, the 7 process corresponds to enhanced radiative decay, while the 7^ process represents the incoherent excitation caused by the exciton phonon interaction (nonelastic photon QD scattering processes are realized through multiphonon transitions), is the cross-dephasing rate induced by phonons, and is the dephasing rate introduced phenomenologically. As is shown in Ref. [3], the relaxation rate 7,,/, = o,7ft +7^ weakly depends 011 A in the detuning range considered below and can be approximated as 7,,/, = kiï2, where k is a coefficient. For sufficiently large driven strengths, fi the cross-dephasing term in Eq. (3) can be
neglected. This term rapidly oscillates in the interaction representation with Hamiltonian (2) and gives only corrections of the second and higher orders in 7^/fi.
After the canonical transformation pi = u+pu, where u = cxp [—0(«+ — #~)/2], the master equation (1) is transformed into
Hi = 11 ' H11 Ai = 11 ' Au =
+ / A|/>i.
. + du ■ tir — = es' ot
(4)
rT
£>[*H
where e = \/A2 + iï2
is the Hamiltonian of the QD in the frame rotating at the frequency u>l, and
721 + 7pfc nr 712 + 7pfc nr + Ap =-—!- D[s }p +-—!— D[s+]p -
T'i = 1 (721 + 712 + rph + 7'pft )(! + cos"2 +
+ \ (721 + 7,7ft - 712 - 7ph ) COS 9 + Jij sill2 9,
D[S~']P — lph(s+Ps+ + 8 P8 ) (3) Tt = i(-}2i + 012 + %h + 0;tft )(! + cos"2
is the relaxation superoperator. Here, are components of the pseudospin operator, describing the QD state and satisfying the commutation relations = 2,s: and [st,.s±] = ±.s,=t, A = u>o — '^l, ft = = i)o oxp [—W(T)/2], where i)0 is the bare Rabi frequency that describes the coherent exciton pumping from the laser field, and oxp[—W(T)] is the Debye Waller factor, which takes the effect of acoustic phonons on the coherent laser QD interaction into account and depends on the bath temperature T and the electron phonon coupling. This factor describes the so-called elastic processes in which the momentum received by the QD from the exciting photon is transmitted to the whole crystal without emission or absorption of a phonon. In addition, D[0]p = 2OpO+ — 0+0p — p0+0, 021 and O12 are the rates of photon radiative processes from the excited state |2) of the QD to its ground state |1) and
- ^ (O21 + Op/, - 712 - 7ph) cos 9 + ijy sin2 9,
T^ = 11 cos2 B + (721 + 012 + 0,7ft + 7,tft) sin2 9, cos 9 = A/e, sin (9 = il/e.
We obtain the relaxation superoperator in the rotating wave approximation (RWA). In this case, because , F-|-, F^ -C e for the strong laser QD interaction, the nondiagonal terms that contain the products of spin operator pairs (s±,s'), (s+,s+), and (n~ ,n~) are neglected in the structure of the operator. The analogous form of the relaxation superoperator was obtained in fll 13] in describing lasing and amplification effects in superconducting qubits.
The solution of Eq. (4) can be written as
pi(f) = exp [(-/Li + Ai)t] pi(0).
(5)
The suporoporator L\ acts in accordance with the rule L\X = [Hi,X], The density matrix p(t) in the laboratory frame is given by
p(t) =
(6)
where pi(t) is defined by Eq. (4) and pi(0) = ufp(0)ui. Moreover, if the QD is in the ground state, then p(0) = 1/2 — s~ and we have
h ) sin (9 — s~ cos 9.
The following relations can be verified by direct calculation:
exp [(—¿Li + Ai)i] s^ = exp [(=pie — T±)t] s±, exp [(—/Li + Ai )t] = oxp(-F||f)«\ exp [(—¿Li + Ai )t] a = = {1 + 2<To [1 - oxp(—Fyf )]
(7)
where a = const,
(T0 =
r+-rt r„
721 + 7ph ~ 712 ~ 7ph T h
cos 9,
T|| = + rT = ô|| + (-U - 7||) sin2 9, = ^(r4+rt+I\,) = (7±-7||) sin2 9,
(8)
7|| = 7i| + 7ph, 7± = 7±
1
7 ph
7|| = 712 + 721, 7± = 2(712 + 721 + >/), 7Ph =fph+fph-
Using Eqs. (4) (8), we obtain the density matrix in the laboratory frame
Piab{t) = ^ + | {[sin(9(cos(9 + 1) x
x exp (—i(u>L + £)t - Fj_f) + sin0(eos0 - 1) x x exp (—i(u>L ~ £)t - r_i_f)] + H.c.} -
■ sin 9
r —r
cos 9 exp( — F11 i )+—^—- [l— oxp(—Fyf )]
x [,s+ oxp(—¿u;£Î)+H.c.] — I oxp(—Fj_i) sin2 0 cos et ■ cos(9oxp(—Fui)
cos 9
r, - rt
x [1-oxp(-F||i)]
0)
We now find the two-time correlation function
9{1\t) = <*"(*)*+( 0)) = {s-(t))g+{ 0)p(0) = = ^ sin2 0cxp(—iujLt — Fyi) + j(l + cos9)2 x
x exp [—+ ujl )t - Fj_f] + ^(1 - cos 9f x
x exp[-i(e-ujL)t-T±t] (10) and the spectral density of the emitted radiation
1
SM = - Re / dtelu"{.s-(t).s+(0)) =
71 J 0
1
2tt
Fii sin2 9
_ Fj_(l + cos9)'2/2
rjj + (u; - u;z)2 +
F±(l - cos»)2/2
r2
(u.1 ■
C0Lf-
. (11)
It follows from Eqs. (8) and (10) that the respective decay rates of the oscillations at the side (ujl ±e) and central (ujl) frequencies are
T± = ^ (71| +lph + ) + J (71| +7/>ft —)
T|| = 711
1
n2
(12)
• 7ph ~ 2 (7"|| + 7ph ~ V) A2 + Q2'
According to Eq. (11), the full width at half maximum (FWHM) of the side and central lines of the Mollow triplet are respectively equal to 2Fj_ and 2F||.
We conclude from Eqs. (8) and (12) that the rates of the transverse 77 and longitudinal 7y relaxations determine the character of the detuning dependences of the decay rates for oscillations at the side (ujl ±e) and central (ujl) frequencies. If 77 > 7y, for a fix
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