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Published paper
Ramsay, A.J., Godden, T.M., Boyle, S.J., Gauger, E.M., Nazir, A., Lovett, B.W.,
Fox, A.M., Skolnick, M.S. (2010) Phonon-induced rabi-frequency renormalization
of optically driven single InGaAs/GaAs quantum dots, Physical Review Letters,
105 (17), Art no.177402
http://dx.doi.org/10.1103/PhysRevLett.105.177402
Phonon induced Rabi frequency renormalization of optically driven single
InGaAs/GaAs quantum dots
A. J. Ramsay,
1, ∗
T. M. Godden,
1
S. J. Boyle,
1
E. M. Gauger,
2
A. Nazir,
3
B. W. Lovett,
2
A. M. Fox,
1
and M. S. Skolnick
1
1
Department of Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, United Kingdom
2
Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom
3
Department of Physics and Astronomy, University College London, London, WC1E 6BT, United Kingdom
(Dated: July 21, 2010)
We study optically driven Rabi rotations of a quantum dot exciton transition between 5 and
50 K, and for pulse-areas of up to 14π. In a high driving field regime, the decay of the Rabi
rotations is nonmonotonic, and the period decreases with pulse-area and increases with temperature.
By comparing the experiments to a weak-coupling model of the exciton-phonon interaction, we
demonstrate that the observed renormalization of the Rabi frequency is induced by fluctuations in
the bath of longitudinal acoustic phonons, an effect that is a phonon analogy of the Lamb-shift.
PACS numbers: 78.67.Hc, 42.50.Hz, 71.38.-k
The decoherence of a two-level system or qubit in the
solid-state is often understood in terms of its interaction
with a reservoir of bosons [1] or half-integer spins [2]. An
important issue with respect to both decoherence physics
and the performance of quantum logic operations is how
these interactions are modified by a driving field. In the
case of bosons, this issue has received considerable theo-
retical attention [1], but as far as we are aware, the only
experimental studies to support this work are restricted
to the field of superconducting qubits [3, 4]. Here we
use Rabi rotation measurements of quantum dot exci-
tons [5–7] to study the dephasing of a solid-state driven
two-level system coupled to a boson bath, where recently
we identified longitudinal acoustic phonons as the prin-
cipal cause of the intensity damping of optically driven
Rabi rotations in InAs/GaAs quantum dots [8].
In this letter, we report experimental evidence for
a phonon-induced shift in the Rabi frequency of an
optically-driven excitonic transition in a semiconductor
quantum dot. A single laser pulse drives a Rabi rota-
tion between the ground and neutral exciton states of a
single InGaAs/GaAs quantum dot, where the pulse-area,
the time-integral of the Rabi frequency, is controlled via
the incident power. Compared to a two-level atom pic-
ture, the resulting Rabi rotations are modified by the
exciton-phonon interaction. We observe key signatures
of the dynamics of a driven two-level system coupled to
a boson bath. Firstly, the period of the Rabi rotations
decreases with the pulse-area, and increases with temper-
ature implying a renormalization of the Rabi frequency as
theoretically anticipated in refs. [9–13]. This shift in the
Rabi energy is a result of fluctuations in the phonon bath,
and is an effect analogous to the Lamb-shift observed in
the spectra of atomic hydrogen. Secondly, the damp-
ing of the Rabi rotations is nonmonotonic and exhibits a
roll-off behavior as the Rabi energy starts to exceed the
bandwidth ~ω
c
of the exciton-phonon coupling.
Since our previous work [8], we have greatly improved
our experimental setup by swapping a cold-finger cryo-
stat for a helium bath cryostat with the sample mounted
on low temperature piezoelectric stages. This provides
access to lower temperatures (4.2 K vs 15 K), higher me-
chanical stability, and enables the use of a shorter focal
length objective. Overall this increases the range of the
pulse-area from 6π to 14π, providing the bandwidth in
the Rabi energy needed to resolve the new physics re-
ported here. The experiments are performed on a single
quantum dot embedded in an n-i-Schottky diode struc-
ture. Uncapped test specimens exhibit an ensemble of
InGaAs/GaAs dots with heights of 3-4 nm, and base di-
ameters of 25-30 nm. The measurements presented here
are from a single dot emitting at 951 nm in the high en-
ergy tail of the dot distribution, and energetically distant
from the wetting layer that emits at 861 nm. The biex-
citon binding energy is 1.9 meV. The dot is excited with
a single laser pulse, and the final occupation of the ex-
citon state is measured using photocurrent detection [6].
A background photocurrent proportional to the incident
power is subtracted from all data, attributed to absorp-
tion of scattered light by other dots in the sample [15].
Further details of the sample, and the optical system can
be found in ref. [16].
Figure 1 presents a Rabi rotation measurement at a
temperature of 12.5 K, where the qualitative features
of the data are clearly expressed. A single laser pulse
with a Gaussian envelope resonantly excites the neutral
exciton transition. Circular polarization and a narrow
0.2-meV FWHM spectral width are used to suppress ex-
citation of the two-photon biexciton transition [17], and
of multi-excitons [18]. The bare Rabi frequency Ω(t) of
the laser pulse is described by Ω(t) =
Θ
2τ
√
π
exp (−(
t
2τ
)
2
),
where the pulse-duration τ = 4 ps, and the pulse-area
Θ =
R
∞
−∞
Ω(t)dt. The photocurrent P C, which is pro-
portional to the final occupation of the exciton state fol-
lowing resonant excitation, is measured as a function of
the square-root of the incident power, which is propor-
2
0 2 4 6 8 10 12
0
2
4
6
8
Photocurr ent (pA )
pulse-area: ( )
T =12.5 K
FIG. 1: (◦ )A Rabi rotation measurement at 12.5 K. The full
lines are calculated using Eqs. (1,2). (red) (K(Ω) is real,
and P = 1) In a low driving field regime (Θ < 6π), the
data are described by an Ω
2
damping, but at higher fields
this model overdamps the oscillation. (green) (K(Ω) is real,
P(Ω) = e
−Ω
2
/ω
2
c
) The roll-off in the damping is reproduced
by including the frequency dependence of the exciton-phonon
form-factor P(Ω). With a constant rotation frequency, this
model provides a ‘clock-signal’ against which the nonlinearity
of the rotation angle with Θ is clearly observed. (blue)(K(Ω)
is complex) By including the imaginary part of K(Ω) the vari-
ation of rotation angle with Θ is reproduced.
tional to the pulse-area Θ. The photocurrent exhibits
a damped oscillation with pulse-area. To help illustrate
the main qualitative features of the data, calculations of
various models of the driven dephasing are presented in
fig. 1 as color-coded lines. The traces are scaled to fit
the data at low pulse-area and will be discussed shortly.
To analyze the data, we use a model described by a
pair of Bloch-equations:
˙s
y
= {Ω + =[K(Ω, T )]}s
z
− {Γ
∗
2
+ <[K(Ω, T )]}s
y
(1)
˙s
z
= −Ωs
y
(2)
where s = (s
x
, s
y
, s
z
) is the Bloch-vector of the two-
level system in the rotating-frame of the laser, composed
of the crystal ground, and the exciton states. s
z
rep-
resents the population inversion, and s
y
the imaginary
component of the excitonic dipole. The photocurrent is
a measure of the final occupation of the exciton state:
P C ∝ (1 + s
z
)/2. In the case of on-resonance excitation,
where the initial state is s = (0, 0, −1), as discussed here,
the s
x
equation is decoupled from Eqs. (1,2) and does
not influence the dynamics of the measured s
z
-basis. Γ
∗
2
is a phenomenological dephasing rate used to account for
electron-tunneling, and other possible sources of driving-
field independent dephasing. The complex resp onse func-
tion K(Ω, T ) describes the Rabi-frequency dependence
of the exciton-phonon interaction. The real part < gives
rise to a rate of dephasing, and the imaginary part =
shifts the rotation frequency of the Bloch-vector; < and
= satisfy a Kramers-Kronig relationship.
Eqs. (1,2) are derived using a weak-coupling
Born-Markov approximation [13] that treats
the exciton-phonon interaction Hamiltonian
H
I
= |XihX|
P
q
~(g
q
b
†
q
+g
∗
q
b
q
) as a perturbation to sec-
ond order, where b
†
q
, b
q
are the creation and annihilation
operators for a longitudinal acoustic phonon of wave-
vector q. The material dependent coupling strengths
g
q
are given in ref. [19]. K(Ω, T ) =
R
∞
0
dt e
iΩt
˜
K(t, T ),
where
˜
K(t, T ) is the time-domain response of the
exciton-phonon system:
˜
K(t, T ) =
Z
∞
0
dωJ(ω)coth(
~ω
2k
B
T
) cos ωt, (3)
where J(ω) =
P
q
|g
q
|
2
δ(ω − ω
q
) is the spectral density
of the exciton-phonon interaction, and can be approxi-
mated by J(ω) ≈
~A
πk
B
ω
3
P
2
(ω). A is defined such that
lim
Ω→0
<[K] = AT Ω
2
[8]. The form-factor, which is the
Fourier transform of the probability distribution of the
carriers, is approximated as P(ω) ≈ e
−ω
2
/2ω
2
c
[13].
A physical interpretation of the model is as follows.
The laser couples the crystal-ground and exciton states
to form optically dressed states separated by a Rabi split-
ting. Emission and absorption of LA-phonons with an
energy equal to the Rabi-splitting results in a relaxation
between the dressed states that leads to a rate of dephas-
ing that is a function of the instantaneous Rabi frequency.
To satisfy a Kramers-Kronig relationship, the dephasing
is accompanied by a shift to the Rabi splitting induced
by fluctuations of the LA-phonon bath.
Now we return to fig. 1, to consider the qualitative fea-
tures of the Rabi rotations using the results of Eqs. (1,2)
to help identify these features. Values of A = 11.2 fs K
−1
,
and ~ω
c
= 1.44 meV have been used, found from fits to
fig. 2(a), discussed below. (i) The red-trace is calculated
by neglecting the imaginary part of K, and approximat-
ing the form-factor as P ≈ 1. The pulse-area used in
the calculations is scaled to provide best-fits to the data.
For low pulse-areas of Θ < 6π, the data are well de-
scribed by the red-trace, which is consistent with our
previous work [8]. In the low-driving field, high tempera-
ture regime (Ω < ω
c
, 2k
B
T À ~Ω) the rate of dephasing
is <[K] ≈ AT Ω
2
, which is the expected behavior of a two-
level system coupled to a three-dimensional b oson bath
[1]. However at higher pulse-areas, the calculation is over
damped, high-lighting the nonmonotonic character of the
decay. (ii) The green-trace is calculated by again neglect-
ing the imaginary part of K, but now including the fre-
quency dependence of the form-factor P(Ω). In this way
the envelope of the Rabi rotation is reproduced. Hence,
the decay is explained by a rate of dephasing that ex-
hibits a cut-off behavior, where for driving fields greater
than the cut-off energy (~Ω > ~ω
c
), the phonon-bath can
no longer follow the driving field, leading to a reduction
in the rate of dephasing. This calculation assumes that
the rotation frequency of the Bloch-vector is equal to
3
the Rabi frequency, and thus provides a periodic-signal
against which the nonlinear dependence of the Rabi ro-
tation angle with pulse-area is clearly observed. (iii) The
blue-trace is calculated using the full model. Now both
the envelope and the variation of Rabi rotation angle
with pulse-area Θ are reproduced, demonstrating that
the nonlinear rotation angle results from the imaginary
part of the same exciton-phonon response that describes
the damping.
To further confirm that the nonmonotonic decay and
the nonlinear rotation angle are a result of the exciton-
phonon interaction, a set of Rabi rotation measurements
as a function of temperature [21] was performed. The
data are presented in fig. 2(a). The Rabi rotations are
offset for clarity, and plotted as a function of the pulse-
area deduced from the fits. For each temperature, the
laser is tuned to resonance with the exciton transition. At
low temperatures, oscillations can be observed over the
full range of measured pulse-area. As the temperature
increases, the damping and the rotation period increase.
At 35 K the period of rotation is approximately 23%
larger than at 5 K.
To compare the model to the experimental data pre-
sented in fig. 2(a), we make fits of the entire data-set to
Eqs. (1,2). This is achieved as follows. First we calculate
all of the Rabi rotations using trial values for A and ~ω
c
.
To accommodate small variations in the dot-laser cou-
plings between measurements at different temperatures,
the calculation is scaled to the data by finding the values
of a = Θ/
√
P , and η that minimize the root-mean square
error for each temperature. a is a measure of the dot-laser
coupling, and η is the photocurrent corresponding to one
exciton. A genetic search algorithm [22] is then used to
find the values of A and ~ω
c
that provide the best fit of
the entire data set. Note that only two parameters are
used to fit the shape of the Rabi rotation, which is char-
acterized by three qualitative features. A is determined
by the strength of the damping, and ~ω
c
by the cut-off
in the damping. The nonlinearity of the rotation angle,
which depends on both A and ~ω
c
, provides a test of the
consistency of the model. By treating A and ~ω
c
as global
fitting parameters, the temperature dependence of these
features is tested against the model, and the values of A
and ~ω
c
are determined with greater accuracy compared
to using a series of local fits. Good fits to the entire data-
set between 5-50 K are achieved. This confirms that the
weak-coupling model used here provides a good descrip-
tion of the experimental data. Note that Γ
∗
2
= 0.025 ps
−1
is not used as an adjustable fitting parameter.
Based on fits to the data in fig. 2(a), we extract values
of A = 10.8 − 11.9 fs.K
−1
, and ~ω
c
= 1.37 − 1.46 meV.
The value of A is consistent with a calculation using bulk
GaAs material parameters [8]. The cut-off energy ~ω
c
corresponds to a spherical [23] wavefunction of ψ(x) ∝
e
−x
2
/2d
2
, with d =
√
2c
s
/ω
c
= 3.25 − 3.46 nm, where c
s
is the speed of sound. This corresponds to a probability
0 20 40
2.0
2.5
3.0
3.5
4.0
P
2
( )
Temperature (K)
(b)
0 2 4 6 8 10 12 14
0
4
8
12
16
20
24
28
32
36
40
44
48
52
56
60
64
68
72
76
80
0.0 0.5 1.0 1.5 2.0
Bare pulse-area ( )
Phot ocurrent (pA)
10 K
15 K
20 K
25 K
30 K
35 K
40 K
45 K
50 K
5 K
(a)
Peak Rabi ener gy
p
(m eV)
0 1 2 3 4
-50
0
50
100
Im[K ] ( eV)
Rabi energy ( meV)
T = 0 K
(c)
FIG. 2: (a) Temp erature dependence of the Rabi rotations.
Red-lines are calculated using A = 11.2 fs.K
−1
, ~ω
c
=
1.44 meV, and Γ
∗
2
= 0.025 ps
−1
, where A and ~ω
c
are ex-
tracted from a fit to all data. The bottom axis is scaled to
the bare pulse-area Θ, while the top axis indicates the peak
Rabi energy of the laser pulse. (b) Square-root of the incident
p ower
√
P
2π
corresp onding to the first minimum plotted vs
temp erature.
√
P
2π
is scaled to the pulse-area Θ for the case
of T = 5 K. The trace is calculated assuming the scaling
parameter a = 1. (c) Calculation of the Lamb-shift, =[K] at
T = 0 K, as a function of Rabi energy.
4
distribution of ∼ 5.5 nm FWHM, which is reasonable for
an InAs/GaAs quantum dot.
Figure 2(b), presents the square-root of the incident
power used to reach the first minimum in the Rabi rota-
tion
√
P
2π
, scaled to the pulse-area at 5 K, and plotted
against the temperature. The line is calculated assuming
that the scaling parameter a is constant. There is good
agreement between the model and the data, with a less
than 9% departure of any data point from the calculation.
This implies that the increase in the period with temper-
ature shown in fig. 2(b) is due to the interaction with
phonons, and that variations in the dot-laser coupling a
due to the thermo-mechanical stability of the alignment
are small. On close inspection of the fits in fig. 2(a) (not
shown) for T ≥ 40 K, the weak-coupling model over esti-
mates the damping reducing the quality of the fits. Since
non-Markovian effects should be more influential at low
temperature [25], we cautiously suggest that at higher
temperatures, a model which uses the renormalized Rabi
frequency for calculating the rate of damping (e.g. as ac-
complished by the strong-coupling approach of Ref. [26])
may provide a better description.
For a number of reasons only LA-phonons can explain
the data. In a low-driving field regime (Θ < 6π), the
Rabi rotations can be describ ed by a rate of dephasing
Γ
2
= K
2
Ω
2
, where K
2
is proportional to temperature [8].
This is characteristic of LA-phonons, since other mech-
anisms such as LO-phonons or the wetting layer would
be characterized by an activation energy. The temper-
ature gradient A is consistent with a calculation based
on LA-phonons using literature values for bulk GaAs [8].
The wetting layer model, as presented in ref. [24], can-
not describe the cut-off behavior in the damping, or the
nonlinearity of the Rabi rotation angle. Furthermore, the
spectral density of the dot-environment interaction, J(Ω)
can be deduced from the Rabi rotation data, and is only
consistent with an LA-phonon mechanism.
To conclude, we have studied the interplay between
an optically driven quantum dot exciton and a reservoir
of LA-phonons using Rabi rotation measurements. The
thermal and vacuum fluctuations of the phonon bath
give rise to a temperature and driving-field dependent
rate of dephasing and a renormalization of the Rabi en-
ergy, which are connected by a Kramers-Kronig relation-
ship. The vacuum contribution, analogous to the photon-
induced Lamb-shift observed in the spectra of atomic hy-
drogen, can be inferred from the fits to the data by calcu-
lating =[K] at 0 K. This is presented in the fig. 2(c). The
Lamb-shift term depends on the Rabi frequency, and has
a value of −49±3 µeV at the first minimum. At low Rabi
frequencies this can account for up to a 4.5 ± 0.3% de-
crease in the effective Rabi rotation frequency. At 20 K,
where k
B
T ∼ ~ω
c
, the thermal and vacuum contribu-
tions to the shift =[K] are of a similar size, indicating
that for T ¿ 20 K the shift to the Rabi rotation fre-
quency is dominated by the vacuum contribution. In
comparison to the photon-induced Lamb-shift, here the
Rabi splitting of the optically dressed states can be tuned
with a laser from zero to a few meV, probing the spec-
tral density of the exciton-phonon coupling. By contrast,
optical transitions have energies in the eV range, which
is large compared with the tuning range of the transi-
tion. The phonon fluctuation-induced shift to the Rabi
frequency observed here contrasts with the case of quan-
tum wells [27], where a Rabi frequency renormalization
results from an increase in the excitonic dipole due to
many-body Coulomb effects.
The authors thank the EPSRC (UK) EP/G001642,
and the QIPIRC UK for financial support. AN is sup-
ported by the EPSRC, and BWL by the Royal Society.
We thank H. Y. Liu and M. Hopkinson for sample growth;
R. Golestanian, G. A. Gehring and D. P. S. McCutcheon
for useful discussions.
∗
Electronic address: a.j.ramsay@shef.ac.uk
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