Stabilization of photon collapse and revival
dynamics by a non-Markovian phonon bath
Alexander Carmele1,2,3, Andreas Knorr1and Frank Milde1
1Institut f¨
ur Theoretische Physik, Nichtlineare Optik und Quantenelektronik,
Technische Universit¨
at Berlin, Hardenbergstraße 36, EW 7-1, D-10623 Berlin,
Germany
2Institute for Quantum Optics and Quantum Information, Austrian Academy of
Sciences, Technikerstrasse 25, 6020 Innsbruck, Austria
E-mail: ale[email protected]
New Journal of Physics 15 (2013) 105024 (13pp)
Received 2 August 2013
Published 23 October 2013
Online at http://www.njp.org/
doi:10.1088/1367-2630/15/10/105024
Abstract. Solid state-based light emitters such as semiconductor quantum
dots (QDs) have been demonstrated to be versatile candidates to study the
fundamentals of light–matter interaction. In contrast to optics with isolated
atomic systems, in the solid-state dissipative processes are induced by the
inherent coupling to the environment and are typically perceived as a major
obstacle toward stable performances in experiments and applications. In this
theoretical model study we show that this is not necessarily the case. In fact,
in certain parameter regimes, the memory of the solid-state environment can
enhance coherent quantum optical effects. In particular, we demonstrate that the
non-Markovian coupling to an incoherent phonon bath can exhibit a stabilizing
effect on the coherent QD cavity-quantum electrodynamics by inhibiting
irregular oscillations and allowing for regular collapse and revival patterns. For
self-assembled GaAs/InAs QDs at low photon numbers we predict dynamics
that deviate dramatically from the well-known atomic Jaynes–Cummings
model. Even if the required sample parameters are not yet available in recent
experimental achievements, we believe our proposal opens the way to a
systematic and deliberate design of photon quantum effects via specifically
engineered solid-state environments.
3Author to whom any correspondence should be addressed.
Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence.
Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal
citation and DOI.
New Journal of Physics 15 (2013) 105024
1367-2630/13/105024+13$33.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
2
Semiconductor quantum dots (QDs), coupled to a photonic cavity, exhibit signatures of strong
coupling [1–8] when the electron–photon interaction outrivals the combined dipole decay rate
and cavity loss. In contrast to fundamental atom–photon interfaces [9], a key factor to the
understanding of observed phenomena in the semiconductor cavity-quantum electrodynamics
(cQED) regime is the interaction of electrons with the QD host material. The inherent many-
body properties of a solid-state, in particular the electron–phonon interaction, lead to often
undesired decoherence, where the superposition of otherwise distinct quantum states becomes
lost and information about the system is carried away into the host material surrounding of
the embedded QD [10,11]. In principle, this dephasing mechanism seems to put limits on
the prospects of the wide field of semiconductor-based quantum information processing and
quantum communication [12]. However, it has been shown that in photosynthetic systems,
for example, the quantum-coherent excitation transfer in Fenna–Matthews–Olsen antenna
complexes is not only resistant to dephasing [13], but is actually supported by the dephasing
processes induced by the thermal fluctuations of the molecule [14,15]. This sparked a
lively debate over its implications for evolutionary biology, energy technologies and quantum
information [16].
It is also known that lattice vibrations in semiconductor nano structures can give rise to new
effects and dynamics [17] not known in atomic quantum optics, nor in Markovian treatments
of the semiconductor environment, e.g. phonon-mediated off-resonant cavity feeding [18–21],
formation of phonon-assisted Mollow triplets [22,23] and temperature-dependent vacuum Rabi
splittings seen in cavity emission spectra experimentally [24] and theoretically [25,26]. Very
recently impressive experiments were realized which could steer the transition from Markovian
to non-Markovian dynamics and thus the flow of information between an open system and its
environment [27].
Expanding on these developments we report on how the non-Markovian coupling to an
environment can be counter-intuitively exploited to support coherent quantum optical features.
For a solid-state emitter, for illustration described as a semiconductor QD, strongly coupled
to a nanocavity, the memory effects of an incoherent bath of harmonic oscillators induce
astonishing pattern formations in the coherent collapse and revival phenomenon of the photon
dynamics in the cavity. The collapse and revival phenomenon results from the interference
of different Rabi frequencies (Jaynes–Cummings ladder contributions) and its fundamentals
were investigated both theoretically [28] and experimentally [29,30]. The presented effect
further challenges the widely adopted notion that the environmental coupling, responsible for
decoherence, constitutes a substantial drawback that needs to be overcome, in order to build
robust solid-state devices using existing semiconductor fabrication technologies. The very same
coupling to the surrounding does not destroy, but rather facilitates and stabilizes the otherwise
fragile quantum effect of collapse and revival, in particular at the few-photon level, and hence
lifts the restriction of the high photon numbers needed in atomic systems to initiate stable
collapse and revivals, i.e. for longer times and more cycles.
To provide a basis for discussion of solid-state effects on an equal footing, we start by
introducing an approach to a two-level system interacting with (i) photons and (ii) phonons,
which encompasses the well-known Jaynes–Cummings model (JCM) and non-Markovian pure
dephasing effects, respectively.
For illustration of the effect, we consider an InAs/GaAs self-assembled QD with well
separated quantum confined electronic states, i.e. valence (v) and conduction (c) shell. This
effective two-level system is embedded in a single-mode photonic cavity and can interact
New Journal of Physics 15 (2013) 105024 (http://www.njp.org/)
3
Figure 1. Investigated system. A two-level QD with states v and c in a photonic
cavity is coupled to (i) a single cavity mode (blue) via Mand (ii) a phonon bath
(orange) via gq.
with both, the quantized lattice vibrations of the QD host matrix and the cavity photons, see
figure 1. Particularly, we will focus on the interaction of electrons with longitudinal-acoustic
(LA) phonons via deformation potential coupling, dominating at low temperatures [31,32].
The investigation will explore the temporal dynamics of the cavity photon density, since it
is experimentally extractable via output coupling [30] or accessible from cavity transmission
measurements [33].
The dynamics of the relevant expectation values is derived via the Heisenberg equation
−
i
¯
h∂tˆ
O(t)=[ˆ
H,ˆ
O]−.(1)
The Hamiltonian of our model includes the free energy of non-interacting electrons, cavity
photons and bulk phonons:
ˆ
H0=
v,c
X
i¯
hωiˆa†
iˆai+¯
hω0ˆc†ˆc+X
q¯
hωqˆ
b†
qˆ
b†
q(2)
with electron creation (annihilation) operators ˆai(ˆa†
i)in shell i=v,c.Usually, the respective
quantized energies ¯
hωi=εiare obtained within the effective mass approximation. The
frequency of the fundamental cavity mode is ω0and ˆc(ˆc†)create (annihilate) a corresponding
photon. Phonons with wave vector qare described by operators ˆ
bqand ˆ
b†
q. Their modes have a
linear dispersion ωq=cs·|q|, where csis the speed of sound. The electron–photon interaction
is treated within the dipole and rotating wave approximation [34]
ˆ
Hel−pt =−¯
hM(ˆa†
vˆacˆc†+ˆa†
cˆavˆc), (3)
where we assume an optical coupling strength of M=314 µeV, which lies within reach
of current QD and cavity fabrication technology but is not achieved as of yet. We show a
comparison with parameters of a state-of-the-art sample [35] in the end of the paper. Finally, the
band diagonal interaction of bulk phonons with the conduction shell QD electrons is given by
ˆ
Hel−pn =X
q
gqˆa†
cˆac(ˆ
b†
q+ˆ
bq). (4)
Note, that the phonon bath is initially in equilibrium with the unexcited QD system. The
coupling elements gq=gc
q−gv
qare given in the appendix. In this section, also details are given
on how to derive a general set of coupled differential equations, which allows one to determine
New Journal of Physics 15 (2013) 105024 (http://www.njp.org/)
4
the photon dynamics N(t)=hˆc†ˆci(t). The relevant dynamical quantities needed are the photon-
assisted ground and excited state density Gm(t)=hˆa†
vˆavˆc†mˆcmi(t)and Xm(t)=hˆa†
cˆacˆc†mˆcmi(t),
as well as the photon-assisted polarization Pm(t)=hˆa†
vˆacˆc†m+1 ˆcmi(t). The number of involved
photons is labeled by m. We assume that only one electron is present in the QD and G0+X0=1
holds. Thus the dynamics of the cavity photons is directly related to the dynamics of the QD
and can be obtained via
N(t)=G1(t)+X1(t). (5)
In our simulation we focus on a situation that starts with an inverted QD, i.e. Gm(t0)=0 and
X0(t0)=1, and a cavity prepared in a coherent photon state with a mean photon number of
N(t0)=3.5 [29]. For now, to explore the undisturbed, basic interaction effects, we assume in
our study a lossless cavity κ=0µeV and neglect the radiative coupling of the QD to non-cavity
modes γrad =0µeV. In the end of this paper we will discuss the influence of these additional loss
channels on the dynamics and explore possible parameter regimes. Further details on parameters
and initial conditions are found in the appendix.
Before investigating the coupled electron–photon–phonon dynamics and presenting our
key results, we will briefly recapitulate the underlying physics in quantum optics with
isolated atoms as active systems electron–photon dynamics solely, where the collapse and
revival phenomenon is observed but also exhibit restrictions for its observation. Then we show
how the solid-state surrounding can reduce certain limitations encountered in isolated atomic
systems.
First, we benchmark our model and derive the exact solution of the coherent-state JCM [28]
by neglecting the electron–phonon interaction and all other loss channels. For a QD and cavity
on resonance figure 2(a) displays the corresponding Rabi oscillations in the mean cavity photon
number N=hˆc†ˆci. The first few picoseconds show the excitation transfer between the QD
and the cavity. Its collapse at 10 ps and subsequent revival within the first 20 ps constitutes
the essential feature of the collapse and revival phenomenon, visually supported by the solid
gray line. However, due to the low initial photon number of N(t0)=3.5, the first collapse
and revival cycle is not completed and after 30 ps the system drifts into irregular fluctuations,
visually supported by the vanishing dashed gray line. For a cavity mode assumed to be on
resonance with the fundamental QD transition this behavior can be explained as follows.
For a photon field in a Fock number state with precisely m photons, the photon density
will oscillate between the ground (N=m) and excited (N=m+ 1) with the quantized Rabi
frequency m=M√m+ 1. In contrast, only a mean photon number can be calculated for
an initially coherent field mode. Here, the typical Jaynes–Cummings solutions of the exact
number states are superimposed in an average over the Poissonian distribution P(m)=
exp(−|α|2)|α|2m/m!, which describes a coherent photon state and exhibits a peak at the mean
photon number |α|2=N, see bar charts for N=3.5 and 100 in figure 3, and forms a quantum
wave packet having certain reoccurrence times. Consequently, collapse and revivals occur [28].
This behavior relates directly to the quantized nature of the cavity field [c,c†]6=0. Typically,
only discrete frequencies maround the mean photon number Ninterfere, which are initially (at
t0) prepared in a definite state and therefore start a correlated dynamics. With increasing time
the superimposed discrete Rabi frequencies exhibit an increasing phase shift and destructive
interference is inevitable, see figure 2(a) at around 10 ps. However, this collapse is followed
by a revival. The dynamics of this revival phenomenon strongly depends on the mean
photon number. Figure 3shows, that for low mean photon numbers the neighboring Rabi
New Journal of Physics 15 (2013) 105024 (http://www.njp.org/)
5
Figure 2. Dynamics of the cavity occupation. (a) Reproduced JCM solution
without phonons shows only a single collapse with an incomplete revival.
(b) Including phonons with a lattice temperature of 50 K (orange) many-cycle
collapse and revivals become possible.
Figure 3. Poissonian distribution Pmto find mphotons for different mean
photon numbers. The orange line shows the Rabi frequency m=M√m+ 1.
For N=3.5 (light blue) two neighboring photon states differ in their respective
frequency by 11%, while for N=100 (dark blue) only by 0.5%. The inset shows
the much more stable collapse and revival for N=100, without phonons.
frequencies mdiffer substantially (11%) and their superposition gets out of phase to a point
where the initially correlated state is not recovered and the dynamics becomes highly irregular in
a quasi-irreversible manner [28], as seen in our simulation in figure 2(a). In contrast, if the mean
photon number is high, the impact of spontaneous emission processes is weak (√m+ 1 ≈√m),
i.e. the phase shift of the superimposed Rabi frequencies is negligible (only 0.5% difference in
New Journal of Physics 15 (2013) 105024 (http://www.njp.org/)
6
Figure 4. Electron–phonon coupling in QDs. (a) Phonon-induced decay of the
QD coherence in absence of a cavity and hence without coupling to photons.
(b) Absolute value of the corresponding electron–phonon memory kernel.
Independent of temperature the kernel vanishes after 4 ps.
m) and the initially correlated state is recovered. As a result, high photon numbers allow for
many regular collapse and revival events in an atomic cQED system, see inset of figure 3.4
Now, we turn to the specifics of the light–matter coupling in a solid-state environment
inducing electron–phonon interaction and show that high photon numbers are not necessarily
required to see many collapse and revival events. When including phonons, the photon density
shows a strongly modified time evolution, see figure 2(b). Remarkably, the coupling to the
phonon bath facilitates regular coherent interference signatures. Depending on the temperature,
collapse and revivals survive for much longer times, e.g. for 50 K over 150 ps before fading out.
The counter-intuitive behavior of an incoherent bath of phonons supporting regular
coherent features and suppressing seemingly random oscillations can be understood by
investigating in detail the coherent polarization Pmof the coupled electron–photon–phonon
system, which drives the dynamics of Xmand vice versa.
As an instructive example, consider a simpler case of a QD that interacts solely with its
host material and not with the quantum light field, first. Here, the band-diagonal coupling to the
phonon bath will lead to so-called pure dephasing, i.e. a destruction of the phase coherence of
the bare initial two-level systems polarization p=hˆa†
vˆaciwithout relaxation of the excited state
occupation. Consider a QD that is excited by a delta-like laser pulse at time t0=0. Figure 4
shows how phonons induce an ultra-fast temperature-dependent decay of the polarization on
4Before going into an in-depth discussion of the phenomenon a few words concerning the ambiguous term
coherence might be in order. On one hand it stands for the electronic coherence/polarization between the two QD
levels v, c (later also a photon-assisted coherence), which is at least partially lost when interacting with the phonons
of the solid-state environment. On the other hand the term coherent describes the nature/statistics of the cavity
photon field (in contrast to, e.g. a Fock number state). When the electronic system of the QD is strongly coupled
to such a coherent photon field the coherent effect of collapse and revival can arise, where an excitation transfer
between the cavity photon number and the electronic occupation (not coherence!) of the QD occurs. Although
the phonon coupling diagonal in the electronic states leads to a loss of electronic coherence (but not to a loss of
the electronic occupation) in the QD, the signatures of the coherent phenomenon of collapse and revival in the
photon number persists and can even be stabilized. This is what we refer to when speaking of phonons supporting
coherence effects.
New Journal of Physics 15 (2013) 105024 (http://www.njp.org/)
7
Figure 5. Coherent polarization dynamics. The top panel (a) shows irregular
behavior of the JCM solution (blue) without phonons. For comparison the
dynamics including γphen =65 µeV is shown as a black line on top of the
blue (undamped) JCM curve, not leading to many-cycle collapse and revivals.
The second panel (b) shows the phonon-induced dephasing of P0(orange) and
resulting pattern formation. The bottom panel shows in a zoom how a phase shift
occurs when phonons are considered.
a picosecond time scale. In contrast to a pure exponential decay, as obtained by introducing
a simple T2-time, the initial dephasing in figure 4(a) reveals the non-Markovian nature of
the system–bath coupling [36] by a quadratic decay at t=0 and a clearly non-exponential
behavior. Indeed, the memory kernel of the electron–phonon interaction [37,38] persists for
up to 4 ps, independent of the bath temperature, see figure 4(b). When including additional
phenomenological decays like κand γrad the memory kernel will be altered. For small losses
the system can retain its non-Markovian nature, but for κand γrad in the range of some tens of
µeV losses will dominate and the dynamics becomes Markovian, see discussion at the end of
this paper and figure 6.
Turning back to our cavity-QED model, the same decoherence happens to the photon-
assisted polarizations Pm=hˆa†
vˆacˆc†m+1 ˆcmivia their coupling to the phonon system, see
equation (A.3) in the appendix. These coherences Pmare the sources for the QD electron
dynamics and, via equation (5), for the photon number N, too. Exemplary, the temporal
dynamics for m=0, i.e. the photon-assisted polarization hˆa†
vˆacˆc†i, is shown in figure 5.
When no phonons are considered figure 5(a) undamped and, at later times, irregular
oscillations (blue line) occur after a first collapse and revival event, continuously driving
the photon-assisted electron densities Xm. Next, when only a simple phenomenological
dephasing constant of γphen =65 µeV for the electronic polarization is used [39], resembling
New Journal of Physics 15 (2013) 105024 (http://www.njp.org/)
8
Figure 6. Cavity photon dynamics for different parameter regimes. Panels (a)
and (b) show the dynamics for the same electron–photon coupling strength Mas
used in figure 2, but including a small cavity loss κand a radiative loss of the QD
occupation γrad. Panels (c) and (d) show the dynamics for a parameter set taken
from recent experiments [35].
the Markovian contribution, the oscillation amplitude gets strongly damped, as the blue line in
figure 5(a) shows. Since the constant damping affects the coherence at all times any reemerging
fluctuations are suppressed.
In stark contrast, figure 5(b) (orange line) shows how the phonons coherent dynamics.
Within the first picoseconds a similar damping compared to γphen occurs. However, due to the
temporally finite interaction memory kernel, see figure 4(b), and the continuous driving via
the photon field, Pmcan revive. Therefore, the initial collapse and revival event is repeated and
subsequent revivals can arise, only with a decreased amplitude, since the phonon bath introduces
a loss channel in the system.
The zoom below figure 5(b) demonstrates the fundamentally different nature of the full
non-Markovian system–bath coupling (orange line), as a clear phase shift in the dynamics
occurs, compare with the blue and black line. Each time a coherence builds up, the phonon-
induced dephasing will set in, prolonging the revival time, but still allowing for later recovery.
Consequently, only the non-Markovian nature of the electron–phonon coupling allows the
collapse and revival phenomena to persist and a corresponding pattern within the photon density
arises.
Under what terms and conditions do we expect to observe this effect? Firstly, our study is
valid within a perturbation theory for the phonons and constitutes an approximative method, as
noted by comparison of figure 4(b) with the exact independent Boson model. However, since
we are still within reasonable limits, we believe that exact numerical calculation available [40]
in the future may obtain modifications, but retain our key finding of enhanced collapse and
revivals. Secondly, in experimental samples further decay channels are present, e.g. radiative
and cavity losses, which will lead to an additional damping of the cavity photon number and
QD occupation X0. As undesirable their influence is in terms of experimental realizations, we
want to stress that they do not affect the principle mechanisms underlying the stabilization of
the collapse and revivals. As for many effects in quantum optics, the decisive quantity here is
the ratio between Mand κ+γrad. If the damping is much weaker than the radiative coupling, i.e.
M(κ +γrad), the stabilization of the collapse and revivals is still observed, see figures 6(a)
New Journal of Physics 15 (2013) 105024 (http://www.njp.org/)
9
and (b). When much higher losses occur, i.e. M(κ +γrad), the cavity photon number will be
damped to a point where no Rabi oscillations are observable let alone collapse and revivals,
see figures 6(c) and (d). Current semiconductor cQED technology would need a substantial
improvement to resolve the presented effect as figures 6(c) and (d) show, where the values for
M, κ and γrad are taken from [35]. In particular we predict a considerably large effect in the
few photon limit, in contrast to atomic cQED, as long as none of the interaction dominates the
dynamics alone. A strong Mis always favorable to the observation of collapse and revivals, but
not a necessity:
(i) a weaker Mcan be compensated for by a higher photon number N(under constant
phononic properties of the material) or
(ii) if Mis reduced and Nis kept constant, the range of the phonon coupling in reciprocal
space must be adjusted via the geometry of the QD, its material composition, etc.
Surprisingly, a very strong electron–photon coupling will inhibit the phonon-induced
regular coherence features, since the electron–bath interaction becomes negligible compared
to the electron–photon coupling [40]. Likewise, high mean photon numbers will outrival
the effects of the phonon bath, too. However, throughout our simulations we found that the
stabilization of collapse and revivals is very robust for a wide range of different parameters
and scenarios. Therefore, with improving sample technology in the future, it should constitute
a fundamental phenomenon inherent in the cQED of semiconductor-based devices. Within a
quantum well–cavity system, for example, a much higher electron–photon coupling strength
Mcan be achieved, while at the same time operating with higher cavity photon numbers [41].
The phenomenon’s robustness relies on the complex interplay between the electron–phonon
and electron–photon dynamics. This interplay is governed primarily by the photon
number Nand the spectral density of the phonon modes. It further depends on the coupling
strengths, Mand gq, and the temperature of the lattice.
Since the crucial electron–phonon coupling and the corresponding dephasing mechanisms
depend on the material parameters, the temperature of the bulk and geometry of the QD,
semiconductor technology is not only capable of engineering collapse and revivals with
predefined, desired properties, but predestined. Investigating quantum optics with QDs
might even advance our understanding of energy transfer in more complex situations, e.g.
photosynthesis.
To summarize, the influence of a bath of bulk LA phonons on the coherent dynamics of a
QD two-level system in the cQED regime was explored. The well-known random oscillations in
the electronic occupation and cavity photon density, typical in the atomic JCM for low photon
numbers, where altered significantly. Instead, due to LA phonons inducing a non-Markovian
dephasing and thus influencing the coherence time, regular collapse and revivals arise and in an
ideal (lossless) scenario last for up to 200 ps depending on temperature.
Acknowledgments
We thank S M¨
uller, M Gl¨
assl and M V Axt for fruitful discussions of our numerical results.
This work was financially supported by the Deutsche Forschungsgemeinschaft within the
Sonderforschungsbereich 787 ‘Nanophotonik’. AC gratefully acknowledges support from the
Alexander-von-Humboldt foundation through the Feodor-Lynen program.
New Journal of Physics 15 (2013) 105024 (http://www.njp.org/)
10
Appendix. Methods
A.1. Numerical parameters
Phonon coupling element |gq|=|gq
v−gq
c|, where gq
i=q¯
hq
2ρcV Die−q2¯
h
4miωi. Used parameters:
photon order m=27, sound velocity of GaAs cs=5110 m s−1, deformation potentials Dv=
−4.8 eV,Dc=−14.6 eV, effective masses: mc=0.067, mv=0.45, confinement energies
¯
hωc=0.030 eV, ¯
hωv=0.024 eV and mass density of GaAs ρ=5370 kg m−3.
A.2. Dynamics
The general set of coupled equations that determine the electron dynamics and through
equation (5) of the paper the photon dynamics as well, are obtained via a method of
induction [42], when using the complete Hamiltonian ˆ
H=ˆ
H0+ˆ
Hel−t+ˆ
Hel−pn and equation (1)
of the paper:
∂tGm=−2mκGm+ 2γrad Xm+
i
M[ˆ
Pm+1 −(Pm+1)∗+m Pm−1−m(Pm−1)∗],(A.1)
∂tXm=−(2mκ+ 2γrad)Xm−
i
M Pm+1 +
i
M(Pm+1)∗,(A.2)
∂tPm=−
i
(1 −
i
(2m+ 1)κ −
i
γrad −
i
γphen)Pm−
i
Mm Xm−1+Xm+1 −Gm+1(A.3)
−
i
X
q
g∗
qBm
+(q)+gqBm
−(q).
The (photon-assisted) ground state density Gm=hˆa†
vˆavˆc†mˆcmiis solely driven by the photon-
assisted polarization Pm=hˆa†
vˆacˆc†m+1 ˆcmivia the electron–photon coupling, as is the excited
state density Xm=hˆa†
cˆa†
cˆc†mˆcmi. The polarization experiences a free rotation due to a
possible detuning 1=ωg−ω0of the QD gap energy ωg=ωc−ωvand cavity resonance.
A phenomenological cavity loss κand dipole decay rate γrad is included, too. We can
furthermore account for additional broadenings, observed in experiments, by introducing
γphen [32]. Since κand γrad reduce the amplitude of all involved quantities equally, it has no
influence on our principle findings of the pronounced collapse and revivals, but much relevance
to its experimental observation.
Besides the spontaneous emission of photons (∝m Xm−1), crucial for Rabi oscillations,
equation (A.3) includes a coupling to phonon–photon-assisted polarizations Bm
+(q)=
hˆa†
vˆacc†m+1cmˆ
b†
qiand Bm
−(q)=hˆa†
vˆacˆc†m+1 ˆcmˆ
bqi, too. The dynamics of B±is solved within a
Born approximation, where photon-assisted electronic and phononic variables factorize. This
yields a closed set in terms of the electron–phonon coupling
∂tBm
+(q)=
i
(1 −
i
(2m+ 1)κ −
i
γrad +ωq)Bm
+(q)−
i
g∗
qnqPm(q), (A.4)
∂tBm
−(q)=
i
(1 −
i
(2m+ 1)κ −
i
γrad −ωq)Bm
−(q)−
i
gq(nq+ 1)Pm(q), (A.5)
where the mean phonon number nq=hˆ
b†
qˆ
bqioccurs. For temperatures well below 100 K, as
considered here, a second-order Born approximation is well validated to solve the occurring
hierarchy problem in the electron–phonon coupling, as can be seen by comparison to exactly
New Journal of Physics 15 (2013) 105024 (http://www.njp.org/)
11
solvable models [43,44] and experiments. To explore the high-temperature regime, terms
beyond second-order must be included or other approaches employed [31].
The phonons of the homogeneous semiconductor material are modeled as a reservoir
of harmonic oscillators, which introduces a temperature dependence to the dynamics via the
thermal Bose–Einstein distribution function nq(T)=(exp[¯
hωq/kBT]−1)−1. The coupling to
the bath will affect the coherent properties of the photon-assisted polarizations Pmover time
and lead to a non-Markovian pure dephasing, which cannot be accounted for by introducing the
simple, constant dephasing time T2=γ−1
phen alone.
The complete set of equations (A.1)–(A.5) is, however, not closed in terms of the
photon number (m). Therefore, depending on the coupling strength M, a sufficient high
order in mis taken into account to reach convergence in the calculations. In the absence of
electron–phonon coupling the obtained solution is thus numerically exact and renders the
JCM [45].
To have a fair comparison to the JCM, we accounted for the polaron shift in the QD
resonance 1pol =Pq|gq|2/ωqby accordingly detuning the cavity, so that both, QD and cavity,
are on resonance again. If 1pol would not have been compensated, a higher Rabi frequency and
smaller amplitude would have resulted. We carefully checked our model for detuning effects.
Including a strong phenomenological dephasing γphen allows for a Fourier transformation. A
spectral analysis of P0(ω) showed that the involved Rabi frequencies in both, the phonon-free
JCM and the phonon cases, are essentially in the same spectral range.
A.3. Initial conditions
The initial state of the photon expectation values is derived in dependence of the (given photon
statistics and) mean photon number N. The cavity is prepared in a coherent photon state
with a mean photon number N=3.5 and thus hˆc†mˆcmi=PnNn(eNn!)−1hn|ˆc†mˆcm|ni= Nm.
Initially the QD is inverted, i.e. Gm(t0)=0.0 and X0(t0)=1.0. The complete density operator
ρconsiders the electron- ρel and photon system ρpt, and reservoirs ρr(LA phonons and
dissipative photon modes). Initially at time t0,ρfactorizes ρ(t0)=ρel(t0)⊗ρpt(t0)⊗ρr(t0).
Consequently, the photon-assisted excited state factorizes Xm(t0)=X0(t0)Nm. No coherence
is present Pm(t0)=Bm
±(t0)=0. The initial state of the phononic bath is determined by the
temperature of the system.
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