Document [original]

Nanophotonics 2019; 8(5): 655–683

Review article

Alexander Carmele* and Stephan Reitzenstein

Non-Markovian features in semiconductor

quantum optics: quantifying the role of phonons

in experiment and theory

https://doi.org/10.1515/nanoph-2018-0222

Received December 16, 2018; revised February 22, 2019; accepted

February 25, 2019

Abstract: We discuss phonon-induced non-Markovian and

Markovian features in QD-based quantum nanooptics.

We cover lineshapes in linear absorption experiments,

phonon-induced incoherence in the Heitler regime,

and memory correlations in two-photon coherences. To

qualitatively and quantitatively understand the underly-

ing physics, we present several theoretical models that

capture the non-Markovian properties of the electron–

phonon interaction accurately in different regimes. Exam-

ples are the Heisenberg equation of motion approach, the

polaron master equation, and Liouville propagator tech-

niques in the independent boson limit and beyond via the

path integral method. Phenomenological modeling over-

estimates typically the dephasing due to the finite mem-

ory kernel of phonons and we give instructive examples

of phonon-mediated coherence such as phonon-dressed

anticrossings in Mollow physics, robust quantum state

preparation, cavity feeding, and the stabilization of the

collapse and revival phenomenon in the strong coupling

limit of cavity quantum electrodynamics.

Keywords: phonons; semiconductor quantum dots; open

quantum system; quantum optics.

1 Introduction

Since the seminal demonstration of optically [1] and elec-

trically triggered [2] single-photon emission, deterministic

generation of entangled photon pairs [3, 4], near-unity

indistinguishable photons [5], and strong coupling to a

microcavity [6] with semiconductor quantum dots (QDs)

[7,8], acting as active quantum light emitters, there has

been a steadily increasing number of research activities

to establish semiconductor systems, in particular QDs,

as a key element for modern photonic quantum tech-

nologies [9–19]. Focusing on the goal to implement QDs

in quantum sensing, metrology, and quantum cryptogra-

phy and to establish them as “artificial atoms” and ideal

candidates for solid-state quantum bits (qubits) and scal-

able quantum information processing [13, 20–22], it has

become clear that QDs cannot be considered as isolated

quantum emitters but interact intrinsically with their sem-

iconductor bulk environment. This coupling has usually

detrimental impact as it leads to decoherence of the con-

fined exciton acting via superposition states as a qubit

[23, 24]. Interestingly, it can also be of positive impact

and facilitate, for instance, attractive phonon-mediated

resonant excitation schemes [25–27] or phonon-induced

quantum coherences [28, 29].

A major phonon coupling mechanism stems from the

surrounding host material and its lattice vibrations [8,

30, 31]. The associate distortion of the underlying crystal

structure leads inevitably to electron–phonon interac-

tion and subsequently to substantial new forms of deco-

herence unknown in standard atom–molecular optics

(AMO). Because the phonon reservoir is structured, that

is, the coupling is frequency-dependent, the influence of

phonons on the quantum optical properties of QDs is of

non-Markovian nature [23, 32–34]. Thus, to quantify and

unravel the underlying physics, non-Markovian dissipa-

tive processes need to be taken into account. However,

non- Markovianity refuses a time-local formulation, and

therefore typical quantum dissipative treatments become

unavailable [35, 36] and advanced, or perturbative

methods need to be employed [37–41].

Before we discuss the electron–phonon interaction

in more detail, we want to briefly comment on how the

term “non-Markovian” and “Markovian” is used in this

*Corresponding author: Alexander Carmele, Technische Universität

Berlin, Institut für Theoretische Physik, Nichtlineare Optik und

Quantenelektronik, Hardenbergstraße 36, 10623 Berlin, Germany,

e-mail: ale[email protected]u-berlin.de. https://orcid.org/0000-0001-8108-

2973

Stephan Reitzenstein: Technische Universität Berlin, Institut für

Festkörperphysik, Optoelektronik und Quantenbauelemente,

Hardenbergstraße 36, 10623 Berlin, Germany

Public License.

656 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

review. We adapt the definition of Markovian and non-

Markovian processes as formulated among others in Refs.

[37, 41]. A system–reservoir interaction is Markovian if the

reservoir correlation time vanishes or can be assumed to

be negligible for the given set of observables of interest.

For example, the correlation function of a reservoir con-

sisting of harmonic oscillators with annihilation (crea-

tion) operators (†

)

q with bosonic commutation relation

†

[, ]=( )bb

′−′

can be written as:

††

() ()

[]

||[(21)cos[( )] sin[

()]]

qq qq

itit itit

qq q

RtRt gg ebbe bb

gn tt

ωω ωω

ωω

′′

−+ −

′′

∗

′′ ′

′

〈〉

=〈〉+ 〈〉

′

=+−− −

′′

∑

qq qq qq

(1)

where †

() (exp[]exp[

])

tgbitb

itωω

=−+

∑

qq q

q and an

initial thermal equilibrium at 1/β = kBT has been assumed

with

†1

[exp()1] =,

bb n

βω −

〈〉=−

qq  the Bose–Einstein dis-

tribution, and (†)(†)

0.bb〈〉

qq If now the relaxation time of

the reservoir is much smaller than the evolution time of

the observed system, that is, Re[〈R(t)R(t′)〉] = Γδ(t–t′), the

system’s evolution does not depend on former system–

reservoir interaction and the process is Markovian [41].

A specific example of a vanishing correlation time is the

radiative decay into free space, where the vacuum field

amplitude is nearly constant in the regime of optical sys-

tem–reservoir interaction and justifies within the small-

bandwidth assumption a Lindblad formulation of the

process as shown in standard quantum optics books [35–

38]. In this case, the dissipative processes can be included

via time-independent Lindblad superoperators [35].

A standard example for non-Markovian processes is

the electron–phonon interaction in semiconductors [7,

23, 24, 42], where the timescale of the interaction is in the

same regime as the corresponding mode structure (ps),

and the environment correlation time is finite. In this

case, a backflow of information between environment

and system takes place and leads to non-Lorentzian line-

shapes (e.g. [43]), longer Wigner delay times [44], reap-

pearance of Rabi oscilllations (e.g. [33, 45]), or allows for

robust state preparation protocols (e.g. [25, 27, 46–48]),

phonon-mediated quantum coherences (e.g. [28, 29, 49]),

and incoherent excitation processes such as cavity feeding

(e.g. [50, 51]), pulse area–dependent damping, renormali-

zation of Rabi oscillations [23, 45], and excitation-induced

dephasing of Mollow triplet sidebands [52–54] as will be

discussed in detail below. Thus, when we refer to non-

Markovian processes, we mean in general a process that

cannot be described with a time-independent, global,

Lindblad-based master equation as the paradigm for

delta-correlated white noise [35–38, 55].

In the following, we review exemplarily recent work

on non-Markovian and Markovian signatures of electron–

phonon dynamics in semiconductor QDs [7, 42]. In par-

ticular, we focus on most mature InGaAs/GaAs QDs (QDs)

with dominant coupling to longitudinal acoustic (LA) and

longitudinal optical phonons (LO) via the deformation or

respective Fröhlich coupling element [24]. For detailed

discussion of semiconductor nanostructures, and the

effort of microscopical calculations to quantify coupling

mechanism in semiconductors, we refer to relevant books

[8, 56] and reviews [57–60].

For the sake of compactness, we do also not include

explicitly results in the research efforts of higher-dimen-

sional semiconductor nanostructures such as quantum

wires, quantum wells, or mesoscopic bulk systems [61,

62]. There is a wide range of exciting phenomena in these

systems because of the ubiquitous non-Markovian elec-

tron–phonon and electron–electron dynamics. To name a

few, quantum wires show phonon-enabled thermal con-

ductivity [63] based on a universal quantum of thermal con-

ductance [64]. Pronounced non-Markovian decoherence

is demonstrated in localized nanotube excitons [32], and

also phonon-assisted Anderson localization phenomena

have been investigated [65]. In quantum wells, coherent

acoustic oscillations are studied [66], and nonequilibrium

cooling effects bottlenecked by non-Markovian phonon

dynamics have been discussed [67]. Four-wave mixing

techniques allow to study and characterize giant excitonic

resonances [68, 69], and via two-dimensional coherent

spectroscopy techniques, incoherent exciton–phonon

Green’s function can be probed and extracted in disor-

dered quantum wells [70, 71].

Moreover, optical and electronic two-dimensional

spectroscopy has drawn a lot interest recently, as non-

Markovian, anomalous lineshapes and relaxation/scatter-

ing processes can be characterized and studied in depth.

For example, the study of signatures of spatially correlated

noise and nonsecular effects [72] has been performed, the

read-out of Rabi oscillations in QDs [73], exciton coher-

ence at room temperature [74] investigated, systematic

study of dephasing processes including quadratic elec-

tron–phonon coupling for elevated temperatures [75, 76]

and phonon sidebands in transition metal dichalcoge-

nides have furthermore been demonstrated [77]. Despite

the exciting results in higher-dimensional nanostructures

and two-dimensional coherent spectroscopy, we focus

in the following on a single material platform, semicon-

ductor QDs. This allows us to discuss in detail instructive

experimental examples in which the phonon interaction

is the dominant source of decoherence and theoretical

models, which are capable to capture their specific details

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 657

and can partially be treated even analytically. These

models are, however, not limited to the QD case and are

used for other material platforms such as quantum wells,

quantum wires, or atomic-thin two-dimensional systems

as well [61, 78, 79].

2 Structure of the review and

Hamiltonians

This review covers two different interaction schemes,

electron–light (e – l) and electron–phonon (e – p) dynam-

ics, and is basically structured by increasing complexity.

To increase readability, we introduce here the correspond-

ing Hamiltonians, which are used throughout the review.

The electron–light interaction is treated either semiclassi-

cally

− quantized in a mode continuum

con

− or within

a cavity or single mode description

cav

− The Hamiltoni-

ans read:

22 21 12

()(),

H t

∆σ Ωσ σ

−=+ + (2)

††

21 22

0122

()

cav

Hcc gc c

ωσ

−=+

 (3)

††

21 22 12 21

(),

con

Hdccgcgc

ωω

ωσ ωω

σσ

−=+ ++

∫

 (4)

where the rotating-wave approximation has been applied,

and Ω(t) is the Rabi frequency, including the exter-

nal driving field E(t) with frequency ωL and the dipole

moment of the QD d12 = d21, assumed in the following as

real number, between the conduction 2 and valence band

1with a bandgap energy of ħω21. In the semiclassical case

(),

− the Hamiltonian is written in the rotating frame of

the laser frequency, leading to a detuning of Δ = ω21–ωL.

The operator σij flips the state |j〉 to |i〉, whereas (†

)(†)

annihilates (creates) a photon in the corresponding mode

with bosonic commutation relation †

]

().

ωω δω ω

′

=−

′

The interaction strength between photons and electron is

denoted g, assuming a Wigner–Weisskopf-like coupling

with an approximate constant vacuum field amplitude

[35, 36].

The second interaction we consider is the electron–

phonon interaction, either in a semiclassical limit sc

−

or for longitudinal acoustical LA

− or longitudinal optical

phonons

− The Hamiltonians read:

()

Ftσ

−= (5)

††

22 12

[],

bb gb bωσ

−=+ +

∑∑

qqq

 (6)

††

22 12

[],

ep LO

bb fb bωσ

−=++

∑∑

 (7)

where F(t) denotes a stochastic force acting on the QD,

the electron-longitudinal acoustic, and

the elec-

tron-longitudinal optical phonon coupling element.

Throughout the review, the standard GaAs phonon bulk

parameters are used. For example, in case of an approxi-

mative spherical geometry, the acoustical phonon cou-

pling element 12 11

ggg=−

qqq

reads, where

ii i

−



 (8)

and typical parameters for InGaAs/GaAs QDs are given

with sound velocity of GaAs cs = 0.00511 nm/fs, defor-

mation potentials D1 = −5.38 eV, D2 = −11.68 eV, effec-

tive masses: m2 = 0.063, m1 = 0.45, confinement energies

ħω2 = 0.040 eV, ħω1 = 0.02 eV, and mass density of GaAs

ρ = 5370 kg/m3 [23, 24, 29, 51, 56, 80]. For the longitudinal

optical phonons, the Fröhlich coupling applies and reads:

ω−

=− ′



εε (9)

with 1/ε′ = 1/ε∞–1/εs = 0.0119347 with the static dielec-

tric constant εs = 12.53 and the high frequency dielectric

constant ε∞ = 10.9, and the reciprocal dielectric constant

1/ε0 = 18.1e2/(eV nm). The longitudinal optical phonon fre-

quency is in GaAs ħωLO = 36.4meV.

This review is structured in five parts. After the intro-

duction (Section 1) and this section (Section 2), we discuss

nonequilibrium phonon dynamics in the semiclassical

regime in Section 3. Here, the light field is treated classi-

cally with

− and acts as an excitation source and a way

to probe the system’s dynamics. Here, we discuss the elec-

tron–phonon and electron–light interaction in different

regimes, from weak coupling and weak driving to strong

coupling and strong driving, and important theoretical

tools are presented and explained in detail.

In Section 4, we include also quantized electron–

light interaction in cavity quantum electrodynamics

cav

−

or for a mode continuum

con

− and show how phonons

contribute to quantum optical phenomena, render-

ing the field of semiconductor quantum optics exciting

and novel. This section is structured from the single-

and two-photon regime to the many-photon dynamics,

including non-Markovian phonons as the main source

for decoherence.

We then conclude the review in Section 5 and give

a short outlook on future directions. Please refer to the

658 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

given Sections for a detailed discussion of the presented

material.

3 Nonequilibrium phonon dynamics

in semiclassical light–matter

interaction

Experimental data on quantum emitters in a solid-state

environment, as discussed in the introduction, show

the necessity of non-Markovian decoherence models.

Modeling the required intrinsic memory of the electron–

phonon interaction is theoretically a demanding but

also very rewarding task as it provides the link between

experimental data and their non-Markovian description.

Analyzing decoherence mechanisms in solid-state devices

is therefore crucial to understand the underlying physics

and to optimize their functionality, in particular when

quantum properties are addressed. It is also an important

part in the pursuit to realize reliable protocols in future

quantum communication networks, which are based on

coherent light–matter interfaces and the exchange of

quantum information via single photons and entangled

photon pairs with high indistinguishability [81, 82]. Here,

light–matter interaction between two-level emitters and

the coherent light fields of an external laser is a well-

established approach to investigate the time evolution of

the polarization density within a medium [83, 84]. This

time evolution can be governed by semiclassical, Maxwell

type of dynamics, or takes place deep in the quantum

regime of single-emitter–light interaction. However, in

this section, we discuss nonequilibrium phonon dynam-

ics in the semiclassical regime.

The achievements in fabrication of high-quality QDs

acting as close-to-ideal two-level emitters in the solid

state[22, 85] established the possibility to investigate the

light–matter interaction in different regimes. This section

is structured from the weak electron–phonon coupling

and weak driving limit (linear) to the strong electron–-

phonon coupling and strong driving regime (Mollow

regime).

We start in Section 3.1 with the discussion of the

impact of lattice distortion processes in absorption experi-

ments [24, 83, 86]. In this linear regime, the QD emission

can be expressed via the absorption coefficient α(ω) and

calculated exactly via the independent boson model

[37, 87]. Phonon-induced non-Lorentzian lineshapes are

unraveled in the absorption coefficient, and a clear devia-

tion from Gaussian white noise correlation is observed

and successfully explained with wavefunctions based on

8 band k · p theory and with inclusion of non-Markovian

phonon effects [43].

We continue in Section 3.2with the weak driving or

Heitler regime in which we investigate the emission and

scattering of a single QD under pulsed excitation [44]. This

regime allows us to explore an intriguing quantum optical

effect namely the Wigner time delay known from atomic

physics [88–90]. This delay originates form the phase

shift between the exciting and emitted field because of

the finite dwelling time of the excitation in the QD before

the QD relaxates. Phonon-induced incoherent processes

explain maximal achievable Wigner delay times τW(Δ) in

the nontrivial detuning dependence between the laser and

the QD transition [44], which cannot be described by con-

ventional optical Bloch equations [89]. Here, we discuss

semiconductor Bloch equations with phonon contribu-

tions via the cluster expansion approach [79, 91], which

allows perturbatively to take into account nonequilibrium

and non-Markovian phonon dynamics. We also show that

for low temperatures and weak electron–phonon cou-

pling, the perturbative cluster expansion solution agrees

well with the solution from the exact independent boson

model, given in Section 3.1.

In the weak electron–phonon coupling but strong

driving limit (Section 3.3), non-Markovian features are

also captured in the perturbative cluster expansion

approach [23, 79, 92–95]. In this limit, ultralong dephas-

ing times can be explained [31] and the recurrence of Rabi

oscillations for large pulse areas [23, 33, 46] is seen. This

method allows a fast and good approximation to quantify

influences of the microscopic properties of the nanostruc-

ture under investigation [78, 96]. As an example for signa-

tures typically not present in Lindblad (Markovian) master

equation simulations, we present in detail damped Rabi

oscillations via longitudinal acoustic phonons. Within

the same theoretical framework but in fourth-order per-

turbation theory, phonon-assisted state preparation pro-

tocols are discussed. Those protocols are robust against

the underlying geometry of the nanostructure; however,

the wavefront of the outgoing acoustic excitations differs

strongly for different geometries [27, 46, 96].

Beyond the weak electron–phonon coupling but still

in the perturbative regime (Section 3.4), master equation

approaches within a dressed-state basis are typically used

[35, 37, 97]. We first discuss the Markovian limit [98–101] in

Section 3.4 and give an analytical solution for the power

spectrum of a resonantly and optically driven QD sub-

jected to Markovian pure dephasing [102]. The Markovian

limit, however, cannot capture incoherent excitation pro-

cesses via phonon feeding [50]. To model such processes

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 659

correctly, a Polaron master equation is a feasible model

to include as much information as possible from the non-

equilibrium phonon dynamics in second-order perturba-

tion theory [51, 97, 103, 104] and shows already very good

agreement with experimental data [50]. We derive the

polaron master equation explicitly and show that for the

weak coupling limit in secular approximation, the cluster

expansion solution and polaron master equation dynam-

ics agree well as long as the exciting laser field amplitude

is time-independent.

We conclude this section with exact solutions of the

electron–phonon interaction. In case of a dispersionless

phonon mode, such as longitudinal optical (LO) phonons

in the Einstein approximation, an inductive equation of

motion scheme leads to numerically exact solutions,

cf.Section 3.5. As an example, we show phonon-assisted

anticrossings in the Mollow regime, which give experi-

mental access to the Huang-Rhys factor. This shows that

the equation of motion approach based on the semicon-

ductor Bloch equations [61, 79, 91] is not limited to the

weak coupling regime in principle. If the dispersion rela-

tion is nearly constant, as for LO phonons, an inductive

equation of motion approach can be used to take into

account higher-order phonon processes [28, 105, 106].

In this case, the equation of motion approach becomes

a numerically exact method and allows to compute LO

phonon–assisted Mollow triplet spectra. Because of the

strong driving limit, anticrossings between polaronic

and polaritonic dressed states occur and the electron–

LO phonon coupling strength can be spectrally obtained

because of the emerging splitting [107]. In case of linear,

or nonconstant dispersion, this method also still possible

to employ is numerically very expensive and therefore not

feasible anymore.

In the limit of time-dependent pulses and/or strong

coupling to phonons with nonconstant dispersion relation

(Section 3.6), perturbative approaches and master equa-

tion models do not capture the phonon-induced dynam-

ics accurately anymore [108]. Given the non-Markovian

and therefore entangled system–bath dynamics, density

matrix renormalization group techniques [109, 110] or

exact diagonalization become the only choice. They allow

for numerical expensive but exact treatments. Examples

are matrix product state evolution techniques [111–114] or

the real-time path integral method [33, 115] in which the

time evolution is discretized and the dissipative quantum

kinetics becomes solvable because of the finite memory

of the dissipative kernel, here of the acoustic phonons.

Within the path integral method technique, excellent

agreement between theory and experiment has been

demonstrated for state preparation protocols [25] and

the description of phonon-induced dephasing of coupled

QD microcavity systems in the strong coupling regime of

cavity quantum electrodynamics [116, 117].

3.1 Non-Lorentzian lineshapes (independent

boson model)

For weak coherent pumping, a system is well described

in the linear response regime [83]. Via a probe field, the

absorption of the nanostructure, here, for example, an

ensemble of QDs is quantified via the Beer–Lambert law

[118]. The absorption coefficient reveals resonances of

the emitter sample, but, more importantly, the interac-

tion with the environment is probed also via the detun-

ing dependence of the absorption. The corresponding

lineshape gives access to important information about

the kind of coupling (e.g. Fröhlich, deformation, or

piezoelectric interaction between electrons and phonons)

and the linewidth allows one to conclude about the effec-

tive electron–phonon coupling strength [7, 8, 12, 56]. A

characteristic quantity to connect theory to experiment is

the absorption coefficient, which can be expressed via the

susceptibility

(, ) Im[(, )].rrα

ωχω

In the semiclassical description, we assume an

incoming electromagnetic field E(r, t) described via Max-

well’s equations including material contributions. The

dynamics of the electric component is described via the

homogeneous equation: ∇ × E = −∂tB, and the inhomo-

geneous ∇ × B = μ0j + ∂tE/c2 with the speed of light in the

vacuum c0 = (ε0μ0)−1/2. Averaging over microscopic degrees

of freedom below the optical wavelengths, assuming an

electrically neutral sample without a macroscopic current,

and neglecting magnetic contributions, we yield for the

averaged current: j(r, t) ≡ ∂tP(r, t) [78, 83, 118]. Deriving the

wave equation in frequency domain for the electric field,

we obtain:

(, ) (, )

,rr

∆ωµω ω



+=−





(10)

where we assume a purely transversal wave and we

neglect in the following out of notation convenience the

background material refractive index. If the sample is

isotropic, homogeneous, and linear, the polarization is

related to the electric field: P(r, ω) = ε0χ(r, ω)E(r, ω) and

we derive the solution of the field intensity (propagating

in z-direction):

(, ) |(, )| ||exp[ ()]

||exp[ 2Im[ ]],

zrEikkz

Ekz

ωω

∗

== −

=−

(11)

660 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

with

(, )

,kk

nr ω= the refraction index of the inves-

tigated sample (, ) 1(,

)nr

ωχ

ω=+ and k0 = ω/c0. The

imaginary part of the wave number is directly propor-

tional to the imaginary part of the susceptibility. The sus-

ceptibility in the linear regime is connected to the dipole

density, whichis microscopically given via the transition

amplitudes multiplied by their dipole momentum and

the average dipole density in the sample with emitter

density n0:

(, ) ()

()

ij ij

rn r

ωρ

ω=

∑

(12)

with ρij(t) = 〈i | ρ(t) | j〉. Therefore, the density matrix

equation gives access to the linear and, also, nonlin-

ear response of the system via solving the Liouville–von

Neumann equation

() [, ].it Hρρ=





In the following, a sample consisting of QDs in the

two-level approximation is assumed, which is probed via

an incoming (linear polarized) field with an amplitude

parallel to the dipole moment of the two-level emitters

[119]. The associated Hamiltonian

− is given in Eq. (2).

The noncoherent contributions are considered by the

decoherence inducing Hamiltonian

()

H t

− given in Eq. (5)

with a c-valued stochastic force F(t). The dynamics of the

transition amplitudes reads in general:

12 12 22 11

() () 1|[(), ( )]|2

iititHttρ∆ρΩρΩ

ρρ

−

∂=

−+−〈 〉

 (13)

In the linear regime (no population change is

induced), we set ρ11(t) ≈ ρ11(0) = 1 and a time-independent

weak driving field Ω(t) = Ω0. Using a phenomenological

decoherence model, we choose a stochastic force con-

tribution to the energy splitting between ground |1〉 and

excited state |2〉, that is, ω21 + F(t) [38, 120]. The formal

solution of the transition dynamics in the linear regime

reads then with 0

()

tdtFt

′′

∫:

()

() ()

12 12 0110

() (0)(0)

itit it it

te idte

∆ξ ∆ξ

ρρΩρ

+−

−

′′

=−

′

∫ (14)

Because of the decoherence-inducing contributions

ξ(t), the equation must be averaged [120]. Assuming the

white noise limit, we average with a Gaussian probabil-

ity distribution, taking only two-body correlations into

account [120]. In this limit, we characterize the correla-

tion function as 〈〈F(t)F(t′)〉〉 = 2γδ(t–t′) with 〈〈F(t)〉〉 = 0 and

obtain 〈〈e±iξ(t′)〉〉 = exp[−γt]. Given the dephasing, the solu-

tion of the transition dynamics is obtained after a simple

integration:

12 12 011

() (0)(0)

it t

it te

te ii

∆γ

ρρΩρ ∆γ

−+

−



−

〈〈

〉〉 =−





−+ (15)

Differentiating with respect to time and taking the

Fourier transform into account, we find an expression for

the polarization density in the linear regime, projected

along the direction of the dipole:

0121

012

(, ) [()()]

)(

Prnd

ωρωρω

Ωρ Ωρ

ω∆ γω

∆γ

=〈〈〉〉+〈〈 〉〉



=−



−− +−



(16)

We have already assumed a linearly polarized elec-

trical field with a real amplitude parallel to the dipole:

Ω0 = d21E0/ħ. Ignoring the off-resonant parts of the suscep-

tibility, we find with ρ11(0) = 1:

012

021

(, )

()

nd i

rγ

χω ωω γ

−+

ε (17)

which connects to the absorption of the incoming

electric field via the Beer–Lambert law: I(z, ω) = | E0 | 2

exp(−2Im[χ]z). The higher the dipole density, the larger the

individual dipole moments and closer the incoming wave

is resonant with the transitions, the stronger is the absorp-

tion of the incoming wave into the sample.

The aforementioned absorptive behavior is experi-

mentally accessible via luminescence signals. In the limit

of Fermi’s golden rule and after subtracting a constant

offset, the absorption is proportional to the luminescence

signal. This assumption is valid up to second order in the

time-evolution operator and within the classical Condon

approximation for semiclassical light–matter interaction

[83]. These requirements are nevertheless fulfilled in most

of the cases, and absorption and luminescence lineshapes

are mirror images of each other. In the phenomenologi-

cal model discussed above, a Lorentzian lineshape is

obtained, cf. Eq. (16). However, comparing the experi-

mental luminescence signal of a single QD nonresonantly

driven via current injection in the p-i-n diode structure, cf.

Figure 1A and B, strong deviations from a Lorentzian oscil-

lator model are visible. The line broadening has a temper-

ature-dependent shoulder, which cannot be explained

with a white noise model as outlined above. In contrast,

the asymmetric lineshape is a direct consequence of the

underlying non-Markovian dynamics between the QD and

lattice vibrations [24, 30, 73, 76, 93, 121, 122].

To model the microscopic interaction between the

weakly driven QD and its semiconductor host matrix

more accurately, we choose the quantized electron–

phonon Hamiltonian

()

H t

− given in Eq. (6) already in

the interaction picture with the free evolution governed

by †

Hbb

ω=

∑

 [24, 93, 123]. To solve the quantum

mechanical decoherence model, we have to take into

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 661

account the quantum mechanical character of the deco-

herence Hamiltonian. The full solution of the density

matrix evolution reads in the Liouville space:

{}

() exp(

)(

tT Lt dtρ ρ







∫ (18)

with time-ordering operator T and Liouvillean superoper-

ator L(t)ρ = (−i/ħ)[H(t), ρ]. To derive the systems dynamics,

we trace out the phonon degrees of freedom:

{}

{()} () Tr exp(

)(

BssB

ttTT Lt dtρρ ρ







∫ (19)

where the time-ordering operators ensure that the corre-

sponding von-Neumann equation is still solved. To evalu-

ate this solution, typically a path integral method is used,

cf. Section 3.6. Here, we assume a vanishing light–matter

coupling and stay in the linear regime. In this case, Eq. (19)

can be solved analytically for ()

()

tHt

−

= [24, 37, 124].

To solve the quantum mechanical model in the linear

regime, one traces out first the electronic degrees of

freedom:

†

2| ()|1 2|[(), ( )]|1

() () 2| ()|1

tiHt t

igbt bt t

ρρ

−

〈〉=−

〈〉



=− +〈 〉



∑q

(20)

This equation can be formally solved via the Dyson

series with 〈2 | ρ(t) | 1〉 = PB(t):

{}

() ex

)/ (0)

tLA

tT idtH

−



=−





∫

Assuming discrete time steps Δt, we can write

() exp(0)

exp[ ]exp[]exp[ ](

Nt TiHP

iH iH iH P

∆







=−











=−

−−

∑



(21)

after performing a Suzuki–Trotter decomposition and

using the abbreviation

(1)

Nt LA

dt Ht

∆

∆−

−

=∫ (22)

The time-order is now trivially fulfilled and therefore,

the time-ordering operator is omitted. The necessary con-

dition for the analytical solution in this case is that the

commutator of two Hamiltonians is a c-value and not an

operator anymore:

(1)2

1121

(1)(2)

]

|sin[( )]

Nt Nt

NN q

Nt Nt

HH idtdtg tt

∆∆

∆∆ ω

−

−−−

=− −

∑

∫∫ q

(23)

This result means that the commutator of the

individual Hamiltonian with the commutator of the

Hamiltonians vanishes [[Hi + Hj], [Hi, Hj]] = 0, and there-

fore the Baker–Campbell–Hausdorff formula can be used

exp[A]exp[B] = exp[A + B]exp[[A, B]/2] to obtain in the

limit of Δt → 0:

200

() exp()/ (0)

exp[(), ()

Bep

tt LA LA

ep ep

tidt Ht P

dt dt HtHt

−

−−



=−











∫

∫∫



 (24)

–8

10–3

10–2

10–1

–6 –4

–2 0

∆

Energy (meV)

77 K

50 K

35 K

15 K

Experiment

8 band k · p

Gaussian

Norm. intensity

10–3

10–2

10–1

246–8–6–4–20

∆Energy (meV)

T = 15 K

Figure 1: Semilogarithmic plot of the electroluminescence intensity of a single QD.

(A) With increasing temperature the line broadening increases due to stronger acoustic phonon interaction. (B) Comparison of the measured

lineshape with two calculated spectra, using alternatively a simple Gaussian (green dotted curve) or a realistic 8 band k · p wave function

662 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

Because the time-ordering has been taken into account

already, one can trace out the phonons and the independ-

ent boson model is given via P(t): = TrB[PB(t)]

() exp[ Im[()] Re[()]],

Ptiti tt

ωφ φ

=+ −

(25)

() [(21)(1 cos[ ]) sin( )]

Iqqq q

tn titi t

ωω ω

=+−+−

∑

(26)

Obviously, the time-order is fully taken via the

imaginary phase whereas the temperature-dependent

dephasing stems alone from the part left untouched by

the time-ordering. Therefore, neglecting the time-order

completely will yield the wrong result interestingly only if

phase-dependent quantities are studied.

Taking the Fourier transform, we yield ρ12(ω) and

ρ21(ω) and the susceptibility includes then non-Markovian

effects because of the quantum character of the struc-

tured bosonic reservoir. The semiconductor host matrix

is considered in the coupling elements, as well as the

temperature dependence mainly governed by the phonon

frequencies from the dispersion relation, which in princi-

ple can be calculated ab initio. In Figure 1B, a microscopic

wavefunction is used to model the correct, experimentally

observed lineshape [43].

To conclude this section, a non-Markovian dephas-

ing model was applied to describe the emission spectra of

a single QD embedded in an electrically pumped device,

which is based on a p-i-n diode design and an oxide aper-

ture to spatially restrict the current flow. Despite the com-

plexity of the device design and the electrical excitation

scheme, the agreement between the analytical solvable

model and experiment is striking. This together with the

possibility to include microscopic parameters into the

model for the description of experimental results proves

the strength of the non- Markovian dephasing model. Note-

worthy, the lineshape of the calculated spectra depends

only on the electron–phonon coupling matrix elements.

Moreover, the sensitivity of the choice of electronic wave

function is clearly visible, comparing a Gaussian wave

function with an 8 band k · p theory [58, 59]. This is an

important feature of the advanced model and highlights

its usefulness for further optimization and technological

fine-tuning of single-QD quantum devices.

3.2 Phonon-enhanced Wigner time delay

(Cluster expansion)

Beyond the linear regime but still in the weak excitation

limit, the Heitler regime includes nonlinear contributions

as it studies the excited state dynamics ρ22(t) = 〈σ22(t)〉 as

a figure of merit in strong contrast to the simpler linear

regime, which addresses ρ12(t) = 〈σ21(t)〉. In the Heitler

regime, the incoming light is mainly coherently scattered,

that is proportional to |ρ12(t) | 2, but in dependence on the

pulse length and pulse area, a part of the incoming laser

excitation is absorbed ρ11(t) < 1, converted into electronic

excitation ρ22(t) > 0 before being emitted back via incoher-

ent scattering. The process of reemission needs a finite time

between the incoming pulse and outgoing emission, and

the delay between the maximum of the excitation pulse and

emission pulse is called Wigner delay or dwell time.

The Wigner delay is ultimately limited by the coher-

ence time

1/ =1/(2)1/TTT∗

+ [89] where T1 denotes the

radiative lifetime and 2

T∗

is the coherence timescale. If no

or negligible decoherence is present ( 2

∗

→∞

) such as in

certain atomic systems [88, 90], the maximum Wigner

delay reads 2T1 as the signal stems from radiative relaxa-

tion, and every relaxation induces a partial decoherence

between the electronic levels. Because the Wigner delay is

limited by the decoherence process in case of solid-state

emitters, it is an interesting figure of merit to quantify the

effective 2

T∗

time. In particular when probing the spectral

response, the detuning dependence of the Wigner delay

reveals clearly a non-Markovian and phonon coupling–

dependent feature as we discuss in the following [44].

Because the pulse induces population dynamics of

the electronical system, we need to solve the optical Bloch

equations. Limiting our analysis of the electronic system

to a two-level system, which is in good agreement with

the experimental situation of studying InGaAs QD with

close to ideal quantum properties at low temperature, we

restrict the dynamics first to the Lindblad master equation

case with

12 22

[(), ] []

[],

Ht γ

ρρσ

ρσ

−

=− ++





(27)

with D[A]ρ = 2AρA+ − A+Aρ − ρA+A as the Lindblad super-

operator. Dissipative contributions are taken into account

via this Lindblad formalism and can be derived via sec-

ond-order perturbation theory based on st

− for the pure

dephasing γp and

con

− for the radiative decay Γ, cf. Section

3.4 for an exemplarily derivation of the pure dephasing

Lindblad. The dynamics of the system reads:

22 22 12

2Im[ () ]t

ρ Γρ

Ωρ

=− +

 (28)

12 12 22

(/2) ()(2 1),

iit

ρ∆Γ γρ Ωρ

=− −− −

 (29)

where we use for the excitation pulse

()

() exp

tπ

ΩΩττ



−

=−





with the amplitude ΩL and

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 663

Δ = ω21–ωL. The Heitler regime is given in the low excita-

tion limit, when no Rabi oscillations will be induced and

the coherence ρ12 is driven weakly and is mainly governed

by the detuning and dephasing dynamics γ = Γ/2 + γp.

Therefore, we yield after assuming quasisteady dynamics

12 12 22

(0)()(21)( )/()ti

ρρΩρ ∆γ

∆γ

≈≈ −− +

 and the exci-

tation density reads: 2 ρ22 ≈ [1 +(Δ2 + γ2)Γ/(4γΩ2)]−1. Because

the detection is proportional to the coherence time of the

electronic signal, the phase shift of the transition dynam-

ics is given in the adiabatic limit. To describe the Wigner

time delay in the presence of decoherence, we are not

interested in the amplitude but in the phase, which is pro-

portional to φ(ω) = atan[Δ/γ] + π/2 of the field amplitude

Δ–iγ. The derivative of the phase with respect to the fre-

quency allows us to find the Wigner delay in the steady-

state limit:

ωγ∆

+ (30)

which for Δ = 0, γp = 0 yields the maximum of the Wigner

delay, namely max

2/ 2.

Tτ

However, if we compare the

Markovian theory (green and blue lines) with experimental

data (black dots) in Figure 2, a Lindblad-based pure dephas-

ing mechanism reproduces either the small or comparably

large detuning limit but misses the asymmetry between

positive and negative detunings. Importantly, it is not valid

in both limits for the same phenomenological γp value,

whereas the radiative decay Γ and the pulse Ω(t) are fixed.

In contrast to the phenomenological decoherence model,

the non- Markovian model, cf. Figure2 (red line) based on

the semiconductor Bloch equation approach reproduces the

experimental data for both limits and also exhibits the slight

asymmetry between positive and negative detunings with

respect to the laser frequency [44]. We will now discuss how

to obtain the non-Markovian system response.

In the weak coupling limit, the experimental data

are modeled by semiconductor Bloch equations in the

Heisenberg picture

d[, ]

sc la

el ep

iAHHA

−−

−=

+ from Eq. (2)

and (6) with corresponding nonequilibrium phonon

contributions [28, 61, 79, 91] and A a quantum mechanical

operator such as σ22. In this context, it is important to note

that lattice vibrations in semiconductor nanostructures

give rise to new effects not encountered in typical atomic

quantum optics [125, 126]. These features stem from the

non-, that is, sub- or super-Ohmian spectral density of

the semiconductor electron–phonon interaction [7, 57,

84, 93]. Corresponding Lindblad-based master equation

treatments are derived via Markovian-, Born-, and secular

approximation and neglect frequency-dependent system–

bath interaction strengths. Furthermore, master equation

approaches fail in this case because of the time- dependent

pulse, which enforces a time-reordering procedure [127].

Here, we model the dynamics via a Born factorization

approach, which is valid up to a temperature of 60K and

in the weak driving limit, cf. Ref. [108], relevant for the

description of the data presented in Figure 2.

When solving the semiconductor Bloch equation in

the Heisenberg picture, the one-electron assumption is

considered 〈σ11〉 = 1–〈σ22〉. The phonon dynamics is treated

non-Markovianly in the bath assumption limit, that is,

second-order Born factorization and the derived set of

equations of motion reads (including a Markovian radia-

tive decay constant Γ) with 〈A(t)〉 = Tr[ρ(0)A(t)]:

22 22 12

22Im[(

)],

σΓ

σΩ

σ〈〉=− 〈〉

+〈

〉 (31)

12 12 22

†

12 12 12 12

() ()(2

iit

igbgb

σΓ∆σ Ωσ

σσ

∗

〈〉=− +〈〉−

〈〉

−

−〈〉+ 〈〉

∑

(32)

12 22 22

†

12 12

()()(2

biib it b

bigbb

σΓ∆ω σΩ σ

∗

〈〉=− ++ 〈〉

−〈〉

−〈 〉− 〈〉〈〉

qqq

(33)

††

12 12 22

†

12 12

()()(2

)

biib it

ig bb

σΓ∆ω σΩ σ

∗∗

〈〉=− +− 〈〉−〈〉−

〈〉

−〈 〉〈 〉

(34)

†

22 22 12 12

(2 )()(

)

bibitb b

σΓωσΩσ σ

∗

〈〉=− +〈 〉− 〈〉−〈 〉

+〈〉

q (35)

Figure 2: Time delay as a function of spectral detuning between the

laser and the TLS.

The red line and black dots correspond, respectively, to the non-

Markovian simulations and the experiment as discussed in the

main text. Green and blue lines show the simulation obtained

via the Markovian approximation. Inset: Integrated intensity of

Physical Society, reprinted from [44].

664 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

12 22

bibig

tωσ

∗

〈〉=− 〈〉−〈〉

qq (36)

The phonon occupation number is given by the Bose–

Einstein distribution:

†1

=[exp( /( )) 1]

bb kTω−

〈〉

−

qq  (37)

Before discussing the Heitler regime, we show the

validity of the second-order truncation in the weak cou-

pling limit. In Figure 3, we compare the solution of the

independent boson model from Eq. (25) (green) with the

perturbative solution in second order (orange) for differ-

ent temperatures (from top to bottom, 4, 50, 77, 100K).

Assuming an initial delta pulse Ω(t) = δ(t)/2, the dynam-

ics of the set of equations [Eq. (31–36)] reduces to three

equations [Eq. (32–34)] and can also be solved analytically

[24, 93]. The resulting dynamics agrees well with the ana-

lytically exact solution up to 60K. If the cluster expansion

solution is expanded up to the fourth order, the agreement

is even better and holds up to 150K [108].

Having shown the validity of the perturbative

approach, we can discuss the role of phonons in the

Heitler regime. In the calculations, the radiative decay

time Γ = 650 ps−1 is taken from the experiment, and we

choose a phonon coupling strength so that we reproduce

the maximum of the Wigner delay on resonance at Δ = 0

for the experimentally given pulse width of τ = 1.05 ns.

Via the detuning dependence, the Wigner delay probes

the amount of phonon-induced incoherence in the signal

and is a good figure of merit to unravel and to monitor

the electron–phonon interaction [50, 107]. Clearly, the

non-Markovian theory interpolates between both limits

(large and small detuning), cf. Figure 2 (red, solid line),

and reproduces also the small temperature-dependent

asymmetry because of the preference of the system to emit

rather than to absorb phonons at low temperatures. This

can be explained with Eq. (33) and (34), as in the low tem-

perature limit mainly †

〈〉

q contributes because of spon-

taneous phonon emission rather than induced absorption

and stimulated emission. Because the phonon frequency

enters with a different sign into the dynamics of both of

the phonon-assisted coherences, the pure dephasing is

blue-detuned effectively larger, which leads subsequently

to an efficiently larger Wigner delay.

Applying this theory is therefore a key to describe

an exciting optical effect known up until now only from

atomic physics. Together with experimental data on the

Wigner time delay of a single two-level emitter repre-

sented by a high-quality QD, it provides important access

to the underlying decoherence processes and associated

timescales, which will be of importance to tailor such

quantum emitters for applications in quantum techno-

logy, which requires a high degree of coherence in the

generation, transfer, and interfacing of single photons in

solid-state quantum devices.

3.3 Phonon-assisted damping of Rabi

oscillations and state preparation

(cluster expansion)

As have been discussed in the previous section, in the weak

coupling regime for the electron–phonon coupling, quan-

tified here via 12

,gq

a factorization approach already mirrors

qualitatively and quantitatively the system’s dynamics

very accurately. A typical factorization approach for time-

dependent and time-independent pulses is the Born fac-

torization, typically within the Heisenberg picture of the

operator of interest A, for example,

††

Ab bAbb〈〉

≈〈 〉〈

〉

[37,38]. To go beyond the Born limit, a systematic correla-

tion expansion approach can be applied [60, 61, 79, 91, 128].

Here, the factorization is accompanied with a correction,

and the dynamics of this correction is taken into account

up to arbitrary high orders: †††

Ab bAbb Ab b

〈〉

=〈 〉〈 〉+

〈〉

qq qq

[79, 129]. For the weak coupling limit and nonentan-

gled system reservoir dynamics, this cluster expansion

approach gives reliable quantitative agreement for weakly

correlated many-body systems [79]. Already on a second-

order level, the quantum kinetic dephasing dynamics

of optically induced nonlinearities in GaAs QDs can be

Figure 3: Comparison for different temperatures (from top to

bottom 4, 50, 77, 100K) between the independent boson solution

from Eq.(25) (green) and the second-order cluster expansion

solution from Eq. (32) with Ω(t) = δ(t)/2 (orange).

For low temperatures and typical semiconductor parameter as given

in the text, the solutions agree well up to around 60K.

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 665

calculated accurately and in agreement with experiments

for so far experimentally accessible pulse strengths.

Evaluating the set of equations of motion [Eq. (31–

36)] beyond the Heitler regime, Rabi oscillations occur,

cf. Figure 4. Here, acoustical phonons renormalize the

Rabi energy and result in damping that depends strongly

on the input pulse strength, which is not included in a

Markovian treatment [23, 130–133]. This can be seen by

comparing the dynamics induced by a Markovian, time-

independent dephasing (green line) with the non-Mark-

ovian, phonon-induced dephasing (orange line) in the

excited state density, cf. Figure 4. The Markovian dephas-

ing dynamics overestimates the influence of the phonon

strongly and acts continuously with the same damping

strength. In contrast, acoustical phonons attack initially

the coherences strongly but saturate after few ps, which

leads to an almost constant Rabi oscillation amplitude.

This shows that Markovian and non-Markovian treat-

ment of decoherence processes lead to qualitative dif-

ferent behavior and are difficult to compare on the same

footing. Note also that the results can be obtained in a

rate equation approach using corresponding dressed

states [134].

Another example is all-optical state preparation [27].

The cluster expansion techniques allows one to investi-

gate the impact of different geometries of the QD in state-

preparation protocols. For example, the electron–phonon

coupling of a spherical QD and a more realistic lens-

shaped QD can explicitly be compared [96]. Interestingly,

the numerical analysis shows that the QD (electronic)

dynamics is hardly influenced on the actual nanostruc-

ture geometry, cf. Figure 5D, and depends mainly on the

smallest dimension, which governs the electron–phonon

interaction. For instance, comparing the state prepara-

tion dynamics for different geometries (lens-shaped QD A,

spherical B,C with different radii) of the QD’s excited state,

the qualitative dependence on the pulse area is the same. In

this regard, it is noteworthy that the strongly lens-shaped

QD A has nearly exact the same pulse area dependence

as the spherical QD C despite different spectral densities

for the electron–phonon interaction strength. This result

allows one to map the electronic dynamics of lens-shaped

QDs with spectral densities derived via a spherical geom-

etry when studying the phonon influence on the elec-

tronic system. In contrast to the electronic kinetics that

are mainly governed by the exciting laser field, the actual

nanostructure geometry has a very strong impact on the

spatiotemporal properties of the phonon dynamics, cf.

Figure 5A and C. An example is given, where for a lens-

shaped QD, the phonon emission is strongly concentrated

along the direction of the smallest axis of the QD, which

0.8

0.6

0.4

0.2

100 150

Excited state density

t (ps)

200

Non-markovian

Markovian

Pulse (×100)

Figure 4: Rabi oscillations of a QD under time-dependent driving

(Gaussian pulse, black line).

Non-Markovian electron–phonon interaction (orange line) leads to a

renormalization of the Rabi frequency and recovers partial coherence

after the first ps in contrast to Markovian damping (greenline).

–20

2.5 ps 2.5 ps

Dot A

Dot B

Dot C

Θ (π)

z (nm)

Occupation

0.5

Figure 5: (D) Final occupation of the excited QD state for a strongly

lens-shaped QD (A) and two spherical shaped QDs (B, C). Despite

different geometries QD, A and C show quantitative and qualitative

the same dependence on the pulse area. (A) and (C) The relative

volume change after an excitation with a 2π pulse for (A) a lens-

shaped and (C) a spherical shaped QD. The outgoing phonon waves

differ strongly in directionality. Reprinted figure with permission

666 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

is important for a phonon-mediated coupling of different

QD [135, 136]. Therefore, the QD shape plays an important

role in determining the properties of the created phonons

and possible application for phonon lasing and sensing

with solid-state nanostructures [137–139]. Also, it has

been shown that adding a high chirp rate to ultrashort

laser pulses, the QD can be decoupled from the phononic

environment and thus a reappearance of rapid adiabatic

passage can be established [46].

3.4 Phonon-assisted incoherent excitation

processes (Polaron master equation)

The Heisenberg equation of motion method is in general

valid in the weak coupling limit. This is not the case for

the strong coupling and high temperature limit, where

most perturbative approaches fail. However, in case of a

time-independent pump, the Hamiltonian in Eq. (2) can

be rewritten in the polaron frame, and a more conveni-

ent non-Lindblad type of master equation can be derived

[50, 127]. The main goal of the polaron transformation is

to trace out the degrees of freedom of the phonon reser-

voir in a second-order Born theory but to keep as much

information as possible about the electron–phonon inter-

action. In second-order perturbation theory, the reduced

density matrix TrB{p(t)} = ρs(t) in the interaction picture

and in second-order Born approximation reads:

()=[ , ()]Tr{[, (

)]}

sc I

sels BI

tHtHtρ

ρρ

−

−− ⊗



 (38)

Tr {[ , [(), (

)]

]}

BI Is B

dHHtττρτρ

−−

−⊗

∫

 (39)

As a first approach, one can choose a stochastic force

via

− In this case, the interaction Hamiltonian reads

HI(τ) = ħσ22(τ)F(t + τ) and the trace over the bath degrees

of freedom is a Gaussian average TrB{… ρB} → 〈〈…〉〉. As

the electronic operators are not affected by the statistical

average and c-values commute, we obtain:

22 22

[, [ (), ()]]

(/2) ()[, [ (),

()]]

II s

HH t

τρ τ

γδτσ στ

ρτ

〈〈 −−〉〉

=−

− (40)

where we used the white noise correlation for the stochas-

tic force 〈〈F(t1)F(t2)〉〉 = γρδ(t1–t2)/2 and 〈〈F(t)〉〉 = 0. In this

limit, the master equation reads as given in Eq. (27):

22 12 21

22 22

() [(), (

)]

[()()].

ps s

ti t

ρ∆σΩσσρ

γσρσ ρ

=− ++

+−

(41)

Such a Markovian limit is valid in many experimen-

tal setups and driving scenarios, as long as the dynam-

ics of the chosen observable evolves on a much smaller

timescale than the environmental correlation times. For

example, Figure 6 depicts excitation-dependent reso-

nance fluorescence emission spectra for different excita-

tion, which were obtained in a unique experiment using a

high-β QD microlaser as cw pump [102].

Acting as excitation the electrically driven high-β

microlaser drives with its output field, a semiconductor

QD. This QD acts as a two-level system within an advanced

fiber-coupled experimental setup based on two cryostats,

which host the QD microlaser and single QD sample,

respectively. The experimental data on the resonantly

driven QD are obtained at cyrogenic temperatures of 7K

to minimize electron and hole escape from the QDs and

phonon-induced decoherence γρ. However, the emerging

Mollow spectra for increasing excitation powers show

driving-induced pure dephasing, which we model quan-

titatively by our model Eq. (41). Indeed, the Markovian

Figure 6: Excitation-dependent resonance fluorescence emission

spectra and photon autocorrelation function under cw excitation by

a state-of-the-art high-β QD microlaser.

Increasing excitation power leads toward a Mollow-triplet like

emission spectrum. Reprinted figure from [102].

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 667

theory (solid lines) reproduces well the experimental data

for optimum T2 values around 500 ps. The Mollow spec-

trum, derived from the power spectrum formula S(ω),

reads:

21 12

() limRe()( ),

() ()

4()

()

PR P

CRSC

tt e

ωτ

ωτσσ τ

ΓΓ

ωπδω ω

ΩΓΓωωΓ

ΓωωΩ ΓωωΩ

∞−

→∞



=〈+〉





=−+

+−+

+−+−−−

+−++−−

∫

(42)

where the following abbreviations are used: ΓR = 2Γ,

ΓP = Γ + γρ, and Γ± = (ΓR ± ΓP)/2,

ΩΩ

Γ−

=−

and the

oscillator strength functions are introduced:

ΩΓΓ

=− + (43)

RRP

ZΓΓ

ΩΩΓΓ



=−





(44)

Although a Markovian model may apply to special

situations, more insight into the decoherence processes is

feasible to overcome temperature and single-photon rep-

etition rate limitations. As shown above, Markovian, that

is, global, frequency-, and time-independent dephasing

dynamics do not always model the dynamics correctly,

and it is necessary to go beyond the Born–Markovian and

secular limit of the master equation.

To include as much information as possible in sec-

ond-order perturbation theory, we transform the Hamil-

tonian for a time-independent Rabi frequency Ω(t) = Ω0

in Eq. (2) into the polaron frame [51, 97, 103, 104]. We use

the electron–phonon interaction Hamiltonian and apply a

unitary transform, that is, diagonalization of the phonon

part of the Hamiltonian, via Up = exp[σ22(R†–R)] and

gb ω=

∑

q The Hamiltonian in the polaron frame

can be calculated via 0

p[ ]exp[] [, ]

xxyn

∞

−=

∑

and

[x, y]0 = y, [x, y]n = [x, [x, y]n-1]:

†

22 22

UUσσ

= (45)

††

12 21 12 21

()cosh[]()

UURR

σσ

−+

†

12 21

sinh[](),RRσσ+−

− (46)

††††

12 12

22 22

()

bU bb bb

σσ

ωω



=− ++





qq qq qq (47)

†† †12

22 22 22

() ()

bU bbσσσω

+=+−

qq qq (48)

The transformed Hamiltonian is split into an interac-

tion Hamiltonian:

†

12 21

/( )( cosh

[])

RRσσ

ΩΩ

−−



†

21 12

()sinh[],RR

Ωσ σ

+−

− (49)

with †

exp[ ]exp[(0)/2

]

ΩΩ Ωφ

=〈 −〉=− where φ(t)

denotes the phonon correlation of the diagonalized

electron–phonon interaction:

() |/|cothcos()sin(

tg ti t

φωω





=−











∑q



(50)

The corresponding Hamiltonian of the free evolution

reads:

†

0221221

bb∆σ Ωσ

σω

=+ ++

∑qq

 (51)

with

∆ω

ωω

=−−∑

q Additionally the Franck–

Condon renormalization 12 12

()Ωσ

σ+ is introduced to

the free evolution and subtracted from the interac-

tion Hamiltonian to ensure in second-order perturba-

tion theory in HI(t) a vanishing first-order contribution:

Tr{[ (), () ]} 0,

Is B

Ht t

ρρ

⊗=

and the resulting non- Markovian

master equation reads:

()

(

)]

ρ∆σΩρ

=− +

{()[ , ()()]h.a.}

iiis

dG XX t

Ωττ τρ

=+ −

−−

∑∫ (52)

with

†

12 21

σσ

=+= and

†

21 12

()Xi X

σσ

−−

=−= to allow

for a compact formulation of the master equation.

The corresponding phonon Green’s functions read

G+(t) = cosh[φ(t)]–1 and G–(t) = sinh[φ(t)]. Interestingly,

this trace-preserving master equation (after setting

ρ(t–τ) = ρ(t)) simulates the complex decoherence dynam-

ics in more detail for time-independent system dynamics

and poses a feasible alternative to time-convolutionless

techniques [37, 140]. In the limit of a vanishing coupling

element, one recovers the system dynamics without

any contributions from the environment, as G±(t) ≡ 0

and φ(t) = 0, it follows ΩR = Ω. The time dynamics of

Xi(τ) is given via the electronic part of H0. The general

668 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

solution for a detuned, driven two-level system reads with

Ω∆=+

00 0

() sin( )cos() sin(

tX tX tX

Ω∆

ηη η

ηη

−−

=+− (53)

022

() [1 cos( )] [4 cos(

)]

sin( ),

Xt tX t

Ω∆ ηΩ

∆η

ηη

∆η

−

=− ++

+ (54)

() cos( )[1cos()][1cos()]

2sin( ),

∆Ω

∆

ηη η

ηη

Ωη

−



=−−+−





−

(55)

where 0

(0)

= are the initial values and Xz = σ22–σ11 is

given for completeness. Another interesting limiting case

assumes Xi(τ) ≈ Xi(0), that is, the system’s dynamics is too

slow and the environmental time correlation dominates

the dissipative part of the master equation. As

+==1

and 0

() ,

Xt X

≈ we can write

{}

()[, ( )()] h.a.

iiis

GXXtττρ

=+ −

−+

∑

2Re[ ()][ () ()

]

is is i

GtXtXτρ ρ

=+ −

≈−

∑

12 21 21 12

Re[()()][2 () 2()2()]

sss

GG ttt

ττσρ σσρσ ρ

+−

=− ++−

12 12 21 21

Re[()()] 2()2() ,

GG tt

ττσρ σσ

ρσ

−+



−−



(56)

which leads to the following master equation:

22 21

() [, ( )] ()([

]

[])()

ti Xt t

ρ∆σΩρΓ σ

σρ

=− ++

12 12 21 21

()(()())

tt t

Γσρσ σρ σ

−

(57)

with the time-dependent damping and dephasing

coefficients:

()

()=Re( 1)

eφτ

Ω

Γτ



−





∫ (58)

()

() Re

(1).

tdeφτ

Ω

Γτ

−



=−





∫ (59)

In Figure 7, we plot the solutions obtained from the

polaron master equation in Eq. (57) in the weak driving

limit Ω = 50μeV and the solution of the cluster expansion

of second order from the set of equations in Eq. (31–36)

for standard GaAs material parameter and T = 77 K. In

the secular approximation Γ−(t) = 0 and without the Rabi

energy renormalization

ΩΩ=

in H0 (left), the solutions

agree well. Deviations with nonsecular terms and the

renormalized Rabi energy become apparent even in the

weak coupling, weak driving limit. However, the solu-

tion has been obtained in the very weak and resonant

driving scenario. The advantage of the polaron master

equation is the possibility to take into account dressed

state dynamics in the case, when the time dependence of

Xi(τ) becomes nonnegligible, however only up to driving

strengths below the cut-off frequency as the polaron

master equation does not allow to simulate dynamical

decoupling scenarios.

An exemplary result is plotted in Figure 8. The experi-

mental data points are obtained from self-assembled

In(Ga,As)/GaAs QDs grown by metal–organic vapor-phase

epitaxy. The corresponding sample includes a single layer

of QDs centered in a 1-λ-thick planar GaAs cavity sur-

rounded by alternating λ/4 periods of AlAs/GaAs as 4 top

and 20 bottom distributed Bragg reflectors to enhance the

photon extraction efficiency. A narrow band (500 kHz)

tunable Ti:sapphire cw laser served as an excitation source,

and the integrated intensity of an incoherently driven

QD is compared with theory [50]. This detailed study of a

0.8

0.6

0.4

0.2

0050

t (ps)

Secular

Cluster expansion

Polaron MEQ

Non-secular

Excited state density

0.8

0.6

0.4

0.2

t (ps)

Excited state densityExcited state density

Figure 7: Comparison between the polaron master equation and cluster expansion solution in the second-order and weak driving limit

(ħΩ = 50μeV).

The solutions agree well without nonsecular terms and Rabi energy renormalization (left) for temperatures up to T = 77K.

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 669

phonon-assisted incoherent excitation mechanism of single

QDs allows one to explore the underlying acoustic phonon

bath dynamics and shows very good agreement with the

polaron master equation theory. Please note that the experi-

mental data points (black spheres) are only well produced,

if the full phonon-induced quantum kinetics is taken into

account such as in a polaron master equation model (red)

with a constant external driving field and emerging addi-

tional incoherent excitation channels. Therefore, the QD

coupling to the phonon reservoir does not only introduce

pure dephasing and an enhancement of radiative decay pro-

cesses, but also accounts for new and in Markovian models

not included incoherent excitation/scattering channels if

spectral detuning becomes important. The applied phonon-

assisted incoherent excitation provides a unique excitation

mechanism of a semiconductor QD and can be used as an

effective new tool to map the characteristic features of the

phonon bath present in such a solid-state quantum emitter

system. Additional, it serves as an interesting quasiresonant

excitation scheme for the triggered generation of single

photons with high indistinguishability [26].

3.5 Phonon anticrossings in the Mollow

regime (inductive equations of motion)

In addition to the perturbative models above, we address

now numerically exact solutions. We start with the

Heisenberg equation of motion approach, which can

be used to calculate nonequilibrium phonon dynamics

up to arbitrary order if an inductive equation of motion

method is used. For example, for longitudinal optical

(LO) phonons, the dispersion is constant in the Einstein

approximation: ω(q) ≡ ωLO [24, 106, 141]. This allows for an

exact treatment of the electron–LO phonon interaction

[28, 105, 106, 141]. Because of the constant dispersion,

we may write the Hamiltonian in Eq. (7) in the interaction

picture as

††

() [()()]

LO LO

tfbt bt Be Be

ωω

−

−=+

∑

This leads to the commutation relation [B(t), B†(t)] = | f | 2

with

|| ||,ff

∑

q which corresponds to the renormal-

ized harmonic oscillator picture. This collective operator

leads to a numerically exact solvable set of equations.

Using the Heisenberg equation of motion for a nonexplicit

time-dependent operator:

d[, ]

sc lo

el ep

iAHHA

−−

−=

+ from

Eqs. (2) and (7), a set of differential equations, defining

〈An,m〉 = 〈AB†nBm〉, can be derived. As an example, we give

the full set of equations of motion for a driven QD with LO

phonon interaction:

12 21 12

,, 1,

22 12 12 12

21,

,0 12 12

(()/2)

()(2 )(

)

(1 )| |,

nm nm

nm nm nm nm

inmi

it if

inff

σω ωΓσ

Ωσ

σσ

δσ

−

〈〉=− −− −〈〉

−〈〉−〈〉−〈〉+

〈〉

−− 〈〉

∑

(60)

22 22

,1,,

12 12 22 22

21,

,0 12 22

2,1

,0 12 22

d(( ))

2Im[ () ](

)

(1 )| |

(1 )|

nm nm

nm nm nm

in m

tif

infg

imff

σΓ ωσ

Ωσ

σσ

δσ

−

〈〉=− −− 〈〉

+〈〉− 〈〉+〈 〉

+− 〈〉

−− 〈〉

∑

(61)

,0 12 12

2,1

,0 12 12

d() (1 )| |

(1 )||.

nm nm

LO n

in minf f

imfg

ωδ σ

δσ

−

〈〉=− 〈〉+−

〈〉

−− 〈〉

∑

(62)

Figure 8: Integrated QD intensity derived from a frequency scan (black dots, experiment) and theory with (red) and without (blue) phonon-

included processes on the basis of the polaron master equation for different temperatures (A) and (B).

670 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

This set of equation is complete and allows one

to solve the dynamics of the coupled phonon-emitter

system and calculate the associated emission spectra, cf.

Figure 9. Note that with given initial conditions and for

a fixed time interval, hierarchies only up to a certain N

in the phonon interaction contribute. This maximum N

is tested until convergence is reached, depending on the

time interval and coupling strength of the system. This

inductive equation of motion method has been applied,

for example, to calculate the luminescence spectrum of a

strongly driven QD with LO phonon satellite peaks [107].

Because of the present electron–LO phonon interaction,

the typical Mollow triplet is changed and additional side

peaks appear at the frequency of the LO phonon satel-

lite peaks 36 meV. In Figure 9, an emission scheme is

depicted showing that besides the strong drive-induced

Mollow sideband, also LO phonon emission and absorp-

tion occurs and introduces Raman-like features into the

spectrum under coherent excitation. Another impor-

tant aspect is the fact that the phonon emission and

absorption is strongly temperature-dependent. Here,

the temperature is included in the initial conditions

for the expectation values: 〈B†n(0)Bm(0)〉 = δnmn![exp(βħ

ωLO)–1]−1 and 1/β = kBT. For low temperatures, because of

the LO phonon frequency in the range of for example,

ħωLO = 36 meV, spontaneous emission dominates,

leading to strong Stokes contributions. Given the set of

equations of motions, the dynamics of the coupled laser-

QD system can be calculated and because of the included

Markovian radiative decay constant Γ, a steady state is

inevitably reached. Via a two-time correlation function,

using the quantum regression theorem [35, 36, 38], the

power spectrum S(ω) is obtained. In Figure 9, the power

spectrum of a QD under cw excitation Ω(t) = const is

plotted for different Rabi energies Ω. Clearly, for low-

excitation amplitudes, the Mollow triplet can be identi-

fied and an additional LO phonon satellite peak triplet

becomes visible, where both scale linearly with the Rabi

energy. Interestingly, if the excitation strength becomes

comparable with the LO phonon energy, anticrossings

occur and both, the electronic and phononic triplet inter-

fere, forming a new Eigenstate [107]. This has the inter-

esting practical consequence that the splitting allows

one to obtain independently and spectrally resolved

the electron–phonon interaction strength without

relying on the delicate measurement of the relative peak

heights [43, 142, 143] to estimate the Huang–Rhys factor

HR LO

F f ω=

∑

3.6 Phonon-assisted population inversion

(real-time path integral)

Because of the intrinsic non-Markovian nature of the elec-

tron–phonon dynamics, perturbation approaches such as

reviewed above break down either in the high tempera-

ture or strong coupling limit. With exception of phonon

modes with constant dispersion, in which the Heisenberg

inductive equations of motion are numerically exact,

only the real-time path integral method is capable to treat

the coupling of QDs to a continuum of acoustic phonons

exactly for time-dependent excitation scenarios. This

method relies on slicing the time evolution into discrete

steps and is similar to recently widely used time-evolving

block decimation or matrix-product state methods [111,

112, 114, 115, 144]. These models take into account not only

the non-Markovian features of the system dynamics but

also the growing degree of entanglement between system

Figure 9: Scheme and power spectrum of a two-level system (e.g. QD) interacting coherently with strong external laser field.

The emission consists of triplets centered at the driving laser frequency (Mollow triplet) and at the Raman frequencies (only Stokes

Society, reprinted from [107].

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 671

and reservoir states because of excitation exchange and

information backflow.

The real-time path integral method is derived from

the Caldeira–Leggett model [110] and has been reviewed

widely within the well-known spin-boson problem [33, 115,

145]. The application to the QD–phonon kinetics has been

successfully applied to exciton [33] and biexciton systems

[47]. Interestingly, the real-time path integral method

has also successfully used to describe phonon-induced

dephasing in a coherently coupled QD microcavity systems

in the regime of cavity quantum electrodynamics (cQED)

with efficient numerical protocols [116, 117].

Starting point is the general solution of the Liouville–

von Neumann equation:

{}{}

†

() exp[ () ](0) exp[ ()

tT iHtdtTiHtdt



=−

′′ ′′





∫∫

(63)

In contrast to Section 3.1, we take light–matter inter-

action into account: ()=(

sc LA

tHtH

−−

+ Choosing a final

QD state, we are interested in

〉〈 ′ at time t = NΔt, and

summing overall possible initial system and phonon and

final phonon states in the coherent state representation,

we yield:

|()| , |(, 0)|,

NN NN

dz dz

iρπππ

′

〈〉

=〈

〉

′∫∫∫

(64)

†

00 00 00

, |(0)|, , | (, 0)| ,

zi zi ziUt zi

〈〉〈〉

′′ ′′ ′ (65)

We proceed as before when we derived the independ-

ent boson model in Eq. (25) and trace out the system

degrees of freedom first. We use a stroboscopic evolution

in time slices Δt, applying the Suzuki–Trotter decom-

position and trivially fulfilling the time-order in doing

so. Then, we insert between the time slices (N-1)-times

the system identity 1,2

=〉

〈

∑

and yield following

matrix elements:

|exp[()]||exp[ ()]|

exp[ (, )],

nnne

ep n

iiHn ii iH ni

iH ni

−−

−

−−

〈− 〉=〈− 〉

− (66)

where we assumed Δt to be small that the electron–

phonon and light–matter Hamiltonian commute to first

order, and we exploit the fact that electron–phonon inter-

action is diagonal in the system states. The light–matter

interaction reads explicitly (for resonant excitation):

|exp[()]|n

neln i

iiHn

−

−−

〈−

〉=

cos[ ]sin[]

||,

sin[ ]cos[]

fif

if f−



−

=〈

〉



−





(67)

reducing the system dynamics to c-values per time slice

and (1)

=().

nnt

ftdt

∆

∆Ω

−

∫ Now, the electron–phonon inter-

action is reduced to a Gaussian problem and the Fey-

nman–Vernon influence functional can be derived. It

is convenient to use the coherent state representation,

inserting (N-1)-times the phonon subspace identity, the

evolution reads:

00 1

, |(, 0)|, [

]

NiS

NN i

ziUt zi

−

〈〉

=∏∫D (68)

with S as the phonon action and formally as a path inte-

gral with corresponding trajectories for Gaussian degrees

of freedom. Taking the extremum, solving for the classical

equation of motion for the phonon degrees of freedom,

and tracing out the phonon reservoir, one yields the

reduced density matrix evolution:

()

|()| |(0)|

NN ii

itiMMe

iiρρ

∗−

′

−−

′

〈〉=〈〉

′′

∏∏ (69)

where the impact of the phonon reservoir is included via:

12 12 12

=()[ ()

()]

nt mt

nm nn

nt mt

tdti itti tti

∆∆

φφ

∗

−− −−−−

′′

∫∫ (70)

for k ≠ m as an example with the phonon correlation φ(t)

given in Eq. (50). Note, the solution assumes a real phonon

coupling element and a two-level system in which the

phonons couple to the excited state only. We recover the

solution of the independent boson model when Ω(t) ≡ 0

and

,ii≠′

=1 =1

2| ()|1 2| (0)|1

IBM

Nt eρ∆ ρ

−

〈〉=〈〉

∏∏ (71)

leading numerically to solution given in Section 3.1 in

Eq.(25).

In principle, the problem is solved with Eq. (69).

However, the evaluation is numerically still expensive and

can be reduced strongly by taking into account an “on fly”

screening [33, 146, 147] because of the fact that for longi-

tudinal acoustic phonon coupling (or other superohmic

environments), the memory falls off rapidly. Further-

more, boundary conditions need to be imposed to recover

known exact solutions [33, 146, 147]. This exact method

can also be used to benchmark perturbative approaches.

In Figure 10 and in Ref. [108], the cluster expansion solu-

tion of the second-order (orange line) is compared with

the numerically exact solution from the real-time path

integral method (green line). As a figure of merit, the real

part of the microscopic coherence is calculated for a QD

672 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

exposed to a weak cw driving. Due to the electron-phonon

interaction, the Eigenstate of the system change and a

real part of the coherence emerges in dependence on the

driving and the electron-phonon coupling strength. This

real part saturates fast and captures intrinsical phonon-

induced effects.

Clearly, for weak coupling 12

(),gq

that is, the standard

semiconductor GaAs parameters, i.e. a < 2, the path inte-

gral solution, and the cluster expansion are in agreement

for low temperatures, e.g. T = 4K. For a stronger coupling,

e.g.

4,α

≥ the deviation is visible and a clear breakdown of

the cluster expansion solution is recognizable, and unre-

alistical oscillations may occur (not shown).

Full demonstration of the reliability of the path inte-

gral method has been achieved in population inversion

protocols where the laser pulses are tuned within the

neutral exciton phonon sideband [25]. This unconven-

tional method achieves the inversion by rapid thermaliza-

tion of the optically dressed states for which incoherent

phonon-assisted relaxation processes are necessary. In

Figure 11, the exciton population TrB[〈2 | ρ(t) | 2〉] is plotted

(experiment b; theory c). The experimental data have been

obtained from a layer of InGaAs/GaAs QDs embedded in

the intrinsic region of an n-i-Schottky diode structure at

42 K, where the measured photocurrent is the quantity

of interest to measure the final occupation of the exciton

state. We observe a very good overall agreement between

experiment and theory, and all qualitative features are

reproduced and confirmed by microscopic input para-

meters. The population inversion arises due to the incoher-

ent phonon-induced relaxation between optically dressed

states, and the occurring phonon scattering becomes an

enabling factor in the high driving limit.

4 Nonequilibrium phonon dynamics

in quantized light–matter

interaction

Quantum optical experiments based on solid-state

quantum emitter platforms have made significant

advances over the past decade. Manifestly atomlike

024

t [ps]

α = 4

α = 2

Cluster expansion 2nd order

α = 0.5

Path integral solution

–0.01

–0.02

–0.03

Re [P]

–0.04

Figure 10: Comparison of the real time path-integral solution with

the cluster expansion time-trace for cw excitation.

Clearly, for weak coupling (α < 2), the cluster expansion solution

of second-order (orange) captures the correct steady-state of the

real part of the microscopic coherence P < σ12. However, for stronger

coupling and elevated temperatures deviations arise, and only the

path-integral solution (green) predicts the correct experimental

values.

Figure 11: Experimentally obtained (left, B) exciton population versus the pulse area and laser detuning. The path integral solution is given

in (C, right) and in very good agreement with the measured signal. The inset (C) shows the calculated values without electron–phonon

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 673

properties have been demonstrated for semiconduc-

tor QDs, and further progress is expected from coupling

those semiconductor nanostructures to microcavities as

development of in situ lithography techniques advances

rapidly [148, 149]. Related progress in the fabrication of

QD microcavities paves the way to use the Purcell effect

in the optical regime to build efficient and deterministic

sources of single photons with tunable photon statistics

[150, 151]. Exploiting such phenomena, QDs coupled to a

cavity mode become even more atomlike [6, 100, 152–155].

Many well-known features of AOM physics have already

been demonstrated on solid state–based platform such

as Rabi splitting [6] or spectral Jaynes–Cummings ladder

signatures [153, 156, 157]. In contrast to atoms, solid-state

environments are also tailorable and provide the possi-

bility to position nanostructures on demand such as QDs

permanently in a high-

microcavity [20], cf. Figure12.

Independent on natural given atomic features, the size

and geometry of the QD are controlled properties, for

example, the coupling strength and the confinement

energies [58]. Those systems are scalable and perform as

material platforms for future technological applications,

including single-photon emitters [1, 158–160]. However,

compared with atomic systems, quantum emission fea-

tures in semiconductor nanostructures are ultimately

accompanied with decoherence and scattering in the

semiconductor environment [161, 162], as also we have

reviewed in Section 3(A–F).

In this section, we selectively review phonon-induced

phenomena in cQED based on QDs as active medium and

where the quantized nature of the light field enters in the

description of the dynamics. The following examples are

ordered from the single- via the two- to the many- photon

regime. First (Section 4.1), we extend the theoretical

toolbox by introducing the time-convolutionless method

to model the Jaynes–Cummings physics in the single-

excitation limit via

cav

− in Eq. (3) and show that acoustic

phonons introduce dephasing but also a Rabi frequency

renormalization via LA

− [117, 163]. In this single-photon

regime, two aspects of the semiconductor environment

in QD-based cQED are given, for example, disadvanta-

geous impact of phonons on the goal to reach the strong

coupling regime [163]. Furthermore, and also in the sin-

gle-photon limit, phonon-mediated cavity feeding is dis-

cussed, an interesting effect that strongly deviates from

the idealized “artificial atom” model typically applied

[80, 164–166].

In the two-photon limit (Section 4.2), we discuss the

emission of photon pairs into free space, based on

con

− In

this quantum optically nonlinear regime, we discuss the

effect of colored noise in st

− on the visibility of subse-

quently emitted photons in a Hong-Ou-Mandel setup. This

colored noise can be interpreted as semiclassically treated

phonon influence as in Section 3.1when the susceptibil-

ity has been derived. We demonstrated that changing the

pulse separation in a two-pulse sequence allows one to

monitor environmental correlations and unravels the

intrinsic memory kernel [167]. Overall, the semiconductor

environment is inevitably involved in excitation processes

and renders coherent excitation processes partially inco-

herent, which are nevertheless crucial for a quantitative

understanding of experimental results. However, as men-

tioned already above, the same environment, which has

detrimental effects on the coherence, may also in certain

scenarios become advantageous and supports quantum

optical properties.

As an example of phonon-assisted coherence

increase, we discuss in Section 4.3 the collapse and

revival phenomenon known from AMO physics based

on the Jaynes–Cummings model Hamiltonian

cav

− Here,

we use the inductive equations of motion approach from

Section 3.5 to model a QD strongly coupled to microcavity

mode, which is initialized in a coherent state. This coher-

ent state leads to the collapse and revival phenomenon

[36, 168, 169]. Typically, for low photon numbers, the

collapse and revival patterns vanish fast into an irregu-

lar oscillatory behavior, but a QD coupled to an acoustic

Figure 12: Scanning electron micrograph of a pillar with a diameter

of about 0.8mm.

A combination of electron beam lithography and reactive dry etching

allows one to obtain micropillars with close to vertical and defect-

free sidewalls. Figure reprinted from [6].

674 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

phonon reservoir exhibits a stabilization via LA

− and the

collapse and revival phenomenon survives much longer in

the presence of phonon-mediated coherence support [29].

Therefore, we show before concluding this review that

phonon nonequilibrium dynamics may even be supported

and enhanced quantum coherences.

4.1 Phonon-induced decoherence

inQD-cQED

There is a wide range of examples how scattering processes

limit the performance of semiconductor nanostructures [7,

11, 12, 154]. To take into account the underlying non-Mark-

ovian physics of the phonon environment, simulations

must rely either on higher-order perturbative Markovian

approaches, time-convolutionless techniques, highly

numerically expensive nonequilibrium Green’s function

models, or exact diagonalization. Here, the influence of the

non-Markovian is calculated using a time- convolutionless

approach [37, 140]. Given the dynamics of the reduced

density matrix in second order of the electron–phonon

interaction Eq. (39), the system density matrix is approxi-

mated in a timelocal form by setting ρ(t–τ) ≈ p(t). Using the

full Hamiltonian H including the cavity–QD interaction

cav

− the dynamics in the single excitation limit yield the

following set of equations of motion, if the photon-assisted

ground state is abbreviated with

†

11 11

ccσσ

22 22 12

2Im[ ],

gσΓ

σσ

〈〉=− 〈〉

+〈

〉 (72)

12 12

11 22

[/2/2()]

[()] [()]

iAt

Bt ig Ct ig

σΓκ∆ σ

σσ

〈〉=− ++−〈〉

−+〈〉++

〈〉

(73)

11 11 12

=2Im

[]

dt gσκ

σσ

〈〉−〈 〉−

〈〉

(74)

where a Markovian radiative decay and a cavity loss is

assumed

[]cκρ

D and a detuning between cavity mode

and the QD is assumed with Δ = ω21–ωc. The influence of

the acoustic phonons is included in the time-dependent

coefficients A, B, C, obtained after inserting a unity in the

time-local master equation:

22 22

() [()1|()|1()2|()|

2],

dτφ τστφτστ

∗

=〈−〉−〈

−〉

∫ (75)

() ()1| ()|2

tdτφ τστ

∗

=〈

−〉

∫ (76)

() ()1| ()|2

dτφ τστ=〈

−〉

∫ (77)

with 〈i | σ22(−τ) | j〉 = 〈i | U(τ, 0) | 2〉〈2 | U(−τ, 0) | j〉 the time-

evolution matrix element of the unperturbed system with

time evolution operator: U(t, 0) = exp[−igt(σ12 + σ21)–iΔt],

and φI(t) is given in Eq. (26). A(t) includes the polaron shift

and the dephasing dynamics via the imaginary and the

real part, respectively. B(t) and C(t) renormalize the inter-

action strength between the cavity mode and the QD via

the independent boson phonon correlation function. This

is an important feature of non-Markovian open quantum

system dynamics, as a system–reservoir coupling intro-

duces an increase of entanglement and thermalization at

the same time, leading to a loss of information and basi-

cally new “dressed” system states. In the Markovian limit

φ(t) → γpδ(t), we recover a Lindblad type of interaction, in

which B(t) and C(t) are vanishing because of 〈i | j〉 = δij and

A(t) reduces to −γM in which all phonon characteristics are

gone and a phenomenological pure dephasing constant

remains.

The formulation with finite time kernel, however,

allows one to investigate the phonon impact on the strong

coupling regime of the QD-cQED without overestimat-

ing the dephasing rate, which is the case for a Markovian

decay. A figure of merit for the strong coupling regime is

a nonmonotonic decrease of the initial QD occupation

〈σ22(0)〉 = 1with 〈σ22(t1)〉 ≥ 〈σ22(t2)〉 for t2 > t1. Figure 13 shows

the parameter space where strong coupling and weak cou-

pling resides depending on the bare coupling constant g

and the temperature. The temperature enters in the phonon

correlation function via φ(t) ≡ φ(t, T). From evaluating

the dynamics in the single-excitation regime, it becomes

apparent that the higher the temperature, the stronger

must be the cavity–QD coupling to yield nonmonotonous

decay dynamics of the QD excitation [34]. Remarkably,

the coupling–renormalization has a strong effect (without

renormalization dashed lines, with renormalization solid

Figure 13: Parameter space where strong coupling and weak

coupling resides for QD-based cQED depending on the bare

coupling constant and the temperature.

Non-Markovian phonon interaction leads to a renormalization and

affects strongly the transition from weak to strong coupling (solid

lines with renormalization, dashed lines without). Reprinted figure

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 675

lines). This energy renormalization is also important in

explaining asymmetries in life times with respect to the

QD–cavity detuning in cavity feeding scenarious.

A pronounced increase in a nonresonant QD–cavity

coupling toward elevated temperatures is a strong indi-

cation of phonon-mediated relaxation processes [51, 54,

164]. The spectral mismatch between the cavity mode and

the emitter is then bridged by either phonon emission

if the emitter is blue detuned from the cavity mode, or

phonon absorption if the emitter field is red detuned. This

asymmetry is included in the phonon correlation func-

tion φ(t) and not included in the Markovian limit, where

φ(t) → γMδ(t). As for low temperatures, nq → 0 processes

accompanied by phonon emission (nq + 1) are favored over

those obtained by phonon absorption (nq), and spectral

asymmetries become visible and pronounce in this tem-

perature regime. In Figure 14, the non-Markovian relaxa-

tion dynamics of a detuned QD–cavity QED platform is

investigated [80] and given in the single excitation as:

)

11 22

gde∆τ φτ

στ

∞−



〈〉

≈〈

〉





∫ (78)

if instead of the phonon interaction, the cavity–QD inter-

action is taken as a perturbation up to second order in

the reduced but timelocal density matrix, cf. Eq. (39).

This transition strength from excited to ground state can

be understood as a generalization of Fermi’s golden rule

[37, 51]. A model without phonons (dashed line) cannot

reproduce the effective lifetimes of the coupled QD–cavity

system as a function of the detuning Δ (experimental

data, dots). The non-Markovian simulations (solid) were

computed with a measured quality factor Q = 2900 and a

cavity–QD coupling strength ħg = 45μeV. Also, the asym-

metric dip of the lifetime with respect to the detuning is

not reproduced without phonon dynamics, showing that

phonon-mediated feeding cannot be captured with a

broadened zero-phonon linewidth. For higher excitation

manifolds, it can furthermore be shown that the cavity

feeding effect depends nontrivially on the detunings, for

example, maximal efficiency is obtained for detunings

corresponding to transition energies between cavity-

dressed states with excitation numbers larger than one

[116]. Quantum correlations, such as entanglement and

indistinguishability, however, still decohere because of

phonon-induced noise, as will be discussed next.

4.2 Hong-Ou-Mandel effect: photon

indistinguishability

A good figure of merit to characterize the noise robustness

of a quantum emitter is the two-photon coherence such as

in Hong-Ou-Mandel (HOM) type of experiments [170]. If the

photon–photon correlation function of two photons sub-

sequently emitted from a single emitter [5] is strongly anti-

bunched after interfering in a Hanbury Brown and Twiss

setup, the quantum emitter is time-translational robust

and shows therefore no noise. Such quantum optical exper-

iments probe therefore the indistinguishability of emitted

photons and are the basis for entanglement distribution

via Bell-state measurements in long-distance quantum

communication networks [81, 82]. Because the visibility

on HOM interference experiments is influenced by the

dephasing of the quantum emitters, they are well suited to

explore decoherence on the nanoseconds timescale under

variation of, for instance, the temperature with high sensi-

tivity. In the theoretical description of HOM experiments,

we extend the single-mode cavity Hamiltonian in Eq. (3) to

a multimode description

con

− in Eq. (4). Furthermore, we

model the phonon contribution with a stochastic force F(t)

via st

− in Eq. (5) to solve the problem analytically. To probe

decoherence on a nanoseconds timescale, the QD is excited

with a two-pulse sequence with variable pulse separation

Figure 14: Extracted decay times of a QD–cavity system as a function

of detuning.

Experimental data (dots) are in good agreement with non-Markovian

phonon theory (solid line) in comparison without phonons (dashed

lines). Figure (A) shows that neglecting the phonon contributions

leads only in the small and large detuning limit the experimentally

correct excitonic lifetimes, whereas including phonons allows to

correctly mediate between these limits. In (B), there are two sets of

data. Squares refer to the fast and circles to the slow components of

the luminescence decay. Reprinted figure with permission from [80].

676 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

δt. The first pulse creates an excitation via resonant p-shell

excitation, which radiatively decays under emission of a

photon from the s-shell of the QD. After a time delay δt in

which the QD may experience a reconfigured charge envi-

ronment slightly shifting its energy levels, a second pulse

creates another excited state within the QD and another

photon is subsequently emitted. The corresponding two-

photon wave function reads:

() ()

()()

() †

|(, 0) (, )|vac

(, )

titit t

EEt

tigd dt

ωω ΦΓ

Ψδ

τω

−−

∞−− −

′′′

−

〉= ∞∞ 〉

=− ′

∫∫ (79)

We distinguish channels with the labels: ωL for long

ωS for short, that is, the photons are distinguishable via

their spatial traveling mode in the detection path until

they superpose at the HOM beam splitter. Note the differ-

ence in the lower limit of the integrals (0, δt) and in the

integrated noise signals ()

(),

tdtFt

ττ

′′

∫ and the radia-

tive decay constant is given with Γ = g2π. If the first photon

now takes a longer way to the 50:50 HOM beam splitter

with single-photon detectors A and B at its two output

ports and the second photon travels the shorter route,

both photons can interfere

() ()

()

ASL

EEE

±±±

() ()

()

BLS

EEE

±±±

=−

(80)

In this case, the detected photon–photon correlation

gives a direct measure for the degree of indistinguish-

ability in terms of the HOM visibility as discussed in the

following.

The unnormalized HOM two-photon correlation func-

tion reads:

(2)()(

(, ) |( )()| ()|

DD BD AD

tEtE

ττΨ

+=〈〈

+〉〉〉

2(2

)(

,, )

21Re[

(2 /)

Γτ ξτδ

−+ −

=〈

〈− 〉〉 (81)

with the noise contribution ξ(tD, τ, δt) = Ф0(tD + τ) + Фδt(tD)–

Фδt(tD + τ) – Ф0(tD). The two-photon correlation function

vanishes in case of indistinguishable photon emission

events, that is, for vanishing noise ξ ≡ 0 or infinite cor-

related noise: Фδt(t) = Ф0(t). This is the expected result,

which implies that if the emitter is not subjected to a

varying environment or negligible environment influence

at all, the emitted photons are indistinguishable as the

emission event is time-translational invariant. However,

in the typical experimental setting with finite noise

contribution, for example, because of charge noise by

access electron and holes under nonresonant excitation,

fluctuating surface charges, and/or electron–phonon

scattering, decoherence occurs and reduces the HOM vis-

ibility. The figure of merit and experimentally accessible

quantity is again the HOM visibility:

(2)

(, )

()1d

(/2)

tτ

δτ

∞∞

=−

∫∫ (82)

Note, the pulse delay and noise contribution are still

present in the two-photon correlation. If a delta-correlated

white noise is assumed, that is, in the situation when the

noise correlation of different emission events vanishes

identically, we obtain:

Markovian

(, )|1

Vt γΓ

δτ Γγ

γΓ

=− =

(83)

This is an interesting result that deserves further

explanation. Firstly we note that here the photon indis-

tinguishability is independent from the radiative decay

constant (as a Markovian radiative decay process has

been assumed) but depends strongly on the pure dephas-

ing processes within the emission processes. However, for

a Markovian decoherence process, the reduced visibility

does not depend on the delay between the two pulses in

a two-pulse sequence. Thus, a non-Markovian treatment

needs to established to fully capture the complexity of the

two-pulse experiment and its underlying physics. To illus-

trate and substantiate this statement, we consider experi-

mental results on the HOM visibility of a semiconductor

QD presented in Figure 15A and B, where the QD is excited

in a two-photon sequence with different pulse delays δt.

The HOM visibility clearly decreases with increasing pulse

Figure 15: Two-photon interference visibilities of consecutively

emitted single photons in dependence on the pulse separation δt.

Experimental data for (A) the neutral exciton state and (B) the

charged exciton state are quantitatively described by a theoretical

model assuming a non-Markovian noise correlation, leading to

spectral diffusion on a nanosecond timescale. Reprinted figure with

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 677

separation, and a Markovian model cannot be applied to

model this result. In contrast, a colored noise model with

a phenomenologically assumed correlation length of the

emission events allows us to characterize the emitter for

given visibility data. For this purpose, we include this

dependence as a finite memory effect with specific corre-

lation time of τc:

()

24 24

() () (min[, ] max[ , ])

tt tt tt

ΦΦ Γ

−

〈〈

〉〉

=−

′ (84)

This kind of noise correlation stems from a non-Mark-

ovian low-frequency noise [171–173] and shows plateau-

like behavior for temporal pulse distances sufficiently

short in comparison with the experimentally extracted

memory depth. With the correlation length parameter, we

derive for the visibility:

(, )

(1 exp[ (/)])

δτ γδ

τΓ

=−− + (85)

Therefore, for vanishing pure dephasing the visibil-

ity is 1, that is, consecutively emitted photons are Fourier

transform limited and coalesce at the beamsplitter into

a perfect coherent two-photon state. Further dephasing

could also be suppressed by strict resonant excitation

[159], electrical charge control [22], or suitable surface pas-

sivation [174]. A detailed understanding of effects limiting

the HOM visibility in two-pulse experiments is of vital

interest for the further development of state-of-the-art

QD-based quantum light sources toward the implemen-

tation of quantum circuits, for example, boson sampling

and advanced quantum communication protocols such

as measurement device independent quantum key distri-

bution and the quantum repeater. This challenging goal

can only be achieved by engineering and operating the

sources to effectively suppress phonon-induced dephas-

ing and spectral diffusion at timescales well beyond the

nanosecond range in the future.

4.3 Phonon-enhanced coherence

Lattice vibrations in semiconductor QDs give rise to new

effects not known in atomic quantum optics such as

phonon-mediated off-resonant cavity feeding (Section

4.1), formation of phonon-assisted Mollow triplets [107]

and temperature-dependent vacuum Rabi splittings

in cavity emission spectra [117]. Expanding on these

developments, another example is phonon-induced

quantum optical coherence [29, 49]. Here, the memory

effects of the phonon bath are investigated with respect

to its impact on quantum optical pattern formations in

the coherent collapse and revival phenomenon. This

expands the non-Markovian investigation beyond the

single excitation limit, as higher-order photon manifolds

are involved.

The basis of the investigation is the inductive Heisen-

berg equation of motion approach [28, 105], presented in

Section 3.5. The cavity–QD dynamics

()

cav

− is calculated

within a numerically exact approach, while the coupling

to the longitudinal acoustic phonon reservoir

()

− is

treated for every photon manifold at second-order Born

level. The corresponding equations of motion read for res-

onant interaction between QD and cavity photons Δ = −Δp

to compensate for the polaron shift:

††

†1

22Im

[]

mm mm mm

cc mccmgcc

κσ

−

〈〉=− 〈〉−〈 〉 (86)

†† †1

22 22 12

22Im

[]

mm mm

cc mccmgccσκ

σσ

〈〉=− 〈〉+〈 〉 (87)

†1 †1 †

12 12 22

[(21)]

mm mm

cc micc igmccσκ∆σ σ

〈〉=− +−〈〉−〈

〉

†1 1†

)

mm mm

ig cc ccσ++

−〈

〉−

〈〉

†1 †

12 12

().

ig ccbbσ+

−〈 +〉

∑

(88)

In the present objective to study the impact of elec-

tron-phonon interaction, the most important contribution

is proportional to the phonon- and photon-assisted tran-

sition dynamics †1 †

().

ccbbσ+

〈+

〉

This transition facili-

tates photon number-dependent dephasing and blocks

the mixing between different photon number manifolds

to stabilize the collapse and revival dynamics. The equa-

tion of motion read, for example:

†1

††1†

12 12

[(21)]

mm mm

ccbmii ccbσκω∆σ

〈〉=− +−−〈

〉

††

12 12

ig bb

σ+

−〈

〉〈 〉

qq (89)

where the Born factorization has been applied to second

order, neglecting contributions proportional to 12

.gg∝q

This set of equations of motion corresponds to Section

3.2 for higher-order photon correlations. The set is,

however, not closed in terms of the photon number (m).

Therefore, depending on the coupling strength g, a suf-

ficient high order in m is taken into account to reach

convergence in the calculations. Let us focus on a sit-

uation that starts with an inverted QD and a coherent

678 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

photon state with a mean photon number N = 3.5 [168].

The obtained solution is numerically exact and renders,

in the absence of electron–phonon coupling, the well-

known Jaynes–Cummings model solutions [28]. Because

of the small mean photon number, the collapse and

revival phenomenon without acoustical phonons known

from the Jaynes–Cummings model [169] is visible only

for short times t < 10 ps, cf. Figure 16 (orange line in both

upper and lower panel). For longer times t > 15 ps, the

pattern of oscillations becomes irregular and the effect

of rephasing vanishes. Excitingly, with phonons for tem-

peratures up to 50K, the phenomenon is stabilized for

much longer times than without phonons (green lines,

upper panels for a temperature of 4K, lower panel for

50K). The underlying physics is that decoherence mani-

festly keeps the different photon-assisted Rabi frequen-

cies within the cavity from mixing and allows much

longer rephasing dynamics in phase space. This effect

is purely due to the frequency-dependent coupling

from the QD to the phonon bath and not included in

any Markovian simulations, showing hereby the excit-

ing possibilities provided by non-Markovian, quantum

kinetics–induced decoherence. Interestingly, it is even

stable against cavity loss but strongly depends on the

strong coupling limit between cavity and QD, that is,

the cooperativity must be high enough to allow for Rabi

oscillations in the microcavity after initialization of the

coherent photon field [29, 175]. The experimental chal-

lenge lies, however, in the preparation of the initial state,

as a perfect coherent superposition initially is essential,

however, subjected to losses and transient effect during

the preparation stage [168]. These problems have ren-

dered the collapse and revival phenomenon difficult to

observe on any platforms. However, if these difficulties

can be overcome, QDs with acoustical phonon coupling

are ideal candidates to observe long and lasting rephas-

ing, collapse, and revival dynamics, as standard GaAs

parameters have been used.

5 Conclusion

In this review, we discussed non-Markovian features in

QD-based optical experiments. To understand limits and

features of semiconductor nanostructures, it is neces-

sary to unravel the microscopic system–environment

dynamics. Electron–phonon interaction is a prominent

coupling mechanism, and several examples have been

given in which the phonon dynamics cannot be reduced

to a Lindblad type of interaction. Phenomenological mod-

eling overestimates typically the dephasing because of the

finite memory kernel of phonons, and interesting intrinsi-

cal features such as polaron-dressed states and phonon-

mediated coherence become inaccessible. Several

theoretical models, however, have been presented, which

allow to a high-degree of precision to understand quan-

titatively and qualitatively the physics in semiconductor

quantum optics, which in the opinion of the authors have

just been started because of advances in experiments and

theory to flourish and will continue in doing so. Of par-

ticular interest is the investigation of the electron–pho-

non dynamics beyond the harmonic approximation. As

nonequilibrium phonon dynamics is already interesting

and partially coherence preserving in the harmonic limit,

new effects are to be expected beyond lifetime broad-

ening mechanisms if the electron–phonon interaction

is treated beyond second order in its lattice vibrations.

Also, detailed investigation on phononic reservoir engi-

neering is on the verge to be realizable in experiments.

0.5

015

4 K

Without phonons

With phonons

Excited state density

t (ps)

0.5

015

50 K

Excited state density

t (ps)

Figure 16: The excited state density 〈σ22(t)〉 for a cavity initialized

with mean coherent photon number of N = 3.5.

The Jaynes–Cummings model solution (orange line) without

phonons exhibits only a single collapse with an incomplete revival.

Including acoustical phonons (green line) with a lattice temperature

of 4K and 50K many cycles of collapse and revival are visible.

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 679

Forexample, the impact of acoustic cavities on electronic

coherences, phonon lasing, and even on surface-reflected

acoustic quantum feedback enables the investigation of

exotic light–matter interaction. Because of the high degree

of nanotechnological control, collective effects become

reachable and superphononance as an acoustic equiva-

lent to superradiance will, maybe, be observed soon. Of

fundamental interest is the design of new coupling mech-

anism and the impact on nonequilibrium steady states

and the question arises whether electron–phonon inter-

action guarantees ergodicity on a microscopical level or

not. Here, semiconductor quantum optics may shed light

because of nonequilibrium electron–phonon dynamics

on the very principles of quantum thermodynamics, heat

transport, and possible quantum memory platforms. The

fast progress of nanodesign has just begun to address

long-standing questions in many-body physics, and, in

the belief of the authors, QDs will play their role in this

pursuit of understanding fundamental limits of light–

matter interaction because of their universal applicability.

Acknowledgments: The research leading to the presented

results has received funding from the European Research

Council under the European Unions Seventh Framework

ERC Grant Agreement No. 615613. We also express grati-

tude to the DFG (German Science Foundation) for contin-

uous funding of our research via the projects RE2974/5-1,

RE2974/9-1, RE2974/12-1, and CRC 787. AC gratefully

acknowledges support from the Deutsche Forschungsge-

meinschaft (DFG) through the project B1 of the SFB 910

and from the European Unions Horizon 2020 research and

innovation program under the SONAR grant Agreement

No. 734690 and Andreas Knorr for fruitful discussions.

References

[1] Michler P, Kiraz A, Becher C, etal. A quantum dot single-photon

turnstile device. Science 2000;290:2282–5.

[2] Yuan Z, Kardynal BE, Stevenson RM, etal. Electrically driven

single-photon source. Science 2002;295:102–5.

[3] Akopian N, Lindner N, Poem E, etal. Entangled photon

pairs from semiconductor quantum dots. Phys Rev Lett

2006;96:130501.

[4] Hafenbrak R, Ulrich S, Michler P, Wang L, Rastelli A, Schmidt

O. Triggered polarization-entangled photon pairs from a single

quantum dot up to 30 k. New J Phys 2007;9:315.

[5] Santori C, Fattal D, Vučković J, Solomon GS, Yamamoto Y.

Indistinguishable photons from a single-photon device. Nature

2002;419:594.

[6] Reithmaier JP, Sek G, Löffler A, etal. Strong coupling in a single

quantum dot–semiconductor microcavity system. Nature

2004;432:197.

[7] Woggon U. Optical properties of semiconductor quantum dots.

Berlin, Heidelberg, Springer, 1997.

[8] Bimberg D, Grundmann M, Ledentsov NN. Quantum dot hetero-

structures. Chichester, John Wiley & Sons, 1999.

[9] Kiraz A, Atatüre M, Imamoğlu A. Quantum-dot

single-photonsources: prospects for applications in

linear optics quantum-information processing. Phys Rev A

2004;69:032305.

[10] Michler P. Single semiconductor quantum dots, vol. 231. Hei-

delberg, Berlin, Springer, 2009.

[11] Michler P. Quantum dots for quantum information technolo-

gies, vol. 237. Berlin, Heidelberg, Springer, 2017.

[12] Jahnke F. Quantum optics with semiconductor nanostructures.

Oxford, Woodhead Publishing, 2012.

[13] Rodgers P. Nanoscience and technology. London, UK,

Macmillan Publishers Ltd., 2009.

[14] Yu-Ming H, Yu H, Yu-Jia W, etal. On-demand semiconductor

single-photon source with near-unity indistinguishability. Nat

Nanotechnol 2013;8:213.

[15] Peter E, Hours J, Senellart P, etal. Phonon sidebands in exciton

and biexciton emission from single GaAs quantum dots. Phys

Rev B 2004;69:041307.

[16] Mendach S, Songmuang R, Kiravittaya S, Rastelli A, Benyoucef

M, Schmidt OG. Light emission and wave guiding of quantum

dots in a tube. Appl Phys Lett 2006;88:111120.

[17] Schmidt OG. Lateral alignment of epitaxial quantum dots.

Heidelberg, Berlin, Springer Science & Business Media, 2007.

[18] Zhang J, Wildmann JS, Ding F, etal. High yield and

ultrafastsources of electrically triggered entangledphoton

pairs based on strain-tunable quantum dots. Nat Commun

2015;6:10067.

[19] Schlehahn A, Fischbach S, Schmidt R, etal. A stand-alone

fiber-coupled single-photon source. Sci Rep 2018;8:1340.

[20] Yoshie T, Scherer A, Hendrickson J, etal. Vacuum rabi splitting

with a single quantum dot in a photonic crystal nanocavity.

Nature 2004;432:200.

[21] Bose R, Cai T, Choudhury KR, Solomon GS, Waks E. All-optical

coherent control of vacuum rabi oscillations. Nat Photonics

2014;8:858–64.

[22] Senellart P, Solomon G, White A. High-performance semicon-

ductor quantum-dot single-photon sources. Nat Nanotechnol

2017;12:1026.

[23] Förstner J, Weber C, Danckwerts J, Knorr A. Phonon-induced

damping of rabi oscillations in semiconductor quantum dots.

Phys Status Solidi B 2003;238:419–22.

[24] Krummheuer B, Axt VM, Kuhn T, D’Amico I, Rossi F. Pure

dephasing and phonon dynamics in GaAs- and GaN-

based quantum dot structures: interplay between material

parameters and geometry. Phys Rev B 2005;71:235329.

[25] Quilter JH, Brash AJ, Liu F, etal. Phonon-assisted population

inversion of a single InGaAs/GaAs quantum dot by pulsed laser

excitation. Phys Rev Lett. 2015;114:137401.

[26] Müller M, Bounouar S, Jöns KD, Glässl M, Michler P.

On-demand generation of indistinguishable polarization-

entangled photon pairs. Nat Photonics 2014;8:224.

[27] Reiter D, Kuhn T, Glässl M, Axt V. The role of phonons for exci-

ton and biexciton generation in an optically driven quantum

dot. J Phys Condens Matter 2014;26:423203.

[28] Carmele A, Richter M, Chow WW, Knorr A. Antibunching of

thermal radiation by a room-temperature phonon bath: a

680 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

numerically solvable model for a strongly interacting light-

matter-reservoir system. Phys Rev Lett 2010;104:156801.

[29] Carmele A, Knorr A, Milde F. Stabilization of photon collapse

and revival dynamics by a non-markovian phonon bath. New J

Phys 2013;15:105024.

[30] Besombes L, Kheng K, Marsal L, Mariette H. Acoustic phonon

broadening mechanism in single quantum dot emission. Phys

Rev B 2001;63:155307.

[31] Borri P, Langbein W, Schneider S, etal. Ultralong dephasing

time in ingaas quantum dots. Phys Rev Lett 2001;87:157401.

[32] Galland C, Högele A, Türeci HE, Imamoğlu A. Non-markovian

decoherence of localized nanotube excitons by acoustic phon-

ons. Phys Rev Lett 2008;101:067402.

[33] Vagov A, Croitoru M, Axt VM, Kuhn T, Peeters F. Nonmonotonic

field dependence of damping and reappearance of rabi oscilla-

tions in quantum dots. Phys Rev Lett 2007;98:227403.

[34] Madsen KH, Ates S, Lund-Hansen T, etal. Observation of non-

markovian dynamics of a single quantum dot in a micropillar

cavity. Phys Rev Lett 2011;106:233601.

[35] Carmichael HJ. Statistical methods in quantum optics 1. Berlin,

Heidelberg, Springer Science & Business Media, 2009.

[36] Scully MO, Zubairy MS. Quantum optics. Am J Phys 1999;67:648.

[37] Breuer H-P, Petruccione F. The theory of open quantum sys-

tems. Oxford, Oxford University Press, 2002.

[38] Gardiner C, Zoller P, Zoller P. Quantum noise: a handbook of

markovian and non- markovian quantum stochastic meth-

ods with applications to quantum optics, vol. 56. Berlin,

Heidelberg, Springer Science & Business Media, 2004.

[39] Breuer H-P, Laine E-M, Piilo J, Vacchini B. Colloquium: non-

markovian dynamics in open quantum systems. Rev Mod Phys

2016;88:021002.

[40] Cerrillo J, Cao J. Non-markovian dynamical maps: numeri-

cal processing of open quantum trajectories. Phys Rev Lett

2014;112:110401.

[41] de Vega I, Alonso D. Dynamics of non-markovian open quantum

systems. Rev Mod Phys 2017;89:015001.

[42] Michler P. Single quantum dots: fundamentals, applications

and new concepts, vol. 90. Berlin, Heidelberg, Springer

Science & Business Media, 2003.

[43] Stock E, Dachner M-R, Warming T, etal. Acoustic and optical

phonon scattering in a single In(Ga)As quantum Dot. Phys Rev

B 2011;83:041304.

[44] Strauß M, Carmele A, Schneider J, etal. Wigner time delay

induced by a single quantum dot. Phys Rev Lett 2019;122:107401.

[45] Ramsay A, Godden T, Boyle S, etal. Phonon-induced rabi-fre-

quency renormalization of optically driven single InGaAs/GaAs

quantum dots. Phys Rev Lett 2010;105:177402.

[46] Kaldewey T, Lüker S, Kuhlmann AV, etal. Demonstrating the

decoupling regime of the electron-phonon interaction in a

quantum dot using chirped optical excitation. Phys Rev B

2017;95:241306.

[47] Bounouar S, Müller M, Barth A, Glässl M, Axt V, MichlerP.

Phonon-assisted robust and deterministic two-photon

biexciton preparation in a quantum dot. Phys Rev B

2015;91:161302.

[48] Ardelt P-L, Hanschke L, Fischer KA, etal. Dissipative prepara-

tion of the exciton and biexciton in self-assembled quantum

dots on picosecond time scales. Phys Rev B 2014;90:241404.

[49] Iles-Smith J, Nazir A. Quantum correlations of light and matter

through environmental transitions. Optica 2016;3:207–11.

[50] Weiler S, Ulhaq A, Ulrich SM, etal. Phonon-assisted incoherent

excitation of a quantum dot and its emission properties. Phys

Rev B 2012;86:241304.

[51] Hohenester U. Cavity quantum electrodynamics with semicon-

ductor quantum dots: role of phonon-assisted cavity feeding.

Phys Rev B 2010;81:155303.

[52] Monniello L, Tonin C, Hostein R, etal. Excitation-induced

dephasing in a resonantly driven InAs/GaAs quantum dot. Phys

Rev Lett 2013;111:026403.

[53] Ulrich S, Ates S, Reitzenstein S, Löffler A, Forchel A, Michler P.

Dephasing of triplet-sideband optical emission of a resonantly

driven InAs/GaAs quantum dot inside a microcavity. Phys Rev

Lett 2011;106:247402.

[54] Roy C, Hughes S. Phonon-dressed mollow triplet in the regime

of cavity quantum electrodynamics: excitation-induced dephas-

ing and nonperturbative cavity feeding effects. Phys Rev Lett

2011;106:247403.

[55] Gardiner C, Zoller P. Quantum noise. Berlin Heidelberg New

York, Springer, 1991.

[56] Bányai L, Koch SW. Semiconductor quantum dots, vol. 2. Singa-

pore, River Edge, NJ, World Scientific, 1993.

[57] Grundmann M, Stier O, Bimberg D. InAs/GaAs pyramidal quan-

tum dots: strain distribution, optical phonons, and electronic

structure. Phys Rev B 1995;52:11969–81.

[58] Stier O, Grundmann M, Bimberg D. Electronic and optical

properties of strained quantum dots modeled by 8-band k · p

theory. Phys Rev B 1999;59:5688–701.

[59] Schliwa A, Winkelnkemper M, Bimberg D. Impact of size,

shape, and composition on piezoelectric effects and elec-

tronic properties of In(Ga)As\GaAs quantum dots. Phys Rev B

2007;76:205324.

[60] Chow WW, Jahnke F. On the physics of semiconductor quantum

dots for applications in lasers and quantum optics. Prog. Quant

Electron 2013;37:109–84.

[61] Haug H Koch SW. Quantum theory of the optical and elec-

tronic properties of semiconductors, 5th ed. Singapore, World

Scientific Publishing Company, 2009.

[62] Brandes T. Coherent and collective quantum optical effects in

mesoscopic systems. Phys Rep 2005;408:315.

[63] Sakaki H. Quantum wires, quantum boxes and related struc-

tures: physics, device potentials and structural requirements.

Surf Sci 1992;267:623–9.

[64] Schwab K, Henriksen E, Worlock J, Roukes ML. Measure-

ment of the quantum of thermal conductance. Nature

2000;404:974.

[65] Gornyi I, Mirlin A, Polyakov D. Interacting electrons in disor-

dered wires: Anderson localization and low-T transport. Phys

Rev Lett 2005;95:206603.

[66] Sun C-K, Liang J-C, Yu X-Y. Coherent acoustic phonon oscilla-

tions in semiconductor multiple quantum wells with piezoelec-

tric fields. Phys Rev Lett 2000;84:179.

[67] Lugli P, Goodnick S. Nonequilibrium longitudinal-optical

phonon effects in GaAs-AlGaAs quantum wells. Phys Rev Lett

1987;59:716.

[68] Kim D-S, Shah J, Cunningham J, etal. Giant excitonic resonance

in time-resolved four-wave mixing in quantum wells. Phys Rev

Lett 1992;68:1006.

[69] Gammon D, Rudin S, Reinecke T, Katzer D, Kyono C. Phonon

broadening of excitons in GaAs/Al x Ga 1- x As quantum wells.

Phys Rev B 1995;51:16785.

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 681

[70] Richter M, Singh R, Siemens M, Cundiff ST. Deconvolu-

tion of optical multidimensional coherent spectra. Sci. Adv.

2018;4:eaar7697.

[71] Singh R, Richter M, Moody G, Siemens ME, Li H, Cundiff ST.

Localization dynamics of excitons in disordered semiconductor

quantum wells. Phys Rev B 2017;95:235307.

[72] Lim J, Ing DJ, Rosskopf J, etal. Signatures of spatially correlated

noise and non-secular effects in two-dimensional electronic

spectroscopy. J Chem Phys 2017;146:024109.

[73] Wigger D, Schneider C, Gerhardt S, etal. Rabi oscillations of a

quantum dot exciton coupled to acoustic phonons: coherence

and population readout. Optica 2018;5:1442–50.

[74] Cassette E, Pensack RD, Mahler B, Scholes GD. Room- temperature

exciton coherence and dephasing in two-dimensional nano-

structures. Nat Commun 2015;6:6086.

[75] Liu A, Almeida DB, Bae WK, Padilha LA, Cundiff ST. Vibrational

coupling modifies spectral diffusion in core-shellcolloidal

quantum dots. arXiv preprint arXiv:1806.06112, 2018.

[76] Suzuki T, Singh R, Moody G, etal. Dephasing of InAs quantum

dot p-shell excitons studied using two-dimensional coherent

spectroscopy. Phys Rev B 2018;98:195304.

[77] Christiansen D, Selig M, Berghäuser G, etal. Phonon

sidebands in monolayer transition metal dichalcogenides.

Phys Rev Lett 2017;119:187402.

[78] Malic E, Knorr A. Graphene and carbon nanotubes:

ultrafastoptics and relaxation dynamics. Weinheim, John Wiley

& Sons, 2013.

[79] Kira M, Koch SW. Semiconductor quantum optics. Cambridge,

Cambridge University Press, 2011.

[80] Hohenester U, Laucht A, Kaniber M, etal. Phonon-assisted

transitions from quantum dot excitons to cavity photons. Phys

Rev B 2009;80:201311.

[81] Briegel H-J, Dür W, Cirac JI, Zoller P. Quantum repeaters: the

role of imperfect local operations in quantum communication.

Phys Rev Lett 1998;81:5932–5.

[82] Kimble HJ. The quantum internet. Nature 2008;453:1023.

[83] Mukamel S. Principles of nonlinear optical spectroscopy, vol.

29. New York, Oxford University Press, 1995.

[84] Axt VM, Mukamel S. Nonlinear optics of semiconductor and

molecular nanostructures; a common perspective. Rev Mod

Phys 1998;70:145–74.

[85] Warburton RJ. Single spins in self-assembled quantum dots.

Nat Mater 2013;12:483.

[86] Matthiesen C, Vamivakas AN, Atatüre M. Subnatural

linewidth single photons from a quantum dot. Phys Rev Lett

2012;108:093602.

[87] May V, Kühn O. Charge and energy transfer dynamics in

molecular systems. Weinheim, John Wiley & Sons, 2008.

[88] Bourgain R, Pellegrino J, Jennewein S, etal. Direct measure-

ment of the wigner time delay for the scattering of light by a

single atom. Opt Lett 2013;38:1963–5.

[89] Sondermann M, Leuchs G. The phase shift induced by a single

atom in free space. J Eur Opt Soc – Rapid 2013;8.

DOI: 10.2971/jeos.2013.13052.

[90] Palatchi C, Dahlstrm JM, Kheifets AS, etal. Atomic delay in

helium, neon, argon and krypton. J Phys B At Mol Opt Phys

2014;47:245003.

[91] Kira M, Koch S. Many-body correlations and excitonic effects

in semiconductor Spectroscopy. Prog Quantum Electron

2006;30:155–296.

[92] Čížek J. On the use of the cluster expansion and the technique

of diagrams in calculations of correlation effects in atoms

and molecules. In: LeFebvre R, Moser C, eds. Advances in

Chemical Physics. Hoboken, NJ, USA, John Wiley & Sons, Ltd.,

1969, 35–89.

[93] Krummheuer B, Axt VM, Kuhn T. Theory of pure dephasing

and the resulting absorption line shape in semiconductor

quantum dots. Phys Rev B 2002;65:195313.

[94] Richter M, Carmele A, Sitek A, Knorr A. Few-photon model of

the optical emission of semiconductor quantum dots. Phys

Rev Lett 2009;103:087407.

[95] Rozbicki E, Machnikowski P. Quantum kinetic theory of pho-

non-assisted excitation transfer in quantum dot molecules.

Phys Rev Lett 2008;100:027401.

[96] Lüker S, Kuhn T, Reiter DE. Phonon impact on optical control

schemes of quantum dots: role of quantum dot geometry and

symmetry. Phys Rev B 2017;96:245306.

[97] Wilson-Rae I, Imamoğlu A. Quantum dot cavity-qed in the

presence of strong electron-phonon interactions. Phys Rev B

2002;65:235311.

[98] Laussy F, Valle ED, Mnchen T, etal. 9 – Luminescence spectra

of quantum dots in microcavities. In: Jahnke F, ed. Quantum

Optics with Semiconductor. Oxford, Woodhead Publishing,

2012, 293–331.

[99] Laussy FP, Laucht A, del Valle E, Finley JJ, Villas-Bôas JM.

Luminescence spectra of quantum dots in microcavities. III.

Multiple quantum dots. Phys Rev B 2011;84:195313.

[100] del Valle E. Microcavity quantum electrodynamics. Saar-

brcken, VDM Verlag, 2010.

[101] del Valle E, Laussy FP. Mollow triplet under incoherent pump-

ing. Phys Rev Lett 2010;105:233601.

[102] Kreinberg S, Grběsić T, Strauß M, etal. Quantum-optical

spectroscopy of a two-level system using an electrically driven

micropillar laser as a resonant excitation source. Light Sci

Appl 2018;7:Article number: 41.

[103] McCutcheon DPS, Dattani NS, Gauger EM, Lovett BW, Nazir A.

Quantum dot rabi rotations beyond the weak exciton-phonon

coupling regime. Phys Rev B 2011;84:081305.

[104] Manson R, Roy-Choudhury K, Hughes S. Polaron master equa-

tion theory of pulse-driven phonon-assisted population inver-

sion and single-photon emission from quantum-dot excitons.

Phys Rev B 2016;93:155423.

[105] Kabuss J, Carmele A, Richter M, Chow WW, Knorr A. Inductive

equation of motion approach for a semiconductor qd-qed:

Coherence induced control of photon statistics. Phys Status

Sol B 2011;248:872–8.

[106] Axt V, Herbst M, Kuhn T. Coherent control of phonon quantum

beats. Superlattices Microstruct 1999;26:117–28.

[107] Kabuss J, Carmele A, Richter M, Knorr A. Microscopic

equation-of-motion approach to the multiphonon assisted

quantum emission of a semiconductor quantum dot. Phys Rev

B 2011;84:125324.

[108] Glässl M, Vagov A, Lüker S, etal. Long-time dynamics and sta-

tionary nonequilibrium of an optically driven strongly confined

quantum dot coupled to phonons. Phys Rev B 2011;84:195311.

[109] Schollwöck U. The density-matrix renormalization group in

theage of matrix product states. Ann Phys 2011;326:96–192.

[110] Caldeira AO, Leggett AJ. Path integral approach to quantum

brownian motion. Phys A Stat Mech Appl 1983;121:587–616.

[111] Vidal G. Phys Rev Lett 2004;93:040502.

682 A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics

[112] del Pino J, Schröder FAYN, Chin AW, Feist J, Garcia-Vidal FJ.

Tensor network simulation of non-markovian dynamics in

organic polaritons. Phys Rev Lett 2018;121:227401.

[113] Strathearn A, Kirton P, Kilda D, Keeling J, Lovett BW. Efficient

non-markovian quantum dynamics using time-evolving matrix

product operators. Nat Commun 2018;9:3322.

[114] Droenner L, Naumann NL, Schöll E, Knorr A, Carmele A. Quan-

tum Pyragas control: selective control of individual photon

probabilities. Phys Rev A 2019;99:023840.

[115] Makri N. Quantum dissipative dynamics: a numerically exact

methodology. J Phys Chem A 1998;102:4414–27.

[116] Cygorek M, Barth AM, Ungar F, Vagov A, Axt VM. Nonlinear

cavity feeding and unconventional photon statistics in solid-

state cavity qed revealed by many-level real-time path-integral

calculations. Phys Rev B 2017;96:201201.

[117] Hopfmann C, Musiał A, Strauß M, etal. Compensation of

phonon-induced renormalization of vacuum rabi splitting in

large quantum dots: towards temperature-stable strong cou-

pling in the solid state with quantum dot-micropillars. Phys

Rev B 2015;92:245403.

[118] Jackson JD. Classical electrodynamics. New York, John Wiley &

Sons, 2012.

[119] Allen L, Eberly JH. Optical resonance and two-level atoms, vol.

28. New York, Dover Publication, Courier Corporation, 1987.

[120] Gardiner C. Stochastic methods, vol. 4. Berlin, Springer, 2009.

[121] Jakubczyk T, Delmonte V, Fischbach S, etal. ACS Photonics

2016;3:2461–6.

[122] Fras F, Mermillod Q, Nogues G, etal. Impact of phonons on

dephasing of individual excitons in deterministic quantum dot

microlenses. Nat Photonics 2016;10:155.

[123] Madelung O. Introduction to solid-state theory, vol. 2. Berlin,

Heidelberg, Springer Science & Business Media, 2012.

[124] Pullerits T, Monshouwer R, van Mourik F, van Grondelle R.

Temperature dependence of electron-vibronic spectra of pho-

tosynthetic systems. Computer simulations and comparison

with experiment. Chem Phys 1995;194:395–407.

[125] Cohen-Tannoudji C, Guéry-Odelin D. Advances in atomic phys-

ics: an overview. Singapore, World Scientific Publishing Co.

Pte. Ltd., 2011.

[126] Gardiner C, Zoller P. The quantum world of ultra-cold atoms

and light book II: the physics of quantum-optical devices.

Singapore, World Scientific, 2015, 1–524.

[127] McCutcheon DP, Nazir A. Quantum dot rabi rotations beyond

the weak exciton–phonon coupling regime. New J Phys

2010;12:113042.

[128] Kabuss J, Werner S, Hoffmann A, Hildebrandt P, Knorr A,

Richter M. Theory of time-resolved raman scattering and fluo-

rescence emission from semiconductor quantum dots. Phys

Rev B 2010;81:075314.

[129] Carmele A, Knorr A, Richter M. Photon statistics as a probe

for exciton correlations in coupled nanostructures. Phys Rev B

2009;79:035316.

[130] Ramsay AJ, Gopal AV, Gauger EM, etal. Damping of

excitonrabi rotations by acoustic phonons in optically

excitedInGaAs/GaAs quantum dots. Phys Rev Lett

2010;104:017402.

[131] Glässl M, Croitoru MD, Vagov A, Axt VM, Kuhn T. Influence of

the pulse shape and the dot size on the decay and reappear-

ance of rabi rotations in laser driven quantum dots. Phys Rev

B 2011;84:125304.

[132] Machnikowski P, Jacak L. Resonant nature of phonon-induced

damping of rabi oscillations in quantum dots. Phys Rev B

2004;69:193302.

[133] Krügel A, Axt VM, Kuhn T. Transition from jaynes-cummings to

autler-townes ladder in a quantum dot–microcavity system.

Phys Rev B 2006;73:035302.

[134] Reiter DE. Time-resolved pump-probe signals of a continu-

ously driven quantum dot affected by phonons. Phys Rev B

2017;95:125308.

[135] Naumann NL, Droenner L, Chow WW, Kabuss J, Carmele

A. Solid-state-based analog of optomechanics. JOSA B

2016;33:1492–501.

[136] Droenner L, Naumann NL, Kabuss J, Carmele A. Collective

enhancements in many-emitter phonon lasing. Phys Rev A

2017;96:043805.

[137] Kabuss J, Carmele A, Brandes T, Knorr A. Optically driven quan-

tum dots as source of coherent cavity phonons: a proposal for a

phonon laser scheme. Phys Rev Lett 2012;109:054301.

[138] Kepesidis KV, Bennett SD, Portolan S, Lukin MD, Rabl P.

Phonon cooling and lasing with nitrogen-vacancy centers in

diamond. Phys Rev B 2013;88:064105.

[139] Czerniuk T, Wigger D, Akimov AV, etal. Picosecond control of

quantum dot laser emission by coherent phonons. Phys Rev

Lett 2017;118:133901.

[140] Richter M, Knorr A. A time convolution less density matrix

approach to the nonlinear optical response of a coupled

system–bath complex. Ann Phys 2010;325:711–47.

[141] Stauber T, Zimmermann R, Castella H. Electron-phonon

interaction in quantum dots: a solvable model. Phys Rev B

2000;62:7336–43.

[142] Heitz R, Mukhametzhanov I, Stier O, Madhukar A, Bimberg

D. Enhanced polar exciton-lo-phonon interaction in quantum

dots. Phys Rev Lett 1999;83:4654–7.

[143] Heitz R, Born H, Guffarth F, etal. Existence of a phonon

bottleneck for excitons in quantum dots. Phys. Rev. B

2001;64:241305.

[144] Droenner L, Finsterhölzl R, Heyl M, Carmele A. Stabilizing

a discrete time crystal against dissipation. arXiv preprint

arXiv:1902.04986, 2019.

[145] Leggett AJ, Chakravarty S, Dorsey AT, Fisher MPA, Garg A,

Zwerger W. Dynamics of the dissipative two-state system. Rev

Mod Phys 1987;59:1.

[146] Vagov A, Croitoru M, Axt V, Kuhn T, Peeters F. High pulse area

undamping of rabi oscillations in quantum dots coupled to

phonons. Phys Status Sol B 2006;243:2233–40.

[147] Vagov A, Croitoru M, Axt V, Machnikowski P, Kuhn T. Dynam-

ics of quantum dots with strong electron phonon coupling:

Correlation expansion vs. path integrals. Phys Status Sol B

2011;248:839–42.

[148] Dousse A, Lanco L, Suffczyński J, etal. Controlled light-matter

coupling for a single quantum dot embedded in a pillar

microcavity using far-field optical lithography. Phys Rev Lett

2008;101:267404.

[149] Gschrey M, Gericke F, Schüßler A, etal. In situ electron-beam

lithography of deterministic single-quantum-dot mesa-

structures using low-temperature cathodoluminescence

spectroscopy. Appl Phys Lett 2013;102:251113.

[150] Sotier F, Thomay T, Hanke T, etal. Femtosecond few-fermion

dynamics and deterministic single-photon gain in a quantum

dot. Nat Phys 2009;5:352.

A. Carmele and S. Reitzenstein: Non-Markovian features in semiconductor quantum optics 683

[151] Flagg EB, Muller A, Robertson J, etal. Resonantly driven

coherent oscillations in a solid-state quantum emitter. Nat

Phys 2009;5:203.

[152] Laussy FP, Del Valle E, Tejedor C. Strong coupling of quantum

dots in microcavities. Phys Rev Lett 2008;101:083601.

[153] Kasprzak J, Reitzenstein S, Muljarov EA, etal. Up on the

jaynes–cummings ladder of a quantum-dot/microcavity sys-

tem. Nat Mater 2010;9:304.

[154] Kavokin A. Microcavities, 21. New York, Oxford University

Press Inc., 2017.

[155] Lodahl P, Mahmoodian S, Stobbe S. Interfacing single pho-

tons and single quantum dots with photonic nanostructures.

Rev Mod Phys 2015;87:347–400.

[156] Schneebeli L, Kira M, Koch SW. Characterization of strong light-

matter coupling in semiconductor quantum-dot microcavities via

photon-statistics spectroscopy. Phys Rev Lett 2008;101:097401.

[157] Hopfmann C, Carmele A, Musiał, etal. Transition from

jaynes-cummings to autler-townes ladder in a quantum

dot–microcavity system. Phys. Rev. B 2017;95:035302.

[158] Lounis B, Orrit M. Single-photon sources. Rep Prog Phys

2005;68:1129.

[159] Ding X, He Y, Duan Z-C, etal. On-demand single photons with

high extraction efficiency and near-unity indistinguishability

from a resonantly driven quantum dot in a micropillar. Phys

Rev Lett 2016;116:020401.

[160] Somaschi N, Giesz V, De Santis L, etal. Near-optimal single-

photon sources in the solid state. Nat Photonics 2016;10:340.

[161] Iles-Smith J, McCutcheon DP, Nazir A, Mørk J. Phonon scatter-

ing inhibits simultaneous near-unity efficiency and indistin-

guishability in semiconductor single-photon sources. Nat

Photonics 2017;11:521.

[162] Calic M, Gallo P, Felici M, etal. Phonon-mediated coupling of

InGaAs/GaAs quantum-dot excitons to photonic crystal cavi-

ties. Phys Rev Lett 2011;106:227402.

[163] Kaer P, Nielsen TR, Lodahl P, Jauho A-P, Mørk J. Non-markovian

model of photon-assisted dephasing by electron-phonon

interactions in a coupled quantum-dot–cavity system. Phys

Rev Lett 2010;104:157401.

[164] Hughes S, Yao P, Milde F, etal. Influence of electron-acoustic

phonon scattering on off-resonant cavity feeding within a

strongly coupled quantum-dot cavity system. Phys Rev B

2011;83:165313.

[165] Winger M, Volz T, Tarel G, etal. Explanation of photon cor-

relations in the far-off-resonance optical emission from a

quantum-dot–cavity system. Phys Rev Lett 2009;103:207403.

[166] Carmele A, Kabuss J, Chow WW. Highly detuned rabi oscil-

lations for a quantum dot in a microcavity. Phys Rev B

2013;87:041305.

[167] Thoma A, Schnauber P, Gschrey M, etal. Exploring dephasing

of a solid-state quantum emitter via time- and temperature-

dependent hong-ou-mandel experiments. Phys Rev Lett

2016;116:033601.

[168] Rempe G, Walther H, Klein N. Observation of quantum

collapse and revival in a one-atom maser. Phys Rev Lett

1987;58:353–6.

[169] Eberly JH, Narozhny NB, Sanchez-Mondragon JJ. Periodic

spontaneous collapse and revival in a simple quantum model.

Phys Rev Lett 1980;44:1323–6.

[170] Hong CK, Ou ZY, Mandel L. Measurement of subpicosecond

time intervals between two photons by interference. Phys Rev

Lett 1987;59:2044–6.

[171] Galperin YM, Altshuler BL, Bergli J, Shantsev DV. Non-

gaussian low-frequency noise as a source of qubit

decoherence. Phys Rev Lett 2006;96:097009.

[172] Laikhtman BD. General theory of spectral diffusion and

echodecay in glasses. Phys Rev B 1985;31:490–504.

[173] Eberly JH, Wódkiewicz K, Shore BW. Noise in strong laser-

atom interactions: phase telegraph noise. Phys Rev A

1984;30:2381–9.

[174] Liu J, Konthasinghe K, Davanço M, etal. Single self-assembled

InAs/GaAs quantum dots in photonic nanostructures: the role

of nanofabrication. Phys Rev Appl 2018;9:064019.

[175] Glässl M, Sörgel L, Vagov A, Croitoru MD, Kuhn T, Axt VM.

Interaction of a quantum-dot cavity system with acoustic pho-

nons: Stronger light-matter coupling can reduce the visibility

of strong coupling effects. Phys Rev B 2012;86:035319.