#  The 1D photonic wire as a single-photon source: Implications of cQED to a
phonon bath of reduced dimensionality

José Ferreira Neto<EMAIL_ADDRESS>Department of Electrical and Photonics
Engineering, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
Matias Bundgaard-Nielsen  Department of Electrical and Photonics Engineering,
Technical University of Denmark, 2800 Kgs. Lyngby, Denmark  NanoPhoton -
Center for Nanophotonics, Technical University of Denmark, Ørsteds Plads 345A,
2800 Kgs. Lyngby, Denmark  Niels Gregersen  Department of Electrical and
Photonics Engineering, Technical University of Denmark, 2800 Kgs. Lyngby,
Denmark  Luca Vannucci  Department of Electrical and Photonics Engineering,
Technical University of Denmark, 2800 Kgs. Lyngby, Denmark

(August 28, 2024)

###### Abstract

While the semiconductor quantum dot placed in a solid-state material allows
for deterministic emission of single photons, the photon indistinguishability
is strongly influenced by the intrinsic coupling to lattice vibrations,
phonons, of the solid-state environment. This work investigates the phonon-
induced decoherence for a quantum dot placed in the one-dimensional system of
a homogeneous cylindrical nanowire. Such a structure supports multiple
longitudinal phonon branches, and we consider both a linear and a quadratic
coupling of the emitter to these modes. Under a polaron approach, we initially
derive an analytical expression for the 1D pure dephasing rate, which leads to
a reduced pure dephasing rate compared with bulk. By implementing these
results into a full cavity quantum electrodynamic model, we demonstrate that
multimode coupling is necessary to correctly predict the indistinguishability
in a 1D system, which may otherwise be significantly underestimated.

##  I Introduction

The triggered emission of high-quality single photons is a key component of
optical quantum technologies [1]. Over the past decades, tremendous progress
has been made in developing quantum light sources spanning various physical
platforms [2]. For applications in quantum information processing, one aims to
simultaneously maximize the photon collection efficiency $\varepsilon$ and the
photon indistinguishability $\mathcal{I}$. While spontaneous parametric down-
conversion offers a viable method for generating indistinguishable single
photons, its inherent probabilistic nature severely limits the collection
efficiency $\varepsilon$ [3]. As an alternative, the semiconductor quantum dot
(QD) has emerged as a promising deterministic platform allowing for pure and
efficient single-photon generation [4, 5, 6].

Unlike atoms, QDs are inexorably coupled to quantized lattice vibrations,
phonons, of the host medium resulting in reduced indistinguishability [7].
Even though placing the QD inside an optical cavity can generally increase
both $\varepsilon$ and $\mathcal{I}$ through Purcell enhancement and cavity
filtering [7], the dimensionality of the hosting material strongly governs the
phonon-induced decoherence process [8]. This behavior is dictated by the
phonon spectral density $J(\omega)$, which generally scales with the phonon
frequency to the power of the phonon bath dimensionality, i.e.,
$J(\omega)\propto\omega^{s}$ [11, 9, 10, 12, 13].

Seminal work by Besombes et al. [14] first elucidated the subject of non-
Lorentzian QD emission and its temperature dependence, attributing these
effects to lattice relaxation caused by exciton-acoustic-phonon coupling in
bulk media. Subsequently, a microscopic theory of optical transitions in QDs
with carrier-phonon interactions was formulated [15, 16]. Since then, the
theory for electron-acoustic phonon interactions in homogeneous bulk material
($s=3$) has been extensively studied [23, 17, 18, 22, 7, 25, 8, 20, 21, 27,
28, 26, 19, 24]. While the bulk model is widely used for structures like the
micropillar single-photon source (SPS) [23, 29], its validity becomes
questionable for geometries of reduced dimensionality. For instance, we expect
the bulk model to break down for structures such as the photonic nanowire
[30], the photonic crystal (PhC) waveguide [31], and the photonic hourglass
[26] due to quantum confinement effects [32, 33], which restrict the motion of
electron-hole pairs. On the other hand, phonon signatures for two-dimensional
materials ($s=2$) have only recently been investigated [37, 40, 35, 34, 36,
38, 39], while a one-dimensional phonon spectral density ($s=1$) has been
utilized in [44, 42, 43, 41].

In this work, we address the 1D case ($s=1$) of the photonic nanowire by
introducing a fully microscopic theory. We calculate an analytical solution
for the quantized vibration modes of an infinite free-standing cylinder. In
turn, we introduce mode-dependent exciton-phonon coupling between an embedded
on-axis QD and higher-order phonon modes, which we then adopt in a cavity
quantum electrodynamic (cQED) model. We determine the two-photon
indistinguishability $\mathcal{I}$ for two different configurations: (i) the
QD in the bare structure and (ii) a case where the optical cavity effect
induced by the nanowire structure is taken into account. The latter is
analyzed through a weak phonon master equation (ME) [25], which is compared
with numerically exact tensor network (TN) calculations [47, 46, 45, 50, 49,
48]. We demonstrate that the Ohmic coupling scenario, where
$J(\omega)\propto\omega$, fails to accurately describe the solid-state
environment of a 1D photonic wire. Lastly, our results indicate that non-
Markovian effects lead to improved photon indistinguishability within this
system.

This paper is structured as follows: Sec. II outlines the theoretical model
for QD-light emission in a cylindrical nanowire taking into account higher-
order mechanical vibration modes. Sec. III covers the calculation of the one-
dimensional pure dephasing rate. In Sec. IV, we present our results for the
photon indistinguishability with and without cavity effects. Sec. V discusses
the key insights of our findings, followed by concluding remarks in Sec. VI.
The Appendix provides further details on our model, as well as a benchmarking
of our tensor network results.

##  II Model

###  II.1 Hamiltonian

$\hbar\omega_{i,\mathbf{k}}$$\hbar\omega_{\mathrm{ph}}$$|X\rangle$$|G\rangle$…$J(\omega)$

Figure 1: Sketch of a cylindrical wire with an embedded QD and its
corresponding energy level diagram. Here, $\omega_{i,\mathrm{\mathbf{k}}}$
corresponds to the frequency of the emitted phonon from phonon branch $i$ and
wave vector $\mathrm{\mathbf{k}}$, whereas $\omega_{\mathrm{ph}}$ is the
frequency of the emitted photon.

We consider the two-level system of a QD in a 1D nanowire coupled to a
continuum of longitudinal acoustic (LA) phonons, sketched in Fig. 1. The GaAs
wire is modeled as a cylindrical structure oriented along the $z$-axis, with a
radius denoted by $R$ and length $L$, where $L\gg R$. We assume that the wire
maintains its zinc-blende crystal structure unaffected by dimensional
confinement effects and that the $z$-axis aligns with the [001]-direction,
such that the wire can be treated as isotropic [51].

We investigate the impact of phonon-induced transitions on single-photon
emission using an independent boson model with the polaron transformation. The
total Hamiltonian $\mathcal{H}$ is given by [16, 52]

$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{\mathrm{ph}}+\mathcal{H}_{\mathrm{lin}}+\mathcal{H}_{\mathrm{quad}},$
(1) $\mathcal{H}_{0}=\hbar\omega_{X}\sigma^{{\dagger}}\sigma,$ (2)
$\mathcal{H}_{\mathrm{ph}}=\sum_{i,\mathrm{\bf{k}}}\hbar\omega_{i,\mathrm{\bf{k}}}{b^{{\dagger}}_{i,\mathrm{\bf{k}}}}b_{i,\mathrm{\bf{k}}},$
(3) $\mathcal{H}_{\mathrm{lin}}=\sum_{i,\mathrm{\bf{k}}}\hbar
g_{i,\mathrm{\bf{k}}}(b_{i,\mathrm{\bf{k}}}^{\dagger}+b_{i,\mathrm{\bf{k}}})\sigma^{{\dagger}}\sigma,$
(4)
$\mathcal{H}_{\mathrm{quad}}=\sum_{i,j}\sum_{\mathrm{\bf{k}},\mathrm{\bf{k^{\prime}}}}\hbar
f_{i,j,\mathrm{\bf{k}},\mathrm{\bf{k^{\prime}}}}(b_{i,\mathrm{\bf{k}}}^{\dagger}+b_{i,\mathrm{\bf{k}}})(b_{j,\mathrm{\bf{k^{\prime}}}}^{\dagger}+b_{j,\mathrm{\bf{k^{\prime}}}})\sigma^{{\dagger}}\sigma.$
(5)

Here, $\mathcal{H}_{0}$ is the Hamiltonian for a two-level emitter with ground
state $|G\rangle$ and excited state (single exciton) $|X\rangle$, $\omega_{X}$
is the bare exciton frequency, $\sigma=|G\rangle\langle X|$ is the lowering
operator, $\mathcal{H}_{\mathrm{ph}}$ represents a multimode acoustic phonon
bath with $b_{i,\mathrm{\bf{k}}}^{\dagger}$ ($b_{i,\mathbf{k}}$) the creation
(annihilation) operator for the phonon mode $i$ with momentum
$\hbar\mathbf{k}$, whereas $\mathcal{H}_{\mathrm{lin}}$ and
$\mathcal{H}_{\mathrm{quad}}$ correspond to the linear and quadratic phonon
coupling, respectively [16, 52]. The linear and quadratic constants,
$g_{i,\mathrm{\bf{k}}}$ and
$f_{i,j,\mathrm{\bf{k}},\mathrm{\bf{k^{\prime}}}}$, are calculated from the
matrix elements of the electron-phonon coupling, as discussed in Appendices A
and B.

Under the polaron transformation extended to multiple phonon branches, the
linear coupling can be eliminated $\mathcal{U}_{\rm
P}\mathcal{H}\mathcal{U}^{\dagger}_{\rm P}$, with $\mathcal{U}_{\rm
P}=\exp[\sigma^{\dagger}\sigma\sum_{i,\mathbf{k}}\omega_{i,\mathbf{k}}^{-1}g_{i,\mathbf{k}}(b^{\dagger}_{i,\mathbf{k}}-b_{i,\mathbf{k}})]$
[11]. The transformed Hamiltonian reads
$\widetilde{\mathcal{H}}=\hbar\widetilde{\omega}_{X}\sigma^{\dagger}\sigma+\mathcal{H}_{\rm
ph}+\mathcal{H}_{\rm quad}$, where the polaron-shifted exciton frequency
$\widetilde{\omega}_{X}=\omega_{X}-\mathcal{D}$, with
$\mathcal{D}=\int_{0}^{\infty}\differential{\omega}\omega^{-1}J(\omega)$.

Following, for the case of multiple phonon branches, the coupling to phonons
is completely characterized by the phonon spectral density
$J(\omega)=\sum\nolimits_{i,\mathbf{k}}\left|g_{i,\mathbf{k}}\right|^{2}\delta(\omega-\omega_{i,\mathbf{k}})$,
where the index $i$ represents the total number of phonon branches and the
summation over $\mathbf{k}$ yields the contribution from different phonon wave
vectors [11]. To assess the coherence evolution of our two-level system as it
interacts with the solid-state environment, also referred to as the optical
polarization between electrons and holes, we define the physical observable
$\langle\sigma_{x}\rangle=\mathrm{Re[e^{\Phi(\tau)}]}$ [53]. The phonon
propagator $\Phi(\tau)$ is described by

$\Phi(\tau)=\int_{0}^{\infty}\mathrm{d}\omega\frac{J(\omega)}{{\omega}^{2}}\left[\frac{\cos(\omega\tau)-1}{\tanh(\beta\hbar\omega/2)}-i\sin(\omega\tau)\right],$
(6)

Figure 2: (a) The first 15 dispersion curves for propagating harmonic waves in
a cylindrical rod with $R=25\ \mathrm{nm}$, obtained as solutions of the
linear Pochhammer-Chree equation. For later convenience, we indicate with a
circle marker the wave vector at which the mode velocity of each mode was
extracted for the calculation of the pure dephasing rate presented in Section
III. (b) The corresponding exciton coupling matrix elements for the first 15
longitudinal modes.

where $\beta=(k_{\mathrm{B}}T)^{-1}$ is the inverse temperature and
$k_{\mathrm{B}}$ is the Boltzmann constant. We then straightforwardly obtain
that

$\Phi(\tau)=\sum_{i,\mathbf{k}}\frac{\left|g_{i,\mathbf{k}}\right|^{2}}{{\omega_{i,\mathbf{k}}}^{2}}\left[\frac{\cos(\omega_{i,\mathbf{k}}\tau)-1}{\tanh(\beta\hbar\omega_{i,\mathbf{k}}/2)}-i\sin(\omega_{i,\mathbf{k}}\tau)\right].$
(7)

###  II.2 Quantized vibration modes

For a free-standing isotropic, cubic medium of cylindrical geometry, we obtain
the dispersion relation curves of the longitudinal modes predicted by the
Pochhammer-Chree equation [55, 54, 56]. In Fig. 2(a), we show multiple phonon
branches obtained for a geometry with a cylindrical cross-section. With this
analytical result, we can now evaluate the coupling between the QD and each
vibration mode [57]. We only include longitudinal phonon modes, since a QD
situated at the nanowire center is unaffected by flexural vibrations, whereas
torsional modes can be disregarded as they do not modulate the bandgap of the
two-level system via deformation potential, at least to a first-order
approximation [58, 59, 60]. The results for the coupling constants
$g_{i,\mathbf{k}}$ are displayed in Fig. 2(b).

Figure 3: Time evolution of the QD coherence for a nanowire radius $R=25\
\mathrm{nm}$ evaluated at different temperatures $T$ $\in$
[$4\text{\,}\mathrm{K}$, $40\text{\,}\mathrm{K}$] (swept in steps of
$2\text{\,}\mathrm{K}$).

We present the coherence dynamics computed from Eq. (7) for a nanowire radius
$R=25\ \mathrm{nm}$ with respect to temperature $T$ in Fig. 3. The curves
exhibit strong signatures of non-Markovianity, as revealed by their
accentuated nonmonotonic modulated exponential decay at short timescales, and
a vanishing value in the long time limit, i.e.,
$\lim_{\tau\rightarrow\infty}\mathrm{Re[e^{\Phi(\tau)}]}=0$. This differs
markedly from bulk behavior, which is characterized by a fast initial decay
(within picoseconds) followed by a stationary coherence
$\expectationvalue{\sigma_{x}}>0$ [53, 61]. This fact causes the Franck-Condon
factor
$B=\exp\quantity[-\frac{1}{2}\int_{0}^{\infty}\differential{\omega}\omega^{-2}J(\omega)\coth(\beta\hbar\omega/2)]$
to be ill-defined for phonon bath dimensionalities $s\leq 2$ [11], since the
integrand function $\omega^{-2}J(\omega)\coth(\beta\hbar\omega/2)$ diverges in
such cases. To fully account for the behavior of our system, the phonon
propagator evolution $\Phi(\tau)$ should be captured until its coherence
vanishes entirely (see complete discussion in Appendix E). While a $R=25\
\mathrm{nm}$ radius is too small to support a guided optical HE11 mode, we
present the corresponding coherence evolution for the $R=100\ \mathrm{nm}$
radius, similar to the ideal photonic nanowire [30], in the Supplementary
Information (SI).

##  III 1D pure dephasing rate calculation

Following [18], we generalize the calculation of the pure dephasing rate
induced by quadratic phonon coupling to the case of multiple phonon branches.
We have

$\displaystyle\gamma^{1\mathrm{D}}_{\mathrm{pd}}=$
$\displaystyle\mathrm{Re}\Bigg{[}\sum_{i,j}\sum_{\mathbf{k,k^{\prime}}}\int_{0}^{\infty}\mathrm{d}\tau\
|f_{i,j,\mathbf{k,k^{\prime}}}|^{2}$ (8)
$\displaystyle\qquad\times\langle\mathcal{B}_{i,\mathbf{k}}(\tau)\mathcal{B}_{j,\mathbf{k^{\prime}}}(\tau)\mathcal{B}_{i,\mathbf{k}}(0)\mathcal{B}_{j,\mathbf{k^{\prime}}}(0)\rangle\Bigg{]},$

where
$\mathcal{B}_{i,\mathbf{k}}(\tau)=b_{i,\mathbf{k}}^{\dagger}e^{ic_{i}k\tau}+b_{i,\mathbf{k}}e^{-ic_{i}k\tau}$,
with $k=|\mathbf{k}|$, and the sum is extended to all combinations between
phonon branches $i$ and $j$. Here, $c_{i}$ corresponds to the velocity
associated with mode $i$. Whereas only the first phonon mode is taken into
account in the bulk model, we now assign mode-dependent velocities to each of
the phonon branches. These mode-dependent phase velocities are calculated from
the slope of the dispersion relation curves in Fig. 2(a) evaluated at the wave
vector that maximizes the squared matrix coupling element
$|f_{i,j,\mathbf{k,k^{\prime}}}|^{2}$. This approach will become more evident
as we take a closer look into $|f_{i,j,\mathbf{k,k^{\prime}}}|^{2}$.

Employing Wick’s theorem to calculate the multi-operator expectation value in
Eq. (8), we assume that the dominant contribution originates from instances
with different phonon wave vectors $\mathbf{k}$ and $\mathbf{k^{\prime}}$,
which allows to factorize the expectation value [18]. Applying the commutation
relations
$\commutator{b_{i,\mathbf{k}}}{b_{j,\mathbf{k^{\prime}}}^{\dagger}}=\delta_{ij}\delta_{\mathbf{k}\mathbf{k^{\prime}}}$,
and using $\int_{0}^{\infty}\differential{\tau}e^{\pm
i(c_{i}k-c_{j}k^{\prime})\tau}=\pi
c_{j}^{-1}\delta(k^{\prime}-\frac{c_{i}}{c_{j}}k)$, we are left with the final
expression

$\displaystyle\gamma^{1\mathrm{D}}_{\mathrm{pd}}=$
$\displaystyle\pi\left(\frac{L}{2\pi}\right)^{2}\sum_{i,j}\frac{1}{c_{i}}\int\mathrm{d}k\
|f_{\mathbf{k},c_{ij}\mathbf{k}}|^{2}\\{n_{i}(k)[1+n_{j}(c_{ij}k)]$ (9)
$\displaystyle+n_{j}(c_{ij}k)[1+n_{i}(k)]\\},$

with $c_{ij}={c_{i}}/{c_{j}}$ the ratio between the mode-dependent velocities
and $n_{i}(k)=\\{\mathrm{exp}[\beta\hbar\omega_{i}(k)]-1\\}^{-1}$ the bosonic
occupation number for the phonon branch $i$ with a wave vector module $k$, at
temperature $T$.

Figure 4: Contour plots $|f_{i,j,\mathbf{k,k^{\prime}}}|^{2}$ for $R=25\
\mathrm{nm}$ for different modes ($i$ and $j$) superimposed with the dashed
white lines defined by $k^{\prime}=\frac{c_{i}}{c_{j}}k$. All plots are
normalized with dark blue (yellow) representing 0 (1) arbitrary units.

We use the following procedure to define the mode-dependent velocities
$c_{i}$, which is based on the inspection of the matrix coupling elements
$|f_{i,j,\mathbf{k,k^{\prime}}}|^{2}$ shown in Fig. 4 and Fig. A in the SI.
For $i=j$, we observe that $|f_{i,i,\mathbf{k,k^{\prime}}}|^{2}$ vanishes
almost everywhere except for a small region around $k=k^{\prime}=k_{i}$, cf.
the exemplary cases $i=3$ and $i=11$ in Fig. 4. This suggests to define the
mode-dependent velocity $c_{i}$ as the slope of the phonon branch $i$ at
$k_{i}$, i.e., $c_{i}=\evaluated{\derivative{\omega_{i}(k)}{k}}_{k_{i}}$. The
corresponding points are indicated in Fig. 2 with circle markers, and the
values of $c_{i}$ and $k_{i}$ are reported in the SI for the first 40 modes.
Similarly, for $i\neq j$, we observe that
$|f_{i,j,\mathbf{k,k^{\prime}}}|^{2}$ is nonzero only in the region around
$(k,k^{\prime})=(k_{i},k_{j})$, with the same values for $k_{i}$ and $k_{j}$
as defined above. We observe that our choice is consistent with the constraint
of $k^{\prime}=\frac{c_{i}}{c_{j}}k$ imposed by the delta function
$\delta(k^{\prime}-\frac{c_{i}}{c_{j}}k)$, which is verified in Fig. 4 by
tracing a white dashed line. This fact is also consistent with the
interpretation of the phonon-induced pure dephasing in terms of virtual
transitions into higher confined states [16, 18, 8], restricted here to the
first excited electron and hole level, implying that the phonon modes involved
must have approximately the same energy $\hbar c_{i}k\approx\hbar
c_{j}k^{\prime}$. Note also that $|f_{i,i,\mathbf{k,k^{\prime}}}|^{2}$ is
symmetric under the simultaneous exchange $i\leftrightarrow j$ and
$k\leftrightarrow k^{\prime}$.

The convergence of the pure dephasing rate with the number of phonon modes is
presented in Fig. 5, and we observe that $\sim$ 15 modes are required to
achieve convergence for $R=25\ \mathrm{nm}$. An increased nanowire radius
requires a higher number of phonon modes for convergence, and for the case of
$R=100\ \mathrm{nm}$, we estimate that $\sim$ 35 modes are needed. In our
study, a total of 40 vibration modes were used for both $R=25\ \mathrm{nm}$
and $R=100\ \mathrm{nm}$.

Figure 5: Convergence analysis of the 1D pure dephasing rate
$\gamma^{1\mathrm{D}}_{\mathrm{pd}}$ as a function of the number of included
phonon modes.

Subsequently, we compare our 1D rates with predictions from a 3D bulk model
given by [18, 8]

$\gamma^{\mathrm{3D}}_{\mathrm{pd}}(T)=\frac{\alpha^{2}\mu}{{\omega_{c}}^{4}}\int\nolimits_{0}^{\infty}\mathrm{d}\omega\
\omega^{10}e^{-2\omega^{2}/\omega_{c}^{2}}\ n(\omega)[n(\omega)+1],$ (10)

where the cutoff frequency $\omega_{c}=\sqrt{2}c_{s}/d$, $d$ is the
confinement length of the QD ($d=a_{e}=a_{h}$), and
$n(\omega)=[\mathrm{exp}(\beta\hbar\omega)-1]^{-1}$ is the Bose-distributed
phonon occupation number. In this bulk model,
$\alpha=(4\pi^{2}\hbar\rho_{m}c_{s}^{5})^{-1}(D_{e}-D_{h})^{2}$,
$\mu=\pi\hbar^{2}(D_{e}-D_{h})^{-4}(\Delta_{e}^{-1}D_{e}^{2}+\Delta_{h}^{-1}D_{h}^{2})^{2}$,
$D_{e}$ ($D_{h}$) is the electron (hole) deformation potential, $\rho_{m}$ is
the mass density, and
$\Delta_{\lambda}=\hbar^{2}/(a_{\lambda}^{2}m_{\lambda})$, while the subscript
$\lambda$ $\in$ {$e$, $h$}. Thus, $a_{\lambda}$ and $m_{\lambda}$ denote the
confinement length and the effective mass, respectively, for electrons ($e$)
and holes ($h$).

Figure 6: Comparison between the pure dephasing rate $\gamma_{\mathrm{pd}}$
calculated for a 1D phonon spectral density, with $R=25\ \mathrm{nm}$ and
$R=100\ \mathrm{nm}$, and the bulk prediction.

We present the pure dephasing rates in Fig. 6 for both $R=25\ \mathrm{nm}$ and
$R=100\ \mathrm{nm}$ as a function of temperature, comparing them to
predictions from the bulk model. We observe a tendency of the 1D pure
dephasing rate for small $R$ to bulklike behavior as the nanowire radius $R$
increases, in agreement with expectations. Specific values for
$\gamma^{1\mathrm{D}}_{\mathrm{pd}}$ for $T=$ $4\text{\,}\mathrm{K}$ and $T=$
$20\text{\,}\mathrm{K}$ for both $R=25\ \mathrm{nm}$ and $R=100\ \mathrm{nm}$
are given in Table 1, along with the bulk model predictions.

$T$ |  $\gamma_{\mathrm{pd}}^{\mathrm{1D}}$ for $R=25\ \mathrm{nm}$ |  $\gamma_{\mathrm{pd}}^{\mathrm{1D}}$ for $R=100\ \mathrm{nm}$ |  $\gamma_{\mathrm{pd}}^{\mathrm{3D}}$ (Bulk)
---|---|---|---
$4\text{\,}\mathrm{K}$ | 0.34 MHz | 2.62 MHz | 26.15 MHz
$20\text{\,}\mathrm{K}$ | 11.26 MHz | 84.29 MHz | 806.86 MHz
Table 1: 1D pure dephasing rates at two different temperatures and predictions
from the bulk model. Figure 7: Normalized emission spectra for two different
temperatures, with $R=25\ \mathrm{nm}$ and $R=100\ \mathrm{nm}$.

##  IV Indistinguishability

### Bare wire calculation

In the absence of a cavity, the emission spectrum of a QD embedded into a bare
wire (see Appendix C for the full derivation and detailed discussion) is given
by

$S(\omega)=\frac{2}{\Gamma_{\mathrm{rad}}}\
\mathrm{Re}\left[\int_{0}^{\infty}\mathrm{d}\tau\
e^{\Phi(\tau)}e^{-\frac{1}{2}(\Gamma_{\mathrm{rad}}+\gamma^{1\mathrm{D}}_{\mathrm{pd}})\tau}e^{-i(\omega-\omega_{0})\tau}\right],$
(11)

where $\Gamma_{\mathrm{rad}}$ = 1 GHz is the spontaneous decay rate. The
derived emission spectra from Fig. 7 display remarkable non-Lorentzian
features, evidenced by the zero-phonon line (ZPL) accompanied by distinct
phonon satellite peaks. The positions of these peaks match the positions in
energy where a new phonon branch appears [57]. For both radii, the emission
spectrum at lower temperatures ($T=$ $4\text{\,}\mathrm{K}$) shows more
pronounced asymmetry when compared to higher temperatures ($T=$
$20\text{\,}\mathrm{K}$), and this asymmetry arises from the preferential
nature of phonon emission over phonon absorption at low bath temperatures.
Moreover, as the radius increases, the energy separation between the phonon
satellite peaks gets increasingly smaller. Consequently, their collective
contribution merges into continuous phonon sidebands (PSBs) as the associated
phonon spectral density becomes very much bulklike.

In Fig. 8, we present the indistinguishability computed using three different
theoretical models, for the two nanowire radii and for bulk as a function of
the temperature.

Under the assumption of Markovian QD-light interaction and a $\delta$-shaped
laser excitation, while considering a perfectly antibunched photon output with
stationary noise, the two-photon indistinguishability can be calculated as
[60, 8, 62]

$\mathcal{I}=\Gamma_{\rm rad}\int_{0}^{\infty}\mathrm{d}\tau\ e^{-\Gamma_{\rm
rad}\tau}|P(\tau)|^{2},$ (12)

where the coherence function $|P(\tau)|=\exp(-\Gamma_{\mathrm{ZPL}}\tau)$ by
assuming the complete suppression of the phonon sidebands and coupling to the
first phonon branch only [8]. Here, $\Gamma_{\mathrm{ZPL}}$ is the total decay
rate arising from exciton-phonon interaction, which depends on the phonon bath
dimensionality. We then straightforwardly obtain that the corresponding photon
indistinguishability, plotted using full curves in Fig. 8, is given by

$\mathcal{I}=\frac{\Gamma_{\mathrm{rad}}}{\Gamma_{\mathrm{rad}}+2\Gamma_{\mathrm{ZPL}}}.$
(13)

Figure 8: Indistinguishability for the bare wire. The solid curves assume
complete suppression of the phonon sidebands/satellite peaks and only consider
coupling to the first phonon branch
$\omega_{1,\mathbf{k}}=c_{s}|\rm{\mathbf{k}}|$, without accounting for cavity
coupling effects. The dotted curves correspond to our bare wire calculations
determined by Eq. (11), which are compared with tensor network results (empty
circles). Details on the error bars for the tensor network results are given
in Appendix E.

In the 3D bulk case, the broadening of the ZPL is exclusively due to the
quadratic exciton-phonon coupling [16]. Consequently, the total decay rate
$\Gamma_{\mathrm{ZPL}}=\Gamma^{3\mathrm{D}}_{\mathrm{ZPL}}=\gamma^{\mathrm{3D}}_{\mathrm{pd}}$
is determined solely by the pure dephasing rate owing to virtual phonon
transitions, as described in Eq. (10). For sufficiently low temperatures, the
average phonon energy $k_{\mathrm{B}}T$ is too small to induce virtual phonon
transitions, leading to negligible pure dephasing rate of the ZPL, i.e.,
$\gamma^{\mathrm{3D}}_{\mathrm{pd}}\ll\Gamma_{\mathrm{rad}}$ and consequently
$\mathcal{I}\approx 1$. In contrast, in 1D systems, there is additional
broadening from linear coupling. This results in
$\Gamma^{\mathrm{1D}}_{\mathrm{ZPL}}=\Gamma_{\mathrm{lin}}^{\mathrm{1D}}+\Gamma^{\mathrm{3D}}_{\mathrm{ZPL}}$,
where
$\Gamma_{\mathrm{lin}}^{\mathrm{1D}}=(2A\rho_{m}c_{s}^{3}\hbar^{2})^{-1}(D_{e}-D_{h})^{2}(1-2\nu)^{2}k_{\mathrm{B}}T$
[8, 57], with the wire cross-sectional area $A=\pi R^{2}$.

Additionally, Fig. 8 presents results in dotted curves from our own model
taking into account multiple mechanical modes. Here, the indistinguishability
is calculated with the two-color spectrum (see Appendix C) from Eq. (11). Our
predictions take lower values than the results from Eq. (13), and we attribute
this difference to the effect of the PSBs/phonon replicas, which is fully
taken into account in our model but not in the model given by Eq. (13). These
initial results appear to confirm previous indications [8] that the reduced
dimensionality of the 1D wire is indeed strongly detrimental to the
indistinguishability.

$D_{e}$ [eV] |  $D_{h}$ [eV] |  $\rho_{m}$ [kg/m3] |  $v_{l}$ [m/s] |  $v_{t}$ [m/s] |  $c_{s}$ [m/s] | $\nu$ |  $a_{e}$ [$\mathrm{nm}$] |  $a_{h}$ [$\mathrm{nm}$] |  $m_{e}$ [$m_{0}$] |  $m_{h}$ [$m_{0}$]
---|---|---|---|---|---|---|---|---|---|---
-15.93 | -8.77 | 5370 | 4780 | 2560 | 4126.14 | 0.2989 | 10 | 10 | 0.067 | 0.51
Table 2: Material parameters for GaAs, used to calculate the electron-phonon
coupling parameters, employed throughout this paper [22, 57]. Here, $v_{l}$
($v_{t}$) is the longitudinal (transverse) sound velocity. The bulk sound
velocity was calculated from
$c_{s}=v_{t}\sqrt{(3v_{l}^{2}-4v_{t}^{2})/(v_{l}^{2}-v_{t}^{2})}$, while
$\nu=(v^{2}_{l}-2v^{2}_{t})/(2v^{2}_{l}-2v^{2}_{t})$ is the Poisson ratio, and
$m_{0}$ is the free electron mass.

Finally, we also present indistinguishability computed using a numerically
exact tensor network (empty circles in Fig. 8). We observe good agreement for
$R=25\ \mathrm{nm}$, confirming the validity of our model Eq. (11). However,
for $R=100\ \mathrm{nm}$ at $T=$ $20\text{\,}\mathrm{K}$, the tensor network
predicts lower indistinguishability, and this deviation arises due to the
finite memory time of the tensor network as discussed in detail in Appendix E.

### Cavity quantum electrodynamic calculation

Placing the QD in an optical cavity allows for enhancement of the zero-phonon
line and suppression of the phonon sidebands through a cavity funneling effect
[7, 63], thus enhancing the indistinguishability. However, when including a
cavity, the known solution to the independent boson model employed in the
previous section is no longer valid. Significant effort has been put into
developing perturbative master equations [64, 25, 22, 7]. However, the use of
master equations in lower dimensional systems is less explored. While the
polaron master equation has been widely used to model non-Markovian phonon
effects for QDs in bulk, it is not valid in low dimensional systems due to the
vanishing Franck-Condon factor $B$ [11].

Instead, we employ a weak master equation [25, 65] and a numerically exact
tensor network [47, 46, 45, 50, 49, 48]. The weak master equation is a
perturbative approach that neglects the non-Markovian effects of phonons. It
establishes a comparison that can point out the nontrivial results of the
tensor network, which, on the other hand, offers exact results, albeit with a
substantial computational overhead. This is especially true when there is a
large separation of timescales between phonon and emitter-cavity dynamics. For
a micropillar structure with distributed Bragg reflectors or a photonic
crystal cavity, typical light-matter coupling rates and cavity decay rates are
$\hbar g=20-250$ µ$\mathrm{eV}$ and $\hbar\kappa=0.01-10\ \mathrm{meV}$ [22],
respectively.

In this work, we choose $\hbar g=40$ µ$\mathrm{eV}$, and the characteristic
timescale can be estimated by the two quantities:
$\kappa/(4g^{2})=1.03-1028.45\ \mathrm{ps}$ and $1/\kappa=65.82-0.06\
\mathrm{ps}$, indicating that the emitter-cavity timescale is of several
hundreds of picoseconds. However, the fast phonon dynamics, with a timescale
of $\sim 1\ \mathrm{ps}$, requires very small time steps compared with the
long simulation times needed for the emitter-cavity dynamics. We explore this
regime conventionally unavailable for tensor network methods by using the code
and algorithm developed in Ref. [50], which utilizes a periodic process tensor
structure to reduce the numerical complexity of simulations with many time
steps. The cavity-emitter coupling is described by the Jaynes-Cummings
Hamiltonian [25]

$\mathcal{H}_{\mathrm{S}}=\hbar\omega_{X}\sigma^{\dagger}\sigma+\hbar\omega_{\rm
cav}a^{\dagger}a+\hbar g(\sigma^{\dagger}a+a^{\dagger}\sigma),$ (14)

where $\omega_{X}$ ($\omega_{\rm cav}$) is the bare exciton (cavity mode)
frequency, $a^{\dagger}$ ($a$) is the cavity photon creation (annihilation)
operator, and $g$ the light-matter coupling strength introduced above. Next,
we will consider the cavity to be on resonance with the phonon-shifted emitter
transition $\tilde{\omega}_{X}$. The cavity mode is leaky with a decay rate
$\kappa$, included through a Lindblad term. For more details, see Appendix D.

Figure 9: Spectral densities of the phonon environment $J(\omega)$ from
different models: Predictions from our multi-mode 1D phonon model for $R=25\
\mathrm{nm}$ and $R=100\ \mathrm{nm}$ and from the expression
$J(\omega)=\alpha\omega^{s}\exp(-\omega^{2}/\omega_{c}^{2})$ [11] for
$s=\\{1,3\\}$ with $\omega_{c}=\\{1.05\ \mathrm{THz},0.58\ \mathrm{THz}\\}$
and $\alpha=\\{0.012,0.049\ \mathrm{ps}^{2}\\}$, respectively.

We will calculate the indistinguishability for four different cases. These
include: a nanowire of radius (i) $R=25\ \mathrm{nm}$ and (ii) $R=100\
\mathrm{nm}$, where we have calculated the spectral density $J(\omega)$
numerically from the exciton coupling $g_{i,\mathrm{\bf{k}}}$; (iii) the bulk
case ($s=3$), where we use the material parameters stated in Table 2; and (iv)
an Ohmic model where $J(\omega)\propto\omega$ ($s=1$). The latter is
considered here to test the validity of our model, and is obtained with a
cutoff frequency $\omega_{c}=1.05\ \mathrm{THz}$ and imposing that the
integral $\int J(\omega)\ \mathrm{d}\omega\approx 0.013$ has the same value as
the case for the $R=25\ \mathrm{nm}$ nanowire. The phonon spectral densities
corresponding to these four cases are shown in Fig. 9.

Figure 10: Indistinguishability as a function of the cavity decay rate for
$\hbar g=40$ µ$\mathrm{eV}$. (a) At a temperature $T=$ $4\text{\,}\mathrm{K}$
and (b) at $T=$ $20\text{\,}\mathrm{K}$. Details on the generation of the
error bars can be found in Appendix E. Figure 11: Real component of the
environmental correlation function for the four phonon environments considered
at $T=4\ \mathrm{K}$. The dashed vertical line indicates the memory time for
$R=100\ \mathrm{nm}$ (carried out until $\tau=12\ \mathrm{ps}$), whereas the
other correlation functions are plotted over the depicted range of the memory
time.

In Fig. 10(a), we present the indistinguishability at $4\text{\,}\mathrm{K}$
for the four cases as a function of the cavity decay rate predicted using the
weak master equation and the tensor network. We use the pure dephasing rates
from Fig. 6, where for $s=1$, the pure dephasing is the same as for the $R=25\
\mathrm{nm}$ nanowire. We observe that the weak master equation describes the
indistinguishability of the $s=1$ and $s=3$ cases in good agreement with the
tensor network. For the nanowire with $R=25\ \mathrm{nm}$, the non-Markovian
effects due to long-lived phonons are very strong, and the weak master
equation strongly underestimates the indistinguishability. This is in
consonance with observations of the quantum regression theorem having
significant limitations in this regime, resulting in an overestimated
influence of the phonon environment [66].

The presence of long-lived phonon correlations is clearly evidenced by
considering the real component of the environmental correlation function
$C_{\mathrm{ZZ}}(\tau)$, an analog of the coherence evolution of a two-level
system within the independent boson model, also dictated by the phonon
spectral density. For the exact expression of $C_{\mathrm{ZZ}}(\tau)$, cf.
Appendix D. As shown in Fig. 11, it is evident that longer-lived correlations
are present for both $R=25\ \mathrm{nm}$ and $R=100\ \mathrm{nm}$. For $R=100\
\mathrm{nm}$, the long-lived effects are, however, neglected as they exceed
the memory time of the tensor network algorithm. Namely, the longer-lived
correlations extend to thousands of picoseconds, which can be associated with
the many satellite peaks observed in the spectral density.

The tensor network results for $R=100\ \mathrm{nm}$ indicate a transition
toward bulk behavior, as the indistinguishability approaches the curve
corresponding to $s=3$ in Fig. 10(a). The finite memory time of the tensor
network only allows it to correctly capture the bulklike features, and the
transition to bulk behavior is thus not unexpected. Still, it confirms the
validity of our 1D nanowire phonon model for large radius. From the
corresponding results at $20\text{\,}\mathrm{K}$ presented in Figs. 10(b), we
verify that similar physics is observed at a higher temperature despite
reduced indistinguishability. However, as the temperature of the bath
increases, the weak-coupling treatment becomes an even less accurate
approximation due to the increasing relevance of multiphonon processes and the
misestimation of the Rabi frequency renormalization [67, 53].

Figure 12: Emission spectra obtained from tensor network calculations for
different cavity decay rates are contrasted with the bare spectrum (shaded in
blue) for a nanowire radius $R=25\ \mathrm{nm}$ at $T=$
$4\text{\,}\mathrm{K}$.

To investigate the effect of the cavity on the emission spectrum, we present
example spectra in Fig. 12 from which the reported indistinguishabilities were
derived using the tensor network approach. Our focus is on a nanowire with a
radius $R$ = 25 $\mathrm{nm}$ at $4\text{\,}\mathrm{K}$, as indicated by the
empty blue circles in Fig. 10(a). When the cavity decay rate $\kappa$ is
sufficiently large, we observe that most phonon satellite peaks are recovered,
while only finer features of the bare spectrum are suppressed [cf. Fig.
12(c)]. However, as $\kappa$ decreases, the optical cavity acts as a spectral
funnel that enhances emission into the ZPL by reducing contributions from
neighboring incoherent phonon peaks [7].

The cQED parameters maximizing the indistinguishability are given by {$\hbar
g$, $\hbar\kappa$, $\hbar\gamma$} = {40 µ$\mathrm{eV}$, 105 µ$\mathrm{eV}$,
0.66 µ$\mathrm{eV}$} for $\gamma=\Gamma_{\mathrm{rad}}=1$ GHz with
corresponding spectrum in Fig. 12(b). For decreasing $\kappa$, we achieve
strong coupling, characterized by $g$ $\gg$ {$\kappa$, $\gamma$}. In this
regime, phonon modes can mediate transitions between the polariton branches,
leading to decoherence in the emission process, as presented in Fig. 12(a)
[22].

##  V Discussion

We introduce a microscopic model for the 1D photonic wire, accounting for
exciton coupling to several phonon branches while incorporating cavity
coupling effects. Our results highlight the key role of Purcell enhancement
and non-Markovian effects in determining the indistinguishability of photons
from solid-state quantum emitters embedded in a geometry of reduced
dimensionality. The indistinguishability is found to be higher than what is
anticipated from models that neglect the cavity coupling effect.

We show that, at the nanoscale, the evolution of QD coherence cannot be
accurately captured by uniquely considering the coupling of the exciton to the
fundamental longitudinal vibration mode. Thus, to fully describe our quantum
emitter interacting with a 1D phonon environment, it becomes crucial to
account for multimode phonon coupling, moving beyond the picture of Markovian
decay (cf. Fig. 3).

In Ref. [8], the pure dephasing rate for one-dimensional structures
$\gamma^{1\mathrm{D}}_{\mathrm{pd}}$ was assumed to be the same as bulk
$\gamma^{3\mathrm{D}}_{\mathrm{pd}}$ due to the significant mismatch between
the waveguide size and the QD. However, it is well known that the 1D coherence
dynamics strongly differs from the 3D bulk case, making one wary of the
validity of the bulk model. Therefore, we deliberately calculated a one-
dimensional pure dephasing rate, where we obtained
$\gamma^{1\mathrm{D}}_{\mathrm{pd}}(R,T)<\gamma^{3\mathrm{D}}_{\mathrm{pd}}(T)$
for a nanowire radius $R$ at temperature $T$, on account of the reduced number
of degrees of freedom. In the limit where $R$ becomes sufficiently big, we can
expect to recover the bulk prediction for the pure dephasing rate, as
suggested by Fig. 6. While important, this task is not fully explored in this
paper, as our primary interest is in geometries of reduced dimensionality.

One of the challenges in performing a 1D pure dephasing calculation is the
need to include a sufficient number of vibration modes $i$ for obtaining
accurate results. The computational complexity of calculating
$\gamma^{1\mathrm{D}}_{\mathrm{pd}}$ scales with $i^{2}$. While this quadratic
growth is manageable for a moderate number of phonon modes, it becomes
cumbersome for larger nanowire radii.

Indistinguishability from a bare wire of radius $R=25$ nm and 100 nm is found
to be much lower than in bulk (see Fig. 8), as results derived from Eq. (13)
must be regarded as upper bound predictions. This follows from the fact that
Eq. (13) is obtained by filtering out incoherent emission from the phonon
satellite peaks [68], which increases $\mathcal{I}$ at the expense of reducing
the photon collection efficiency $\varepsilon$ [7]. However, for scalable
quantum computation with single photons, both $\mathcal{I}$ and $\varepsilon$
must be increased as close as possible to unity simultaneously, making the
filtering unpractical.

For the case where cavity effects are exploited, we compared results from a
weak coupling master equation with those obtained using tensor network
calculations, cf. Fig. 10. While the master equation approach successfully
captures the overall trend of photon indistinguishability as a function of
cavity linewidth at a fixed cavity coupling strength, it underestimates
$\mathcal{I}$ in a 1D wire due to its inherent limitations [66, 53, 67].
Conversely, tensor network calculations, though computationally demanding,
offer more reliable predictions.

We thus show that, when making calculations of low-dimensional systems, one
has to consider the microscopic details of the phonons. For cylindrical
nanowires, approximating the phonons with an Ohmic spectral density ($s=1$)
overestimates the phonon influence. Notably, the nanowire spectral density
leads to long-lived phonon interactions, which, through coherent exchange with
the emitter, result in greater indistinguishability compared to the Ohmic
spectral density, where such long-lived modes and coherent exchange are
absent. More specifically, we attribute the physical interpretation of this
result to two phenomena. First, the assumption of an Ohmic spectral density
leads to complete coherence loss in the long time limit due to the
impossibility of benefiting from coherence trapping, as the dephasing is
effectively Markovian for $s\leq 1$ [69]. Second, the long-lived phonon
interaction allows the emitter to regain coherence lost through an emitted
phonon by reabsorbing the phonon at a later time. This information backflow is
indicative of non-Markovianity, which is reflected by the nonmonotonic
behavior of $\langle\sigma_{x}\rangle$ and
$\mathrm{Re}[C_{\mathrm{ZZ}}(\tau)]$, presented in this paper. We can verify
that for $s=1$ (green curve in Fig. 11), the evolution is monotonic, as
nonmonotonicity requires $s>2$ to enter the non-Markovian regime [69].

Using a numerically exact tensor network, we have demonstrated that our model,
which incorporates multimode coupling, yields significantly better
indistinguishability, providing substantial benefits for SPS applications.
Lastly, it should be emphasized that actual indistinguishability measurements
are sensitive to losses and decoherence that may occur both during the emitter
excitation and emission. We have assumed an instantaneous laser pulse
excitation with 100% state preparation fidelity, thereby initializing the
emitter in the excited state at $\tau=0$. As a result, only decoherence during
the emission process (i.e., via phonon-assisted transitions) was considered. A
detailed modeling of the excitation scheme, which may introduce additional
limitations to the indistinguishability through time-jitter or imperfect
exciton preparation [70, 71, 72, 73], is beyond the scope of this work.

##  VI Conclusions

In summary, by solving the equations for LA phonon modes in a homogeneous
cylindrical nanowire, given by their nontrivial dispersion relations, we have
derived a closed-form expression for the linear and quadratic electron-phonon
coupling, where the latter enabled the development of an analytical expression
for the 1D pure dephasing rate $\gamma^{1\mathrm{D}}_{\mathrm{pd}}$. Assuming
that a QD embedded in a free-standing 1D cylindrical waveguide is exclusively
coupled to the fundamental LA phonon mode, i.e., only considering the first
phonon branch ($\omega_{1,\mathbf{k}}=c_{s}|\rm{\mathbf{k}}|$), Markovian
decay is observed over long timescales [57, 8]. Conversely, considering the QD
to be additionally coupled to higher-order modes, the decay becomes highly
non-Markovian. We underscore the importance of going beyond the assumption of
Markovian decay in a 1D photonic wire, where memory and backaction effects in
the reservoir can increase photon indistinguishability [74]. Our results hint
at the crossover between the physics of bulk media and nanoscale structures,
paving the way for the development of solid-state quantum technologies with
enhanced coherence properties. In future follow-up work, it would be of
interest to investigate phonon modeling in inhomogeneous media [75] and
explore how reduced dimensionality impacts the deformation potential [76],
thereby influencing single-photon characteristics.

###### Acknowledgements.

J.F.N. is thankful to Devashish Pandey and Resul Al for the fruitful
discussions at the early stage of this work. This paper was supported by the
European Union’s Horizon 2020 Research and Innovation Programme under the
Marie Skłodowska-Curie Grant (Agreement No. 861097), and the Danish National
Research Foundation through the NanoPhoton-Center for Nanophotonics, Grant No.
DNRF147. The authors also acknowledge support from the European Research
Council (ERC-CoG “Unity”, Grant No. 865230).

##  Appendix A Linear electron-phonon coupling

The diagonal matrix element for linear electron-phonon coupling via
deformation potential is given by [77, 78]

$M^{0,0}_{\lambda,i,\mathrm{\bf{k}}}=\frac{D_{\lambda}}{\sqrt{N}}\langle\psi_{\lambda,0}|\mathbf{\bm{\nabla}}\cdot\mathbf{u}_{i,\mathrm{\bf{k}}}(\mathbf{r})|\psi_{\lambda,0}\rangle,$
(15)

where the subscript $\lambda$ $\in$ {$e$, $h$} refers to electrons ($e$) and
holes ($h$), $|\psi_{\lambda,0}\rangle$ corresponds to the ground state wave
function, $D_{\lambda}$ is the associated deformation potential,
$\mathbf{u}_{i,\mathrm{\bf{k}}}(\mathbf{r})$ is the normalized displacement
field of the phonon mode $i$, and $N$ corresponds to the total number of ions
in the one-dimensional lattice. In cylindrical coordinates and with the
quantized wire modes, the coupling matrix element becomes

$M^{0,0}_{\lambda,i,\mathrm{\bf{k}}}=D_{\lambda}\frac{R}{2}\sqrt{\frac{\hbar}{V\rho_{m}\
\omega_{i,\mathrm{\bf{k}}}}}\frac{\chi_{i,\mathrm{\bf{k}}}}{\sqrt{\sigma_{i,\mathrm{\bf{k}}}}}(k^{2}+{k_{l}}^{2})\
F_{\lambda,i,\mathrm{\bf{k}}}^{0,0}.$ (16)

Here, $V\rho_{m}=\mathcal{M}N$, where the volume of the cylindrical wire is
given by $V=\pi R^{2}L$, recall that $\rho_{m}$ denotes the mass density,
$\mathcal{M}$ is the ion mass, and $F_{\lambda,i,\mathrm{\bf{k}}}^{0,0}$ is
the form factor

$F_{\lambda,i,\mathrm{\bf{k}}}^{0,0}=\matrixelement{\psi_{\lambda,0}}{J_{0}(k_{l}\sqrt{x^{2}+y^{2}})\
e^{ikz}}{\psi_{\lambda,0}}.$ (17)

It follows from the normalization condition of acoustic-phonon modes in
quantum wires that [54, 79]

$\displaystyle\sigma_{i,\mathrm{\bf{k}}}=\Bigg{\\{}$
$\displaystyle\frac{k_{t}^{4}}{k^{2}}\frac{R^{2}}{2}[J_{1}^{2}(k_{t}R)+J_{0}^{2}(k_{t}R)]-2k_{t}^{2}\chi_{i,\mathrm{\bf{k}}}\frac{R}{k_{l}^{2}-k_{t}^{2}}[k_{l}J_{1}(k_{l}R)J_{0}(k_{t}R)-k_{t}J_{1}(k_{t}R)J_{0}(k_{l}R)]$
$\displaystyle+2k_{t}k_{l}\chi_{i,\mathrm{\bf{k}}}\frac{R}{k_{l}^{2}-k_{t}^{2}}[k_{t}J_{0}(k_{t}R)J_{1}(k_{l}R)-k_{l}J_{0}(k_{l}R)J_{1}(k_{t}R)]+k^{2}\chi_{i,\mathrm{\bf{k}}}^{2}\frac{R^{2}}{2}[J_{1}^{2}(k_{l}R)+J_{0}^{2}(k_{l}R)]$
$\displaystyle+k_{t}^{2}\frac{R^{2}}{2}[J_{1}^{2}(k_{t}R)-J_{0}(k_{t}R)J_{2}(k_{t}R)]+k_{l}^{2}\chi_{i,\mathrm{\bf{k}}}^{2}\frac{R^{2}}{2}[J_{1}^{2}(k_{l}R)-J_{0}(k_{l}R)J_{2}(k_{l}R)]\Bigg{\\}},$
(18)

where $J_{0}$ and $J_{1}$ correspond to the lowest- and first-order Bessel
functions of the first kind, respectively. Also, for the case of free-surface
boundary condition (FSBC), i.e., unrestricted displacements at the boundary
between the cylindrical wire and the vacuum, $\chi_{i,\mathrm{\bf{k}}}$ is
given by

$\chi_{i,\mathrm{\bf{k}}}=\frac{k_{t}(k_{t}^{2}-k^{2})}{2k^{2}k_{l}}\frac{J_{1}(k_{t}R)}{J_{1}(k_{l}R)}.$
(19)

In addition, $k_{l,t}$ is defined as

$k_{l,t}=\sqrt{\frac{\omega_{i,\mathrm{\bf{k}}}^{2}}{v_{l,t}^{2}}-k^{2}},$
(20)

where, as a reminder, $v_{l,t}$ represents the longitudinal and transverse
sound velocities, respectively. The longitudinal waves are coupled axial and
radial modes with the associated quantized wave vectors $k_{l,t}$.

Considering an isotropic harmonic potential for electrons and holes, with the
confinement length $a_{\lambda}$, the ground state wave function is

$\psi_{\lambda,0}(\mathbf{r})=\frac{1}{\pi^{3/4}{a_{\lambda}}^{3/2}}\
\mathrm{exp}\left(-\frac{x^{2}+y^{2}+z^{2}}{2{a_{\lambda}}^{2}}\right),$ (21)

In cylindrical coordinates $(\varrho,\varphi,z)$ the form factor is then

$\displaystyle F_{\lambda,i,\mathbf{k}}^{0,0}=$
$\displaystyle\frac{1}{\pi^{3/2}{a_{\lambda}}^{3}}\
\int_{0}^{2\pi}\mathrm{d}\varphi\int_{0}^{\infty}\varrho\
\mathrm{exp}\left(\frac{-\varrho^{2}}{{a_{\lambda}}^{2}}\right)\
J_{0}(k_{l}\varrho)\ \mathrm{d}\varrho\ $ (22)
$\displaystyle\times\int_{-\infty}^{\infty}\mathrm{exp}\left(\frac{-z^{2}}{{a_{\lambda}}^{2}}+ikz\right)\
\mathrm{d}z,$

where the radial integral results in

$\displaystyle\int_{0}^{\infty}\varrho\
\mathrm{exp}\left(\frac{-\varrho^{2}}{{a_{\lambda}}^{2}}\right)\
J_{0}(k_{l}\varrho)\ \mathrm{d}\varrho=\frac{a_{\lambda}^{2}}{2}\
\exp\left(\frac{-a_{\lambda}^{2}k_{l}^{2}}{4}\right),$ (23)

and the integral over the vertical coordinate $z$ corresponds to

$\int_{-\infty}^{\infty}\mathrm{exp}\left(\frac{-z^{2}}{{a_{\lambda}}^{2}}+ikz\right)\
\mathrm{d}z=a_{\lambda}\sqrt{\pi}\
\mathrm{exp}\left(\frac{-a_{\lambda}^{2}k^{2}}{4}\right).$ (24)

Combining the results above, the form factor
$F_{\lambda,i,\mathrm{\bf{k}}}^{0,0}$ is given by

$\displaystyle
F_{\lambda,i,\mathrm{\bf{k}}}^{0,0}=\exp\left(\frac{-a_{\lambda}^{2}k_{l}^{2}}{4}\right)\
\exp\left(\frac{-a_{\lambda}^{2}k^{2}}{4}\right).$ (25)

Following, making use of the definition from Eq. (20), we are left with

$\displaystyle
F_{\lambda,i,\mathrm{\bf{k}}}^{0,0}=\exp\left(\frac{-a_{\lambda}^{2}\omega_{i,\mathrm{\mathbf{k}}}^{2}}{4v_{l}^{2}}\right).$
(26)

Ultimately, the difference between electron and hole contributions results in
the exciton coupling matrix elements, i.e., $\hbar g_{i,\mathbf{k}}$ =
$M^{0,0}_{e,\mathrm{\bf{k}}}-M^{0,0}_{h,\mathrm{\bf{k}}}$. These results are
displayed in Fig. 2(b) of the main text.

##  Appendix B Quadratic electron-phonon coupling

We begin by defining the coupling constants $f_{i,j,\mathbf{k,k^{\prime}}}$
between phonon modes $i$ and $j$ as

$f_{i,j,\mathbf{k,k^{\prime}}}=\frac{1}{\hbar}\sum_{\lambda=e,h}\sum_{\nu\neq
0}\frac{M_{\lambda,i,\mathrm{\mathbf{k}}}^{0,\nu}M_{\lambda,i,\mathrm{\mathbf{k^{\prime}}}}^{\nu,0}}{\Delta_{\lambda}},$
(27)

with

$\Delta_{\lambda}=E_{\lambda,\nu}-E_{\lambda,0}=\frac{\hbar^{2}}{a_{\lambda}^{2}m_{\lambda}},$
(28)

where $E_{\lambda,\nu}$ is the energy of the $\nu$-th state, and the sum
involves virtual transitions to higher-lying electronic states, whereas
$m_{\lambda}$ is the effective electron/hole mass.

The coupling matrix element between the excitonic ground state and the first
excited state is

$M_{\lambda,i,\mathrm{\mathbf{k}}}^{0,\nu}=\frac{D_{\lambda}}{\sqrt{N}}\langle\psi_{\lambda,0}|\mathbf{\bm{\nabla}}\cdot\mathbf{u}_{i,\mathbf{k}}(\mathbf{r})|\psi_{\lambda,\nu}\rangle,$
(29)

where the superscript $\nu$ $\in$ {0,1,…} indicates the bound states in the QD
confinement potential. Analogously to (16), we have

$M_{\lambda,i,\mathrm{\mathbf{k}}}^{0,\nu}=D_{\lambda}\frac{R}{2}\sqrt{\frac{\hbar}{V\rho_{m}\
\omega_{i,\mathrm{\bf{k}}}}}\frac{\chi_{i,\mathrm{\bf{k}}}}{\sqrt{\sigma_{i,\mathrm{\bf{k}}}}}(k^{2}+{k_{l}}^{2})\
F_{\lambda,i,\mathrm{\bf{k}}}^{0,\nu},$ (30)

with the form factor between two different electronic states

$F_{\lambda,i,\mathrm{\bf{k}}}^{0,\nu}=\matrixelement{\psi_{\lambda,0}}{J_{0}(k_{l}\sqrt{x^{2}+y^{2}})e^{ikz}}{\
\psi_{\lambda,\nu}}.$ (31)

Thereafter, we restrict the sum on $\nu$ to the first excited state [18]. For
a 3D isotropic harmonic oscillator, the latter is the 3-fold degenerate state
$\nu=(1,m)$, with $m\in\\{-1,0,+1\\}$, and the wave functions correspond to

$\psi_{\lambda,(1,m)}(\mathbf{r})=\frac{2\sqrt{2}}{\sqrt{3}\pi^{1/4}{a_{\lambda}}^{5/2}}\
|\mathbf{r}|\
\mathrm{exp}\left(\frac{-|\mathbf{r}|^{2}}{2{a_{\lambda}}^{2}}\right)\
Y^{m}_{l=1}(\theta,\phi),$ (32)

where the spherical harmonics for $l=1$ read

$Y_{l=1}^{m=-1}(\theta,\phi)=\frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{-i\phi}\sin(\theta)=\frac{1}{2}\sqrt{\frac{3}{2\pi}}\frac{x-iy}{r},$
(33)
$Y_{l=1}^{m=0}(\theta,\phi)=\frac{1}{2}\sqrt{\frac{3}{\pi}}\cos(\theta)=\frac{1}{2}\sqrt{\frac{3}{\pi}}\frac{z}{r},$
(34)
$Y_{l=1}^{m=+1}(\theta,\phi)=-\frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{i\phi}\sin(\theta)=-\frac{1}{2}\sqrt{\frac{3}{2\pi}}\frac{x+iy}{r},$
(35)

with $r=|\mathrm{\mathbf{r}}|=\sqrt{x^{2}+y^{2}+z^{2}}$. One can verify that
only $F_{\lambda,i,\mathrm{\bf{k}}}^{0,(1,m=0)}$ has a non-vanishing
contribution given by

$\displaystyle F_{\lambda,i,\mathrm{\bf{k}}}^{0,(1,m=0)}=$
$\displaystyle\frac{\sqrt{2}}{\pi^{3/2}{a_{\lambda}}^{4}}\int_{0}^{\infty}\varrho\,\exp\left(-\frac{\varrho^{2}}{{a_{\lambda}}^{2}}\right)\
J_{0}(k_{l}\varrho)\ \differential{\varrho}$ (36)
$\displaystyle\times\int_{-\infty}^{\infty}\ z\
\mathrm{exp}\left(\frac{-z^{2}}{{a_{\lambda}}^{2}}+ikz\right)\
\mathrm{d}z\int_{0}^{2\pi}\mathrm{d}\varphi.$

The radial integral remains precisely the same as given by Eq. (23), whereas
the integral over the vertical coordinate $z$ reads

$\int_{-\infty}^{\infty}z\
\mathrm{exp}\left(\frac{-z^{2}}{{a_{\lambda}}^{2}}+ikz\right)\
\mathrm{d}z=\frac{i}{2}\sqrt{\pi}\ a_{\lambda}^{3}\ k\
\mathrm{exp}\left(\frac{-a_{\lambda}^{2}k^{2}}{4}\right).$ (37)

The form factor $F_{\lambda,i,\mathrm{\bf{k}}}^{0,(1,m=0)}$ is then

$F_{\lambda,i,\mathrm{\bf{k}}}^{0,(1,m=0)}=\frac{i\sqrt{2}}{2}a_{\lambda}k\
F_{\lambda,i,\mathrm{\bf{k}}}^{0,0},$ (38)

with $F_{\lambda,i,\mathrm{\bf{k}}}^{0,0}$ as defined in (26).

##  Appendix C Bare wire calculation

Formally, one can define the emission power spectrum

$\displaystyle S(\omega)$
$\displaystyle=\langle\sigma^{{\dagger}}(\omega)\sigma(\omega)\rangle$ (39)
$\displaystyle=\int_{-\infty}^{\infty}\mathrm{d}t\int_{-\infty}^{\infty}\mathrm{d}\tau\
e^{-i\omega\tau}\ \langle\sigma^{{\dagger}}(t+\tau)\sigma(t)\rangle.$

We can equivalently write

$S(\omega)=2\
\mathrm{Re}\left[\int_{0}^{\infty}\mathrm{d}t\int_{0}^{\infty}\mathrm{d}\tau\
e^{-i\omega\tau}\ \langle\sigma^{{\dagger}}(t+\tau)\sigma(t)\rangle\right],$
(40)

where

$\langle\sigma^{{\dagger}}(t+\tau)\sigma(t)\rangle=g^{(1)}(t,\tau)=\Delta(\tau)\
\tilde{g}^{(1)}(t,\tau),$ (41)

which corresponds to the first-order dipole correlation function. Plugging
this definition in Eq. (40), we obtain

$S(\omega)=2\
\mathrm{Re}\left[\int_{0}^{\infty}\mathrm{d}t\int_{0}^{\infty}\mathrm{d}\tau\
\Delta(\tau)\ \tilde{g}^{(1)}(t,\tau)e^{-i\omega\tau}\right],$ (42)

where $\Delta(\tau)=e^{\Phi(\tau)}$ is the free phonon correlation function.
Using the quantum regression theorem, while assuming that the quantum emitter
is initially excited at time $t=0$, the correlation function in the polaron
frame can be written as

$\tilde{g}^{(1)}(t,\tau)=e^{-\Gamma_{\mathrm{rad}}t}e^{-\frac{1}{2}(\Gamma_{\mathrm{rad}}+\gamma^{1\mathrm{D}}_{\mathrm{pd}})\tau}e^{i\omega_{0}\tau},$
(43)

with $\Gamma_{\mathrm{rad}}$ the rate of spontaneous emission (radiative rate)
of the QD exciton. Then, the emission spectrum takes the form

$S(\omega)=\frac{2}{\Gamma_{\mathrm{rad}}}\
\mathrm{Re}\left[\int_{0}^{\infty}\mathrm{d}\tau\
e^{\Phi(\tau)}e^{-\frac{1}{2}(\Gamma_{\mathrm{rad}}+\gamma^{1\mathrm{D}}_{\mathrm{pd}})\tau}e^{-i(\omega-\omega_{0})\tau}\right],$
(44)

as given by Eq. (11) in the main text. Additionally, we can conveniently
define $G(\omega)$ as

$G(\omega)=\int_{0}^{\infty}\mathrm{d}\tau\
e^{\Phi(\tau)}e^{-\frac{1}{2}(\Gamma_{\mathrm{rad}}+\gamma^{1\mathrm{D}}_{\mathrm{pd}})\tau}e^{-i(\omega-\omega_{0})\tau}.$
(45)

Ultimately, we can compactly write the emission power as

$S(\omega)=\frac{2}{\Gamma_{\mathrm{rad}}}\ G_{\mathrm{R}}(\omega),$ (46)

where $G_{\mathrm{R}}(\omega)$ stands for the real part of $G(\omega)$.

We can write the bare spectrum without loss of generality as [7]

$S(\omega,\nu)=F(\omega,\nu)+F(\nu,\omega)^{\ast},$ (47)

with

$F(\omega,\nu)=\int_{0}^{\infty}\mathrm{d}t\int_{0}^{\infty}\mathrm{d}\tau\
e^{i(\nu-\omega)t}e^{-i\omega\tau}\langle\sigma^{{\dagger}}(t+\tau)\sigma(t)\rangle.$
(48)

Here, however, we are not allowed to separate the contribution from the zero-
phonon line and the phonon sidebands due to the assumed dimensionality of the
phonon bath ($s=1$). Once again, making use of Eq. (41) and that
$\Delta(\tau)=e^{\Phi(\tau)}$, we obtain

$\displaystyle F(\omega,\nu)=$ $\displaystyle\ \int_{0}^{\infty}\mathrm{d}t\
e^{i(\nu-\omega)t}e^{-\Gamma_{\mathrm{rad}}t}$ (49) $\displaystyle\hskip
1.99997pt\times\int_{0}^{\infty}\mathrm{d}\tau\
e^{\Phi(\tau)}e^{-\frac{1}{2}(\Gamma_{\mathrm{rad}}+\gamma^{1\mathrm{D}}_{\mathrm{pd}})\tau}e^{-i(\omega-\omega_{0})\tau}.$

The first integral above is trivial. By resorting to Eq. (45), we can now
conveniently rewrite our expression as

$\displaystyle F(\omega,\nu)=\frac{1}{\Gamma_{\mathrm{rad}}-i(\nu-\omega)}\
G(\omega).$ (50)

Next, by taking the complex conjugate of both sides of Eq. (50) and
interchanging the variables $\omega$ and $\nu$, we arrive at

$F(\nu,\omega)^{\ast}=\frac{1}{\Gamma_{\mathrm{rad}}-i(\nu-\omega)}\
G(\omega)^{\ast}.$ (51)

Plugging Eqs. (50) and (51) into Eq. (47), we are left with

$\displaystyle S(\omega,\nu)$
$\displaystyle=\frac{G(\omega)+G(\omega)^{\ast}}{\Gamma_{\mathrm{rad}}-i(\nu-\omega)}=\frac{2G_{\mathrm{R}}(\omega)}{\Gamma_{\mathrm{rad}}-i(\nu-\omega)},$
(52)

and by setting $\omega=\nu$

$S(\omega,\omega)=\frac{2}{\Gamma_{\mathrm{rad}}}\ G_{\mathrm{R}}(\omega).$
(53)

Comparing Eq. (46) and Eq. (53), one can retrieve the emission power spectrum

$S(\omega,\omega)=S(\omega).$ (54)

Now that we have unequivocally introduced the variables above, we can
calculate the indistinguishability $\mathcal{I}$ of two-photon interference in
frequency space. To do so, we employ the definition from the two-color
spectrum [7]

$\mathcal{I}=P_{\mathrm{D}}^{-2}\int_{-\infty}^{\infty}\mathrm{d}\omega\int_{-\infty}^{\infty}\mathrm{d}\nu\
|S(\omega,\nu)|^{2},$ (55)

where the frequency-integrated spectral power $P_{\mathrm{D}}$ reads

$P_{\mathrm{D}}=\int_{-\infty}^{\infty}\mathrm{d}\omega\ S(\omega,\omega).$
(56)

##  Appendix D Cavity quantum electrodynamic calculation

In the main text, we employ a weak master equation for comparison with the
results obtained from the tensor network. In this Appendix, we provide details
on the utilized master equation. We derive a second-order master equation for
the density operator $\rho$ of the emitter-cavity system [25]. The system
Hamiltonian is given in Eq. (14), which includes a temperature-dependent pure
dephasing rate due to the quadratic phonon interaction
$\gamma^{1\mathrm{D}}_{\mathrm{pd}}(T)$ and a cavity decay rate $\kappa$
leading to

$\dot{\rho}=-\frac{i}{\hbar}[\mathcal{H}_{S},\rho]+2\gamma^{1\mathrm{D}}_{\mathrm{pd}}(T)\mathcal{L}_{\sigma^{\dagger}\sigma}[\rho]+\kappa\mathcal{L}_{a}[\rho]+\mathcal{K}[\rho],$
(57)

where $\mathcal{L}_{O}[\rho]=O\rho
O^{\dagger}-\frac{1}{2}\anticommutator{O^{\dagger}O}{\rho}$ is the Lindblad
dissipator and $\mathcal{K}[\rho]$ corresponds to the phononic dissipator
given by [53, 25]

$\displaystyle\mathcal{K}[\rho]=$ $\displaystyle\
-\int_{0}^{\infty}\mathrm{d}\tau\
\left(C_{\mathrm{ZZ}}(\tau)\left[O_{\mathrm{Z}},\tilde{O}_{\mathrm{Z}}(-\tau)\rho_{\mathrm{S}}(t)\right]\right.$
(58) $\displaystyle\hskip
3.00003pt\left.+C_{\mathrm{ZZ}}(-\tau)\left[\rho_{\mathrm{S}}(t)\tilde{O}_{\mathrm{Z}}(-\tau),O_{\mathrm{Z}}\right]\right),$

with $O_{\mathrm{Z}}=\sigma^{\dagger}\sigma$ and
$\tilde{O}_{\mathrm{Z}}(\tau)=\mathrm{e}^{i\mathcal{H}_{S}\tau/\hbar}O_{\mathrm{Z}}\mathrm{e}^{-i\mathcal{H}_{S}\tau/\hbar}$.
The environmental correlation function is described by [53, 25]

$C_{\mathrm{ZZ}}(\tau)=\int_{0}^{\infty}\mathrm{d}\omega\
J(\omega)\left[\frac{\cos(\omega\tau)}{\tanh(\beta\hbar\omega/2)}-i\sin(\omega\tau)\right],$
(59)

where $\beta=(k_{\mathrm{B}}T)^{-1}$, with $k_{\mathrm{B}}$ the Boltzmann
constant. The indistinguishability $\mathcal{I}$ is calculated as in Appendix
C by replacing the two-time emitter correlation with the cavity two-time
correlation $\expectationvalue{a^{\dagger}(t+\tau)a(t)}$, considering an
initially excited emitter.

##  Appendix E Tensor network

In this Appendix, we provide details on the accuracy of the tensor network.
For details on the theory and implementation of the network used, we refer to
Ref. [50]. In the tensor network, three numerical parameters determine the
accuracy: the time step $\Delta t$, the Singular Value Truncation threshold
$\epsilon$ of the Matrix Product Operator compression, and the memory time
$\tau_{\mathrm{mem}}$. The memory time $\tau_{\mathrm{mem}}$ should be chosen
to capture all environment features. In reality, it means that the real part
of the environmental correlation function $\mathrm{Re}[C_{\mathrm{ZZ}}(\tau)]$
is zero after $\tau_{\mathrm{mem}}$. The environmental correlation function is
given as in Eq. (59) [50]. In Fig. 11 of the main text, we show the
correlation functions of the four phonon environments plotted over the range
of their memory time $\tau_{\mathrm{mem}}$, which are
$\tau_{\mathrm{mem}}=\\{7\ \mathrm{ps},14\ \mathrm{ps},80\ \mathrm{ps},12\
\mathrm{ps}$} for $s=\\{1,3\\}$, nanowire radius $R=25\ \mathrm{nm}$, and
$R=100\ \mathrm{nm}$, respectively.

For $R=100\ \mathrm{nm}$, we have made an exception since we plot it over
$\tau=100\ \mathrm{ps}$, but the cutoff is $\tau_{\mathrm{mem}}=12\
\mathrm{ps}$ as indicated by the vertical line. To estimate the impact of the
truncation parameter $\epsilon$, we do each calculation twice using first a
larger epsilon $\epsilon_{\mathrm{R}}$ and secondly a smaller
$\epsilon_{\mathrm{F}}$, where the subscripts denote the result of the rough
(R) or fine (F) calculation. Comparing the two indistinguishabilities, we thus
estimate the relative error as
$\delta_{\epsilon}=\absolutevalue{\mathcal{I}_{\mathrm{F}}-\mathcal{I}_{\mathrm{R}}}/\mathcal{I}_{\mathrm{F}}$.
For $s=\\{1,3\\}$ and the nanowire radius $R={100\ \mathrm{nm}}$, we use
$\epsilon_{\mathrm{R}}=10^{-7}$ and $\epsilon_{\mathrm{F}}=10^{-8}$, whereas
for $R=25\ \mathrm{nm}$ we use $\epsilon_{\mathrm{R}}=10^{-6}$ and
$\epsilon_{\mathrm{F}}=5\times 10^{-7}$.

When calculating the nanowire spectral density, it should be noted that we
have employed the numerical trick of
$J(\omega)=\sum_{i,\mathrm{\mathbf{k}}}|g_{i,\mathrm{\mathbf{k}}}|^{2}\delta(\omega-\nobreak\omega_{i,\mathrm{\mathbf{k}}})\approx\sum_{i,\mathrm{\mathbf{k}}}|g_{i,\mathrm{\mathbf{k}}}|^{2}(\sqrt{\pi}\sigma)^{-1}\exp[(\omega-\omega_{i,\mathrm{\mathbf{k}}})^{2}/\sigma^{2}]$,
the delta function is thus approximated as a Gaussian with a finite width
$\sigma$. We do this to smoothen the spectral density. This is much like the
method employed in Ref. [37]. The smoothening is done to make the phonon
coherence finite, thus rendering the calculations with the numerical tensor
network feasible. However, unlike in [37], $\sigma$ is not a physical
parameter but rather just a numerical trick.

As depicted in Fig. 8 of the main text, we confirmed that this smoothening
still allows for an accurate description of the nanowire phonons by comparing
the bare wire calculations with the tensor network [47, 46, 45, 50, 49, 48]
and obtaining very good agreement between the two methods. The deviation for
$R=100\ \mathrm{nm}$ at $T=$ $20\text{\,}\mathrm{K}$ stems from the finite
memory of the tensor network and not the smoothening. The finite memory time
of the tensor network makes it only capture the bulklike features of the
spectral density, whereas the longer-lived thin peaks in the spectral density
are not well described.

We estimate the error due to the smoothening of the spectral density in a
similar fashion as for the truncation threshold $\epsilon$ by considering
$\sigma_{\mathrm{R}}=0.1\ \mathrm{THz}$ and $\sigma_{\mathrm{F}}=0.05\
\mathrm{THz}$, and estimating the relative error $\delta_{\mathrm{smooth}}$.
For the nanowire simulations, we thus estimate the tensor network error as
$\delta_{\mathrm{tot}}=\sqrt{\delta_{\epsilon}^{2}+\delta_{\mathrm{smooth}}^{2}}$,
whereas for $s=\\{1,3\\}$ it is simply given by
$\delta_{\mathrm{tot}}=\delta_{\epsilon}$. These errors are shown as the error
bars in Figs. 8 and 10 with a size defined by the product
$\mathcal{I}\delta_{\mathrm{tot}}$. Finally, estimating the error due to the
finite discretization $\Delta t$ is more challenging. Changing $\Delta t$ also
changes the requirement for convergence on the truncation cutoff $\epsilon$
[49]. We thus assured in an initial study that we observed no significant
changes for $\Delta t=0.4\ \mathrm{ps}$ when a small enough $\epsilon$ was
chosen.

## References

  * [1] C. Couteau, S. Barz, T. Durt, T. Gerrits, J. Huwer, R. Prevedel, J. Rarity, A. Shields, and G. Weihs, “Applications of single photons to quantum communication and computing”, Nat. Rev. Phys. $\mathbf{5}$, 326–338 (2023).
  * [2] M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited Review Article: Single-photon sources and detectors”, Rev. Sci. Instrum. $\mathbf{82}$ (7), 071101 (2011).
  * [3] M. Tanida, R. Okamoto, and S. Takeuchi, “Highly indistinguishable heralded single-photon sources using parametric down conversion”, Opt. Express $\mathbf{20}$, 15275-15285 (2012).
  * [4] N. Somaschi, V. Giesz, L. De Santis, J. C. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Antón, J. Demory, C. Gómez, I. Sagnes, N. D. Lanzillotti-Kimura, A. Lemaître, A. Auffèves, A. G. White, L. Lanco, and P. Senellart, “Near-optimal single-photon sources in the solid state”, Nat. Photonics $\mathbf{10}$, 340–345 (2016).
  * [5] P. Senellart, G. Solomon, and A. White, “High-performance semiconductor quantum-dot single-photon sources”, Nat. Nanotechnol. $\mathbf{12}$, 1026–1039 (2017).
  * [6] T. Heindel, J.-H. Kim, N. Gregersen, A. Rastelli, and S. Reitzenstein, “Quantum dots for photonic quantum information technology”, Adv. Opt. Photon. $\mathbf{15}$, 613-738 (2023).
  * [7] J. Iles-Smith, D. P. S. McCutcheon, A. Nazir, and J. Mørk, “Phonon scattering inhibits simultaneous near-unity efficiency and indistinguishability in semiconductor single-photon sources”, Nat. Photonics $\mathbf{11}$, 521–526 (2017).
  * [8] P. Tighineanu, C. L. Dreeßen, C. Flindt, P. Lodahl, and A. S. Sørensen, “Phonon decoherence of quantum dots in photonic structures: Broadening of the zero-phonon line and the role of dimensionality”, Phys. Rev. Lett. $\mathbf{120}$, 257401 (2018).
  * [9] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative two-state system”, Rev. Mod. Phys. $\mathbf{59}$, 1 (1987).
  * [10] V. Hizhnyakov, “Zero-phonon lines of systems with different dimensions and unconventional vibronic interactions”, J. Phys.: Condens. Matter $\mathbf{24}$, 104011 (2012).
  * [11] Y. Chassagneux, A. Jeantet, T. Claude, and C. Voisin, “Effect of phonon-bath dimensionality on the spectral tuning of single-photon emitters in the Purcell regime”, Phys. Rev. B $\mathbf{97}$, 205124 (2018).
  * [12] M. Reitz, C. Sommer, B. Gurlek, V. Sandoghdar, D. Martin-Cano, and C. Genes, “Molecule-photon interactions in phononic environments”, Phys. Rev. Res. $\mathbf{2}$, 033270 (2020).
  * [13] R. Doll, P. Hänggi, S. Kohler, and M. Wubs, “Fast initial qubit dephasing and the influence of substrate dimensions on error correction rates”, Eur. Phys. J. B $\mathbf{68}$, 523–527 (2009).
  * [14] L. Besombes, K. Kheng, L. Marsal, and H. Mariette, “Acoustic phonon broadening mechanism in single quantum dot emission”, Phys. Rev. B $\mathbf{63}$, 155307 (2001).
  * [15] B. Krummheuer, V. M. Axt, and T. Kuhn, “Theory of pure dephasing and the resulting absorption line shape in semiconductor quantum dots”, Phys. Rev. B $\mathbf{65}$, 195313 (2002).
  * [16] E. A. Muljarov and R. Zimmermann, “Dephasing in quantum dots: quadratic coupling to acoustic phonons”, Phys. Rev. Lett. $\mathbf{93}$, 237401 (2004).
  * [17] P. Kær, P. Lodahl, A.-P. Jauho, and J. Mørk, “Microscopic theory of indistinguishable single-photon emission from a quantum dot coupled to a cavity: The role of non-Markovian phonon-induced decoherence”, Phys. Rev. B $\mathbf{87}$, 081308(R) (2013).
  * [18] A. Reigue, J. Iles-Smith, F. Lux, L. Monniello, M. Bernard, F. Margaillan, A. Lemaître, A. Martinez, D. P. S. McCutcheon, J. Mørk, R. Hostein, and V. Voliotis, “Probing electron-phonon interaction through two-photon interference in resonantly driven semiconductor quantum dots”, Phys. Rev. Lett. $\mathbf{118}$, 233602 (2017).
  * [19] T. Grange, N. Somaschi, C. Antón, L. De Santis, G. Coppola, V. Giesz, A. Lemaître, I. Sagnes, A. Auffèves, and P. Senellart, “Reducing phonon-induced decoherence in solid-state single-photon sources with cavity quantum electrodynamics”, Phys. Rev. Lett. $\mathbf{118}$, 253602 (2017).
  * [20] S. Gerhardt, J. Iles-Smith, D. P. S. McCutcheon, Y.-M. He, S. Unsleber, S. Betzold, N. Gregersen, J. Mørk, S. Höfling, and C. Schneider, “Intrinsic and environmental effects on the interference properties of a high-performance quantum dot single-photon source”, Phys. Rev. B $\mathbf{97}$, 195432 (2018).
  * [21] A. Morreau and E. A. Muljarov, “Phonon-induced dephasing in quantum-dot–cavity QED”, Phys. Rev. B $\mathbf{100}$, 115309 (2019).
  * [22] E. V. Denning, J. Iles-Smith, N. Gregersen, and J. Mørk, “Phonon effects in quantum dot single-photon sources”, Opt. Mater. Express $\mathbf{10}$, 222–239 (2020).
  * [23] B.-Y. Wang, E. V. Denning, U. M. Gür, C.-Y. Lu, and N. Gregersen, “Micropillar single-photon source design for simultaneous near-unity efficiency and indistinguishability”, Phys. Rev. B $\mathbf{102}$, 125301 (2020).
  * [24] E. V. Denning, M. Bundgaard-Nielsen, and J. Mørk, “Optical signatures of electron-phonon decoupling due to strong light-matter interactions”, Phys. Rev. B $\mathbf{102}$, 235303 (2020).
  * [25] M. Bundgaard-Nielsen, J. Mørk, and E. V. Denning, “Non-Markovian perturbation theories for phonon effects in strong-coupling cavity quantum electrodynamics”, Phys. Rev. B $\mathbf{103}$, 235309 (2021).
  * [26] B. Gaál, M. A. Jacobsen, L. Vannucci, J. Claudon, J.-M. Gérard, and N. Gregersen, “Near-unity efficiency and photon indistinguishability for the ‘hourglass’ single-photon source using suppression of the background emission”, Appl. Phys. Lett. $\mathbf{121}$, 170501 (2022).
  * [27] A. J. Brash and J. Iles-Smith, “Nanocavity enhanced photon coherence of solid-state quantum emitters operating up to 30 K”, Mater. Quantum. Technol. $\mathbf{3}$, 045001 (2023).
  * [28] D. Pandey and M. Wubs, “Collective photon emission in solid state environments: Concatenating non-Markovian and Markovian dynamics”, Phys. Rev. Res. $\mathbf{6}$, 033044 (2024).
  * [29] X. Ding, Y. He, Z.-C. Duan, N. Gregersen, M.-C. Chen, S. Unsleber, S. Maier, C. Schneider, M. Kamp, S. Höfling, C.-Y. Lu, and J.-W. Pan, “On-demand single photons with high extraction efficiency and near-unity indistinguishability from a resonantly driven quantum dot in a micropillar”, Phys. Rev. Lett. $\mathbf{116}$, 020401 (2016).
  * [30] J. Claudon, J. Bleuse, N. S. Malik, M. Bazin, P. Jaffrennou, N. Gregersen, C. Sauvan, P. Lalanne, and J.-M. Gérard, “A highly efficient single-photon source based on a quantum dot in a photonic nanowire”, Nat. Photonics $\mathbf{4}$, 174-177 (2010).
  * [31] R. Uppu, F. T. Pedersen, Y. Wang, C. T. Olesen, C. Papon, X. Zhou, L. Midolo, S. Scholz, A. D. Wieck, A. Ludwig, and P. Lodahl, “Scalable integrated single-photon source”, Sci. Adv. $\mathbf{6}$, eabc8268 (2020).
  * [32] C. Galland, A. Högele, H. E. Türeci, and A. Imamoğlu, “Non-Markovian decoherence of localized nanotube excitons by acoustic phonons”, Phys. Rev. Lett. $\mathbf{101}$, 067402 (2008).
  * [33] E. Malic and A. Knorr, Graphene and Carbon Nanotubes: Ultrafast Relaxation Dynamics and Optics (John Wiley & Sons Ltd, 2013).
  * [34] P. Khatri, I. J. Luxmoore, and A. J. Ramsay, “Phonon sidebands of color centers in hexagonal boron nitride”, Phys. Rev. B $\mathbf{100}$, 125305 (2019).
  * [35] E. V. Denning, M. Wubs, N. Stenger, J. Mørk, and P. T. Kristensen, “Quantum theory of two-dimensional materials coupled to electromagnetic resonators”, Phys. Rev. B $\mathbf{105}$, 085306 (2022).
  * [36] J. A. Preuss, D. Groll, R. Schmidt, T. Hahn, P. Machnikowski, R. Bratschitsch, T. Kuhn, S. M. de Vasconcellos, and D. Wigger, “Resonant and phonon-assisted ultrafast coherent control of a single hBN color center”, Optica $\mathbf{9}$, 522-531 (2022).
  * [37] M. K. Svendsen, S. Ali, N. Stenger, K. S. Thygesen, and J. Iles-Smith, “Signatures of non-Markovianity in cavity QED with color centers in two-dimensional materials”, Phys. Rev. Res. $\mathbf{5}$, L032037 (2023).
  * [38] V. N. Mitryakhin, A. Steinhoff, J.-C. Drawer, H. Shan, M. Florian, L. Lackner, B. Han, F. Eilenberger, S. A. Tongay, K. Watanabe, T. Taniguchi, C. Antón-Solanas, A. Predojević, C. Gies, M. Esmann, and C. Schneider, “Engineering the impact of phonon dephasing on the coherence of a WSe2 single-photon source via cavity quantum electrodynamics”, Phys. Rev. Lett. $\mathbf{132}$, 206903 (2024).
  * [39] C. Piccinini, A. Paralikis, J. F. Neto, A. A. Madigawa, P. Wyborski, V. Remesh, L. Vannucci, N. Gregersen, and B. Munkhbat, “High-purity and stable single-photon emission in bilayer WSe2 via phonon-assisted excitation”, arXiv.2406.07097.
  * [40] L. Vannucci, J. F. Neto, C. Piccinini, A. Paralikis, N. Gregersen, and B. Munkhbat, “Single-photon emitters in WSe2: Critical role of phonons on excitation schemes and indistinguishability”, Phys. Rev. B $\mathbf{109}$, 245304 (2024).
  * [41] J. Kikas, A. Suisalu, V. Zazubovich, and P. Vois, “Anomalous impurity spectra in low‐temperature glasses”, J. Chem. Phys. $\mathbf{104}$, 4434–4440 (1996).
  * [42] D. Reichman, R. J. Silbey, and A. Suárez, “On the nonperturbative theory of pure dephasing in condensed phases at low temperatures”, J. Chem. Phys. $\mathbf{105}$, 10500–10506 (1996).
  * [43] J. Liu and D. Segal, “Dissipation engineering of nonreciprocal quantum dot circuits: An input-output approach”, Phys. Rev. B $\mathbf{102}$, 125416 (2020).
  * [44] P. Laferrière, A. Yin, E. Yeung, L. Kusmic, M. Korkusinski, P. Rasekh, D. B. Northeast, S. Haffouz, J. Lapointe, P. J. Poole, R. L. Williams, and D. Dalacu, “Approaching transform-limited photons from nanowire quantum dots using excitation above the band gap”, Phys. Rev. B $\mathbf{107}$, 155422 (2023).
  * [45] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, “Non-Markovian quantum processes: Complete framework and efficient characterization”, Phys. Rev. A $\mathbf{97}$, 012127 (2018).
  * [46] A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett, “Efficient non-Markovian quantum dynamics using time-evolving matrix product operators”, Nat. Commun. $\mathbf{9}$, 3322 (2018).
  * [47] M. R. Jørgensen and F. A. Pollock, “Exploiting the causal tensor network structure of quantum processes to efficiently simulate non-Markovian path integrals”, Phys. Rev. Lett. $\mathbf{123}$, 240602 (2019).
  * [48] The TEMPO collaboration, OQuPy: A Python 3 package to efficiently compute non-Markovian open quantum systems (2020).
  * [49] M. Cygorek, M. Cosacchi, A. Vagov, V. M. Axt, B. W. Lovett, J. Keeling, and E. M. Gauger, “Simulation of open quantum systems by automated compression of arbitrary environments”, Nat. Phys. $\mathbf{18}$, 662–668 (2022).
  * [50] M. Cygorek, J. Keeling, B. W. Lovett, and E. M. Gauger, “Sublinear scaling in non-Markovian open quantum systems simulations”, Phys. Rev. X $\mathbf{14}$, 011010 (2024).
  * [51] S. Yu, K. W. Kim, M. A. Stroscio, and G. J. Iafrate, “Electron-acoustic-phonon scattering rates in cylindrical quantum wires”, Phys. Rev. B $\mathbf{51}$, 4695(R) (1995).
  * [52] D. Bimberg, Semiconductor nanostructures (Springer, 2008).
  * [53] A. Nazir and D. P. S. McCutcheon, “Modelling exciton–phonon interactions in optically driven quantum dots”, J. Phys. Condens. Matter $\mathbf{28}$, 103002 (2016).
  * [54] M. A. Stroscio, K. W. Kim, S. Yu, and A. Ballato, “Quantized acoustic phonon modes in quantum wires and quantum dots”, J. Appl. Phys. $\mathbf{76}$, 4670–4675 (1994).
  * [55] S. V. Kuznetsov, “Pochhammer-Chree waves: Spectral analysis of axially symmetric modes”, J. Appl. Mech. Eng. $\mathbf{7}$, No. 1 (2018).
  * [56] D. Royer and T. Valier-Brasier, Elastic Waves in Solids 1: Propagation (ISTE Ltd, 2022).
  * [57] G. Lindwall, A. Wacker, C. Weber, and A. Knorr, “Zero-phonon linewidth and phonon satellites in the optical absorption of nanowire-based quantum dots”, Phys. Rev. Lett. $\mathbf{99}$, 087401 (2007).
  * [58] P. Stepanov, M. Elzo-Aizarna, J. Bleuse, N. S. Malik, Y. Curé, E. Gautier, V. Favre-Nicolin, J.-M. Gérard, and J. Claudon, “Large and uniform optical emission shifts in quantum dots strained along their growth axis”, Nano Lett. $\mathbf{16}(5)$, 3215–3220 (2016).
  * [59] M. Munsch, A. V. Kuhlmann, D. Cadeddu, J.-M. Gérard, J. Claudon, M. Poggio, and R. J. Warburton, “Resonant driving of a single photon emitter embedded in a mechanical oscillator”, Nat. Commun. $\mathbf{8}$, 76 (2017).
  * [60] A. Artioli, S. Kotal, N. Gregersen, P. Verlot, J.-M. Gérard, and J. Claudon, “Design of quantum dot-nanowire single-photon sources that are immune to thermomechanical decoherence”, Phys. Rev. Lett. $\mathbf{123}$, 247403 (2019).
  * [61] B. Krummheuer, V. M. Axt, T. Kuhn, I. D’Amico, and F. Rossi, “Pure dephasing and phonon dynamics in GaAs- and GaN-based quantum dot structures: Interplay between material parameters and geometry”, Phys. Rev. B $\mathbf{71}$, 235329 (2005).
  * [62] A. Kiraz, M. Atatüre, and A. Imamoğlu, “Quantum-dot single-photon sources: Prospects for applications in linear optics quantum-information processing”, Phys. Rev. A $\mathbf{69}$, 032305 (2004).
  * [63] T. Grange, G. Hornecker, D. Hunger, J.-P. Poizat, J.-M. Gérard, P. Senellart, and A. Auffèves, “Cavity-funneled generation of indistinguishable single photons from strongly dissipative quantum emitters”, Phys. Rev. Lett. $\mathbf{114}$, 193601 (2015).
  * [64] I. Wilson-Rae and A. Imamoğlu, “Quantum dot cavity-QED in the presence of strong electron-phonon interactions”, Phys. Rev. B $\mathbf{65}$, 235311 (2002).
  * [65] D. P. S. McCutcheon, N. S. Dattani, E. M. Gauger, B. W. Lovett, and A. Nazir, “A general approach to quantum dynamics using a variational master equation: Application to phonon-damped Rabi rotations in quantum dots”, Phys. Rev. B $\mathbf{84}$, 119903 (2011).
  * [66] M. Cosacchi, T. Seidelmann, M. Cygorek, A. Vagov, D. E. Reiter, and V. M. Axt, “Accuracy of the quantum regression theorem for photon emission from a quantum dot”, Phys. Rev. Lett. 127, 100402 (2021).
  * [67] D. P. S. McCutcheon and A. Nazir, “Quantum dot Rabi rotations beyond the weak exciton–phonon coupling regime”, New J. Phys. 12, 113042 (2010).
  * [68] C. L. Phillips, A. J. Brash, D. P. S. McCutcheon, J. Iles-Smith, E. Clarke, B. Royall, M. S. Skolnick, A. M. Fox, and A. Nazir, “Photon statistics of filtered resonance fluorescence”, Phys. Rev. Lett. $\mathbf{125}$, 043603 (2020).
  * [69] C. Addis, G. Brebner, P. Haikka, and S. Maniscalco, “Coherence trapping and information backflow in dephasing qubits”, Phys. Rev. A 89, 024101 (2014).
  * [70] C. Gustin and S. Hughes, “Efficient pulse-excitation techniques for single photon sources from quantum dots in optical cavities”, Adv. Quantum Technol. $\mathbf{3}$, 1900073 (2020).
  * [71] E. Schöll, L. Schweickert, L. Hanschke, K. D. Zeuner, F. Sbresny, T. Lettner, R. Trivedi, M. Reindl, S. F. C. da Silva, R. Trotta, J. J. Finley, J. Vučković, K. Müller, A. Rastelli, V. Zwiller, and K. D. Jöns, “Crux of using the cascaded emission of a three-level quantum ladder system to generate indistinguishable photons”, Phys. Rev. Lett. $\mathbf{125}$, 233605 (2020).
  * [72] Y. Wei, S. Liu, X. Li, Y. Yu, X. Su, S. Li, X. Shang, H. Liu, H. Hao, H. Ni, S. Yu, Z. Niu, J. Iles-Smith, J. Liu, and X. Wang , “Tailoring solid-state single-photon sources with stimulated emissions”, Nat. Nanotechnol. $\mathbf{17}$, 470–476 (2022).
  * [73] L. Vannucci and N. Gregersen, “Highly efficient and indistinguishable single-photon sources via phonon-decoupled two-color excitation”, Phys. Rev. B $\mathbf{107}$, 195306 (2023).
  * [74] B. Bylicka, D. Chruściński, and S. Maniscalco, “Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective”, Sci. Rep. $\mathbf{4}$, 5720 (2014).
  * [75] F. Grosse and R. Zimmermann, “Electron-phonon interaction in embedded semiconductor nanostructures”, Phys. Rev. B $\mathbf{75}$, 235320 (2007).
  * [76] F. Murphy-Armando, G. Fagas, and J. C. Greer, “Deformation potentials and electron-phonon coupling in silicon nanowires”, Nano Lett. $\mathbf{10}(3)$, 869-873 (2010).
  * [77] T. Takagahara, “Theory of exciton dephasing in semiconductor quantum dots”, Phys. Rev. B $\mathbf{60}$, 2638 (1999).
  * [78] H. Bruus and K. Flensberg, Many-body quantum theory in condensed matter physics: an introduction (Oxford Univ. Press, 2004).
  * [79] M. A. Stroscio and M. Dutta, Phonons in nanostructures (Cambridge Univ. Press, 2001).