# Polaritonic Bottleneck in Colloidal Quantum Dots

Kaiyue Peng  Department of Chemistry, University of California, Berkeley,
California 94720, United States<EMAIL_ADDRESS>Eran Rabani
Department of Chemistry, University of California, Berkeley, California 94720,
United States<EMAIL_ADDRESS>
###### Abstract

Controlling the relaxation dynamics of excitons is key to improve the
efficiencies of semiconductor–based applications. Confined semiconductor
nanocrystals (NCs) offer additional handles to control the properties of
excitons, for example, by changing the size or the shape of the NC, resulting
in a mismatch between the excitonic gaps and the phonon frequencies. This has
led to the hypothesis of a significant slowing down of exciton relaxation in
strongly confined NC, but in practice due to increasing exciton–phonon
coupling and rapid multiphonon relaxation channels, the relaxation of excitons
depends only weakly on the size or shape of the NC. In this work, we focus on
elucidating the nonradiative relaxation of excitons in NC placed in an optical
cavity. We find that as the cavity photon energy decreases and/or the
cavity–exciton coupling increases, the relaxation to thermal equilibrium is
significantly slowed down, resulting in a polariton–induced phonon bottleneck,
with relaxation timescales that are slower by orders of magnitude compared to
the cavity–free case.

###### keywords:

Nanocrystals, Quantum Dots, Polariton, Micro–cavity, Phonon Bottleneck

Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley,
California 94720, United States The Sackler Center for Computational Molecular
and Materials Science, Tel Aviv University, Tel Aviv, Israel 69978

##  1 Introduction

The electronic and optical properties of semiconductor quantum dot (QD)
nanocrystals (NCs) have been widely studied over the past decades, leading to
the development of colloidal–based optoelectronic devices and remarkable
advancements in new commercialized technologies.1, 2, 3 A key to the rational
design of NC–based technologies with increased quantum yields and reduced
thermal losses relies on controlling the radiative, nonradiative, dephasing,
and energy transfer channels, governed by exciton–exciton and exciton–phonon
couplings.4, 5 One of the most important relaxation channels limiting the
efficiency of NC–based devices involves the nonradiative relaxation of
excitons to form a band edge excitation. It has been argued that due to a
mismatch between the excitonic energy gaps and the lattice vibrations, the
relaxation of excitons in strongly confined NCs would be extremely slow,
resulting in a "phonon bottleneck", despite enhancements of the exciton–phonon
couplings.6 However, measurements of the cooling process in NCs have yielded
conflicting results, where most time–resolved measurements do not support the
existence of a phonon bottleneck and the relaxation of excitons occurs on
sub–picosecond timescales for various NC materials and sizes.7, 8, 9 This
unexpected result was recently rationalized theoretically signifying the role
of multiphonon processes in NCs, resulting in ultrafast exciton relaxation to
the band edge.10

Controlling the nonradiative relaxation timescales and relaxation pathways of
excitons has been central for improved quantum efficiencies.11, 12, 13 The use
of heterostructure core–shell NCs, for example, results in slower relaxation
rates compared to core NCs, mainly due to the reduction of the couplings
between excitons and surface phonon modes,14 with relaxation rates that are
slower by a factor of $5$.10 An alternative approach to modify the electronic
and vibronic properties of the NC is by strongly coupling to light, for
example, inside a cavity (see Fig. 1 for an illustration). The properties of
such hybrid exciton–photon states (polaritonic states) can be tuned without
changing the size and composition of the NC, by modulating the cavity
parameters such as the cavity photon energy and/or the coupling strength
between excitons and photons.15, 16, 17, 18 This approach has been applied for
J–aggregates, two–dimensional quantum wells, and bulk materials,19, 20, 21,
22, 16 and is also popular for controlling the electronic properties of
molecular systems, and in particular how the products in chemical reactions
can be tuned by coupling with light, with reported reactions being selective,
slower or faster in optical cavities.23, 24 In addition, such hybrid states
exhibit unique properties that make them relevant for a wide range of
applications, including lasers,25, 26, 27, 28, 29 Bose–Einstein
condensates,30, 31, 32 and quantum bits.33, 34, 35

Figure 1: An illustration of a nanocrystal QD in an optical cavity.

In this work, we consider a CdSe NC in a cavity and study the relaxation
pathways and timescales of polaritonic states as we tune the cavity photon
energy ($\hbar\omega_{c}$) and cavity–exciton coupling ($\hbar g$). We
parametrize the Pauli–Fierz Hamiltonian36 to describe a lossless micro–cavity
confined single photon mode coupled to a manifold of excitonic states. The
latter were calculated within the atomistic semiempirical pseudopotential
model combined with the Bethe–Salpeter equation (BSE), accounting for
electron–hole correlations.14 We perform a small polaron transformation 10, 37
on the Pauli–Fierz Hamiltonian to rescale the polariton–phonon coupling
(obtained from the semiempirical pseudopotential model) and use a quantum
master equation with memory approximated to second–order in this coupling
(Redfield equation). This transformation allows us to account for multiphonon
processes within the lowest–order perturbation scheme.

We find that the relaxation to thermal equilibrium is hindered by the coupling
to the cavity mode, which can be controlled by changing the micro–cavity
photon energy or by changing the cavity–exciton coupling. We explore the role
of temperature and cavity parameters on the timescales of relaxation and find
that as the cavity photon energy decreases and/or the cavity–exciton coupling
increases, the relaxation to thermal equilibrium is significantly slowed down.
This polaritonic–induced phonon bottleneck is attributed to the increase in
the polaritonic gap of the two lowest polaritonic states and to changes in the
polariton-phonon coupling strength.

##  2 Model Hamiltonian

We adopt the following model Hamiltonian to describe the ground state and a
manifold of excitonic states coupled to the vibrational modes of a nanocrystal
QD, with exciton–phonon coupling (EXPC) expanded to the lowest order in the
vibrational normal modes:14, 10

$\displaystyle H_{\text{QD}}$
$\displaystyle=E_{g}\left|\psi_{g}\right\rangle\left\langle\psi_{g}\right|+\sum_{n}E_{n}\left|\psi_{n}\right\rangle\left\langle\psi_{n}\right|+$
$\displaystyle+$
$\displaystyle\sum_{\alpha}\hbar\omega_{\alpha}b_{\alpha}^{\text{$\dagger$}}b_{\alpha}+\sum_{nm\alpha}V_{nm}^{\alpha}\left|\psi_{m}\right\rangle\left\langle\psi_{n}\right|q_{\alpha}.$
(1)

In the above equation, $E_{g}$ is the energy of the ground state
$\left|\psi_{g}\right\rangle$ and $E_{n}$ is the energy of an exciton in state
$\left|\psi_{n}\right\rangle$. The phonon frequencies $\omega_{\alpha}$ and
phonon modes
$q_{\alpha}=\sqrt{\frac{\hbar}{2\omega_{\alpha}}}\left(b_{\alpha}^{\text{$\dagger$}}+b_{\alpha}\right)$
were obtained by diagonalizing the dynamical matrix of the NC calculated using
the Stillinger–Weber force field.38 The excitonic states and the EXPC
elements, $V_{nm}^{\alpha}$, were calculated using the semiempirical
pseudopotential methods 39 combined with the BSE 40, including only
spin–allowed bright or dim states.14, 5 We ignore the EXPC between the ground
state and all the other excitonic states, which would lead to nonradiative
relaxation of excitons to the ground state on timescales much longer than the
current interest. This parametrization procedure was validated against optical
stokes shift measurements,14 single–particle photoluminescence lineshape
broadening,41 and most recently to the relaxation dynamics of excitons in both
core and core/shell NCs.42, 10

The nanocrystal QD is placed in an optical cavity modeled by a single–mode
Pauli−Fierz Hamiltonian:43

$\displaystyle H_{\text{cav}}$
$\displaystyle=\hbar\omega_{c}a^{\dagger}a+\sum_{n}\hbar
g_{n}\left(a^{\dagger}\left|\psi_{g}\right\rangle\left\langle\psi_{n}\right|+h.c.\right)$
$\displaystyle+$
$\displaystyle\frac{\hbar}{\omega_{c}}\left[\text{$\sum_{n}g_{n}^{2}$}\left|\psi_{g}\right\rangle\left\langle\psi_{g}\right|+\sum_{nm}g_{n}g_{m}\left|\psi_{n}\right\rangle\left\langle\psi_{m}\right|\right],$
(2)

where $a$ ($a^{\dagger}$) is the destruction (creation) operator of a photon
with frequency $\omega_{c}$ and $g_{n}$ is coupling strength between the
cavity mode and an exciton in state $\left|\psi_{n}\right\rangle$. The latter
is given by
$g_{n}=\sqrt{\frac{\omega_{c}}{2\hbar\varepsilon\mathcal{V}}}\bm{\mu}_{n}\cdot\bm{k}$,
where
$\bm{\mu}_{n}=\left\langle\psi_{n}\right|\hat{\bm{\mu}}\left|\psi_{g}\right\rangle$
is the transition dipole moment from the ground state to exciton
$\left|\psi_{n}\right\rangle$ (obtained from the BSE calculation 44) and
$\bm{k}$ is a unit vector of the polarization direction of the electromagnetic
field. Finally, $\varepsilon$ is the effective permittivity inside the cavity,
and $\mathcal{V}$ is the effective cavity quantization volume.

The total Hamiltonian of the nanocrystal QD coupled to the cavity mode can be
represented using the excitonic–photonic basis set, namely,
$\left\\{\left|\psi_{g};1\right\rangle,\left|\psi_{n};0\right\rangle\right\\}$,
where $0,1$ represent the number of photons (we limit the discussion to
single–excitation manifold only). In this basis, the total Hamiltonian can be
written as a sum of two terms,
$H_{\text{PF}}=H_{\text{QD}}+H_{\text{cav}}=H_{\text{S}}+H_{\text{ph}}$, where

$\displaystyle H_{\text{S}}$
$\displaystyle=\left(\hbar\omega_{c}+\sum_{n}\frac{\hbar}{\omega_{c}}g_{n}^{2}\right)\left|\psi_{g},1\right\rangle\left\langle\psi_{g},1\right|+$
$\displaystyle+$
$\displaystyle\sum_{n}E_{n}\left|\psi_{n},0\right\rangle\left\langle\psi_{n},0\right|+$
$\displaystyle+$
$\displaystyle\frac{\hbar}{\omega_{c}}\sum_{nm}g_{n}g_{m}\left|\psi_{n},0\right\rangle\left\langle\psi_{m},0\right|$
$\displaystyle+$ $\displaystyle\sum_{n}\hbar
g_{n}\left(\left|\psi_{n},0\right\rangle\left\langle\psi_{g},1\right|+\left|\psi_{g},1\right\rangle\left\langle\psi_{n},0\right|\right),$
(3)

and as before, the Hamiltonian describing the phonon modes is given by:

$H_{\text{ph
}}=\sum_{\alpha}\hbar\omega_{\alpha}b_{\alpha}^{\text{$\dagger$}}b_{\alpha}+\sum_{nm\alpha}V_{mn}^{\alpha}\left|\psi_{m},0\right\rangle\left\langle\psi_{n},0\right|q_{\alpha}.$
(4)

The hybrid eigenstates of the $H_{\text{S}}$ are called polaritonic states and
can be obtained by diagonalizing $H_{\text{S}}$. The total Hamiltonian
($\tilde{H}_{\text{PF}}$) is then given in terms of the polaritonic states,
$\left|\varphi_{n}\right\rangle$, as:

$\displaystyle\tilde{H}_{\text{PF}}$
$\displaystyle=\sum_{n}\tilde{E}_{n}\left|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right|+\sum_{\alpha}\hbar\omega_{\alpha}b_{\alpha}^{\text{$\dagger$}}b_{\alpha}$
$\displaystyle+$
$\displaystyle\sum_{mn\alpha}\tilde{V}_{mn}^{\alpha}\left|\varphi_{m}\right\rangle\left\langle\varphi_{n}\right|q_{\alpha},$
(5)

where
$\left|\varphi_{n}\right\rangle=c_{ng}\left|\psi_{g},1\right\rangle+\sum_{m}c_{nm}\left|\psi_{m},0\right\rangle$
is a polaritonic state with energy $\tilde{E}_{n}$. $\tilde{V}_{mn}^{\alpha}$
represents the coupling matrix element between two polaritonic states
$\left|\varphi_{m}\right\rangle$ and $\left|\varphi_{n}\right\rangle$ via
phonon mode $\alpha$.

The above Hamiltonian has an identical form to that of a bare NC Hamiltonian,
with the additional complexity of describing the dressed ground state
($\left|\psi_{g},1\right\rangle$). To describe the relaxation dynamics of the
polaritonic states generated by this Hamiltonian, we followed the procedure
developed in Ref. 10. The first step involves a polaron transformation, which
not only rescales the coupling between the polaritonic states and the phonon
modes, but also allows for the description of multi–phonon relaxation within
the lowest–order perturbation theory. Such multi–phonon relaxation processes
are central in describing the ultrafast dynamics of hot excitons. The polaron
transformation of the polaritonic Hamiltonian is similar to the one used for
excitons, namely, ${\cal H}=e^{S}\tilde{H}_{\text{PF}}e^{-S}$, where
$S=-\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right|$
and $p_{\alpha}$ is the momentum operator of vibrational mode $\alpha$. The
total transformed Hamiltonian is then given by:

$\displaystyle{\cal H}$
$\displaystyle=\sum_{n}\left(\tilde{E}_{n}-\lambda_{n}\right)\left|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right|+\sum_{\alpha}\hbar\omega_{\alpha}b_{\alpha}^{\text{$\dagger$}}b_{\alpha}+$
$\displaystyle+$ $\displaystyle\sum_{n\neq
m}\left(\sum_{\alpha}W_{nm}^{\alpha}q_{\alpha}-\lambda_{nm}\right)\left|\varphi_{n}\right\rangle\left\langle\varphi_{m}\right|$
(6)

where
$\lambda_{n}=\frac{1}{2}\sum_{\alpha}\left(\tilde{V}_{nn}^{\alpha}\right)^{2}/\omega_{\alpha}^{2}$
is the reorganization energy (polaron shift) of the polaritonic state
$\left|\varphi_{n}\right\rangle$,
$\lambda_{nm}=\frac{1}{2}\sum_{\alpha}W_{nm}^{\alpha}\left(\tilde{V}_{mm}^{\alpha}+\tilde{V}_{nn}^{\alpha}\right)/\omega_{\alpha}^{2}$,
and the rescaled couplings between the polaritonic states and the vibrational
mode $\alpha$ is given by $W=e^{S}\tilde{V}e^{-S}$, with matrix elements:

$\displaystyle W_{nm}^{\alpha}$
$\displaystyle=\exp\left(-\frac{i}{\hbar}\sum_{\gamma}\frac{p_{\gamma}\tilde{V}_{nn}^{\gamma}}{\omega_{\gamma}^{2}}\right)$
$\displaystyle\times$
$\displaystyle\tilde{V}_{nm}^{\alpha}\exp\left(+\frac{i}{\hbar}\sum_{\gamma}\frac{p_{\gamma}\tilde{V}_{mm}^{\gamma}}{\omega_{\gamma}^{2}}\right).$
(7)

The dynamics of the polaritonic excitation is approximated using the lowest
order perturbation theory in the rescaled coupling term, $W_{nm}^{\alpha}$.
The equation of motion for the reduced density matrix representing the Hilbert
space of the polaritonic states is given by the non–Markovian time–local
secular Redfield equations. A detailed derivation and description of the
non–Markovian equations is given in Supporting Information.

##  3 Results and Discussion

Figure 2: (a) The calculated linear absorption spectrum (top) and the density
of excitonic states (bottom) for a $\text{Cd}_{435}\text{Se}_{435}$ NC. The
vertical lines in the top panel indicate the magnitude of the oscillator
strength of the transition from the ground state to that excitonic state. (b)
Phonon–mediated hot exciton 1P–1S cooling simulated for CdSe NCs. The
inclusion of multiphonon processes allows the systems to relax to thermal
equilibrium faster, circumventing the phonon bottleneck. Figure 3: Polaritonic
energy relaxation for a $\text{Cd}_{435}\text{Se}_{435}$ NC inside a cavity.
Panel (a) shows the dependence on the coupling strength for $\hbar
g_{\text{X}_{2}}=0$ (black dashed), $\hbar g_{\text{X}_{2}}=20$ (light blue),
$\hbar g_{\text{X}_{2}}=50$ (green), and $\hbar g_{\text{X}_{2}}=70$ meV (dark
green) for $\hbar\omega_{c}=2.02$ eV (slightly below the absorption onset) and
$T=300\text{\,K}$. Panel (b) shows the dependence on the photon energy for
$\hbar\omega_{c}$ range from $2.02-2.20$ eV (in steps of $0.06$ eV), $\hbar
g_{\text{X}_{2}}=50$ meV, and $T=300\,\text{K}$. Panel (c) shows the
dependence on temperature for $T=50$K (dark blue), $150$K (blue), $300$K
(orange), and $400$K (red), $\hbar\omega_{c}=2.06$ eV, and $\hbar
g_{\text{X}_{2}}=50$ meV.

We consider a $\text{Cd}_{435}\text{Se}_{435}$ nanocrystal with a diameter of
$D\approx 4\thinspace\text{nm}$. The single particle states near the top of
the valence band and the bottom of the conduction band were generated using
the filter–denationalization technique on a real–space grid basis with a total
number of grid points exceeding $N_{g}>10^{6}$ with a grid spacing of $\approx
0.65\thinspace a_{0}$. $80$ electron (unoccupied) and $140$ hole (occupied)
states were calculated to construct the Bethe–Salpeter matrix. This was
sufficient to converge the excitonic energies and transition dipole moments
for the $\approx 100$ lowest excitonic states, but only the lowest $\approx
50$ were necessary to accurately describe the relaxation dynamics starting
from the $\text{X}_{1}$ exciton (see Fig. 2). In Fig. 2(a) we plot the
computed linear absorption spectrum of the $\text{Cd}_{435}\text{Se}_{435}$
nanocrystal alongside the density of excitonic states. The linear absorption
spectrum shows several distinct features in agreement with the optical
measurements (we label the main transitions as 1S and 1P following the
literature convention).5 The density of excitonic states is relatively high
due to the dense spectrum of hole states, but only a few excitonic states are
"bright", as indicated by the oscillator strength of the individual transition
also shown in Fig. 2(a). Most of the excitonic states are "dim" with a small
oscillator strength.

The cooling dynamics of excitons for a $4\thinspace\text{nm}$ CdSe NC in the
absence of a cavity mode was recently studied both theoretically and
experimentally.10, 42 A mismatch between excitonic energy gaps and phonon
frequencies has led to the hypothesis of a phonon bottleneck and extremely
slow cooling, however, recent 2D electronic measurements revealed ultrafast
excitonic relaxation on timescales of a few tens of femtoseconds, providing no
evidence for a phonon bottleneck.42 The absence of a phonon bottleneck was
rationalized by an Auger–assisted mechanism, which emerged naturally from the
work of Jasrasaria and Rabani, however, it is the high density of vibrational
modes and the coupling of excitons to both acoustic and optical modes that
mainly contributes to the fast exciton cooling and the absence of a phonon
bottleneck.10

Figure 4: (a) Polaritonic or excitonic energies for different cavity modes
with a cavity–exciton coupling strength of $\hbar g_{\text{X}_{2}}=50$ meV and
$T=300\text{\,K}$. The photon fraction is depicted by the orange color for
each photon energy ($\hbar\omega_{c}$ range from $2.02-2.20$ eV, in increments
of $0.06$ eV). (b) A semi–log plot of the transition rate between the two
lowest polaritonic states as a function of the transition energy gap between
them, for three different temperatures and for $\hbar\omega_{c}=2.06$ eV. The
dashed curve shows this transition gap for $\text{Cd}_{435}\text{Se}_{435}$ at
$\hbar\omega_{c}=2.06$ eV and $g_{\text{X}_{2}}=50$ meV. (c) The spectral
density computed for the lowest polaritonic state as a function of phonon
frequency for $4$ different photon energies. We use the same color code as in
panel (a).

In Fig. 2(b) we show the cooling dynamics from the 1P excitons when only
single phonon processes are allowed (green curve) and for the multiphonon
relaxation case (black curve). Since the initial state here is the 1P exciton,
one requires more excitonic states to converge the dynamics, the details of
which are given in Ref. 10. We plot the average excitonic energy, $\Delta
E\left(t\right)=\left\langle
H_{\text{S}}\right\rangle\left(t\right)-\left\langle
H_{\text{S}}\right\rangle_{{\rm eq}}$,45 as a function of time, where
$H_{\text{S}}$ is given in Eq. (3) for $g_{n}=0$. When only single phonon
transitions are allowed, the 1P exciton relaxes on timescales that are much
longer than observed experimentally.46, 9 In this limit, the relaxation
proceeds through a cascade of excitonic transitions with gaps that are smaller
than the highest optical frequency. The slower long–time relaxation observed
in this limit is a result of the sparse density of excitons near the band
edge. For even smaller NCs ($D<3\thinspace\text{nm}$), the excitonic gaps
become larger than the largest optical phonon frequency and the system cannot
relax to thermal equilibrium in the limit of single–phonon processes. When
multiphonon channels are allowed (black curve, Fig. 2(b)), the decay occurs on
much faster timescales and agrees well with recent measurements for both core
and core/shell NCs.42 The inclusion of multiphonon processes allows for
efficient relaxation of excitons across energy gaps that are larger than the
largest optical frequency, opening multiphonon relaxation channels that
compete with the single phonon paths, even for gaps that are smaller than the
optical frequencies. The unexpected contribution of multiphonon channels to
the relaxation of excitons was used to rationalize the absence of a phonon
bottleneck.

In Fig. 3 we plot the relaxation dynamics for the same NC studied in Fig. 2
placed in an optical cavity starting from $\text{X}_{1}$. We consider several
different values of the coupling strengths (panel (a)), several different
photon energies (panel (b)), and several different temperatures (panel (c)).
For weak coupling to the cavity mode ($\hbar g_{\text{X}_{2}}<20$ meV), the
relaxation timescales are similar to those computed for the isolated NC
(dashed black curve). As the coupling to the cavity mode increases above
$\hbar g_{\text{X}_{2}}>50$ meV, a slower timescale appears and the relaxation
dynamics is characterized by a multi–exponential decay. A similar behavior was
observed when either the photon energy is significantly below the onset of the
NC absorption (panel (b)) or at lower temperatures (panel (c)). We note in
passing that an analogous behavior was recently reported for a 2D quantum well
system, where a significant slowing down of the relaxation dynamics was
observed by manipulating the energy of the photon mode.20

To understand the emergence of this polaritonic–induced phonon bottleneck, we
plot in Fig. 4(a) the polaritonic energies ($\tilde{E_{n}}$) in the absence of
a cavity (excitonic energies, $E_{n}$) and for $4$ different photon energies.
The cavity coupling is set to $\hbar g_{\text{X}_{2}}=50$ meV. The lower
polaritonic (LP) state has a significant photon fraction as depicted by the
orange portion of the LP energy bar. Most of the polaritonic states above the
LP have a vanishing fraction of the cavity mode (dim states couple very weakly
to the cavity), except for a few bright states with non–vanishing photonic
character. The polaritonic density of states for different cavity–exciton
coupling strengths is plotted in the Supporting Information (see Fig. S2(a)).

As the gap ($\delta E$) between the two lowest polaritonic states increases
beyond the highest optical frequency, by either lowering the photon energy or
increasing the coupling to the cavity, the relaxation to the lowest
polaritonic state can only occur via multiphonon emission, with transition
rates that are exponentially suppressed, as shown in Fig. 4(b). However, the
increase in the gap between the two lowest polaritonic states with decreasing
photon energy or increasing coupling, by itself, is not sufficient to account
for the entire behavior shown in Fig. 3 panels (a) and (b).

In Fig. 4(c) we plot the spectral density,
$J_{\text{LP}}\left(\omega\right)=\frac{1}{2}\sum_{\alpha}\left(\frac{\tilde{V}_{\text{LP}}^{\alpha}}{\omega_{\alpha}}\right)^{2}\delta\left(\omega-\omega_{\alpha}\right)$,
for LP for different values of the photon energy. The spectral density is a
measure of the strength of the polariton–phonon coupling to each phonon mode.
As the cavity mode energy decreases below the absorption onset of the NC, the
photon fraction of the LP state increases and the overall polariton–phonon
coupling decreases. This, together with the opening of the gap $\delta E$
leads to a significant slowing down of the phonon emission rate and the
emergence of a polaritonic–induced phonon bottleneck.

In the Supporting Information, we plot the spectral density for different
values of the cavity–exciton coupling strengths (Fig. S2(b)). As the
cavity–exciton coupling increases, the photon fraction of the LP state
slightly decreases (as can be seen in Fig. S2(a)), opposite to the behavior
observed in Fig. 4(a) for different photon energies. As a result, the overall
polariton–phonon coupling increases with increasing cavity–exciton couplings.
This enhanced vibronic coupling competes with the slowing down of the
relaxation due to the opening of the gap $\delta E$, but the dependence is
rather mild to circumvent the bottleneck phenomena, as can be seen in
population dynamics for different couplings strength shown in Fig. 3(a).

To better quantify the onset of the polaritonic phonon bottleneck, we further
analyze the transition rate between the two lowest polaritonic states, LP and
$\text{LP}+1$, as the bottleneck phenomenon is mainly correlated with the
relaxation between these two states. The decay rate between the two lowest
polaritonic states with an energy spacing of $\delta E$ can be approximated by
(see Supporting Information):

$\varGamma\left(\delta E\right)\approx\frac{k_{{\rm
B}}T}{\hbar}\sqrt{\frac{\pi\Lambda^{2}}{\eta\left(T\right)}}e^{-\delta
E^{2}/4\eta\left(T\right)},$ (8)

where
$\Lambda=\sum_{\alpha}\left|\frac{\tilde{V}^{\alpha}}{\omega_{\alpha}}\right|^{2}$
and $\eta\left(T\right)$ is a thermal weighted average measure of the
polariton–phonon couplings:

$\displaystyle\eta\left(T\right)$
$\displaystyle=-\frac{\hbar^{2}}{\Lambda}\sum_{\alpha}\left|\tilde{V}^{\alpha}\right|^{2}+$
$\displaystyle+$ $\displaystyle 2k_{{\rm
B}}T\left(\lambda_{\text{LP}}+\lambda_{\text{LP+1}}-\sum_{\alpha}\frac{1}{\omega_{\alpha}^{2}}\tilde{V}_{\text{LP}}^{\alpha}\tilde{V}_{\text{LP+1}}^{\alpha}\right).$
(9)

In the above equation, $\lambda_{\text{LP}}$ and $\lambda_{\text{LP+1}}$ are
the reorganization energies of the lowest two polaritonic states,
respectively, $\tilde{V}_{\text{LP}}^{\alpha}$ and
$\tilde{V}_{\text{LP+1}}^{\alpha}$ are the diagonal couplings between the
lowest two polaritonic states and phonon mode $\alpha$, respectively, and
$\tilde{V}^{\alpha}$ is the coupling between the two lowest polaritonic states
to phonon mode $\alpha$.

Figure 5: Plot of $\gamma_{\rm ph}=\delta E^{2}/4\eta\left(T\right)$ vs. $T$
for $\hbar g_{\text{X}_{2}}=50$ meV and for $\hbar\omega_{c}$ range from
$2.02-2.20$ eV, in increments of $0.06$ eV. The dash black line is for
$\gamma_{\rm ph}=1$.

The dependence of transition rate on the $\delta E$ and $T$ is governed by the
exponential term ($e^{-\delta E^{2}/4\eta\left(T\right)}$), while the
pre–exponential factor ($\frac{k_{{\rm
B}}T}{\hbar}\sqrt{\frac{\pi\Lambda^{2}}{\eta\left(T\right)}}$) depends weakly
on the optical cavity properties and temperature. We define a "bottleneck
parameter", $\gamma_{\rm ph}=\delta E^{2}/4\eta\left(T\right)$, and assume
that the onset of a polaritonic phonon bottleneck occurs $\gamma_{\rm ph}>1$
(dashed black curve in Fig. 5). In Fig. 5 we plot $\gamma_{\rm ph}$ as a
function of the temperature for $\hbar g_{\text{X}_{2}}=50$ meV and for all
the values of the photon energy considered in this work. For different
cavity–exciton coupling strengths, a similar diagram is shown in the
Supporting Information, Fig. S4. The phonon bottleneck regime seems to appear
for low cavity photon energies and for strong cavity–exciton coupling, while
the temperatures can be used to drive the system in or out of a bottleneck
regime.

##  4 Conclusion

In this work, we explored the relaxation of hot excitons in confined
semiconductor NCs placed inside a micro–cavity. We adopted the pseudopotential
model combined with the Bethe–Salpeter equation to describe the excitons and
their coupling to the lattice phonons, and generalized the Pauli–Fierz
Hamiltonian to account for the coupling of the excitons to a single cavity
mode. Using the polaron–transformed Redfield equations within the polaritonic
representation, we observe orders of magnitude inhibition of thermalization of
polaritons by tuning the hybrid system, giving rise to a cavity–induced phonon
bottleneck. The transition to the bottleneck regime depends on the cavity
photon mode, cavity–exciton coupling, and temperature. Our predictions provide
fundamental insights into the nonradiative relaxation dynamics of confined
excitons and how to control thermalization using optical cavities.

This work was supported by the NSF-BSF International Collaboration in the
Division of Materials Research program, NSF grant number DMR2026741.
Computational resources were provided in part by the National Energy Research
Scientific Computing Center (NERSC), a U.S. Department of Energy Office of
Science User Facility operated under contract no. DEAC02- 05CH11231. We would
like to thank Dr. Dipti Jasrasaria for helpful discussions.

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