Supporting Information: Polariton assisted incoherent to coherent excitation energy transfer between colloidal quantum dots

I Nanocrystal Configurations

We consider $3$ quantum dot (QD) systems. The $3$ nm CdSe core with a $2$ -monolayer CdS shell QD contains a total of $753$ Cd atoms, $252$ Se atoms, and $501$ S atoms. The $3.9$ nm CdSe core with a $3$ -monolayer CdS shell QD contains a total of $1788$ Cd atoms, $483$ Se atoms, and $1305$ S atoms. The $3.9$ nm CdSe core with a $4$ -monolayer CdS shell QD contains a total of $2637$ Cd atoms, $483$ Se atoms, and $2154$ S atoms.

The Stillinger–Weber force field [zhou2013stillinger] was utilized to determine the equilibrium structure and normal vibrational modes using LAMMPS.[plimpton1995fast] Following minimization, the outermost layer was replaced by ligands with a modification of the pseudopotential to represent the passivation layer.[rabani1999electronic]

II Polaron transformation

The total Hamiltonian in polaritonic basis:

\displaystyle\tilde{H}

\displaystyle=\sum_{n}\tilde{E}_{n}\left|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right|+\sum_{\alpha}\hbar\omega_{\alpha}b_{\alpha}^{\dagger}b_{\alpha}+\sum_{\alpha nm}\tilde{V}_{nm}^{\alpha}\left|\varphi_{n}\right\rangle\left\langle\varphi_{m}\right|q_{\alpha}.

(S1)

Apply a unitary polaron transformation:[jasrasaria2023circumventing, peng2023polaritonic]

$\displaystyle{\cal H}$	$\displaystyle=e^{S}\tilde{H}e^{-S}=\exp\left(-\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)\tilde{H}\exp\left(\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)$	(S2)
	$\displaystyle=\exp\left(-\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)\left[\sum_{\alpha}\hbar\omega_{\alpha}b_{\alpha}^{\dagger}b_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)$	(S3)
	$\displaystyle+\exp\left(-\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)\left[\sum_{n}\tilde{E}_{n}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right]\exp\left(\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)$	(S4)
	$\displaystyle+\exp\left(-\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)\left[\sum_{\alpha nm}\tilde{V}_{nm}^{\alpha}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{m}\right\|q_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right).$	(S5)

Use the relationship:

$\displaystyle U(\lambda)$	$\displaystyle=e^{-\lambda\frac{\partial}{\partial q}}=e^{-\lambda\frac{ip}{\hbar}},$	(S6)
$\displaystyle U(\lambda)qU^{\dagger}(\lambda)$	$\displaystyle=e^{-\lambda\frac{ip}{\hbar}}qe^{\lambda\frac{ip}{\hbar}}=q-\lambda,$	(S7)
$\displaystyle U(\lambda)pU^{\dagger}(\lambda)$	$\displaystyle=e^{-\lambda\frac{ip}{\hbar}}pe^{\lambda\frac{ip}{\hbar}}=p.$	(S8)

Eq. S3 transforms the phonon bath part:

	$\displaystyle\exp\left(-\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)\left[\sum_{\alpha}\hbar\omega_{\alpha}b_{\alpha}^{\dagger}b_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)$
	$\displaystyle=\exp\left(-\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)\left[\sum_{\alpha}\left(\frac{p_{\alpha}^{2}}{2m}+\frac{1}{2}\omega_{\alpha}^{2}q_{\alpha}^{2}\right)\right]\exp\left(\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)$
	$\displaystyle=\sum_{\alpha}\left[\frac{p_{\alpha}^{2}}{2m}+\frac{1}{2}\omega_{\alpha}^{2}\left(q_{\alpha}-\frac{\sum_{n}\left(\tilde{V}_{nn}^{\alpha}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)}{\omega_{\alpha}^{2}}\right)^{2}\right]$
	$\displaystyle=\sum_{\alpha}\hbar\omega_{\alpha}b_{\alpha}^{\dagger}b_{\alpha}+\sum_{n}\left[\sum_{\alpha}\frac{\left(\tilde{V}_{nn}^{\alpha}\right)^{2}}{2\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right]-\sum_{n}\left(\sum_{\alpha}\tilde{V}_{nn}^{\alpha}q_{\alpha}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right).$		(S9)

Eq. S4 transforms the diagonal system part, which is unchanged:

	$\displaystyle\exp\left(-\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)\left[\sum_{n}\tilde{E}_{n}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right]\exp\left(\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)$
	$\displaystyle=\sum_{n}\tilde{E}_{n}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|.$		(S10)

Eq. S5 transforms the polariton-phonon interaction part:

	$\displaystyle\exp\left(-\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)\left[\sum_{\alpha nm}\tilde{V}_{nm}^{\alpha}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{m}\right\|q_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)$
	$\displaystyle=\sum_{nm}\left[\exp\left(-\sum_{\alpha}\frac{i\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\left[\sum_{\alpha}\tilde{V}_{nm}^{\alpha}q_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\tilde{V}_{mm}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\right]\left\|\varphi_{n}\right\rangle\left\langle\varphi_{m}\right\|$
	$\displaystyle=\sum_{n}\left(\sum_{\alpha}\tilde{V}_{nn}^{\alpha}q_{\alpha}-\sum_{\alpha}\frac{\left(\tilde{V}_{nn}^{\alpha}\right)^{2}}{\omega_{\alpha}^{2}}\right)\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|$
	$\displaystyle+\sum_{n\neq m}\exp\left(-\sum_{\alpha}\frac{i\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\left[\sum_{\alpha}\tilde{V}_{nm}^{\alpha}q_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\tilde{V}_{mm}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\left\|\varphi_{n}\right\rangle\left\langle\varphi_{m}\right\|.$		(S11)

Then the transformed total Hamiltonian is given by:

	$\displaystyle{\cal H}$	$\displaystyle=\sum_{n}\left(\tilde{E}_{n}-\sum_{\alpha}\frac{\left(\tilde{V}_{nn}^{\alpha}\right)^{2}}{2\omega_{\alpha}^{2}}\right)\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|+\sum_{\alpha}\hbar\omega_{\alpha}b_{\alpha}^{\dagger}b_{\alpha}$
		$\displaystyle+\sum_{n\neq m}\exp\left(-\sum_{\alpha}\frac{i\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\left[\sum_{\alpha}\tilde{V}_{nm}^{\alpha}q_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\tilde{V}_{mm}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\left\|\varphi_{n}\right\rangle\left\langle\varphi_{m}\right\|.$		(S12)

We label reorganization energy $\lambda_{n}=\frac{1}{2}\sum_{\alpha}\left(\tilde{V}_{nn}^{\alpha}\right)^{2}/\omega_{\alpha}^{2}$ and the transformed polariton-phonon coupling $W_{nm}=\exp\left(-\sum_{\alpha}\frac{i\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\left[\sum_{\alpha}\tilde{V}_{nm}^{\alpha}q_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\tilde{V}_{mm}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)$ , then total Hamiltonian $\mathcal{H}$ can be written as:

\displaystyle{\cal H}

\displaystyle=\overset{{\scriptstyle\mathcal{H}_{{\rm S}}}}{\overbrace{\sum_{n}\left(\tilde{E}_{n}-\lambda_{n}\right)\left|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right|}}+\overset{{\scriptstyle\mathcal{H}_{\text{B}}}}{\overbrace{\sum_{\alpha}\hbar\omega_{\alpha}b_{\alpha}^{\dagger}b_{\alpha}}+}\overset{{\scriptstyle\mathcal{H}_{\text{I}}}}{\overbrace{\sum_{n\neq m}W_{nm}\left|\varphi_{n}\right\rangle\left\langle\varphi_{m}\right|}}.

(S13)

III Time–local Redfield Equations

We use polaritonic basis $\left|\varphi_{n}\right\rangle$ , to express the reduced density matrix and the system Hamiltonian:

	$\displaystyle\sigma_{nm}\left(t\right)=\left\langle\varphi_{n}\right\|\sigma\left(t\right)\left\|\varphi_{m}\right\rangle,$		(S14)
	$\displaystyle\left\langle\varphi_{n}\right\|\left[\mathcal{H}_{\text{S}},\sigma\left(t\right)\right]\left\|\varphi_{m}\right\rangle/\hbar=\omega_{nm}\sigma_{nm}.$		(S15)

Using second order perturbation in $\mathcal{H}_{\text{I}}$ , the equation of motion of the reduced density matrix is given by:[nitzan2006chemical]

$\displaystyle\frac{\partial\sigma_{nm}}{\partial t}$	$\displaystyle=-i\omega_{nm}\sigma_{nm}-\frac{i}{\hbar}\left[\sum_{l\neq n}\left\langle W_{nl}\right\rangle_{{\rm B}}\sigma_{lm}-\sum_{l\neq m}\sigma_{nl}\left\langle W_{lm}\right\rangle_{{\rm B}}\right]$
	$\displaystyle+\sum_{kl}\int_{0}^{t}d\tau\left[-M_{nk,kl}\left(\tau\right)e^{i\omega_{mk}\tau}\sigma_{lm}\left(t-\tau\right)-M_{kl,lm}\left(-\tau\right)e^{i\omega_{ln}\tau}\sigma_{nk}\left(t-\tau\right)\right.$
	$\displaystyle\left.+M_{nk,lm}\left(\tau\right)e^{i\omega_{ln}\tau}\sigma_{kl}\left(t-\tau\right)+M_{nk,lm}\left(-\tau\right)e^{i\omega_{mk}\tau}\sigma_{kl}\left(t-\tau\right)\right],$	(S16)

where

	$\displaystyle\text{if }n\neq m\text{ and }k\neq l:\leavevmode\nobreak\$	$\displaystyle M_{nm,kl}\left(\tau\right)=\frac{1}{\hbar^{2}}\left[\left\langle W_{nm}\left(\tau\right)W_{kl}(0)\right\rangle_{{\rm B}}-\left\langle W_{nm}\right\rangle_{{\rm B}}\left\langle W_{kl}\right\rangle_{{\rm B}}\right],$		(S17)
	else:	$\displaystyle M_{nm,kl}\left(\tau\right)=0.$		(S18)

Employing a time local approximation, with $e^{i\omega_{nm}(t-\tau)}\sigma_{nm}\left(t-\tau\right)\approx e^{i\omega_{nm}t}\sigma_{nm}\left(t\right)$ ,[nitzan2006chemical] we arrive at:

$\displaystyle\frac{\partial\sigma_{nm}}{\partial t}$	$\displaystyle=-i\omega_{nm}\sigma_{nm}-\frac{i}{\hbar}\left[\sum_{l\neq n}\left\langle W_{nl}\right\rangle_{{\rm B}}\sigma_{lm}-\sum_{l\neq m}\sigma_{nl}\left\langle W_{lm}\right\rangle_{{\rm B}}\right]$
	$\displaystyle+\sum_{kl}\int_{0}^{t}d\tau\left[-M_{nk,kl}\left(\tau\right)e^{i\omega_{lk}\tau}\sigma_{lm}\left(t\right)-M_{kl,lm}\left(-\tau\right)e^{i\omega_{lk}\tau}\sigma_{nk}\left(t\right)\right.$
	$\displaystyle\left.+M_{nk,lm}\left(\tau\right)e^{i\omega_{kn}\tau}\sigma_{kl}\left(t\right)+M_{nk,lm}\left(-\tau\right)e^{i\omega_{ml}\tau}\sigma_{kl}\left(t\right)\right].$	(S19)

Here we could see that, the so-called Redfield tensor: $R_{nm,kl}\left(\omega,t\right)=\int_{0}^{t}d\tau M_{nm,kl}\left(\tau\right)e^{i\omega\tau}$ depends on time.

IV Calculating $\left\langle W_{nm}\left(t\right)W_{kl}\left(0\right)\right\rangle_{\text{B}}$

We represent the bath position ( $q_{\alpha}$ ) and momentum ( $p_{\alpha}$ ) operators using raising and lowering operators ( $b_{\alpha}^{\dagger}$ and $b_{\alpha}$ , respectively):

	$\displaystyle q_{\alpha}$	$\displaystyle=\sqrt{\frac{\hbar}{2\omega_{\alpha}}}\left(b_{\alpha}^{\dagger}+b_{\alpha}\right),$		(S20)
	$\displaystyle p_{\alpha}$	$\displaystyle=i\sqrt{\frac{\hbar\omega_{\alpha}}{2}}\left(b_{\alpha}^{\dagger}-b_{\alpha}\right).$		(S21)

Recall the dressed phonon coupling element, $W_{nm}$ ,

W_{nm}=\exp\left(-\frac{i}{\hbar}\sum_{\gamma}\frac{p_{\gamma}\tilde{V}_{nn}^{\gamma}}{\omega_{\gamma}^{2}}\right)\left[\sum_{\alpha}\tilde{V}_{nm}^{\alpha}q_{\alpha}\right]\exp\left(+\frac{i}{\hbar}\sum_{\xi}\frac{p_{\xi}\tilde{V}_{mm}^{\xi}}{\omega_{\xi}^{2}}\right).

(S22)

In the above equation, $\tilde{V}_{nm}^{\alpha}$ , the coupling matrix element between two polaritonic states $\left|\varphi_{n}\right\rangle$ and $\left|\varphi_{m}\right\rangle$ and phonon mode $\alpha$ . $\tilde{V}_{nm}^{\alpha}$ was obtained by applying the unitary transformation $U$ that diagonalizes $H_{{\rm S}}$ to $\tilde{V}^{\alpha}=U^{\dagger}V^{\alpha}U$ , for each mode $\alpha$ (see main text). The corresponding correlation function, $\left\langle W_{nm}\left(t\right)W_{kl}\left(0\right)\right\rangle_{\text{B}}$ (Eq. (9)) can then be expressed as :

$\displaystyle\left\langle W_{nm}\left(t\right)W_{kl}\left(0\right)\right\rangle_{\text{B}}$	$\displaystyle=\left\langle\left\{\exp\left(-\sum_{\alpha}\frac{i\tilde{V}_{nn}^{\alpha}p_{\alpha}\left(t\right)}{\hbar\omega_{\alpha}^{2}}\right)\left[\sum_{\alpha}\tilde{V}_{nm}^{\alpha}q_{\alpha}\left(t\right)\right]\exp\left(\sum_{\alpha}\frac{i\tilde{V}_{mm}^{\alpha}p_{\alpha}\left(t\right)}{\hbar\omega_{\alpha}^{2}}\right)\right\}\right.$
	$\displaystyle\left.\left\{\exp\left(-\sum_{\alpha}\frac{i\tilde{V}_{kk}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\left[\sum_{\alpha}\tilde{V}_{kl}^{\alpha}q_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\tilde{V}_{ll}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\right\}\right\rangle_{\text{B}}$
	$\displaystyle=\left\langle e^{\Omega_{n}\left(t\right)}L^{\left(nm\right)}\left(t\right)e^{-\Omega_{m}\left(t\right)}e^{\Omega_{k}}L^{\left(kl\right)}e^{-\Omega_{l}}\right\rangle_{\text{B}},$	(S23)

where

	$\displaystyle q_{\alpha}\left(t\right)$	$\displaystyle=\sqrt{\frac{\hbar}{2\omega_{\alpha}}}\left(b_{\alpha}^{\dagger}e^{i\omega_{\alpha}t}+b_{\alpha}e^{-i\omega_{\alpha}t}\right),$		(S24)
	$\displaystyle p_{\alpha}\left(t\right)$	$\displaystyle=i\sqrt{\frac{\hbar\omega_{\alpha}}{2}}\left(b_{\alpha}^{\dagger}e^{i\omega_{\alpha}t}-b_{\alpha}e^{-i\omega_{\alpha}t}\right),$		(S25)

so

	$\displaystyle\Omega_{n}\left(t\right)$	$\displaystyle=\sum_{\alpha}\tilde{d}_{\alpha}^{\left(n\right)}\left(a_{\alpha}^{\dagger}e^{i\omega_{\alpha}t}-a_{\alpha}e^{-i\omega_{\alpha}t}\right),$		(S26)
	$\displaystyle L^{\left(nm\right)}\left(t\right)$	$\displaystyle=\sum_{\alpha}\tilde{c}_{\alpha}^{\left(nm\right)}\left(a_{\alpha}^{\dagger}e^{i\omega_{\alpha}t}+a_{\alpha}e^{-i\omega_{\alpha}t}\right),$		(S27)

where

	$\displaystyle\tilde{d}_{\alpha}^{\left(n\right)}$	$\displaystyle=\frac{\tilde{V}_{nn}^{\alpha}}{\hbar\omega_{\alpha}^{2}}\sqrt{\frac{\hbar\omega_{\alpha}}{2}},$		(S28)
	$\displaystyle\tilde{c}_{\alpha}^{\left(nm\right)}$	$\displaystyle=\tilde{V}_{nm}^{\alpha}\sqrt{\frac{\hbar}{2\omega_{\alpha}}}.$		(S29)

Using the commutation relationship:

\left[\left(\alpha a_{j}^{\dagger}+\beta a_{j}\right),e^{\left(\gamma a_{j}^{\dagger}+\delta a_{j}\right)}\right]=\left[\left(\alpha a_{j}^{\dagger}+\beta a_{j}\right),\left(\gamma a_{j}^{\dagger}+\delta a_{j}\right)\right]e^{\left(\gamma a_{j}^{\dagger}+\delta a_{j}\right)},

(S30)

the correlation function can be expressed as

\left\langle W_{nm}\left(t\right)W_{kl}\left(0\right)\right\rangle_{\text{B}}=\left[h\left(t\right)+g\left(t\right)\right]f\left(t\right),

(S31)

where

	$\displaystyle h\left(t\right)$	$\displaystyle=\sum_{\alpha}\left\{-\left(\tilde{d}_{\alpha}^{(k)}-\tilde{d}_{\alpha}^{(l)}\right)\tilde{c}_{\alpha}^{(nm)}e^{i\omega_{\alpha}t}n_{\alpha}+\left(\tilde{d}_{\alpha}^{(k)}-\tilde{d}_{\alpha}^{(l)}\right)\tilde{c}_{\alpha}^{(nm)}e^{-i\omega_{\alpha}t}\left(n_{\alpha}+1\right)-\tilde{c}_{\alpha}^{(nm)}\left(\tilde{d}_{\alpha}^{(n)}+\tilde{d}_{\alpha}^{(m)}\right)\right\}\times$
		$\displaystyle\times\sum_{\alpha}\left\{\tilde{c}_{\alpha}^{(kl)}\left(\tilde{d}_{\alpha}^{(n)}-\tilde{d}_{\alpha}^{(m)}\right)e^{i\omega_{\alpha}t}n_{\alpha}-\tilde{c}_{\alpha}^{(kl)}\left(\tilde{d}_{\alpha}^{(n)}-\tilde{d}_{\alpha}^{(m)}\right)e^{-i\omega_{\alpha}t}\left(n_{\alpha}+1\right)-\tilde{c}_{\alpha}^{(kl)}\left(\tilde{d}_{\alpha}^{(k)}+\tilde{d}_{\alpha}^{(l)}\right)\right\},$		(S32)

g\left(t\right)=\sum_{\alpha}\tilde{c}_{\alpha}^{(nm)}\tilde{c}_{\alpha}^{(kl)}\left[\left(n_{\alpha}+1\right)e^{-i\omega_{\alpha}t}+n_{\alpha}e^{i\omega_{\alpha}t}\right],

(S33)

	$\displaystyle f\left(t\right)$	$\displaystyle=\prod_{\alpha}\exp\left\{-\left[\left(\tilde{d}_{\alpha}^{(n)}-\tilde{d}_{\alpha}^{(m)}\right)^{2}+\left(\tilde{d}_{\alpha}^{(k)}-\tilde{d}_{\alpha}^{(l)}\right)^{2}+2\left(\tilde{d}_{\alpha}^{(n)}-\tilde{d}_{\alpha}^{(m)}\right)\left(\tilde{d}_{\alpha}^{(k)}-\tilde{d}_{\alpha}^{(l)}\right)\cos(\omega_{\alpha}t)\right]\left(n+1/2\right)\right.$
		$\displaystyle+\left.i\left(\tilde{d}_{\alpha}^{(n)}-\tilde{d}_{\alpha}^{(m)}\right)\left(\tilde{d}_{\alpha}^{(k)}-\tilde{d}_{\alpha}^{(l)}\right)\sin\left(\omega_{\alpha}t\right)\right\},$		(S34)

and $n=\left\langle a^{\dagger}a\right\rangle=\left(e^{\beta\hbar\omega}-1\right)^{-1}$ .

Similarly, the average $\left\langle W_{nm}\right\rangle_{\text{B}}$ is given by:

\left\langle W_{nm}\right\rangle_{\text{B}}=\left\langle e^{\Omega_{n}}L^{(nm)}e^{-\Omega_{m}}\right\rangle_{\text{B}}=\sum_{\alpha}\left(-\tilde{c}_{\alpha}^{(nm)}\tilde{d}_{\alpha}^{(n)}-\tilde{c}_{\alpha}^{(nm)}\tilde{d}_{\alpha}^{(m)}\right)\prod_{\alpha}\exp\left\{-\left(\tilde{d}_{\alpha}^{(n)}-\tilde{d}_{\alpha}^{(m)}\right)^{2}\left(n_{\alpha}+\frac{1}{2}\right)\right\}.

(S35)

V Effective Coupling

V.1 Three level system

We consider a three level system representing a donor, acceptor coupled directly and both are coupled to a bridge state (cavity). The system Hamiltonian is written as:

H=\left(\begin{array}[]{ccc}E_{B}&g&g\\ g&E_{D}&J\\ g&J&E_{A}\end{array}\right),

(S36)

where the donor and acceptor states with the energies $E_{D}$ and $E_{A}$ are directly coupled by $J$ . The bridge state couples to the donor and acceptor with coupling strength $g$ . The goal is to rewrite the eigenvalue problem

\left[\left(\begin{array}[]{ccc}E_{B}&g&g\\ g&E_{D}&J\\ g&J&E_{A}\end{array}\right)-\boldsymbol{I}\lambda\right]\left(\begin{array}[]{c}d_{B}\\ d_{D}\\ d_{A}\end{array}\right)=0.

(S37)

in the donor-acceptor subspace. The three eigenvalues (we assume for clarity that $E_{D}=E_{A}=E$ ) are given by:

$\displaystyle\lambda_{1}$	$\displaystyle=E-J,$	(S38)
$\displaystyle\lambda_{2}$	$\displaystyle=\frac{1}{2}\left(E_{B}+E+J-\sqrt{8g^{2}+\left(E_{B}-E-J\right)^{2}}\right),$	(S39)
$\displaystyle\lambda_{3}$	$\displaystyle=\frac{1}{2}\left(E_{B}+E+J+\sqrt{8g^{2}+\left(E_{B}-E-J\right)^{2}}\right).$	(S40)

The corresponding eigenvectors can be obtained from:

$\displaystyle\left(E_{B}-\lambda\right)d_{B}+gd_{D}+gd_{A}$	$\displaystyle=0,$	(S41)
$\displaystyle gd_{B}+\left(E-\lambda\right)d_{D}+Jd_{A}$	$\displaystyle=0,$	(S42)
$\displaystyle gd_{B}+Jd_{D}+\left(E-\lambda\right)d_{A}$	$\displaystyle=0.$	(S43)

Replacing $d_{B}$ in the last two equations by the solution of the first one, $d_{B}=\frac{gd_{D}+gd_{A}}{\left(\lambda-E_{B}\right)}$ , we find:

	$\displaystyle\left[\frac{g^{2}}{\left(\lambda-E_{B}\right)}+E-\lambda\right]d_{D}+\left[\frac{g^{2}}{\left(\lambda-E_{B}\right)}+J\right]d_{A}$	$\displaystyle=0,$		(S44)
	$\displaystyle\left[\frac{g^{2}}{\left(\lambda-E_{B}\right)}+J\right]d_{D}+\left[\frac{g^{2}}{\left(\lambda-E_{B}\right)}+E-\lambda\right]d_{A}$	$\displaystyle=0.$		(S45)

which can be rewritten in a similar form to an eigenvalue problem in the Hilbert space of the donor and acceptor:

\left(\begin{array}[]{cc}E_{D}^{\text{eff}}-\lambda&J_{\text{eff}}\\ J_{\text{eff}}&E_{A}^{\text{eff}}-\lambda\end{array}\right)\left(\begin{array}[]{c}d_{D}\\ d_{A}\end{array}\right)=0,

(S46)

where

\displaystyle E_{D}^{\text{eff}}

\displaystyle=E_{A}^{\text{eff}}=\frac{g^{2}}{\left(\lambda-E_{B}\right)}+E,

(S47)

and the effective coupling

J_{\text{eff}}=\frac{g^{2}}{\left(\lambda-E_{B}\right)}+J

(S48)

depend on $\lambda.$ For the two eigenvalues $\lambda_{2}$ and $\lambda_{3}$ we find:

\displaystyle J_{\text{eff}}^{(2,3)}

\displaystyle=J+\frac{2g^{2}}{\left(E+J-E_{B}\right)\left(1\mp\sqrt{1+2\left(\frac{2g}{E+J-E_{B}}\right)^{2}}\right)}.

(S49)

For small values $g\ll J$ , $J_{\text{eff}}^{(2,3)}\rightarrow J$ and for large values of $g\gg J$ , $J_{\text{eff}}^{(2,3)}\rightarrow\mp\frac{g}{\left(\sqrt{2}\right)}$ . This implies that for large coupling of the donor and acceptor to the bridge results in an effective direct coupling between the donor and acceptor that scales with $g$ .

	$\displaystyle\exp\left(-\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)\left[\sum_{\alpha}\hbar\omega_{\alpha}b_{\alpha}^{\dagger}b_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)$
	$\displaystyle=\exp\left(-\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)\left[\sum_{\alpha}\left(\frac{p_{\alpha}^{2}}{2m}+\frac{1}{2}\omega_{\alpha}^{2}q_{\alpha}^{2}\right)\right]\exp\left(\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)$
	$\displaystyle=\sum_{\alpha}\left[\frac{p_{\alpha}^{2}}{2m}+\frac{1}{2}\omega_{\alpha}^{2}\left(q_{\alpha}-\frac{\sum_{n}\left(\tilde{V}_{nn}^{\alpha}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)}{\omega_{\alpha}^{2}}\right)^{2}\right]$
	$\displaystyle=\sum_{\alpha}\hbar\omega_{\alpha}b_{\alpha}^{\dagger}b_{\alpha}+\sum_{n}\left[\sum_{\alpha}\frac{\left(\tilde{V}_{nn}^{\alpha}\right)^{2}}{2\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right]-\sum_{n}\left(\sum_{\alpha}\tilde{V}_{nn}^{\alpha}q_{\alpha}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right).$		(S9)

	$\displaystyle\exp\left(-\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)\left[\sum_{\alpha nm}\tilde{V}_{nm}^{\alpha}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{m}\right\|q_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\sum_{n}\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|\right)$
	$\displaystyle=\sum_{nm}\left[\exp\left(-\sum_{\alpha}\frac{i\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\left[\sum_{\alpha}\tilde{V}_{nm}^{\alpha}q_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\tilde{V}_{mm}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\right]\left\|\varphi_{n}\right\rangle\left\langle\varphi_{m}\right\|$
	$\displaystyle=\sum_{n}\left(\sum_{\alpha}\tilde{V}_{nn}^{\alpha}q_{\alpha}-\sum_{\alpha}\frac{\left(\tilde{V}_{nn}^{\alpha}\right)^{2}}{\omega_{\alpha}^{2}}\right)\left\|\varphi_{n}\right\rangle\left\langle\varphi_{n}\right\|$
	$\displaystyle+\sum_{n\neq m}\exp\left(-\sum_{\alpha}\frac{i\tilde{V}_{nn}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\left[\sum_{\alpha}\tilde{V}_{nm}^{\alpha}q_{\alpha}\right]\exp\left(\sum_{\alpha}\frac{i\tilde{V}_{mm}^{\alpha}p_{\alpha}}{\hbar\omega_{\alpha}^{2}}\right)\left\|\varphi_{n}\right\rangle\left\langle\varphi_{m}\right\|.$		(S11)

Supporting Information: Polariton assisted incoherent to coherent excitation energy transfer between colloidal quantum dots

I Nanocrystal Configurations

II Polaron transformation

III Time–local Redfield Equations

IV Calculating ⟨Wn​m​(t)​Wk​l​(0)⟩B\left\langle W_{nm}\left(t\right)W_{kl}\left(0\right)\right\rangle_{\text{B}}

V Effective Coupling

V.1 Three level system

IV Calculating $\left\langle W_{nm}\left(t\right)W_{kl}\left(0\right)\right\rangle_{\text{B}}$