v\rangle^{\frac{1}{2}}w^{|\beta|}\partial_{\beta}^{\alpha}(\frac{\partial_{\tau}\overline{G}}{\sqrt{\mu}})\|\big{)}\|\langle
v\rangle^{-\frac{1}{2}}w^{|\beta|}\partial^{\alpha}_{\beta}f\|$
$\displaystyle\leq\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}f\|_{\sigma,|\beta|}^{2}+C_{\eta}\epsilon^{\frac{7}{5}+\frac{1}{15}a}(\delta+\epsilon^{a}\tau)^{-\frac{4}{3}}+C_{\eta}k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}\mathcal{D}_{2}(\tau),$

according to (1.10), (5.25), (2.14), Lemma 5.3 and the elementary
inequalities. Hence, for $|\alpha|+|\beta|\leq 2$ and $|\beta|\geq 1$, we have
by the above related estimates and $\eta>0$ small enough that

$\displaystyle\frac{1}{2}\frac{d}{d\tau}\|\partial_{\beta}^{\alpha}f\|^{2}_{2,|\beta|}+c\epsilon^{a-1}\|\partial^{\alpha}_{\beta}f\|^{2}_{\sigma,|\beta|}$
$\displaystyle\leq
C\epsilon^{a-1}\|\partial^{\alpha}f\|_{\sigma}^{2}+C\epsilon^{1-a}\big{\\{}\|\partial^{\alpha}\partial_{y}f\|_{\sigma}^{2}+\|\partial^{\alpha}(\widetilde{u}_{y},\widetilde{\theta}_{y})\|^{2}\big{\\}}$
$\displaystyle\quad+C\epsilon^{1-a}\|\partial_{y}\partial_{\beta-
e_{1}}^{\alpha}f\|_{\sigma,|\beta-
e_{1}|}^{2}+C\epsilon^{a-1}\sum_{|\beta_{1}|<|\beta|}\|\partial^{\alpha}_{\beta_{1}}f\|_{\sigma,|\beta_{1}|}^{2}$
$\displaystyle\quad+C\epsilon^{\frac{7}{5}+\frac{1}{15}a}(\delta+\epsilon^{a}\tau)^{-\frac{4}{3}}+C(\eta_{0}+k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a})\mathcal{D}_{2}(\tau).$
(3.72)

Notice that the coefficients on the third line of (3.3) is large and
$|\beta_{1}|<|\beta|$. We will use the induction in $|\beta|$ to control this
term. By the suitable linear combinations, we can obtain

$\displaystyle\sum_{|\alpha|=0}^{1}C_{|\alpha|}\sum_{j=1}^{2-|\alpha|}C_{j}\sum_{|\beta|=j}\frac{d}{d\tau}\|\partial_{\beta}^{\alpha}f\|_{2,|\beta|}^{2}+c\epsilon^{a-1}\sum_{|\alpha|+|\beta|\leq
2,|\beta|\geq 1}\|\partial_{\beta}^{\alpha}f\|_{\sigma,|\beta|}^{2}$
$\displaystyle\leq C\epsilon^{a-1}\sum_{|\alpha|\leq
1}\|\partial^{\alpha}f\|_{\sigma}^{2}+C\epsilon^{1-a}\sum_{1\leq|\alpha|\leq
2}\big{\\{}\|\partial^{\alpha}f\|_{\sigma}^{2}+\|\partial^{\alpha}(\widetilde{\rho},\widetilde{u},\widetilde{\theta})\|^{2}\big{\\}}$
$\displaystyle\quad+C\epsilon^{\frac{7}{5}+\frac{1}{15}a}(\delta+\epsilon^{a}\tau)^{-\frac{4}{3}}+C(\eta_{0}+k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a})\mathcal{D}_{2}(\tau).$
(3.73)

Here we have required that

$\displaystyle\epsilon^{a-1}\geq\epsilon^{1-a},\quad\mbox{that is}\quad a\leq
1.$ (3.74)

Integrating (3.3) with respect to $\tau$ and using (3.2) with (3.3) gives

$\displaystyle\sum_{|\alpha|+|\beta|\leq 2,|\beta|\geq
1}\big{\\{}\|\partial_{\beta}^{\alpha}f\|_{2,|\beta|}^{2}+\epsilon^{a-1}\int^{\tau}_{0}\|\partial_{\beta}^{\alpha}f\|_{\sigma,|\beta|}^{2}ds\big{\\}}$
$\displaystyle\leq C\epsilon^{a-1}\sum_{|\alpha|\leq
1}\int^{\tau}_{0}\|\partial^{\alpha}f\|_{\sigma}^{2}ds+C\epsilon^{1-a}\sum_{1\leq|\alpha|\leq
2}\int^{\tau}_{0}\big{\\{}\|\partial^{\alpha}f\|_{\sigma}^{2}+\|\partial^{\alpha}(\widetilde{\rho},\widetilde{u},\widetilde{\theta})\|^{2}\big{\\}}ds$
$\displaystyle\quad+Ck^{\frac{1}{3}}\epsilon^{\frac{6}{5}-\frac{4}{5}a}+C(\eta_{0}+k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a})\int^{\tau}_{0}\mathcal{D}_{2}(s)ds.$
(3.75)

This completes the proof of the weighted derivative estimates of the function
$f$.

##  4\. Stability and convergence rate

Based on the energy estimates derived in Section 3, in this section we are now
in a position to complete the

Proof of Theorem 1.1: By a suitable linear combination of (3.32), (3.2) and
(3.3), we can obtain

$\displaystyle\mathcal{E}_{2}(\tau)+\int^{\tau}_{0}\|\sqrt{\bar{u}_{1y}}(\widetilde{\rho},\widetilde{u}_{1},\widetilde{\theta})\|^{2}ds+\int^{\tau}_{0}\mathcal{D}_{2}(s)ds$
$\displaystyle\hskip 28.45274pt\leq
Ck^{\frac{1}{3}}\epsilon^{\frac{6}{5}-\frac{4}{5}a}+C(\eta_{0}+k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}+k^{\frac{1}{12}}\epsilon^{\frac{3}{5}a-\frac{2}{5}})\int^{\tau}_{0}\mathcal{D}_{2}(s)ds.$
(4.1)

Here $\mathcal{E}_{2}(\tau)$ and $\mathcal{D}_{2}(\tau)$ are defined by (2.2)
and (2.2), respectively. At the moment, one has to require that the second
term on the right hand side of (4) should be absorbed by the left hand side.
Thus, this leads us to impose

$\displaystyle\frac{3}{5}-\frac{2}{5}a\geq
0\quad\mbox{and}\quad\frac{3}{5}a-\frac{2}{5}\geq 0,\quad\mbox{that
is}\quad\frac{2}{3}\leq a\leq\frac{3}{2}.$ (4.2)

Due to (3.74) and (4.2), we need to require that $\frac{2}{3}\leq a\leq 1$ for
the choice of the parameter $a$ in the scaling transformation (2.1) which we
start with. Hence, by using the smallness of $k>0$, $\eta_{0}>0$ and
$\epsilon>0$, we have from (4) that

$\displaystyle\mathcal{E}_{2}(\tau)+\int^{\tau}_{0}\|\sqrt{\bar{u}_{1y}}(\widetilde{\rho},\widetilde{u}_{1},\widetilde{\theta})\|^{2}ds+\frac{1}{2}\int^{\tau}_{0}\mathcal{D}_{2}(s)ds\leq
Ck^{\frac{1}{3}}\epsilon^{\frac{6}{5}-\frac{4}{5}a}<\frac{1}{2}k^{\frac{1}{6}}\epsilon^{\frac{6}{5}-\frac{4}{5}a}.$
(4.3)

Then (4.3) implies that for $a\in[\frac{2}{3},1]$ and
$\tau_{1}\in(0,+\infty)$, one has

$\sup_{0\leq\tau\leq\tau_{1}}\mathcal{E}_{2}(\tau)<\frac{1}{2}k^{\frac{1}{6}}\epsilon^{\frac{6}{5}-\frac{4}{5}a},$
(4.4)

which is strictly stronger than (3.4). Thus the a priori assumption (3.4) can
be closed. Therefore, by the uniform a priori estimates and the local
existence of the solution, the standard continuity argument gives the
existence and uniqueness of global solutions to the Landau equation (1.1) with
initial data (3.1). Moreover, the desired estimate (1.27) holds true.

We are going to justify the convergence rate as in (1.28). By (4.4), (2.2) and
the Sobolev imbedding theorem, we get

$\sup_{0\leq\tau\leq+\infty}\\{\|(\widetilde{\rho},\widetilde{u},\widetilde{\theta})(\tau,y)\|_{L_{y}^{\infty}}+\|f(\tau,y,v)\|_{L^{\infty}_{y}L^{2}_{v}}\\}\leq
Ck^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}.$ (4.5)

On the other hand, we have by using (5), Lemma 5.3 and
$\delta=\frac{1}{k}\epsilon^{\frac{3}{5}-\frac{2}{5}a}$ with $\frac{2}{3}\leq
a\leq 1$ that

$\sup_{0\leq\tau\leq+\infty}\|\frac{\overline{G}(\tau,y,v)}{\sqrt{\mu}}\|_{L_{y}^{\infty}L_{v}^{2}}\leq
C\epsilon^{1-a}\sup_{0\leq\tau\leq+\infty}(\|\bar{u}_{1y}\|_{L^{\infty}_{y}}+\|\bar{\theta}_{y}\|_{L^{\infty}_{y}})\leq
Ck^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}.$ (4.6)

It follows from (4.5) and (4.6) that

$\displaystyle\sup_{0\leq\tau\leq+\infty}\|\frac{F-M_{[\bar{\rho},\bar{u},\bar{\theta}]}}{\sqrt{\mu}}\|_{L_{x}^{\infty}L_{v}^{2}}$
$\displaystyle\leq
C\sup_{0\leq\tau\leq+\infty}\Big{\\{}\|\frac{M-M_{[\bar{\rho},\bar{u},\bar{\theta}]}}{\sqrt{\mu}}\|_{L_{y}^{\infty}L_{v}^{2}}+\|f\|_{L_{y}^{\infty}L_{v}^{2}}+\|\frac{\overline{G}}{\sqrt{\mu}}\|_{L_{y}^{\infty}L_{v}^{2}}\Big{\\}}$
$\displaystyle\leq Ck^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a},$
(4.7)

where we have used the facts that $F=M+\overline{G}+\sqrt{\mu}f$ and (3.5). By
Lemma 5.2 and $\delta=\frac{1}{k}\epsilon^{\frac{3}{5}-\frac{2}{5}a}$ with $k$
independent of $\epsilon$ satisfying $\epsilon\ll k$, we have for $t>0$ that

$\displaystyle\|(\bar{\rho},\bar{u},\bar{\theta})(t,x)-(\rho^{R},u^{R},\theta^{R})(\frac{x}{t})\|_{L^{\infty}_{x}}\leq
C\frac{1}{k}t^{-1}\epsilon^{\frac{3}{5}-\frac{2}{5}a}\\{\ln(1+t)+|\ln\epsilon|\\}.$
(4.8)

With (4) and (4.8) in hand, for any given constant $l>0$ and all
$t\in[l,+\infty)$, there exists a constant $C_{l,k}>0$, independent of
$\epsilon$, such that

$\displaystyle\|\frac{F(t,x,v)-M_{[\rho^{R},u^{R},\theta^{R}](x/t)}(v)}{\sqrt{\mu}}\|_{L_{x}^{\infty}L_{v}^{2}}$
$\displaystyle\leq\|\frac{F-M_{[\bar{\rho},\bar{u},\bar{\theta}]}}{\sqrt{\mu}}\|_{L_{x}^{\infty}L_{v}^{2}}+\|\frac{M_{[\bar{\rho},\bar{u},\bar{\theta}]}-M_{[\rho^{R},u^{R},\theta^{R}]}}{\sqrt{\mu}}\|_{L_{x}^{\infty}L_{v}^{2}}$
$\displaystyle\leq C_{l,k}\epsilon^{\frac{3}{5}-\frac{2}{5}a}|\ln\epsilon|.$

This gives (1.28) and then completes the proof of Theorem 1.1.∎

##  5\. Appendix

In this section, we will give some basic estimates, which have been used in
the previous energy estimates. We first list some properties for the
rarefaction wave defined by (1.22) and (1.23). Then, we give some properties
of the Burnett functions and the fast velocity decay of $\overline{G}$ to
overcome the slow time decay of the term
$\|(\bar{u}_{y},\bar{\theta}_{y})\|^{2}$ in the term $P_{1}(v_{1}M_{y})$ in
(2.9). Lastly, we recall some basic properties of the collision operators and
prove some linear and nonlinear estimates in the previous energy analysis.

We now give the properties of the solution $\overline{\omega}_{\delta}(t,x)$
to Burgers equation (1.22) and the smooth approximate 3-rarefaction wave
$(\bar{\rho},\bar{u},\bar{\theta})(t,x)$ constructed by (1.23). Their proofs
can be found in [22, 30, 43].

######  Lemma 5.1.

The Burgers equation (1.22) has a unique smooth global solution
$\overline{\omega}_{\delta}(t,x)$ such that
(1) $\omega_{-}<\overline{\omega}_{\delta}(t,x)<\omega_{+}$,
$\partial_{x}\overline{\omega}_{\delta}(t,x)>0$, $\forall~{}x\in\mathbb{R}$,
$t\geq 0$.
(2) The following estimates hold for any $t>0$, $\delta>0$ and
$p\in[1,+\infty]$

$\displaystyle\|\partial_{x}\overline{\omega}_{\delta}(t,x)\|_{L^{p}(\mathbb{R}_{x})}\leq
C(\omega_{+}-\omega_{-})^{\frac{1}{p}}(\delta+t)^{-1+\frac{1}{p}},$
$\displaystyle\|\partial^{j}_{x}\overline{\omega}_{\delta}(t,x)\|_{L^{p}(\mathbb{R}_{x})}\leq
C\delta^{-j+1+\frac{1}{p}}(\delta+t)^{-1},\quad j\geq 2.$

(3) There exists a constant $\delta_{0}\in(0,1)$ such that for
$\delta\in(0,\delta_{0})$ and $t>0$

$\|\overline{\omega}_{\delta}(t,x)-\omega^{R}(\frac{x}{t})\|_{L^{\infty}(\mathbb{R}_{x})}\leq
C\delta t^{-1}\\{\ln(1+t)+|\ln\delta|\\}.$

######  Lemma 5.2.

The smooth approximate 3-rarefaction wave
$(\bar{\rho},\bar{u},\bar{\theta})(t,x)$ defined in (1.23) satisfying
(i) $\bar{u}_{2}=\bar{u}_{3}=0$, $\bar{u}_{1x}>0$, and
$\bar{\theta}_{x}=\sqrt{\frac{2}{5}}\bar{\theta}^{\frac{1}{2}}\bar{u}_{1x}$,
$\forall~{}x\in\mathbb{R}$, $t\geq 0$.
(ii) The following estimates hold for any $t>0$, $\delta>0$ and
$p\in[1,+\infty]$

$\displaystyle\|\partial_{x}(\bar{\rho},\bar{u}_{1},\bar{\theta})(t,x)\|_{L^{p}(\mathbb{R}_{x})}\leq
C(\omega_{+}-\omega_{-})^{\frac{1}{p}}(\delta+t)^{-1+\frac{1}{p}},$
$\displaystyle\|\partial_{x}^{j}(\bar{\rho},\bar{u}_{1},\bar{\theta})(t,x)\|_{L^{p}(\mathbb{R}_{x})}\leq
C\delta^{-j+1+\frac{1}{p}}(\delta+t)^{-1},\quad j\geq 2.$

(iii) There exists a constant $\delta_{0}\in(0,1)$ such that for
$\delta\in(0,\delta_{0})$ and $t>0$

$\|(\bar{\rho},\bar{u},\bar{\theta})(t,x)-(\rho^{R},u^{R},\theta^{R})(\frac{x}{t})\|_{L^{\infty}(\mathbb{R}_{x})}\leq
C\delta t^{-1}\\{\ln(1+t)+|\ln\delta|\\}.$

Since the scaling transformation $y=\epsilon^{-a}x$ and $\tau=\epsilon^{-a}t$
is considered through the proof, the following lemma is equivalent to Lemma
5.2 (ii), which will be used frequently in the previous energy estimates.

######  Lemma 5.3.

The smooth approximate 3-rarefaction wave
$(\bar{\rho},\bar{u},\bar{\theta})(t,x)$ defined in (1.23) satisfying

$\displaystyle\|\partial_{y}(\bar{\rho},\bar{u}_{1},\bar{\theta})(\epsilon^{a}\tau,\epsilon^{a}y)\|_{L^{p}(\mathbb{R}_{y})}\leq
C\epsilon^{a(1-\frac{1}{p})}(\delta+\epsilon^{a}\tau)^{-1+\frac{1}{p}},$
$\displaystyle\|\partial_{y}^{j}(\bar{\rho},\bar{u}_{1},\bar{\theta})(\epsilon^{a}\tau,\epsilon^{a}y)\|_{L^{p}(\mathbb{R}_{y})}\leq
C\epsilon^{a(j-\frac{1}{p})}\delta^{-j+1+\frac{1}{p}}(\delta+\epsilon^{a}\tau)^{-1},\quad
j\geq 2,$

for any $\tau>0$, $\delta>0$ and $p\in[1,+\infty]$.

We remark that the temporal derivatives of
$(\bar{\rho},\bar{u}_{1},\bar{\theta})(t,x)$ in Lemma 5.2 (ii) and 5 obviously
hold in terms of Euler system (1.24) and the elementary inequalities.

Recall the Burnett functions, cf. [3, 4, 8, 18, 44], defined as

$\hat{A}_{j}(v)=\frac{|v|^{2}-5}{2}v_{j}\quad\mbox{and}\quad\hat{B}_{ij}(v)=v_{i}v_{j}-\frac{1}{3}\delta_{ij}|v|^{2}\quad\mbox{for}\quad
i,j=1,2,3.$ (5.1)

Noting that $\hat{A}_{j}M$ and $\hat{B}_{ij}M$ are orthogonal to the null
space $\mathcal{N}$, we can define functions $A_{j}(v)$ and $B_{ij}(v)$ such
that $P_{0}A_{j}=0$, $P_{0}B_{ij}=0$ and

$A_{j}(\frac{v-u}{\sqrt{R\theta}})=L^{-1}_{M}[\hat{A}_{j}(\frac{v-u}{\sqrt{R\theta}})M]\quad\mbox{and}\quad
B_{ij}(\frac{v-u}{\sqrt{R\theta}})=L^{-1}_{M}[\hat{B}_{ij}(\frac{v-u}{\sqrt{R\theta}})M].$
(5.2)

The following lemma is borrowed from [11, Lemma 6.2]. Readers also refer to
[3, 4, 18, 44].

######  Lemma 5.4.

The Burnett functions have the following properties:

  * •

$-\langle\hat{A}_{i},A_{i}\rangle$ is positive and independent of i;

  * •

$\langle\hat{A}_{i},A_{j}\rangle=0$ for any $i\neq j$;
$\langle\hat{A}_{i},B_{jk}\rangle=0$ for any i, j, k;

  * •

$\langle\hat{B}_{ij},B_{kj}\rangle=\langle\hat{B}_{kl},B_{ij}\rangle=\langle\hat{B}_{ji},B_{kj}\rangle$,
which is independent of i, j, for fixed k, l;

  * •

$-\langle\hat{B}_{ij},B_{ij}\rangle$ is positive and independent of i, j when
$i\neq j$;

  * •

$\langle\hat{B}_{ii},B_{jj}\rangle$ is positive and independent of i, j when
$i\neq j$;

  * •

$-\langle\hat{B}_{ii},B_{ii}\rangle$ is positive and independent of i;

  * •

$\langle\hat{B}_{ij},B_{kl}\rangle=0$ unless either $(i,j)=(k,l)$ or $(l,k)$,
or i=j and k=l;

  * •

$\langle\hat{B}_{ii},B_{ii}\rangle-\langle\hat{B}_{ii},B_{jj}\rangle=2\langle\hat{B}_{ij},B_{ij}\rangle$
holds for any $i\neq j$.

In terms of the properties of Burnett functions, the viscosity coefficient
$\mu(\theta)$ and heat conductivity coefficient $\kappa(\theta)$ can be
represented by

$\displaystyle\mu(\theta)=$
$\displaystyle-R\theta\int_{\mathbb{R}^{3}}\hat{B}_{ij}(\frac{v-u}{\sqrt{R\theta}})B_{ij}(\frac{v-u}{\sqrt{R\theta}})dv>0,\quad
i\neq j,$ $\displaystyle\kappa(\theta)=$
$\displaystyle-R^{2}\theta\int_{\mathbb{R}^{3}}\hat{A}_{j}(\frac{v-u}{\sqrt{R\theta}})A_{j}(\frac{v-u}{\sqrt{R\theta}})dv>0.$
(5.3)

Notice that these coefficients are positive smooth functions depending only on
$\theta$.

The following lemma is borrowed from [11, Lemma 6.1], which is about the fast
velocity decay of the Burnett functions.

######  Lemma 5.5.

Suppose that $U(v)$ is any polynomial of
$\frac{v-\hat{u}}{\sqrt{R}\hat{\theta}}$ such that
$U(v)\widehat{M}\in(\ker{L_{\widehat{M}}})^{\perp}$ for any Maxwellian
$\widehat{M}=M_{[\widehat{\rho},\widehat{u},\widehat{\theta}]}(v)$ where
$L_{\widehat{M}}$ is as (1.15). For any $\varepsilon\in(0,1)$ and any multi-
index $\beta$, there exists constant $C_{\beta}>0$ such that

$|\partial_{\beta}L^{-1}_{\widehat{M}}(U(v)\widehat{M})|\leq
C_{\beta}(\widehat{\rho},\widehat{u},\widehat{\theta})\widehat{M}^{1-\varepsilon}.$

In particular, if the assumptions of (3.5) hold, there exists constant
$C_{\beta}>0$ such that

$|\partial_{\beta}A_{j}(\frac{v-u}{\sqrt{R\theta}})|+|\partial_{\beta}B_{ij}(\frac{v-u}{\sqrt{R\theta}})|\leq
C_{\beta}M^{1-\varepsilon}.$ (5.4)

Now, we shall turn to summarize some refined estimates for the collision
operators $\mathcal{L}$ and $\Gamma$ defined as (2.7). We first recall the
properties of the linearized operators $\mathcal{L}$. Note that the null space
$\mathcal{N}_{1}$ of $\mathcal{L}$ is spanned by the functions
$\\{\sqrt{\mu},v\sqrt{\mu},|v|^{2}\sqrt{\mu}\\}$ in [9, 19]. Moreover, for any
$g\in\mathcal{N}^{\perp}_{1}$, there exists $\sigma_{1}>0$ such that

$-\langle\mathcal{L}g,g\rangle\geq\sigma_{1}|g|^{2}_{\sigma}.$ (5.5)

In addition, the weighted coercivity estimates on the linearized operators
$\mathcal{L}$ can be stated as follows.

######  Lemma 5.6.

Let $|\beta|>0$ and $w$ defined in (2.12). Then for any $\eta>0$ , there
exists $C_{\eta}>0$ such that

$-\langle\partial_{\beta}\mathcal{L}g,w^{2|\beta|}\partial_{\beta}g\rangle\geq|\partial_{\beta}g|^{2}_{\sigma,|\beta|}-\eta\sum_{|\beta_{1}|\leq|\beta|}|\partial_{\beta_{1}}g|_{\sigma,|\beta_{1}|}^{2}-C_{\eta}|g|_{\sigma}^{2}.$
(5.6)

###### Proof.

The proof of (5.6) can be found in [19] and we omit the proof here for
brevity. ∎

In what follows we recall the weighted estimates on the nonlinear collision
operators $\Gamma$.

######  Lemma 5.7.

Let $w$ defined in (2.12) and $\ell\geq 0$, for arbitrarily large constant
$b>0$, one has

$\displaystyle|\langle\partial^{\alpha}\Gamma(g_{1},g_{2}),\partial^{\alpha}g_{3}\rangle|\leq
C\sum_{|\alpha_{1}|\leq|\alpha|}|\langle
v\rangle^{-b}\partial^{\alpha_{1}}g_{1}|_{2}|\partial^{\alpha-\alpha_{1}}g_{2}|_{\sigma}|\partial^{\alpha}g_{3}|_{\sigma},$
(5.7)

and

$\displaystyle|\langle\partial^{\alpha}_{\beta}\Gamma(g_{1},g_{2}),w^{2\ell}\partial^{\alpha}_{\beta}g_{3}\rangle|\leq
C\sum_{|\alpha_{1}|\leq|\alpha|}\sum_{|\beta^{\prime}|\leq|\beta_{1}|\leq|\beta|}|\langle
v\rangle^{-b}\partial^{\alpha_{1}}_{\beta^{{}^{\prime}}}g_{1}|_{2}|\partial^{\alpha-\alpha_{1}}_{\beta-\beta_{1}}g_{2}|_{\sigma,\ell}|\partial^{\alpha}_{\beta}g_{3}|_{\sigma,\ell}.$
(5.8)

###### Proof.

The proof of (5.7) and (5.8) can be found in [39, Proposition 1]. ∎

Finally, we prove some linear and nonlinear estimates, which are used in
section 3. The first estimates involving the linear terms
$\Gamma(\frac{M-\mu}{\sqrt{\mu}},f)$ and $\Gamma(f,\frac{M-\mu}{\sqrt{\mu}})$.

######  Lemma 5.8.

Let $|\alpha|+|\beta|\leq 2$ with $|\beta|\geq 1$ and $w$ defined in (2.12).
Suppose that $\mathcal{E}_{2}(\tau)\leq
k^{\frac{1}{6}}\epsilon^{\frac{6}{5}-\frac{4}{5}a}$ and
$\delta=\frac{1}{k}\epsilon^{\frac{3}{5}-\frac{2}{5}a}$ for
$a\in[\frac{2}{3},1]$ as well as the assumption (3.5) holds. If we choose
$\eta_{0}$ in (3.5) and $k$ in (3.3) small enough, for any small $\eta>0$, we
get

$\displaystyle\epsilon^{a-1}|(\partial^{\alpha}_{\beta}\Gamma(\frac{M-\mu}{\sqrt{\mu}},f),w^{2|\beta|}\partial^{\alpha}_{\beta}h)|+\epsilon^{a-1}|(\partial^{\alpha}_{\beta}\Gamma(f,\frac{M-\mu}{\sqrt{\mu}}),w^{2|\beta|}\partial^{\alpha}_{\beta}h)|$
$\displaystyle\hskip 28.45274pt\leq
C\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|^{2}_{\sigma,|\beta|}+C_{\eta}(\eta_{0}+k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a})\mathcal{D}_{2}(\tau).$
(5.9)

Moreover, for $|\alpha|\leq 1$, one has

$\displaystyle\epsilon^{a-1}$
$\displaystyle|(\partial^{\alpha}\Gamma(\frac{M-\mu}{\sqrt{\mu}},f),\partial^{\alpha}h)|+\epsilon^{a-1}|(\partial^{\alpha}\Gamma(f,\frac{M-\mu}{\sqrt{\mu}}),\partial^{\alpha}h)|$
$\displaystyle\leq
C\eta\epsilon^{a-1}\|\partial^{\alpha}h\|^{2}_{\sigma}+C_{\eta}(\eta_{0}+k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a})\mathcal{D}_{2}(\tau).$
(5.10)

###### Proof.

We only consider the first term on the left-hand side of (5.8) and the second
term of (5.8) can be handled in the same way. Since $|\alpha|+|\beta|\leq 2$
with $|\beta|\geq 1$, one has $|\alpha|\leq 1$. First of all, we have from
(5.8) that

$\displaystyle\epsilon^{a-1}|(\partial^{\alpha}_{\beta}\Gamma(\frac{M-\mu}{\sqrt{\mu}},f),w^{2|\beta|}\partial^{\alpha}_{\beta}h)|$
$\displaystyle\leq
C\epsilon^{a-1}\sum_{|\alpha_{1}|\leq|\alpha|}\sum_{|\beta^{\prime}|\leq|\beta_{1}|\leq|\beta|}\int_{\mathbb{R}}|\langle
v\rangle^{-b}\partial^{\alpha_{1}}_{\beta^{{}^{\prime}}}(\frac{M-\mu}{\sqrt{\mu}})|_{2}|\partial^{\alpha-\alpha_{1}}_{\beta-\beta_{1}}f|_{\sigma,|\beta-\beta_{1}|}|\partial^{\alpha}_{\beta}h|_{\sigma,|\beta|}dy,$
(5.11)

due to the fact that $w^{2|\beta|}\leq w^{2|\beta-\beta_{1}|}$ for
$|\beta-\beta_{1}|\leq|\beta|$. For any $\bar{\beta}\geq 0$ and $m>0$, there
exists a small constant $\varepsilon_{1}>0$ such that

$\displaystyle|\langle
v\rangle^{m}\partial_{\bar{\beta}}(\frac{M-\mu}{\sqrt{\mu}})|^{2}_{\sigma}+|\langle
v\rangle^{m}\partial_{\bar{\beta}}(\frac{M-\mu}{\sqrt{\mu}})|^{2}_{2}\leq
C_{m}\sum_{|\bar{\beta}|\leq|\beta^{\prime}|\leq|\bar{\beta}|+1}\int_{\mathbb{R}^{3}}\mu^{-\varepsilon_{1}}|\partial_{\beta^{\prime}}(\frac{M-\mu}{\sqrt{\mu}})|^{2}dv.$

For $\eta_{0}>0$ in (3.5), there exists a suitably large constant $R>0$ such
that

$\displaystyle\int_{|v|\geq
R}\mu^{-\varepsilon_{1}}|\partial_{\beta^{\prime}}(\frac{M-\mu}{\sqrt{\mu}})|^{2}dv\leq
C\eta^{2}_{0},$

and

$\displaystyle\int_{|v|\leq
R}\mu^{-\varepsilon_{1}}|\partial_{\beta^{\prime}}(\frac{M-\mu}{\sqrt{\mu}})|^{2}dv\leq
C(|\rho-1|+|u-0|+|\theta-\frac{3}{2}|)^{2}\leq C\eta^{2}_{0}.$

Thus for any $\bar{\beta}\geq 0$ and $m>0$, we deduce from the above related
estimates that

$\displaystyle|\langle
v\rangle^{m}\partial_{\bar{\beta}}(\frac{M-\mu}{\sqrt{\mu}})|^{2}_{\sigma}+|\langle
v\rangle^{m}\partial_{\bar{\beta}}(\frac{M-\mu}{\sqrt{\mu}})|^{2}_{2}\leq
C\eta^{2}_{0}.$ (5.12)

Notice that $|\alpha_{1}|\leq|\alpha|\leq 1$ in (5). If $|\alpha_{1}|=0$, we
use (5.12), (2.2) and the smallness of $\eta_{0}$ to get

$\displaystyle\epsilon^{a-1}\int_{\mathbb{R}}|\langle
v\rangle^{-b}\partial^{\alpha_{1}}_{\beta^{{}^{\prime}}}(\frac{M-\mu}{\sqrt{\mu}})|_{2}|\partial^{\alpha-\alpha_{1}}_{\beta-\beta_{1}}f|_{\sigma,|\beta-\beta_{1}|}|\partial^{\alpha}_{\beta}h|_{\sigma,|\beta|}dy$
$\displaystyle\leq
C\eta_{0}\epsilon^{a-1}\|\partial^{\alpha}_{\beta-\beta_{1}}f\|_{\sigma,|\beta-\beta_{1}|}\|\partial^{\alpha}_{\beta}h\|_{\sigma,|\beta|}$
$\displaystyle\leq\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|_{\sigma,|\beta|}^{2}+C_{\eta}\eta_{0}\mathcal{D}_{2}(\tau).$
(5.13)

If $|\alpha_{1}|=|\alpha|=1$, by the one-dimensional Sobolev imbedding theorem
and (3.5), we can deduce from Lemma 5.3 and the Cauchy inequality that

$\displaystyle\epsilon^{a-1}\int_{\mathbb{R}}|\langle
v\rangle^{-b}\partial^{\alpha_{1}}_{\beta^{{}^{\prime}}}(\frac{M-\mu}{\sqrt{\mu}})|_{2}|\partial^{\alpha-\alpha_{1}}_{\beta-\beta_{1}}f|_{\sigma,|\beta-\beta_{1}|}|\partial^{\alpha}_{\beta}h|_{\sigma,|\beta|}dy$
$\displaystyle\leq
C\epsilon^{a-1}\int_{\mathbb{R}}|\partial^{\alpha_{1}}(\rho,u,\theta)||\partial_{\beta-\beta_{1}}f|_{\sigma,|\beta-\beta_{1}|}|\partial^{\alpha}_{\beta}h|_{\sigma,|\beta|}dy$
$\displaystyle\leq
C\epsilon^{a-1}\|\partial^{\alpha_{1}}(\rho,u,\theta)\|\|\partial_{\beta-\beta_{1}}f\|^{\frac{1}{2}}_{\sigma,|\beta-\beta_{1}|}\|\partial_{y}\partial_{\beta-\beta_{1}}f\|^{\frac{1}{2}}_{\sigma,|\beta-\beta_{1}|}\|\partial^{\alpha}_{\beta}h\|_{\sigma,|\beta|}$
$\displaystyle\leq\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|^{2}_{\sigma,|\beta|}+C_{\eta}\epsilon^{a-1}\\{\epsilon^{a}(\delta+\epsilon^{a}\tau)^{-1}+\mathcal{E}_{2}(\tau)\\}\|\partial_{\beta-\beta_{1}}f\|_{\sigma,|\beta-\beta_{1}|}\|\partial_{y}\partial_{\beta-\beta_{1}}f\|_{\sigma,|\beta-\beta_{1}|}$
$\displaystyle\leq\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|^{2}_{\sigma,|\beta|}+C_{\eta}(\epsilon^{a}\delta^{-1}+k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a})\mathcal{D}_{2}(\tau).$
(5.14)

Here we have used the fact that $\mathcal{E}_{2}(\tau)\leq
k^{\frac{1}{6}}\epsilon^{\frac{6}{5}-\frac{4}{5}a}\leq
k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}$ by (3.4). Due to (3.3),
one has

$\epsilon^{a}\delta^{-1}=k\epsilon^{a}\epsilon^{-\frac{3}{5}+\frac{2}{5}a}\leq
k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}.$ (5.15)

It follows from (5.15), (5), (5) and (5) that

$\epsilon^{a-1}|(\partial^{\alpha}_{\beta}\Gamma(\frac{M-\mu}{\sqrt{\mu}},f),w^{2|\beta|}\partial^{\alpha}_{\beta}h)|\leq
C\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|^{2}_{\sigma,|\beta|}+C_{\eta}(\eta_{0}+k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a})\mathcal{D}_{2}(\tau).$
(5.16)

On the other hand, similar arguments as (5.16) imply

$\epsilon^{a-1}|(\partial^{\alpha}_{\beta}\Gamma(f,\frac{M-\mu}{\sqrt{\mu}}),w^{2|\beta|}\partial^{\alpha}_{\beta}h)|\leq
C\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|^{2}_{\sigma,|\beta|}+C_{\eta}(\eta_{0}+k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a})\mathcal{D}_{2}(\tau).$
(5.17)

This ends the proof of (5.8) in terms of (5.16) and (5.17). By (5.7) and the
similar arguments as (5.16) and (5.17), we can prove that (5.8) holds and we
omit the details for brevity. This completes the proof of Lemma 5.8. ∎

The second estimates are concerned with the nonlinear term
$\Gamma(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}})$.

######  Lemma 5.9.

Under the assumptions of Lemma 5.8 and let $|\alpha|+|\beta|\leq 2$ with
$|\beta|\geq 1$. Then for any $\eta>0$, one has

$\displaystyle\epsilon^{a-1}|(\partial^{\alpha}_{\beta}\Gamma(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}}),w^{2|\beta|}\partial^{\alpha}_{\beta}h)|$
$\displaystyle\leq
C\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|^{2}_{\sigma,|\beta|}+C_{\eta}\epsilon^{\frac{7}{5}+\frac{1}{15}a}(\delta+\epsilon^{a}\tau)^{-\frac{4}{3}}+C_{\eta}k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}\mathcal{D}_{2}(\tau).$
(5.18)

Moreover, for $|\alpha|\leq 1$, it holds that

$\displaystyle\epsilon^{a-1}|(\partial^{\alpha}\Gamma(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}}),\partial^{\alpha}h)|\leq
C\eta\epsilon^{a-1}\|\partial^{\alpha}h\|^{2}_{\sigma}+C_{\eta}\epsilon^{\frac{7}{5}+\frac{1}{15}a}(\delta+\epsilon^{a}\tau)^{-\frac{4}{3}}+C_{\eta}k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}\mathcal{D}_{2}(\tau).$
(5.19)

###### Proof.

Recall $G=\overline{G}+\sqrt{\mu}f$, a simple computation shows that

$\Gamma(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}})=\Gamma(\frac{\overline{G}}{\sqrt{\mu}},\frac{\overline{G}}{\sqrt{\mu}})+\Gamma(\frac{\overline{G}}{\sqrt{\mu}},f)+\Gamma(f,\frac{\overline{G}}{\sqrt{\mu}})+\Gamma(f,f).$
(5.20)

From (5.8), one can see that

$\displaystyle\epsilon^{a-1}|(\partial^{\alpha}_{\beta}\Gamma(\frac{\overline{G}}{\sqrt{\mu}},\frac{\overline{G}}{\sqrt{\mu}}),w^{2|\beta|}\partial^{\alpha}_{\beta}h)|$
$\displaystyle\leq
C\epsilon^{a-1}\sum_{|\alpha_{1}|\leq|\alpha|}\sum_{|\beta^{\prime}|\leq|\beta_{1}|\leq|\beta|}\int_{\mathbb{R}}|\partial^{\alpha_{1}}_{\beta^{{}^{\prime}}}(\frac{\overline{G}}{\sqrt{\mu}})|_{2,|\beta^{\prime}|}|\partial^{\alpha-\alpha_{1}}_{\beta-\beta_{1}}(\frac{\overline{G}}{\sqrt{\mu}})|_{\sigma,|\beta-\beta_{1}|}|\partial^{\alpha}_{\beta}h|_{\sigma,|\beta|}dy.$
(5.21)

In view of (2.3), (5.1) and (5.2), we can write

$\displaystyle\overline{G}=\epsilon^{1-a}\frac{\sqrt{R}\bar{\theta}_{y}}{\sqrt{\theta}}A_{1}(\frac{v-u}{\sqrt{R\theta}})+\epsilon^{1-a}\bar{u}_{1y}B_{11}(\frac{v-u}{\sqrt{R\theta}}),$

which implies that for $\beta_{1}=(1,0,0)$,

$\partial_{\beta_{1}}\overline{G}=\epsilon^{1-a}\Big{\\{}\frac{\sqrt{R}\bar{\theta}_{y}}{\sqrt{\theta}}\partial_{v_{1}}A_{1}(\frac{v-u}{\sqrt{R\theta}})(\frac{1}{\sqrt{R\theta}})+\bar{u}_{1y}\partial_{v_{1}}B_{11}(\frac{v-u}{\sqrt{R\theta}})\frac{1}{\sqrt{R\theta}}\Big{\\}},$
(5.22)

and

$\displaystyle\partial_{y}\overline{G}$
$\displaystyle=\epsilon^{1-a}\Big{\\{}\frac{\sqrt{R}\bar{\theta}_{yy}}{\sqrt{\theta}}A_{1}(\frac{v-u}{\sqrt{R\theta}})-\frac{\sqrt{R}\bar{\theta}_{y}\theta_{y}}{2\sqrt{\theta^{3}}}A_{1}(\frac{v-u}{\sqrt{R\theta}})$
$\displaystyle\quad-\frac{\sqrt{R}\bar{\theta}_{y}}{\sqrt{\theta}}\nabla_{v}A_{1}(\frac{v-u}{\sqrt{R\theta}})\cdot\frac{u_{y}}{\sqrt{R\theta}}-\frac{\sqrt{R}\bar{\theta}_{y}\theta_{y}}{\sqrt{\theta}}\nabla_{v}A_{1}(\frac{v-u}{\sqrt{R\theta}})\cdot\frac{v-u}{\sqrt{2R\theta^{3}}}$
$\displaystyle\quad+\bar{u}_{1yy}B_{11}(\frac{v-u}{\sqrt{R\theta}})-\frac{\bar{u}_{1y}u_{y}}{\sqrt{R\theta}}\cdot\nabla_{v}B_{11}(\frac{v-u}{\sqrt{R\theta}})-\frac{\bar{u}_{1y}\theta_{y}(v-u)}{2\sqrt{R\theta^{3}}}\cdot\nabla_{v}B_{11}(\frac{v-u}{\sqrt{R\theta}})\Big{\\}}.$
(5.23)

By using (5.4) and the similar expansion as (5.22) and (5), for any
$|\bar{\alpha}|\geq 1$ and $|\bar{\beta}|\geq 0$, we can obtain

$|\langle
v\rangle^{m}\partial_{\bar{\beta}}(\frac{\overline{G}}{\sqrt{\mu}})|_{2,|\bar{\beta}|}+|\langle
v\rangle^{m}\partial_{\bar{\beta}}(\frac{\overline{G}}{\sqrt{\mu}})|_{\sigma,|\bar{\beta}|}\leq
C\epsilon^{1-a}|(\bar{u}_{1y},\bar{\theta}_{y})|,$ (5.24)

and

$|\langle
v\rangle^{m}\partial^{\bar{\alpha}}_{\bar{\beta}}(\frac{\overline{G}}{\sqrt{\mu}})|_{2,|\bar{\beta}|}+|\langle
v\rangle^{m}\partial^{\bar{\alpha}}_{\bar{\beta}}(\frac{\overline{G}}{\sqrt{\mu}})|_{\sigma,|\bar{\beta}|}\leq
C\epsilon^{1-a}\\{|\partial^{\bar{\alpha}}(\bar{u}_{1y},\bar{\theta}_{y})|+\cdot\cdot\cdot+|(\bar{u}_{1y},\bar{\theta}_{y})||\partial^{\bar{\alpha}}(u,\theta)|\\},$
(5.25)

due to the fact that $|\langle
v\rangle^{m}w^{|\bar{\beta}|}\mu^{-\frac{1}{2}}M^{1-\varepsilon}|_{2}\leq C$
for any $m\geq 0$ and $\varepsilon>0$ small enough .

With the help of (5.24) and (5.25), we get from the Sobolev imbedding theorem
that

$\displaystyle\epsilon^{a-1}\int_{\mathbb{R}}|\partial^{\alpha_{1}}_{\beta^{{}^{\prime}}}(\frac{\overline{G}}{\sqrt{\mu}})|_{2,|\beta^{\prime}|}|\partial^{\alpha-\alpha_{1}}_{\beta-\beta_{1}}(\frac{\overline{G}}{\sqrt{\mu}})|_{\sigma,|\beta-\beta_{1}|}|\partial^{\alpha}_{\beta}h|_{\sigma,|\beta|}dy$
$\displaystyle\leq
C\epsilon^{a-1}\int_{\mathbb{R}}\Big{\\{}\epsilon^{1-a}\\{|\partial^{\alpha_{1}}(\bar{u}_{1y},\bar{\theta}_{y})|+|(\bar{u}_{1y},\bar{\theta}_{y})||\partial^{\alpha_{1}}(u,\theta)|\\}$
$\displaystyle\qquad\times\epsilon^{1-a}\\{|\partial^{\alpha-\alpha_{1}}(\bar{u}_{1y},\bar{\theta}_{y})|+|(\bar{u}_{1y},\bar{\theta}_{y})||\partial^{\alpha-\alpha_{1}}(u,\theta)|\\}|\partial^{\alpha}_{\beta}h|_{\sigma,|\beta|}\Big{\\}}dy$
$\displaystyle\leq
C\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|_{\sigma,|\beta|}^{2}+C_{\eta}\epsilon^{\frac{7}{5}+\frac{1}{15}a}(\delta+\epsilon^{a}\tau)^{-\frac{4}{3}}+C_{\eta}k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}\mathcal{D}_{2}(\tau),$
(5.26)

according to (5), (3.4) and $|\alpha_{1}|\leq|\alpha|\leq 1$. This together
with (5) give that

$\displaystyle\epsilon^{a-1}|(\partial^{\alpha}_{\beta}\Gamma(\frac{\overline{G}}{\sqrt{\mu}},\frac{\overline{G}}{\sqrt{\mu}}),w^{2|\beta|}\partial^{\alpha}_{\beta}h)|$
$\displaystyle\leq
C\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|_{\sigma,|\beta|}^{2}+C_{\eta}\epsilon^{\frac{7}{5}+\frac{1}{15}a}(\delta+\epsilon^{a}\tau)^{-\frac{4}{3}}+C_{\eta}k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}\mathcal{D}_{2}(\tau).$
(5.27)

For the second term of (5.20), by using (5.8), (5.25) and the Sobolev
imbedding theorem, one can deduce from (3.3) and (3.4) that

$\displaystyle\epsilon^{a-1}|(\partial^{\alpha}_{\beta}\Gamma(\frac{\overline{G}}{\sqrt{\mu}},f),w^{2|\beta|}\partial^{\alpha}_{\beta}h)|$
$\displaystyle\leq
C\epsilon^{a-1}\sum_{|\alpha_{1}|\leq|\alpha|}\sum_{|\beta_{1}|\leq|\beta|}\int_{\mathbb{R}}\epsilon^{1-a}\Big{\\{}|\partial^{\alpha_{1}}(\bar{u}_{1y},\bar{\theta}_{y})|$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad+|(\bar{u}_{1y},\bar{\theta}_{y})||\partial^{\alpha_{1}}(u,\theta)|\Big{\\}}|\partial^{\alpha-\alpha_{1}}_{\beta-\beta_{1}}f|_{\sigma,|\beta-\beta_{1}|}|\partial^{\alpha}_{\beta}h|_{\sigma,|\beta|}dy$
$\displaystyle\leq\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|^{2}_{\sigma,|\beta|}+C_{\eta}k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}\mathcal{D}_{2}(\tau).$
(5.28)

Since the third term of (5.20) shares the same estimates as (5.28). Thus, we
still deal with the last term of (5.20). In view of (5.8), the imbedding
theorem and (3.4), one has

$\displaystyle\epsilon^{a-1}|(\partial^{\alpha}_{\beta}\Gamma[f,f],w^{2|\beta|}\partial^{\alpha}_{\beta}h)|$
$\displaystyle\leq
C\epsilon^{a-1}\sum_{|\alpha_{1}|\leq|\alpha|}\sum_{|\beta^{\prime}|\leq|\beta_{1}|\leq|\beta|}\int_{\mathbb{R}}|\partial^{\alpha_{1}}_{\beta^{{}^{\prime}}}f|_{2,|\beta^{\prime}|}|\partial^{\alpha-\alpha_{1}}_{\beta-\beta_{1}}f|_{\sigma,|\beta-\beta_{1}|}|\partial^{\alpha}_{\beta}h|_{\sigma,|\beta|}dy$
$\displaystyle\leq\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|^{2}_{\sigma,|\beta|}+C_{\eta}\sqrt{\mathcal{E}_{2}(\tau)}\mathcal{D}_{2}(\tau)$
$\displaystyle\leq\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|^{2}_{\sigma,|\beta|}+C_{\eta}k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}\mathcal{D}_{2}(\tau).$
(5.29)

Collecting the estimates of (5.27), (5.28) and (5.29), we can obtain

$\displaystyle\epsilon^{a-1}|(\partial^{\alpha}_{\beta}\Gamma(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}}),w^{2|\beta|}\partial^{\alpha}_{\beta}h)|$
$\displaystyle\leq
C\eta\epsilon^{a-1}\|\partial^{\alpha}_{\beta}h\|^{2}_{\sigma,|\beta|}+C_{\eta}\epsilon^{\frac{7}{5}+\frac{1}{15}a}(\delta+\epsilon^{a}\tau)^{-\frac{4}{3}}+C_{\eta}k^{\frac{1}{12}}\epsilon^{\frac{3}{5}-\frac{2}{5}a}\mathcal{D}_{2}(\tau).$

This completes the proof of (5.18). One can deduce (5.19) by employing (5.7)
and the similar arguments as the above related estimates. This ends the proof
of Lemma 5.9. ∎

Acknowledgment: The research of Renjun Duan was partially supported by the
General Research Fund (Project No. 14302817) from RGC of Hong Kong and a
Direct Grant from CUHK. The research of Hongjun Yu was supported by the GDUPS
2017 and the NNSFC Grant 11371151.

Conflict of Interest: The authors declare that they have no conflict of
interest.

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# Cavity Jahn-Teller Polaritons in Molecules

Krishna R. Nandipati<EMAIL_ADDRESS>Theoretische
Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, Im
Neuenheimer Feld 229, 69120 Heidelberg, Germany  Oriol Vendrell
<EMAIL_ADDRESS>Theoretische Chemie, Physikalisch-Chemisches
Institut, Universität Heidelberg, Im Neuenheimer Feld 229, 69120 Heidelberg,
Germany

###### Abstract

We investigate Jahn-Teller (JT) polaritons, which emerge from the interaction
of the two normal-incidence electromagnetic modes with perpendicular
polarizations in a Fabry-Perot cavity resonator with JT active systems. These
JT polaritons are characterized by a mixed $(+/-)$–circular electromagnetic
polarization that originates from the molecular JT vibronic coupling of the
material subsystem. Consequently, the exchange of photonic and vibronic
angular momenta can be very efficient; exciting the cavity-JT system with
short, polarized light pulses results in a dynamical and oscillatory response
of the polarization in the cavity medium. Due to the photonic-vibronic
coupling, we show how the cavity polarization direction becomes frequency
dependent and does not necessarily coincide with the polarization direction of
the external fields used to drive the system.

The formation of polaritons and identification of their light-matter
composition are key to understanding cavity-controlled processes in molecules
Hutchison  _et al._ (2012); Schwartz  _et al._ (2013); Herrera and Owrutsky
(2020); Ribeiro  _et al._ (2018); Kowalewski  _et al._ (2016); Galego  _et
al._ (2016); Flick  _et al._ (2017); Feist  _et al._ (2018); Vendrell (2018a);
Herrera and Spano (2016, 2017); Vendrell (2018b); Dunkelberger  _et al._
(2022); Morigi  _et al._ (2007) and materials Ebbesen (2016); Orgiu  _et al._
(2015); Wang  _et al._ (2019). Polaritons with mixed electronic, vibrational
and photonic character promise a new handle to achieve cavity-controlled
photophysics and photochemistry of single molecules and of molecular ensembles
Herrera and Spano (2016, 2017); Vendrell (2018b). Recently, the possibility to
control and exploit the photonic angular momentum and the helicity of the
cavity photons has received much attention Gautier  _et al._ (2022); Hübener
_et al._ (2021); Feis  _et al._ (2020). Tuning the circular polarization of
the polaritons can have profound implications in cavity-molecular processes
Hübener  _et al._ (2021); Shelykh  _et al._ (2009), for instance, enantio-
selectivity Yoo and Park (2015); Feis  _et al._ (2020) and polariton ring
currents Sun  _et al._ (2022). This has lead to the design of various schemes
that rely on the use of special mirrors and cavity configurations for
controlling the polarization and helicity of cavity modes Hübener  _et al._
(2021); Feis  _et al._ (2020); Abasahl  _et al._ (2013). While the mechanisms
of vibronic coupling involving molecules and a single cavity mode are well
understood Herrera and Spano (2016, 2017); Vendrell (2018b); Ribeiro  _et al._
(2018); Dunkelberger  _et al._ (2022), vibronic interactions involving
circular cavity-modes have, to the best of our knowledge, not been addressed
so far. In highly symmetric molecular systems, electronic-state degeneracies
can be lifted by vibrational distortions of the molecular scaffold Bersuker
(2006). Hence the natural question arises, to what extent do vibronic effects
couple otherwise non-interacting circularly-polarized cavity modes.

Molecular systems with an $n$-fold rotational symmetry axis $C_{n\geq 3}$
present doubly degenerate electronic states belonging to two-dimensional
irreducible representations of $E$ symmetry. One of these $E$ subspaces always
transforms according to the $x$ and $y$ functions in real space, the two
components being labeled $E_{x}$ and $E_{y}$. There are two properties of
these $E$ electronic states that need to be considered: first, from simple
selection rules, the two orthogonal polarization directions of light
propagating parallel to the $C_{n}$ symmetry axis couple the totally symmetric
$A$ ground electronic state of the molecules with the corresponding $E_{x/y}$
component of electronically excited states. Examples are found in the planar
triazine or benzene molecules, featuring $C_{3}$ and $C_{6}$ rotation axes
perpendicular to the molecular plane, respectively. Second, inevitably, the
presence of vibronic coupling between $E_{x}$ and $E_{y}$ states in the
molecules caused by displacements along vibrational modes creates vibronic
states with mixed electronic character, i.e., the well-known Jahn-Teller (JT)
effect Bersuker (2006); Englman (1972); Longuet-Higgins  _et al._ (1958);
Köppel  _et al._ (1984); Whetten  _et al._ (1986).

We consider now that molecules with these characteristics lie with their
$C_{n}$ axes perpendicular to the mirrors of a Fabry-Perot (FP) cavity
resonator (cf. Figure 1). This configuration of the molecules and propagation
direction of the FP electromagnetic modes will result in the formation of JT
polaritons, where the vibronically coupled electronic states of the molecules
couple with the two orthogonal polarization directions of the normal-incidence
cavity modes, thus mixing them. The theoretical description of the JT
polaritonic states, their mixed polarization character, and the effect of the
latter on the dynamics of the cavity polarization degree of freedom under
external perturbations, is the main subject of our investigation.

Without loss of generality, we base our description of the mechanism of
photonic-vibronic mixing of the two cavity polarization directions by
considering the paradigmatic $(E\times e)$ JT Hamiltonian, where the $E$
electronic states are coupled by the doubly degenerate $e$ vibrational modes
Jahn and Teller (1937); Longuet-Higgins  _et al._ (1958); Englman (1972);
Bersuker (2006); Köppel  _et al._ (1984). The $(E\times e)$ coupling mechanism
occurs, for example, in molecules with a $C_{3}$ symmetry axis. The general
properties of the $(E\times e)$ JT Hamiltonian have been well understood
Moffitt and Liehr (1957); Longuet-Higgins  _et al._ (1958); Sturge (1968);
Englman (1972); Bersuker (2006) in molecular spectroscopy as a premise to
approach complex multi-mode vibronic interactions in polyatomic systems Köppel
_et al._ (1984), thus making it an ideal model for investigating molecular
non-adiabatic effects with cavity modes.

Figure 1:  Schematic of a JT active system situated inside the FP cavity: The
symmetry axis of the JT system, represented by the triangular shaped molecule,
and the wave vector $\vec{k}$ of the $x(y)$-polarized cavity light point along
the z-axis. The cavity is schematically represented by mirrors (in gray)
separated by a distance $L$. The $x(y)$-polarized cavity mode interacts via
dipole coupling with the $E_{x}$($E_{y}$) degenerate electronic excitation of
the JT system.

_Cavity Jahn-Teller Hamiltonian –_ The cavity-JT (CJT) Hamiltonian is
comprised of the JT molecular system plus the two normal-incidence modes of
the FP cavity with polarization directions $(\vec{e}_{x},\vec{e}_{y})$,
$\hat{H}=\hat{H}_{JT}+\hat{H}_{C}$, where $\hat{H}_{JT}$ is the linear
$(E\times e)$ JT Hamiltonian and $\hat{H}_{C}$ is the cavity-Hamiltonian with
light-matter coupling (see schematic in Figure 1). $\hat{H}_{JT}$ is
represented in the diabatic basis of the excited $E$ electronic states
$|E_{x(y)}(\bm{Q}_{0})\rangle$, which are energetically well separated from
the ground electronic state $|A(\bm{Q}_{0})\rangle$. These electronic states
are defined to be the eigenstates of $\hat{H}_{JT}-\hat{T}_{N}$, the clamped-
nuclei Hamiltonian, at the reference nuclear geometry $\bm{Q}_{0}$. For
simplicity, in the following we drop the indication $\bm{Q}_{0}$ inside the
diabatic electronic state kets.

In the electronic $\\{|E_{+}\rangle,|A\rangle,|E_{-}\rangle\\}$ basis, the
molecular JT Hamiltonian reads

$\displaystyle\hat{H}_{JT}$
$\displaystyle=\left(\hat{T}_{N}+\frac{\omega}{2}\rho^{2}\right)\mathbf{1}_{e}$
(1) $\displaystyle+\epsilon\left(|E_{+}\rangle\langle
E_{+}|+|E_{-}\rangle\langle E_{-}|\right)$
$\displaystyle+\kappa\,\rho\,\left(e^{i\phi}|E_{-}\rangle\langle
E_{+}|+\text{h.c.}\right),$

where $\hat{T}_{N}$ is the vibrational kinetic energy, $(\rho,\phi)$
correspond to the polar representation of the $e$ vibrational modes, $\rho
e^{i\phi}=Q_{x}+iQ_{y}$, and $E_{r}$ refers to the complex basis
$|E_{\pm}\rangle=\left(|E_{x}\rangle\pm i|E_{y}\rangle\right)/\sqrt{2}$.
$\mathbf{1}_{e}$ is the unit operator in the space of the electronic states,
$\omega$, $\epsilon$ and $\kappa$ are the frequency of the $e$ modes, the
energy of the $E$ electronic states at the reference geometry $\bm{Q}_{0}$,
and the linear JT coupling parameter, respectively. This representation of
$\hat{H}_{JT}$ makes particularly transparent that the vibronic coupling
results in an exchange of angular momentum between the electronic subspace and
the pseudo-rotation $\phi$ of the molecular scaffold, where the angular
momentum perpendicular to the $(x,y)$-plane for the pseudo-rotation of the
vibrational modes is given as

$\displaystyle\hat{L}_{z}=-i\hbar\frac{\partial}{\partial\phi},$ (2)

and where one can introduce an electronic angular momentum-like operator
within the $E$-subspace,

$\displaystyle\hat{S}_{z}=\hbar\left(|E_{+}\rangle\langle
E_{+}|-|E_{-}\rangle\langle E_{-}|\right).$ (3)

The vibronic angular momentum of the molecular JT subsystem is defined as
$\hat{J}_{\text{JT}}=2\hat{L}_{z}+\hat{S}_{z}$, where the factor $2$ in front
of $\hat{L}_{z}$ follows from the $\pi$-radians periodicity of the vibrational
pseudo-rotation instead of $2\pi$ Longuet-Higgins  _et al._ (1958). It is a
simple exercise to check that $[\hat{H}_{JT},\hat{J}_{\text{JT}}]=0,$ the
well-known symmetry of the linear $(E\times e)$ JT Hamiltonian resulting in
the double degeneracy of the vibronic spectrum. It is worth noting here that
the spectrum of the _quadratic_ JT Hamiltonian is also doubly degenerate due
to the remaining symmetry, although $\hat{J}_{\text{JT}}$ ceases to be a
conserved quantity Longuet-Higgins  _et al._ (1958).

The coupling of each cavity polarization to the corresponding electronic
excitation is now considered within the Condon approximation of constant
transition dipole Tannor (2007), and within the rotating wave approximation
Crisp (1991)

$\displaystyle\hat{H}_{C}$
$\displaystyle=\hbar\omega_{c}\left(\hat{a}_{+}^{\dagger}\hat{a}_{+}+\hat{a}_{-}^{\dagger}\hat{a}_{-}\right)$
(4) $\displaystyle+\frac{\Omega}{2}\left(\hat{a}_{+}^{\dagger}|A\rangle\langle
E_{+}|+\hat{a}_{-}^{\dagger}|A\rangle\langle E_{-}|+\text{h.c.}\right).$

Here $\hbar\omega_{c}$ is the photon energy of the cavity modes, and
$\Omega/2$ is the coupling strength between the molecule and the cavity modes
in energy units. Hence, at zero detuning the Rabi splitting takes on the value
$\Omega$ in units of energy. For the representation of $\hat{H}_{C}$, we have
introduced circular cavity modes as linear combinations of the $(x,y)$ linear
polarizations, $\hat{a}_{\pm}=(\hat{a}_{x}\mp i\hat{a}_{y})/\sqrt{2}$ and
$\hat{a}_{\pm}^{\dagger}=(\hat{a}_{x}^{\dagger}\pm
i\hat{a}_{y}^{\dagger})/\sqrt{2}$, where $\hat{a}_{\pm}^{(\dagger)}$
annihilate (create) cavity photons with $\pm\hbar$ angular momentum Crisp
(1991). Here we should be reminded that the $(+/-)$–circular modes of a cavity
with conventional mirrors have well-defined angular momentum (and thus a well-
defined direction of rotation of the electric and magnetic fields in the plane
of the cavity as seen by an external observer) but have no helicity, the
projection of the angular momentum of a particle onto its linear momentum
Gautier  _et al._ (2022); Hübener  _et al._ (2021). Thus, we will refer the
angular momenta associated with these circular cavity modes in the following
to the cavity polarizations, but we caution the reader that the cavity modes
we are considering are not chiral.

After having introduced circular cavity modes, the photonic angular momentum
of the FP cavity perpendicular to the $(x,y)$ reads Crisp (1991)

$\displaystyle\hat{l}_{z}$
$\displaystyle=\hbar\left(\hat{a}_{+}^{\dagger}\hat{a}_{+}-\hat{a}_{-}^{\dagger}\hat{a}_{-}\right).$
(5)

Introducing the total vibronic-photonic angular momentum
$\hat{J}=2\hat{L}_{z}+\hat{S}_{z}+\hat{l}_{z}$ and using the JT commutation
relation introduced above one finds out that $\hat{J}$ is conserved for the
CJT Hamiltonian, $[\hat{H},\hat{J}]=0.$ It is straightforward to show that
this commutation relation holds as well for molecular ensembles, where then
$\hat{H}=\sum_{M}(\hat{H}_{JT}^{(M)}+\hat{H}_{C}^{(M)})$ and
$\hat{J}_{\text{tot}}=\sum_{M}(2\hat{L}_{z}^{(M)}+\hat{S}_{z}^{(M)})+\hat{l}_{z}$.
Thus, the vibronic-photonic eigenstates of the CJT Hamiltonian can be cast as
eigenstates of the total vibronic plus photonic angular momenta of the system
and can be characterized by the quantum number $j=2L+s+l$. Here $L$ is the
angular quantum number of the vibrational pseudo-rotation, $s=0,\pm 1$
represents the electronic angular momentum of the $A$ and $E_{\pm}$ electronic
states, respectively, and $l=n_{+}-n_{-}$ is the angular momentum quantum
number of the cavity photons.

Figure 2: JT polaritonic PESs as a function of the $e$ normal mode coordinates
(dimensionless), $x=\rho cos(\phi)$ and $y=\rho sin(\phi)$, plotted for
$\Delta$ =0 and $\Omega=0.75$ eV. The PESs are color-coded with the molecular
and photonic contributions at a given $(x,y)-$The color bar on the right shows
the photonic contribution on a 0 to 1 scale, where 1 indicates 100% of
photonic character.

_Jahn-Teller Polaritons–_ We are primarily interested in describing the strong
light-matter coupling regime, where light-matter coupling dominates over the
vibronic non-adiabatic coupling, i.e. the Rabi splitting $\Omega$ is at least
of the order of the width of the JT spectrum of the bare molecule. Hence, we
introduce the basis of polaritonic states of the electronic-photonic subsystem
(in the single-excitation subspace),

$\displaystyle|\Lambda_{\pm}^{(+1)}\rangle$
$\displaystyle=\left(|A,1_{+},0_{-}\rangle\pm|E_{+},0_{+},0_{-}\rangle\right)/\sqrt{2}$
(6) $\displaystyle|\Lambda_{\pm}^{(-1)}\rangle$
$\displaystyle=\left(|A,0_{+},1_{-}\rangle\pm|E_{-},0_{+},0_{-}\rangle\right)/\sqrt{2}$

The superscript $p=\pm 1$ in $|\Lambda_{\pm}^{(p)}\rangle$ indicates the
combined electronic-photonic angular momentum, i.e. $p=s+l$. The subscript
$\pm$ indicates the upper/lower polaritonic state. In its matrix
representation and in the polaritonic states basis (in the given order)
$\\{|\Lambda_{+}^{(+1)}\rangle,|\Lambda_{+}^{(-1)}\rangle,|A,0_{+},0_{-}\rangle,|\Lambda_{-}^{(+1)}\rangle,|\Lambda_{-}^{(-1)}\rangle\\}$,
the CJT Hamiltonian reads

$\displaystyle\hat{H}=\left(\hat{T}_{N}+\frac{\omega}{2}\rho^{2}\right)\mathbf{1}_{5\times
5}+\frac{\kappa\rho}{2}\left[\begin{array}[]{ccccc}0&e^{-i\phi}&0&0&-e^{-i\phi}\\\
e^{i\phi}&0&0&-e^{i\phi}&0\\\ 0&0&0&0&0\\\ 0&-e^{-i\phi}&0&0&e^{-i\phi}\\\
-e^{i\phi}&0&0&e^{i\phi}&0\\\
\end{array}\right]+\left[\begin{array}[]{ccccc}\Sigma+\Omega/2&0&0&\Delta&0\\\
0&\Sigma+\Omega/2&0&0&\Delta\\\ 0&0&0&0&0\\\ \Delta&0&0&\Sigma-\Omega/2&0\\\
0&\Delta&0&0&\Sigma-\Omega/2\\\ \end{array}\right].$ (17)

The second term in in Eq. (17) decribes the vibronic coupling between the two
upper and the two lower polaritonic basis states. The upper-left $2\times 2$
submatrix describes the vibronic mixing of the upper $|\Lambda_{+}^{(\pm
1)}\rangle$, whereas the lower-right $2\times 2$ submatrix describes the
vibronic mixing of the lower $|\Lambda_{-}^{(\pm 1)}\rangle$ polaritonic
states, respectively. The third term corresponds to the matrix representation
of $\hat{H}_{C}$ (cf. Eq. 4), where $\Sigma=(\hbar\omega_{c}+\epsilon)/2$, and
$\Delta=(\hbar\omega_{c}-\epsilon)/2$ is half the cavity detuning.

The diagonalization of the clamped-nuclei CJT Hamiltonian,
$\hat{H}-\hat{T}_{N}$, as a function of the vibrational displacements
$(\rho,\phi)$ results in the coupled JT polaritonic potential energy surfaces
(PESs), which we refer to henceforth as JT polaritons, and which are
represented in Fig. 2 for $\Delta=0$ and $\Omega=0.75$ eV. The two upper JT
polaritonic surfaces are connected by a conical intersection (CI) that has
been inherited from their molecular contribution. The coloring of the JT
polaritonic surfaces indicates their photonic (red) and molecular (blue)
contributions, with the strongest light-matter mixing at the CI. Likewise, a
CI connects the two lower JT polaritonic surfaces. This picture makes it clear
that the molecular non-adiabatic coupling is ultimately responsible for the
mixing of cavity photons with positive and negative angular momentum (or
circular polarization).

_Spectrum and properties of the cavity Jahn-Teller Hamiltonian –_
Diagonalization of the CJT Hamiltonian $\hat{H}$ (cf. Eq. (17)) yields the
doubly degenerate eigenstates supported by JTP belonging to $j=\pm 1$ blocks:

$\displaystyle|k_{j}\rangle$
$\displaystyle=\sum_{p,r=+,-}|\Xi_{p,r}^{(k_{j})}\rangle\otimes|\Lambda_{r}^{(p)}\rangle,$
(18)

where $|\Xi_{p,r}^{(k_{j})}\rangle$ is the vibrational contribution (since the
total electronic and photonic angular momentum $p$ can only take the values
$\pm 1$, we refer to it as $\pm$ henceforth). The expectation value of the
angular momentum of the cavity photons in the eigenstate $|k_{j}\rangle$ is
given by

$\displaystyle\mathcal{P}_{k}^{(j)}=\langle k_{j}|\hat{l}_{z}|k_{j}\rangle$
$\displaystyle=\langle\Xi_{+,+}^{(k_{j})}|\Xi_{+,+}^{(k_{j})}\rangle+\langle\Xi_{+,-}^{(k_{j})}|\Xi_{+,-}^{(k_{j})}\rangle$
(19)
$\displaystyle-\langle\Xi_{-,+}^{(k_{j})}|\Xi_{-,+}^{(k_{j})}\rangle-\langle\Xi_{-,-}^{(k_{j})}|\Xi_{-,-}^{(k_{j})}\rangle.$

By symmetry, $\mathcal{P}_{k}^{(1)}=-\mathcal{P}_{k}^{(-1)}$.

Figure 3:  (a) The spectrum of the ($E\times e$) JT model of sym-triazine (in
black) and the spectrum of the same system coupled to the FP cavity (in
orange) in Fig. 1. The $\omega_{c}$ is set at the energy of the most optically
bright state of the bare JT spectrum (indicated with vertical blue line) of
the triazine, which is lying below the CI energy $\epsilon$ (marked $\times$
on the energy axis). (b) The net polarizations presented by the most optically
bright $k$ state within $j=-1$ block as a function of the JT coupling
($\kappa/\omega$), at selected cavity-molecule couplings
($\Omega/\omega_{c}$).

The cavity angular momentum (or cavity polarization) $\mathcal{P}_{k}^{(j)}$
is strongly dependent on the JT coupling, which determines the mixing of both
polarization directions in each eigenstate. We illustrate the dependency of
the cavity polarization on a vibronic coupling model with parameters
$\omega=0.003$ a.u. ($\approx 660$ cm-1) and $\epsilon$ = 7 eV (cf. Eq. 1)
which are in the typical range for JT active vibrational modes and vertical
electronic transitions organic molecules such as benzene ($D_{6h}$) Worth
(2007) and sym-triazine ($D_{3h}$) Whetten  _et al._ (1986).

Figure 3a presents the spectrum of the CJT model of sym-triazine with the JT
coupling, $\kappa/\omega$ = 2.2 Whetten  _et al._ (1986) and the cavity
coupling, $\Omega/\omega_{c}$ = 0.05. The states in the spectrum are doubly
degenerate, characterised by $j=\pm 1$ and the spectrum splits into lower
polariton (LP) and upper polariton (UP) branches that are separated by
$\Omega$. The photonic angular momentum $\mathcal{P}_{k}^{(-1)}$ for the most
optically bright $k_{-1}$ state as a function of JT coupling
($\kappa/\omega$), at different cavity-molecule couplings ($\Omega/\omega_{c}$
$\geq$ 0.5 eV) are shown in Figure 3b. The magnitude of the cavity
polarization $|\mathcal{P}|$ is strongly affected by the JT coupling
$\kappa/\omega$. At strong vibronic couplings, $\kappa/\omega$ $>1$, the net
polarization in the cavity is almost suppressed to 10%, even at the strongest
cavity coupling $\Omega/\omega_{c}$. Note that $|\mathcal{P}|$ for $\kappa=0$
is determined by the cavity-molecule detuning parameter $\Delta$ (here
$2\Delta$ = 0.15 a.u.).

Figure 4:  (a) The spectrum of the CJT system for strong JT coupling
$(\kappa/\omega)$ and cavity coupling $(\Omega/\omega_{c})$. The spectrum is
superimposed with the spectral representation of 5 fs (shaded in orange) and
400 fs (shaded in yellow, purple, cyan, red and blue, from left to right) RCPs
with resonant frequencies at, above and below the CI (marked $\times$ on the
energy axis). (b, c) Time-dependent cavity polarization
($\tilde{P}$=$\mathcal{P}_{k_{-1}}(t)/\hbar/p_{ex}(t_{f})$) generated by the
pulses shown in (a), respectively. Here the line(s) color follows that of the
excitation pulse(s) in (a). The time-dependent net polarizations are
normalized by the total excitation probability $p_{ex}(t_{f})$ after the pulse
(cf. main text).

_Interaction of the cavity Jahn-Teller system with circularly polarized light
–_ We now couple the CJT system to an external circularly polarized (CP) pulse
propagating along the $z$-direction, and consider a stronger JT coupling,
$\kappa/\omega\approx 8.8$, than before. In this strong coupling regime, the
electronic states are highly mixed, which in turn contributes to a strong
polarization mixing in the cavity. We consider a cavity coupling of
$\Omega/\omega_{c}$ $\approx 0.1$. Depending on the bandwidth of the CP pulse,
it can be made resonant with a single $|k_{j}\rangle$ state, or a group of
states. The interaction of the CP pulse with the CJT system is treated in the
electric dipole approximation,
$-(\hat{\mu}_{x}\mathcal{E}_{x}(t)+\hat{\mu}_{y}\mathcal{E}_{y}(t))$. The
transition dipole operators $\hat{\mu}_{x}$ and $\hat{\mu}_{y}$ are assumed to
couple the cavity ground state with the one-photon states of either $x$ or $y$
polarization, respectively. The left and right CP pulses, namely LCP and RCP,
are taken as Fourier-limited of Gaussian shape. Details are found in Ref.
Nandipati and Vendrell (2021). Dipolar transitions initiated from the GS
$|A,0_{+},0_{-}\rangle$ fulfill the selection rule $\Delta j=\pm 1$ and CP
radiation can only induce transitions $(0\leftrightarrow 1)$ for LCP and
$(0\leftrightarrow-1)$ for RCP Nandipati and Vendrell (2020). We consider the
CP pulses weak enough to remain in first-order perturbation so that the
generated polarization is intensity-independent when normalized by the
excited-state population $p_{ex}(t_{f})=(1-p_{0}(t_{f}))$, where $p_{0}(t)$ is
the population of the absolute ground state and $t_{f}$ is the final time of
the simulation after the pulse is over.

We apply both long (400 fs, shaded in yellow/purple/cyan/red/blue) and short
(5 fs, shaded in orange) RCP pulses whose spectra are shown superimposed in
Figure 4a. At about 7.4 and 7.52 eV, the long pulses (red/blue) are resonant
with the eigenstates featuring successively the largest transition dipole
moments in the UP branch, located above the CI. The bandwidths of the pulses
are narrow enough to target only these eigenstates, which result in a
stationary polarization after their action is over, as indicated by the red
and blue curves in Figure 4b. The most optically bright state presents the net
polarization $\tilde{\mathcal{P}}_{k_{-1}}$ about 0.009 $\hbar$ in clockwise
direction (cf. solid red curve) which is in opposite direction to the
polarization presented by the second-most optically bright state of about
-0.022 $\hbar$ (cf. solid blue curve). The long pulses (yellow/purple/cyan)
targeting the states in the LP branch, located below the CI, result now in
polarizations different polarization directions and with different magnitudes,
as shown in Figure 4b, in the same colors as that of the corresponding pulses.
Thus, different spectral regions of the CJT system feature opposite cavity-
polarization directions and these states can be addressed by pulses of fixed
polarization (e.g. RCP) with different frequencies. This is a direct
consequence of the vibronic coupling effects and of the mixing of the photonic
and vibronic angular momenta. Therefore, an RCP pulse does not necessarily
result in $(-)$ polarization of the cavity, as is always the case in the
$\kappa\to 0$ limit.

Both the LP and UP branches together can be spanned by a pulse as short as 5
fs centered at 7 eV. Since the eigenstates inside the pulse bandwidth have
either $(+)$ or $(-)$ net expectation of the photonic angular momentum, the
resulting polarization is highly oscillatory (cf. Figure 4c). The oscillatory
circular polarization dynamics of the CJT system reaches polarization
amplitudes markedly larger than the maximum polarization achieved by long
pulses targeting single eigenstates.

_Conclusions –_ In summary, we have described the mechanism of formation of
mixed-polarization JT polaritonic states by the interaction of JT-active
systems with the two degenerate cavity modes at normal incidence of a FP
resonator. The upper and lower JT polaritons present CIs that strongly mix the
two polarization directions of the cavity photon. This vibronic coupling
suppresses the maximum degree of photonic angular momentum of individual
eigenstates of the polaritonic system. However, short circularly polarized
external pulses can trigger highly-oscillatory polarization dynamics as a
superposition of the polaritonic eigenstates. These findings can result in
schemes to achieve chiral environments in cavities without resorting to
special types of mirrors. The ultrafast polarization oscillations resulting
from the vibronic-photonic coupling can be verified by pump-probe
spectroscopic measurements involving CP pulses Svoboda  _et al._ (2022);
Baykusheva  _et al._ (2016).

###### Acknowledgements.

This work has been supported by the collaborative research center “SFB 1249:
N-Heteropolyzyklen als Funktionsmaterialen” of the German Research Foundation
(DFG). We thank professor Wolfgang Domcke for useful comments on the
manuscript.

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