$\displaystyle\overset{(c)}{\leq}\frac{\alpha}{\beta}L_{\omega}L_{g}\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|^{2}\right]+\frac{\alpha}{2\beta}L_{\omega}\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|^{2}\right]+\frac{\alpha}{8\beta}L_{\omega}\mathbb{E}\left[\left\|\nabla
J(\theta_{\tau_{\beta}})\right\|^{2}\right]+\frac{\alpha}{\beta}\|z_{0}\|L_{\omega}C_{g}\beta$
$\displaystyle\quad+\frac{\alpha^{2}}{\beta}\tau_{\beta}L_{k}\|z_{0}\|C_{g}+\frac{2\alpha}{\beta}L_{\omega}C_{g}\mathbb{E}\left[\left\|z_{\tau_{\beta}}-z_{0}\right\|\right]+\frac{2\alpha^{2}}{\beta}C_{g}^{2}D_{\omega}\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|\right]$
$\displaystyle=\left(\frac{\alpha}{\beta}L_{\omega}L_{g}+\frac{\alpha}{2\beta}L_{\omega}\right)\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|^{2}\right]+\frac{2\alpha^{2}}{\beta}C_{g}^{2}D_{\omega}\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|\right]+\frac{\alpha}{8\beta}L_{\omega}\mathbb{E}\left[\left\|\nabla
J(\theta_{\tau_{\beta}})\right\|^{2}\right]$ $\displaystyle\quad+\left(\alpha
L_{\omega}C_{g}+\frac{\alpha^{2}}{\beta}\tau_{\beta}L_{k}C_{g}\right)\|z_{0}\|+\frac{2\alpha}{\beta}L_{\omega}C_{g}\mathbb{E}\left[\left\|z_{\tau_{\beta}}-z_{0}\right\|\right],$
(118)

where $(a)$ is from the Mean-Value theorem and
$\hat{\theta}_{\tau_{\beta}}=c\theta_{\tau_{\beta}}+(1-c)\theta_{\tau_{\beta}+1}$
for some $c\in[0,1]$, $(b)$ is from Lemmas 1 and 2, $(c)$ is due to the fact
that
$\left\|\mathbb{E}\left[G_{t+1}(\theta_{0},\omega(\theta_{0}))+\frac{\nabla
J(\theta_{0})}{2}\right]\right\|\leq C_{g}\beta$ for any $t\geq\tau_{\beta}$
and $\|\theta_{\tau_{\beta}}-\theta_{0}\|\leq\alpha\tau_{\beta}C_{g}$, and
$L_{k}=2C_{g}D_{\omega}+\left(L_{J}+\frac{L_{g}^{\prime}}{2}\right)L_{\omega}$
is the Lipschitz constant of
$\nabla\omega(\theta)\left(G_{t+1}(\theta,\omega(\theta))+\frac{\nabla
J(\theta)}{2}\right)$, and $L_{g}^{\prime}$ is the Lipschitz constant of
$G_{t+1}(\theta,\omega(\theta))$.

Our next step is to rewrite the bound in (C.1.2) using $\|z_{\tau_{\beta}}\|$.
Note that from Lemma 4, we have that

$\displaystyle\|z_{0}\|\leq\left\|z_{\tau_{\beta}}-z_{0}\right\|+\left\|z_{\tau_{\beta}}\right\|\leq(1+4\beta\tau_{\beta}C_{\phi}^{2})\left\|z_{\tau_{\beta}}\right\|+4\beta\tau_{\beta}(b_{\max}+L_{\omega}C_{g}).$
(119)

Plugging in (C.1.2), it follows that

$\displaystyle\left|\mathbb{E}\left[z_{\tau_{\beta}}^{\top}\frac{\omega(\theta_{\tau_{\beta}})-\omega(\theta_{\tau_{\beta}+1})}{\beta}\right]\right|$
$\displaystyle\leq\left(\frac{\alpha}{\beta}L_{\omega}L_{g}+\frac{\alpha}{2\beta}L_{\omega}\right)\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|^{2}\right]+\frac{2\alpha^{2}}{\beta}C_{g}^{2}D_{\omega}\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|\right]+\frac{\alpha}{8\beta}L_{\omega}\mathbb{E}\left[\left\|\nabla
J(\theta_{\tau_{\beta}})\right\|^{2}\right]$ $\displaystyle\quad+\left(\alpha
L_{\omega}C_{g}+\frac{\alpha^{2}}{\beta}\tau_{\beta}L_{k}C_{g}\right)\|z_{0}\|+\frac{2\alpha}{\beta}L_{\omega}C_{g}\mathbb{E}\left[\left\|z_{\tau_{\beta}}-z_{0}\right\|\right]$
$\displaystyle\leq\left(\frac{\alpha}{\beta}L_{\omega}L_{g}+\frac{\alpha}{2\beta}L_{\omega}\right)\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|^{2}\right]+\frac{2\alpha^{2}}{\beta}C_{g}^{2}D_{\omega}\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|\right]+\frac{\alpha}{8\beta}L_{\omega}\mathbb{E}\left[\left\|\nabla
J(\theta_{\tau_{\beta}})\right\|^{2}\right]$ $\displaystyle\quad+\left(\alpha
L_{\omega}C_{g}+\frac{\alpha^{2}}{\beta}\tau_{\beta}L_{k}C_{g}\right)\left((1+4\beta\tau_{\beta}C_{\phi}^{2})\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|\right]+4\beta\tau_{\beta}(b_{\max}+L_{\omega}C_{g})\right)$
$\displaystyle\quad+\frac{2\alpha}{\beta}L_{\omega}C_{g}\left(\mathbb{E}\left[4\beta\tau_{\beta}C_{\phi}^{2}\left\|z_{\tau_{\beta}}\right\|\right]+4\beta\tau_{\beta}(b_{\max}+L_{\omega}C_{g})\right)$
$\displaystyle=\left(\frac{\alpha}{\beta}L_{\omega}L_{g}+\frac{\alpha}{2\beta}L_{\omega}\right)\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|^{2}\right]$
$\displaystyle\quad+\left(\frac{2\alpha^{2}}{\beta}C_{g}^{2}D_{\omega}+\left(\alpha
L_{\omega}C_{g}+\frac{\alpha^{2}}{\beta}\tau_{\beta}L_{k}C_{g}\right)(1+4\beta\tau_{\beta}C_{\phi}^{2})+8\alpha\tau_{\beta}L_{\omega}C_{g}C_{\phi}^{2}\right)\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|\right]$
$\displaystyle\quad+\frac{\alpha}{8\beta}L_{\omega}\mathbb{E}\left[\left\|\nabla
J(\theta_{\tau_{\beta}})\right\|^{2}\right]+\left(\alpha
L_{\omega}C_{g}+\frac{\alpha^{2}}{\beta}\tau_{\beta}L_{k}C_{g}\right)\left(4\beta\tau_{\beta}(b_{\max}+L_{\omega}C_{g})\right)$
$\displaystyle\quad+{8\alpha\tau_{\beta}}L_{\omega}C_{g}(b_{\max}+L_{\omega}C_{g})$
$\displaystyle\triangleq
R_{3}\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|^{2}\right]+P_{3}\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|\right]+Q_{3}+\frac{\alpha}{8\beta}L_{\omega}\mathbb{E}\left[\left\|\nabla
J(\theta_{\tau_{\beta}})\right\|^{2}\right],$ (120)

where
$R_{3}=\left(\frac{\alpha}{\beta}L_{\omega}L_{g}+\frac{\alpha}{2\beta}L_{\omega}\right)=\mathcal{O}\left(\frac{\alpha}{\beta}\right)$,
$P_{3}=\bigg{(}\frac{2\alpha^{2}}{\beta}C_{g}^{2}D_{\omega}+\left(\alpha
L_{\omega}C_{g}+\frac{\alpha^{2}}{\beta}\tau_{\beta}L_{k}C_{g}\right)(1+4\beta\tau_{\beta}C_{\phi}^{2})+8\alpha\tau_{\beta}L_{\omega}C_{g}C_{\phi}^{2}\bigg{)}=\mathcal{O}(\alpha\tau_{\beta})$
and $Q_{3}=\left(\alpha
L_{\omega}C_{g}+\frac{\alpha^{2}}{\beta}\tau_{\beta}L_{k}C_{g}\right)\left(4\beta\tau_{\beta}(b_{\max}+L_{\omega}C_{g})\right)+{8\alpha\tau_{\beta}}L_{\omega}C_{g}(b_{\max}+L_{\omega}C_{g})=\mathcal{O}(\alpha\tau_{\beta})$.

Then we combine all three bounds in (C.1.2), (C.1.2) and (C.1.2), and it
follows that

$\displaystyle\left|\mathbb{E}\left[z_{\tau_{\beta}}^{\top}\left(-A_{\theta_{\tau_{\beta}}}z_{\tau_{\beta}}-\frac{1}{\beta}\left(z_{\tau_{\beta}+1}-z_{\tau_{\beta}}\right)\right)\right]\right|$
$\displaystyle\leq(R_{1}+R_{3})\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|^{2}\right]+(P_{1}+P_{2}+P_{3})\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|\right]+(Q_{1}+Q_{2}+Q_{3})$
$\displaystyle\quad+\frac{\alpha}{8\beta}L_{\omega}\mathbb{E}\left[\left\|\nabla
J(\theta_{\tau_{\beta}})\right\|^{2}\right],$ (121)

Finally due to the fact that $x\leq x^{2}+1$, $\forall x\in\mathbb{R}$, it
follows that

$\displaystyle\left|\mathbb{E}\left[z_{\tau_{\beta}}^{\top}\left(-A_{\theta_{\tau_{\beta}}}z_{\tau_{\beta}}-\frac{1}{\beta}\left(z_{\tau_{\beta}+1}-z_{\tau_{\beta}}\right)\right)\right]\right|$
$\displaystyle\leq(R_{1}+R_{3}+P_{1}+P_{2}+P_{3})\mathbb{E}\left[\left\|z_{\tau_{\beta}}\right\|^{2}\right]+(Q_{1}+Q_{2}+Q_{3}+P_{1}+P_{2}+P_{3})$
$\displaystyle\quad+\frac{\alpha}{8\beta}L_{\omega}\mathbb{E}\left[\left\|\nabla
J(\theta_{\tau_{\beta}})\right\|^{2}\right].$ (122)

This completes the proof. ∎

####  C.2 Proof under the Markovian Setting

In this section, we prove Theorem 1 under the Markovian setting.

From the $L_{J}$-smoothness of $J(\theta)$, it follows that

$\displaystyle J(\theta_{t+1})$ $\displaystyle\leq
J(\theta_{t})+\left\langle\nabla
J(\theta_{t}),\theta_{t+1}-\theta_{t}\right\rangle+\frac{L_{J}}{2}\|\theta_{t+1}-\theta_{t}\|^{2}$
$\displaystyle=J(\theta_{t})+\alpha\left\langle\nabla
J(\theta_{t}),G_{t+1}(\theta_{t},\omega_{t})\right\rangle+\frac{L_{J}}{2}\alpha^{2}\|G_{t+1}(\theta_{t},\omega_{t})\|^{2}$
$\displaystyle=J(\theta_{t})-\alpha\left\langle\nabla
J(\theta_{t}),-G_{t+1}(\theta_{t},\omega_{t})-\frac{\nabla
J(\theta_{t})}{2}+G_{t+1}(\theta_{t},\omega(\theta_{t}))-G_{t+1}(\theta_{t},\omega(\theta_{t}))\right\rangle$
$\displaystyle\quad-\frac{\alpha}{2}\|\nabla
J(\theta_{t})\|^{2}+\frac{L_{J}}{2}\alpha^{2}\|G_{t+1}(\theta_{t},\omega_{t})\|^{2}$
$\displaystyle=J(\theta_{t})-\alpha\left\langle\nabla
J(\theta_{t}),-G_{t+1}(\theta_{t},\omega_{t})+G_{t+1}(\theta_{t},\omega(\theta_{t}))\right\rangle$
$\displaystyle\quad+\alpha\left\langle\nabla J(\theta_{t}),\frac{\nabla
J(\theta_{t})}{2}+G_{t+1}(\theta_{t},\omega(\theta_{t}))\right\rangle-\frac{\alpha}{2}\|\nabla
J(\theta_{t})\|^{2}+\frac{L_{J}}{2}\alpha^{2}\|G_{t+1}(\theta_{t},\omega_{t})\|^{2}$
$\displaystyle{\leq}J(\theta_{t})+\alpha L_{g}\|\nabla
J(\theta_{t})\|\|\omega(\theta_{t})-\omega_{t}\|+\alpha\left\langle\nabla
J(\theta_{t}),\frac{\nabla
J(\theta_{t})}{2}+G_{t+1}(\theta_{t},\omega(\theta_{t}))\right\rangle$
$\displaystyle\quad-\frac{\alpha}{2}\|\nabla
J(\theta_{t})\|^{2}+\frac{L_{J}}{2}\alpha^{2}\|G_{t+1}(\theta_{t},\omega_{t})\|^{2}$
$\displaystyle\overset{(a)}{\leq}J(\theta_{t})+\alpha L_{g}\|\nabla
J(\theta_{t})\|\|z_{t}\|+\alpha\left\langle\nabla J(\theta_{t}),\frac{\nabla
J(\theta_{t})}{2}+G_{t+1}(\theta_{t},\omega(\theta_{t}))\right\rangle$
$\displaystyle\quad-\frac{\alpha}{2}\|\nabla
J(\theta_{t})\|^{2}+\frac{L_{J}}{2}\alpha^{2}C_{g}^{2},$ (123)

where $(a)$ is from the fact that $\|\theta_{t+1}-\theta_{t}\|\leq\alpha
C_{g}$. Thus by re-arranging the terms, taking expectation and summing up
w.r.t. $t$ from $0$ to $T-1$, it follows that

$\displaystyle\frac{\alpha}{2}\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]$
$\displaystyle\leq-\mathbb{E}[J(\theta_{T})]+J(\theta_{0})+\alpha
L_{g}\sqrt{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}\sqrt{\sum^{T-1}_{t=0}\mathbb{E}[\|z_{t}\|^{2}]}+\sum^{T-1}_{t=0}\alpha\mathbb{E}[\zeta_{G}(\theta_{t},O_{t})]$
$\displaystyle\quad+L_{J}\alpha^{2}TC_{g}^{2},$ (124)

where $\zeta_{G}(\theta_{t},O_{t})=\left\langle\nabla
J(\theta_{t}),\frac{\nabla
J(\theta_{t})}{2}+G_{t+1}(\theta_{t},\omega(\theta_{t}))\right\rangle$. We
then bound $\zeta_{G}$ in the following lemma.

######  Lemma 6.

For any $t\geq\tau_{\beta}$,

$\displaystyle\mathbb{E}[\zeta_{G}(\theta_{t},O_{t})]\leq
2C_{g}^{2}\beta+2\alpha\tau_{\beta}L_{\zeta}C_{g}.$ (125)

###### Proof.

We only need to consider the case $t=\tau_{\beta}$, the proof for general case
of $t\geq\tau_{\beta}$ is similar, and thus is omitted here. We first have
that

$\displaystyle\zeta_{G}(\theta_{\tau_{\beta}},O_{\tau_{\beta}})$
$\displaystyle=\left\langle\nabla J(\theta_{\tau_{\beta}}),\frac{\nabla
J(\theta_{\tau_{\beta}})}{2}+G_{\tau_{\beta}+1}(\theta_{\tau_{\beta}},\omega(\theta_{\tau_{\beta}}))\right\rangle$
$\displaystyle=\left\langle\nabla J(\theta_{0}),\frac{\nabla
J(\theta_{0})}{2}+G_{\tau_{\beta}+1}(\theta_{0},\omega(\theta_{0}))\right\rangle$
$\displaystyle\quad+\left\langle\nabla J(\theta_{\tau_{\beta}}),\frac{\nabla
J(\theta_{\tau_{\beta}})}{2}+G_{\tau_{\beta}+1}(\theta_{\tau_{\beta}},\omega(\theta_{\tau_{\beta}}))\right\rangle$
$\displaystyle\quad-\left\langle\nabla J(\theta_{0}),\frac{\nabla
J(\theta_{0})}{2}+G_{\tau_{\beta}+1}(\theta_{0},\omega(\theta_{0}))\right\rangle$
$\displaystyle\leq\left\langle\nabla J(\theta_{0}),\frac{\nabla
J(\theta_{0})}{2}+G_{\tau_{\beta}+1}(\theta_{0},\omega(\theta_{0}))\right\rangle+2L_{\zeta}\|\theta_{\tau_{\beta}}-\theta_{0}\|$
$\displaystyle\leq\left\langle\nabla J(\theta_{0}),\frac{\nabla
J(\theta_{0})}{2}+G_{\tau_{\beta}+1}(\theta_{0},\omega(\theta_{0}))\right\rangle+2\alpha\tau_{\beta}L_{\zeta}C_{g},$
(126)

where $L_{\zeta}=2C_{g}(L_{g}^{\prime}+\frac{3L_{J}}{2})$ is the Lipschitz
constant of $\zeta_{G}(\theta,O_{t})$.

Then it follows that

$\displaystyle\mathbb{E}[\zeta_{G}(\theta_{\tau_{\beta}},O_{\tau_{\beta}})]$
$\displaystyle=\mathbb{E}\left[\left\langle\nabla J(\theta_{0}),\frac{\nabla
J(\theta_{0})}{2}+G_{\tau_{\beta}+1}(\theta_{0},\omega(\theta_{0}))\right\rangle\right]+2\alpha
L_{\zeta}C_{g}\tau_{\beta}$ $\displaystyle\leq
2C_{g}^{2}\beta+2\alpha\tau_{\beta}L_{\zeta}C_{g},$ (127)

where the last step follows from the uniform ergodicity of the MDP (Assumption
4). ∎

Plugging the bound in (C.2), it follows that

$\displaystyle\frac{\alpha}{2}\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]$ $\displaystyle\leq J(\theta_{0})-J^{*}+\alpha
L_{g}\sqrt{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}\sqrt{\sum^{T-1}_{t=0}\mathbb{E}[\|z_{t}\|^{2}]}$
$\displaystyle\quad+\alpha^{2}C_{g}^{2}L_{J}T+\alpha\left(T(2C_{g}^{2}\beta+2\alpha\tau_{\beta}L_{\zeta}C_{g})+4\tau_{\beta}C_{g}^{2}\right),$
(128)

and thus

$\displaystyle\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla J(\theta_{t})\|^{2}]$
$\displaystyle\leq\frac{2(J(\theta_{0})-J^{*})}{\alpha}+2L_{g}\sqrt{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}\sqrt{\sum^{T-1}_{t=0}\mathbb{E}[\|z_{t}\|^{2}]}+2\alpha
C_{g}^{2}L_{J}T$
$\displaystyle\quad+2\left(T(2C_{g}^{2}\beta+2\alpha\tau_{\beta}L_{\zeta}C_{g})+4\tau_{\beta}C_{g}^{2}\right).$
(129)

This further implies that

$\displaystyle\frac{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}{T}$
$\displaystyle\leq\frac{2(J(\theta_{0})-J^{*})}{\alpha
T}+2L_{g}\sqrt{\frac{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}{T}}\sqrt{\frac{\sum^{T-1}_{t=0}\mathbb{E}[\|z_{t}\|^{2}]}{T}}+2\alpha
C_{g}^{2}L_{J}$
$\displaystyle\quad+2\left((2C_{g}^{2}\beta+2\alpha\tau_{\beta}L_{\zeta}C_{g})+4C_{g}^{2}\frac{\tau_{\beta}}{T}\right).$
(130)

We plug in the tracking error (C.1), and it follows that

$\displaystyle\frac{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}{T}$
$\displaystyle\leq\frac{2(J(\theta_{0})-J^{*})}{\alpha T}+2\alpha
C_{g}^{2}L_{J}+2\left((2C_{g}^{2}\beta+2\alpha\tau_{\beta}L_{\zeta}C_{g})+4C_{g}^{2}\frac{\tau_{\beta}}{T}\right)$
$\displaystyle\quad+2L_{g}\sqrt{\frac{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}{T}}$
$\displaystyle\quad\cdot\sqrt{{\left(2\|z_{0}\|+2\beta\tau_{\beta}(b_{\max}+L_{\omega}C_{g})\right)^{2}}\left(\frac{1}{Tq}+\frac{\tau_{\beta}}{T}\right)+\frac{\alpha
L_{\omega}}{4q}\frac{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}{T}+\frac{p}{q}}$
$\displaystyle\leq\frac{2(J(\theta_{0})-J^{*})}{\alpha T}+2\alpha
C_{g}^{2}L_{J}+2\left((2C_{g}^{2}\beta+2\alpha\tau_{\beta}L_{\zeta}C_{g})+4C_{g}^{2}\frac{\tau_{\beta}}{T}\right)$
$\displaystyle\quad+2L_{g}\sqrt{\frac{\alpha
L_{\omega}}{4q}}\frac{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}{T}$
$\displaystyle\quad+2L_{g}\sqrt{\frac{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}{T}}\sqrt{{\left(2\|z_{0}\|+2\beta\tau_{\beta}(b_{\max}+L_{\omega}C_{g})\right)^{2}}\left(\frac{1}{Tq}+\frac{\tau_{\beta}}{T}\right)+\frac{p}{q}}.$
(131)

Note that $2L_{g}\sqrt{\frac{\alpha
L_{\omega}}{4q}}=\mathcal{O}\left(\sqrt{\frac{\alpha}{\beta}}\right)$, hence
we can choose $\alpha$ and $\beta$ such that $2L_{g}\sqrt{\frac{\alpha
L_{\omega}}{4q}}\leq\frac{1}{2}$. Hence it follows that

$\displaystyle\frac{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}{T}$
$\displaystyle\leq\frac{4(J(\theta_{0})-J^{*})}{\alpha T}+4\alpha
C_{g}^{2}L_{J}+4\left((2C_{g}^{2}\beta+2\alpha\tau_{\beta}L_{\zeta}C_{g})+4C_{g}^{2}\frac{\tau_{\beta}}{T}\right)$
$\displaystyle\quad+4L_{g}\sqrt{\frac{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}{T}}$
$\displaystyle\cdot\sqrt{{\left(2\|z_{0}\|+2\beta\tau_{\beta}(b_{\max}+L_{\omega}C_{g})\right)^{2}}\left(\frac{1}{Tq}+\frac{\tau_{\beta}}{T}\right)+\frac{p}{q}}$
$\displaystyle\triangleq U\sqrt{\frac{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}{T}}+V,$ (132)

where
$U=4L_{g}\sqrt{{\left(2\|z_{0}\|+2\beta\tau_{\beta}(b_{\max}+L_{\omega}C_{g})\right)^{2}}\left(\frac{1}{Tq}+\frac{\tau_{\beta}}{T}\right)+\frac{p}{q}}=\mathcal{O}\left(\sqrt{\beta\tau_{\beta}+\frac{1}{T\beta}}\right)$
and $V=\frac{4(J(\theta_{0})-J^{*})}{\alpha T}+4\alpha
C_{g}^{2}L_{J}+4\left((2C_{g}^{2}\beta+2\alpha\tau_{\beta}L_{\zeta}C_{g})+4C_{g}^{2}\frac{\tau_{\beta}}{T}\right)=\mathcal{O}\left(\frac{1}{T\alpha}+\alpha\tau_{\beta}+\beta\right)$.
Thus it can be shown that

$\displaystyle\frac{\sum^{T-1}_{t=0}\mathbb{E}[\|\nabla
J(\theta_{t})\|^{2}]}{T}$
$\displaystyle\leq\left(\frac{U+\sqrt{U^{2}+4V}}{2}\right)^{2}$
$\displaystyle\leq U^{2}+2V$
$\displaystyle=16L_{g}^{2}\left({\left(2\|z_{0}\|+2\beta\tau_{\beta}(b_{\max}+L_{\omega}C_{g})\right)^{2}}\left(\frac{1}{Tq}+\frac{\tau_{\beta}}{T}\right)+\frac{p}{q}\right)$
$\displaystyle\quad+\frac{8(J(\theta_{0})-J^{*})}{\alpha T}+8\alpha
C_{g}^{2}L_{J}+8\left((2C_{g}^{2}\beta+2\alpha\tau_{\beta}L_{\zeta}C_{g})+4C_{g}^{2}\frac{\tau_{\beta}}{T}\right)$
$\displaystyle=\mathcal{O}\left(\beta\tau_{\beta}+\frac{1}{T\beta}+\alpha\tau_{\beta}+\frac{1}{T\alpha}\right).$
(133)

This completes the proof.

####  C.3 Choice of Step-sizes

In the proof under the Markovian setting, we first assume
$\beta\tau_{\beta}C_{\phi}^{2}\leq\frac{1}{4}$. The last assumption on the
step-sizes is $\frac{\alpha}{q}\leq\frac{1}{4L_{g}^{2}L_{\omega}}$, where
$q=2\beta\lambda_{v}-2\beta(R_{1}+R_{3}+P_{1}+P_{2}+P_{3})-2\beta^{2}C_{\phi}^{4}=\mathcal{O}(\beta)$.
Note that this assumption can be satisfied by controlling
$\frac{\alpha}{\beta}$ similar to Section B.3, which we omit here. Hence we
set $\beta<\min\left\\{1,\frac{1}{4\tau_{\beta}C_{\phi}^{2}}\right\\}$, and
$\frac{\alpha}{q}\leq\left\\{1,\frac{1}{4L_{g}^{2}L_{\omega}}\right\\}$.

###  Appendix D Experiments

In this section, we provide some numerical experiments on two RL examples: the
Garnet problem [Archibald et al., 1995] and the “spiral” counter example in
[Tsitsiklis and Van Roy, 1997].

####  D.1 Garnet Problem

The first experiment is on the Garnet problem [Archibald et al., 1995], which
can be characterized by $\mathcal{G}(|\mathcal{S}|,|\mathcal{A}|,b,N)$. Here
$b$ is a branching parameter specifying how many next states are possible for
each state-action pair, and these $b$ states are chosen uniformly at random.
The transition probabilities are generated by sampling uniformly and randomly
between 0 and 1. The parameter $N$ is the dimension of $\theta$ to be updated.
In our experiments, we generate a reward matrix uniformly and randomly between
0 and 1. For every state $s$ we randomly generate one feature function
$k(s)\in[0,1]$ using as the input. In both experiments, we use a five-layer
neural network with (1,2,2,3,1) neurons in each layer as the function
approximator. And for the activation function, we use the Sigmoid function,
i.e., $f(x)=\frac{1}{1+e^{-x}}$. We set all the weights and bias of the
neurons as the parameter $\theta\in\mathbb{R}^{23}$.

We consider two sets of parameters: $\mathcal{G}(5,2,5,23)$ and
$\mathcal{G}(3,2,3,23)$. We set the step-size $\alpha=0.01$ and $\beta=0.05$,
and also the discount factor $\gamma=0.95$. In Figures 1 and 2, we plot the
squared gradient norm v.s. the number of samples using 40 Garnet MDP
trajectories, i.e., at each time $t$, we plot $\|\nabla J(\theta_{t})\|^{2}$.
The upper and lower envelopes of the curves correspond to the 95 and 5
percentiles of the 40 curves, respectively. We also plot the estimated
variance of the stochastic update along the iterations in Figures 1(b) and
2(b). Specifically, we first run the algorithm to get a sequence of
$\theta_{t}$ and $\omega_{t}$. Then we generate 500 different trajectories
$O^{i}=(O^{i}_{1},O^{i}_{2},...,O^{i}_{t},...)$ where $i=1,...,500$, and use
them to estimate the variance $\|G^{i}_{t+1}(\theta_{t},\omega_{t})-\nabla
J(\theta_{t})\|^{2}$ and plot
$\frac{\sum^{500}_{i=1}\|G^{i}_{t+1}(\theta_{t},\omega_{t})-\nabla
J(\theta_{t})\|^{2}}{500}$ at each time $t$.

It can be seen from the figures that both gradient norm $\|\nabla
J(\theta_{t})\|$ and the estimated variance converge to zero.

(a) $\|\nabla J(\theta_{t})\|^{2}$.

(b) Estimated variance.

Figure 1: Garnet problem 1: $\mathcal{G}(5,2,5,23)$.

(a) $\|\nabla J(\theta_{t})\|^{2}$.

(b) Estimated variance.

Figure 2: Garnet problem 2: $\mathcal{G}(3,2,3,23)$.

####  D.2 Spiral Counter Example

In our second experiment, we consider the spiral counter example proposed in
[Tsitsiklis and Van Roy, 1997], which is often used to show the TD algorithm
may diverge with nonlinear function approximation. The problem setting is
given in Figure 3. There are three states and each state can transit to the
next one with probability $\frac{1}{2}$ or stay at the current state with
probability $\frac{1}{2}$. The reward is always zero with the discount factor
$\gamma=0.9$. Similar to [Bhatnagar et al., 2009], we consider the value
function approximation:

$\displaystyle
V_{\theta}(s)=(a(s)\cos(k\theta)+b(s)\sin(k\theta))e^{\epsilon\theta},$ (134)

where in Figure 4, $a=[0.94,-0.43,0.18]$ and $b=[0.21,-0.52,0.76]$; and in
Figure 5, $a=[0.21,-0.33,0.29]$ and $b=[0.68,0.41,0.82]$. We let $k=0.866$ and
$\epsilon=0.1$. The step-size are chosen as $\alpha=0.01$ and $\beta=0.05$. In
Figures 4(a) and 5(a), we plot the squared gradient norm v.s. the number of
samples using 40 MDP trajectories. The upper and lower envelopes of the curves
correspond to the 95 and 5 percentiles of the 40 curves. Similarly, we also
plot the estimated variance $\|G_{t+1}(\theta_{t},\omega_{t})-\nabla
J(\theta_{t})\|^{2}$ of the stochastic update along the iterations using 50
samples at each time step. More specifically, we first run the algorithm to
get a sequence of $\theta_{t}$ and $\omega_{t}$. Then we generate 50 different
trajectories $O^{i}=(O^{i}_{1},O^{i}_{2},...,O^{i}_{t},...)$ where
$i=1,...,50$, and use them to estimate the variance
$\|G^{i}_{t+1}(\theta_{t},\omega_{t})-\nabla J(\theta_{t})\|^{2}$ and plot
$\frac{\sum^{50}_{i=1}\|G^{i}_{t+1}(\theta_{t},\omega_{t})-\nabla
J(\theta_{t})\|^{2}}{50}$ at each time $t$.

It can be seen that in both experiments, the gradient norm $\|\nabla
J(\theta_{t})\|$ converges to 0, i.e., the algorithm converges to a stationary
point. The estimated variance also decreases to zero.

Figure 3: Spiral counter example.

(a) $\|\nabla J(\theta_{t})\|^{2}$.

(b) Estimated variance.

Figure 4: Spiral counter example 1:
$a=[0.94,-0.43,0.18]$,$b=[0.21,-0.52,0.76]$.

(a) $\|\nabla J(\theta_{t})\|^{2}$.

(b) Estimated variance.

Figure 5: Spiral counter example 2:
$a=[0.21,-0.33,0.29]$, $b=[0.68,0.41,0.82]$.


The correlation function of a two-dimensional electron gas with anharmonic
potential and Rashba coupling

Rachid<EMAIL_ADDRESS>, El Bouâzzaoui
<EMAIL_ADDRESS>, Abdellatif Kamal2,3‡‡‡abdellatif.kamal@ensam-
casa.ma , Abdelhadi<EMAIL_ADDRESS>and Mohammed El
<EMAIL_ADDRESS>
1Team of Theoretical Physics and high energy, Department of Physics, Faculty
of Sciences, Ibn Zohr University, Agadir, Morocco,

PO Box 8106, Agadir, Morocco

2L.P.M.C. Laboratory, Theoretical Physics Group, Faculty of Sciences, Chouaïb
Doukkali University,

PO Box 20, 24000 El Jadida, Morocco

3Department of Mechanical Engineering, National Higher School of Arts and
Crafts, Hassan II University, Casablanca, Morocco

In this paper, we study a two-dimensional gas of electrons with Rashba spin-
orbit coupling with anharmonic potential. The splitting of the Hamiltonian in
two parts can be recovering the Jaynes-Cummings model, which describes a
system with two states. The dynamics of the rising Pauli operator and the
creation operators are studied. Finally, we examined the behavior of the
correlation function of emission and absorption photons for the strong and
weak coupling.

PACS numbers: 42.50.-p, 71.70.Ej.

Keywords: Anharmonic potential, Rashba coupling, Jaynes-Cummings model,
Correlation function.

##  1 Introduction

Spin electronics, or spintronics, is concerned with the degree of freedom
offered by the electron’s spin to create devices for storing, processing
logic, or transporting information [1]. Since the $1990$s, two axes have
developed in parallel: the spintronics of metals and semiconductors. Metal
spintronics took off with the discovery of giant magnetoresistance in $1988$
[2, 3]. This discovery, which earned A. Fert and P. Grünberg the Nobel Prize
in 2007, demonstrated that electronic transport could be controlled by the
magnetization of ferromagnetic materials, itself controllable by a magnetic
field. A direct application is the spin valve [4].

Semiconductor spintronics [5] is in line with the invention of the transistor
in 1948. The transistor exploits the possibility, specific to semiconductors,
of controlling the electronic properties by doping and applying electric
fields. Semiconductors also have certain advantages for spintronics: the spin
is easily accessible optically. Generally, it has a greater coherence than in
metals. Thus, it was quickly considered to reproduce in semiconductors the
spintronic effects initially demonstrated in metals. Interest has turned to
hybrid ferromagnetic/semiconductor structures [6, 7], and to multifunctional
materials, both semiconductor and magnetic diluted magnetic semiconductors
like $GaMnAs$ [8, 9] or $CdMnTe$ [10]. A third strategy has also emerged,
which consists of exploiting the spin-orbit interaction of semiconductors.

The spin-orbit interaction is a relativistic effect. Whereby an electron was
moving in an electric field sees an effective magnetic field $\vec{B}_{SO}$,
which acts on its spin. This very general mechanism makes it possible to
generate and manipulate the spin of electrons by simply influencing their
movement by electric fields. It, therefore, opens up the possibility of
avoiding the use of ferromagnetic materials or external magnetic fields and of
considering all-electric semiconductor devices for spintronics. Thus, it has
recently been shown that one could generate an accumulation of spin. By the
simple passage of a charge current: this is the spin Hall effect [11, 12]. It
has also been shown that continuous or pulsed electric fields can carry out
coherent rotations of the spin of traveling electrons [13, 14].

Moreover, the Jaynes–Cummings (J-C) model makes it possible to represent the
light-matter interaction insofar as it models a system made up, within an
optical cavity, by an atom at two levels interacting with a quantized mode of
the electromagnetic field. The model offers the advantage of lending itself to
calculations of exact solutions. In addition, the simplicity of the
representation of the interaction does not affect its ability to deal with a
large number of observed phenomena. The appropriate framework for the
formulation and practical testing of the J-C concept is quantum
electrodynamics cavity (QEC), as the fundamental experiments have been carried
out through the use of microwaves and optical cavities [15, 16, 17]. The use
of semiconductor quantum dots in photonic crystal, micro-pillar, or micro-disk
resonators [18], which may be used to investigate the domain of electrodynamic
quantum cavities, is common in solid state systems as well.

Motivated by our research [19], we broadened our previous approach to address
a variety of problems. To do this, we examine a two-coupled harmonic
oscillator operating in a uniform magnetic field $B$ and with Rashba spin-
orbit coupling in order to establish a link with quantum optics and to
investigate its various quantum dynamics. We diagonalize our system using the
creation and annihilation operators to arrive at the solutions to the energy
spectrum. We demonstrate that our system simplifies the Landau issue in the
first instance by examining a strong and a weak $B$. We constructed two
replicas of the Jaynes–Cummings model oscillating at different frequencies
using the Heisenberg image. The following is the structure of the current
paper. In Section 2, we formalize our issue by specifying the Hamiltonian and
selecting a suitable gauge for consideration. In section 3, as a result, we
add a sequence of annihilation and creation operators to diagonalize our
system. This allows us to calculate the precise eigenvalues and eigenstates of
our Hamiltonian, which is useful for many applications. In section 4, We build
a connection with the Jaynes–Cummings model and make a variety of findings. In
section 5, We will investigate the correlation function for photon absorption
or emission, and we will analyze it for two limiting instances in order to
demonstrate the significance of our findings. Finally, we give our conclusions
in section 6.

##  2 System energy

We begin by defining our issue in order to arrive at the Hamiltonian that best
describes the system under discussion. Indeed, when we examine a two-
dimensional gas of electrons with Rashba spin-orbit coupling [19] and an
anharmonic potential [20], the system’s Hamiltonian reads as

$H={p_{x}^{2}\over 2m_{1}}+{p_{y}^{2}\over 2m_{2}}+{k_{1}\over
2}x^{2}+{k_{2}\over 2}y^{2}+{k_{3}\over
2}xy+\alpha\left(\sigma_{y}p_{x}-\sigma_{x}p_{y}\right),$ (1)

where $\alpha$ is the Rashba coupling parameter and
$\vec{\sigma}=\left(\sigma_{x},\sigma_{y},\sigma_{z}\right)$ are the Pauli
matrices. It is easy to incorporate additional phase space variables by
rescaling those found in (1). Indeed, the positions may be defined as:

$P_{x}=\left({m_{2}\over m_{1}}\right)^{1\over 4}p_{x},\quad
P_{y}=\left({m_{1}\over m_{2}}\right)^{1\over 4}p_{y}$ (2)

By substituting them all, we demonstrate that (1) becomes

$H={P_{x}^{2}+P_{y}^{2}\over 2\mu}+{1\over
2}\left(k_{1}x^{2}+k_{2}y^{2}+k_{3}xy\right)+\alpha\left(\sigma_{y}p_{x}-\sigma_{x}p_{y}\right),$
(3)

In this case, $\mu=\sqrt{m_{1}m_{2}}$. Due to the fact that the Hamiltonian
contains an interaction term, it is difficult to conduct a simple examination
of the system’s fundamental properties. Nonetheless, this issue may be
simplified via a translation to new phase space variables

$\left(\begin{array}[]{c}P_{x}\\\ P_{y}\\\
\end{array}\right)=M\left(\begin{array}[]{c}p_{1}\\\ p_{2}\\\
\end{array}\right);\qquad\left(\begin{array}[]{c}x\\\ y\\\
\end{array}\right)=M\left(\begin{array}[]{c}q_{1}\\\ q_{2}\\\
\end{array}\right)$ (4)

Where $M$ is a unitary rotation matrix with a mixing angle, $\theta$ can be
defined by:

$M=\left(\begin{array}[]{cc}\cos{\theta\over 2}&-\sin{\theta\over 2}\\\
\sin{\theta\over 2}&\cos{\theta\over 2}\\\ \end{array}\right)$ (5)

By including the mapping (5) into (3), it becomes clear that $\theta$ must
fulfill the requirement

$\tan\theta={k_{3}\over k_{2}-k_{1}}$ (6)

To get the following Hamiltonian

$H={1\over 2\mu}\left(p_{1}^{2}+p_{2}^{2}\right)+{k\over
2}\left(e^{2\lambda}q_{1}^{2}+e^{-2\lambda}q_{2}^{2}\right)+\alpha\left[\left(\cos{\theta\over
2}\sigma_{y}-\sin{\theta\over 2}\sigma_{x}\right)p_{1}-\left(\sin{\theta\over
2}\sigma_{y}+\cos{\theta\over 2}\sigma_{x}\right)p_{2}\right]$ (7)

where

$k=\sqrt{k_{1}k_{2}-{k_{3}^{2}/4}},\qquad
e^{\lambda}={k_{1}+k_{2}+\sqrt{\left(k_{1}-k_{2}\right)^{2}+k_{3}^{2}}\over
2k}.$ (8)

In addition, the condition $4k_{1}k_{2}>k_{3}^{2}$ must be met. With the
separation of Cartesian variables technique, it is simple to see the algebraic
structure of the Hamiltonian described above. This method indicates that (7)
should be divided into two parts

$H=H_{F}+H_{R}$ (9)

As a result, the free part has the structure

$H_{F}={1\over 2\mu}\left(p_{1}^{2}+p_{2}^{2}\right)+{k\over
2}\left(e^{2\lambda}q_{1}^{2}+e^{-2\lambda}q_{2}^{2}\right)$ (10)

as well as the Rashba coupling with the transformation above, which is given
by:

$H_{R}=\alpha\left[\left(\cos{\theta\over 2}\sigma_{y}-\sin{\theta\over
2}\sigma_{x}\right)p_{1}-\left(\sin{\theta\over 2}\sigma_{y}+\cos{\theta\over
2}\sigma_{x}\right)p_{2}\right].$ (11)

Specifically, we point out that (9) is divided into two separate harmonic
oscillator Hamiltonians, with the Rashba spin–orbit coupling serving as a
supplement. This handy version of the Hamiltonian will assist us in easily
obtaining its diagonalization in a straightforward manner.

##  3 Weyl–Heisenberg symmetry solution

To acquire solutions to the energy spectrum of the Hamiltonian (9), we present
conventional methods that have been developed throughout time. It is
preferable to use annihilation operators on the oscillator instead of directly
utilizing them

$a_{1}={q_{1}\over\ell}e^{\lambda}+{i\ell\over 2\hbar}p_{1},\quad
a_{2}={q_{2}\over\ell}e^{-\lambda}+{i\ell\over 2\hbar}p_{2}$ (12)

where

$[a_{i},a_{j}^{\dagger}]=e^{\lambda}\delta_{ij}.$ (13)

Currently, we deal with two additional ones, which are the linear combination
of $a_{1}$ and $a_{2}$, such that

$\displaystyle a_{d}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}\left(a_{1}-ia_{2}\right)$ (14) $\displaystyle
a_{g}$ $\displaystyle=$
$\displaystyle{1\over\sqrt{2}}\left(a_{1}+ia_{2}\right)$

where $\ell=\sqrt{{2\hbar\omega\over k}}$ denotes the magnetic length and
$\omega=\sqrt{{k\over\mu}}$ represents the frequency of the new system. Note
that bosonic operators $a_{d}$ and $a_{g}$ fulfill the relationship
commutations

$[a_{g},a_{g}^{\dagger}]=[a_{d},a_{d}^{\dagger}]=e^{\lambda}$ (15)

as well as the disappearance of all other relationships. It is possible to
derive useful identities for momentum from the operators listed above

$\displaystyle p_{1}$ $\displaystyle=$ $\displaystyle{\hbar\over
i\sqrt{2}\ell}\left(a_{d}-a_{d}^{\dagger}+a_{g}-a_{g}^{\dagger}\right)$ (16)
$\displaystyle p_{2}$ $\displaystyle=$
$\displaystyle{\hbar\over\sqrt{2}\ell}\left(a_{d}+a_{d}^{\dagger}-a_{g}-a_{g}^{\dagger}\right)$

Also, for the positions

$\displaystyle q_{1}$ $\displaystyle=$ $\displaystyle{\ell\over
2\sqrt{2}}\left(a_{d}+a_{d}^{\dagger}+a_{g}+a_{g}^{\dagger}\right)$ (17)
$\displaystyle q_{2}$ $\displaystyle=$ $\displaystyle{\ell\over
2\sqrt{2}i}\left(-a_{d}+a_{d}^{\dagger}+a_{g}-a_{g}^{\dagger}\right)$

These algebraic structures will be fundamental in resolving various problems
and, more specifically, in diagonalizing the many Hamiltonians that are
derived. We begin by simplifying the formulation of the Rashba Hamiltonian
(11), which is the first step

$H_{R}=\alpha\left(\begin{array}[]{cc}0&-e^{-i{\theta\over
2}}\left(p_{2}+ip_{1}\right)\\\ e^{i{\theta\over
2}}\left(-p_{2}+ip_{1}\right)&0\\\ \end{array}\right)$ (18)

where the phases appearing in the formula of $H_{R}$ are defined by

$e^{\pm i{\theta\over 2}}=\cos{\theta\over 2}\pm i\sin{\theta\over 2}.$ (19)

In terms of the annihilation and creation operators discussed above, the
Hamiltonian $H_{R}$ may be expressed as follows:

$H_{R}=\sqrt{2}\alpha{\hbar\over\ell}\left(\begin{array}[]{cc}0&-e^{-i{\theta\over
2}}\left(a_{d}-a_{g}^{\dagger}\right)\\\ e^{i{\theta\over
2}}\left(-a_{d}^{\dagger}+a_{g}\right)&0\\\ \end{array}\right)$ (20)

The Hamiltonian in the above paragraph may be divided into two commuted parts.
As a result, we have

$H_{R}=H_{R}^{d}+H_{R}^{g}$ (21)

where the first and second parts are given by:

$\displaystyle H_{R}^{g}$ $\displaystyle=$
$\displaystyle\sqrt{2}\alpha{\hbar\over\ell}\left(\begin{array}[]{cc}0&e^{-i{\theta\over
2}}a_{g}^{\dagger}\\\ e^{i{\theta\over 2}}a_{g}&0\\\ \end{array}\right)$ (24)
$\displaystyle H_{R}^{d}$ $\displaystyle=$
$\displaystyle-\sqrt{2}\alpha{\hbar\over\ell}\left(\begin{array}[]{cc}0&e^{-i{\theta\over
2}}a_{d}\\\ e^{i{\theta\over 2}}a_{d}^{\dagger}&0\\\ \end{array}\right)$ (27)

We split the total Hamiltonian into two commuted parts to find energy spectrum
solutions for this issue

$H=H_{d}+H_{g}$ (28)

where the Hamiltonians $H_{g}$ and $H_{d}$ are described by the formula

$\displaystyle H_{g}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\hbar\omega\left(a_{g}^{\dagger}a_{g}+{1\over
2}\cosh\lambda\right)&\sqrt{2}\alpha{\hbar\over l}e^{-i{\theta\over
2}}a_{g}^{\dagger}\\\ \sqrt{2}\alpha{\hbar\over l}e^{i{\theta\over
2}}a_{g}&\hbar\omega\left(a_{g}^{\dagger}a_{g}+{1\over
2}\cosh\lambda\right)\\\ \end{array}\right)$ (31) $\displaystyle H_{d}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\hbar\omega\left(a_{d}^{\dagger}a_{d}+{1\over
2}\cosh\lambda\right)&-\sqrt{2}\alpha{\hbar\over l}e^{-i{\theta\over
2}}a_{d}\\\ -\sqrt{2}\alpha{\hbar\over l}e^{i{\theta\over
2}}a_{d}^{\dagger}&\hbar\omega\left(a_{d}^{\dagger}a_{d}+{1\over
2}\cosh\lambda\right)\\\ \end{array}\right)$ (34)

Where the terms in the diagonal represent the components of free Hamiltonian
part $H_{F}$. To find the solutions of the energy spectrum for the Hamiltonian
(34), we resolve the eigenvalues equations

$H_{d}|\psi_{d}\rangle=E_{d}|\psi_{d}\rangle,\quad
H_{g}|\psi_{g}\rangle=E_{g}|\psi_{g}\rangle$ (35)

where $|\psi_{d}\rangle$ and $|\psi_{g}\rangle$ are the $n_{d}$ and $n_{g}-$th
Landau levels, respectively. The overall value of the system thus takes the
following form

$E=E_{d}+E_{g}$ (36)

Fixation of a certain intra-Landau quantum number and all other eigenstates
are of the type [21, 22]

$|\psi_{d}\rangle={1\over\sqrt{2}}|n_{d}\rangle+{1\over\sqrt{2}}|n_{d}-1\rangle,\quad|\psi_{g}\rangle={1\over\sqrt{2}}|n_{g}\rangle+{1\over\sqrt{2}}|n_{g}-1\rangle.$
(37)

To find the system’s spectrum, we begin with the Hamiltonian $H_{g}$, and the
second may be obtained in the same manner, indeed

$\left(\begin{array}[]{cc}\hbar\omega\left(a_{g}^{\dagger}a_{g}+{1\over
2}\cosh\lambda\right)-E_{g}&\sqrt{2}\alpha{\hbar\over l}e^{-i{\theta\over
2}}a_{g}^{\dagger}\\\ \sqrt{2}\alpha{\hbar\over l}e^{i{\theta\over
2}}a_{g}&\hbar\omega\left(a_{g}^{\dagger}a_{g}+{1\over
2}\cosh\lambda\right)-E_{g}\\\
\end{array}\right)\left(\begin{array}[]{c}|n_{g}\rangle\\\ |n_{g}-1\rangle\\\
\end{array}\right)=0$ (38)

then we obtain

$\left\\{\begin{array}[]{r c c l}\left(\hbar\omega\left(n_{g}+{1\over
2}\cosh\lambda\right)-E_{g}\right)+\sqrt{2}\alpha{\hbar\over
l}\sqrt{n_{g}}e^{-i{\theta\over 2}}&=&0\\\ \sqrt{2}\alpha{\hbar\over
l}\sqrt{n_{g}}e^{i{\theta\over 2}}+\left(\hbar\omega\left(n_{g}-1+{1\over
2}\cosh\lambda\right)-E_{g}\right)&=&0\\\ \end{array}\right.$ (39)

by solving the equations system we get

$E_{g}^{\pm}=\hbar\omega\left(n_{g}+\frac{1}{2}\left(\cosh\lambda-1\right)\pm\sqrt{{2\alpha^{2}\mu\over\hbar\omega}n_{g}+1}\right)$
(40)

and by same way we can find the eigenvalue of $H_{g}$, so

$E_{d}^{\pm}=\hbar\omega\left(n_{d}+\frac{1}{2}\left(\cosh\lambda-1\right)\pm\sqrt{{2\alpha^{2}\mu\over\hbar\omega}n_{d}+1}\right)$
(41)

and the total eigenvalue of the system is

$E=\hbar\omega\left(n_{g}+n_{d}+\cosh\lambda-1\pm\left(\sqrt{{2\alpha^{2}\mu\over\hbar\omega}n_{g}+1}+\sqrt{{2\alpha^{2}\mu\over\hbar\omega}n_{d}+1}\right)\right)$
(42)

##  4 Connection to the Jaynes–Cummings model

Recently, intriguing links between many branches of physics have been
discovered. Between these is the special connection between condensed matter
and high energy physics made possible by the combination of graphene and
quantum electromagnetics, as cited by [23]. This prompted us to seek more
linkages and create additional connections between diverse systems. Thus, By
showing how our system is linked to quantum optics via a mapping between the
required Hamiltonian and the J–C model, we create a connection to another
branch of physics. This may contribute to our understanding of many elements
of quantum optics.

###  4.1 Models equivalence

To demonstrate the significance of our findings so far, using the J–C theory
for an atomic shift in a radiation source with a frequency $\omega$, we show
that our Hamiltonian is formally equivalent to two repetitions of the model.
In order to do this, we use the same methodology as Ackerhalt and Rzazewski
when studying operators’ perturbation theory in the Heisenberg image [24]. For
the second situation, we deduce the appropriate results from our findings in a
straightforward manner. To continue, we must reorganize our Hamiltonian such
that each component is dealt with individually and the corresponding
connection is established. To begin, we divide our Hamiltonian into two halves
and simplify the mathematics by setting $\hbar=1$.

$H=H_{g}+H_{d}$ (43)

in this case, the first half is

$H_{g}=\omega a_{g}^{\dagger}a_{g}+{\omega\over 2}\cosh\lambda+H_{R}^{g}$ (44)

and the second is as follows:

$H_{d}=\omega a_{d}^{\dagger}a_{d}+{\omega\over 2}\cosh\lambda+H_{R}^{d}$ (45)

To begin, let us examine the Hamiltonian (44) and see how it relates to the
Jaynes–Cummings model. In doing so, we demonstrate that the equation (44) may
be expressed as

$H_{g}=\omega M_{g}+\alpha^{\prime}\left(e^{-i{\theta\over
2}}a_{g}^{\dagger}\sigma^{+}+e^{i{\theta\over 2}}a_{g}\sigma^{-}\right)$ (46)

the two operators, $M_{g}$ and $\sigma^{\pm}$, are denoted by and respectively

$\displaystyle M_{g}$ $\displaystyle=$ $\displaystyle
a_{g}^{\dagger}a_{g}+{1\over 2}\cosh\lambda$ (47) $\displaystyle\sigma^{\pm}$
$\displaystyle=$ $\displaystyle{1\over 2}\left(\sigma_{x}\pm
i\sigma_{y}\right)$ (48)

and we have assigned the constant $\alpha^{\prime}$ to the value of

$\alpha^{\prime}=\left(\omega\mu\right)^{1\over 2}\alpha$ (49)

We present the Heisenberg equation of motion for the operators
$a_{g}^{\dagger}$ and $\sigma^{-}$ in order to investigate the dynamics
associated with the Hamiltonian (46). They are as follows:

$\displaystyle{d\over dt}a_{g}^{\dagger}$ $\displaystyle=$ $\displaystyle
i[H_{g},a_{g}^{\dagger}]$ (50) $\displaystyle{d\over dt}\sigma^{-}$
$\displaystyle=$ $\displaystyle i[H_{g},\sigma^{-}].$ (51)

The result of a simple calculation is

$\displaystyle\left(i{d\over dt}+\omega e^{\lambda}\right)a_{g}^{\dagger}$
$\displaystyle=$ $\displaystyle-\alpha^{\prime}e^{{1\over
2}\left(2\lambda+i\theta\right)}\sigma^{-}$ (52) $\displaystyle i{d\over
dt}\sigma^{-}$ $\displaystyle=$
$\displaystyle-\alpha^{\prime}e^{-i{\theta\over 2}}\sigma_{z}a_{g}^{\dagger}$
(53)

using the formula

$\sigma_{z}\sigma^{-}=-\sigma^{-}$ (54)

the equation (53) is transformed into the following:

$i{d\over dt}\sigma^{-}=\alpha^{\prime}e^{-i{\theta\over 2}}a_{g}^{\dagger}$
(55)

Above, we calculated the Heisenberg equations of motion for the operators
$a^{\dagger}$ and $\sigma^{-}$ using the Heisenberg equations of motion. As a
result, we are looking for solutions in the form of

$\sigma^{-}(t)=s_{+}^{g}e^{i\beta_{+}t}+s_{-}^{g}e^{i\beta_{-}t},$ (56)
$a_{g}^{\dagger}(t)=d_{+}^{g}e^{i\beta_{+}t}+d_{-}^{g}e^{i\beta_{-}t}$ (57)

where $r_{\pm}^{g}$, $d_{\pm}^{g}$ and $s_{\pm}^{g}$ are initial time
operators. In order to get the auxiliary equation for $\beta_{1}$, we must
first substitute a trial solution of the type $e^{i\beta t}$ into (55) and
((52) to get

$\beta^{2}_{1}-\omega e^{\lambda}\beta_{1}+\alpha^{\prime 2}e^{\lambda}=0$
(58)

it has two roots, which are represented as follows:

$\beta^{\pm}_{1}=\Omega\pm\omega_{g}$ (59)

in this case, $\Omega={\omega e^{\lambda}\over 2}$, and the values
$\omega_{g}$ are determined by the equation

$\omega_{g}=\sqrt{\Omega^{2}-2\mu\alpha^{2}\Omega}$ (60)

such as the operators constant $s_{\pm}^{g}$ and $d_{\pm}^{g}$, which are
established in terms of the initial time operators $a_{0}^{\dagger}$ and
$\sigma_{0}^{-}$. Indeed, by decomposing $a_{g}^{\dagger}(t)$ and
$\sigma^{-}(t)$, one can get the operator’s constants $s_{\pm}$ and $d_{\pm}$
by setting $t=0$ in (56) and (57). These establish the relationships

$\displaystyle\sigma_{0}^{-}$ $\displaystyle=$ $\displaystyle
s_{+}^{g}+s_{-}^{g}$ (61) $\displaystyle a_{g}^{\dagger}(0)$ $\displaystyle=$
$\displaystyle d_{+}^{g}+d_{-}^{g}$ (62)

If we replace the solutions (56) and (52) with (52) and (55), respectively,
and then evaluate these equations at $t=0$, the initial operators
$s_{\pm}^{g}$ and $d_{\pm}^{g}$ may be represented as

$d_{-}^{g}={-\beta_{1}^{-}a_{g}^{\dagger}(0)-\alpha^{\prime}_{\theta}e^{\lambda}\sigma_{0}^{-}\over
2r_{g}^{+}}$ (63)
$d_{+}^{g}={\beta_{1}^{+}a_{g}^{\dagger}(0)+\alpha^{\prime}_{\theta}e^{\lambda}\sigma_{0}^{-}\over
2r_{g}^{+}}$ (64)
$s_{-}^{g}={\alpha^{\prime}_{-\theta}a_{g}^{\dagger}(0)+\beta_{1}^{+}\sigma_{0}^{-}\over
2r_{g}^{+}}$ (65)
$s_{+}^{g}={-\alpha^{\prime}_{-\theta}a_{g}^{\dagger}(0)+\beta_{1}^{-}\sigma_{0}^{-}\over
2r_{g}^{+}}$ (66)

where $\alpha^{\prime}_{\pm\theta}=\alpha^{\prime}e^{\pm i{\theta\over 2}}$.
After combining all of the variables, we get the final solutions of (52) and
(55). They are as follows:

$\sigma^{-}(t)={e^{i\Omega
t}\over\omega_{g}}\left(i\alpha^{\prime}_{-\theta}\sin(\omega_{g}t)a_{g}^{\dagger}(0)+\left(\Omega\cos(\omega_{g}t)-i\omega_{g}\sin(\omega_{g}t)\right)\sigma^{-}_{0}\right)$
(67)

and

$a_{g}^{\dagger}(t)={e^{i\Omega
t}\over\omega_{g}}\left(\left(i\Omega\sin(\omega_{g}t)+\omega_{g}\cos(\omega_{g}t)\right)a_{g}^{\dagger}(0)+i\alpha^{\prime}_{\theta}e^{\lambda}\sin(\omega_{g}t)\sigma^{-}_{0}\right)$
(68)

They are the precise operator solutions for the fundamental variables
Jaynes–Cummings.

For the second part (45), we are using the same method as previously to get
comparable outcomes. In fact, the operator is introduced

$M_{d}=a_{d}^{\dagger}a_{d}+{1\over 2}\cosh\lambda$ (69)

we may write the Hamiltonian $H_{d}$ as follows:

$H_{d}=\omega M_{d}-\alpha^{\prime}\left(e^{-i{\theta\over
2}}a_{d}\sigma^{+}+e^{i{\theta\over 2}}a_{d}^{\dagger}\sigma^{-}\right).$ (70)

In the same manner, we come up with appropriate solutions

$\sigma^{+}(t)={e^{i\Omega
t}\over\omega_{d}}\left(\alpha^{\prime}_{\theta}\cos(\omega_{d}t)a_{d}^{\dagger}(0)+\left(\Omega\cos(\omega_{d}t)+i\omega_{d}\sin(\omega_{d}t)\right)\sigma^{+}_{0}\right)$
(71)

and

$a_{d}^{\dagger}(t)={e^{i\Omega
t}\over\omega_{d}}\left(\left(-i\Omega\sin(\omega_{d}t)+\omega_{d}\cos(\omega_{d}t)\right)a_{d}^{\dagger}(0)+i\alpha^{\prime}_{-\theta}e^{\lambda}\sin(\omega_{d}t)\sigma^{+}_{0}\right)$
(72)

where the quantity $\omega_{d}$ is determined by the formula

$\omega_{d}=\sqrt{\Omega^{2}+2\mu\alpha^{2}\Omega}$ (73)

Based on our insights, these procedures describe how the second copy of the
J-C model should be produced. We conclude that our system is equivalent to two
J–C type oscillating at two distinct frequencies. Finally, we should point out
that we can simply get the dynamics of the positions from the solutions shown
before. It is true that the formulations of the complex position $z$ in terms
of the annihilation and creation operators are equivalent

$z=e^{i{\theta\over
2}}\left(q_{1}+iq_{2}\right)={\ell\over\sqrt{2}}\left(a_{d}^{\dagger}+a_{g}\right)e^{i{\theta\over
2}}$ (74)

If we assume that the starting velocity is zero at time $t=0$, we may simplify
the formula for $z$. In this case, we get the following equation:

$a_{d}^{\dagger}(0)=a_{g}(0)$ (75)

then the complex position $z$ is

$\displaystyle z(t)$ $\displaystyle=$
$\displaystyle{\ell\over\sqrt{2}}\bigg{(}{e^{i\Omega
t}\over\omega_{d}}\left(\left(-i\Omega\sin(\omega_{d}t)+\omega_{d}\cos(\omega_{d}t)\right)a_{d}^{\dagger}(0)+i\alpha^{\prime}_{-\theta}e^{\lambda}\sin(\omega_{d}t)\sigma^{+}_{0}\right)$
$\displaystyle+$ $\displaystyle{e^{-i\Omega
t}\over\omega_{g}}\left(\left(-i\Omega\sin(\omega_{g}t)+\omega_{g}\cos(\omega_{g}t)\right)a_{g}(0)-i\alpha^{\prime}_{-\theta}e^{\lambda}\sin(\omega_{g}t)\sigma^{+}_{0}\right)\bigg{)}e^{i{\theta\over
2}}$

The next section will be devoted to studying the correlation function for the
absorption or emission of photons in order to provide context for our prior
investigation.

##  5 Correlation functions

The benefit of having calculated operators’ time evolution is that it is now
simple to calculate significant amounts, such as $\zeta$, which in the
Schrodinger image are difficult to calculate. For instance, kindly let us
construct using (72) and its Hermitian conjugate, the correlation function to
destroy a photon at $t^{\prime}$ time and create a photon at $t$ time:

$\zeta=\langle
a_{d}^{\dagger}\left(t\right)a_{d}\left(t^{\prime}\right)\rangle$ (77)

then we have

$\displaystyle\zeta$ $\displaystyle=$
$\displaystyle\langle{1\over{\omega_{d}}^{2}}\left(\left(-i\Omega\sin(\omega_{d}t)+\omega_{d}\cos(\omega_{d}t)\right)a_{d}^{\dagger}(0)+i\alpha^{\prime}_{-\theta}e^{\lambda}\sin(\omega_{d}t)\sigma^{+}_{0}\right)$
$\displaystyle\times$
$\displaystyle\left(\left(i\Omega\sin(\omega_{d}t^{\prime})+\omega_{d}\cos(\omega_{d}t^{\prime})\right)a_{d}(0)-i\alpha^{\prime}_{\theta}e^{\lambda}\sin(\omega_{d}t^{\prime})\sigma^{-}_{0}\right)\rangle$

In the Heisenberg picture, we can calculate the quantity (5) by looking at the
time evolution of the expectation value of the photon number operator, where
absorption (emission) of a photon corresponds to the choice of initial state
$|\psi(0)\rangle=|n_{d},+\rangle(|\psi(0)\rangle=|n_{d}-1,-\rangle)$. Then the
expression of correlation function for absorption of photons by setting
$t^{\prime}=t$ in (5) is written by

$\zeta_{A}=\left(\left({\Omega\over\omega_{d}}\right)^{2}\sin^{2}(\omega_{d}t)+\cos^{2}(\omega_{d}t)\right)n-\left({\alpha^{\prime}e^{\lambda}\over\omega_{d}}\right)^{2}\sin^{2}(\omega_{d}t)$
(79)

in addition to photon emission in the same way

$\zeta_{E}=\left(\left({\Omega\over\omega_{d}}\right)^{2}\sin^{2}(\omega_{d}t)+\cos^{2}(\omega_{d}t)\right)\left(n-1\right)+\left({\alpha^{\prime}e^{\lambda}\over\omega_{d}}\right)^{2}\sin^{2}(\omega_{d}t)$
(80)

In (79), the atom is initially in the ground state such that only absorption
of a photon is possible. The expectation value of $a{\dagger}(t)a(t)$ is
therefore always less than or equal to the number of photons at time $t=0$. In
(80), the atom is initially excited such that only emission of a photon is
possible. The expectation value of $a{\dagger}(t)a(t)$ is therefore always
greater than or equal to the number of photons at time $t=0$. The probability
of absorption or emission can be identified as the magnitude of the second
term on the right-hand side of equations. The probability of emission or
absorption of a photon $P(t)$ [24] is given by

$P(t)=\left({\alpha^{\prime}e^{\lambda}\over\omega_{d}}\right)^{2}\sin^{2}(\omega_{d}t)$
(81)

###  5.1 Strong and weak coupling

####  5.1.1 weak Rashba coupling

in the case when the coupling is very small $\alpha\rightarrow 0$ that is to
say that $\omega_{d}=\Omega$, then the expression of the correlation function
for emission case is written by

$\zeta=n-{4\mu\alpha^{2}\over\omega}\sin^{2}\left({\omega e^{\lambda}\over
2}t\right).$ (82)

It is clear from the previous equation that $\zeta$ is affected by many
variables. The study of this function may help us understand the correlation
behavior of our system. The correlation function for the emission in this
instance is now shown to allow us to draw some inferences. Assume that
$\alpha=0.1,0.2,0.4$ are fixed values. The below curves represents the
variation of $\zeta$ in terms of $\lambda$ parameter.

Figure 1: The correlation function in terms of $\lambda$ parameter for the
first Landau level at a given instant for weak coupling for the values
$\omega=\pi,4\pi,10\pi,80\pi$ and $\alpha=0.1,0.2,0.4$.

We remark in figure (1) in the first hand that the correlation function for
weak Rashba coupling remains almost constant $\zeta\sim 1$ in high frequency
regime $(\omega=80\pi)$ for any value of coupling, and the other hand for low
frequency regime, the amplitude of the oscillation of the correlation function
decreases for the large values of the coupling. The seconde remak is that
$\zeta$ is always positive, which shows that the destruction and creation of
photons in this case increase or decrease together.

####  5.1.2 Strong Rashba coupling

in the case when Rashba coupling is strong the frequency $\omega_{d}$ is
written by

$\omega_{d}\simeq\alpha\sqrt{\mu\omega e^{\lambda}}$ (83)

with this approximation the correlation function become

$\zeta=\left({\omega e^{\lambda}\over
4\alpha^{2}\mu}\sin^{2}\left(\alpha\sqrt{\mu\omega
e^{\lambda}}t\right)+\cos^{2}\left(\alpha\sqrt{\mu\omega
e^{\lambda}}t\right)\right)n-e^{\lambda}\sin^{2}\left(\alpha\sqrt{\mu\omega
e^{\lambda}}t\right)$ (84)

The graphs below illustrate the variation of $\zeta$ in terms of the $\lambda$
parameter for Rashba coupling values $\alpha=5,10,15$.

Figure 2: The correlation function in terms of $\lambda$ parameter for the
first Landau level at a given instant for strong coupling for the values
$\omega=\pi,4\pi,10\pi,80\pi$ and $\alpha=5,10,15$.

The figure (2) shows that the correlation function of photon creation and
annihilation oscillates with a variable frequency depending on $\lambda$ and
that the oscillation amplitude also increases with large values of the
parameter $\lambda$ that of on the one hand and on the other hand it depends
on the coupling and the totality of $\zeta$ is negative, which shows that
there is an antagonistic variation between the destruction and the creation of
the photons.

##  6 Conclusion

We have looked into the fundamental properties of two anharmonic oscillators
with Rashba spin–orbit coupling and discovered some interesting results. In
this work, we demonstrated that it is feasible to get a diagonalized
Hamiltonian by scaling various variables. In fact, this is accomplished via
the application of a unitary transformation. Essentially, this was beneficial
in that it allowed us to get the eigenvalues and their associated
wavefunctions in a straightforward manner by considering the coupling
parameter $\alpha$. This method has specifically been used to the splitting of
the related Hamiltonian into halves. This decomposition was advantageous in
that it allowed for the generation of various spectra and the discovery of
complete solutions for the energy spectrum. In order to establish a link with
quantum optics, we have developed a technique that is based on deriving two
Hamiltonians from a single Hamiltonian. Furthermore, we have also shown that
by dividing the Hamiltonian into halves, it is feasible to reconstruct the J–C
model, which represents a two-state system. This was accomplished by computing
the dynamics of the rising Pauli operator $\sigma^{+}$ and the creation
operators $a^{\dagger}_{d},a^{\dagger}_{g}$ using the Heisenberg dynamics. We
arrived at the appropriate operator solutions for the fundamental J–C
variables by evaluating a multitude of characteristic equations. After that,
we looked at the behaviors of the correlation function of emission and
absorption photons under strong and weak coupling to see how it influences our
correlation function.

## Acknowledgment

We greatly appreciate Mohamed Monkad, director of the Laboratory of Physics of
Condensed Matter (LPMC), Faculty of Sciences, Chouaïb Doukkali University, for
his generous donations. And we would like to express our gratitude to El
Mouhafid Abderrahim for his numerical assistance.

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