small changes
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@ -124,12 +124,13 @@
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}_{\bHod(s)}
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}_{\bHod(s)}
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\end{equation}
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\end{equation}
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\end{block}
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\end{block}
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%
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\begin{block}{Components of the Hamiltonian}
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\begin{block}{Components of the Hamiltonian}
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\begin{subequations}
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\begin{subequations}
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\begin{align}
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\begin{align}
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\bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) + \la^2 \bH^{(2)}(s) + \cdots
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\bH(s) & = \bH^{(0)}(s) + \la \bH^{(1)}(s) + \la^2 \bH^{(2)}(s) + \cdots
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\\
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\\
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\bF(s) &= \bF^{(0)}(s) + \la \bF^{(1)}(s) + \la^2 \bF^{(2)}(s) + \cdots
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\\
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\bC(s) & = \bC^{(0)}(s) + \la \bC^{(1)}(s) + \la^2 \bC^{(2)}(s) + \cdots
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\bC(s) & = \bC^{(0)}(s) + \la \bC^{(1)}(s) + \la^2 \bC^{(2)}(s) + \cdots
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\\
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\\
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\bV(s) & = \bV^{(0)}(s) + \la \bV^{(1)}(s) + \la^2 \bV^{(2)}(s) + \cdots
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\bV(s) & = \bV^{(0)}(s) + \la \bV^{(1)}(s) + \la^2 \bV^{(2)}(s) + \cdots
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@ -220,16 +221,16 @@
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\begin{block}{Diagonal terms}
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\begin{block}{Diagonal terms}
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\begin{equation}
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\begin{equation}
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\dv{\bF^{(1)}}{s} = \bO
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\dv{\bF^{(1)}}{s} = \bO
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\land
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\Leftrightarrow
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\bF^{(1)}(0) = \bO
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\bF^{(1)}(s) = \bF^{(1)}(0)
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\Rightarrow
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\Leftrightarrow
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\boxed{\bF^{(1)}(s) = \bO}
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\boxed{\bF^{(1)}(s) = \bO}
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\end{equation}
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\end{equation}
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\begin{equation}
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\begin{equation}
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\dv{\bC^{(1)}}{s}
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\dv{\bC^{(1)}}{s} = \bO
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\land
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\Leftrightarrow
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\bC^{(1)}(0) = \bO
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\bC^{(1)}(s) = \bC^{(1)}(0)
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\Rightarrow
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\Leftrightarrow
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\boxed{\bC^{(1)}(s) = \bO}
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\boxed{\bC^{(1)}(s) = \bO}
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\end{equation}
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\end{equation}
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\end{block}
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\end{block}
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@ -308,16 +309,8 @@
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- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
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- 2 \bV^{(1)} \bC^{(0)} \bV^{(1),\dag}
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\\
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\\
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\Rightarrow
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\Rightarrow
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F_{pq}^{(2)}(s) = \sum_{(r,v)} \frac{\Delta_p^{(r,v)} + \Delta_q^{(r,v)}}{(\Delta_p^{(r,v)})^2 + (\Delta_q^{(r,v)})^2 } W_{p,(r,v)}^{(1)}(0)W_{q,(r,v)}^{(1)}(0)\qty(1-e^{-(\Delta_{p}^{(r,v)})^2s} e^{-(\Delta_{q}^{(r,v)})^2s})
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F_{pq}^{(2)}(s) = \sum_{(r,v)} \frac{\Delta_p^{(r,v)} + \Delta_q^{(r,v)}}{(\Delta_p^{(r,v)})^2 + (\Delta_q^{(r,v)})^2 } W_{p,(r,v)}^{(1)}(0)W_{q,(r,v)}^{(1)}(0)\qty(1-e^{-(\Delta_{p}^{(r,v)})^2s} e^{-(\Delta_{q}^{(r,v)})^2s}) \\
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\end{gather}
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\qq{with} \Delta_{q}^{(r,v)} = \epsilon_q - \epsilon_r \pm \Omega_v \notag
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\begin{gather}
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\dv{\bC^{(2)}}{s}
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= \bC^{(0)} \bV^{(1)} \bV^{(1),\dag}
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+ \bV^{(1)} \bV^{(1),\dag} \bC^{(0)}
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- 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag}
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\\
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\Rightarrow
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C_{(r,v),(s,t)}^{(2)}(s) = \sum_{p} \frac{-\Delta_p^{(r,v)} - \Delta_p^{(s,t)}}{(\Delta_p^{(r,v)})^2 + (\Delta_p^{(s,t)})^2 } W_{p,(r,v)}^{(1)}(0)W_{p,(s,t)}^{(1)}(0)\qty(1-e^{-(\Delta_{p}^{(r,v)})^2s} e^{-(\Delta_{p}^{(s,t)})^2s})
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\end{gather}
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\end{gather}
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\end{block}
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\end{block}
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\begin{block}{Off-diagonal terms}
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\begin{block}{Off-diagonal terms}
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@ -358,6 +351,21 @@
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\end{block}
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\end{block}
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\end{frame}
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\end{frame}
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%-----------------------------------------------------
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\begin{frame}{Integration of the Second-Order Terms}
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\begin{block}{Diagonal terms}
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\begin{gather}
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\dv{\bC^{(2)}}{s}
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= \bC^{(0)} \bV^{(1)} \bV^{(1),\dag}
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+ \bV^{(1)} \bV^{(1),\dag} \bC^{(0)}
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- 2 \bV^{(1)} \bF^{(0)} \bV^{(1),\dag}
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\\
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\Rightarrow
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C_{(r,v),(s,t)}^{(2)}(s) = \sum_{p} \frac{-\Delta_p^{(r,v)} - \Delta_p^{(s,t)}}{(\Delta_p^{(r,v)})^2 + (\Delta_p^{(s,t)})^2 } W_{p,(r,v)}^{(1)}(0)W_{p,(s,t)}^{(1)}(0)\qty(1-e^{-(\Delta_{p}^{(r,v)})^2s} e^{-(\Delta_{p}^{(s,t)})^2s})
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\end{gather}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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