Trajectories

P_{\ket{\alpha}\!\bra{\alpha}\rightarrow\ket{\beta}\!\bra{\beta}} (t) = C_{\alpha\beta}(t) = \mathrm{Tr}\Big[ \hat{\rho}_{\mathrm{nuc}}\ket{\alpha}\!\bra{\alpha} \mathrm{e}^{\mathrm{i}\hat{H}t}\ket{\beta}\!\bra{\beta} \mathrm{e}^{-\mathrm{i}\hat{H}t} \Big]

P_{\ket{\alpha}\!\bra{\alpha}\rightarrow\ket{\beta}\!\bra{\beta}} (t) = C_{\alpha\beta}(t) = \mathrm{Tr}\Big[ \hat{\rho}_{\mathrm{nuc}}\ket{\alpha}\!\bra{\alpha} \mathrm{e}^{\mathrm{i}\hat{H}t}\ket{\beta}\!\bra{\beta} \mathrm{e}^{-\mathrm{i}\hat{H}t} \Big]

C_{\alpha\beta}(t) \approx \frac{1}{(2\pi)^{F+S}} \iint\iint \rho^W_\mathrm{nuc}(x, p) P_\alpha^W(X,P) P_\beta^W(X(t), P(t))~ \mathrm{d}x\,\mathrm{d}p\,\mathrm{d}X\,\mathrm{d}P

Simple Matter - Hard Cavity

\hat{H} = \left( \begin{array}{cc} \varepsilon_1 & 0 \\ 0 & \varepsilon_2 \\ \end{array}\right) + \sum\limits_{j=1}^{\textcolor{red}{200}} \mu \omega_j \lambda_j \textcolor{red}{R_j} \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array}\right) + \sum\limits_{j=1}^{\textcolor{red}{200}} \hbar \omega_j a^{\dagger}_j a_j

a 3-level version

Hard Matter - Simple Cavity

\begin{align*} \hat{H} = &\left( \begin{array}{cc} 0 & \Delta \\ \Delta & -\varepsilon \\ \end{array}\right) + \frac{1}{2} \left[ \frac{P_s^2}{M_s} + M_s \omega_s^2\left( \begin{array}{cc} R_s^2 & 0 \\ 0 & (R_s - R_\mathrm{A}^0)^2 \end{array}\right) \right]\\ &+ \frac{1}{2} \sum\limits_{j=1}^{\textcolor{red}{100}} \frac{P_j^2}{M_j} + m_j \omega_j^2 \Big(R_j - \frac{c_j}{M_j \omega_j^2}R_s \Big)^2\\ &+ \hbar g_c (a^{\dagger}_c + a_c) \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array}\right) + \hbar \omega_c a^{\dagger}_c a_c \end{align*}

Hard Matter - Simple Cavity

k_{\mathrm{D}\to\mathrm{A}} = \int \mathrm{e}^{\mathrm{i}\varepsilon t} C_{FGR} (t) ~\mathrm{d}t

C_{FGR} (t) = \mathrm{Tr}\Big[\hat{\rho}^{eq} \mathrm{e}^{\mathrm{i}\hat{H}_{\mathrm{D}}t} \hat{\Gamma}_{\mathrm{DA}}\mathrm{e}^{-\mathrm{i}\hat{H}_{\mathrm{A}}t} \hat{\Gamma}_{\mathrm{DA}}\Big]

Accurate and cost-effective cavity-modified quantum dynamics from classical-like trajectories Maximilian Saller , Yifan Lai, Eitan Geva Department of Chemistry, University of Michigan, Ann Arbor

Accurate and cost-effective cavity-modified quantum dynamics from classical-like trajectories

Cavity

Trajectories

Some Mapping methods you may have heard of

Quantum Dynamics

Simple Matter - Hard Cavity

Simple Matter - Hard Cavity

Simple Matter - Hard Cavity

Simple Matter - Hard Cavity

Simple Matter - Hard Cavity

Hard Matter - Simple Cavity

Hard Matter - Simple Cavity

Hard Matter - Simple Cavity

Hard Matter - Simple Cavity

Hard Matter - Simple Cavity

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