Understanding Light-Matter Interaction
2022, IPAM
Matteo, Carlos, Phillip, Aurora, Kangbo
Outline
- Classical Field.
- Quantization: Scalar Field.
- Quantization: Free EM Field.
- Semiclassical Light-Matter Interaction.
- Matter in a Quantum Field.
- Outlook
Save for Open & Excited
- Jaynes-Cumming & Dicke Model
- Superradiance
Classical Field
$$
\begin{align*}
L(t) &= \frac{1}{2} \sum_{i}^N \left(m \dot{\phi}_i(t)^2 - k (\phi_{i+1}(t) - \phi_i(t))^2 \right)\\
\end{align*}
$$
$$
= \frac{a}{2} \sum_i^N \left(\frac{m}{a} \dot{\phi}_i(t)^2 - ka \left(\frac{\phi_{i+1}(t) -
\phi_{i}(t)}{a}
\right)^2 \right)\\
$$
$$
= \int \mathrm{d}\mathbf{x} \frac{1}{2} \left(\mu \dot{\phi}(\mathbf{x}, t)^2 -
Y\left(\frac{\partial \phi(\mathbf{x}, t)}{ \partial
\mathbf{x}}\right)^2 \right)
$$
$$
= \int \mathrm{d}x \underbrace{ \frac{1}{2} \left(\partial_{\mu} \phi(\mathbf{x}, t)
\right)^2}_{\mathcal{L}(\mathbf{x}, t)}
$$
The Lagrangian is guessed to
- Reproduce the classical field EOM.
- Satisfy Lorentz Invariance.
Quantization of a Scalar Field.
Klein-Gordon Field $$ \mathcal{L}(\mathbf{x}, t) = \frac{1}{2} \left(\partial_{\mu} \phi(\mathbf{x}, t)\right)^2 - \frac{1}{2} m^2 \phi(\mathbf{x}, t)^2. $$ EOM $$ (\partial^{\mu} \partial_{\mu} + m^2) \phi(\mathbf{x}, t) = 0. $$Quantization of a Scalar Field.
Canonical Quantization $$ \begin{align*} \phi(\mathbf{x}, t) &\to \hat{\phi}(\mathbf{x}, t)\\ \pi(\mathbf{x}, t) = \frac{\partial \mathcal{L}(\mathbf{x}, t)}{\partial \dot{\phi}(\mathbf{x}, t)} &\to \hat{\pi}(\mathbf{x}, t)\\ [\hat{\phi}(\mathbf{x}, t), \hat{\pi}(\mathbf{x}', t)] &= i \delta^3(\mathbf{x} - \mathbf{x}') \end{align*} $$Normal Modes
The EOM has the solution (real valued) $$ \begin{align*} \phi(\mathbf{x}, t) =& \int \frac{\mathrm{d}^3 k}{(2\pi)^3 2 \omega}\\ & (a(\mathbf{k}) e^{\mathbf{k}\mathbf{x} + i\omega t } + a^{\dagger}(\mathbf{k}) e^{\mathbf{k}\mathbf{x} - i\omega t } )\\ \omega^2 =& \mathbf{k}^2 + m^2 \end{align*} $$Normal Modes
$$ \begin{align} a(\mathbf{k}) &\to \hat{a}(\mathbf{k})\\ a^{\dagger}(\mathbf{k}) &\to \hat{a}^{\dagger}(\mathbf{k})\\ [\hat{\phi}(\mathbf{x}, t), \hat{\pi}(\mathbf{x}', t)] &= i \delta^3(\mathbf{x} - \mathbf{x}')\\ \implies [\hat{a}(\mathbf{k}), \hat{a}^{\dagger}(\mathbf{k}')] &= (2 \pi)^3 2\omega \delta^3(\mathbf{k} - \mathbf{k}') \end{align} $$Quantization of Free EM Field
Coulomb/Transverse/Radiation Gauge Lagrangian $$ \begin{align*} \nabla \cdot \mathbf{A}(\mathbf{x}, t) &= 0\\ A_i(\mathbf{x}, t) &\to \left(\delta_{ij} - \frac{\nabla_i\nabla_j}{\nabla^2}\right) A_j(\mathbf{x}, t) \end{align*} $$ $$ \mathcal{L}(\mathbf{x}, t) = \frac{1}{2} \dot{A}_i \dot{A}_i - \frac{1}{2} \nabla_j A_i \nabla_j A_i + J_i A_i + \mathcal{L}_{coul} $$ The EOMs are the Maxwell equations.Quantization of Free EM Field
Canonical Quantization $$ \begin{align*} A_i(\mathbf{x}, t) &\to \hat{A}_i(\mathbf{x}, t)\\ \Pi_i(\mathbf{x}, t) = \frac{\partial \mathcal{L}}{\partial \dot{A}_i} &\to \hat{\Pi}_i(\mathbf{x}, t)\\ [\hat{A}_i(\mathbf{x}, t), \hat{\Pi}_j(\mathbf{x}', t)] &= i(\delta_{ij} - \frac{\nabla_i \nabla_j}{\nabla^2}) \delta^3(\mathbf{x} - \mathbf{x}') \end{align*} $$Normal Modes
EOM Solution $$ \begin{align*} &\mathbf{A}(\mathbf{x}, t) = \sum_{\lambda=\pm} \int \frac{\mathrm{d}k^3}{(2 \pi)^3 2\omega}\\ &\left(\epsilon_{\lambda}^*(\mathbf{k}) a_{\lambda}(\mathbf{k})e^{i (\mathbf{k}\mathbf{x} - \omega t)} +\epsilon_{\lambda}(\mathbf{k}) a_{\lambda}^{\dagger}(\mathbf{k})e^{-i (\mathbf{k}\mathbf{x} - \omega t)} \right) \end{align*} $$Normal Modes
$$ \begin{align*} a_{\lambda}(\mathbf{k}) &\to \hat{a}_{\lambda}(\mathbf{k})\\ a_{\lambda}^{\dagger}(\mathbf{k}) &\to \hat{a}_{\lambda}^{\dagger}(\mathbf{k})\\ [\hat{A}_i(\mathbf{x}, t), \hat{\Pi}_j(\mathbf{x}', t)] &= \ldots \\ \implies [\hat{a}_{\lambda}(\mathbf{k}), \hat{a}_{\lambda'}^{\dagger}(\mathbf{k}')] &= (2 \pi)^3 2\omega \delta^3(\mathbf{k} - \mathbf{k}')\delta_{\lambda \lambda'} \end{align*} $$Alternative Quantization
$$ \begin{align*} a_{\lambda}(\mathbf{k})e^{-i \omega t} &\to \hat{a}_{\lambda}(\mathbf{k})\\ a_{\lambda}^{\dagger}(\mathbf{k})e^{i \omega t} &\to \hat{a}_{\lambda}^{\dagger}(\mathbf{k})\\ [\hat{A}_i(\mathbf{x}), \hat{\Pi}_j(\mathbf{x}')] &= \ldots \\ \end{align*} $$Light-Matter Interaction
Minimum Coupling (Single Electron) $$ H = \frac{1}{2} \left(\mathbf{p} - \mathbf{A}\right)^2 - \phi $$ The EOM is the Lorentz equation.Light-Matter Interaction
Let $V = -\phi$, $$ \begin{align*} H =& \underbrace{-\frac{1}{2 } \nabla^2 + V}_{H_0} +\frac{1}{2} (2 i \mathbf{A} \cdot \nabla) \\ &+ \underbrace{i (\nabla \cdot \mathbf{A})}_{0 \text{ in Coulomb}} + \frac{1}{2} \mathbf{A}^2 \end{align*} $$Approximations
Radiation Theory: Semiclassical Canonical Quantization $$ \begin{align*} \mathbf{x} &\to \hat{\mathbf{x}}\\ \mathbf{p} &\to \hat{\mathbf{p}} \end{align*} $$ Good Idea for Long Range Interactions.Approximations
Small Systems: The Dipole Approximation $$ A(\mathbf{x}, t) = A(t) $$ Good Idea for Large Molecules?Semiclassical Gauge Transform
General Gauge Transform $$ \psi = V^{\dagger} \tilde{\psi} $$ $$ \begin{align*} i \frac{\partial}{\partial t} (V^{\dagger} \tilde{\psi}) &= H(V^{\dagger} \tilde{\psi})\\ i \frac{\partial}{\partial t} \tilde{\psi} &= \underbrace{\left(V H V^{\dagger} - V \frac{\partial}{\partial t} V^{\dagger}\right)}_{H_V} \tilde{\psi} \end{align*} $$Semiclassical Dipole Gauge Transform
Let $V = \exp\left(-i \chi\right)$, $$ \begin{align*} H_V = \left(\mathbf{p} - (\mathbf{A} - \nabla \chi) \right)^2 - \left(\phi - \frac{\partial}{\partial t} \chi\right) \end{align*} $$ This gives the vector potential gauge transform. $$ \begin{align*} \mathbf{A} &\to \mathbf{A} - \nabla \chi\\ \phi &\to \phi - \frac{\partial}{\partial t} \chi\\ \end{align*} $$Semiclassical Dipole Gauge Transform
Let $\chi(t) = \mathbf{r} \cdot \mathbf{A}(t)$ s.t. $$ \begin{align*} \mathbf{A}(t) - \nabla \chi(t) &= 0\\ - \frac{\partial}{\partial t} \chi(t) &= \mathbf{r} \cdot \underbrace{-\frac{\partial}{\partial t} \mathbf{A}(t)}_{ \mathbf{E}(t)}\\ \end{align*} $$ The Length Gauge $$ H_V = H_0 - \mathbf{r} \cdot \mathbf{E}(t). $$Matter in a Quantum Field Cavity
Single polarization, single mode, Coulomb gauge, and dipole approximation. $$ \mathbf{A}(t) = \underbrace{\left(\frac{2 \pi}{\omega V}\right)^{1/2}}_{s}(a + a^{\dagger}) \epsilon $$ $$ \begin{align*} H =& H_0 + i \mathbf{A} \cdot \nabla + \frac{1}{2} \mathbf{A}^2,\\ H =& H_0 + s (\mathbf{p} \cdot \epsilon) (a + a^{\dagger}) \\ &+ \frac{1}{2} s^2 (a + a^{\dagger})^2 + H_F \end{align*} $$Gauge transform of a Quantum Field
$$ \begin{align*} a &= (2 \omega)^{-1/2} (\omega q + i p)\\ a^{\dagger} &= (2 \omega)^{-1/2} (\omega q - i p)\\ \mathbf{A} &\propto q \epsilon\\ \mathbf{E} &\propto p \epsilon \end{align*} $$ $$ H = H_0 + c_1 (\mathbf{p} \cdot \epsilon) q + \frac{1}{2} \left(p^2 + \omega^2 q^2 \right). $$Gauge transform of a Quantum Field
Let $\chi = \mathbf{r} \cdot \epsilon q$, unlike in the semiclassical case $[p, V] \neq 0$.Length Gauge
$$ \begin{align*} H_V = H_0 + H_F - \mathbf{r} \cdot \mathbf{E} + c_2 \mathbf{r}^2 \end{align*} $$Outlook
- Long Range Interaction.
- Cavity Control of Quantum Systems.
- Boson-Fermion Interactions.