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.