Field quantization for traveling waves
Continuing on the journey on Traveling plane electromagnetic waves, I explore the intricacies of quantizing electromagnetic fields through the lens of quantum optics. As an enthusiast exploring the fundamental aspects of photonics, I find it vital to understand how the complex amplitude of the electric field defines the dimensionless normal variable \alpha_\lambda and its temporal evolution.
The electric field, E_\lambda(t), can be expressed in terms of \alpha_\lambda(t) as:
E_\lambda(t) = i \mathscr{E}^{(1)}_\lambda \alpha_\lambda(t)
where i represents an arbitrary factor chosen for consistency with existing literature in quantum optics. This representation leads us to a differential equation that describes the time evolution of \alpha_\lambda(t):
\frac{\mathrm{d} \alpha_\lambda(t)}{\mathrm{dt}} = -i\omega_\lambda \alpha_\lambda(t)
I define the real and imaginary parts of \alpha_\lambda(t) to simplify later expressions:
\alpha(t) = \frac{1}{\sqrt{2\hbar}}(Q_\lambda + iP_\lambda)
These components allow us to express the energy of a mode, \mathcal{H}_\lambda, in terms of the quantization volume V_\lambda and the one photon amplitude \mathscr{E}^{(1)}_\lambda:
\mathcal{H}_\lambda = \epsilon_0 V_\lambda \left( \mathscr{E}^{(1)}_\lambda \right)^2 |\alpha_\lambda(t)|^2
This relationship not only captures the energy in terms of the physical constants and parameters but also sets a foundation for introducing canonical conjugate variables Q_\lambda and P_\lambda, leading to the Hamiltonian:
\mathcal{H}_\lambda = \frac{\omega_\lambda}{2}(Q_\lambda^2 + P_\lambda^2)
Thus, the field’s quantization can be approached as an harmonic oscillator problem. The canonical quantization of these variables leads to operators \mathbf{Q} and \mathbf{P}, which satisfy the commutation relation [\mathbf{Q}, \mathbf{P}] = i \hbar. The quantized Hamiltonian of the field then takes the form:
\mathbf{H} = \frac{\omega_\lambda}{2}(\mathbf{Q}_\lambda^2 + \mathbf{P}_\lambda^2)
As a scholar dedicated to advancing the understanding of quantum field theory, I further explore the transformations of \mathbf{Q}_\lambda and \mathbf{P}_\lambda into the ladder operators \mathbf{a}_\lambda and \mathbf{a}_\lambda^\dag:
\mathbf{a}_\lambda = \frac{1}{\sqrt{2\hbar}}(\mathbf{Q}_\lambda + i\mathbf{P}_\lambda)
These operators are essential for the field’s quantization, representing the annihilation and creation operators in quantum optics. Through these elements, the Hamiltonian becomes a cornerstone for predicting and analyzing the behavior of quantum fields.
The observables in quantum optics, particularly in the context of electromagnetic waves, can thus be modeled accurately. This modeling not only provides a clearer picture of the theoretical framework but also enhances the practical understanding necessary for experimental quantum optics.
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