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Quantum electrodynamics in a one-dimensional cavity

Quantum electrodynamics (QED) is a fascinating field that combines the principles of quantum mechanics with classical electromagnetism. In this blog post, I discuss the behavior of photon states within a one-dimensional optical cavity and their associated electric fields.

Theoretical framework

The setup consists of a box extending infinitely in the y and z directions and having a finite length L in the x direction. This configuration supports standing wave patterns for photons. The electric field operator for a mode \lambda can be described as follows:

\mathbf{E}_{\lambda,z} = i(\mathbf{a}^\dag_\lambda - \mathbf{a}_\lambda) \sqrt{\frac{\hbar\omega_\lambda}{L\varepsilon_0}} \sin(kx)

Here, \mathbf{a}^\dag_\lambda and \mathbf{a}_\lambda are the photon creation and annihilation operators, respectively, and \hbar represents the reduced Planck’s constant.

Quantum state of the system

Considering a system initially in a superposition state of zero and one photon, we can describe the state vector | \Psi \rangle as:

| \Psi \rangle = \frac{1}{\sqrt 2}\left(e^{-i\frac{1}{2}\omega_1t} | 0\rangle + e^{-i\frac{3}{2}\omega_1t} | 1\rangle \right)

This expression allows us to analyze the time evolution of the quantum state under the influence of the cavity’s boundary conditions.

Expectation value of the electric field

Calculating the expectation value of the electric field gives us insights into the physical observable corresponding to the quantum state. The expectation \langle \mathbf{E}_{\lambda,z} \rangle is given by:

\langle \Psi | \mathbf E_{\lambda,z} | \Psi \rangle = -\sqrt{\frac{\hbar\pi c}{L^2\varepsilon_0}} \sin\left(\frac{\pi}{L}x\right)\sin\left(\frac{c\pi}{L}t\right)

This formula showcases how the electric field varies as a function of both position and time within the cavity.

Multi-mode electric field analysis

In a more complex scenario involving multiple modes, the electric field can be represented as:

\mathbf E_z = i \sum_m \left(\mathbf a^\dag_m - \mathbf a_m\right) \sqrt{\frac{\hbar\omega_m}{L\varepsilon_0}} \sin(k_m x)

I focus on a state where one photon is present in mode 1 and a superposition state exists in mode 2. These calculations help us understand the distribution of energy across different modes and how it affects the overall dynamics of the field within the cavity.

Conclusion

Through this analysis, I have demonstrated the intricate balance of quantum mechanics and electromagnetism that governs the behavior of photons in a one-dimensional cavity. This study not only reinforces the theories of QED but also provides a clear mathematical depiction of photon dynamics in restricted geometries. For more insights into this topic, you can find the details here.