Complex plane representation of electric field time evolution
In this blog post, I explore the time evolution of the electric field average and its dispersion for quasi-classical states. Understanding how these properties evolve in time provides a valuable perspective on the dynamics of quantum fields.
For a quasi-classical state characterized by a complex amplitude \alpha_1, the average electric field can be expressed as:
\langle\boldsymbol \alpha_1|\mathbf e_1\cdot\mathbf{E}_1(\mathbf{0},t)|\boldsymbol \alpha_1\rangle = i\mathscr E^{(1)}_1(\alpha_1 e^{-i\omega_1t} - \bar \alpha_1 e^{i\omega_1t})
Simplifying this expression, we observe a sinusoidal time dependence:
\langle\boldsymbol \alpha_1|\mathbf e_1\cdot\mathbf{E}_1(\mathbf 0, t)|\boldsymbol \alpha_1\rangle = -2i\mathscr E^{(1)}_1 \left|\alpha_1\right| \sin(-\omega_1t + \varphi_1)
This time evolution can be visualized as the rotation of the complex amplitude \alpha_1 in the complex plane at an angular frequency -\omega_1. The projection of this rotating \alpha_1 onto the imaginary axis corresponds to the sinusoidal time dependence of the electric field average.
To represent the dispersion of the electric field, I use a disk centered on the rotating complex amplitude \alpha_1. This disk visually depicts the uncertainty associated with the electric field. Projecting this disk onto the imaginary axis generates a band around the evolving average field. The half-width of this band represents the standard deviation of the field, which remains constant over time for a quasi-classical state and is equal to one-half in units of 2\mathscr E^{(1)}_1.
It’s important to distinguish this time-dependent field evolution from the static quadrature representation. Quadratures are time-independent quantities that can be measured experimentally using balanced homodyne detection. By varying the local oscillator phase, we can measure different quadratures, effectively mapping out the state in the Q-plane at a specific time, typically t=0. This ensemble of quadrature measurements constructs a representation similar to the dispersion disk, but it is a static snapshot, unlike the dynamic time evolution described earlier.
While the graphs presented here are for quasi-classical states, this visualization technique is generally applicable to other quantum states for a single mode, offering a powerful tool for understanding and representing their properties.
For more insights into this topic, you can find the details here.