Quantum optics simulations: visualizing coherent and squeezed states
In this blog post, I present a series of simulations designed to offer visual insights into fundamental concepts within quantum optics. These videos explore the quantum-coherent evolution of the electric field and the dynamic characteristics of squeezed states, specifically addressing both positive and negative squeezing scenarios. By watching these simulations, you can develop a visual intuition for how quantum uncertainty behaves as the phase evolves and how squeezing techniques modify quantum fluctuations.
Visualizing quantum-coherent evolution
The first simulation focuses on the quantum-coherent evolution of an electric field. In quantum optics, a coherent state is often described as the “most classical” state of the electromagnetic field. It can be represented as an eigenstate of the annihilation operator \mathbf{a}_\lambda:
\mathbf{a}_\lambda |\boldsymbol \alpha_\lambda\rangle = \alpha_{\lambda R} |\boldsymbol \alpha_\lambda\rangle
where \alpha_{\lambda R} is a complex number representing the amplitude and phase of the coherent field. The simulation visually represents this state by showing the electric field in the complex plane. You will observe the field rotating at a constant angular speed, maintaining a stable circular uncertainty region. This circular shape is characteristic of a coherent state, indicating equal uncertainty in both quadratures of the electric field. The video clearly illustrates the phase evolution of a quantum optical field in a coherent state.
Quantum Evolution of squeezed states
The subsequent simulations explore squeezed states, which are non-classical states of light where the quantum uncertainty is reduced in one quadrature at the expense of increased uncertainty in the other. Squeezed states are crucial in quantum technologies for enhancing measurement precision and reducing noise.
Positive squeezing
The second video demonstrates a squeezed state with positive squeezing. As the electric field evolves in the complex plane, you can observe that the quantum uncertainty region is no longer circular but elliptical. This ellipse rotates along with the field, and its orientation changes periodically. The simulation shows that the uncertainty oscillates, reaching its minimum at phases 0, 2\pi, 4\pi, \ldots and its maximum at \pi/2, 3\pi/2, 5\pi/2, \ldots. This periodic oscillation illustrates the phase-dependent nature of squeezing dynamics.
Negative squeezing
The third video illustrates a squeezed state with negative squeezing. Similar to the positive squeezing case, the uncertainty region is elliptical and rotates. However, in this scenario, the minima and maxima of the uncertainty are phase-shifted. The simulation reveals that the minimum uncertainty now occurs at phases \pi/2, 3\pi/2, 5\pi/2, \ldots, while the maximum uncertainty is found at 0, 2\pi, 4\pi, \ldots. This phase shift demonstrates how negative squeezing alters the quadrature in which the noise is reduced, showcasing the versatility of squeezed states.
You can find the playlist here.