Two lasers beat note
In this post, I explore the interaction between quantum states and laser beams through a beam splitter, focusing on heterodyne detection to measure beat notes. A beam splitter involves two input modes and two output modes, represented as | \boldsymbol{\Psi}_{12} \rangle and | \Psi_{34} \rangle, connected through a unitary transformation. The input is given by two laser beams:
| \boldsymbol{\Psi}_{12} \rangle = | \boldsymbol{\alpha}_1 \rangle_1 | \boldsymbol{\alpha}_2 \rangle_2 = | \alpha_1 e^{i\phi_1} \rangle_1 | \alpha_2 e^{i\phi_2} \rangle_2
where each beam is described by a single mode quasi-classical state. A photodetector is placed in output channel (4), and the average current is proportional to the photodetection rate:
\langle \mathbf i \rangle(t) = q_eS w^{(1)}(\mathbf r_4,t)
The rate of photodetection is expressed as:
w^{(1)}(\mathbf r_4,t) = s \left\| t\mathbf E_1^{(+)} - r\mathbf E_2^{(+)} | \boldsymbol{\Psi}_{12} \rangle \right\|^2
Here, I focus on how the beat note, created by interference between the laser signals, can be observed. When frequencies of the two beams are close, I can express the photodetection rate in a simplified form:
w^{(1)}(\mathbf r_4,t) = s \left[ \mathscr E^{(1)}_\omega \right]^2 \left| t\boldsymbol{\alpha}_1 e^{-i\omega_1 t} - r\boldsymbol{\alpha}_2 e^{-i\omega_2 t} \right|^2
leading to:
\langle \mathbf i \rangle(t) = q_e \eta \left[ t^2 \Phi_{\text{{photon}}_1} + r^2 \Phi_{\text{{photon}}_2} - 2rt \sqrt{\Phi_{\text{{photon}}_1}\Phi_{\text{{photon}}_2}} \cos\left((\omega_1 - \omega_2) t - \phi_1 + \phi_2 \right) \right]
In this way, the interference between the two beams is captured by the photocurrent, allowing me to detect the beat note. The beat note is crucial for measuring small differences in laser frequencies and is commonly used when stabilizing lasers for specific atomic transitions. For the beat note to be observed, the difference between the frequencies \omega_1 and \omega_2 must be detectable by the photodetector, which currently is in the range of GHz for fast detectors.
An important application of this is in heterodyne detection, where one of the laser signals, | \boldsymbol{\alpha}_2 \rangle, is much stronger than the other and acts as a local oscillator. By controlling the amplitude of this local oscillator, I can effectively tune the detection sensitivity. The photon flux of the weaker signal, \Phi_{\text{{photon}}_1}, can be determined by measuring the amplitude of the beat note.
In this context, the current becomes:
\langle \mathbf i \rangle(t) = q_e \eta \left[ 1 - \cos((\omega_1 - \omega_2)t - \phi_1 + \phi_2) \right]
This method proves useful not only for measuring and controlling the frequency difference between laser sources but also in studying quantum fields. By observing the beat note between a laser and an unknown quantum field, I can infer properties of the unknown field, which is essential in quantum optics experiments.
Heterodyne detection allows me to explore and measure weak quantum signals by combining them with a strong local oscillator. This is particularly useful in targeting atomic transitions, where precise control over laser frequencies is essential. The ability to measure the beat note between two lasers with great accuracy makes this a vital technique for quantum experiments involving lasers and atomic physics.
As I work with these systems, I find that observing beat notes helps me manage the stability and accuracy of laser frequencies. This is critical for a variety of experimental setups, where controlling the laser frequency to target specific transitions is necessary. In particular, locking a primary laser to an atomic transition and using feedback based on the beat note detection allows for fine-tuned control of additional lasers targeting different transitions.
In conclusion, heterodyne detection through beam splitters and the observation of beat notes between laser signals are powerful techniques for controlling and measuring quantum optical systems. Whether used for precise frequency measurements or studying quantum fields, these methods are key to advancing experiments in quantum optics and atomic physics.
For more insights into this topic, you can find the details here.