Two Lasers Beat Note
Let’s consider a beam splitter, in which there are two input modes (1) and (2) and two output modes (3) and (4), with a unitary transformation \mathbf U which link the state | \Psi_{12} \rangle with |\Psi_{34} \rangle.
In this case, we consider two laser beams in the two input mode of the beam splitter:
| \Psi_{12} \rangle = | \alpha_1 \rangle_1 | \alpha_2 \rangle_2 = | \alpha_1 e^{i\phi_1} \rangle_1 | \alpha_2 e^{\phi_2} \rangle_2
and each of the beam is described by a single mode quasi-classical state; there is a photodetector placed in output channel (4). The average current is proportional to the mean rate of photodetection:
\langle \mathbf i \rangle(t) = q_eS w^{(1)}(\mathbf r_4,t)
The rate of photodetection is:
\begin{aligned} w^{(1)}(\mathbf r_4,t) & = s \left\| \mathbf E_4^{(+)} | \Psi_{34} \rangle \right\|^2 \\ & = s \left\| t\,\mathbf E_1^{(+)} - r\,\mathbf E_2^{(+)} | \Psi_{12} \rangle \right\|^2 \\ & = s \left[ \mathscr E^{(1)}_\omega\right]^2 \left\|\left( t\,\alpha_1 e^{-i\omega_1 t} - r\,\alpha_2 e^{-i\omega_2 t} \right) | \Psi_{12} \rangle \right\|^2\\ & = s \left[ \mathscr E^{(1)}_\omega\right]^2 \left| t\,\alpha_1 e^{-i\omega_1 t} - r\,\alpha_2 e^{-i\omega_2 t} \right|^2 \\ & = s \left[ \mathscr E^{(1)}_\omega\right]^2 \left[|t\,\alpha_1|^2 + |r\,\alpha_2|^2 e^{-i\omega_1 t} - 2|rt\,\alpha_1\alpha_2| \cos\left((\omega_1 - \omega_2) t - \phi_1 + \phi_2 \right) \right] \end{aligned}
In this case again, the result is identical to the result that the one that would be obtained for classical field of equivalent amplitude; since the frequencies are close, it is possible to factor out the sensitivity and the value of the electric field.
Using the expressions derived from a freely propagating beam here:
\begin{aligned} & |\alpha_i|^2 = \frac{\Phi}{\hbar \omega_i}T = \Phi_{\text{photon}_i} T \\ & s \left[ \mathscr E^{(1)}_\omega\right]^2 = \frac{\eta}{ST} \end{aligned}
The current becomes:
\begin{aligned} \langle \mathbf i \rangle(t) & = q_eS w^{(1)}(\mathbf r_4,t) \\ & = q_eS s \left[ \mathscr E^{(1)}_\omega\right]^2 \left[|t\,\alpha_1|^2 + |r\,\alpha_2|^2 - 2|rt\,\alpha_1\alpha_2| \cos\left((\omega_1 - \omega_2) t - \phi_1 + \phi_2 \right) \right] \\ & = q_eS \frac{\eta}{ST} \left[t^2\,T\Phi_{\text{photon}_1} + r^2\,T\Phi_{\text{photon}_2} - 2rt\,T \sqrt{\Phi_{\text{photon}_1}\Phi_{\text{photon}_2}} \cos\left((\omega_1 - \omega_2) t - \phi_1 + \phi_2 \right) \right] \\ & = 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] \end{aligned}
This formula only depends from the photon flux and no longer from arbitrary quantities. If the two term that interfere are equal, the visibility is 100\%:
\langle \mathbf i \rangle(t) = q_e \eta \left[1 - \cos\left((\omega_1 - \omega_2) t - \phi_1 + \phi_2 \right) \right]
This is the quantum average of the photocurrent, the quantum fluctuations are the shot noise.
Observing the beat note between two lasers is possible only if the frequencies \omega_1 and \omega_2 fulfill some conditions, as the frequencies for visible light are of the order of 5 \times 10^{14} \text{ Hz}, so if the frequency are randomly chosen, they will likely fall in the same range, which is too high to detected; they are detectable only if the difference is less than the maximum detectable frequency of the detector; which currently, for fast detectors, is of the order of 10^{10} \text{ Hz} (a few tens of Ghz), which is quite small on the scale of lasers; so in practice currently the beat note is observed with the two realizations of the same type of laser.
Measuring the beat note is the way to measure and to control the difference between the two source of laser. This technique is especially useful for targeting various atomic transitions, which typically vary by a few gigahertz; by locking a primary laser to a specific transition, its stability and accuracy can be transferred to additional lasers targeting other transitions through feedback techniques that rely on the beat note detection between the master and other lasers.
Observing the beat note between a laser and an unknown quantum field is way to study a particular quantum field and it is called heterodyne detection.
Let’s consider again the case described above where it is possible to observe the beat note between two semi-classical states, and assume that the signal | \alpha_1 \rangle in input (1) is a small signal to detect, while the signal in input (2) is a strong signal that can be tuned. The beam | \alpha_2 \rangle is called the local oscillator.
If the photon flux \Phi_{\text{photon}_2} is known, then measuring the amplitude of the beat note will allow to determine the photon flux \Phi_{\text{photon}_1}.