Homodyne detection measurements
Balanced homodyne detection is a technique for measuring quadrature observables of a light field. This method is particularly useful because it allows us to access information about the electric field, even in the visible spectrum, and to characterize its quantum fluctuations.
The setup is based on a balanced beam splitter, a device that splits an incoming light beam into two equal parts. In our case, we send the light field we want to analyze (| \boldsymbol{\Psi}_1 \rangle), into one input port of the beam splitter. Into the other input port, we introduce a strong quasi-classical field called the local oscillator (| \boldsymbol {\alpha}_2 \rangle), which has the same frequency.
We then measure the difference between the photocurrents generated by the light exiting the two output ports of the beam splitter. This is the essence of balanced detection.
For a balanced beam splitter, the output fields \mathbf E_3^{(+)} and \mathbf E_4^{(+)} are related to the input fields \mathbf E_1^{(+)} and \mathbf E_2^{(+)} by:
\begin{aligned} \mathbf E_3^{(+)} & = \frac{1}{\sqrt 2}\left(\mathbf E_1^{(+)} + \mathbf E_2^{(+)} \right) \\ \mathbf E_4^{(+)} & = \frac{1}{\sqrt 2}\left(\mathbf E_1^{(+)} - \mathbf E_2^{(+)}\right) \end{aligned}
After some quantum optics calculations, we find that the difference in the average photocurrents, \langle \mathbf i_3 - \mathbf i_4 \rangle(t), is proportional to:
d = q_e \eta \left|\alpha_\lambda\right| \langle \boldsymbol \Psi_1 | \left(e^{-i\varphi_\lambda}\mathbf a_1 + e^{i\varphi_\lambda} \mathbf a_1^\dag \right) |\boldsymbol \Psi_1 \rangle
where \alpha_\lambda = \left|\alpha_\lambda\right|e^{i\varphi_\lambda} describes the local oscillator, and \mathbf a_1 and \mathbf a_1^\dag are the annihilation and creation operators for the input mode, respectively. By adjusting the phase \varphi_\lambda of the local oscillator, we can measure different linear combinations of \mathbf a_1 and \mathbf a_1^\dag. Specifically, for \varphi_\lambda = 0 we measure the Q quadrature, and for \varphi_\lambda = \frac{\pi}{2} we measure the P quadrature. These are the quadrature observables of the light field.
Furthermore, we can also compute the fluctuations of this measurement by looking at the variance of the photocurrent difference. This is related to:
\Delta^2\left(\mathbf N_3 - \mathbf N_4\right) = \langle \boldsymbol {\Psi} | \left(\mathbf N_3 - \mathbf N_4 \right)^2| \boldsymbol {\Psi} \rangle - \left(\langle \boldsymbol {\Psi} | \mathbf N_3 - \mathbf N_4 | \boldsymbol {\Psi} \rangle\right)^2
where \mathbf N_3 and \mathbf N_4 are the photon number operators at the output ports.
A significant advantage of balanced homodyne detection is that it allows us to overcome limitations related to the detector response time. By using a strong local oscillator, the signal becomes large enough to be measured even with detectors that might not be fast enough to directly follow the optical frequencies. This technique is particularly powerful for studying the quantum nature of light and its fluctuations.
For more insights into this topic, you can find the details here.