> [...] A pertinent example of continuous variables comprise the quadrature representation of the optical quantum state, as opposed to the discrete-variable photon-number representation. The wave-like nature of the field results from the coherence between different photon number components of a field with uncertain photon number. Therefore, to study the wave-like nature of the field, one must be able to measure superpositions of photon number states.
> To do so, the field of a weak optical quantum state (signal) comprising a few photons interferes with the field of a bright coherent state (i.e., a local oscillator, LO) on a balanced beam splitter. The two output modes of the beam splitter are then measured with two photodetectors. By calculating the difference between the detector count rates, which is proportional to the field quadrature at a reference phase provided by the local oscillator, the optical quantum state can be characterized [3–5]. The ability to characterize optical quantum states in their phase-space representation [1] makes homodyne detection an essential tool for quantum information processing with continuous variables [6].
> Typically, conventional semiconductor photodiodes are used as the detector in balanced homodyne detection. The high optical flux arising from the LO lifts the optical signal above the electronic noise floor, which is typically in the pW√Hz range at telecommunication wavelengths. As a result, the generated carriers in the photodiode can be integrated over a characteristic response time, resulting in a photocurrent which is proportional to the incident optical flux [7]. For BHD, it is essential that the photodetector output is directly proportional to the intensity (photon flux) of the input light. In this case, we refer to the detector output as linear.