Optimal temporal mode extraction for quantum state engineering via a direct multimode analysis of homodyne data

Abstract

The ability to engineer various quantum states of light is a central requirement for quantum information processing, including quantum communication, computing and metrology. In this endeavor, many experiments use the so-called conditional preparation technique. The dynamic of the heralded states, i.e. its temporal mode, mainly depends on the setup and has been studied theoretically with different strategies [1,2]. However, it remains difficult to extract the optimal mode with a precise description of the experimental setup: it requires very often some additional assumptions. This is why it is important to characterize experimentally the dynamic of the generated states as it might improve the fidelity of the generated state.Some experiments have been realized to study this issue. They are based on photon-counting coincidence. On one hand, this technique can be questionable as the detector is not photon number resolved and, on the other hand, it can also be very difficult to implement for some wavelengths for which the quantum efficiency of photon detectors is low. Here we present a novel approach, based on homodyne measurements, to determine the temporal mode function involved in a conditional preparation. This method, which only uses data from quantum state tomography performed by homodyne measurements, enables to include all the experimental aspects that theory may forget for simplicity and to extract the optimal temporal mode directly from the data.As an illustration, we consider the generation of a single-photon state by conditional measurement operated on a two-mode squeezed state. By using this technique, we demonstrated in [3] a 79% fidelity with a singlephoton state without any correction of detection losses (91% after correction), the highest value reported to date.In our experiment, the heralded state is characterized by homodyne quantum state tomography. For each recorded segment one extracts the quadrature measurement by using a temporal mode . Our goal is to measure the temporal mode occupied by the single photon state. The initial idea uses the fact that the measured electromagnetic field contains some single photon state, vacuum and very small thermal states. We can thus optimize the temporal mode by maximizing the variance of the mode. Analytically this optimal mode corresponds to the eigenfunction of the quadrature correlation function with the highest eigenvalue. Fig. 1 a) eigenvalues of the correlation function for heralded state (red) and vacuum (blue). b) First eigenmode (red) and the theoretical temporal mode (blue).The experimental results shown in Fig 1.a provide the variance of the electromagnetic field in the basis of the eigenmodes of the correlation function. The single-mode character can be clearly seen (i.e. high purity is achieved). This approach enabled to optimize the setup. Thanks to this process, the final results are very closed to the simplest theoretical model, as shown in Fig 1.b for the temporal mode. This works gives a more complete information in terms of multimode decomp...