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Partial Bell-State Measurement with Type-II Parametric Down Conversion: Extracting Phase Information from the Zeropoint Field (I)

Alberto Casado, S. Guerra

Published Feb 21, 2023• 1 citation

Abstract

In this paper, the nexus between the Bell-state measurement and extracting phase information from the zeropoint field is investigated. For this purpose, the Wigner representation in the Heisenberg picture is applied in a Bell-type experiment in which the polarisation-entangled photon pairs generated in a type-II parametric down-conversion do not overlap. The signal intensities at the detectors are calculated in a four-mode approximation, being expressed as functions of the modules and phases of the four zeropoint amplitudes entering the crystal. A general criterion for identifying the correlated detectors is proposed based on the equality of the signal intensities, and without involving the calculation of the joint detection probabilities. In addition, from the analyses in the rectilinear and diagonal basis, it is shown that the distinguishability of the polarisation Bell states, which is in direct correspondence with the joint detection events in each experiment, can be related to the knowledge of the phases of the vacuum field entering the entanglement source, and giving rise to correlated detections. To this purpose, it is conjectured that a detection event is associated with a maximum value of the signal intensity averaged in the modules of the zeropoint amplitudes, as a function of the vacuum phases.

DOI: 10.3390/e25030393

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We apply the Wigner formalism of quantum optics in the Heisenberg picture to study the role of the zeropoint field fluctuations in entanglement swapping produced via parametric down conversion. It is shown that the generation of mode entanglement between two initially non-interacting photons is related to the quadruple correlation properties of the electromagnetic field, through the stochastic properties of the vacuum. The relationship between the process of transferring entanglement and the different zeropoint inputs at the non-linear crystal and the Bell-state analyser is emphasized.

The Wigner formalism in the Heisenberg picture constitutes a bridge that connects Quantum Optics to Stochastic Optics. The vacuum field appears explicitly in the formalism, and the wavelike aspects of light are emphasised. In addition, the zeropoint intensity as a threshold for detection is a common denominator in both theories. In this paper, after summarising the basic rules of the Wigner approach and its application to parametric down-conversion, some new results are presented that delve into the physical meaning of the zeropoint field in optical quantum communication. Specifically, the relationship between Bell-state distinguishability and the number of sets of zeropoint modes that take part in the experiment is analysed in terms of the coupling between the phases of the different fields involved and the subtraction of the zeropoint intensity at the detectors. Additionally, the connection between the compatibility theorem in quantum cryptography and zeropoint field is stressed.

We apply the Wigner formalism of quantum optics to study the role of the zeropoint field fluctuations in entanglement swapping produced via parametric down conversion. It is shown that the generation of mode entanglement between two initially non interacting photons is related to the quadruple correlation properties of the electromagnetic field, through the stochastic properties of the vacuum. The relationship between the process of transferring entanglement and the different zeropoint inputs at the nonlinear crystal and the Bell-state analyser is emphasized.

We apply the Wigner function formalism to partial Bell‐state analysis using polarization entanglement produced in parametric down conversion. Two‐photon statistics at a beam‐splitter are reproduced by a wave‐like description with zeropoint fluctuations of the electromagnetic field. In particular, the fermionic behaviour of two photons in the singlet state is explained from the invariance on the correlation properties of two light beams going through a balanced beam‐splitter. Moreover, we show that a Bell‐state measurement introduces some fundamental noise at the idle channels of the analyzers. As a consequence, the consideration of more independent sets of vacuum modes entering the crystal appears as a need for a complete Bell‐state analysis.

from the point of view of the physical behavior of the detectors. To preserve maximum generality, we adopt a non-factorisable expression of joint detection probabilities (we suspect it is a necessary condition to match quantum correlations, an issue we will examine elsewhere): such non-factorability on (the vacuum amplitudes) is simply an indicator of the need to introduce other (vector of) additional variables in the model, that would possibly correspond to other relevant physical variables related to the laser and the state of the crystal. A conjecture: for the simplest possible model, may possibly correspond to the phase of the laser complex amplitude (or its relative value with respect to ’s). As a correction on initial versions or this paper, such additional -dependence is now introduced directly within the intensities Ii( ) ! Ii(; ) at each (the i-th) detector, which introduces no conceptual change in our initial approach but preserves a reasonable dependence of the probability of detection on the incoming intensity alone: Pi(detj; ) f(I(; )) instead of our former Pi(detj; ) f(I( ); ). For completeness, we also treat some closely related issues, such as: (i) the appearance of (detection probability) enhancement in this framework, (ii) the interpretation of that mysterious \subtraction of the average ZPF intensity at the detector, and (iii) why vacuum uctuations are not directly \observable. Points (i) and (iii) are developed in side-notes [14] and [15], respectively. PACS numbers:

Summary form only given. Entangled photons are the workhorse of quantum information technology and cryptography. So far, the emphasis has been on entanglement of the polarization state (for type II down conversion), or of the wave vectors of the degenerate signal and idler pair (for type I down conversion). A subject of current interest is entanglement in the spatial domain, i.e. nontrivial, spatially extended, correlation between signal and idler. A prime example of this being the discussion regarding the conservation and entanglement of orbital angular momentum in spontaneous parametric down conversion. We report our very first measurements of the signal-idler correlation in type I down conversion, employing a frequency-doubled femtosecond Ti:sapphire laser as our source of pump photons, and a fourth generation intensified CCD as our "multi-element single-photon detector" With pump pulses at /spl lambda/ = 400 nm of approximately 2 nJ energy and 250 fs duration directed at a 0.5 mm long BBO crystal we detect roughly one pair in every four frames.