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Register a new userDetermination of the Wigner function from photon statistics
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We present an experimental realisation of the direct scheme for measuring the Wigner function of a single quantized light mode. In this method, the Wigner function is determined as the expectation value of the photon number parity operator for the phase space displaced quantum state.
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We report a direct measurement of the Wigner function characterizing the quantum state of a light mode. The experimental scheme is based on the representation of the Wigner function as an expectation value of a displaced photon number parity operator. This allowed us to scan the phase space point-by-point, and obtain the complete Wigner function without using any numerical reconstruction algorithms.
The Hong-Ou-Mandel (HOM) experiment was a benchmark in quantum optics, evidencing the quantum nature of the photon. In order to go deeper, and obtain the complete information about the quantum state of a system, for instance, composed by photons, the direct measurement or reconstruction of the Wigner function or other quasi--probability distribution in phase space is necessary. In the present paper, we show that a simple modification in the well-known HOM experiment provides the direct measurement of the Wigner function. We apply our results to a widely used quantum optics system, consisting of the biphoton generated in the parametric down conversion process. In this approach, a negative value of the Wigner function is a sufficient condition for non-gaussian entanglement between two photons. In the general case, the Wigner function provides all the required information to infer entanglement using well known necessary and sufficient criteria. We analyze our results using two examples of parametric down conversion processes taken from recent experiments. The present work offers a new vision of the HOM experiment that further develops its possibilities to realize fundamental tests of quantum mechanics involving decoherence and entanglement using simple optical set-ups.
We consider the Wigner quasi-probability distribution function of a single mode of an electromagnetic or matter-wave field to address the question of whether a direct stochastic sampling and binning of the absolute square of the complex field amplitude can yield a distribution function $tilde{P}_n$ that closely approximates the true particle number probability distribution $P_n$ of the underlying quantum state. By providing an operational definition of the binned distribution $tilde{P}_n$ in terms of the Wigner function, we explicitly calculate the overlap between $tilde{P}_n$ and ${P}_n$ and hence quantify the statistical distance between the two distributions. We find that there is indeed a close quantitative correspondence between $tilde{P}_n$ and $P_n$ for a wide range of quantum states that have smooth and broad Wigner function relative to the scale of oscillations of the Wigner function for the relevant Fock state. However, we also find counterexamples, including states with high mode occupation, for which $tilde{P}_n$ does not closely approximate $P_n$.
Quantum engineering now allows to design and construct multi-qubit states in a range of physical systems. These states are typically quite complex in nature, with disparate, but relevant properties that include both single and multi-qubit coherences and even entanglement. All these properties can be assessed by reconstructing the density matrix of those states - but the large parameter space can mean physical insight of the nature of those states and their coherence can be hard to achieve. Here we explore how the Wigner function of a multipartite system and its visualization provides rich information on the nature of the state, not only at illustrative level but also at the quantitative level. We test our tools in a photonic architecture making use of the multiple degrees of freedom of two photons.
The Wigner function of a dynamical infinite dimensional lattice is studied. A closed differential equation without diffusion terms for this function is obtained and solved. We map atom-photon interaction systems, such as the Jaynes-Cummings model, into this lattice model, where each dressed or polariton state corresponds to a point in the lattice and the conjugate momenta are described by the eigenvalues of the phase operator. The corresponding Wigner function is defined by these two conjugate variables in what we name the polariton phase space. We derive a general propagator of the Wigner function, which is also valid for other hybrid models.
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