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Chinese Physics Letters

Measuring the Wigner Functions of Two-Mode Cavity Fields and Testing the Bell's Inequalities

Zhi‐Ming Zhang

Published Jan 1, 2004• 16 citations
Publisher

Abstract

We propose a scheme for measuring the Wigner function of a two-mode cavity field. The scheme bases on the interaction between the two-mode cavity field and three-level atoms. We find a simple relation between the Wigner function and the atomic population. One can obtain the Wigner function by measuring the atomic population with a micromaser-like experiment and doing a numerical integral. By using the two-mode Wigner function one can obtain the Clauser–Horne combination and test the Bell's inequalities. We test our equations with a two-mode entanglement state and the results are rather good.

DOI
10.1088/0256-307x/21/1/002

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We propose a scheme for measuring the Wigner function of a two-mode cavity field. The scheme bases on the interaction between the two-mode cavity field and three-level atoms. We find a simple relation between the Wigner function and the atomic population. One can obtain the Wigner function by measuring the atomic population with a micromaser-like experiment and doing a numerical integral. By using the two-mode Wigner function one can obtain the Clauser–Horne combination and test the Bell's inequalities. We test our equations with a two-mode entanglement state and the results are rather good.
We suggest a scheme for reconstructing the Wigner function of two-mode fields. The scheme is based on the interaction between a Λ-type three-level atom and two field-modes in spatially-separated cavities. The two field-modes are displaced with two classical coherent fields and the atom passes through the two cavities successively. We find that there is a simple relation between the Wigner function of the cavity fields and the atomic populations. Therefore, we can reconstruct the Wigner function of cavity fields by measuring the atomic populations in the micromaser experiments.
We propose a scheme for directly measuring the Wigner functions of cavity fields. The scheme is based on the Raman interaction between atoms and cavity fields. We find a simple and direct relation between the Wigner function and the atomic population difference. By suitably choosing the interaction time, we find that the Wigner function is just two times of the atomic population difference. Thus, one can obtain the Wigner function by measuring the atomic populations and calculating the population difference.
A scheme for reconstructing the Wigner function of cavity fields is proposed, and this scheme can be realized with the micromaser. We find that there is a simple relation between the Wigner function of cavity fields and the atomic populations. Therefore, we can reconstruct the Wigner function of cavity fields by measuring the atomic populations in the micromaser experiments. We test our theory with the one-photon Fock state and the Schr?dinger cat state, and find that the results are rather good.
We propose a scheme for the reconstruction of a cavity field state. In the scheme the cavity field is first displaced by a microwave source, and then couples with a three-level Λ-type atom in a Raman manner. After an appropriate interaction time the measurement of the difference between the probabilities of the atom being in two ground states directly yields the Wigner function for the field state. The scheme can be easily generalized to measure the Wigner function of an entangled state of two spatially separated single-mode cavities.
We propose a scheme for the reconstruction of a cavity field state. In the scheme the cavity field is first displaced by a microwave source, and then is placed in front of one slit of the two-slit apparatus. Ground state atoms pass through the apparatus, hitting upon the screen far away from the two-slit apparatus. The atom-field interaction is dispersive. The contrast of interference fringes is directly related to the Wigner function for the field state. The scheme can be easily generalized to measure the Wigner function of an entangled state of two spatially separated single-mode cavities.
A scheme is proposed for the reconstruction of two-mode entangled states in cavity QED and ion traps. For a two-mode field we show that the Wigner function can be obtained by measuring the probability of a two-level atom being in ground states after resonant interaction with two classical fields and dispersive interaction with the two-mode cavity field displaced by resonant sources. For the two-dimensional motion of a trapped ion the Wigner function is obtained by measuring the probability of the ion in its ground electronic state after displacing the ion motion and then resonantly exciting the ion.