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Journal of Physics B Atomic Molecular and Optical Physics

Measurement of the Wigner function via atomic beam deflection in the Raman–Nath regime

Ashfaq H. Khosa, Muhammad Suhail Zubairy

Published Nov 23, 2006• 3 citations
Publisher

Abstract

A method for the reconstruction of photon statistics and even the Wigner function of a quantized cavity field state is proposed. The method is based on the measurement of momentum distribution of two-level atoms in the Raman–Nath regime. Both the cases of resonant and off-resonant atom–field interaction are considered. The Wigner function is reconstructed by displacing the photon statistics of the cavity field. This reconstruction method is straightforward and does not need much mathematical manipulation of experimental data.

DOI
10.1088/0953-4075/39/24/005

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We propose a scheme for the measurement of a two-mode entangled field-state in a high-QQQ cavity. The scheme utilizes the momentum distribution spectrum in the Raman-Nath regime of a three-level atom in VVV configuration. Due to the two modes of the electromagnetic field the atom may have xxx interactions with mode AAA, and yyy interactions with mode BBB, causing a complex momentum distribution. The momentum distribution of the atom after interaction with the quantized cavity fields contains the information of the field photon statistics. We reconstruct the joint photon statistics of the entangled field with the help of recorded momentum spectrum. We also propose to reconstruct the Wigner function of a two-mode entangled field state by injecting two coherent states resonant to each mode into the cavity and then measuring the joint photon statistics of the displaced field.
We consider the deflection of a resonant two-level atom by a quantized electromagnetic field using the Jaynes-Cummings Hamiltonian. We show that a joint measurement of the atomic momentum and an appropriate field variable allows us to reconstruct the original photon statistics even for this demolition Hamiltonian. We demonstrate that the momentum distribution of atoms scattered at the nodes of the standing wave also follows the original photon statistics of the field. In this sense a recent experiment on the optical Stern-Gerlach effect [T. Sleator et al., Phys. Rev. Lett. 68, 1996 (1992)] measures the intensity fluctuations of the standing wave.
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 study the interaction of a quantized single-mode standing-wave cavity field with a two-level atom de Broglie wave. For the sake of simplicity we consider the field to be far detuned and the atom to be moving in the Raman - Nath regime. We show that the Wigner function of the field is a superposition of the Wigner functions for the coherent states aligned on an arc in phase space. The back action of the measurement of the atomic internal state leads to the modification of the diagonal as well as of the off-diagonal elements of the field density matrix. We investigate the formation of nonclassical field states via atomic deflection and internal-state measurement and show that the coherence of the field and the interference between the constituent coherent states disappear in the limit of large interaction times. The width of the atomic momentum distribution allows us to determine the mean photon number and the width of the photon distribution. We find that, for an appropriate choice of the initial state of the atomic centre-of-mass motion, the form of the atomic momentum distribution is identical to that of the photon distribution. The field near a node or an antinode acts as a focusing or defocusing lens for atoms, depending on the detuning and on the initial internal state of the atom.