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Physical Review A

Measurement of a negative value for the Wigner function of radiation

Gilles Nogues, Arno Rauschenbeutel, S. Osnaghi, Patrice Bertet, M. Brune, J. M. Raimond, S. Haroche, L. G. Lutterbach, L. Davidovich

Published Oct 11, 2000• 116 citations
Publisher

Abstract

Following a proposal by two of us [L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997)], we have measured the Wigner function at the origin of phase space for a single photon field. Its value is negative, exhibiting the nonclassical nature of this state. The experiment is based on the absorption-free detection of the microwave field stored in a superconducting cavity [G. Nogues et al., Nature (London) 400, 239 (1999)]. Extension to a measurement of the Wigner function over the complete phase space is discussed.

DOI
10.1103/physreva.62.054101

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Following a proposal by two of us [L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997)], we have measured the Wigner function at the origin of phase space for a single photon field. Its value is negative, exhibiting the nonclassical nature of this state. The experiment is based on the absorption-free detection of the microwave field stored in a superconducting cavity [G. Nogues et al., Nature (London) 400, 239 (1999)]. Extension to a measurement of the Wigner function over the complete phase space is discussed.
We have measured the complete Wigner function W of the vacuum and of a single-photon state for a field stored in a high-Q cavity. This experiment implements the direct Lutterbach and Davidovich method [L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997)]] and is based on the dispersive interaction of a single circular Rydberg atom with the cavity field. The nonclassical nature of the single-photon field is exhibited by a region of negative W values. Extensions to other nonclassical cavity field states are discussed.
We present an operational definition of the Wigner function. Our method relies on the Fresnel transform of measured Rabi oscillations and applies to motional states of trapped atoms as well as to field states in cavities. We illustrate this technique using data from recent experiments in ion traps [Phys. Rev. Lett. 76, 1796 (1996)]] and in cavity QED [Nature (London) 403, 743 (2000)]]. The values of the Wigner functions of the underlying states at the origin of phase space are W(|0>)(0)=+1.75 for the vibrational ground state and W(|1>)(0)=-1.4 for the one-photon number state. We generalize this method to wave packets in arbitrary potentials.
We show that the value at the origin of phase space of the Wigner function of entangled GHZ states, involving three modes of the electromagnetic field either in the same cavity or in different ones, is a signature of the quantum character of those states. This multimode Wigner function can be directly measured by a simple extension of a method previously proposed by two of the present authors. A single measurement on an atom which crosses the three modes is enough to distinguish between quantum mechanics and local hidden variable theories.