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Physical Review Letters

Method for Direct Measurement of the Wigner Function in Cavity QED and Ion Traps

L. G. Lutterbach, L. Davidovich

Published Mar 31, 1997• 395 citations
Publisher

Abstract

We show that the Wigner function corresponding to an electromagnetic field in a cavity or a vibronic state of a trapped ion can be directly measured by means of a simple experimental procedure. The proposed method is insensitive to the relatively low detection efficiency in cavity QED experiments and yields a direct physical meaning to the Wigner distribution.

DOI
10.1103/physrevlett.78.2547

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We show that the Wigner function corresponding to an electromagnetic field in a cavity or a vibronic state of a trapped ion can be directly measured by means of a simple experimental procedure. The proposed method is insensitive to the relatively low detection efficiency in cavity QED experiments and yields a direct physical meaning to the Wigner distribution.
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.
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.
Recent experiments in cavity QED have allowed the monitoring of the decoherence process, which is at the heart of the quantum theory of measurement and plays an essential role in the classical limit of quantum mechanics. It is shown here that the technique of Ramsey interferometry, central to those experiments, may be used to determine completely the quantum state of an electromagnetic field in a cavity. This new method, which allows the direct measurement of the Wigner function of the field, is a useful tool for probing the quantum-classical transition and, in particular, the decoherence of superpositions of distinguishable coherent states of the electromagnetic field in the cavity. It is also shown that the realization of a controlled-not gate within the framework of cavity QED is a special case of this proposal, and is related to the measurement of the Wigner function of a one-photon state.