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Fresnel Transform: An Operational Definition of the Wigner Function

Pavel Lougovski, Emilia R. Solano, Z. M. Zhang, Hans‐Otto Walther, Holger Mack, W. P. Schleich

Published Jun 13, 2002• 4 citations
OpenAlex

Abstract

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 [D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996)] and in cavity QED [B. Varcoe et al., Nature 403, 743 (2000)]. With the help of the Fresnel transform we extract the values of the Wigner functions of the underlying states at the origin of phase space. We find W(0)=+1.75 for the vibrational ground state and W(0)=-1.9 for the one-photon number state.

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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 [D. M. Meekhof et al., Phys. Rev. Lett. 76, 1796 (1996)] and in cavity QED [B. Varcoe et al., Nature 403, 743 (2000)]. With the help of the Fresnel transform we extract the values of the Wigner functions of the underlying states at the origin of phase space. We find W(0)=+1.75 for the vibrational ground state and W(0)=-1.9 for the one-photon number state.
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.
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.
Based on the technique of integration within an ordered product (IWOP) of operators, we introduce the Fresnel operator for converting a kind of time-dependent Hamiltonian into the standard harmonic oscillator Hamiltonian. The Fresnel operator with the parameters A, B, C, D corresponds to classical optical Fresnel transformation, these parameters are the solution to a set of coupled partial differential equations set up in the above-mentioned converting process. In this way, the Caldirola–Kanai Hamiltonian has been easily converted into the standard harmonic oscillator Hamiltonian. And then the exact wave function solution of the Schrödinger equation governed by the Caldirola–Kanai Hamiltonian is obtained, which represents a squeezed number state. The corresponding Wigner function is derived by virtue of the Weyl ordered form of the Wigner operator and the order-invariance of Weyl ordered operators under similar transformations.
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.
These lectures review some basic techniques of quantum optics, related to the description of the quantized electromagnetic field in phase space and of the interaction between atoms and photons in cavities. The Wigner function is introduced, and some of the methods for measuring this distribution are reviewed. The combination of phase space methods with cavity QED techniques is shown to lead to experiments which are closely connected to fundamental problems regarding the classical limit of quantum mechanics and the quantum theory of measurement.