Fresnel representation of the Wigner function: an operational approach.

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.

[1]  J. Wheeler.  The Young Feynman , 1989 .

[2]  Fermionic molecular dynamics SCV252SCV133 V2 , 1997, nucl-th/9703014.

[3]  P. Bertet,et al.  Measurement of a negative value for the Wigner function of radiation , 2000 .

[4]  B. Englert,et al.  Parity states in the one-atom maser , 1993 .

[5]  P. Bertet,et al.  Direct measurement of the Wigner function of a one-photon Fock state in a cavity. , 2002, Physical review letters.

[6]  Herbert Walther,et al.  SCHEME FOR DIRECT OBSERVATION OF THE WIGNER CHARACTERISTIC FUNCTION IN CAVITY QED , 1998 .

[7]  King,et al.  Experimental Determination of the Motional Quantum State of a Trapped Atom. , 1996, Physical review letters.

[8]  Kevin Cahill,et al.  DENSITY OPERATORS AND QUASIPROBABILITY DISTRIBUTIONS. , 1969 .

[9]  Beck,et al.  Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum. , 1993, Physical review letters.

[10]  H. Walther,et al.  Preparing pure photon number states of the radiation field , 2000, Nature.

[11]  Banaszek,et al.  Direct probing of quantum phase space by photon counting. , 1996, Physical review letters.

[12]  K. Wódkiewicz,et al.  Operational Approach to Phase-Space Measurements in Quantum Mechanics , 1984 .

[13]  Dreyer,et al.  Quantum Rabi oscillation: A direct test of field quantization in a cavity. , 1996, Physical review letters.

[14]  Walther,et al.  Observation of wave packet motion along quasi-Landau orbits. , 1993, Physical review letters.

[15]  K. Banaszek,et al.  Direct measurement of the Wigner function by photon counting , 1999 .

[16]  D. Leibfried,et al.  Shadows and Mirrors: Reconstructing Quantum States of Atom Motion , 1998 .

[17]  L. Davidovich,et al.  Method for Direct Measurement of the Wigner Function in Cavity QED and Ion Traps , 1997 .

[18]  Willis E. Lamb,et al.  An operational interpretation of nonrelativistic quantum mechanics , 1969 .

[19]  E Solano,et al.  Conditional large Fock state preparation and field state reconstruction in cavity QED. , 2001, Physical review letters.

[20]  King,et al.  Generation of nonclassical motional states of a trapped atom. , 1996, Physical review letters.

[21]  QUANTUM-STATE RECONSTRUCTION IN THE ONE-ATOM MASER , 1998 .

[22]  Vogel,et al.  Unbalanced homodyning for quantum state measurements. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[23]  Salman Habib,et al.  Decoherence, chaos, and the correspondence principle , 1998 .

[24]  Royer.  Measurement of the Wigner function. , 1985, Physical review letters.

[25]  C. Kurtsiefer,et al.  Measurement of the Wigner function of an ensemble of helium atoms , 1997, Nature.

[26]  Nonlinear Jaynes-Cummings dynamics of a trapped ion. , 1995 .

[27]  A. Lvovsky,et al.  Quantum state reconstruction of the single-photon Fock state. , 2001, Physical Review Letters.

[28]  Wilkens,et al.  Nonlinear atomic homodyne detection: A technique to detect macroscopic superpositions in a micromaser. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[29]  Walther,et al.  Observation of quasi-Landau wave packets. , 1994, Physical review letters.

[30]  S. Schiller,et al.  Measurement of the quantum states of squeezed light , 1997, Nature.

[31]  M. Scully,et al.  Distribution functions in physics: Fundamentals , 1984 .