Complete experimental characterization of the quantum state of a light mode via the Wigner function and the density matrix: application to quantum phase distributions of vacuum and squeezed-vacuum states

We have used the recently demonstrated method of optical homodyne tomography (OHT) to measure the Wigner quasiprobability distribution (Wigner function) and the density matrix for both a squeezed-vacuum and a vacuum state of a single spatial-temporal mode of the electromagnetic field. This method consists of measuring a set of probability distributions for many different Hilbert-space representations of the field-quadrature amplitude, using balanced homodyne detection, and then using tomography to obtain the Wigner function. Once the Wigner function is obtained, one can acquire the density matrix, including its complex phase. In the case of a pure state, this technique yields an experimentally determined complex wavefunction, as demonstrated here for the vacuum. The density matrix represents a complete quantum mechanical characterization of the state. From the measured density matrix we have obtained the Pegg–Barnett optical phase distribution, and from the Wigner function, the Wigner optical phase distribution.

[1]  Mandel,et al.  Further investigations of the operationally defined quantum phase. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[2]  J. Noh,et al.  Measurement of the quantum phase by photon counting. , 1991, Physical review letters.

[3]  Generating and detecting short-duration pulses of squeezed light. , 1987, Physical review. A, General physics.

[4]  B. Yurke,et al.  Squeezed-light-enhanced polarization interferometer. , 1987, Physical review letters.

[5]  Leonard Susskind,et al.  Quantum mechanical phase and time operator , 1964 .

[6]  E. Wigner.  On the quantum correction for thermodynamic equilibrium , 1932 .

[7]  Barnett,et al.  Phase properties of the quantized single-mode electromagnetic field. , 1989, Physical review. A, General physics.

[8]  Smith,et al.  Observation of zero-point noise squeezing via a Josephson-parametric amplifier. , 1990, Physical review letters.

[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]  Beck,et al.  Experimental determination of quantum-phase distributions using optical homodyne tomography. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[11]  G. Litfin,et al.  Phase measurement of a microscopic radiation field , 1974 .

[12]  Schleich,et al.  Bifurcation in the phase probability distribution of a highly squeezed state. , 1989, Physical review. A, General physics.

[13]  Pulsed squeezed light. , 1987, Physical review letters.

[14]  P. Dirac.  The Quantum Theory of the Emission and Absorption of Radiation , 1927 .

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

[16]  Lee,et al.  Gauged Q balls. , 1989, Physical review. D, Particles and fields.

[17]  R. F. O'Connell,et al.  The Wigner distribution function—50th birthday , 1983 .

[18]  G. Björk,et al.  II Quantum Mechanical Limit in Optical Precision Measurement and Communication , 1990 .

[19]  R. Gerchberg.  A practical algorithm for the determination of phase from image and diffraction plane pictures , 1972 .

[20]  Physical Number-phase Intelligent and Minimum-uncertainty States of Light , 1990 .

[21]  Vogel,et al.  Phase distribution of a quantum state without using phase states. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[22]  Beck,et al.  Sub-shot-noise correlation of total photon number using macroscopic twin pulses of light. , 1992, Physical review letters.

[23]  Jeffrey H. Shapiro,et al.  Optical communication with two-photon coherent states-Part III: Quantum measurements realizable with photoemissive detectors , 1980, IEEE Trans. Inf. Theory.

[24]  V. Chan,et al.  Noise in homodyne and heterodyne detection. , 1983, Optics letters.

[25]  S. Barnett,et al.  On the Hermitian Optical Phase Operator , 1989 .

[26]  Fluctuation of the Barnett-Pegg phase operator in a coherent state. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[27]  E. C. Kemble.  The Fundamental Principles Of Quantum Mechanics , 1937 .

[28]  M. Nieto.  Phase-difference operator analysis of microscopic radiation-field measurements☆ , 1977 .

[29]  A. Royer.  Measurement of quantum states and the Wigner function , 1989 .

[30]  D. R. Matthys,et al.  Phase-sensitive optical amplifier , 1980 .

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

[32]  L. Ballentine.  Can one detect the state of an individual system? , 1992 .

[33]  A. Huiser,et al.  PHASE RETRIEVAL IN ELECTRON-MICROSCOPY FROM IMAGE AND DIFFRACTION PATTERN , 1976 .

[34]  Slusher,et al.  Squeezing limits at high parametric gains. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[35]  Shapiro,et al.  Quantum phase measurement: A system-theory perspective. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[36]  Vogel,et al.  Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. , 1989, Physical review. A, General physics.

[37]  R. Shelby,et al.  Time dependence of quantum fluctuations in solitons. , 1989, Optics letters.