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Nonlinear Dynamics in Optical Systems

Density Of Probability Distribution For Quantum Electromagnetic Field

Игорь Евгеньевич Проценко

Published Jan 1, 1992• 0 citations
Publisher

Abstract

Based on the first principles of quantum mechanic the density of probability distribution function W(x,p) is created for quantum electromagnetic field (QEF) - Figs. 1, 2. Unlike a well-known distributions of density of quasi-probability (as Wigner function et.c.) W(x,p) can be measured directly in 8-port homodyne experiment or in the experiment we propose. Using W(x,p) one can calculate the probability to have the field quadratures x and p in a chosen area of a phase space, find the minimum in amplitude or phase fluctuations of QEF. W(x,p) predicts the lemniscate-of-ellipse shape of the dispersion of squeezed QEF (Loudon, 1989; Kimble, 1986). Using a methods of quantum optic in experimental study of W(x,p) one can reconstruct the wave function of QEF and, in principle, the wave function of an emitting substance.

DOI
10.1364/nldos.1992.mc27

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Based on the first principles of quantum mechanic the density of probability distribution function W(x,p) is created for quantum electromagnetic field (QEF) - Figs. 1, 2. Unlike a well-known distributions of density of quasi-probability (as Wigner function et.c.) W(x,p) can be measured directly in 8-port homodyne experiment or in the experiment we propose. Using W(x,p) one can calculate the probability to have the field quadratures x and p in a chosen area of a phase space, find the minimum in amplitude or phase fluctuations of QEF. W(x,p) predicts the lemniscate-of-ellipse shape of the dispersion of squeezed QEF (Loudon, 1989; Kimble, 1986). Using a methods of quantum optic in experimental study of W(x,p) one can reconstruct the wave function of QEF and, in principle, the wave function of an emitting substance.
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.
We report measurements of the Wigner distribution and the density matrix of an electromagnetic field mode, for both vacuum and quadrature-squeezed states. As proposed by Vogel and Risken,1 we obtained the Wigner distribution by tomographic inversion of a set of probability distributions of field-quadrature amplitudes measured by using balanced homodyne detection. A Fourier transform of the Wiener function yields the density matrix, which, according to the standard interpretation of quantum mechanics, contains all knowable information about a given quantum system.
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We have experimentally determined all of the quantities involved in the uncertainty relation for the phase and photon number of a mode of the electromagnetic field when the field mode is in a coherent state of small average photon number. This is accomplished by determining the quantum state of the field using optical homodyne tomography, which uses measured distributions of electric-field quadrature amplitude to determine the Wigner function and hence the density matrix. The measured state is then used to calculate the uncertainty product for the number and phase, as well as the expectation value of the commutator of the number and phase operators. The experimental results agree with the quantum-mechanical predictions. We also present measured phase- and photon-number distributions for these weak coherent states, as well as their measured complex wave functions.