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Conference on Lasers and Electro-Optics

What can we learn from two-dimensional optical Wigner functions?

J.-W. Wu, Ping Koy Lam, M. Gray, Hans‐A. Bachor

Published Nov 27, 2002• 0 citations

Abstract

Summary form only given.We show experimentally the reconstructions of Wigner functions for the output states of a laser. These states have classical Wigner functions with Gaussian cross-sections. In order to investigate the spectral properties of the output states, we normalize the Wigner functions in the inverse Radon transform algorithm to a coherent state. In this way, we are able to compare the full spectrum of Wigner functions obtained with different detection frequencies, paying particular attention to Wigner functions near the laser relaxation oscillation. The same normalized Wigner function is also used to measure squeezed optical states from an optical parametric oscillator. When large modulation is introduced to the laser beam, a dc component is introduced in the Wigner function. We show that Wigner functions of incoherent superposition states can be reconstructed using this modulation technique. In some cases, fluctuations can be large enough to produce highly non-Gaussian Wigner functions. We provide examples of such cases and show that optical homodyne tomography can provide essentially new information not detectable with standard quadrature variance measurements. Furthermore, we demonstrate that single sideband analysis used to date will sometimes no longer suffice in the analysis of such optical states. We include the higher order sideband correlations experimentally and discuss the necessity of these higher order sidebands for measuring non-Gaussian quantum states, e.g. Schrodinger's cat and Fock states.

DOI: 10.1109/iqec.1998.680248

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Summary form only given.We show experimentally the reconstructions of Wigner functions for the output states of a laser. These states have classical Wigner functions with Gaussian cross-sections. In order to investigate the spectral properties of the output states, we normalize the Wigner functions in the inverse Radon transform algorithm to a coherent state. In this way, we are able to compare the full spectrum of Wigner functions obtained with different detection frequencies, paying particular attention to Wigner functions near the laser relaxation oscillation. The same normalized Wigner function is also used to measure squeezed optical states from an optical parametric oscillator. When large modulation is introduced to the laser beam, a dc component is introduced in the Wigner function. We show that Wigner functions of incoherent superposition states can be reconstructed using this modulation technique. In some cases, fluctuations can be large enough to produce highly non-Gaussian Wigner functions. We provide examples of such cases and show that optical homodyne tomography can provide essentially new information not detectable with standard quadrature variance measurements. Furthermore, we demonstrate that single sideband analysis used to date will sometimes no longer suffice in the analysis of such optical states. We include the higher order sideband correlations experimentally and discuss the necessity of these higher order sidebands for measuring non-Gaussian quantum states, e.g. Schrodinger's cat and Fock states.

Optical homodyne tomography (OHT) is a tool that allows the reconstruction of Wigner functions for each detection frequency of a propagating optical beam. It can measure probability distribution functions (PDF's) of the field amplitude for any given quadrature of interest. We demonstrate OHT for a range of classical optical states with constant and time varying modulations and show the advantage of OHT over conventional homodyne detection. The OHT simultaneously determines the signal to noise ratio in both amplitude and phase quadratures. We show that highly non-Gaussian Wigner functions can be obtained from incoherent superpositions of optical states.

Wigner function tomography is indispensable for characterizing quantum states, but its commonly used version, balanced homodyne detection, suffers from several weaknesses. First, it requires efficient detection, which is critical for measuring fragile non-Gaussian states, especially bright ones. Second, it needs a local oscillator, tailored to match the spatiotemporal properties of the state under test, and fails for multimode and broadband states. Here we propose Wigner function tomography based on optical parametric amplification followed by direct detection. The method is immune to detection inefficiency and loss, and suitable for broadband, spatially and temporally multimode quantum states. To prove the principle, we experimentally reconstruct the Wigner function of squeezed vacuum occupying a single mode of a strongly multimode state. We obtain a squeezing of −7.5±0.4dB and purity of 0.91 −0.08 +0.09 despite more than 97% loss caused mainly by filtering. Theoretically, we also consider the reconstruction of a squeezed single photon—a bright non-Gaussian state. Due to multimode parametric amplification, the method allows for simultaneous tomography of multiple modes. This makes it a powerful tool for optical quantum information processing.

This chapter begins with a review of classical tomography, which reconstructs hidden structures inside an object by successive transmissions of an X-ray beam at different angles and lateral displacements. By interpreting the Wigner distribution as an analogue of the density distribution in a physical object, the mathematical methods of classical tomography — the Radon transform, the Fourier slice theorem, and the inverse Radon transform — are adapted to perform a high resolution determination of a quantum state of light. In optical homodyne tomography, the classical transmission angle is replaced by the adjustable phase of a local oscillator, and the successive lateral displacements are replaced by a series of measurements of the quadrature operator defined by the local oscillator phase. Experimental data is presented showing various properties of the Wigner distribution for a coherent state.

For the visualization of quantum states, the approach based on Wigner functions can be very effective. Homodyne detection has been extensively used to obtain the density matrix, Wigner functions and tomographic reconstructions of optical fields for many thermal, coherent or squeezed states. Here, we use time-domain optical homodyne tomography for the quantum state recognition and reconstruction of the femtosecond optical field from a nonequilibrium superradiant coherent electron-hole state formed in a semiconductor laser structure. We observe severe deviations from the Poissonian statistics of the photons associated with the coherent laser state when the transformation from lasing to superradiance occurs. The reconstructed Wigner functions show large areas of negative values, a characteristic sign of non-classicality, demonstrating the quantum nature of the generated superradiant emission. The photon number distribution and Wigner function of the SR state are very similar to those of the displaced Fock state

This review covers recent theoretical and experimental efforts to extend the application of the continuous-variable quantum technology of light beyond "Gaussian" quantum states, such as coherent and squeezed states, into the domain of "non-Gaussian" states with negative Wigner functions. Starting with basic Gaussian nonclassicality associated with single- and two-mode vacuum states produced by means of parametric down-conversion and applying a set of standard tools, such as linear interferometry, coherent state injection, and conditional homodyne and photon number measurements, one can implement a large variety of optical states and processes that are relevant in fundamental quantum physics as well as quantum optical information processing. We present a systematic review of these methods, paying attention to both fundamental and practical aspects of their implementation, as well as a comprehensive overview of the results achieved therewith.

The two-dimensional Radon transform of the Wigner quasiprobability is introduced in canonical form and the functions playing a role in its inversion are discussed. The transformation properties of this Radon transform with respect to displacement and squeezing of states are studied and it is shown that the latter is equivalent to a symplectic transformation of the variables of the Radon transform with the contragredient matrix to the transformation of the variables in the Wigner quasiprobability. The reconstruction of the density operator from the Radon transform and the direct reconstruction of its Fock-state matrix elements and of its normally ordered moments are discussed. It is found that for finite-order moments the integration over the angle can be reduced to a finite sum over a discrete set of angles. The reconstruction of the Fock-state matrix elements from the normally ordered moments leads to a new representation of the pattern functions by convergent series over even or odd Hermite polynomials which is appropriate for practical calculations. The structure of the pattern functions as first derivatives of the products of normalizable and non-normalizable eigenfunctions to the number operator is considered from the point of view of this new representation.