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arXiv:1101.2657 · Optics Express

Optical phase-space-time-frequency tomography

Paul Rojas, Rachel A. Blaser, Yong Meng Sua, Kim Fook Lee

Published Apr 4, 2011• 3 citations
arXivHTMLPDF

Abstract

We present a new approach for constructing optical phase-space-time-frequency tomography (OPSTFT) of an optical wave field. This tomography can be measured by using a novel four-window optical imaging system based on two local oscillator fields balanced heterodyne detection. The OPSTFT is a Wigner distribution function of two independent Fourier Transform pairs, i.e., phase-space and time-frequency. From its theoretical and experimental aspects, it can provide information of position, momentum, time and frequency of a spatial light field with precision beyond the uncertainty principle. Besides the distributions of x-p and t-ω, the OPSTFT can provide four other distributions such as x-t, p-t, x-ω and p-ω. We simulate the OPSTFT for a light field obscured by a wire and a single-line absorption filter. We believe that the four-window system can provide spatial and temporal properties of a wave field for quantum image processing and biophotonics.

DOI
10.1364/oe.19.007480

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We present a new approach for constructing optical phase-space-time-frequency tomography (OPSTFT) of an optical wave field. This tomography can be measured by using a novel four-window optical imaging system based on two local oscillator fields balanced heterodyne detection. The OPSTFT is a Wigner distribution function of two independent Fourier Transform pairs, i.e., phase-space and time-frequency. From its theoretical and experimental aspects, it can provide information of position, momentum, time and frequency of a spatial light field with precision beyond the uncertainty principle. Besides the distributions of x-p and t-ω, the OPSTFT can provide four other distributions such as x-t, p-t, x-ω and p-ω. We simulate the OPSTFT for a light field obscured by a wire and a single-line absorption filter. We believe that the four-window system can provide spatial and temporal properties of a wave field for quantum image processing and biophotonics.
The technique of dc-balanced, pulsed homodyne detection for the purpose of determining optical-field statistics on short time scales is analyzed theoretically. Such measurements provide photon-number and phase distributions associated with a repetitive signal light field in a short time window. Time- and space-varying signal and local-oscillator pulses are treated, thus generalizing earlier treatments of photoelectron difference statistics in homodyne detection. Experimental issues, such as the effects of imperfect detector balancing on (time-integrated) dc detection and the consequences of background noise caused by non-mode-matched parts of the multimode signal field, are analyzed. The Wigner, or joint, distribution for the two field-quadrature amplitudes during the sampling time window can be directly determined by tomographic inversion of the measured photoelectron distributions. It is pointed out that homodyne detection provides a new method for the simultaneous measurement of temporal and spectral information. Although the theory is generally formulated, with both signal and local-oscillator fields being quantized, emphasis is placed on the limit of a strong, coherent local-oscillator field, making semiclassical interpretation possible.
Experiments have been performed to determine the Wigner distribution and the density matrix (and for pure states the wave function) of a light mode, by using tomographic inversion of a set of measured probability distributions for quadrature amplitudes.From these measurements the quantum distributions of optical phase and photon number have been obtained.The measurements of quadrature-amplitude distributions for a temporal mode of the electromagnetic field are carried out using balanced homodyne detection.We refer to this new method as optical homodyne tomography.Given the measured density matrix, one can experimentally infer any of the various quantum distributions of optical phase, in particular the Pegg-Barnett (or, equivalently, Shapiro-Shepard) phase distribution, the marginal Wigner distribution, and the Vogel-Schleich operational phase distribntion.We have used this approach to make measurements of the number-plase uncertainty relation for coherent-state fields.The coherent states do not attain the minimum value for the number-phase uncertainty product, as set by the expectation value of the commutator of the number and phase operators; this is true theoretically and experimentally.
We introduce a method for quantum-state tomography for a pair of optical modes, which we call generalized rotations in phase space. Rather than using a dual-mode local oscillator field for implementing balanced-homodyne detection, as in previous methods, a single-mode local oscillator field can be used in our method, greatly simplifying certain aspects of experiments. The signal field is put through a series of general transformations before entering the detection system where quantum statistics are measured.
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.
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 present a direct measurement of the Kirkwood-Rihaczek (KR) distribution for spatial properties of a coherent light beam in terms of position and momentum (angle) coordinates. We employ a two-local oscillator (LO) balanced heterodyne detection (BHD) to simultaneously extract the distribution of the transverse position and momentum of a light beam. The two-LO BHD can measure the KR distribution for any complex wave field (including quantum mechanical wave function) without applying tomography methods (inverse Radon transformation). The transformation of the KR distribution to the Wigner, Glauber-Sudarshan PPP, and Husimi or QQQ distributions in spatial coordinates are illustrated through experimental data. The KR distribution can provide the local information of a wave field, which is suitable for studying particle properties of a quantum system. Meanwhile, the Wigner function is suitable for studying wave properties such as interference, and hence provides nonlocal information of the wave field. The method developed here can be used for exploring the spatial quantum state for information processing and optical phase-space imaging for biomedical applications.