7 Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling

Abstract

We review experimental work on the measurement of the quantum state of optical fields, and the relevant theoretical background. The basic technique of optical homodyne tomography is described with particular attention paid to the role played by balanced homodyne detection in this process. We discuss some of the original single-mode squeezed-state measurements as well as recent developments including: other field states, multimode measurements, array detection, and other new homodyne schemes. We also discuss applications of state measurement techniques to an area of scientific and technological importance-the ultrafast sampling of time-resolved photon statistics. Please cite as: M. G. Raymer and M. Beck, "Experimental quantum state tomography of optical fields and ultrafast statistical sampling," in Quantum State Estimation, Vol. 649, edited by M. G. A. Paris and J. Rehacek (Springer-Verlag, Berlin, 2004), pp. 235-295.function [13][14][15]. However, in the case of mixed states, even for one-dimensional systems, more than two distributions are needed for state reconstruction.Because some of the first experimental work on quantum state measurement [16] analyzed the collected data using a mathematical technique that is very similar to the tomographic reconstruction technique used in medical imaging, and because all techniques are necessarily indirect, a generally accepted term for quantum state measurement has become quantum state tomography(QST.) Systems measured have included angular momentum states of electrons [17], the field [16,[18][19][20] and polarization [21] states of photon pairs, molecular vibrations [22], trapped ions [23], atomic beams [24], and nuclear spins [25]. Indeed, QST has become so prevalent in modern physics that it has been given its own Physics and Astronomy Classification Scheme (PACS) code by the American Institute of Physics: 03.65.Wj, State Reconstruction and Quantum Tomography.The purpose of this article is to review some of the theoretical and experimental work on quantum state measurement. Prior reviews can be found in [1,8,9,26]. We will concentrate here on measurements of the state of the field of an optical beam with an indeterminate number of photons in one or more modes. Measurement of the polarization state of beams with fixed photon number is discussed in the article by Altepeter, James and Kwiat in this volume. Nearly all field-state measurements are based on the technique of balanced homodyne detection, and hence fall under the category of optical homodyne tomography (OHT.) This technique was first suggested by Vogel and Risken [10] and first demonstrated by Smithey et al. [16]. A reasonably complete list of experiments that have measured the quantum states (or related quantities) of optical fields is presented in Table 1.Here we will describe balanced homodyne detection, and its application to OHT. We emphasize pulsed, balanced homodyne detection at zero frequency (DC) in order to model experiments on "wholepulse" detection. Such detection provides information on, for exampl...

Detecting quantum non-Gaussianity via the Wigner function

“…The successful implementation of our criteria corresponds to the measurement of the Wigner function at the origin of the phase space which, in turn, corresponds to the (photon) parity of the state under investigation. This may be obtained with current technology by direct parity measurement [51], or by reconstruction of the photon distribution either by tomographic reconstruction or by the on/off method [52][53][54][55][56][57][58][59][60][61][62][63][64][65]. When the criterion is satisfied, one can confirm that the quantum state at disposal has been generated by means of a highly non-linear process, even in the cases where, perhaps due to inefficient detectors or other types of noise, negativity of the Wigner function can not be detected.…”

Continuous-variable optical quantum-state tomography

“…The meaning of Eqs. ͑11͒ and ͑12͒ is that the BHD detects the state of the electromagnetic field in the spatialtemporal mode defined by the LO pulse ͑Smithey, Beck, Raymer, et al, 1993;Raymer et al, 1995;Raymer and Beck, 2004͒. This allows temporal and spatial selectivity, or gating, of the signal field ͑not the signal intensity͒.…”

“…This allows temporal and spatial selectivity, or gating, of the signal field ͑not the signal intensity͒. This gating technique ͑linear-optical sampling͒ has application in ultrafast signal characterization ͑Dorrer et Raymer and Beck, 2004͒. As usual, the mode's annihilation operator can be expressed as a sum of Hermitian operators â = e i ͑Q + iP ͒ / ͱ 2, called quadrature amplitudes, with 4 ͓Q , P ͔ = i. For zero phase, Q , P are denoted Q , P , respectively ͑so Q = Q cos + P sin ͒, and are analogous to position and momentum variables for a massive harmonic oscillator.…”

“…This leads to a distortion of the quadrature noise behavior which needs to be compensated for in the reconstructed state. We present, without derivation, a generalization of the above expression for detectors with a nonunitary quantum efficiency ͑Raymer et Raymer and Beck, 2004͒, pr͑Q ,͒ = ͳ :…”

Single plane minimal tomography of double slit qubits

“…In the case of mixed states, it is possible to reconstruct the Wigner quasiprobability function from the probability distributions along straight lines in phase space [24,25]. Experimental state reconstruction in a diversity of quantum systems (including molecular vi-brational modes, one-mode and two-mode states of light, trapped ions, and helium atoms) have been reported [20,26].…”