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arXiv:quant-ph/0301135 · Physical Review Letters

Quantum State Tomography of Dissociating Molecules

Esben Skovsen, Henrik Stapelfeldt, Søren Juhl, Klaus Mølmer

Published Aug 28, 2003• 43 citations
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Abstract

Using tomographic reconstruction we determine the complete internuclear quantum state, represented by the Wigner function, of a dissociating I2 molecule based on femtosecond time resolved position and momentum distributions of the atomic fragments. The experimental data are recorded by timed ionization of the photofragments with an intense 20 fs laser pulse. Our reconstruction method, which relies on Jaynes's maximum entropy principle, will also be applicable to time resolved position or momentum data obtained with other experimental techniques.

DOI
10.1103/physrevlett.91.090406

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Using tomographic reconstruction we determine the complete internuclear quantum state, represented by the Wigner function, of a dissociating I2 molecule based on femtosecond time resolved position and momentum distributions of the atomic fragments. The experimental data are recorded by timed ionization of the photofragments with an intense 20 fs laser pulse. Our reconstruction method, which relies on Jaynes's maximum entropy principle, will also be applicable to time resolved position or momentum data obtained with other experimental techniques.
With the ability to directly obtain the Wigner function and density matrix of photon states, quantum tomography (QT) has had a significant impact on quantum optics, quantum computing and quantum information. By an appropriate sequence of measurements on the evolution of each degreeof freedom (DOF), the full quantum state of the observed photonic system can be determined. The first proposal to extend the application of QT to reconstruction of complete quantum states of matter wavepackets had generated enormous interest in ultrafast diffraction imaging and pump-probe spectroscopy of molecules. This interest was elevated with the advent of ultrafast electron and X-ray diffraction techniques using electron accelerators and X-ray free electron lasers to add temporal resolution to the observed nuclear and electron distributions. However, the great interest in this area has been tempered by the illustration of an impossibility theorem, known as the dimension problem. Not being able to associate unitary evolution to every DOF of molecular motion, quantum tomography could not be used beyond 1D and categorically excludes most vibrational and all rotational motion of molecules. Here we present a theoretical advance to overcome the notorious dimension problem. Solving this challenging problem is important to push imaging molecular dynamics to the quantum limit. The new theory has solved this problem, which makes quantum tomography a truly useful methodology in ultrafast physics and enables the making of quantum version of a molecular movie. With the new theory, quantum tomography can be finally advanced to a sufficient level to become a general method for reconstructing quantum states of matter, without being limited in one dimension. Our new concept is demonstrated using a simulated dataset of ultrafast diffraction experiment of laser-aligned nitrogen molecules.
Summary form only given. Experimental techniques for measurement of the quantum state of light have been the subject of intensive investigation for some time. Tomographic techniques have been applied to experiments such as the homodyne measurement of the Wigner function of a single mode of light and of the density matrix of the polarization degrees of freedom of a pair of entangled photons. In this technique, the density matrix (or Wigner function) which characterizes the quantum state of the system being measured is found from a linear transformation of experimental data. There are a number of drawbacks to the method, principally in that the recovered state might not, because of experimental noise, correspond to a physical state. For example, density matrices for any quantum state must be Hermitian, positive semi-definite matrices with unit trace. The tomographically measured matrices often fail to be positive semi-definite. To avoid this problem, the maximum likelihood approach to the estimation of quantum states has been developed. We evaluate two different approaches to the maximum likelihood technique, namely defining the likelihood function f as a weighted sum of squared variances from measured data, and defining it in terms of information content. These are compared with results of standard tomographical schemes.
A single-photon Fock state has been generated by means of conditional preparation from a two-photon state emitted in the process of spontaneous parametric down-conversion. A recently developed high-frequency homodyne tomography technique has been used to completely characterize the Fock state by means of a pulse-to-pulse analysis of the detectors' difference photocurrent. The density matrix elements of the generated state have been retrieved with a final detection efficiency of about 57%. A comparison has been performed between the phase-averaged tomographic reconstructions of the Wigner function as obtained from the measured density-matrix elements and from a direct Abel transform of the homodyne data. The ability of our system to work at the full repetition rate of the pulsed laser (82MHz)(82\phantom{\rule{0.3em}{0ex}}\mathrm{MHz})(82MHz) substantially simplifies the detection scheme, allowing for more ``exotic'' quantum states to be generated and analyzed.