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Physical Review Letters

Experimental Determination of the Quantum-Mechanical State of a Molecular Vibrational Mode Using Fluorescence Tomography

Thomas J. Dunn, Ian A. Walmsley, Shaul Mukamel

Published Feb 6, 1995• 324 citations
Publisher

Abstract

We present a spectroscopic method for the complete characterization of the quantum state of the vibrational mode of a molecule in terms of a phase-space quasiprobability distribution. The distribution for a molecular vibrational mode excited by a short optical pulse is reconstructed from measurements of its time-dependent spectrum of fluorescence.

DOI
10.1103/physrevlett.74.884

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We present a spectroscopic method for the complete characterization of the quantum state of the vibrational mode of a molecule in terms of a phase-space quasiprobability distribution. The distribution for a molecular vibrational mode excited by a short optical pulse is reconstructed from measurements of its time-dependent spectrum of fluorescence.
Emission tomography provides a method for the characterization of quantum states of those degrees of freedom of material quantum systems that are coupled to electronic states of the system. In such systems, the time-dependent fluorescence spectrum contains complete information about the non-electronic degree of freedom. Phase-space or, in certain cases, density matrix representations of the state can be reconstructed from the spectrum using standard methods of computer-assisted tomography or linear algebra. The principles and implementation of this procedure are described in detail and applied to the specific example of the reconstruction of the quantum state of the vibrational mode of a diatomic molecule excited into a non-stationary wavepacket state.
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 propose a method to reconstruct the vibrational quantum state of molecules excited by a general excitation laser pulse. Unlike existing methods, we do not require the molecules before excitation to be in a pure state, allowing us to treat the important case of initially thermally excited molecules. Even if only a single initial level is appreciably populated, initial levels with small populations can still give major contributions to the unknown vibrational state, making it essential to take them into account. In addition to the excitation pulse, the method uses two incident, short laser pulses in a non-co-linear geometry to create four-wave mixing in the molecules. The measurements used in the reconstruction are spectra of the outgoing four-wave mixing pulse at different time delays of the excitation laser pulse. An important point is that the method does not require detailed knowledge of molecular transition moments between excited states nor of any of the incoming laser pulses, but circumvents this requirement by using one or more calibration laser pulses in a separate experiment either before or after the main data are recorded. The only requirements for the calibration laser pulses are that the constant parts of their spectrums should together cover the spectral range of the excitation laser pulse, and the constant part of each should have sufficient spectral overlap with one other calibration pulse to populate two of the same levels. Finally, we discuss the extension of the reconstruction method in this paper to more general situations, hereby presenting the new idea of quantum state reconstruction through perturbations with calibration.
Atoms trapped in an optical lattice have long been a system of interest in the atomic, molecular and optical community, and in recent years much study has been devoted to both short- and long-range coherence in this system, as well as to its possible applications to quantum information processing. Here we demonstrate for the first time a complete determination of the quantum phase space distributions for an ensemble of 85 Rb atoms in such a lattice, including a negative Wigner function for atoms in an inverted state.