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Mysteries, Puzzles, and Paradoxes in Quantum Mechanics

Decoherence, Wigner functions, and the classical limit of quantum mechanics in cavity QED

L. Davidovich

Published Jan 1, 1999• 1 citation
Publisher

Abstract

Recent experiments in cavity QED have allowed the monitoring of the decoherence process, which is at the heart of the quantum theory of measurement and plays an essential role in the classical limit of quantum mechanics. It is shown here that the technique of Ramsey interferometry, central to those experiments, may be used to determine completely the quantum state of an electromagnetic field in a cavity. This new method, which allows the direct measurement of the Wigner function of the field, is a useful tool for probing the quantum-classical transition and, in particular, the decoherence of superpositions of distinguishable coherent states of the electromagnetic field in the cavity. It is also shown that the realization of a controlled-not gate within the framework of cavity QED is a special case of this proposal, and is related to the measurement of the Wigner function of a one-photon state.

DOI
10.1063/1.57851

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Recent experiments in cavity QED have allowed the monitoring of the decoherence process, which is at the heart of the quantum theory of measurement and plays an essential role in the classical limit of quantum mechanics. It is shown here that the technique of Ramsey interferometry, central to those experiments, may be used to determine completely the quantum state of an electromagnetic field in a cavity. This new method, which allows the direct measurement of the Wigner function of the field, is a useful tool for probing the quantum-classical transition and, in particular, the decoherence of superpositions of distinguishable coherent states of the electromagnetic field in the cavity. It is also shown that the realization of a controlled-not gate within the framework of cavity QED is a special case of this proposal, and is related to the measurement of the Wigner function of a one-photon state.
We show that the technique of Ramsey interferometry may be used to completely determine the quantum state of an electromagnetic field in a cavity. Our method, which allows the direct measurement of the Wigner function of the field, is a useful tool for probing the quantum-classical transition and, in particular, the decoherence between coherent superpositions of distinguishable coherent states of the electromagnetic field in the cavity. We also show that the realization of a controlled-not gate within framework of cavity QED is a special case of our proposal, and leads to the measurement of a negative value for the Wigner function of an electromagnetic field.
These lectures review some basic techniques of quantum optics, related to the description of the quantized electromagnetic field in phase space and of the interaction between atoms and photons in cavities. The Wigner function is introduced, and some of the methods for measuring this distribution are reviewed. The combination of phase space methods with cavity QED techniques is shown to lead to experiments which are closely connected to fundamental problems regarding the classical limit of quantum mechanics and the quantum theory of measurement.
Every quantum system subjected to measurements is an open quantum system. The cavity QED system is elegant in that it probes the interaction between two quantum systems, the atom and the field, while its loss mechanisms are well understood and can be externally monitored. The study of cross-correlations in cavity QED is important for understanding how entanglement evolves in open quantum systems. As quantum information science grows we need to learn more about entanglement and how it can be quantified and measured. Correlation functions have been used to compare an electromagnetic field (intensity) of one mode with the electromagnetic field (intensity) of the same mode at a later time or different spatial location. In quantum optics, correlation functions have been calculated and measured to probe the nonclassical field that results from the interaction of a single mode of the electromagnetic field and an ensemble of two-level atoms (the canonical cavity QED system). This field can exhibit antibunching, squeezing, and can violate inequalities required for a classical field. Entanglement in the steady state of a cavity QED system cannot be measured directly with traditional correlation functions (Hanbury-Brown and Twiss type experiments). Cross-correlations, however, interrogate directly both modes of the entangled pair, the transmitted (cavity) and the fluorescent (atom) intensities, and can act as an entanglement witness. This thesis presents the implementation of a cross-correlation measurement in a cavity QED system. The work has required the construction of an apparatus that incorporates laser cooling and trapping with quantum optics to carefully control both the external (center of mass motion) and internal (atomic state) degrees of freedom of a collection of atoms that interact with a single mode of a high finesse Fabry-Perot cavity. We examine theoretically and experimentally a new intensity cross-correlation function which probes the evolution of the cavity field conditioned on the detection of a fluorescent photon from an atom in the cavity. The results open the possibility to generalize the dynamics of entanglement as a physical resource necessary for the nascent quantum information science.