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arXiv:quant-ph/9808062

Realization scheme for continuous fuzzy measurement of energy and the monitoring of a quantum transition

Audretsch, J., Mensky, M. B.

Published Aug 28, 1998• 7 citations
arXivHTMLPDF

Abstract

The continuous fuzzy measurement of energy of a single two-level system driven by a resonant external field is studied. An analysis is given in the framework of the phenomenological restricted path integral approach (RPI) (which reduces effectively to a Schrodinger equation with a complex Hamiltonian) as well as with reference to the microphysical details of a class of concrete physical realizations. Within the RPI approach it is demonstrated that for appropriately adjusted fuzziness, information about the evolution of the state of the system can be read off from the measurement readout E(t). It is shown furthermore how a measurement of this type may be realized by a series of weak and short interactions of the two-level system with a quantum mechanical meter system. After each interaction a macroscopic measuring apparatus causes the meter to transit into one of two states. The result is used to generate the energy readout E(t). In this way a complete agreement with the RPI approach is demonstrated which thus obtains an operational interpretation.

DOI
10.48550/arxiv.quant-ph/9808062

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The continuous fuzzy measurement of energy of a single two-level system driven by a resonant external field is studied. An analysis is given in the framework of the phenomenological restricted path integral approach (RPI) (which reduces effectively to a Schrodinger equation with a complex Hamiltonian) as well as with reference to the microphysical details of a class of concrete physical realizations. Within the RPI approach it is demonstrated that for appropriately adjusted fuzziness, information about the evolution of the state of the system can be read off from the measurement readout E(t). It is shown furthermore how a measurement of this type may be realized by a series of weak and short interactions of the two-level system with a quantum mechanical meter system. After each interaction a macroscopic measuring apparatus causes the meter to transit into one of two states. The result is used to generate the energy readout E(t). In this way a complete agreement with the RPI approach is demonstrated which thus obtains an operational interpretation.
A continuous measurement of energy that is sharp (perfect) leads to the quantum Zeno effect (freezing of the state). Only if the quantum measurement is fuzzy (unsharp), continuous monitoring gives a readout E(t)E(t)E(t) from which information about the dynamical development of the state vector of the system may be obtained in certain cases. This is studied in detail. Fuzziness is thereby introduced with the help of restricted path integrals equivalent to non-Hermitian Hamiltonians. For an otherwise undisturbed multilevel system it is shown that this measurement represents a model of decoherence. If it lasts long enough, the measurement readout discriminates between the energy levels and the von Neumann state reduction is obtained. For a two-level system under resonance influence (which undergoes, in the absence of measurement, Rabi oscillations between the levels) different regimes of measurement are specified depending on its duration and fuzziness: (i) the Zeno regime, where the measurement results in a freezing of the transitions between the levels, and (ii) the Rabi regime, where the transitions are maintained. It is shown that in the Rabi regime at the border to the Zeno regime a correlation exists between the time-dependent measurement readout and the modified Rabi oscillations of the state of the measured system. Possible realizations of continuous fuzzy measurements of energy are sketched.
Decoherence of a quantum system (which then starts to display classical features) results from the interaction of the system with the environment, and is well described in the framework of the theory of continuous quantum measurements (CQM). Reviewed are the various approaches to the CQM theory, and the approach based on the effective complex Hamiltonians is discussed in greater detail. The effective complex Hamiltonian is derived from the restricted path integral, which emphasizes the role of information in the dynamics of the system being measured. The complex Hamiltonian is used for analyzing the CQM of energy in a two-level system. Such measurement is demonstrated to be capable of monitoring the quantum transition, and the back effect of monitoring on the probability of transition is analyzed. The realization of this type of measurement by a long series
Decoherence of a quantum system (which then starts to display classical features) results from the interaction of the system with the environment, and is well described in the framework of the theory of continuous quantum measurements (CQM). Reviewed are the various approaches to the CQM theory, and the approach based on the effective complex Hamiltonians is discussed in greater detail. The effective complex Hamiltonian is derived from the restricted path integral, which emphasizes the role of information in the dynamics of the system being measured. The complex Hamiltonian is used for analyzing the CQM of energy in a two-level system. Such measurement is demonstrated to be capable of monitoring the quantum transition, and the back effect of monitoring on the probability of transition is analyzed. The realization of this type of measurement by a long series of soft observations of the system is presented.
The restricted-path-integral (RPI) theory of continuous quantum measurements including the evolution of the measured systems and phenomenon of decoherence is reviewed. The measured system is considered as an open quantum system but without usage of any model of the measurement (of the measuring medium or the system's environment). The propagator of a measured system (conditioned by the measurement readout) is presented by RPI. In the important special case of monitoring an observable the propagator and the system's wave function satisfy Schroedinger equation with a complex Hamiltonian (depending on the measurement readout). Going over to the non-selective description of the measurement leads to the Lindblad master equation. In case of non-minimally disturbing measurements this gives theory of dissipative systems avoiding difficulties of other approaches. The whole theory is deduced from first principles of quantum mechanics. This proves that quantum mechanics includes theory of measurements and is therefore conceptually closed.
List of Figures. Preface. 1. Introduction. I: The Nature of Quantum Measurements. 2. The Physics Behind Quantum Measurements. 3. Measurement induced uncertainties. II: Continuous Quantum Measurements: Theory. 4. Continuous Quantum Measurements: The General Notion. 5. Fuzzy Continuous Measurements: Phenomenology. 6. Restricted Path Integrals: General Features. III: Continuous Quantum Measurements: Applications. 7. The Monitoring of Energy. 8. Monitoring by a series of observations. 9. The Quantum Zeno effect in a passive setup. IV: Conceptual Problems and Prospects. 10. RPI and the Philosophy of Quantum Mechanics. 11. Conclusion: New Prospects. A. The Formalism of Quantum Mechanics. References. Index.
The restricted-path-integral (RPI) description of a continuous quantum measurement is rederived starting from the description of an open system by the Feynman-Vernon influence functional. For this end the total evolution operator of the compound system consisting of the open system and its environment is decomposed into the sum of partial evolution operators. Accordingly, the influence functional of the open system is decomposed into the integral of partial influence functionals (PIF). If the partial evolution operators or PIF are chosen in such a way that they decohere (do not interfere with each other), then the formalism of RPI effectively arises. The evolution of the open system may then be interpreted as a continuous measurement of this system by its environment. This is possible if the environment is macroscopic or mesoscopic.