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arXiv:quant-ph/9703049 · Physical Review A

Continuous fuzzy measurement of energy for a two-level system

Jürgen Audretsch, Michael B. Mensky

Published Jul 1, 1997• 26 citations
arXivHTMLPDF

Abstract

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 $E(t)$ 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.

DOI
10.1103/physreva.56.44

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•Jan 1, 1998•66 citations
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.
Imposing restrictions on the Feynman paths of the monitored system has in the past been proposed as a universal model-free approach to continuous quantum measurements. Here we revisit this proposition and demonstrate that a Gaussian restriction, resulting in a sequence of many highly inaccurate (weak) von Neumann measurements, is not sufficiently strong to ensure proximity between a readout and the Feynman paths along which the monitored system evolves. Rather, in the continuous limit, the variations of a typical readout become much larger than the separation between the eigenvalues of the measured quantity. Thus, a typical readout is not represented by a nearly constant curve, correlating with one of the eigenvalues of the measured quantity $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}$, even when decoherence or Zeno effect is achieved for the observed two-level system, and does not point directly to the system's final state. We show that the decoherence in a ``free'' system can be seen as induced by a Gaussian random walk with a drift, eventually directing the system towards one of the eigenstates of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}$. A similar mechanism appears to be responsible for the Zeno effect in a driven system, when its Rabi oscillations are quenched by monitoring. Alongside the Gaussian case, which can only be studied numerically, we also consider a fully tractable model with a ``hard wall'' restriction and show the results to be similar.
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.
The influence of continuous measurements of energy with finite accuracy is studied in various quantum systems through a restriction of the Feynman path integrals around the measurement result. The method, which is equivalent to considering an effective Schr\"odinger equation with a non-Hermitian Hamiltonian, allows one to study the dynamics of the wave-function collapse. A numerical algorithm for solving the effective Schr\"odinger equation is developed and checked in the case of a harmonic oscillator. The situations, of physical interest, of a two-level system and of a metastable quantum well are then discussed. In the first case, the Zeno inhibition observed in quantum optics experiments is recovered and extended to nonresonant transitions, and in the second case we propose an observation of the inhibition of spontaneous decay in mesoscopic heterostructures. In all the considered examples, the effect of the continuous measurement of energy is a freezing of the evolution of the system proportional to the accuracy of the measurement itself.
Continuous weak or fuzzy measurement of the Rabi oscillation of a two level atom subjected to a π−π-π−pulse of a resonant light field is simulated numerically. We thereby address the question whether it is possible to measure characteristic features of the motion of the state of a single quantum system in real time. We compare two schemes of continuous measurement: continuous measurement with constant fuzziness and with fuzziness changing in the course of the measurement. Because the sensitivity of the Rabi atom to the influence of the measurement depends on the state of the atom, it is possible to optimize the continuous fuzzy measurement by varying its fuzziness.
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.
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