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arXiv:quant-ph/9907026

Approximate real time visualization of a Rabi transition by means of continuous fuzzy measurement

Jürgen Audretsch, Th. Konrad, Michael B. Mensky

Published Jul 8, 1999• 0 citations
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Abstract

Continuous weak or fuzzy measurement of the Rabi oscillation of a two level atom subjected to a $π-$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.

DOI
10.48550/arxiv.quant-ph/9907026

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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.
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.
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 normalized state $|\ensuremath{\psi}(t)〉{=c}_{1}(t)|1〉{+c}_{2}(t)|2〉$ of a single two-level system performs oscillations under the influence of a resonant driving field. It is assumed that only one realization of this process is available. We show that it is possible to approximately visualize in real time the evolution of the system as far as it is given by ∣c2(t)∣2.|{c}_{2}(t){|}^{2}.∣c2​(t)∣2. For this purpose we use a sequence of particular unsharp measurements separated in time. They are specified within the theory of generalized measurements in which observables are represented by positive operator-valued measures. A realization of the unsharp measurements may be obtained by coupling the two-level system to a meter and performing the usual projection measurements on the meter only.
We address the peculiarities of the quantum measurement process in the course of a continuous weak linear measurement (CWLM). As a tool, we implement an efficient numerical simulation scheme that allows us to generate single quantum trajectories of the measured system state as well as the recorded detector signal, and study statistics of these trajectories with and without post-selection. In this scheme, a linear detector is modelled with a qubit that is weakly coupled to the quantum system measured and is subject to projective measurement and re-initialization after a time interval at each simulation step. We explain the conditions under which the scheme provides an accurate description of CWLM. We restrict ourselves to a qubit non-demolition measurement. The qubit is initially in an equal-weight superposition of two quantum states. In the course of time, the detector signal is accumulated and the superposition is destroyed. The times required to resolve the quantum states and to destroy the superposition are of the same order. We prove numerically a rather counter intuitive fact: the average detector output conditioned on the final state does not depend on time. It looks like from the very beginning, the qubit knows in which state it is. We study statistics of decision times where the decision time is defined as time required for the density matrix along a certain trajectory to reach a threshold where it is close to one of the resulting states. This statistics is useful to estimate how fast a decisive CWLM can be. Basing on this, we devise and study a simple feedback scheme that attempts to keep the qubit in the equal-weight superposition. The detector readings are used to decide in which state the qubit is and which correction rotation to apply to bring it back to the superposition. We show how to optimize the feedback parameters and move towards more efficient feedback schemes.
In quantum physics, measurements give random results and yield a corresponding random backaction on the state of the system subject to measurement. If a quantum system is probed continuously over time, its state evolves along a stochastic quantum trajectory. To investigate the characteristic properties of such dynamics, we perform weak continuous measurements on a superconducting qubit that is driven to undergo Rabi oscillations. From the data we observe a number of striking temporal correlations within the time dependent signals and the quantum trajectories of the qubit, and we discuss their explanation in terms of quantum measurement and photodetection theory.