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arXiv:cond-mat/0211001 · Physical review. B, Condensed matter

Quantum-limited measurement and information in mesoscopic detectors

Aashish A. Clerk, S. M. Girvin, A. Douglas Stone

Published Apr 28, 2003• 131 citations
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Abstract

We formulate general conditions necessary for a linear-response detector to reach the quantum limit of measurement efficiency, where the measurement-induced dephasing rate takes its minimum possible value. These conditions are applicable to both noninteracting and interacting systems. We assess the status of these requirements in an arbitrary noninteracting scattering-based detector, identifying the symmetries of the scattering matrix needed to reach the quantum limit. We show that these conditions are necessary to prevent the existence of information in the detector that is not extracted in the measurement process.

DOI
10.1103/physrevb.67.165324

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We formulate general conditions necessary for a linear-response detector to reach the quantum limit of measurement efficiency, where the measurement-induced dephasing rate takes its minimum possible value. These conditions are applicable to both noninteracting and interacting systems. We assess the status of these requirements in an arbitrary noninteracting scattering-based detector, identifying the symmetries of the scattering matrix needed to reach the quantum limit. We show that these conditions are necessary to prevent the existence of information in the detector that is not extracted in the measurement process.
We study charge detection and controlled dephasing of a mesoscopic system via a quantum dot detector (QDD), where the mesoscopic system and the QDD are capacitively coupled. The QDD is considered to have coherent resonant tunnelling via a single level. It is found that the dephasing rate is proportional to the square of the conductance of the QDD for the Breit-Wigner model, showing that the dephasing is completely different from the shot noise of the detector. The measurement rate, on the other hand, shows a dip near the resonance. Our findings are peculiar especially for a symmetric detector in the following aspect: The dephasing rate is maximum at resonance of the QDD where the detector conductance is insensitive to the charge state of the mesoscopic system. As a result, the efficiency of the detector shows a dip and vanishes at resonance, in contrast to the single-channel symmetric non-resonant detector that has always a maximum efficiency. We find that this difference originates from a very general property of the scattering matrix: The abrupt phase change exists in the scattering amplitudes in the presence of the symmetry, which is insensitive to the detector current but {\em stores} the information of the quantum state of the mesoscopic system.
Using standard linear response relations, we derive the quantum limit on the sensitivity of a generic linear-response position detector, and the noise temperature of a generic linear amplifier. Particular emphasis is placed on the detector's effective temperature and damping effects; the former quantity directly determines the dimensionless power gain of the detector. Unlike the approach used in the seminal work of Caves [Phys. Rev. D 26 1817 (1982)], the linear-response approach directly involves the noise properties of the detector, and allows one to derive simple necessary and sufficient conditions for reaching the quantum limit. Our results have direct relevance to recent experiments on nanoelectromechanical systems, and complement recent theoretical studies of particular mesoscopic position detectors.
We consider a mesoscopic measuring device whose conductance is sensitive to the state of a two-level system. The detector is described with the help of its scattering matrix. Its elements can be used to calculate the relaxation and decoherence times of the system, and determine the characteristic time for a reliable measurement. We derive conditions needed for an efficient ratio of decoherence and measurement times. To illustrate the theory we discuss the distribution function of the efficiency of an ensemble of open chaotic cavities.
We show that a noninvasive,``negative-result measurement'' can be realized in quantum dot systems. The measurement process is studied by applying the Schrödinger equation to the whole system (including the detector). We demonstrate that the possibility of observing a particular state out of coherent superposition leads to collapse of the corresponding nondiagonal density-matrix elements of the measured system. No additional reduction postulate is needed. Experimental consequences of the collapse time and the relativistic requirement are discussed for mesoscopic and optical systems.
Coherent enhancement is a powerful mechanism for improving the sensitivity of a wide range of detectors, but its practical use is often limited by the difficulty of preparing the required quantum states. We show that this difficulty has a fundamental origin: coherent enhancement of a signal interacting with a detector is quantitatively constrained by entanglement. We prove general bounds on how the strength of coherent effects can scale with system size, as a function of the single-mode entanglement entropy of the detector. These bounds smoothly interpolate between the incoherent and fully coherent regimes, and apply both to parameter-estimation problems and to scattering processes. We discuss these results from two complementary perspectives: First, they appear as bounds on the quantum Fisher information of many-body states, which translate directly into limits on parameter sensitivity via the quantum Cramér-Rao bound. Second, they can be interpreted as limits on a class of scattering cross sections, leading to predictions for how minimum detectable interaction strengths scale with target size. Together, these results provide a unified view of coherent enhancement in metrology and scattering experiments, and motivate the development of new techniques for generating entangled detector states.