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arXiv:1002.4984 · Physical Review Letters

Quasiprobabilistic Interpretation of Weak Measurements in Mesoscopic Junctions

Adam Bednorz, Wolfgang Belzig

Published Sep 3, 2010• 56 citations
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Abstract

The impossibility of measuring noncommuting quantum mechanical observables is one of the most fascinating consequences of the quantum mechanical postulates. Hence, to date the investigation of quantum measurement and projection is a fundamentally interesting topic. We propose to test the concept of weak measurement of noncommuting observables in mesoscopic transport experiments, using a quasiprobabilistic description. We derive an inequality for current correlators, which is satisfied by every classical probability but violated by high-frequency fourth-order cumulants in the quantum regime for experimentally feasible parameters.

DOI
10.1103/physrevlett.105.106803

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The impossibility of measuring noncommuting quantum mechanical observables is one of the most fascinating consequences of the quantum mechanical postulates. Hence, to date the investigation of quantum measurement and projection is a fundamentally interesting topic. We propose to test the concept of weak measurement of noncommuting observables in mesoscopic transport experiments, using a quasiprobabilistic description. We derive an inequality for current correlators, which is satisfied by every classical probability but violated by high-frequency fourth-order cumulants in the quantum regime for experimentally feasible parameters.
The impossibility of measuring non-commuting quantum mechanical observables is one of the most fascinating consequences of the quantum mechanical postulates. Hence, to date the investigation of quantum measurement and projection is a fundamentally interesting topic. We propose to test the concept of weak measurement of non-commuting observables in mesoscopic transport experiments, using a quasiprobabilistic description. We derive an inequality for current correlators, which is satisfied by every classical probability but violated by high-frequency fourth-order cumulants in the quantum regime for experimentally feasible parameters. We further address the creation and detection of entanglement in solid-state electronics, which is of fundamental importance for quantum information processing. We propose a general test of entanglement based on the violation of a classically satisfied inequality for continuous variables by 4th or higher order quantum correlation functions. Our scheme provides a way to prove the existence of entanglement in a mesoscopic transport setup by measuring higher order cumulants without requiring the additional assumption of single charge detection.
Quantum transport in mesoscopic conductors is essentially governed by the laws of quantum mechanics. One of the major open questions of quantum mechanics is what happens if noncommuting observables are measured simultaneously. Since current operators at different times do not commute, the high-frequency correlation functions of the current are realization of this fundamental quantum question. We formulate this problem in the context of measurements of finite-frequency current cumulants in a general quantum point contact, which are the subject to ongoing experimental effort. To this end, we present two models of detectors that correspond to a weak time-resolved measurement of the electronic current in a mesoscopic junction. In both cases, the backaction of the detector leads to observable corrections to the current correlations functions involving the so-called noise susceptibilities. As a result, we propose a reinterpretation of environmental corrections to the finite-frequency cumulants as inevitable effect resulting from basic quantum mechanical principles. Finally we make concrete predictions for the temperature, voltage, and frequency dependence of the third cumulant, which could be verified directly using current experimental techniques.
The creation and detection of entanglement in solid state electronics is of fundamental importance for quantum information processing. We prove that second-order quantum correlations can be always interpreted classically and propose a general test of entanglement based on the violation of a classically derived inequality for continuous variables by fourth-order quantum correlation functions. Our scheme provides a way to prove the existence of entanglement in a mesoscopic transport setup by measuring higher order cumulants without requiring the additional assumption of a single-charge detection.
A long-standing problem in quantum mesoscopic physics is which operator order corresponds to noise expressions like <I(-ω)I(ω)>, where I(ω) is the measured current at frequency ω. Symmetrized order describes a classical measurement while nonsymmetrized order corresponds to a quantum detector, e.g., one sensitive to either emission or absorption of photons. We show that both order schemes can be embedded in quantum weak-measurement theory taking into account measurements with memory, characterized by a memory function which is independent of a particular experimental detection scheme. We discuss the resulting quasiprobabilities for different detector temperatures and how their negativity can be tested on the level of second-order correlation functions already. Experimentally, this negativity can be related to the squeezing of the many-body state of the transported electrons in an ac-driven tunnel junction.
The development of solid-state quantum technologies requires the understanding of quantum measurements in interacting, nonisolated quantum systems.In general, a permanent coupling of detectors to a quantum system leads to memory effects that have to be taken into account in interpreting the measurement results.In this thesis we develop a microscopic framework for correlations in non-Markovian weak measurements.A special point of interest are the different orders of quantum mechanical operators that appear naturally in such a scheme.First, we analyze a generic setup of two detectors coupled to a quantum system and derive a compact formula in the weak-measurement limit that interpolates between an instantaneous (text-book type) and almost continuous -detector dynamics-dependent -measurement.A quantum memory effect that we term system-mediated detector-detector interaction is crucial to observe noncommuting observables simultaneously.A response function of the observed system variables introduces an antisymmetric operator order to the total result.We propose a mesoscopic double-dot detector setup in which the memory effect is tunable and that can be used to explore the transition to non-Markovian quantum measurements experimentally.The response and noise functions which we find to describe the system-mediated detector-detector interaction are analyzed in detail.They are compared to the phenomenological "equilibrium order" approach for non-Markovian measurements which was introduced in Ref. [1].We give an overview of the broad spectrum of results beyond "equilibrium order" in the microscopic model in a compendium of different detectors and discuss general mathematical properties of the detector functions.As the quantum measurement problem is not banned from our theory by operation in the weak measurement limit, but deferred from the system of interest to the detectors, we comment on the influence of the detector readout procedure on the results.Finally, we combine the phenomenological proposal for nonsymmetrized correlations in quantum noninvasive measurements [1] and our microscopic derivation of the systemmediated detector-detector interaction of second order to investigate higher-order correlations of non-Markovian weak quantum measurements.The general third-order correlator is introduced and the underlying physical processes are identified and discussed.We find a more complex behavior than in second-order correlations with four types of system operator orders which we denote third cumulant, noise susceptibility I and II and second-order susceptibility.We introduce a convenient diagrammatic description which we illustrate by an example of measuring the third-order correlations of the tunnel current through a barrier.v I would like to express my deep gratitude to all the people who contributed to this thesis.Special thanks to my supervisor Wolfgang Belzig for offering me the opportunity to work in his research group, for the numerous helpful discussions in which I learned a lot and which gave impetus to this work again and again, for being always patient and an excellent guidance during my studies.I also wish to express great thanks to my other collaborators, Adam Bednorz, whose ideas are an important part of the foundation of this work and who was always ready to comment new results and bring things forward, and Christoph Bruder, who was a great help in thinking through results and formulating them in an understandable manner while preparing parts of this work for publication.Many thanks to Maxim Trushin, who was willing to discuss our weak measurement formalism with the example of a graphene coherence measurement, and who contributed the initial idea of a optical detection of pseudospin superposition states as well as the description of the carrier statistics using the Liouville-von Neumann equation and very important parts of the discussion of the results.In this context I also want to acknowledge valuable discussions with Daniele Brida about the optical detection scheme and with Bertrand Reulet about third-order current correlations.Finally, I wish to thank the current and former members of the Quantum Transport Group
Quantum theory is inconsistent with any local hidden variable model as was first shown by Bell. To test Bell inequalities two separated observers extract correlations from a common ensemble of identical systems. Since quantum theory does not allow simultaneous measurements of noncommuting observables, on each system every party measures a single randomly chosen observable out of a given set. Here we suggest a different approach for testing Bell inequalities that is experimentally realizable by current methods. We show that Bell inequalities can be maximally violated even when all observables are measured on each member of the ensemble. This is possible by using weak measurements that produce small disturbance, at the expense of accuracy. However, our approach does not constitute an independent test of quantum nonlocality since the local hidden variables may correlate the noise of the measurement instruments. Nevertheless, by adding a randomly chosen precise measurement at the end of every cycle of weak measurements, the parties can verify that the hidden variables were not interfering with the noise, and thus validate the suggested test.