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arXiv:1209.2977 · Physical Review B

Leggett-Garg inequality in electron interferometers

Clive Emary, Neill Lambert, Franco Nori

Published Dec 27, 2012• 35 citations
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Abstract

We consider the violation of the Leggett-Garg inequality in electronic Mach-Zehnder inteferometers. This setup has two distinct advantages over earlier quantum-transport proposals: Firstly, the required correlation functions can be obtained without time-resolved measurements. Secondly, the geometry of an interferometer allows one to construct the correlation functions from ideal negative measurements, which addresses the noninvasiveness requirement of the Leggett-Garg inequality. We discuss two concrete realizations of these ideas: the first in quantum Hall edge-channels, the second in a double quantum dot interferometer.

DOI
10.1103/physrevb.86.235447

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We consider the violation of the Leggett-Garg inequality in electronic Mach-Zehnder inteferometers. This setup has two distinct advantages over earlier quantum-transport proposals: Firstly, the required correlation functions can be obtained without time-resolved measurements. Secondly, the geometry of an interferometer allows one to construct the correlation functions from ideal negative measurements, which addresses the noninvasiveness requirement of the Leggett-Garg inequality. We discuss two concrete realizations of these ideas: the first in quantum Hall edge-channels, the second in a double quantum dot interferometer.
In the tests for macrorealism proposed by Leggett and Garg, the temporal correlation functions of a dichotomic variable QQQ must be measured in a noninvasive way to rule out alternative classical explanations of Leggett-Garg inequality violations. Ideal negative measurements in which a null result is argued to be a noninvasive determination of the system's state are often used. From a quantum-mechanical perspective, such a measurement collapses the wave function and will therefore typically be found to be invasive under any experimental check. Here, a simple modified ideal negative measurement protocol is described for measuring the correlation functions, which is argued to be noninvasive from both classical and quantum perspectives and hence the noninvasiveness can then be checked experimentally, thereby permitting a quantitative measure of the degree of clumsiness of the measurement. It is also shown how this procedure may be extended to measure higher-order correlation functions, and a number of higher-order conditions for macrorealism are derived.
Ambiguous measurements do not reveal complete information about the system under test. Their quantum-mechanical counterparts are semiweak (or in the limit, weak) measurements and here we discuss their role in tests of the Leggett-Garg inequalities. We show that, while ambiguous measurements allow one to forgo the usual noninvasive measurability assumption, to derive a Leggett-Garg inequality that may be violated, we are forced to introduce another assumption that equates the invasive influence of ambiguous and unambiguous detectors. Based on this assumption, we derive signaling conditions that should be fulfilled for the plausibility of the Leggett-Garg test. We then propose an experiment on a three-level system with a direct quantum-optics realization that satisfies all signaling constraints and violates a Leggett-Garg inequality.
In contrast to the spatial Bell's inequalities which probe entanglement between spatially separated systems, the Leggett–Garg inequalities test the correlations of a single system measured at different times. Violation of a genuine Leggett–Garg test implies either the absence of a realistic description of the system or the impossibility of measuring the system without disturbing it. Quantum mechanics violates the inequalities on both accounts and the original motivation for these inequalities was as a test for quantum coherence in macroscopic systems. The last few years has seen a number of experimental tests and violations of these inequalities in a variety of microscopic systems such as superconducting qubits, nuclear spins, and photons. In this article, we provide an introduction to the Leggett–Garg inequalities and review these latest experimental developments. We discuss important topics such as the significance of the non-invasive measurability assumption, the clumsiness loophole, and the role of weak measurements. Also covered are some recent theoretical proposals for the application of Leggett–Garg inequalities in quantum transport, quantum biology and nano-mechanical systems.
When a physical object is in a quantum superposition, interference among the possible quantum paths affects the outcomes of a measurement. Gaining knowledge about the system with an intermediate measurement smears out, if not completely destroys, the resulting interference pattern. Based on this idea, Leggett and Garg formulated an inequality relating two-time correlation measurements, which establishes a quantitative criterion for discerning quantum superposition on a fundamental ground. Originally proposed to test realism of macroscopic physical objects, the inequality is here used to prove the non-classicality of the motion of a cesium atom, which propagates on a space-time discrete lattice. Making use of ideal negative measurements to test the atom's motion, we obtain a 6 sigma experimental violation of the Leggett-Garg inequality. From a broader perspective, our findings provide rigorous validation of the idea that quantum transport experiments cannot be interpreted in terms of classical trajectories.
The Leggett-Garg inequality is a widely used test of the ``quantumness'' of a system and involves correlations between measurements realized at different times. According to its widespread interpretation, a violation of the Leggett-Garg inequality disproves macroscopic realism and noninvasiveness. Nevertheless, recent results point out that macroscopic realism is a model-dependent notion and that one should always be able to attribute to invasiveness a violation of a Leggett-Garg inequality. This opens some natural questions: How do we provide such an attribution in a systematic way? How can apparent macroscopic realism violation be recast into a dimensional-independent invasiveness model? The present work answers these questions by introducing an operational model where the effects of invasiveness are controllable through a parameter associated with what is called the measurability of the physical system. Such a parameter leads to different generalized measurements that can be associated with the dimensionality of a system, to measurement errors, or to back action.
In an endeavor to better define the distinction between classical macroscopic and quantum microscopic regimes, the Leggett-Garg inequalities were established as a test of macroscopic-realistic theories, which are commonly thought to be a suitable class of descriptions for classical dynamics. The relationship between their violation and nonclassicality is however not obvious. We show that classical states of light, which in the quantum optical sense are any convex sums of coherent states, may not satisfy the Leggett-Garg inequalities. After introducing a simple Mach-Zehnder setup and showing how to obtain a violation with a single photon using negative measurements, we focus on classical states of light, in particular those of low average photon number. We demonstrate how one can still perform negative measurements with an appropriate assignment of variables, and show that the inequalities are violable with coherent states. Finally, we abandon the initial phase reference and demonstrate that the violation is still possible, in particular with thermal states of light, and we investigate the effect of intermediate dephasing.