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Leggett–Garg inequalities for a quantum top affected by classical noise

Jerzy Dajka, Marcin Łobejko, J. Łuczka

Published Aug 1, 2016• 2 citations
Publisher

Abstract

The violation of the Leggett–Garg inequality is studied for a quantum top (with angular momentum $$J_z$$ J z of integer or half-integer size), being driven by classical Gaussian white noise. The form of a longitudinal $$(J_z)$$ ( J z ) or a transverse $$(J_x)$$ ( J x ) coupling of noise to the angular momentum affects both (i) to what extent the Leggett–Garg inequality is violated and (ii) how this violation is influenced by the size j of the spinning top and direction of a coupling (transverse or longitudinal). We introduce j-independent method, using two- dimensional invariant subspace of the system Hilbert space, which allows us to find out strict analytical solution for a noise-free system and with longitudinal coupling and to extract from the whole dynamics effects purely induced by a noise. It is demonstrated that in the semi-classical limit of a large angular momentum j and for the transverse coupling, the Leggett–Garg inequalities become more strongly violated as compared to the deep quantum regime of small j.

DOI
10.1007/s11128-016-1401-1

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The violation of the Leggett–Garg inequality is studied for a quantum top (with angular momentum JzJ_zJz​ J z of integer or half-integer size), being driven by classical Gaussian white noise. The form of a longitudinal (Jz)(J_z)(Jz​) ( J z ) or a transverse (Jx)(J_x)(Jx​) ( J x ) coupling of noise to the angular momentum affects both (i) to what extent the Leggett–Garg inequality is violated and (ii) how this violation is influenced by the size j of the spinning top and direction of a coupling (transverse or longitudinal). We introduce j-independent method, using two- dimensional invariant subspace of the system Hilbert space, which allows us to find out strict analytical solution for a noise-free system and with longitudinal coupling and to extract from the whole dynamics effects purely induced by a noise. It is demonstrated that in the semi-classical limit of a large angular momentum j and for the transverse coupling, the Leggett–Garg inequalities become more strongly violated as compared to the deep quantum regime of small j.
The Leggett-Garg inequalities serve to test whether or not quantum correlations in time can be explained within a classical macrorealistic framework. We apply this test to thermodynamics and derive a set of Leggett-Garg inequalities for the statistics of fluctuating work done on a quantum system unitarily driven in time. It is shown that these inequalities can be violated in a driven two-level system, thereby demonstrating that there exists no general macrorealistic description of quantum work. These violations are shown to emerge within the standard Two-Projective-Measurement scheme as well as for alternative definitions of fluctuating work that are based on weak measurement. Our results elucidate the influences of temporal correlations on work extraction in the quantum regime and highlight a key difference between quantum and classical thermodynamics.
We investigate how discrete internal degrees of freedom in a quasimacroscopic system affect the violation of the Leggett-Garg inequality, a test of macroscopic realism based on temporal correlation functions. As a specific example, we focus on an ensemble of qubits subject to collective and individual noise. This generic model can describe a range of physical systems, including atoms in cavities, electron or nuclear spins in nitrogen-vacancy (NV) centers in diamond, erbium in Y2SiO5{\mathrm{Y}}_{2}{\mathrm{SiO}}_{5}Y2​SiO5​, bismuth impurities in silicon, or arrays of superconducting circuits, to indicate but a few. Such large ensembles are potentially more macroscopic than other systems that have been used so far for testing the Leggett-Garg inequality and open a route toward probing the boundaries of quantum mechanics at macroscopic scales. We find that, because of the nontrivial internal structure of such an ensemble, the behavior of different measurement schemes, under the influence of noise, can be surprising. We discuss which measurement schemes are optimal for flux qubits and NV centers, and some of the technological constraints and difficulties for observing such violations with present-day experiments.
We study violation of the Leggett-Garg inequality for the correlator K3{K}_{3}K3​ in two cases: (i) for a qubit weakly coupled to a thermal decohering (dissipative and/or dephasing) environment and (ii) for a pair of qubits with only one of them coupled to the environment. In the case of a single qubit, we identify conditions depending both on initial state preparation and a measurement protocol when violation of the K3{K}_{3}K3​ inequality becomes dependent only on dephasing and independent of dissipation and temperature. In the case of two qubits, we have found the temperature-independent regime as well as the regime where all three components of the decoherence can control violation of the Leggett-Garg inequality.
The Leggett-Garg inequalities probe the classical-quantum boundary by putting limits on the sum of pairwise correlation functions between classical measurement devices that consecutively measured the same quantum system. The apparent violation of these inequalities by standard quantum measurements has cast doubt on quantum mechanics' ability to consistently describe classical objects. Recent work has concluded that these inequalities cannot be violated by either strong or weak projective measurements [1]. Here I consider an entropic version of the Leggett-Garg inequalities that are different from the standard inequalities yet similar in form, and can be defined without reference to any particular observable. I find that the entropic inequalities also cannot be be violated by strong quantum measurements. The entropic inequalities can be extended to describe weak quantum measurements, and I show that these weak entropic Leggett-Garg inequalities cannot be violated either even though the quantum system remains unprojected, because the inequalities describe the classical measurement devices, not the quantum system. I conclude that quantum mechanics adequately describes classical devices, and that we should be careful not to assume that the classical devices accurately describe the quantum system.