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arXiv:2401.10692 · Physical Review A

Large violation of Leggett-Garg inequalities with coherent-state projectors for a harmonic oscillator and chiral scalar field

Tomoya Hirotani, Akira Matsumura, Yasusada Nambu, Kazuhiro Yamamoto

Published Aug 12, 2024• 1 citation
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Abstract

We investigate violations of Leggett-Garg inequalities (LGIs) for a harmonic oscillator and a $(1+1)$-dimensional chiral scalar field with coherent-state projectors, which is equivalent to a heterodyne-type measurement scheme. For the harmonic oscillator, we found that the vacuum and thermal states violated the LGIs by evaluating the two-time quasiprobability distribution function. In particular, we demonstrate that the value of the two-time quasiprobability reaches $\ensuremath{-}0.123$ for a squeezed coherent-state projector, which is equivalent to $98%$ of the L\"uders bound corresponding to the maximal violation of the LGIs. We also find a violation of the LGIs for the local mode of a quantum chiral scalar field by constructing a coherent-state projector similar to the harmonic-oscillator case. In contrast with the harmonic oscillator, the periodicity in the time direction of the quasiprobability disappears, which is related to the existence of quantum entanglement between the local mode and its complementary degrees of freedom.

DOI
10.1103/physreva.110.022217

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We investigate the violation of the Leggett-Garg inequalities for a harmonic oscillator in various quantum states and with various choices of a projection operator for a dichotomic variable. We focus on the two-time quasiprobability distribution function with a dichotomic variable constructed with the position or momentum operator of a harmonic oscillator. Our results are the generalization of the previous work by Mawby and Halliwell [Phys. Rev. A 107, 032216 (2023)], but the new points are the following. We first obtain the explicit expression for the two-time quasiprobability distribution function for the (thermal) squeezed coherent state. Second, we find the two-time quasiprobability distribution function with the dichotomic variable and the projection operator constructed in terms of the momentum operator. Third, we demonstrate that the violation of the Leggett-Garg inequalities can be boosted by adopting the dichotomic variable and the projection operator defined by a finite range of the position/momentum, in which a larger violation appears for the ground state and the squeezed state. We give an intuitive interpretation when the violation of the Leggett-Garg inequalities appears. We also present a mathematical formula to compute the quasiprobability distribution function by using the integral representation of the Heaviside function, which is useful to generalize it to a quantum field theory.
We investigate the violation of the Leggett-Garg inequalities for a quantum field, focusing on the two-time quasiprobability distribution function of the dichotomic variable with a coarse-grained scalar field. The Leggett-Garg inequalities are violated depending on the quantum state of the field and the size of coarse graining. We demonstrate that the violation of the Leggett-Garg inequalities appears even for the vacuum state and the squeezed state by properly constructing the dichotomic variable and the projection operator.
In this study, we determine a violation of the Leggett-Garg inequalities due to gravitational interaction in a hybrid system consisting of a harmonic oscillator and a spatially localized superposed particle. The violation of the Leggett-Garg inequalities is discussed using the two-time quasiprobability in connection with the entanglement negativity generated by gravitational interaction. It is demonstrated that the entanglement suppresses the violation of the Leggett-Garg inequalities when one of the two times of the quasiprobability t1{t}_{1}t1​ is chosen as the initial time. Further, it is shown that the Leggett-Garg inequalities are generally violated due to gravitational interaction by properly choosing the configuration of the parameters, including t1{t}_{1}t1​ and t2{t}_{2}t2​, which are the times of the two-time quasiprobability. The feasibility of detecting violations of the Leggett-Garg inequalities in hybrid systems is also discussed.
The Leggett-Garg (LG) inequalities were introduced to test for the possible presence of macroscopic quantum coherence. Since such effects may be found in various types of macroscopic oscillators, we consider the application of the LG approach to the one-dimensional quantum harmonic oscillator and also to more general bound systems, using a single dichotomic variable QQQ given by the sign of the oscillator position. We present a simple method to calculate the temporal correlation functions appearing in the LG inequalities for any bound system for which the eigenspectrum is (exactly or numerically) known. We apply this result to the quantum harmonic oscillator for a variety of experimentally accessible states, namely, energy eigenstates, and superpositions thereof. For the subspace of states spanned by only the ground state and first excited state, we readily find substantial regions of parameter space in which the LG inequalities at two, three, and four times can each be independently violated or satisfied. We also find that the violations persist, although are reduced, when the sign function defining QQQ is smeared to reflect experimental imprecision. For higher-energy eigenstates, we find that LG violations diminish, showing the expected classicalization. With a QQQ defined using a more general type of position coarse graining, we find two-time LG violations even in the ground state. We also show that two-time LG violations in a Gaussian state are readily found if the dichotomic variable at one of the times is taken to be the parity operator. To demonstrate the versatility of the approach, we repeat much of the LG analysis for the Morse potential, finding qualitatively similar physical results.
Leggett-Garg (LG) inequalities test whether correlations measured at different times admit a positive joint probability distribution over histories. A violation requires negative values in any joint quasiprobability distribution reproducing those correlations, but such distributions can have negative values even without a violation. Spacetime density kernels extend the density-matrix description to multiple times and are used to study timelike entanglement. We establish a universal bound that relates LG violations to their negative eigenvalue weight. For LG tests involving any number of measurements, we determine the least compatible kernel negativity through an exact relation valid for any system dimension, initial state, dynamics, or measurement rule. This minimum is nonzero if and only if at least one LG inequality is violated. Three-time Lüders LG tests certify at most one eighth of negativity, whereas measurements resolving individual spin levels can approach one half. Including additional incompatible qubit observables gives a further exact minimum that can exceed the LG-only ceiling without increasing the LG violation.
We discuss a model of two nonlinear quantum oscillators mutually coupled by linear interaction and continuously driven by external coherent excitation. For such a system, we analyze temporal correlations. We examine the violation of the Leggett-Garg inequality analysing various scenarios of measurements. These scenarios are based on the projection onto different Bell states. We show that the possibility of violation of the Leggett-Garg inequalities is associated with the use of different projectors.