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arXiv:1906.04900 · Physical Review A

Testing macroscopic local realism using local nonlinear dynamics and time settings

Manushan Thenabadu, G.-L. Cheng, T. Pham, Lisa V. Drummond, L. Rosales-Zárate, M. D. Reid

Published Aug 5, 2020• 14 citations
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Abstract

We show how one may test macroscopic local realism where, differently from conventional Bell tests, all relevant measurements that lead to a violation of a Bell inequality need only distinguish between two macroscopically distinct states of the system being measured. Here, measurements give macroscopically distinct outcomes for some property of the system and do not resolve microscopically (of order $\ensuremath{\hbar}$). To obtain a quantifiable test, we define $N$-scopic local realism where the outcomes are separated by an amount $\ensuremath{\sim}N$. We show for $N$ up to 20 that violations of $N$-scopic local realism are predicted for entangled systems of $N$ bosons at each of two sites. We further demonstrate for arbitrarily large $\ensuremath{\alpha}$ the violation of $\ensuremath{\alpha}$-scopic local realism, for entangled superpositions of coherent states with amplitudes $\ensuremath{\sim}\phantom{\rule{1.0pt}{0ex}}\ensuremath{\alpha}$, infinitely separated in phase space, as $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}$. To achieve the Bell violations, the two separated subsystems evolve dynamically according to a local nonlinear unitary interaction. The tests may be understood in two ways. First, they are macroscopic Bell tests involving spacelike separated measurement events, where the choice of measurement setting at each site corresponds to a choice between two different times of local interaction. The interaction may be seen as the unitary stage of a local measurement process. Second, the proposal is a strong version of a Leggett-Garg test of "macrorealism" that uses two trajectories, where the usual assumption of noninvasive measurability as applied to measurements at one site is replaced by that of macroscopic locality. For $N=1$, tests are feasible and give a way to demonstrate ``no classical trajectories'' without the assumption of noninvasive measurability, using instead that of locality.

DOI
10.1103/physreva.102.022202

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We show how one may test macroscopic local realism where, differently from conventional Bell tests, all relevant measurements that lead to a violation of a Bell inequality need only distinguish between two macroscopically distinct states of the system being measured. Here, measurements give macroscopically distinct outcomes for some property of the system and do not resolve microscopically (of order \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ). To obtain a quantifiable test, we define NNN-scopic local realism where the outcomes are separated by an amount \ensuremath∼N\ensuremath{\sim}N\ensuremath∼N. We show for NNN up to 20 that violations of NNN-scopic local realism are predicted for entangled systems of NNN bosons at each of two sites. We further demonstrate for arbitrarily large \ensuremathα\ensuremath{\alpha}\ensuremathα the violation of \ensuremathα\ensuremath{\alpha}\ensuremathα-scopic local realism, for entangled superpositions of coherent states with amplitudes \ensuremath∼\ensuremathα\ensuremath{\sim}\phantom{\rule{1.0pt}{0ex}}\ensuremath{\alpha}\ensuremath∼\ensuremathα, infinitely separated in phase space, as \ensuremathα\ensuremath→\ensuremath∞\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}\ensuremathα\ensuremath→\ensuremath∞. To achieve the Bell violations, the two separated subsystems evolve dynamically according to a local nonlinear unitary interaction. The tests may be understood in two ways. First, they are macroscopic Bell tests involving spacelike separated measurement events, where the choice of measurement setting at each site corresponds to a choice between two different times of local interaction. The interaction may be seen as the unitary stage of a local measurement process. Second, the proposal is a strong version of a Leggett-Garg test of "macrorealism" that uses two trajectories, where the usual assumption of noninvasive measurability as applied to measurements at one site is replaced by that of macroscopic locality. For N=1N=1N=1, tests are feasible and give a way to demonstrate ``no classical trajectories'' without the assumption of noninvasive measurability, using instead that of locality.
We consider tests of Leggett-Garg's macrorealism and of macroscopic local realism, where for spacelike separated measurements the assumption of macroscopic noninvasive measurability is justified by that of macroscopic Bell locality. We give a mapping between the Bell and Leggett-Garg experiments for microscopic qubits based on spin-1/21/21/2 eigenstates and gedanken experiments for macroscopic qubits based on two macroscopically distinct coherent states $|\ensuremath{\alpha}\ensuremath{\rangle}$ and $|\ensuremath{-}\ensuremath{\alpha}\ensuremath{\rangle}$ (as $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}$). In this mapping, the unitary rotation ${U}_{\ensuremath{\theta}}$ of the Stern-Gerlach analyzer or polarizing beam splitter is realized by a local interaction $H=\mathrm{\ensuremath{\Omega}}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}^{4}$ where $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}$ is the number of quanta. By adjusting the time of interaction, one alters the measurement setting $\ensuremath{\theta}$. We thus predict violations of Leggett-Garg and Bell inequalities in a macroscopic regime where coarse-grained measurements $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ need only discriminate between two macroscopically distinct coherent states. To interpret the violations, we distinguish between different definitions of macroscopic realism. Deterministic macroscopic local realism (dMR) assumes the system is in a state with a definite outcome ${\ensuremath{\lambda}}_{\ensuremath{\theta}}$ for the measurement $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ prior to the unitary rotation ${U}_{\ensuremath{\theta}}$, and is negated by the violations. Weak macroscopic realism (wMR) assumes a definite outcome for systems prepared in a superposition ${\ensuremath{\psi}}_{\text{pointer}}$ of two macroscopically distinct eigenstates of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$, after the unitary rotation ${U}_{\ensuremath{\theta}}$. We find that wMR can be viewed consistently with the violations and with the predictions of quantum mechanics for two tests of wMR. A model is presented in which wMR holds and the macroscopic violations emerge over the course of the unitary dynamics occurring at both sites. Implications for the realism of the micro-macro state generated during a quantum measurement are discussed. Finally, we point out an EPR-type paradox, which demonstrates inconsistency between wMR and the completeness of quantum mechanics at a microscopic level.
Macroscopic Realism (MR) says that a macroscopic system is always determinately in one or other of the macroscopically distinguishable states available to it. The Leggett-Garg (LG) inequality was derived to allow experimental test of whether or not this doctrine is true; it is also often thought of as a temporal version of a Bell-inequality. Despite recent interest in the inequality, controversy remains regarding what would be shown by its violation. Here we resolve this controversy, which arises due to an insufficiently general and model-independent approach to the question so far. We argue that LG's initial characterisation of MR does not pick out a particularly natural realist position, so we articulate an operationally well-defined and well-motivated position in its place. We show that much weaker conditions than LG's are sufficient to derive the inequality: in the first instance, its violation only demonstrates that certain measurements fail to be non-disturbing at the operational level. We articulate three distinct species of MR-ist position, and argue that it is only the first of these which can be refuted by LG inequality violation. This first position is an attractive one, so ruling it out remains of interest, however. A crucial role is played in LG's argument by the assumption of noninvasive measurability. We show that this notion is ambiguous between the weaker notion of disturbance at the operational level, and the stronger notion of invasiveness at the ontic level of properties of the system. Ontic noninvasiveness would be required to rule out MR per se but this property is not entailed by MR, and its presence cannot be established in a model-independent way. It follows that despite the formal parallels, Bell's and LG's inequalities are not methodologically on a par. We close with some reflections on the implications of our analysis for the pedagogy of quantum superposition.
Macroscopic realism (MR) per se specifies that a system which has two macroscopically distinct states available to it (such as a cat being dead or alive) is at all times predetermined to be in one or other of those two states. A minimal assumption of a macroscopic realistic theory therefore is the validity of a hidden variable ${\ensuremath{\lambda}}_{M}$ that predetermines the outcome (whether dead or alive) of a measurement $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{M}$ distinguishing the two states. Proposals to test MR generally introduce a second premise to further qualify the meaning of MR. Thus, we consider a model, macroscopic local realism (MLR), where the second premise is that measurements at one location cannot cause an instantaneous macroscopic change $\ensuremath{\delta}$ to the results of measurements made on a second system at another location. To provide a practical test, we define the intermediate concept of $\ensuremath{\delta}$-scopic local realism ($\ensuremath{\delta}$-LR), where $\ensuremath{\delta}\ensuremath{\ne}0$ can be quantified, but need not be macroscopic. By considering the amplification of quantum fluctuations, we show how negation of $\ensuremath{\delta}$-LR is possible using fields violating a continuous variable Bell inequality. A modified Bell-Clauser-Horne-Shimony-Holt inequality is derived that tests $\ensuremath{\delta}$-LR, and a quantitative proposal given for experiments based on polarization entanglement. In the proposal, $\ensuremath{\delta}$ is the magnitude of the quantum noise scaled by an adjustable coherent amplitude $\ensuremath{\alpha}$ that can also be considered part of the measurement apparatus. Thus, $\ensuremath{\delta}$ is large in an absolute sense, but scales inversely with the square root of the system size, which is proportional to ${|\ensuremath{\alpha}|}^{2}$. We discuss how the proposed experiment gives a realization of a type of Schr\"odinger-cat experiment without problems of decoherence.
We construct quantifiable generalizations of Leggett-Garg tests for macro- and mesoscopic realism and noninvasive measurability that apply when not all outcomes of measurement can be identified as arising from one of two macroscopically distinguishable states. We show how quantum mechanics predicts a negation of the Leggett-Garg premises for strategies involving ideal negative-result, weak, and minimally invasive (``nonclumsy'') projective measurements on dynamical entangled systems, as might be realized with Bose-Einstein condensates in a double-well potential, path-entangled NOON states, and atom interferometers. Potential loopholes associated with each strategy are discussed.
While quantum theory has been tested to an incredible degree on microscopic scales, quantum effects are seldom observed in our everyday macroscopic world. The curious results of applying quantum mechanics to macroscopic objects are perhaps best illustrated by Erwin Schrödinger's famous thought experiment, where a cat can be put into a superposition state of being both dead and alive. Obviously, these quantum predictions are in stark contradiction to our common experience. Even with plenty of theoretical explanations put forward to explain this discrepancy, a large number of questions about the frontier between the quantum and the classical world remain unanswered. To distinguish between classical and quantum behavior, two fundamental concepts inherent to classical physics have been established over the years: The world view of local realism limits the power of classical experiments to establish correlations over space, while the world view of macroscopic realism (or macrorealism) restricts temporal correlations. Necessary conditions for both world views have been formulated in the form of Bell and Leggett-Garg inequalities, and Bell inequalities have been shown to be violated by quantum mechanics through increasingly conclusive experiments. Furthermore, many challenging steps towards convincing violations of macrorealism have been taken in a number of recent experiments. In the first part of this thesis, conditions for macrorealism are analyzed in detail. Two necessary conditions for macrorealism, the original Leggett-Garg inequality and the recently proposed no-signaling in time condition, are presented. It is then shown that a combination of no-signaling in time conditions is not only necessary but also sufficient for the existence of a macrorealistic description. Finally, an operational formulation of no-signaling in time, in terms of positive-operator valued measurements and Hamiltonians, is derived. In the next part, we argue that these results lead to a suitable definition of classical behavior. In particular, we provide a formalism to judge the classicality of measurements and time evolutions. We then proceed to apply it to a number of exemplary measurement operators and Hamiltonians. Finally, we argue for the importance of spontaneously realized Hamiltonians in our intuition of classical behavior. Next, differences between local realism and macrorealism are analyzed. For this purpose, the probability polytopes for spatially and temporally separated experiments are compared, and a fundamental difference in the power of quantum mechanics to build both types of correlations is discovered. This result shows that Fine's theorem, which states that a set of Bell inequalities is necessary and sufficient for local realism, is not transferable to macrorealism. Thus, (Leggett-Garg) inequalities are in principle not well-suited for tests of macrorealism, as they can never form a necessary and sufficient condition, and unnecessarily restrict the violating parameter space. No-signaling in time is both better suited and more strongly motivated from the underlying physical theory. In the final part of this thesis, a concrete experimental setup for implementing quantum experiments with macroscopic objects is proposed. It consists of a superconducting micro-sphere in the Meißner state, which is levitated by magnetic fields. Through its expelled magnetic field, the sphere's center-of-mass motion couples to a superconducting quantum circuit. Properly tuned, ground state cooling can be realized, since the sphere's motion is extremely well isolated from the surrounding environment. This setup therefore is a promising candidate for the observation of quantum effects in macroscopic systems.
We show violations of Leggett-Garg inequalities to be possible for single-mode cat states evolving dynamically in the presence of a nonlinear quantum interaction arising from, for instance, a Kerr medium. In order to prove the results, we derive a generalized version of the Leggett-Garg inequality involving different cat states at different times. The violations demonstrate failure of the premise of macrorealism as defined by Leggett and Garg, provided extra assumptions associated with experimental tests are valid. With the additional assumption of stationarity, violations of the Leggett-Garg inequality are predicted for the multicomponent cat states observed in the Bose-Einstein condensate and superconducting circuit experiments of Greiner et al. [Nature (London) 419, 51 (2002)] and Kirchmair et al. [Nature (London) 495, 205 (2013)]. The violations demonstrate a mesoscopic quantum coherence, by negating that the system can be in a classical mixture of mesoscopically distinct coherent states. Higher orders of nonlinearity are also studied and shown to give strong violation of Leggett-Garg inequalities. Loopholes for testing macrorealism are discussed.