Why we do not see large macroscopic objects in entangled states? There are two ways to approach this question. The first is dynamic: the coupling of a large object to its environment cause any entanglement to decrease considerably. The second approach, which is discussed in this paper, puts the stress on the difficulty to observe a large scale entanglement. As the number of particles n grows we need an ever more precise knowledge of the state, and an ever more carefully designed experiment, in order to recognize entanglement. To develop this point we consider a family of observables, called witnesses, which are designed to detect entanglement. A witness W distinguishes all the separable (unentangled) states from some entangled states. If we normalize the witness W to satisfy |tr(Wρ)| \leq 1 for all separable states ρ, then the efficiency of W depends on the size of its maximal eigenvalue in absolute value; that is, its operator norm ||W||. It is known that there are witnesses on the space of n qbits for which ||W|| is exponential in n. However, we conjecture that for a large majority of n-qbit witnesses ||W|| \leq O(\sqrt{n logn}). Thus, in a non ideal measurement, which includes errors, the largest eigenvalue of a typical witness lies below the threshold of detection. We prove this conjecture for the family of extremal witnesses introduced by Werner and Wolf (Phys. Rev. A 64, 032112 (2001)).
We show that quantum mechanics predicts a contradiction with local hidden variable theories for photon number measurements which have limited resolving power, to the point of imposing an uncertainty in the photon number result which is macroscopic in absolute terms. We show how this can be interpreted as a failure of a new premise, macroscopic local realism.
In the Schrödinger-cat gedanken experiment a cat is in a quantum superposition of two macroscopically distinct states, alive and dead.The paradoxical interpretation of quantum mechanics is that the cat is not in one state or the other, alive or dead, immediately prior to its measurement. Because of this apparent defiance of macroscopic reality, quantum superpositions of states macroscopically distinct have generated much interest. Here we address the issue of proving a contradiction with macroscopic reality objectively, through the testable predictions of quantum mechanics. We consider the pre…
In this paper, we propose a feasible scheme for generating the Schrödinger cat states of a macroscopic mechanical resonator in pulsed cavity optomechanics. Starting with cooling the mechanical oscillator to its ground state, a red and a blue pulses with different powers are simultaneously employed to drive the cavity to achieve squeezed mechanical states. Subsequently, a second red pulse is utilized to generate the macroscopic mechanical quantum superpositions, conditioned on the detection of cavity output photons. Finally, after being stored in the resonator for a period of time, the mechanic…
We describe a set of measurement protocols for performing non-classicality tests and the verification of entangled superposition states of macroscopic continuous variable systems, such as nanomechanical resonators. Following earlier works, we first consider a setup where a two-level system is used to indirectly probe the motion of the mechanical system via Ramsey measurements and discuss the application of this methods for detecting non-classical mechanical states. We then show that the generalization of this techniques to multiple resonator modes allows the conditioned preparation and the det…
We discuss how the optomechanical coupling provided by radiation pressure can be used to cool macroscopic collective degrees of freedom, as vibrational modes of movable mirrors. Cooling is achieved using a phase-sensitive feedback-loop which effectively overdamps the mirrors motion without increasing the thermal noise. Feedback results able to bring macroscopic objects down to the quantum limit. In particular, it is possible to achieve squeezing and entanglement.
Why microscopic objects exhibit wave properties (are delocalized), but macroscopic do not (are localized)? Traditional quantum mechanics attributes wave properties to all objects. When complemented with a deterministic collapse model (Quantum Stud.: Math. Found. 3, 279 (2016)) quantum mechanics can dissolve the discrepancy. Collapse in this model means contraction and occurs when the object gets in touch with other objects and satisfies a certain criterion. One single collapse usually does not suffice for localization. But the object rapidly gets in touch with other objects in a short time, le…
A model of repeated quantum measurements of magnetic flux in superconducting circuits manifesting tunneling is discussed. The perturbation due to the previous measurements of magnetic flux is always present unless quantum nondemolition measurements are performed. By replacing the classical notion of noninvasivity with this condition, temporal Bell-like inequalities allows one to test the observability at the macroscopic level of the conflict between realism and quantum theory.
We investigate a hybrid electro-mechanical system consisting of a pair of charged macroscopic mechanical oscillators coupled to a small ensemble of Rydberg atoms. The resonant dipole-dipole coupling between an internal atomic Rydberg transition and the mechanics allows cooling to its motional ground state with a single atom despite the considerable mass imbalance between the two subsystems. We show that the rich electronic spectrum of Rydberg atoms, combined with their high degree of optical control, paves the way towards implementing various quantum-control protocol for the mechanical oscilla…
In this study, we solve analytically the Schrodinger equation for a macroscopic quantum oscillator as a central system coupled to a large number of environmental micro-oscillating particles. Then, the Langevin equation is obtained for the system using two approaches: Quantum Mechanics and Bohmian Theory. Our results show that the predictions of the two theories are inherently different in real conditions. Nevertheless, the Langevin equation obtained by Bohmian approach could be reduced to the quantum one, when the vibrational frequency of the central system is high enough compared to the maxim…
We study the connection between dissipation and reality in macroscopic quantum systems. We present the following scenario; if we consider the dynamics of a `partial' wave function, the dissipation is represented as a nonlocal term and it causes destructive interference to suppress the quantum fluctuation. Using the variational method, we confirm that this dissipation term is a reasonable extension of the standard (Schrödinger) description for isolated systems, from which we also derive the classical action. Consequently, in macroscopic systems, the states whose time-integrated dissipation take…
On the basis of our recent model of a one-dimensional (1D) completed scattering we argue that Leggett's principles of macroscopic realism must and can be extended onto the level of single electrons and atoms. These principles need three quite feasible innovations in quantum mechanics (QM): (1) at the conceptual level, QM must treat a pure time-dependent one-particle state to involve two or more macroscopically distinct alternatives for a particle as a pure {\it combined} one - the intermediate link between a pure {\it elementary} state (indecomposable into macroscopically distinct parts) and s…
We propose a new approach to justify the use of the microcanonical ensemble for isolated macroscopic quantum systems. Since there are huge number of independent observables in a macroscopic system, we cannot see all of them. Actually what we observe can be written in a rather simple combination of local observables. Considering this limitation, we show that almost all states in a energy shell are practically indistinguishable from one another, and hence from the microcanonical ensemble. In particular, the expectation value of a macroscopic observable is very close to its microcanonical ensembl…
Certifying Macroscopic Quantum Mechanics via Hypothesis Testing with Finite Data
ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（光子科学研究所，巴塞罗那科技研究院）；Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）；ICREA – Institucio Catalana de Recerca i Estudis Avançats（加泰罗尼亚研究与高等学院）
Rapid experimental progress has recently allowed the use of light to prepare macroscopic mechanical objects into nearly pure quantum states. This research field of quantum optomechanics opens new doors toward testing quantum mechanics, and possibly other laws of physics, in new regimes. In the first part of this paper, I will review a set of techniques of quantum measurement theory that are often used to analyze quantum optomechanical systems. Some of these techniques were originally designed to analyze how a classical driving force passes through a quantum system, and can eventually be detect…
To find the Hermitian phase operatorof a single-mode electromagnetic field in quantum mechanics, the Schroedinger representation is extended to a larger Hilbert space augmented by states with infinite excitation by nonstandard analysis. The Hermitian phase operator is shown to exist on the extended Hilbert space. This operator is naturally considered as the controversial limit of the approximate phase operators on finite dimensional spaces proposed by Pegg and Barnett. The spectral measure of this operator is a Naimark extension of the optimal probability operator-valued measure for the phase …
Quantum superposition in mechanical systems is not only key evidence for macroscopic quantum coherence, but can also be utilized in modern quantum technology. Here we propose an efficient approach for creating macroscopically distinct mechanical superposition states in a two-mode optomechanical system. Photon hopping between the two cavity modes is modulated sinusoidally. The modulated photon tunneling enables an ultrastrong radiation-pressure force acting on the mechanical resonator, and hence significantly increases the mechanical displacement induced by a single photon. We study systematica…
Macroscopic realism, the classical world view that macroscopic objects exist independently of and are not influenced by measurements, is usually tested using Leggett-Garg inequalities. Recently, another necessary condition called no-signaling in time (NSIT) has been proposed as a witness for non-classical behavior. In this paper, we show that a combination of NSIT conditions is not only necessary but also sufficient for a macrorealistic description of a physical system. Any violation of macroscopic realism must therefore be witnessed by a suitable NSIT condition. Subsequently, we derive an ope…
We propose a simple method to generate quantum entanglement between two macroscopic mechanical resonators in a two-cavity optomechanical system. This entanglement is induced by the radiation pressure of a single photon hopping between the two cavities. Our results are analytical, so that the entangled states are explicitly shown. Up to local operations, these states are two-mode three-component states, and hence the degree of entanglement can be well quantified by the concurrence. By analyzing the system parameters, we find that, to achieve a maximum average entanglement, the system should wor…
We show that the optomechanical coupling between an optical cavity mode and the two movable cavity end mirrors is able to entangle two different macroscopic oscillation modes of the mirrors. This continuous variable entanglement is maintained by the light bouncing between the mirrors and is robust against thermal noise. In fact, it could be experimentally demonstrated using present technology.
It is possible to construct a classical, macroscopic system which has a mathematical structure that is exactly the same as that of a quantum mechanical system and which can be put into a state which is identical to quantum mechanical entanglement. This paper presents a simple example, including a way in which the system can be measured to violate Bell's inequalities. This classical simulation of a quantum system allows us to visualize entanglement and also helps us to see what aspects of quantum mechanical systems are truly nonclassical.
We propose a deterministic scheme for establishing hybrid Einstein-Podolsky-Rosen (EPR) entanglement channel between a macroscopic mechanical oscillator and a magnon mode in a distant yttrium-iron-garnet (YIG) sphere across about ten gigahertz of frequency difference. The system consists of a driven electromechanical cavity which is unidirectionally coupled to a distant electromagnonical cavity inside which a YIG sphere is placed. We find that far beyond the sideband-resolved regime in the electromechanical subsystem, stationary phonon-magnon EPR entanglement can be achieved. This is realized …
A quantum emitter coupled to a nano-mechanical oscillator is a hybrid system where a macroscopic degree of freedom is coupled to a purely quantum system. Recent progress in nanotechnology has led to the realization of such devices by embedding single artificial atoms like quantum dots or superconducting qubits into vibrating wires or membranes, opening up new perspectives for quantum information technologies and for the exploration of the quantum-classical boundary. In this letter, we show that the quantum emitter can be turned into a strikingly efficient light-controlled source of mechanical …
We propose the quantum mechanical description of complex systems should be performed using two types of causality relation: the ordering relation ($x\prec y$) and the subset relation ($A\subseteq B$). The structures with two ordering operations, called the causal sites, have been already proposed in context of quantum gravity (Christensen and Crane, 2005). We suggest they are also common to biological physics and may describe how the brain works. In the spirit of the Penrose ideas we identify the geometry of the space-time with universal field of consciousness. The latter has its evident count…
Macroscopic mechanical qubits are fundamental both to test the classical-quantum boundary and present suitable candidates for quantum information processing. Motivated by these, we propose a feasible probabilistic scheme to generate an on-demand mechanical qubit, as well as Schrödinger's cat and Fock number states. In order to accomplish this proposal, we study an open dispersive spin-mechanical system in the absence of any external driving. The procedure is solely based on spin post-selection in the weak coupling regime. Through this scheme we demonstrate that the achieved superposition is cl…
We argue that the B^0 B^0 bar state generated in the decay of Upsilon(4S) is well suited for performing tests of Einstein-Podolsky-Rosen correlations, i.e., quantum-mechanical interference effects over macroscopic distances. Using measurements of the ratio R = (#like-sign dilepton events)/(#opposite-sign dilepton events) and of the B_H - B_L mass difference we show that already presently existing data strongly favour the contribution of the interference term to R, as it is required by the rules of quantum mechanics.
A two boundary quantum mechanics without time ordered causal structure is advocated as consistent theory. The apparent causal structure of usual "near future" macroscopic phenomena is attributed to a cosmological asymmetry and to rules governing the transition between microscopic to macroscopic observations. Our interest is a heuristic understanding of the resulting macroscopic physics.
In this didactical note I review in depth the rationale for using generalised canonical distributions in quantum statistics. Particular attention is paid to the proper definitions of quantum entropy and quantum relative entropy, as well as to quantum state reconstruction on the basis of incomplete data. There are two appendices in which I outline how generalised canonical distributions link to the conventional formulation of statistical mechanics, and how classical probability calculus emerges at the macroscopic level.
We present and analyze a proposal for a macroscopic quantum delayed-choice experiment with massive mechanical resonators. In our approach, the electronic spin of a single nitrogen-vacancy impurity is employed to control the coherent coupling between the mechanical modes of two carbon nanotubes. We demonstrate that a mechanical phonon can be in a coherent superposition of wave and particle, thus exhibiting both behaviors at the same time. We also discuss the mechanical noise tolerable in our proposal and predict a critical temperature below which the morphing between wave and particle states ca…
We propose a scheme to prepare macroscopic quantum superpositions of motion in optomachanical nano- or micromechanical oscillators quadratically coupled to an intracavity field. The nonlinear optomechanical coupling leads to an effective degenerate three-wave mixing interaction between the mechanical and cavity modes. The quantum superpositions result from the combined effects of the interaction and cavity dissipation. We show analytically and confirm numerically that various deterministic quantum superpositions can be achieved, depending on initial mechanical state. The effect of mechanical d…
The existence of quantum correlations affects both microscopic and macroscopic systems. On macroscopic systems they are difficult to observe and usually irrelevant for the system's evolution due to the frequent energy exchange with the environment. The world-wide network of gravitational-wave (GW) observatories exploits optical as well as mechanical systems that are highly macroscopic and largely decoupled from the environment. The quasi-monochromatic light fields in the kilometre-scale arm resonators have photon excitation numbers larger than $10^{19}$, and the mirrors that are quasi-free fal…
We study the process of observation (measurement), within the framework of a `perspectival' (`relational', `relative state') version of the modal interpretation of quantum mechanics. We show that if we assume certain features of discreteness and determinism in the operation of the measuring device (which could be a part of the observer's nerve system), this gives rise to classical characteristics of the observed properties, in the first place to spatial localization. We investigate to what extent semi-classical behavior of the object system itself (as opposed to the observational system) is ne…
For an isolated macrosystem classical state parameters $ζ(t)$ are introduced inside a quantum mechanical treatment. By a suitable mathematical representation of the actual preparation procedure in the time interval $[T,t_0]$ a statistical operator is constructed as a solution of the Liouville von Neumann equation, exhibiting at time $t$ the state parameters $ζ(t')$, $t_0\leq t' \leq t$, and {\it preparation parameters} related to times $T \leq t'\leq t_0$. Relation with Zubarev's non-equilibrium statistical operator is discussed. A mechanism for memory loss is investigated and time evolution b…
In this paper we analyze the performance of the Quantum Adiabatic Evolution algorithm on a variant of Satisfiability problem for an ensemble of random graphs parametrized by the ratio of clauses to variables, $γ=M/N$. We introduce a set of macroscopic parameters (landscapes) and put forward an ansatz of universality for random bit flips. We then formulate the problem of finding the smallest eigenvalue and the excitation gap as a statistical mechanics problem. We use the so-called annealing approximation with a refinement that a finite set of macroscopic variables (versus only energy) is used, …
The author cannot grant that Quantum Mechanics satisfies the axioms described here when repeated measurements are at stake. This does not compromise the validity of the conclusions of the previous article "A glance beyond the quantum model", Proc. Roy. Soc. Lond. A 466, 881-890 (2009).
Quantum entanglement of mechanical systems emerges when distinct objects move with such a high degree of correlation that they can no longer be described separately. Although quantum mechanics presumably applies to objects of all sizes, directly observing entanglement becomes challenging as masses increase, requiring measurement and control with a vanishingly small error. Here, using pulsed electromechanics, we deterministically entangle two mechanical drumheads with masses of 70 pg. Through nearly quantum-limited measurements of the position and momentum quadratures of both drums, we perform …
Quantum entanglement in mechanical systems is not only a key signature of macroscopic quantum effects, but has wide applications in quantum technologies. Here we proposed an effective approach for creating strong steady-state entanglement between two directly coupled mechanical oscillators (or a mechanical oscillator and a microwave resonator) in a modulated optomechanical system. The entanglement is achieved by combining the processes of a cavity cooling and the two-mode parametric interaction, which can surpass the bound on the maximal stationary entanglement from the two-mode parametric int…
In this paper, we propose a continuous measurement and feedback scheme to achieve strong Einstein-Podolsky-Rosen (EPR) steering and Bell nonlocality of two macroscopic mechanical oscillators in cavity optomechanics. Our system consists of two optomechanical cavities in which two cavity fields are coupled to each other via a nondegenerate parametric downconversion. The two cavity output fields are subject to continuous Bell-like homodyne detection and the detection currents are fed back to drive the cavity fields. It is found that when the feedback is absent, the two mechanical oscillators can …
Some predictions of quantum mechanics are in contrast with the macroscopic realm of everyday experience, in particular those originated by the Heisenberg uncertainty principle, encoded in the non-commutativity of some measurable operators. Nonetheless, in the last decade opto-mechanical experiments have actualized macroscopic mechanical oscillators exhibiting such non-classical properties. A key indicator is the asymmetry in the strength of the motional sidebands generated in an electromagnetic field that measures interferometrically the oscillator position. This asymmetry is a footprint of th…
This study is aimed at investigating the feasibility of generating quantum entanglement between macroscopic mechanical mirrors in optomechanical systems while under continuous measurement and feedback control. We carefully derive a covariance matrix for mechanical mirrors in a steady state, employing the Kalman filtering problem with an assumed dominant cavity photon dissipation, such that the common and differential modes of the mirrors are squeezed by the action of measuring the output light beams. We demonstrate that entanglement between the mechanical mirrors is generated when the states o…
Quantum physics has intrigued scientists and philosophers alike, because it challenges our notions of reality and locality--concepts that we have grown to rely on in our macroscopic world. It is an intriguing open question whether the linearity of quantum mechanics extends into the macroscopic domain. Scientific progress over the last decades inspires hope that this debate may be decided by table-top experiments.
The investigation of macroscopic quantum phenomena is a current active area of research that offers significant promise to advance the forefronts of both fundamental and applied quantum science. Utilizing the exquisite precision and control of quantum optics provides a powerful toolset for generating such quantum states where the types and 'size' of the states that can be generated are set by the resourcefulness of the protocol applied. In this work we present a new scheme for 'growing' macroscopic superposition states of motion of a mechanical oscillator via cavity quantum optomechanics. The …
The Aharonov-Bohm (AB) effect is a purely quantum mechanical effect. The original (classified as Type-I) AB-phase shift exists in experimental conditions where the electromagnetic fields and forces are zero. It is the absence of forces that makes the AB-effect entirely quantum mechanical. Although the AB-phase shift has been demonstrated unambiguously, the absence of forces in Type-I AB-effects has never been shown. Here, we report the observation of the absence of time delays associated with forces of the magnitude needed to explain the AB-phase shift for a macroscopic system.
Franson-type nonlocal quantum correlation based on the particle nature of quantum mechanics has been intensively studied for both fundamental physics and potential applications of quantum key distribution between remotely separated parties over the last several decades. Recently, a coherence theory of deterministic quantum features has been applied for Franson-type nonlocal correlation [arXiv:2102.06463] to understand its quantumness in a purely classical manner, where the resulting features are deterministic and macroscopic. Here, nearly sub-Poisson distributed coherent photon pairs obtained …
We describe a new and experimentally feasible protocol for performing fundamental tests of quantum mechanics with massive objects. In our approach a single two level system is used to probe the motion of a nanomechanical resonator via multiple Ramsey interference measurements. This scheme enables the measurement of modular variables of macroscopic continuous variable systems and we show that correlations thereof violate a Leggett-Garg inequality and can be applied for tests of quantum contextuality. Our method can be implemented with a variety of different solid state or photonic qubit-resonat…
We analyze a modified version of the Coleman-Hepp model, that is able to take into account energy-exchange processes between the incoming particle and the linear array made up of $N$ spin-1/2 systems. We bring to light the presence of a Wiener dissipative process in the weak-coupling, macroscopic ($N \rightarrow \infty$) limit. In such a limit and in a restricted portion of the total Hilbert space, the particle undergoes a sort of Brownian motion, while the free Hamiltonian of the spin array serves as a Wiener process. No assumptions are made on the spectrum of the Hamiltonian of the spin syst…
We study the dynamics of a bipartite quantum system in a way such that its formal description keeps holding even if one of its parts becomes macroscopic: the problem is related with the analysis of the quantum-to-classical crossover, but our approach implies that the whole system stays genuinely quantum. Aim of the work is to understand 1) if, 2) to what extent, and possibly 3) how, the evolution of a macroscopic environment testifies to the coupling with its microscopic quantum companion. To this purpose we consider a magnetic environment made of a large number of spin-1/2 particles, coupled …
Optomechanical systems offer unique opportunities to explore macroscopic quantum state and related fundamental problems in quantum mechanics. Here, we propose a quantum optomechanical system involving exchange interaction between spin angular momentum of light and a torsional oscillator. We demonstrate that this system allows coherent control of the torsional quantum state of a torsional oscillator on the single photon level, which facilitates efficient cooling and squeezing of the torsional oscillator. Furthermore, the torsional oscillator with a macroscopic length scale can be prepared in Sc…
Entanglement is a vital property of multipartite quantum systems, characterised by the inseparability of quantum states of objects regardless of their spatial separation. Generation of entanglement between increasingly macroscopic and disparate systems is an ongoing effort in quantum science which enables hybrid quantum networks, quantum-enhanced sensing, and probing the fundamental limits of quantum theory. The disparity of hybrid systems and the vulnerability of quantum correlations have thus far hampered the generation of macroscopic hybrid entanglement. Here we demonstrate, for the first t…
Reduction is shown to be a possible consequence of the basic principles of quantum mechanics, involving no branching of the quantum state of the universe. The key feature of a measurement is attributed to the creation of macroscopic germs of ionized tracks, bubbles, sparks and so on, in a superposed quantum state, the main mechanism being a breaking and a restoration of ``classicality'' in nearby macroscopic objects and their small patterns. A new significance of quantum probabilities, involving no absolute randomness, is also proposed.
I argue that conventional estimates of the criterion for classical behavior of a macroscopic body are incorrect, because they do not take into account the locality of interactions, which characterizes the behavior of all systems described approximately by local quantum field theory. Black holes are the only localized objects which do not have such a description. The deviations from classical behavior of a macroscopic body, except for those which can be described as classical uncertainties in the initial values of macroscopic variables,are {\it exponentially} small as a function of the volume o…
We show that for macroscopic measurements which cannot reveal full information about microscopic states of the system, the monogamy of Bell inequality violations present in quantum mechanics implies that practically all correlations between macroscopic measurements can be described by local realistic models. Our results hold for sharp measurement and arbitrary closed quantum systems.
