The quantum mechanical description of the world leads to predictions that defy our daily experi- ence when dealing with macroscopic objects. Schrodinger's famous cat illustrates most graphically the paradoxical situations that one encounters when extrapolating the quantum rules to systems possessing macroscopically distinguishable states. Leggett and Garg formulated this question qual- itatively by deriving a series of inequalities based upon the premises of macroscopic realism (1, 2) which have been recently tested within several microscopic systems (3-7) and micrometer-sized superconducting circuits (8). Here we present an experimental violation of Legget-Garg inequali- ties (LGI) with millimeter-sized crystals whose state can be thought of as a cat of cats. Through the reversible transfer of polarization state of single photons and the collective excitation states of crystals, we prepared superposition states of two macroscopic crystals separated by a distance in the mm range. The phase relation between the states in the superposition is governed by the dy- namics of a large number of individual atoms and can be controlled with a frequency detuned and polarization dependent atomic frequency comb (9-13). The quantum character of the considered global state is therefore governed by a macroscopic quantity, which allows for an additional level of complexity in the system. The observed violation of a LGI confirms the persistence of quantum coherence effects within macroscopic objects which can be seen with a naked eye.
The quantum mechanical description of the world leads to predictions that defy our daily experi- ence when dealing with macroscopic objects. Schrodinger's famous cat illustrates most graphically the paradoxical situations that one encounters when extrapolating the quantum rules to systems possessing macroscopically distinguishable states. Leggett and Garg formulated this question qual- itatively by deriving a series of inequalities based upon the premises of macroscopic realism (1, 2) which have been recently tested within several microscopic systems (3-7) and micrometer-sized superconducting circuits (8). Here we present an experimental violation of Legget-Garg inequali- ties (LGI) with millimeter-sized crystals whose state can be thought of as a cat of cats. Through the reversible transfer of polarization state of single photons and the collective excitation states of crystals, we prepared superposition states of two macroscopic crystals separated by a distance in the mm range. The phase relation between the states in the superposition is governed by the dy- namics of a large number of individual atoms and can be controlled with a frequency detuned and polarization dependent atomic frequency comb (9-13). The quantum character of the considered global state is therefore governed by a macroscopic quantity, which allows for an additional level of complexity in the system. The observed violation of a LGI confirms the persistence of quantum coherence effects within macroscopic objects which can be seen with a naked eye.
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-1/2 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.
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.
According to the world view of macrorealism, the properties of a given system exist prior to and independent of measurement, which is incompatible with quantum mechanics. Leggett and Garg put forward a practical criterion capable of identifying violations of macrorealism, and so far experiments performed on microscopic and mesoscopic systems have always agreed with quantum mechanics. However, a macrorealist can always assign the cause of such violations to the perturbation that measurements effect on such small systems, and hence a definitive test would require using noninvasive measurements, preferably on macroscopic objects, where such measurements seem more plausible. However, the generation of truly macroscopic quantum superposition states capable of violating macrorealism remains a big challenge. In this work we propose a setup that makes use of measurements on the polarization of light, a property that has been extensively manipulated both in classical and quantum contexts, hence establishing the perfect link between the microscopic and macroscopic worlds. In particular, we use Leggett-Garg inequalities and the criterion of no signaling in time to study the macrorealistic character of light polarization for different kinds of measurements, in particular with different degrees of coarse graining. Our proposal is noninvasive for coherent input states by construction. We show for states with well-defined photon number in two orthogonal polarization modes, that there always exists a way of making the measurement sufficiently coarse grained so that a violation of macrorealism becomes arbitrarily small, while sufficiently sharp measurements can always lead to a significant violation.
The `Schrödinger's cat' thought experiment highlights the counterintuitive facet of quantum theory that entanglement can exist between microscopic and macroscopic systems, producing a superposition of distinguishable states like the fictitious cat that is both alive and dead. The hallmark of entanglement is the detection of strong correlations between systems, for example by the violation of Bell's inequality. Using the CHSH variant of the Bell test, this violation has been observed with photons, atoms, solid state spins, and artificial atoms in superconducting circuits. For larger, more distinguishable states, the conflict between quantum predictions and our classical expectations is typically resolved due to the rapid onset of decoherence. To investigate this reconciliation, one can employ a superposition of coherent states in an oscillator, known as a cat state. In contrast to discrete systems, one can continuously vary the size of the prepared cat state and therefore its dependence on decoherence. Here we demonstrate and quantify entanglement between an artificial atom and a cat state in a cavity, which we call a `Bell-cat' state. We use a circuit QED architecture, high-fidelity measurements, and real-time feedback control to violate Bell's inequality without post-selection or corrections for measurement inefficiencies. Furthermore, we investigate the influence of decoherence by continuously varying the size of created Bell-cat states and characterize the entangled system by joint Wigner tomography. These techniques provide a toolset for quantum information processing with entangled qubits and resonators. While recent results have demonstrated a high level of control of such systems, this experiment demonstrates that information can be extracted efficiently and with high fidelity, a crucial requirement for quantum computing with resonators.
The Leggett-Garg inequality (LGI), based on the notions of realism and noninvasive measurability, is applied in the context of a linear harmonic oscillator. It is found that merely through observing at various instants which region of the potential well, the oscillating quantum object is in, the LGI can be violated without taking recourse to any ancillary quantum system. Strikingly, this violation reveals an unexplored nonclassicality of the state which is considered the most "classical-like" of all quantum states, namely the Schrödinger coherent state. In the macrolimit, the extent to which such nonclassicality persists for large values of mass and classical amplitudes of oscillation is quantitatively investigated. It is found that while for any given mass and oscillator frequency, a significant quantum violation of LGI can be obtained by suitably choosing the initial peak momentum of the coherent state wave packet, as the mass is sufficiently increased, actual observability of this violation becomes increasingly difficult. A feasible experimental setup for testing the predicted quantum mechanical violation of LGI is suggested using a trapped nano-object of ∼106−109 amu mass.