Macroscopic Superposition Research Articles

The emergent Copenhagen interpretation of quantum mechanics

We introduce a new and conceptually simple interpretation of quantum mechanics based on reduced density matrices of sub-systems from which the standard Copenhagen interpretation emerges as an effective description of macroscopically large systems. This interpretation describes a world in which definite measurement results are obtained with probabilities that reproduce the Born rule. Wave function collapse is seen to be a useful but fundamentally unnecessary piece of prudent book keeping which is only valid for macro-systems. The new interpretation lies in a class of modal interpretations in that it applies to quantum systems that interact with a much larger environment. However, we show that it does not suffer from the problems that have plagued similar modal interpretations like macroscopic superpositions and rapid flipping between macroscopically distinct states. We describe how the interpretation fits neatly together with fully quantum formulations of statistical mechanics and that a measurement process can be viewed as a process of ergodicity breaking analogous to a phase transition. The key feature of the new interpretation is that joint probabilities for the ergodic subsets of states of disjoint macro-systems only arise as emergent quantities. Finally we give an account of the EPR–Bohm thought experiment and show that the interpretation implies the violation of the Bell inequality characteristic of quantum mechanics but in a way that is rather novel. The final conclusion is that the Copenhagen interpretation gives a completely satisfactory phenomenology of macro-systems interacting with micro-systems.

Newtonian semiclassical gravity in the Ghirardi–Rimini–Weber theory with matter density ontology

We propose a Newtonian semiclassical gravity theory based on the GRW collapse theory with matter density ontology (GRWm), which we term GRWmN. The theory is proposed because, as we show, the standard Newtonian semiclassical gravity theory based on the Schroedinger-Newton equations does not have a consistent Born rule probability interpretation for gravitationally self-interacting particles and implies gravitational cat states for macroscopic mass superpositions. By contrast, we show that GRWmN has a consistent statistical description of gravitationally self-interacting particles and adequately suppresses the cat states for macroscopic superpositions. Two possible routes to experimentally testing GRWmN are also considered. We conclude with a discussion of possible variants of GRWmN, what a general relativistic extension would involve, and various objections that might be raised against semiclassical gravity theories like GRWmN.

Reservoir engineering of a mechanical resonator: generating a macroscopic superposition state and monitoring its decoherence

A deterministic scheme for generating a macroscopic superposition state of a nanomechanical resonator is proposed. The nonclassical state is generated through a suitably engineered dissipative dynamics exploiting the optomechanical quadratic interaction with a bichromatically driven optical cavity mode. The resulting driven dissipative dynamics can be employed for monitoring and testing the decoherence processes affecting the nanomechanical resonator under controlled conditions.

Measurement- and comparison-based sizes of Schrödinger cat states of light

We extend several measurement-based definitions of effective "cat-size" to coherent state superpositions with branches composed of either single coherent states or tensor products of coherent states. These effective cat-size measures depend on determining the maximal quantum distinguishability of certain states associated with the superposition state: e.g., in one measure, the maximal distinguishability of the branches of the superposition is considered as in quantum binary decision theory; in another measure, the maximal distinguishability of the initial superposition and its image after a one-parameter evolution generated by a local Hermitian operator is of interest. The cat-size scaling with the number of modes and mode intensity (i.e., photon number) is compared to the scaling derived directly from the Wigner function of the superposition and to that estimated experimentally from decoherence. We also apply earlier comparison-based methods for determining macroscopic superposition size that require a reference GHZ state. The case of a hierarchical Schr\"{o}dinger cat state with branches composed of smaller superpositions is also analyzed from a measurement-based perspective.

Size of quantum superpositions as measured with classical detectors

We propose a criterion which defines whether a superposition of two photonic components is macroscopic. It is based on the ability to discriminate these components with a particular class of "classical" detectors, namely a photon number measurement with a resolution coarse-grained by noise. We show how our criterion can be extended to a measure of the size of macroscopic superpositions by quantifying the amount of noise that can be tolerated and taking the distinctness of two Fock states differing by N photons as a reference. After applying our measure to several well-known examples, we demonstrate that the superpositions which meet our criterion are very sensitive to phase fluctuations. This suggests that quantifying the macroscopicity of a superposition state through the distinguishability of its components with "classical" detectors is not only a natural measure but also explains why it is difficult to observe superpositions at the macroscopic scale.

Interference of macroscopic beams on a beam splitter: phase uncertainty converted into photon-number uncertainty

Squeezed-vacuum twin beams, commonly generated through parametric down-conversion, are known to have perfect photon-number correlations. According to the Heisenberg principle, this is accompanied by a huge uncertainty in their relative phase. By overlapping bright twin beams on a beam splitter, we convert phase fluctuations into photon-number fluctuations and observe this uncertainty as a typical ‘U-shape’ of the output photon-number distribution. This effect, although reported for atomic ensembles and giving hope for phase super-resolution, has never been observed for light beams. The shape of the normalized photon-number difference distribution is similar to the one that would be observed for high-order Fock states. It can be also mimicked by classical beams with artificially mixed phase, but without any perspective for phase super-resolution. The probability distribution at the beam splitter output can be used for filtering macroscopic superpositions at the input.

Certifiability criterion for large-scale quantum systems

Can one certify the preparation of a coherent, many-body quantum state by measurements with bounded accuracy in the presence of noise and decoherence? Here, we introduce a criterion to assess the fragility of large-scale quantum states, which is based on the distinguishability of orthogonal states after the action of very small amounts of noise. States which do not pass this criterion are called asymptotically incertifiable. We show that if a coherent quantum state is asymptotically incertifiable, there exists an incoherent mixture (with entropy at least log 2) which is experimentally indistinguishable from the initial state. The Greenberger–Horne–Zeilinger states are examples of such asymptotically incertifiable states. More generally, we prove that any so-called macroscopic superposition state is asymptotically incertifiable. We also provide examples of quantum states that are experimentally indistinguishable from highly incoherent mixtures, i.e. with an almost-linear entropy in the number of qubits. Finally, we show that all unique ground states of local gapped Hamiltonians (in any dimension) are certifiable.

A unified measure of macroscopic quantum superposition

In this study, we use a decoherence-based measure of interference to quantify the effective degree of quantum superposition. We show that a type of measure of the effective degree of quantum superposition can be considered from a unified theoretical framework. We use three decoherence processes to define a corresponding decoherence-based measure of quantum superposition, including amplitude damping, phase damping and quantum Brownian motion. With these definitions, we compare the effective degree of quantum superposition achieved for the different experiments. We also investigate the effective degree of quantum superposition prepared from a mixed state through the reservoir engineering technique.

Generation of macroscopic quantum superpositions of optomechanical oscillators by dissipation

We propose a scheme that exploits the combined effects of nonlinear dynamics and dissipation to generate macroscopic quantum superpositions in massive optomechanical oscillators. The effective degenerate three-wave mixing interaction between the mechanical and optical cavity modes that results from quadratic optomechanical coupling, together with the cavity dissipation, can result in the existence of dark macroscopic superpositions. We show analytically and confirm numerically that various quantum superpositions can be achieved deterministically, depending on the initial state of the mechanics. The effects of mechanical damping are also studied in detail and quantified by the negativity of the Wigner function. The present scheme can be realized in various optomechanical systems with current technology.

Dynamics of a two-level system under the simultaneous influence of a spin bath and a boson bath

We study dynamics of a two-level system coupled simultaneously to a pair of dissimilar reservoirs, namely, a spin bath and a boson bath, which are connected via finite interbath coupling. It is found that the steady-state population transfer in the two-level system increases with its coupling to the spin bath, while optimal transfer occurs at intermediate coupling in the transient process. If the two-level system is strongly coupled to the spin bath, the population transfer is unidirectional barring minor population oscillations of minute amplitudes. If the spin bath is viewed as an atomic ensemble, robust generation of macroscopic superposition states exists against parameter variations of the two-level system and the boson bath.

The Copenhagen interpretation as an emergent phenomenon

The Copenhagen interpretation has been remarkably successful but seems at odds with the underlying linearity of quantum mechanics. We show how it can emerge in a simple way from the underlying microscopic quantum world governed by Schrödinger’s equation without the need for observers or their brains. In order to achieve this, we assemble pieces of various pre-existing ideas. Firstly, we adopt a relational approach and use the eigenvectors of the reduced density matrix of a quantum sub-system, or equivalently the Schmidt decomposition, to define the ‘internal state’ of a sub-system. Previous work has identified serious objections to such an interpretation because it apparently leads to macroscopic superpositions and physically unacceptable instabilities near degeneracies. We show that both these problems are solved if the sub-system consists of a large number of coarse-grained degrees of freedom as one expects in order to make contact with the classical world. We further argue that coarse graining is a necessary ingredient because measuring devices have both finite spatial and temporal resolutions. What results is an interpretation in which both decoherence and coarse graining play key roles and from which the rules of the Copenhagen interpretation are seen to emerge in realistic situations that include the measurement of the position of a particle and a decay process.

Macroscopic superpositions in Bose-Josephson junctions: Controlling decoherence due to atom losses

We study how macroscopic superpositions of coherent states produced by the nondissipative dynamics of binary mixtures of ultracold atoms are affected by atom losses. We identify different decoherence scenarios for symmetric or asymmetric loss rates and interaction energies in the two modes. In the symmetric case the quantum coherence in the superposition is lost after a single loss event. By tuning appropriately the energies we show that the superposition can be protected, leading to quantum correlations useful for atom interferometry even after many loss events.

Are Collapse Models Testable via Flavor Oscillations?

Collapse models predict the spontaneous collapse of the wave function, in order to avoid the emergence of macroscopic superpositions. In their mass-dependent formulation, they claim that the collapse of any system’s wave function depends on its mass. Neutral K, D, B mesons are oscillating systems that are given by Nature as superposition of two distinct mass eigenstates. Thus they are unique laboratory for testing collapse models that are sensitive to the mass. In this paper we derive—for the single mesons and bipartite entangled mesons—the effect of the mass-proportional CSL (Continuous Spontaneous Localization) collapse model on the dynamics on neutral mesons. We compare the theoretical prediction with experimental data from different accelerator facilities.

Multi-phonon relaxation and generation of quantum states in a nonlinear mechanical oscillator

The dissipative quantum dynamics of an anharmonic oscillator is investigated theoretically in the context of carbon-based nano-mechanical systems. In the short-time limit, it is known that macroscopic superposition states appear for such oscillators. In the long-time limit, single- and multi-phonon dissipation lead to decoherence of the non-classical states. However, at zero temperature, as a result of two-phonon losses the quantum oscillator eventually evolves into a non-classical steady state. The relaxation of this state due to thermal excitations and one-phonon losses is numerically and analytically studied. The possibility of verifying the occurrence of the non-classical state is investigated and signatures of the quantum features arising in a ring-down setup are presented. The feasibility of the verification scheme is discussed in the context of quantum nano-mechanical systems.

Quantum biology at the cellular level—Elements of the research program

Quantum biology is emerging as a new field at the intersection between fundamental physics and biology, promising novel insights into the nature and origin of biological order. We discuss several elements of QBCL (quantum biology at cellular level) – a research program designed to extend the reach of quantum concepts to higher than molecular levels of biological organization. We propose a new general way to address the issue of environmentally induced decoherence and macroscopic superpositions in biological systems, emphasizing the ‘basis-dependent’ nature of these concepts. We introduce the notion of ‘formal superposition’ and distinguish it from that of Schroedinger's cat (i.e., a superposition of macroscopically distinct states). Whereas the latter notion presents a genuine foundational problem, the former one contradicts neither common sense nor observation, and may be used to describe cellular ‘decision-making’ and adaptation. We stress that the interpretation of the notion of ‘formal superposition’ should involve non-classical correlations between molecular events in a cell. Further, we describe how better understanding of the physics of Life can shed new light on the mechanism driving evolutionary adaptation (viz., ‘Basis-Dependent Selection’, BDS). Experimental tests of BDS and the potential role of synthetic biology in closing the ‘evolvability mechanism’ loophole are also discussed.

Emergence of the Macroscopic Quantum Superposition State in Microtubules

Many researchers conceive communication in Microtubules (MTs), and established theoretical models to show both classical and quantum information processing. In this paper, we studied the usually neglected interactions between the electronic dipole of water molecules in microtubules and the quantized electromagnetic radiation field. We find that the emergence collective coherent radiation, and it can turn into macroscopic quantum superposition state when passing through MTs. This could have a fundamental role in the quantum information processing.

Filtering of the absolute value of photon-number difference for two-mode macroscopic quantum superpositions

We discuss a device capable of filtering out two-mode states of light with mode populations differing by more than a certain threshold, while not revealing which mode is more populated. It would allow engineering of macroscopic quantum states of light in a way which is preserving specific superpositions. As a result, it would enhance optical phase estimation with these states as well as distinguishability of "macroscopic" qubits. We propose an optical scheme, which is a relatively simple, albeit non-ideal, operational implementation of such a filter. It uses tapping of the original polarization two-mode field, with a polarization neutral beam splitter of low reflectivity. Next, the reflected beams are suitably interfered on a polarizing beam splitter. It is oriented such that it selects unbiased polarization modes with respect to the original ones. The more an incoming two-mode Fock state is unequally populated, the more the polarizing beam splitter output modes are equally populated. This effect is especially pronounced for highly populated states. Additionally, for such states we expect strong population correlations between the original fields and the tapped one. Thus, after a photon-number measurement of the polarizing beam splitter outputs, a feed-forward loop can be used to let through a shutter the field, which was transmitted by the tapping beam splitter. This happens only if the counts at the outputs are roughly equal. In such a case, the transmitted field differs strongly in occupation number of the two modes, while information on which mode is more populated is non-existent (a necessary condition for preserving superpositions).

Colloquium: Multiparticle quantum superpositions and the quantum-to-classical transition

This work reports on an extended research endeavor focused on the theoretical and experimental realization of a macroscopic quantum superposition (MQS) made up of photons. This intriguing, fundamental quantum condition is at the core of a famous argument conceived by Schr\"odinger in 1935. The main experimental challenge to the actual realization of this object resides generally in unavoidable and uncontrolled interactions with the environment, i.e., ``decoherence,'' leading to the cancellation of any evidence of the quantum features associated with the macroscopic system. The present scheme is based on a nonlinear process, ``quantum-injected optical parametric amplification,''which, by a linearized cloning process maps the quantum coherence of a single-particle state, i.e., a microqubit, onto a macroqubit consisting of a large number $M$ of photons in quantum superposition. Since the adopted scheme was found resilient to decoherence, a MQS demonstration was carried out experimentally at room temperature with $M\ensuremath{\ge}{10}^{4}$. The result led to an extended study of quantum cloning, quantum amplification, and quantum decoherence. The related theory is outlined and several experiments are reviewed, such as the test of the ``no-signaling theorem'' and the dynamical interaction of the photon MQS with a Bose-Einstein condensate. In addition, the consideration of the microqubit-macroqubit entanglement regime is extended to macroqubit-macroqubit conditions. The MQS interference patterns for large $M$ are revealed in the experiment and bipartite microqubit-macroqubit entanglement was also demonstrated for a limited number of generated particles: $M\ensuremath{\precsim}12$. Finally, the perspectives opened by this new method for further studies on quantum foundations and quantum measurement are considered.

Macroscopic superpositions via nested interferometry: finite temperature and decoherence considerations

Recently, there has been much interest in optomechanical devices for the production of macroscopic quantum states. Here we focus on a proposed scheme for achieving macroscopic superpositions via nested interferometry. We consider the effects of finite temperature on the superposition produced. We also investigate in detail the scheme's feasibility for probing various novel decoherence mechanisms.

Are Cloned Quantum States Macroscopic?

We study quantum states produced by optimal phase covariant quantum cloners. We argue that cloned quantum superpositions are not macroscopic superpositions in the spirit of Schrödinger's cat, despite their large particle number. This is indicated by calculating several measures for macroscopic superpositions from the literature, as well as by investigating the distinguishability of the two superposed cloned states. The latter rapidly diminishes when considering imperfect detectors or noisy states and does not increase with the system size. In contrast, we find that cloned quantum states themselves are macroscopic, in the sense of both proposed measures and their usefulness in quantum metrology with an optimal scaling in system size. We investigate the applicability of cloned states for parameter estimation in the presence of different kinds of noise.