The size of macroscopic superposition states in flux qubits

The question as to whether or not quantum mechanics is applicable to the macroscopic scale has motivated efforts to generate superposition states of macroscopic numbers of particles and to determine their effective size. Superpositions of circulating current states in flux qubits constitute candidate states that have been argued to be at least mesoscopic. We present a microscopic analysis that reveals the number of electrons participating in these superpositions to be surprisingly but not trivially small, even though differences in macroscopic observables are large.Comment: 7 pages, no figure

Electronic structure of superposition states in flux qubits

Flux qubits, small superconducting loops interrupted by Josephson junctions, are successful realizations of quantum coherence for macroscopic variables. Superconductivity in these loops is carried by $\sim 10^6$ -- $10^{10}$ electrons, which has been interpreted as suggesting that coherent superpositions of such current states are macroscopic superpositions analogous to Schr\"odinger's cat. We provide a full microscopic analysis of such qubits, from which the macroscopic quantum description can be derived. This reveals that the number of microscopic constituents participating in superposition states for experimentally accessible flux qubits is surprisingly but not trivially small. The combination of this relatively small size with large differences between macroscopic observables in the two branches is seen to result from the Fermi statistics of the electrons and the large disparity between the values of superfluid and Fermi velocity in these systems.Comment: Minor cosmetic changes. Published version

A measurement-based measure of the size of macroscopic quantum superpositions

Recent experiments claiming formation of quantum superposition states in near macroscopic sys- tems raise the question of how the sizes of general quantum superposition states in an interacting system are to be quantified. We propose here a measure of size for such superposition states that is based on what measurements can be performed to probe and distinguish the different branches of the state. The measure allows comparison of the effective size for superposition states in very different physical systems. It can be applied to a very general class of superposition states and reproduces known results for near-ideal cases. Comparison with a prior measure based on analy- sis of coherence between branches indicates that significantly smaller effective superposition sizes result from our measurement-based measure. Application to a system of interacting bosons in a double-well trapping potential shows that the effective superposition size is strongly dependent on the relative magnitude of the barrier height and interparticle interaction.Comment: 21 pages, 20 figures. Accepted by Phys. Rev. A. Replaced old version with accepted version. Significant changes and improvements, particularly to section on 1-particle measurement

Decoherence of Highly Mixed Macroscopic Quantum Superpositions

It is known that a macroscopic quantum superposition (MQS), when it is exposed to environment, decoheres at a rate scaling with the separation of its component states in phase space. This is more or less consistent with the well known proposition that a more macroscopic quantum state is reduced more quickly to a classical state in general. Effects of initial mixedness, however, on the subsequent decoherence of MQSs have been less known. In this paper, we study the evolution of a highly mixed MQS interacting with an environment, and compare it with that of a pure MQS having the same size of the central distance between its component states. Although the decoherence develops more rapidly for the mixed MQS in short times, its rate can be significantly suppressed after a certain time and becomes smaller than the decoherence rate of its corresponding pure MQS. In an optics experiment to generate a MQS, our result has a practical implication that nonclassicality of a MQS can be still observable in moderate times even though a large amount of noise is added to the initial state.Comment: 6 pages, 4 figure

The GEMS Approach to Stationary Motions in the Spherically Symmetric Spacetimes

We generalize the work of Deser and Levin on the unified description of Hawking radiation and Unruh effect to general stationary motions in spherically symmetric black holes. We have also matched the chemical potential term of the thermal spectrum of the two sides for uncharged black holes.Comment: Latex file, 12 pages, no figure; v2: typos fixed; v3: minor corrections, final version published in JHE

The Fulling-Unruh effect in general stationary accelerated frames

We study the generalized Unruh effect for accelerated reference frames that include rotation in addition to acceleration. We focus particularly on the case where the motion is planar, with presence of a static limit in addition to the event horizon. Possible definitions of an accelerated vacuum state are examined and the interpretation of the Minkowski vacuum state as a thermodynamic state is discussed. Such athermodynamic state is shown to depend on two parameters, the acceleration temperature and a drift velocity, which are determined by the acceleration and angular velocity of the accelerated frame. We relate the properties of Minkowski vacuum in the accelerated frame to the excitation spectrum of a detector that is stationary in this frame. The detector can be excited both by absorbing positive energy quanta in the "hot" vacuum state and by emitting negative energy quanta into the "ergosphere" between the horizon and the static limit. The effects are related to similar effects in the gravitational field of a rotating black hole.Comment: Latex, 39 pages, 5 figure

Quantum dynamics of local phase differences between reservoirs of driven interacting bosons separated by simple aperture arrays

We present a derivation of the effective action for the relative phase of driven, aperture-coupled reservoirs of weakly-interacting condensed bosons from a (3+1)-D microscopic model with local U(1) gauge symmetry. We show that inclusion of local chemical potential and driving velocity fields as a gauge field allows derivation of the hydrodynamic equations of motion for the driven macroscopic phase differences across simple aperture arrays. For a single aperture, the current-phase equation for driven flow contains sinusoidal, linear, and current-bias contributions. We compute the renormalization group (RG) beta function of the periodic potential in the effective action for small tunneling amplitudes and use this to analyze the temperature dependence of the low-energy current-phase relation, with application to the transition from linear to sinusoidal current-phase behavior observed in experiments by Hoskinson et al. \cite{packard} for liquid $^{4}$He driven through nanoaperture arrays. Extension of the microscopic theory to a two-aperture array shows that interference between the microscopic tunneling contributions for individual apertures leads to an effective coupling between apertures which amplifies the Josephson oscillations in the array. The resulting multi-aperture current-phase equations are found to be equivalent to a set of equations for coupled pendula, with microscopically derived couplings.Comment: 16 pages, 5 figures v2: typos corrected, RG phase diagram correcte

Drivers of declining CO2 emissions in 18 developed economies

Global emissions of carbon dioxide (CO 2 ) from fossil fuels and industry increased by 2.2% per year on average between 2005 and 2015 1 . Global emissions need to peak and decline rapidly to limit climate change to well below 2 °C of warming 2,3 , which is one of the goals of the Paris Agreement 4 . Untangling the reasons underlying recent changes in emissions trajectories is critical to guide efforts to attain those goals. Here we analyse the drivers of decreasing CO 2 emissions in a group of 18 developed economies that have decarbonized over the period 2005–2015. We show that within this group, the displacement of fossil fuels by renewable energy and decreases in energy use explain decreasing CO 2 emissions. However, the decrease in energy use can be explained at least in part by a lower growth in gross domestic product. Correlation analysis suggests that policies on renewable energy are supporting emissions reductions and displacing fossil fuels in these 18 countries, but not elsewhere, and that policies on energy efficiency are supporting lower energy use in these 18 countries, as well as more widely. Overall, the evidence shows that efforts to reduce emissions are underway in many countries, but these efforts need to be maintained and enhanced by more stringent policy actions to support a global peak in emissions followed by global emissions reductions in line with the goals of the Paris Agreement 3