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

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

Global Carbon Budget 2015

Accurate assessment of anthropogenic carbon dioxide (CO2) emissions and their redistribution among the atmosphere, ocean, and terrestrial biosphere is important to better understand the global carbon cycle, support the development of climate policies, and project future climate change. Here we describe data sets and a methodology to quantify all major components of the global carbon budget, including their uncertainties, based on the combination of a range of data, algorithms, statistics, and model estimates and their interpretation by a broad scientific community. We discuss changes compared to previous estimates as well as consistency within and among components, alongside methodology and data limitations. CO2 emissions from fossil fuels and industry (E-FF) are based on energy statistics and cement production data, while emissions from land-use change (E-LUC), mainly deforestation, are based on combined evidence from land-cover-change data, fire activity associated with deforestation, and models. The global atmospheric CO2 concentration is measured directly and its rate of growth (G(ATM)) is computed from the annual changes in concentration. The mean ocean CO2 sink (S-OCEAN) is based on observations from the 1990s, while the annual anomalies and trends are estimated with ocean models. The variability in S-OCEAN is evaluated with data products based on surveys of ocean CO2 measurements. The global residual terrestrial CO2 sink (S-LAND) is estimated by the difference of the other terms of the global carbon budget and compared to results of independent dynamic global vegetation models forced by observed climate, CO2, and land-cover change (some including nitrogen-carbon interactions). We compare the mean land and ocean fluxes and their variability to estimates from three atmospheric inverse methods for three broad latitude bands. All uncertainties are reported as +/- 1 sigma, reflecting the current capacity to characterise the annual estimates of each component of the global carbon budget. For the last decade available (20052014), E-FF was 9.0 +/- 0.5 GtC yr(-1) E-LUC was 0.9 +/- 0.5 GtC yr(-1), GATM was 4.4 +/- 0.1 GtC yr(-1), S-OCEAN was 2.6 +/- 0.5 GtC yr(-1), and S LAND was 3.0 +/- 0.8 GtC yr(-1). For the year 2014 alone, E FF grew to 9.8 +/- 0.5 GtC yr(-1), 0.6% above 2013, continuing the growth trend in these emissions, albeit at a slower rate compared to the average growth of 2.2% yr(-1) that took place during 2005-2014. Also, for 2014, E-LUC was 1.1 +/- 0.5 GtC yr(-1), G(ATM) was 3.9 +/- 0.2 GtC yr(-1), S-OCEAN was 2.9 +/- 0.5 GtC yr(-1), and S-LAND was 4.1 +/- 0.9 GtC yr(-1). G(ATM) was lower in 2014 compared to the past decade (2005-2014), reflecting a larger S-LAND for that year. The global atmospheric CO2 concentration reached 397.15 +/- 0.10 ppm averaged over 2014. For 2015, preliminary data indicate that the growth in E-FF will be near or slightly below zero, with a projection of 0.6 [ range of 1.6 to C 0.5] %, based on national emissions projections for China and the USA, and projections of gross domestic product corrected for recent changes in the carbon intensity of the global economy for the rest of the world. From this projection of E-FF and assumed constant E LUC for 2015, cumulative emissions of CO2 will reach about 555 +/- 55 GtC (2035 +/- 205 GtCO(2)) for 1870-2015, about 75% from E FF and 25% from E LUC. This living data update documents changes in the methods and data sets used in this new carbon budget compared with previous publications of this data set (Le Quere et al., 2015, 2014, 2013). All observations presented here can be downloaded from the Carbon Dioxide Information Analysis Center (doi: 10.3334/CDIAC/GCP_2015)

Global carbon budget 2019

Accurate assessment of anthropogenic carbon dioxide (CO2) emissions and their redistribution among the atmosphere, ocean, and terrestrial biosphere-the "global carbon budget"-is important to better understand the global carbon cycle, support the development of climate policies, and project future climate change. Here we describe data sets and methodology to quantify the five major components of the global carbon budget and their uncertainties. Fossil CO2 emissions (EFF) are based on energy statistics and cement production data, while emissions from land use change (ELUC), mainly deforestation, are based on land use and land use change data and bookkeeping models. Atmospheric CO2 concentration is measured directly and its growth rate (GATM) is computed from the annual changes in concentration. The ocean CO2 sink (SOCEAN) and terrestrial CO2 sink (SLAND) are estimated with global process models constrained by observations. The resulting carbon budget imbalance (BIM), the difference between the estimated total emissions and the estimated changes in the atmosphere, ocean, and terrestrial biosphere, is a measure of imperfect data and understanding of the contemporary carbon cycle. All uncertainties are reported as ±1σ. For the last decade available (2009-2018), EFF was 9:5±0:5 GtC yr-1, ELUC 1:5±0:7 GtC yr-1, GATM 4:9±0:02 GtC yr-1 (2:3±0:01 ppm yr-1), SOCEAN 2:5±0:6 GtC yr-1, and SLAND 3:2±0:6 GtC yr-1, with a budget imbalance BIM of 0.4 GtC yr-1 indicating overestimated emissions and/or underestimated sinks. For the year 2018 alone, the growth in EFF was about 2.1% and fossil emissions increased to 10:0±0:5 GtC yr-1, reaching 10 GtC yr-1 for the first time in history, ELUC was 1:5±0:7 GtC yr-1, for total anthropogenic CO2 emissions of 11:5±0:9 GtC yr-1 (42:5±3:3 GtCO2). Also for 2018, GATM was 5:1±0:2 GtC yr-1 (2:4±0:1 ppm yr-1), SOCEAN was 2:6±0:6 GtC yr-1, and SLAND was 3:5±0:7 GtC yr-1, with a BIM of 0.3 GtC. The global atmospheric CO2 concentration reached 407:38±0:1 ppm averaged over 2018. For 2019, preliminary data for the first 6-10 months indicate a reduced growth in EFF of C0:6% (range of.0:2% to 1.5 %) based on national emissions projections for China, the USA, the EU, and India and projections of gross domestic product corrected for recent changes in the carbon intensity of the economy for the rest of the world. Overall, the mean and trend in the five components of the global carbon budget are consistently estimated over the period 1959-2018, but discrepancies of up to 1 GtC yr-1 persist for the representation of semi-decadal variability in CO2 fluxes. A detailed comparison among individual estimates and the introduction of a broad range of observations shows (1) no consensus in the mean and trend in land use change emissions over the last decade, (2) a persistent low agreement between the different methods on the magnitude of the land CO2 flux in the northern extra-tropics, and (3) an apparent underestimation of the CO2 variability by ocean models outside the tropics. This living data update documents changes in the methods and data sets used in this new global carbon budget and the progress in understanding of the global carbon cycle compared with previous publications of this data set (Le Quéré et al., 2018a, b, 2016, 2015a, b, 2014, 2013). The data generated by this work are available at https://doi.org/10.18160/gcp-2019 (Friedlingstein et al., 2019). © 2019 by the authors

Global Carbon Budget 2020

Accurate assessment of anthropogenic carbon dioxide (CO$_{2}$) emissions and their redistribution among the atmosphere, ocean, and terrestrial biosphere in a changing climate – the “global carbon budget” – is important to better understand the global carbon cycle, support the development of climate policies, and project future climate change. Here we describe and synthesize data sets and methodology to quantify the five major components of the global carbon budget and their uncertainties. Fossil CO$_{2}$ emissions (E$_{FOS}$) are based on energy statistics and cement production data, while emissions from land-use change (E$_{LUC}$), mainly deforestation, are based on land use and land-use change data and bookkeeping models. Atmospheric CO$_{2}$ concentration is measured directly and its growth rate (G$_{ATM}$) is computed from the annual changes in concentration. The ocean CO$_{2}$ sink (S$_{OCEAN}$) and terrestrial CO$_{2}$ sink (S$_{LAND}$) are estimated with global process models constrained by observations. The resulting carbon budget imbalance (B$_{IM}$), the difference between the estimated total emissions and the estimated changes in the atmosphere, ocean, and terrestrial biosphere, is a measure of imperfect data and understanding of the contemporary carbon cycle. All uncertainties are reported as ±1σ. For the last decade available (2010–2019), E$_{FOS}$ was 9.6 ± 0.5 GtC yr$^{-1}$ excluding the cement carbonation sink (9.4 ± 0.5 GtC yr−1 when the cement carbonation sink is included), and E$_{LUC}$ was 1.6 ± 0.7 GtC yr$^{-1}$. For the same decade, G$_{ATM}$ was 5.1 ± 0.02 GtC yr$^{-1}$ (2.4 ± 0.01 ppm yr$_{-1}$), S$_{OCEAN}$ 2.5 ±  0.6 GtC yr$^{-1}$, and S$_{LAND}$ 3.4 ± 0.9 GtC yr$^{-1}$, with a budget imbalance B$_{IM}$ of −0.1 GtC yr$^{-1}$ indicating a near balance between estimated sources and sinks over the last decade. For the year 2019 alone, the growth in E$_{FOS}$ was only about 0.1 % with fossil emissions increasing to 9.9 ± 0.5 GtC yr$^{-1}$ excluding the cement carbonation sink (9.7 ± 0.5 GtC yr$^{-1}$ when cement carbonation sink is included), and E$_{LUC}$ was 1.8 ± 0.7 GtC yr$^{-1}$, for total anthropogenic CO$_{2}$ emissions of 11.5 ± 0.9 GtC yr$^{-1}$ (42.2 ± 3.3 GtCO$_{2}$). Also for 2019, G$_{ATM}$ was 5.4 ± 0.2 GtC yr$^{-1}$ (2.5 ± 0.1 ppm yr$^{-1}$), S$_{OCEAN}$ was 2.6 ± 0.6 GtC yr$^{-1}$, and S$_{LAND}$ was 3.1 ± 1.2 GtC yr$^{-1}$, with a B$_{IM}$ of 0.3 GtC. The global atmospheric CO$_{2}$ concentration reached 409.85 ± 0.1 ppm averaged over 2019. Preliminary data for 2020, accounting for the COVID-19-induced changes in emissions, suggest a decrease in E$_{FOS}$ relative to 2019 of about −7 % (median estimate) based on individual estimates from four studies of −6 %, −7 %, −7 % (−3 % to −11 %), and −13 %. Overall, the mean and trend in the components of the global carbon budget are consistently estimated over the period 1959–2019, but discrepancies of up to 1 GtC yr$^{-1}$ persist for the representation of semi-decadal variability in CO$_{2}$ fluxes. Comparison of estimates from diverse approaches and observations shows (1) no consensus in the mean and trend in land-use change emissions over the last decade, (2) a persistent low agreement between the different methods on the magnitude of the land CO$_{2}$ flux in the northern extra-tropics, and (3) an apparent discrepancy between the different methods for the ocean sink outside the tropics, particularly in the Southern Ocean. This living data update documents changes in the methods and data sets used in this new global carbon budget and the progress in understanding of the global carbon cycle compared with previous publications of this data set (Friedlingstein et al., 2019; Le Quéré et al., 2018b, a, 2016, 2015b, a, 2014, 2013). The data presented in this work are available at https://doi.org/10.18160/gcp-2020 (Friedlingstein et al., 2020)

Global Carbon Budget 2018

Accurate assessment of anthropogenic carbon dioxide (CO2) emissions and their redistribution among the atmosphere, ocean, and terrestrial biosphere – the “global carbon budget” – is important to better understand the global carbon cycle, support the development of climate policies, and project future climate change. Here we describe data sets and methodology to quantify the five major components of the global carbon budget and their uncertainties. Fossil CO2 emissions (EFF) are based on energy statistics and cement production data, while emissions from land use and land-use change (ELUC), mainly deforestation, are based on land use and land-use change data and bookkeeping models. Atmospheric CO2 concentration is measured directly and its growth rate (GATM) is computed from the annual changes in concentration. The ocean CO2 sink (SOCEAN) and terrestrial CO2 sink (SLAND) are estimated with global process models constrained by observations. The resulting carbon budget imbalance (BIM), the difference between the estimated total emissions and the estimated changes in the atmosphere, ocean, and terrestrial biosphere, is a measure of imperfect data and understanding of the contemporary carbon cycle. All uncertainties are reported as ±1σ. For the last decade available (2008–2017), EFF was 9.4±0.5 GtC yr−1, ELUC 1.5±0.7 GtC yr−1, GATM 4.7±0.02 GtC yr−1, SOCEAN 2.4±0.5 GtC yr−1, and SLAND 3.2±0.8 GtC yr−1, with a budget imbalance BIM of 0.5 GtC yr−1 indicating overestimated emissions and/or underestimated sinks. For the year 2017 alone, the growth in EFF was about 1.6 % and emissions increased to 9.9±0.5 GtC yr−1. Also for 2017, ELUC was 1.4±0.7 GtC yr−1, GATM was 4.6±0.2 GtC yr−1, SOCEAN was 2.5±0.5 GtC yr−1, and SLAND was 3.8±0.8 GtC yr−1, with a BIM of 0.3 GtC. The global atmospheric CO2 concentration reached 405.0±0.1 ppm averaged over 2017. For 2018, preliminary data for the first 6–9 months indicate a renewed growth in EFF of +2.7 % (range of 1.8 % to 3.7 %) based on national emission projections for China, the US, the EU, and India and projections of gross domestic product corrected for recent changes in the carbon intensity of the economy for the rest of the world. The analysis presented here shows that the mean and trend in the five components of the global carbon budget are consistently estimated over the period of 1959–2017, but discrepancies of up to 1 GtC yr−1 persist for the representation of semi-decadal variability in CO2 fluxes. A detailed comparison among individual estimates and the introduction of a broad range of observations show (1) no consensus in the mean and trend in land-use change emissions, (2) a persistent low agreement among the different methods on the magnitude of the land CO2 flux in the northern extra-tropics, and (3) an apparent underestimation of the CO2 variability by ocean models, originating outside the tropics. This living data update documents changes in the methods and data sets used in this new global carbon budget and the progress in understanding the global carbon cycle compared with previous publications of this data set (Le Quéré et al., 2018, 2016, 2015a, b, 2014, 2013). All results presented here can be downloaded from https://doi.org/10.18160/GCP-2018

Global Carbon Budget 2018

Accurate assessment of anthropogenic carbon dioxide (CO2) emissions and their redistribution among the atmosphere, ocean, and terrestrial biosphere – the “global carbon budget” – is important to better understand the global carbon cycle, support the development of climate policies, and project future climate change. Here we describe data sets and methodology to quantify the five major components of the global carbon budget and their uncertainties. Fossil CO2 emissions (EFF) are based on energy statistics and cement production data, while emissions from land use and land-use change (ELUC), mainly deforestation, are based on land use and land-use change data and bookkeeping models. Atmospheric CO2 concentration is measured directly and its growth rate (GATM) is computed from the annual changes in concentration. The ocean CO2 sink (SOCEAN) and terrestrial CO2 sink (SLAND) are estimated with global process models constrained by observations. The resulting carbon budget imbalance (BIM), the difference between the estimated total emissions and the estimated changes in the atmosphere, ocean, and terrestrial biosphere, is a measure of imperfect data and understanding of the contemporary carbon cycle. All uncertainties are reported as ±1σ. For the last decade available (2008–2017), EFF was 9.4±0.5 GtC yr−1, ELUC 1.5±0.7 GtC yr−1, GATM 4.7±0.02 GtC yr−1, SOCEAN 2.4±0.5 GtC yr−1, and SLAND 3.2±0.8 GtC yr−1, with a budget imbalance BIM of 0.5 GtC yr−1 indicating overestimated emissions and/or underestimated sinks. For the year 2017 alone, the growth in EFF was about 1.6 % and emissions increased to 9.9±0.5 GtC yr−1. Also for 2017, ELUC was 1.4±0.7 GtC yr−1, GATM was 4.6±0.2 GtC yr−1, SOCEAN was 2.5±0.5 GtC yr−1, and SLAND was 3.8±0.8 GtC yr−1, with a BIM of 0.3 GtC. The global atmospheric CO2 concentration reached 405.0±0.1 ppm averaged over 2017. For 2018, preliminary data for the first 6–9 months indicate a renewed growth in EFF of +2.7 % (range of 1.8 % to 3.7 %) based on national emission projections for China, the US, the EU, and India and projections of gross domestic product corrected for recent changes in the carbon intensity of the economy for the rest of the world. The analysis presented here shows that the mean and trend in the five components of the global carbon budget are consistently estimated over the period of 1959–2017, but discrepancies of up to 1 GtC yr−1 persist for the representation of semi-decadal variability in CO2 fluxes. A detailed comparison among individual estimates and the introduction of a broad range of observations show (1) no consensus in the mean and trend in land-use change emissions, (2) a persistent low agreement among the different methods on the magnitude of the land CO2 flux in the northern extra-tropics, and (3) an apparent underestimation of the CO2 variability by ocean models, originating outside the tropics. This living data update documents changes in the methods and data sets used in this new global carbon budget and the progress in understanding the global carbon cycle compared with previous publications of this data set (Le Quéré et al., 2018, 2016, 2015a, b, 2014, 2013). All results presented here can be downloaded from https://doi.org/10.18160/GCP-2018