Non-Markovian Dynamics and Entanglement of Two-level Atoms in a Common Field

We derive the stochastic equations and consider the non-Markovian dynamics of a system of multiple two-level atoms in a common quantum field. We make only the dipole approximation for the atoms and assume weak atom-field interactions. From these assumptions we use a combination of non-secular open- and closed-system perturbation theory, and we abstain from any additional approximation schemes. These more accurate solutions are necessary to explore several regimes: in particular, near-resonance dynamics and low-temperature behavior. In detuned atomic systems, small variations in the system energy levels engender timescales which, in general, cannot be safely ignored, as would be the case in the rotating-wave approximation (RWA). More problematic are the second-order solutions, which, as has been recently pointed out, cannot be accurately calculated using any second-order perturbative master equation, whether RWA, Born-Markov, Redfield, etc.. This latter problem, which applies to all perturbative open-system master equations, has a profound effect upon calculation of entanglement at low temperatures. We find that even at zero temperature all initial states will undergo finite-time disentanglement (sometimes termed "sudden death"), in contrast to previous work. We also use our solution, without invoking RWA, to characterize the necessary conditions for Dickie subradiance at finite temperature. We find that the subradiant states fall into two categories at finite temperature: one that is temperature independent and one that acquires temperature dependence. With the RWA there is no temperature dependence in any case.Comment: 17 pages, 13 figures, v2 updated references, v3 clarified results and corrected renormalization, v4 further clarified results and new Fig. 8-1

The Rotating-Wave Approximation: Consistency and Applicability from an Open Quantum System Analysis

We provide an in-depth and thorough treatment of the validity of the rotating-wave approximation (RWA) in an open quantum system. We find that when it is introduced after tracing out the environment, all timescales of the open system are correctly reproduced, but the details of the quantum state may not be. The RWA made before the trace is more problematic: it results in incorrect values for environmentally-induced shifts to system frequencies, and the resulting theory has no Markovian limit. We point out that great care must be taken when coupling two open systems together under the RWA. Though the RWA can yield a master equation of Lindblad form similar to what one might get in the Markovian limit with white noise, the master equation for the two coupled systems is not a simple combination of the master equation for each system, as is possible in the Markovian limit. Such a naive combination yields inaccurate dynamics. To obtain the correct master equation for the composite system a proper consideration of the non-Markovian dynamics is required.Comment: 17 pages, 0 figures

Relativistic Quantum Information in Detectors-Field Interactions

We review Unruh-DeWitt detectors and other models of detector-field interaction in a relativistic quantum field theory setting as a tool for extracting detector-detector, field-field and detector-field correlation functions of interest in quantum information science, from entanglement dynamics to quantum teleportation. We in particular highlight the contrast between the results obtained from linear perturbation theory which can be justified provided switching effects are properly accounted for, and the nonperturbative effects from available analytic expressions which incorporate the backreaction effects of the quantum field on the detector behaviour.Comment: 21 pages, 3 figures. Prepared for the special focus issue on RQ

Impact of primary kidney disease on the effects of empagliflozin in patients with chronic kidney disease: secondary analyses of the EMPA-KIDNEY trial

Background: The EMPA KIDNEY trial showed that empagliflozin reduced the risk of the primary composite outcome of kidney disease progression or cardiovascular death in patients with chronic kidney disease mainly through slowing progression. We aimed to assess how effects of empagliflozin might differ by primary kidney disease across its broad population. Methods: EMPA-KIDNEY, a randomised, controlled, phase 3 trial, was conducted at 241 centres in eight countries (Canada, China, Germany, Italy, Japan, Malaysia, the UK, and the USA). Patients were eligible if their estimated glomerular filtration rate (eGFR) was 20 to less than 45 mL/min per 1·73 m2, or 45 to less than 90 mL/min per 1·73 m2 with a urinary albumin-to-creatinine ratio (uACR) of 200 mg/g or higher at screening. They were randomly assigned (1:1) to 10 mg oral empagliflozin once daily or matching placebo. Effects on kidney disease progression (defined as a sustained ≥40% eGFR decline from randomisation, end-stage kidney disease, a sustained eGFR below 10 mL/min per 1·73 m2, or death from kidney failure) were assessed using prespecified Cox models, and eGFR slope analyses used shared parameter models. Subgroup comparisons were performed by including relevant interaction terms in models. EMPA-KIDNEY is registered with ClinicalTrials.gov, NCT03594110. Findings: Between May 15, 2019, and April 16, 2021, 6609 participants were randomly assigned and followed up for a median of 2·0 years (IQR 1·5–2·4). Prespecified subgroupings by primary kidney disease included 2057 (31·1%) participants with diabetic kidney disease, 1669 (25·3%) with glomerular disease, 1445 (21·9%) with hypertensive or renovascular disease, and 1438 (21·8%) with other or unknown causes. Kidney disease progression occurred in 384 (11·6%) of 3304 patients in the empagliflozin group and 504 (15·2%) of 3305 patients in the placebo group (hazard ratio 0·71 [95% CI 0·62–0·81]), with no evidence that the relative effect size varied significantly by primary kidney disease (pheterogeneity=0·62). The between-group difference in chronic eGFR slopes (ie, from 2 months to final follow-up) was 1·37 mL/min per 1·73 m2 per year (95% CI 1·16–1·59), representing a 50% (42–58) reduction in the rate of chronic eGFR decline. This relative effect of empagliflozin on chronic eGFR slope was similar in analyses by different primary kidney diseases, including in explorations by type of glomerular disease and diabetes (p values for heterogeneity all >0·1). Interpretation: In a broad range of patients with chronic kidney disease at risk of progression, including a wide range of non-diabetic causes of chronic kidney disease, empagliflozin reduced risk of kidney disease progression. Relative effect sizes were broadly similar irrespective of the cause of primary kidney disease, suggesting that SGLT2 inhibitors should be part of a standard of care to minimise risk of kidney failure in chronic kidney disease. Funding: Boehringer Ingelheim, Eli Lilly, and UK Medical Research Council

Impact of primary kidney disease on the effects of empagliflozin in patients with chronic kidney disease: secondary analyses of the EMPA-KIDNEY trial

Background: The EMPA-KIDNEY trial showed that empagliflozin reduced the risk of the primary composite outcome of kidney disease progression or cardiovascular death in patients with chronic kidney disease mainly through slowing progression. We aimed to assess how effects of empagliflozin might differ by primary kidney disease across its broad population. Methods: EMPA-KIDNEY, a randomised, controlled, phase 3 trial, was conducted at 241 centres in eight countries (Canada, China, Germany, Italy, Japan, Malaysia, the UK, and the USA). Patients were eligible if their estimated glomerular filtration rate (eGFR) was 20 to less than 45 mL/min per 1·73 m2, or 45 to less than 90 mL/min per 1·73 m2 with a urinary albumin-to-creatinine ratio (uACR) of 200 mg/g or higher at screening. They were randomly assigned (1:1) to 10 mg oral empagliflozin once daily or matching placebo. Effects on kidney disease progression (defined as a sustained ≥40% eGFR decline from randomisation, end-stage kidney disease, a sustained eGFR below 10 mL/min per 1·73 m2, or death from kidney failure) were assessed using prespecified Cox models, and eGFR slope analyses used shared parameter models. Subgroup comparisons were performed by including relevant interaction terms in models. EMPA-KIDNEY is registered with ClinicalTrials.gov, NCT03594110. Findings: Between May 15, 2019, and April 16, 2021, 6609 participants were randomly assigned and followed up for a median of 2·0 years (IQR 1·5-2·4). Prespecified subgroupings by primary kidney disease included 2057 (31·1%) participants with diabetic kidney disease, 1669 (25·3%) with glomerular disease, 1445 (21·9%) with hypertensive or renovascular disease, and 1438 (21·8%) with other or unknown causes. Kidney disease progression occurred in 384 (11·6%) of 3304 patients in the empagliflozin group and 504 (15·2%) of 3305 patients in the placebo group (hazard ratio 0·71 [95% CI 0·62-0·81]), with no evidence that the relative effect size varied significantly by primary kidney disease (pheterogeneity=0·62). The between-group difference in chronic eGFR slopes (ie, from 2 months to final follow-up) was 1·37 mL/min per 1·73 m2 per year (95% CI 1·16-1·59), representing a 50% (42-58) reduction in the rate of chronic eGFR decline. This relative effect of empagliflozin on chronic eGFR slope was similar in analyses by different primary kidney diseases, including in explorations by type of glomerular disease and diabetes (p values for heterogeneity all >0·1). Interpretation: In a broad range of patients with chronic kidney disease at risk of progression, including a wide range of non-diabetic causes of chronic kidney disease, empagliflozin reduced risk of kidney disease progression. Relative effect sizes were broadly similar irrespective of the cause of primary kidney disease, suggesting that SGLT2 inhibitors should be part of a standard of care to minimise risk of kidney failure in chronic kidney disease. Funding: Boehringer Ingelheim, Eli Lilly, and UK Medical Research Council

Effects of empagliflozin on progression of chronic kidney disease: a prespecified secondary analysis from the empa-kidney trial

Background: Sodium-glucose co-transporter-2 (SGLT2) inhibitors reduce progression of chronic kidney disease and the risk of cardiovascular morbidity and mortality in a wide range of patients. However, their effects on kidney disease progression in some patients with chronic kidney disease are unclear because few clinical kidney outcomes occurred among such patients in the completed trials. In particular, some guidelines stratify their level of recommendation about who should be treated with SGLT2 inhibitors based on diabetes status and albuminuria. We aimed to assess the effects of empagliflozin on progression of chronic kidney disease both overall and among specific types of participants in the EMPA-KIDNEY trial. Methods: EMPA-KIDNEY, a randomised, controlled, phase 3 trial, was conducted at 241 centres in eight countries (Canada, China, Germany, Italy, Japan, Malaysia, the UK, and the USA), and included individuals aged 18 years or older with an estimated glomerular filtration rate (eGFR) of 20 to less than 45 mL/min per 1·73 m2, or with an eGFR of 45 to less than 90 mL/min per 1·73 m2 with a urinary albumin-to-creatinine ratio (uACR) of 200 mg/g or higher. We explored the effects of 10 mg oral empagliflozin once daily versus placebo on the annualised rate of change in estimated glomerular filtration rate (eGFR slope), a tertiary outcome. We studied the acute slope (from randomisation to 2 months) and chronic slope (from 2 months onwards) separately, using shared parameter models to estimate the latter. Analyses were done in all randomly assigned participants by intention to treat. EMPA-KIDNEY is registered at ClinicalTrials.gov, NCT03594110. Findings: Between May 15, 2019, and April 16, 2021, 6609 participants were randomly assigned and then followed up for a median of 2·0 years (IQR 1·5-2·4). Prespecified subgroups of eGFR included 2282 (34·5%) participants with an eGFR of less than 30 mL/min per 1·73 m2, 2928 (44·3%) with an eGFR of 30 to less than 45 mL/min per 1·73 m2, and 1399 (21·2%) with an eGFR 45 mL/min per 1·73 m2 or higher. Prespecified subgroups of uACR included 1328 (20·1%) with a uACR of less than 30 mg/g, 1864 (28·2%) with a uACR of 30 to 300 mg/g, and 3417 (51·7%) with a uACR of more than 300 mg/g. Overall, allocation to empagliflozin caused an acute 2·12 mL/min per 1·73 m2 (95% CI 1·83-2·41) reduction in eGFR, equivalent to a 6% (5-6) dip in the first 2 months. After this, it halved the chronic slope from -2·75 to -1·37 mL/min per 1·73 m2 per year (relative difference 50%, 95% CI 42-58). The absolute and relative benefits of empagliflozin on the magnitude of the chronic slope varied significantly depending on diabetes status and baseline levels of eGFR and uACR. In particular, the absolute difference in chronic slopes was lower in patients with lower baseline uACR, but because this group progressed more slowly than those with higher uACR, this translated to a larger relative difference in chronic slopes in this group (86% [36-136] reduction in the chronic slope among those with baseline uACR <30 mg/g compared with a 29% [19-38] reduction for those with baseline uACR ≥2000 mg/g; ptrend<0·0001). Interpretation: Empagliflozin slowed the rate of progression of chronic kidney disease among all types of participant in the EMPA-KIDNEY trial, including those with little albuminuria. Albuminuria alone should not be used to determine whether to treat with an SGLT2 inhibitor. Funding: Boehringer Ingelheim and Eli Lilly