Johannes Kroes

Rafael Alfonso-Cristancho

Aruna Bansal

Emmanuelle Berret

Kristina Bieksiene

Arnaud Bourdin

Luisa Brussino

Diogo Canhoto

Cristina Cardini

Gulfem Celik

Johannes A Kroes1

Rafael Alfonso-Cristancho2

Aruna T Bansal3

Emmanuelle Berret4

Kristina Bieksiene5

Luisa Brussino7

Diogo Canhoto8

Cristina Cardini9

Gulfem Celik10

Zsuzsanna Csoma11

Barbro Dahlén12

Ebru Damadoglu13

Katrien Eger

Lisa Gauquelin15

Bilun Gemicioglu

Ozlem Goksel17

Sophie Graff

Enrico Heffler19

Hendrik B Hofstee21

Peter Howarth

Rupert W Jakes

Fabienne Jaun

Virginija Kalinauskaite-Zukauske5

Peter Kopač24

Namhee Kwon22

Claudia C Loureiro

Victor Lozoya García27

Matthew Masoli

Mariana Paula Rezelj24

Luis Pérez De Llano29

Sanja Popović-Grle30

David Ramos-Barbón31

Ana Sà Sousa32

Konstantinos Samitas33

Florence Schleich

Concetta Sirena9

Sabina Skrgat25

Eleftherios Zervas

George Zichnalis36

Elisabeth H Bel14

Jacob K Sont21

Simone Hashimoto

Anneke Ten

Evaluation of real-world mepolizumab use in severe asthma across Europe: the SHARP experience with privacy-preserving federated analysis

Introduction

Meta-analyses of randomised controlled trials (RCTs) have shown that biological therapies targeting interleukin (IL)-5 are effective in patients with severe eosinophilic asthma and lead to significant improvements in exacerbation rates, oral corticosteroid (OCS) use, asthma control and health-related quality of life [1][START_REF] Li | The efficacy and safety of reslizumab for inadequately controlled asthma with elevated blood eosinophil counts: a systematic review and meta-analysis[END_REF][START_REF] Tian | Efficacy and safety of benralizumab for eosinophilic asthma: a systematic review and meta-analysis of randomized controlled trials[END_REF]. However, these RCTs are performed in highly selected populations under standardised and fully controlled conditions typically differen from those in clinical practice [START_REF] Brown | Randomised controlled trials in severe asthma: selection by phenotype or stereotype[END_REF][START_REF] Richards | Treatment eligibility of real-life mepolizumab-treated severe asthma patients[END_REF]. Therefore, real-world evidence is an indispensable complement to RCTs as it helps to elucidate which patients are prescribed these therapies and how effective they are in terms of relevant clinical end-points [START_REF] Sherman | Real-world evidencewhat is it and what can it tell us?[END_REF].

Many European countries have established registries of patients with severe asthma in order to collect real-world data on the impact of novel biological treatments [START_REF] Van Bragt | Characteristics and treatment regimens across ERS SHARP severe asthma registries[END_REF][START_REF] Paoletti | Biologics in severe asthma: the role of real-world evidence from registries[END_REF]. Unfortunately, each single country usually has a limited number of included patients, restricting the ability to deliver generalisable evidence and answer important research questions. Therefore, combining patient data from multiple countries and institutes is required in order to generate more robust and meaningful outcomes by increasing sample size and statistical power.

The European Respiratory Society (ERS) clinical research collaboration SHARP (Severe Heterogeneous Asthma Registry, Patient-centered) was set up to harmonise severe asthma management across Europe and unravel heterogeneity in a patient-centred way [START_REF] Djukanovic | The Severe Heterogeneous Asthma Research Collaboration, Patient-centred (SHARP) ERS clinical research collaboration: a new dawn in asthma research[END_REF]. An objective of SHARP is to produce real-world evidence on a pan-European scale by linking together all available data from the national severe asthma registries that are part of the SHARP network. However, linking data from pre-existing registries is challenging, due to unavoidable discrepancies between the data collection models and limitations on the transfer of privacy-sensitive data [START_REF] Bentzen | Balancing protection and free movement of personal data: the new European Union general data protection regulation[END_REF]. For that reason, SHARP leveraged a federated analytics platform that enabled privacy-preserving analysis of distributed datasets and could deliver accurate results without revealing sensitive data [START_REF] Wirth | Privacy-preserving data sharing infrastructures for medical research: systematization and comparison[END_REF][START_REF]Standardized Data: the OMOP Common Data Model[END_REF][START_REF] Kent | Common problems, common data model solutions: evidence generation for health technology assessment[END_REF].

The first research question that SHARP aimed to answer using this federated analytics platform, and which would also serve as proof of principle for this novel approach, was to assess the real-world effectiveness of mepolizumab in patients with severe asthma in Europe. Mepolizumab was the first anti-IL-5 biologic for severe eosinophilic asthma available in most European countries [START_REF] Brusselle | Biologic therapies for severe asthma[END_REF]. Its clinical efficacy has been demonstrated in multiple randomised, double-blind clinical trials showing roughly a halving of the rate of severe asthma exacerbations, a significant reduction in maintenance oral glucocorticoid use and improved health-related quality of life [START_REF] Pavord | Mepolizumab for severe eosinophilic asthma (DREAM): a multicentre, double-blind, placebo-controlled trial[END_REF][START_REF] Ortega | Mepolizumab treatment in patients with severe eosinophilic asthma[END_REF][START_REF] Chupp | Efficacy of mepolizumab add-on therapy on health-related quality of life and markers of asthma control in severe eosinophilic asthma (MUSCA): a randomised, double-blind, placebo-controlled, parallel-group, multicentre, phase 3b trial[END_REF][START_REF] Bel | Oral glucocorticoid-sparing effect of mepolizumab in eosinophilic asthma[END_REF]. Other prospective and closely monitored studies also showed that unselected patients with severe eosinophilic asthma who were prescribed mepolizumab in a real-life setting showed similar improved outcomes [START_REF] Harrison | Real-world mepolizumab in the prospective severe asthma REALITI-A study: initial analysis[END_REF][START_REF] Schleich | Real-world experience with mepolizumab: does it deliver what it has promised?[END_REF].

The present study was designed to evaluate the real-world use of mepolizumab in patients with severe asthma who had been included and followed up in national disease registries of several European countries since the introduction of mepolizumab. The specific aims of the study were to characterise patients receiving mepolizumab, evaluate the number or patients switching or discontinuing mepolizumab treatment, and assess the effectiveness of mepolizumab in patients with severe asthma across Europe using a federated analysis approach. We hypothesised that mepolizumab treatment would 1) reduce frequent (two or more per year) severe exacerbations and 2) lower maintenance use and daily dose of OCS after 1 year (range 11-18 months).

Methods

Study population and design

This was a real-world observational study involving the extraction and analysis of patientlevel data from nonstandardised severe asthma registries from 10 countries in Europe. Most European registries included patients who fulfilled the severe asthma criteria according to the ERS/American Thoracic Society (ATS) guidelines [START_REF] Chung | International ERS/ATS guidelines on definition, evaluation and treatment of severe asthma[END_REF], or national asthma guidelines, and in some cases other patients who attended specialist asthma centres were included as well based on clinical judgement of the treating specialist. The final results were obtained from adult severe asthma patients who initiated mepolizumab between 1 January 2016 and 31 December 2021 and had a follow-up visit available at 1 year (range 11-18 months) after mepolizumab initiation, or at the time of stopping mepolizumab, if sooner. Patients were excluded if they had received another biological treatment for severe asthma in the 12 months prior to inclusion, in order to remove the potential confounding effect of other biological treatments. We distinguished two separate study periods, as it was likely that the coronavirus disease 2019 (COVID-19) pandemic would have influenced the treatment of patients with mepolizumab in terms of initiation, modification and discontinuation of concomitant treatments, as well as outcomes such as the number of asthma exacerbations [START_REF] Eger | The effect of the COVID-19 pandemic on severe asthma care in Europe: will care change for good?[END_REF]. The first period was pre-pandemic and was defined by the initiation and follow-up of mepolizumab treatment between 1 January 2016 and 31 March 2020. The second period spanned the pandemic and was defined by initiation and/or follow-up of mepolizumab treatment between 1 April 2020 and 31 December 2021.

Patients' informed consent for using their data for international research purposes was collected at registry enrolment; the respective national medical ethics committees approved the study protocol.

Data source

10 individual SHARP registries agreed to participate in the study and to have their data used for federated analyses. The database field names of each national registry were harmonised to concepts via the Observational Medical Outcomes Partnership Common Data Model developed by the Observational Health Data Sciences and Informatics community [START_REF]Standardized Data: the OMOP Common Data Model[END_REF][START_REF] Biedermann | Standardizing registry data to the OMOP common data model: experience from three pulmonary hypertension databases[END_REF]. A federated analysis information technology platform (FAP) was then developed and implemented by SHARP in order to generate summary statistics from the harmonised registries, ensuring that individual patient data would remain at the local sites. The process of, and experiences with, the registry mapping, development and implementation of the FAP has been described separately [START_REF] Kroes | Blueprint for harmonizing unstandardized disease registries to allow federated data analysis: prepare for the future[END_REF].

Study outcomes

Two time points were considered: initiation of mepolizumab (baseline) and 11-18 months after baseline (follow-up). Clinical comparisons were drawn between the baseline and followup.

Primary outcomes

Co-primary study outcomes included 1) change in frequent (two or more per year) severe exacerbations; and 2) change in maintenance use and daily dose of OCS after 1 year (range 11-18 months) of mepolizumab treatment.

Secondary outcomes

Secondary outcomes included the description of 1) characteristics of patients prescribed mepolizumab in the 10 registries; and 2) rates of discontinuation of mepolizumab and/or switching to another biologic.

Study variables and definitions

Study variables included demographics, pulmonary function (forced expiratory volume in 1s), comorbidities (nasal polyposis), inflammatory markers (blood eosinophils, fraction of exhaled nitric oxide, total immunoglobulin (Ig)E), exacerbation rate, OCS use, OCS maintenance dose and rates of patients switching or discontinuing mepolizumab treatment. Severe asthma exacerbations were defined by at least one of the following criteria: 1) patient reported use of OCS courses (if not on maintenance OCS); 2) patient-reported doubling of maintenance dose of OCS for ⩾3 days; and 3) patient-reported unscheduled emergency visits or hospitalisation for asthma. Frequent exacerbations were defined as two or more exacerbations per year. This categorical variable was chosen to maximise the availability of this outcome variable due to different methods of recording annual exacerbation rate across registries.

Maintenance OCS dose was expressed as the prednisolone-equivalent daily maintenance dose of OCS (mg•day -1 ).

Statistical analysis

Continuous variables were expressed as mean±SD or median (interquartile range) as appropriate, and categorical variables as n (%). Generalised estimating equations (GEE) were used to derive a parameter estimate for the difference between time windows (baseline and follow-up). The outcomes of interest were regressed upon time-window (pre-/post-), and a sandwich estimator was used to correct the standard errors for within-person correlation, where present. It is noted that there was a mixture of paired (baseline plus follow-up) observations and unpaired (baseline or follow-up) observations. An inverse variance weighted, fixed-effects meta-analysis was used to combine results and estimate effect sizes across participating registries, and results were presented in a forest plot. The pre-pandemic and pandemic groups were analysed separately. To describe patients switching or discontinuing mepolizumab treatment, a tabular summary was generated, by registry. The discontinuation date was set at 28 days after the last known prescription. Discontinuation was also considered to have occurred if there was a break of ⩾90 days between prescriptions; the 90 days were measured from the end of the last known prescription plus 28 days. All statistical analyses were performed using R Statistical Software (version 4.1.2) [START_REF] Core | R: A Language and Environment for Statistical Computing[END_REF].

Results

Patients

Figure 1 shows the flow chart of inclusion in the study. Out of 5871 patients with severe asthma included in the 10 national registries, 2109 initiated mepolizumab treatment. A total of 912 patients met the additional inclusion criteria of consent for an international study; initiation of treatment between 1 January 2016 and 31 December 2021 and no biological treatment for severe asthma for 1 year prior. 671 patients initiated mepolizumab and had follow-up data before the COVID-19 pandemic (pre-pandemic group) and 241 patients either initiated mepolizumab or were followed-up during the pandemic (pandemic group) (figure 1). Tables 1 (pre-pandemic) and 2 (pandemic) summarise the characteristics of the study patients from each national registry at initiation of mepolizumab treatment for the two time periods. The majority of patients had at least two exacerbations per year and the use of maintenance OCS ranged from 26.3% to 80% in the pre-pandemic group and from 10% to 50% in the pandemic group. Although no patient had received biological treatment for severe asthma in the year prior to inclusion in this analysis, six patients in the pre-pandemic group and one in the pandemic group had earlier received another biologic for severe asthma.

Exacerbation rate

Annual exacerbation rate data were available from 369 patients in the pre-pandemic group and 194 patients in the pandemic group (figure 2). The odds of having experienced two or more exacerbations per year after mepolizumab initiation was significantly reduced for both the pre-pandemic group (OR 0.18, 95% CI 0.13-0.25; p<0.001) and the pandemic group (OR 0.08, 95% CI 0.05-0.13; p<0.001). Heterogeneity was clearly present in the pre-pandemic group (I2=69.4%), but not in the pandemic group (I2<0.05%).

Maintenance OCS use

Figure 3 shows the odds of receiving maintenance OCS therapy at follow-up. Data on maintenance OCS use were available for 449 patients in the pre-pandemic group and 138 patients in the pandemic group. For the pre-pandemic group, the odds of patients receiving maintenance OCS at follow-up was significantly reduced from baseline (OR 0.75, 95% CI 0.61-0.92; p=0.005), whereas for the pandemic group the effect was not statistically significant (OR 0.91, 95% CI 0.67-1.23; p=0.527). Heterogeneity was present in both analyses (I2>80%). The reduction in maintenance OCS dose is shown in figure 4. Data on daily OCS maintenance dose were available for 161 patients in the pre-pandemic group. The maintenance OCS dose was significantly different from baseline (mean -3.93 mg•day-1, 95% CI -5.24-2.62; p<0.001). For the pandemic group, the dose of maintenance OCS at follow-up was not significantly different from baseline (mean -0.88 mg•day -1 , 95% CI-1.91-0.15; p=0.096). However, the available data were extremely limited, with only two countries contributing (Italy n=15, Turkey n=19). Again, heterogeneity was observed in both metaanalyses, with I2>60%.

Characteristics of patients prescribed mepolizumab

The baseline characteristics of patients in the analysis set are shown graphically by country in supplementary figure S1. These figures show marked heterogeneity in characteristics, both within individual registries between the periods before and during the pandemic, and between registries. For example, the proportion of previous smokers ranges from [START_REF] Kent | Common problems, common data model solutions: evidence generation for health technology assessment[END_REF].3% to 43.3% in the pre-pandemic group and from 25% to 56.2% in the pandemic group. Similarly, body mass index, age and total IgE show marked differences between periods and between registries.

Rates of patients switching or discontinuing mepolizumab

475 (71%) patients in the pre-pandemic group continued mepolizumab treatment at follow-up, while 291 (91%) patients in the pandemic group continued mepolizumab treatment (tables 3 and 4, respectively). There were marked differences between countries in the number of patients who stopped biological treatment or switched to another biologic. One-third of patients in France stopped biological therapy in the pre-pandemic period and patients in the French and Dutch registries most frequently switched to another anti-IL-5-targeting biologic (23% and 26%, respectively, in the pre-pandemic period).

Discussion

In this study, SHARP succeeded in linking existing patient-level data from 10 different national registries for severe asthma to evaluate the use of mepolizumab therapy across Europe. With the use of a federated analysis approach, the study shows that treatment with mepolizumab significantly reduces severe exacerbations, as well as maintenance OCS use in our sample of patients with severe asthma. In addition, the results show substantial heterogeneity among patients initiating mepolizumab, and rates of switching and discontinuing mepolizumab treatments across European countries. The registry data used for this study were pre-existing, and turned out to be far from complete for current purposes. This fact, in combination with strict requirement for defined entry and outcome data to be available to enable the federated analysis to be undertaken, resulted in a high analysis dropout rate. Thus, the results of this study, while in line with those of RCTs, should be interpreted with these limitations in mind.

The present real-world study supports and complements the results from RCTs and previous real-world studies on the effectiveness of mepolizumab therapy [START_REF] Richards | Treatment eligibility of real-life mepolizumab-treated severe asthma patients[END_REF][START_REF] Pavord | Mepolizumab for severe eosinophilic asthma (DREAM): a multicentre, double-blind, placebo-controlled trial[END_REF][START_REF] Ortega | Mepolizumab treatment in patients with severe eosinophilic asthma[END_REF][START_REF] Chupp | Efficacy of mepolizumab add-on therapy on health-related quality of life and markers of asthma control in severe eosinophilic asthma (MUSCA): a randomised, double-blind, placebo-controlled, parallel-group, multicentre, phase 3b trial[END_REF][START_REF] Bel | Oral glucocorticoid-sparing effect of mepolizumab in eosinophilic asthma[END_REF][START_REF] Thomas | Mepolizumab and oral corticosteroid stewardship: data from the Australian Mepolizumab Registry[END_REF][START_REF] Silver | Real-world reductions in oral corticosteroid use in the USA following mepolizumab therapy for severe asthma[END_REF][START_REF] Elsey | Outcomes over the first two years of treatment with mepolizumab in severe asthma[END_REF][START_REF] Pilette | REALITIa study: real-world oral corticosteroid-sparing effect of mepolizumab in severe asthma[END_REF]. While the RCTs involve large numbers of patients from around the world, most real-world studies were conducted with data from a single institution or national registry. In addition, these studies often included small numbers of patients [START_REF] Li | Real-world effectiveness of mepolizumab in severe eosinophilic asthma: a systematic review and meta-analysis[END_REF]. We are aware of only one real-world study that used data from patients with severe asthma from different countries [START_REF] Harrison | Real-world mepolizumab in the prospective severe asthma REALITI-A study: initial analysis[END_REF]. In contrast to our study, patients in this study were prospectively followed over time, capturing data from clinical practice and recorded in a standardised way. Our study, which made use of data already collected by clinicians in 10 different European countries, not only complements the findings from the previously published studies, but also reflects the effectiveness of mepolizumab in daily clinical practice. Registries have been recognised as an important tool to provide real-world evidence [START_REF] Paoletti | Biologics in severe asthma: the role of real-world evidence from registries[END_REF]. By connecting several registries, our study further seeks to generate real-world evidence in large populations.

In our study, we used a relatively new approach to link privacy-sensitive data from clinical disease registries. That process was not without its challenges. The complexity and labour intensive nature of harmonising privacy-sensitive data from different sources has been extensively described by BIEDERMANN et al. [START_REF] Biedermann | Standardizing registry data to the OMOP common data model: experience from three pulmonary hypertension databases[END_REF]. In their study, three registries of pulmonary hypertension patients were linked and the data analysed in a federated manner. The harmonisation process in our study was even more complex, as all registries had a different data model. This makes our study the first to have harmonised and used nonstandardised real-world disease registries to obtain real-world evidence. The harmonisation process has been completed and the present proof-of-principle study has demonstrated that this federated approach can produce valid results. The platform can now be further used to obtain real-world evidence to help guide better treatment and care to the many thousands of patients with severe asthma in Europe.

In addition to the exceptional method by which we have linked registries in a privacyprotective way, our research is unique in several respects. Successfully analysing patient-data that have been collected in clinical practice and entered into local registries by clinicians from 10 different countries without the involvement or monitoring of contract research organisations including pharmaceutical companies is unprecedented to date. This is probably the best method to get as close as possible to daily clinical practice and to compare treatment practices across countries. Another strength of our study is that our large-scale approach has allowed us to analyse data from the time periods before and during the pandemic separately, so that we could avoid bias due to the alterations in circulating pathogens and changed circumstances of care for severe asthma patients.

Nevertheless, this study has its limitations. First, there are the limitations inherent to conducting real-world studies, for example lack of a control group. Due to the lack of a control group, it is not certain that the observed effectiveness of mepolizumab is caused solely by mepolizumab treatment or by other contributing factors. Second, many patients' data could not be included in the analysis because data were missing or incomplete, or because the moment of data collection was not recorded. In addition, some patients had not given informed consent to use their data for research outside the institution in which they were treated. This feature of real-world data might have led to a selection bias between countries and may limit the generalisability of our study results. Nevertheless, we observe that the health outcomes of well-designed clinical trials can also be detected in a highly imperfect real-world setting. Third, treatment outcomes could only be evaluated for patients still on mepolizumab after 1 year of follow-up, which may have led to a selection bias, potentially overestimating our results and limiting the generalisability of our results. Furthermore, a potential limitation of the analysis is that patients considered at baseline overlapped with, but were not exactly the same as those considered at follow-up. Where paired data was available, the correlation was appropriately accounted for in the analysis. Due to reasons of statistical power, the GEE analyses were not adjusted for possible confounders. Future larger studies might be able to repeat the analyses and include covariates. Finally, there was the effect of the COVID-19 pandemic. This led to many hurdles in data collection and forced us to split the analysis into two periods. Although this reduced the statistical power of the study, we still were able to demonstrate the effect of mepolizumab in reducing exacerbations in both the prepandemic and pandemic time periods.

Our study shows that mepolizumab treatment leads to a reduction in the number of exacerbations and in the use of maintenance OCS in a real-world population. While this outcome was expected, it was reassuring to observe this finding in a setting alternative to RCTs or prospective observational studies, as patients included in an RCT are strictly selected, and may differ significantly from real-world populations. Our results imply that physicians should not be concerned that the effect of mepolizumab in their patients with severe eosinophilic asthma will be below expectations, even if they are less rigorously selected in terms of factors that were exclusion criteria in the RCTs. Of note was the consistent reduction in exacerbations across the countries despite the heterogeneity in the patient populations, with respect to baseline demographics.

The characteristics of real-world patients not only differ from those in the RCTs, but our study also found that patients prescribed mepolizumab differed considerably between the different European countries. This heterogeneity may be due to differences in reimbursement practices for biological therapies, to recommendations in national guidelines and therefore the eligibility criteria for biologics, or to preferences and choices of individual physicians. The differences in patient characteristics treated with mepolizumab before and during the pandemic may have influenced the extent to which some patients could or could not cope with remote care [START_REF] Eger | The effect of the COVID-19 pandemic on severe asthma care in Europe: will care change for good?[END_REF]. The heterogeneity of our population also illustrates that the definition of "severe asthma" does not appear to be used unambiguously. A revision of the international ERS/ATS definition of severe asthma may therefore be required. Interestingly, we also observed differences in OCS tapering strategies, and this important medical practice also requires greater attention and harmonisation. Fortunately, an important first step in the right direction was recently taken with the PONENTE study [START_REF] Menzies-Gow | Oral corticosteroid elimination via a personalised reduction algorithm in adults with severe, eosinophilic asthma treated with benralizumab (PONENTE): a multicentre, open-label, single-arm study[END_REF][START_REF] Korn | Development of methodology for assessing steroidtapering in clinical trials for biologics in asthma[END_REF].

A remarkable finding of our study is the difference between countries in the proportion of patients who continue or discontinue mepolizumab, or switch to another biological treatment. These switchers are likely to be patients who have partially responded to mepolizumab treatment, but have not yet shown full normalisation of all outcome parameters [START_REF] Eger | Long-term therapy response to anti[END_REF]. The fact that such a switch has not been observed in all countries probably reflects different availability of alternative biologics and contrasting practices and levels of acceptance of what is a beneficial outcome. These findings define an area in which greater insight is needed. Until now the best treatment with biologics could only be found based on physician knowledge and experience. It would be less burdensome for all parties if switching of biologics were not necessary and we had good predictors of response. By continuing to use the federated analysis approach and by further optimising national databases and enriching them with biological samples, finding reliable predictors will very likely become possible in the future. More so, if we apply artificial intelligence, machine learning and federated learning to clinical outcome data. The SHARP federated analysis platform is optimally suited for this.

In this study, SHARP demonstrated the real-world effectiveness of mepolizumab in patients with severe asthma from 10 different European countries. Mepolizumab reduced asthma exacerbations and OCS use consistent with evidence generated by RCTs. We observed heterogeneity in characteristics of patients receiving mepolizumab and in rates of switching or discontinuing treatment across countries, signalling the need for further alignment of asthma management across European countries. Our study can be seen as a successful proof of principle as to whether a federated analysis approach can be used to link privacy-sensitive data from different sources. It can thus serve as an example for other clinical research collaborations with a similar ambition. While there is still some room for improvement regarding completeness and quality of data, the SHARP federated analysis platform has great potential for future pan-European real-world severe asthma studies using patient-level data in a privacy-protected way.

Acknowledgements:

This study was conducted on behalf of the Severe Heterogeneous Asthma Registry, Patientcentred (SHARP) (a full contributor acknowledgement list is presented in the supplementary material). lecture honoraria and advisory board participation from AstraZeneca, GSK and Berlin-Chemie outside the submitted work. R. Alfonso-Cristancho reports support for the present work from GSK, as an employee and shareholder. G. Celik reports consulting fees and lecture honoraria from Novartis and GSK outside the submitted work. J.K. Sont reports grants from AstraZeneca; outside the submitted work. All other authors have nothing to disclose.

Support statement:

This study was supported by GlaxoSmithKline (study identifier: 212821).

Funding information for this article has been deposited with the Crossref Funder Registry.

Conflict of interest:


P L Krapivsky
email: pkrapivsky@gmail.com

J M Luck

K Mallick
email: kirone.mallick@ipht.fr

Return probability of N fermions released from a 1D confining potential

We consider N non-interacting fermions prepared in the ground state of a 1D confining potential and submitted to an instantaneous quench consisting in releasing the trapping potential. We show that the quantum return probability of finding the fermions in their initial state at a later time falls off as a power law in the long-time regime, with a universal exponent depending only on N and on whether the free fermions expand over the full line or over a half-line. In both geometries the amplitudes of this power-law decay are expressed in terms of finite determinants of moments of the one-body bound-state wavefunctions in the potential. These amplitudes are worked out explicitly for the harmonic and square-well potentials. At large fermion numbers they obey scaling laws involving the Fermi energy of the initial state. The use of the Selberg-Mehta integrals stemming from random matrix theory has been instrumental in the derivation of these results.

Introduction

Dynamical properties of a quantum system subject to a quench, i.e., a sudden modification of its Hamiltonian, have become an active field of research (see [START_REF] Calabrese | Time dependence of correlation functions following a quantum quench[END_REF][START_REF] Caux | Constructing the Generalized Gibbs ensemble after a quantum quench[END_REF][START_REF] Fagotti | Reduced density matrix after a quantum quench[END_REF], and [START_REF] Polkovnikov | Nonequilibrium dynamics of closed interacting quantum systems[END_REF][START_REF] Eisert | Quantum many-body systems out of equilibrium[END_REF][START_REF] Gogolin | Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems[END_REF][START_REF] D'alessio | From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics[END_REF][START_REF] Borgonovi | Quantum chaos and thermalization in isolated systems of interacting particles[END_REF] for reviews). This renewed interest has been largely motivated by experimental progress in cold atom physics, where the evolution of a condensate after removing a trapping potential can be scrutinized. Amongst many possible observables, the return probability of a quantum system to its initial state (also referred to as fidelity or quantum Loschmidt echo), originally proposed by Peres as an indicator of chaos in quantum systems [START_REF] Peres | Quantum Theory: Concepts and Methods[END_REF], is fully relevant to monitor the dynamics after a quench. The decay of the quantum return probability with time -that can intuitively be related to decoherence -is found to be of various types: generically exponential in chaotic systems [START_REF] Peres | Stability of quantum motion in chaotic and regular systems[END_REF], it can be shown by semi-classical arguments to be slower in regular systems [START_REF] Jalabert | Environment-independent decoherence rate in classically chaotic systems[END_REF]. The situation is more subtle, though, as various time scales are involved [START_REF] Manfredi | Loschmidt echo in a system of interacting electrons[END_REF], and a crossover occurs around the Ehrenfest time (see [START_REF] Gorin | Dynamics of Loschmidt echoes and fidelity decay[END_REF] for a review). For many-particle systems, decay laws ranging from power-law to super-exponential have been predicted [START_REF] García-Calderón | Time evolution of decay of two identical quantum particles[END_REF][START_REF] Del Campo | Long-time behavior of many-particle quantum decay[END_REF][START_REF] Del Campo | Exact quantum decay of an interacting many-particle system: The Calogero-Sutherland model[END_REF][START_REF] Longhi | Many-particle quantum decay and trapping: The role of statistics and Fano resonances[END_REF][START_REF] Marchewka | Role of quantum statistics in multi-particle decay dynamics[END_REF][START_REF] Stéphan | Local quantum quenches in critical one-dimensional systems: Entanglement, the Loschmidt echo, and light-cone effects[END_REF][START_REF] Stéphan | Return probability after a quench from a domain-wall initial state in the spin-1/2 XXZ chain[END_REF].

In a recent work [START_REF] Krapivsky | Quantum return probability of a system of N non-interacting lattice fermions[END_REF] we studied the quantum return probability for a system of N free fermions hopping on a discrete infinite or semi-infinite lattice, launched from a compact configuration where the fermions occupy neighboring sites. We showed that this probability decays algebraically in time, with an exponent that exhibits an intriguing dependence on the parity of N , due to the combined effects of quantum interferences and discreteness. We also determined the exact decay amplitudes, thanks to a mapping to the Selberg-Mehta integrals of random matrix theory [START_REF] Mehta | Random Matrices[END_REF].

The aim of the present work is to investigate the return probability of N fermions after free expansion in a different setting, namely when the system is prepared in the lowest energy state in a 1D trapping potential. We first consider the case where the fermions expand over the full line (section 2). At time t = 0 the confining potential is instantaneously removed. We show that the return probability falls off as a negative power of time, with a universal exponent N 2 . The associated amplitude is expressed in terms of a finite determinant of moments of the one-body bound-state wavefunctions in the potential (sections 2.1 and 2.2). It is evaluated exactly in the cases of harmonic (section 2.3) and square-well (section 2.4) potentials. When the fermion number N is large, the return probability is shown to assume a scaling form involving the ratio N/(E F t), where E F is the Fermi energy of the initial state (section 2.5). All these results are then extended in section 3, with the same setup, to the situation where the fermions expand only over a semi-infinite line. The universal decay exponent of the return probability is now N (2N + 1). Section 4 contains a brief discussion of our findings. In Appendix A we give a self-consistent treatment of the problem of non-colliding classical random walkers, using an approach that parallels the quantum calculations. Appendix B and Appendix C are respectively devoted to Mehta integrals and to the Barnes G-function.

Non-interacting fermions released on the infinite line

In this section we consider a system of N non-interacting spinless fermions on the infinite continuous line. The system is prepared in the lowest energy state in an arbitrary confining potential. At time t = 0, a quantum quench is performed by releasing the confining potential. We are interested in the return probability R N (t) that the particles are back to their initial state at a later time t, and especially in the asymptotic decay of this probability in the long-time regime.

Generalities

The one-body Hamiltonian reads

H = p 2 2 + V (x), (2.1)
with p = -i d/dx. The Planck constant and the fermion mass have been set to unity. The potential V (x) is confining, i.e., V (x) → +∞ as x → ±∞. Let E n (n = 0, 1, . . .) be the ordered eigenvalues of H and ψ n (x) the associated normalized wavefunctions obeying

- 1 2 ψ ′′ n (x) + V (x)ψ n (x) = E n ψ n (x). (2.2)
We consider the situation where the system is prepared in its lowest energy state. The fermions therefore occupy the N lowest bound states (n = 0, . . . , N -1). The corresponding many-body wavefunction is the Slater determinant ‡

Ψ(0)|x = 1 √ N ! det (ψ m-1 (x n )), (2.3)
with x = (x 1 , . . . , x N ). At time t = 0 the confining potential is released, and so the fermions undergo free expansion over the infinite line. The return probability reads

R N (t) = |A N (t)| 2 , (2.4)
with

A N (t) = Ψ(0)|Ψ(t) . (2.5)
The free expansion dynamics is conveniently described in momentum space. In analogy with the tight-binding case studied in [START_REF] Krapivsky | Quantum return probability of a system of N non-interacting lattice fermions[END_REF], we have

A N (t) = ∞ -∞ • • • ∞ -∞ N n=1 dq n 2π e -1 2 itq 2 n | Ψ(0)|q | 2 , (2.6)
Ψ(0)|q = 1 √ N ! det ( ψ m-1 (q n )), (2.7)
with q = (q 1 , . . . , q N ) and

ψ m (q) = ∞ -∞ ψ m (x) e -iqx dx. (2.8)
We are mainly interested in the asymptotic decay of R N (t). In the long-time regime, the integral entering (2.6) is dominated by the region where all the momenta q n are small. In the latter region, the determinant in (2.7) can be estimated by expanding each wavefunction in momentum space as with

ψ m (q) = k≥0 M m,k q k , ( 2
M m,k = (-i) k k! ∞ -∞ ψ m (x) x k dx. (2.10)
The leading contribution is obtained by truncating the expansion (2.9) at order k = N -1. The array M thus becomes a square matrix of size N × N , and the sum over its row index can be read as a matrix product. We thus obtain

det ( ψ m-1 (q n )) ≈ det N k=1 M m-1,k-1 q k-1 n = C N ∆ N (q), (2.11)
where the amplitude

C N = det (M m-1,k-1 ) (2.12)
depends on the confining potential, whereas the universal second factor

∆ N (q) = det (q m-1 n ) = 1≤m<n≤N (q n -q m ) (2.13)
is the Vandermonde determinant of the momenta.

In the long-time regime, the expression (2.6) for the amplitude A N (t) thus simplifies to

A N (t) ≈ |C N | 2 N ! ∞ -∞ • • • ∞ -∞ N n=1 dq n 2π e -1 2 itq 2 n ∆ 2 N (q). (2.14)
The key observation is that the above integral is the analytical continuation to a = 1 2 it of the Mehta integral (B.2). We thus obtain the following prediction for the asymptotic decay of the quantum return probability in the long-time regime:

R N (t) ≈ |C N | 4 G(N + 1) 2 (2π) N t N 2 , (2.15)
where G denotes the Barnes G-function (see Appendix C). The universal decay exponent N 2 can be recovered by the following heuristic argument [START_REF] Krapivsky | Quantum return probability of a system of N non-interacting lattice fermions[END_REF]. The wavefunction of N non-interacting fermions can be estimated by expressing that the particles spread on a ballistic scale L(t) ∼ t and that the wavefunction vanishes when two particles occupy the same position. Thus one can write

|Ψ(x, t)| ∼ K N (t) 1≤m<n≤N |x m -x n | (2.16) if |x i | ≤ L(t)
, whereas Ψ essentially vanishes otherwise. Dimensional analysis implies that the normalization scales as K N (t) ∼ t -N 2 /2 , whence the exponent N 2 derived above by more quantitative means for the return probability. The expression (2.15) only depends on the confining potential V (x) through the prefactor C N , given by (2.12). Hereafter we shall calculate C N exactly for harmonic and square-well potentials in sections 2.3 and 2.4. We shall also estimate its large-N asymptotics for a general confining potential V (x) in section 2.5, using a heuristic semi-classical analysis.

For a single particle, (2.15) reads

R 1 (t) ≈ |M 00 | 4 2πt , (2.17)
where

M 00 = ∞ -∞ ψ 0 (x) dx (2.18)
is the integral of the ground-state wavefunction. This quantity can be used to define the spatial extent ℓ of the ground state by setting

|M 00 | = √ ℓ. The resulting expression, R 1 (t) ≈ ℓ 2 2πt , (2.19)
can be read as R 1 (t) ∼ ℓ/L(t), where the dynamical length L(t) ∼ t/ℓ represents the ballistic spreading of the quantum particle, whose momentum scale p ∼ 1/ℓ is dictated by the uncertainty principle. An alternative interpretation of (2. [START_REF] Stéphan | Local quantum quenches in critical one-dimensional systems: Entanglement, the Loschmidt echo, and light-cone effects[END_REF]) is that it exhibits the formal diffusive scaling of the Schrödinger equation.

Considerations about symmetry

In the situation where the confining potential is symmetric, i.e., V (-x) = V (x), the eigenstates have a definite parity, i.e.,

ψ n (-x) = (-1) n ψ n (x), (2.20)
and so the matrix element M m,k vanishes if m + k is odd. The expression (2.12) of C N therefore splits into the product of two determinants corresponding to each parity sector, namely For a symmetric potential, we have c

C 2p = c (even) p c (odd) p , (2.21)
C 2p+1 = c (even) p+1 c (odd) p , (2.22)
(even) 1 = M 00 , c (odd) 1
= M 11 , hence C 2 = M 00 M 11 , and so

R 2 (t) ≈ |M 00 M 11 | 4 4π 2 t 4 .
(2.26)

Harmonic potential

In this section we consider the case of a harmonic potential well:

V (x) = ω 2 x 2 2 . (2.27)
The quantum harmonic oscillator is a textbook example of an exactly solvable system. This is seemingly the only example where the full time dependence of the return probability can be worked out explicitly.

The energy levels read

E n = n + 1 2 ω (n = 0, 1, . . .).
(2.28)

The corresponding wavefunctions have essentially the same form in position space and in momentum space, namely

ψ n (x) = (ω/π) 1/4 √ 2 n n! H n (x √ ω) e -ωx 2 /2 , (2.29)
ψ n (q) = (-i) n (4π/ω) 1/4 √ 2 n n! H n (q/ √ ω) e -q 2 /(2ω) , (2.30)
where

H n (z) = ⌊n/2⌋ k=0 (-1) k n! k!(n -2k)! (2z) n-2k (2.31)
are the Hermite polynomials [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas[END_REF]. Using (2.30), and introducing the notation z n = q n / √ ω, the expression (2.7) can be recast as

Ψ(0)|q = - i √ 2 N (N -1)/2 (4π/ω) N/4 G(N + 2) × exp - 1 2 N n=1 z 2 n det (H m-1 (z n )). (2.32)
The latter determinant can be evaluated as follows. By subtracting from the mth line a suitably chosen linear combination of the previous ones, the Hermite polynomial H m-1 (z n ) can be replaced by its leading term (2z n ) m-1 . The determinant is therefore proportional to a Vandermonde determinant:

det (H m-1 (z n )) = 2 N (N -1)/2 ∆ N (z), (2.33)
with z = (z 1 , . . . , z N ). The expression (2.6) of the amplitude A N (t) therefore reads

A N (t) = 2 N (N -1)/2 π N/2 G(N + 2) × ∞ -∞ • • • ∞ -∞ N n=1 dz n e -(1+ 1 2 iωt)z 2 n ∆ 2 N (z). (2.34)
The above integral is a Mehta integral of the form (B.2) at any finite time t. This is a very peculiar feature of the harmonic oscillator. We thus obtain the remarkably simple exact expressions § which hold for all fermion numbers N and all times t. The return probability therefore exhibits the power-law decay

A N (t) = 1 + iωt 2 -N 2 /2 , (2.35) R N (t) = 1 + ω 2 t 2 4 -N 2 /2 , ( 2
R N (t) ≈ 2 ωt N 2 , ( 2.37)
with exponent N 2 , in agreement with (2.15), and a simple prefactor.

Anticipating the analysis of the large-N regime presented in section 2.5, we recast the above formula in terms of the Fermi energy E F , defined as the energy of the last occupied one-particle state. In the case of the harmonic oscillator, we have

E F ≈ N ω, hence R N (t) ∼ 2N E F t N 2
.

(2.38)

Square-well potential

In this section we consider the case of a square-well potential:

V (x) = 0 (|x| < L/2), +∞ (|x| > L/2).
(2.39)

The particles are thus confined between two impenetrable walls at x = ±L/2. This is another textbook example of an exactly solvable system. The energy levels read

E n = (n + 1) 2 π 2 2L 2 (n = 0, 1, . . .). (2.40)
The corresponding wavefunctions are as follows.

• Even sector (n = 2p, p = 0, 1, . . .):

ψ 2p (x) = 2 L cos (2p + 1)πx L , (2.41)
ψ 2p (q) = (-1) p π √ 2L 2(2p + 1) (2p + 1) 2 π 2 -q 2 L 2 cos qL 2 .
(2.42)

• Odd sector (n = 2p -1, p = 1, 2, . . .):

ψ 2p-1 (x) = 2 L sin 2pπx L , (2.43)
ψ 2p-1 (q) = (-1) p iπ √ 2L 4p 4p 2 π 2 -q 2 L 2 sin qL 2 . (2.44)
At variance with the case of the harmonic oscillator, the return probability cannot be evaluated exactly at finite time.

The prefactor C N entering the asymptotic decay law (2.15) of R N (t) can however be evaluated exactly, for all values of the fermion number N . The key point of the derivation resides in the following observation. The determinants c (even) p and c (odd) p , introduced in (2.23), (2.24), can be simplified along the lines of the derivation of (2.33). In the even sector, it is legitimate to replace cos(qL/2) by unity in the expression (2.42) of ψ 2p (q), as this amounts to subtracting from the rows of the matrix M 2k,2l a suitably chosen linear combination of the previous ones. We denote by M the matrix simplified in this way. Similarly, for the odd sector, sin(qL/2) can be linearized to qL/2 in the expression (2.44) of ψ 2p-1 (q). In the even sector, we thus obtain

c (even) p = det ( M 2k,2l ) 0≤k,l≤p-1 , M 2k,2l = (-1) k 2 √ 2 ((2k + 1)π) 2l+1 L 2l+1/2 ,
(2.45) and so

c (even) p = 2 3p/2 π p 2 (2p -1)!! d (even) p L p(p-1/2) , (2.46)
with

d (even) p = (-1) p(p-1)/2 det ((2k + 1) -2l ) 0≤k,l≤p-1 .
(2.47)

In the odd sector, we have

c (odd) p = det ( M 2k-1,2l-1 ) 1≤k,l≤p , M 2k-1,2l-1 = (-1) k i √ 2 (2kπ) 2l-1 L 2l-1/2 , (2.48)
and so

c (odd) p = (-i) p 2 p/2 (2π) p 2 p! d (odd) p L p(p+1/2) , (2.49)
with

d (odd) p = (-1) p(p-1)/2 det (k -2(l-1) ) 1≤k,l≤p . (2.50)
The determinants d

c (even) p = 2 3p 2 π p 2 p! (2p)! 2p G(2p + 2) p! L p(p-1/2) , (2.51) c (odd) p = (-i) p (2π) p 2 p! 2p G(2p + 2) p! L p(p+1/2) .
(2.52)

Inserting the above results into (2.21), (2.22), we get

|C N | = G(N + 2) N ! N Γ( N 2 + 1) 2L π N 2 /2 , (2.53)
irrespective of the parity of N .

We thus obtain the following asymptotic decay law for the return probability

R N (t) ≈ K N L 2 π 2 t N 2 , (2.54)
where the exponent N 2 is in agreement with (2.15), and the prefactor reads

K N = 2 2N 2 G(N + 1) 6 (2π) N N ! 4(N -1) Γ( N 2 + 1) 4 .
(2.55)

In particular

K 1 = 32 π 3 , K 2 = 4 π 2 , K 3 = 2 21 3 12 π 5 .
(2.56)

When the fermion number N becomes large, the leading behavior of K N can be derived by using the asymptotic expansions (C.4) and (C.5). Keeping only terms in N 2 , we obtain

ln K N ≈ -N 2 ln N -2 ln 2 + 1 2 .
(2.57)

The expression (2.54) therefore simplifies to

R N (t) ∼ 4L 2 e 1/2 π 2 N t N 2 .
(2.58)

Anticipating again the analysis of section 2.5, we recast the above formula in terms of the Fermi energy E F . For a square-well potential, the Fermi energy grows as E F ≈ N 2 π 2 /(2L 2 ) and the above formula can be rewritten as

R N (t) ∼ 2e -1/2 N E F t N 2
.

(2.59)

Scaling at large N

We now focus our attention onto the regime where the fermion number N is large. By observing the formulas (2.38) and (2.59) found for the harmonic and the squarewell potential, it is tempting to propose the following scaling Ansatz for the return probability for an arbitrary confining potential V (x):

R N (t) ∼ BN E F t N 2 . (2.60)
This scaling law is meant to hold in the regime where the long-time limit is taken before the limit of a large fermion number. In the denominator, E F is the Fermi energy, i.e., the energy of the last occupied one-particle state. The occurrence of the dimensionless combination E F t is quite natural. In the numerator, B appears as a numerical constant of order unity, which depends on the confining potential V (x).

The above results give B = 2 for the harmonic oscillator and B = 2e -1/2 = 1.213061 for the square-well potential. The Ansatz (2.60) is corroborated by the following heuristic semi-classical analysis for a general symmetric power-law confining potential,

V (x) = g|x| a , (2.61)
with arbitrary growth exponent a > 0. Consider a highly excited bound state in this potential, with energy E n (n ≫ 1). The wavefunction ψ n (x) exhibits two turning points at x = ±x n , such that E n = gx a n . The energy E n is given by the semi-classical quantization formula [START_REF] Bender | Advanced Mathematical Methods for Scientists and Engineers[END_REF] 2

xn 0 2(E n -gx a ) dx ≈ n + 1 2 π.
(2.62) Some algebra leads to

x n ≈ λn √ g β , E n ≈ g β (λn) 2(1-β) , (2.63)
with

β = 2 a + 2 , λ = π 2 Γ 3 2 + 1 a Γ 1 + 1 a .
(2.64)

We have therefore E F ≈ g β (λN ) 2 (1-β) . The wavefunction ψ n (x) is rapidly oscillating in the allowed region (|x| < x n ), and exponentially decaying in the forbidden regions (|x| > x n ). Its amplitude is maximal in the transition regions near the turning points. As a consequence, for large values of the integers m and n, the integral entering the expression (2.10) of M m,k can be expected to be dominated by the vicinity of the turning points. This heuristic argument yields the rough estimate

M m,k ∼ x k m k! ; (2.65)
and hence

ln |M m,k | ≈ k β ln λm √ g -ln k + 1 .
(2.66)

The leading scaling behavior of C N can be read off from the above estimate by replacing m by N in the argument of the first logarithm and summing the resulting expression over k. This yields

ln |C N | ≈ N 2 2 (β -1) ln N + β ln λ √ g + • • • . (2.67)
Inserting this estimate into (2.15), we find that the return probability indeed obeys the scaling law (2.60). The case of a square-well potential is recovered in the β → 0 limit. The above line of reasoning is however too crude to predict the numerical constant B.

Non-interacting fermions released on the semi-infinite line

We now consider the same problem on the semi-infinite line (x > 0). We assume that there is an impenetrable wall at the origin. The system is prepared in the lowest energy state in the presence of an arbitrary confining potential V w (x) acting for x > 0. At time t = 0 the confining potential is released but the wall at the origin is kept, so that the particles expand over the semi-infinite line. We are again interested in the return probability R w N (t), and especially in its asymptotic decay.

Generalities

The one-body Hamiltonian reads

H w = p 2 2 + V w (x) (x > 0), (3.1)
with Dirichlet boundary condition at the origin. Let E w n (n = 0, 1, . . .) be the ordered eigenvalues of H w and ψ w n (x) the associated normalized wavefunctions. At time t = 0 the confining potential is released, and so the fermions undergo free expansion over the semi-infinite line. The return probability reads

R w N (t) = |A w N (t)| 2 , (3.2)
with

A w N (t) = Ψ w (0)|Ψ w (t) . (3.3)
The free expansion dynamics is again best described in momentum space. We have

A w N (t) = ∞ 0 • • • ∞ 0 N n=1 dq n π e -1 2 itq 2 n | Ψ w (0)|q | 2 , (3.4)
The superscript 'w' reminds of the permanent presence of an impenetrable wall at the origin.

with

Ψ w (0)|q = 1 √ N ! det ( ψ w m-1 (q n )) (3.5)
and

ψ w m (q) = √ 2 ∞ 0 ψ w m (x) sin qx dx. (3.6)
In the long-time regime, the integral entering (3.4) is again dominated by the region where all the momenta q n are small. In the latter region, the determinant in (3.5) can be estimated by expanding each wavefunction in momentum space as

ψ w m (q) = √ 2 k≥0 M w m,k q 2k+1 , (3.7)
with

M w m,k = (-1) k (2k + 1)! ∞ 0 ψ w m (x) x 2k+1 dx. (3.8)
The leading contribution is again obtained by truncating the expansion (3.7) at order k = N -1. We thus obtain det (

ψ w m-1 (q n )) ≈ 2 N/2 C w N N n=1 q n ∆ N (q 2 ), (3.9)
where

C w N = det (M w m-1,k-1 ), (3.10)
and q 2 = (q 2 1 , . . . , q 2 N ). In the long-time regime, the expression (3.3) for the amplitude A w N (t) thus simplifies to

A w N (t) ≈ 2 N |C w N | 2 N ! ∞ 0 • • • ∞ 0 N n=1 dq n π q 2 n e -1 2 itq 2 n ∆ 2 N (q 2 ). (3.11)
The above integral is proportional to the analytical continuation to a = 1 2 it of the Mehta integral (B.6). We thus obtain the following asymptotic prediction for the return probability in the long-time regime:

R w N (t) ≈ |C w N | 4 G(2N + 2) π N N ! t N (2N +1) .
(3.12)

The return probability again exhibits a universal power-law decay, with exponent N (2N + 1). This exponent can also be predicted by means of a heuristic argument (see [START_REF] Krapivsky | Quantum return probability of a system of N non-interacting lattice fermions[END_REF]). The dependence of the above prediction on the confining potential V w (x) is entirely contained in the prefactor C w N , given by (3.10). For one single particle, (3.12) reads

R w 1 (t) ≈ 2|M w 00 | 4 πt 3 , (3.13)
with

M w 00 = ∞ 0 x ψ w 0 (x) dx. (3.14)
This integral can be used to define the spatial extent ℓ w of the ground state by setting |M w 00 | = (ℓ w ) 3/2 . The resulting expression,

R 1 (t) ≈ 2 (ℓ w ) 6 πt 3 , (3.15)
again reflects the formal diffusive scaling of the Schrödinger equation.

Considerations about symmetry

It is worth comparing the one-sided situation, i.e., free expansion on the semi-infinite line after preparation in the confining potential V w (x) for x > 0, with the two-sided situation (section 2) with potential

V (x) = V w (|x|), (3.16)
obtained by symmetrizing V w (x).

The correspondence between both situations goes as follows. The Dirichlet boundary condition at the origin implies that the wavefunctions ψ w n (x) essentially coincide with the odd wavefunctions of the two-sided problem, namely

ψ w n (x) = √ 2 ψ 2n+1 (x) (3.17)
for x > 0 and n = 0, 1, . . ., where the factor √ 2 ensures the correct normalizations. In momentum space, with the definitions (2.8) and (3.6), this reads

ψ w n (q) = i ψ 2n+1 (q). (3.18)
We have therefore

M w m,k = i √ 2 M 2m+1,2k+1 . (3.19)
In particular

C w N = i √ 2 N c (odd) N .
(3.20)

Harmonic potential

In this section we consider the case of a harmonic half-well:

V w (x) = ω 2 x 2 2 (x > 0). (3.21)
This will again be the only example where the full time dependence of the return probability R w N (t) can be worked out explicitly. We shall exploit the correspondence underlined in section 3.2. The two-sided situation is that of a harmonic well, studied in section 2.3. Using (3.18), as well as (2.30) and (C.10), and introducing the notation z n = q n / √ ω, the expression (3.5) can be recast as

Ψ w (0)|q = - 1 2 N (N -1)/2 (2π/ω) N N !G(2N + 2) 1/4 × exp - 1 2 N n=1 z 2 n det (H 2m-1 (z n )). (3.22)
The latter determinant can be simplified along the lines of the derivation of (2.33).

We thus obtain

det (H 2m-1 (z n )) = 2 N 2 N i=1 z n ∆ N (z 2 ), (3.23) with z 2 = (z 2 1 , . . . , z 2 N ).
The expression (3.4) of the amplitude A w N (t) therefore reads

A w N (t) = 2 N (2N +1)/2 π N N ! G(2N + 2) × ∞ -∞ • • • ∞ -∞ N n=1 dz n z 2 n e -(1+ 1 2 iωt)z 2 n ∆ 2 N (z 2 ). (3.24)
The above integral is a Mehta integral of the form (B.6) at any finite time t. We again obtain the remarkably simple exact expressions ¶

A w N (t) = 1 + iωt 2 -N (2N +1)/2 , (3.25) R w N (t) = 1 + ω 2 t 2 4 -N (2N +1)/2 , (3.26)
for all fermion numbers N and all times t. The return probability therefore exhibits the power-law decay

R w N (t) ≈ 2 ωt N (2N +1) , (3.27)
with exponent N (2N + 1), in agreement with (3.12), and a simple prefactor.

Square-well potential

In this section we consider the case of the square-well potential:

V (x) = 0 (0 < x < L/2), +∞ else. (3.28)
The particles are thus confined between two impenetrable walls at x = 0 and x = L/2. The right wall is removed instantaneously at time t = 0, whereas the left one is maintained permanently. This setting can be viewed as a special case of the problem studied long ago by Doescher and Rice [START_REF] Doescher | Infinite square-well potential with a moving wall[END_REF], namely a square well with a wall moving at constant velocity. The present problem of an instantaneous release corresponds to the infinite velocity limit. We shall again make use of the correspondence underlined in section 3.2. The two-sided situation is that considered in section 2.4. Inserting the expression (2.52) of c (odd) N into (3.20), we readily obtain

C w N = 1 (2π) N 2 N ! 2N G(2N + 2) 2 N N ! L N (N +1/2) , (3.29)
and finally

R w N (t) ≈ K w N L 2 4π 2 t N (2N +1) , (3.30)
where the exponent N (2N + 1) is again in agreement with (3.12), and the prefactor reads

K w N = π N G(2N + 2) 3 N ! 8N +3 . ( 3
.31) ¶ The normalization A w N (0) = 1 again provides a useful check of the formalism.

In particular

K w 1 = 8π, K w 2 = 3 6 π 2 2 4 , K w 3 = 2 9 5 6 π 3 3 12 . (3.32)
When the fermion number N becomes large, the leading decay law of K w N can again be derived by using the asymptotic expansions (C.4) and (C.5). Keeping only terms in N 2 , we obtain

ln K w N ≈ -2N 2 ln N -3 ln 2 + 1 2 . ( 3.33)
The expression (3.30) therefore simplifies to

R w N (t) ∼ 2L 2 e 1/2 π 2 N t N (2N +1)
.

(3.34)

Scaling at large N

In the regime where the fermion number N is large, along the lines of section 2.5, we propose the following scaling Ansatz for the return probability:

R w N (t) ∼ B w N E w F t N (2N +1) . (3.35)
Let us again begin by revisiting the two exactly solvable examples considered above. For the harmonic oscillator (section 3.3), the decay of the return probability is given by (3.27), whereas E w F ≈ 2N ω. This is in agreement with the above Ansatz, with B w = 4, whereas we had B = 2 in the two-sided situation. For the squarewell potential (section 3.4), the decay of the return probability is given by (3.34) at large N , whereas E w F ≈ 2N 2 π 2 /L 2 . This, too, is in agreement with the above Ansatz, with B w = 4e -1/2 , whereas we had B = 2e -1/2 in the two-sided situation.

The relation

B w = 2B (3.36)
between the constants pertaining to the one-sided and two-sided situations, in the sense of section 3.2, is in fact quite general. This can be shown as follows. Assume the Ansatz (2.60) holds in the two-sided situation, in the presence of the symmetrized potential (3.16). Using (2.15), this yields the estimate

ln |C N | ≈ N 2 4 ln B -ln E F (N ) + 3 2 , (3.37)
where we have emphasized the dependence of the Fermi energy on the fermion number N . Consider now the one-sided situation. First, the correspondence (3.17) implies that the Fermi energy reads approximately

E w F (N ) ≈ E F (2N ). (3.38) Second, (3.20) yields ln |C w N | ≈ ln |c (odd) N | ≈ 1 2 ln |C 2N |. (3.39)
The last estimate is obtained by expressing that both sectors equally contribute to the expression (2.21) of C 2N , to leading order for large N . Combining the two above results with (3.12) and (3.37), we readily obtain the aforementioned relation (3.36).

Discussion

We have investigated the quantum return probability of a system of N non-interacting fermions prepared in the ground state of a 1D confining potential and submitted to an instantaneous quench consisting in releasing the trapping potential. Our main finding is that this return probability falls off as a power law in the long-time regime, with a universal exponent which only depends on the fermion number and on the geometry, equal to N 2 when the fermions expand over the full line and N (2N + 1) when the free expansion is constrained to take place over a half-line. These universal exponents are however not robust with respect to interactions. For the Calogero-Sutherland model, an exactly solvable interacting many-particle system on the full line [START_REF] Del Campo | Exact quantum decay of an interacting many-particle system: The Calogero-Sutherland model[END_REF], the decay exponent is known to be N (1 + λ(N -1)), where the coupling constant λ allows to interpolate between free bosons for λ = 0, where the exponent is simply N , and hardcore bosons (equivalent to non-interacting fermions in one dimension) for λ = 1, where the exponent N 2 is recovered. Table 1 presents a comparison of the decay exponent in both geometries with exponents defined similarly in two other situations, namely tight-binding lattice fermions launched from a compact configuration, investigated in our recent work [START_REF] Krapivsky | Quantum return probability of a system of N non-interacting lattice fermions[END_REF], and non-colliding classical random walkers, whose survival and return probabilities are derived in Appendix A. There are both analogies and differences between these three situations. In all cases, the dependence of the exponent on the particle number N is a quadratic polynomial with simple coefficients, and its growth law at large N is twice larger in the half-line geometry than on the full line. The qualitative differences between continuum and lattice fermions can be explained in terms of the symmetries of their respective dispersion relations. In the continuum, the quadratic dispersion relation E = q 2 /2 has a single minimum at q = 0, where the group velocity v = dE/dq = q vanishes. On the one-dimensional lattice, the band structure of a tight-binding particle, E = 2 cos q, possesses two inequivalent stationary points where the group velocity v = -2 sin q vanishes, namely q = 0, as before, and q = π. In the long-time regime of the free expansion phase, both stationary points in momentum space are roughly equally populated [START_REF] Krapivsky | Quantum return probability of a system of N non-interacting lattice fermions[END_REF]. This band-structure effect has two noticeable consequences on the decay exponent: it is roughly twice smaller than its counterpart in the continuum and exhibits a rather unexpected dependence on the parity of N . In the case of non-colliding classical walkers, considered in Appendix A, there are of course no interferences. Lattice and continuum random walks share the same universal continuum limit, namely Brownian motion, and especially the same decay exponents.

The amplitudes of the power-law decay of the quantum return probability of N continuum fermions in both geometries have been shown to depend on the confining potential only through the quantities C N and C w N , expressed in (2.12) and (3.10) as N × N determinants of moments of the one-body bound-state wavefunctions in the potential. These amplitudes have been worked out explicitly for the harmonic and square-well potentials (see (2.37), (2.54) and (3.27), (3.30)). The return probabilities have also been demonstrated to simplify at large fermion numbers, where they obey scaling laws involving the ratio N/(E F t), with E F being the Fermi energy of the initial state (see (2.60), (3.35)).

Finally, the investigations pursued in [START_REF] Krapivsky | Quantum return probability of a system of N non-interacting lattice fermions[END_REF] and in the present work reveal similarities between the dynamics of free fermionic systems and random matrix theory. This resemblance, which has also been put forward recently in a static context [START_REF] Dean | Non-interacting fermions at finite temperature in a d-dimensional trap: Universal correlations[END_REF], is essentially due to the effective repulsion felt both by the eigenvalues of a random matrix Model Exponent (full line) Exponent (half-line)

Continuum fermions N 2 N (2N + 1) Lattice fermions N even N odd 1 2 N 2 1 2 (N 2 + 1) N (N + 1) N 2 + N + 1 Classical walkers 1 4 N (N + 1) 1 2 N (N + 1)
Table 1. Decay exponent of the return probability for a system of N particles moving either on the full line or on a half-line. First row: Non-interacting fermions in the continuum, prepared in the ground state of a confining potential and instantaneously released (body of this work). Second row: Non-interacting tightbinding lattice fermions launched from a compact configuration (Reference [START_REF] Krapivsky | Quantum return probability of a system of N non-interacting lattice fermions[END_REF]).

There, the exponent depends on the parity of the fermion number N . Third row: Non-colliding classical walkers (Appendix A of this work). There, the exponent governs the power-law fall-off of the return probability conditioned on survival.

and by 1D fermions. More specifically, Selberg-Mehta integrals stemming from random matrix theory have been instrumental in deriving most key results. The outcomes often involve the Barnes G-function, which is also ubiquitous in random matrix theory, whereas the scaling in N 2 of the exponents listed in table 1 is reminiscent of the scaling of the free energy of matrix models.

where p(x, y, t) is the transition kernel (probability or probability density) for one single walker. The above determinantal formula holds for several kinds of microscopic realizations of a random walk, either discrete or continuous. The only condition is that no particle can jump over another one. Here are two important examples.

• For Brownian particles, with diffusion coefficient D = 1/2, the transition probability density reads

p(x, y, t) = e -(y-x) 2 /(2t) √ 2πt . (A.2)
• For particles executing continuous-time random walks on the lattice of integers, with jumps to neighboring sites at unit rate, so that again D = 1/2, the transition probability reads p(x, y, t) = e -t I y-x (t), (A.3)

where I y-x is the modified Bessel function. In the continuum limit, i.e., for t large and distances |y -x| not much greater than the diffusive scale √ t, the discrete expression (A.3) becomes the continuous one (A.2).

For a long time and for fixed (i.e., bounded) initial positions x i , and arbitrary final positions y j , using (A.2) allows us to simplify (A.1) to

P (x, y, t) ≈ (2πt) -N/2 det e xiyj /t exp   - 1 2t j y 2 j   . (A.4)
Furthermore, the determinant can be evaluated along the lines of the derivation of (2.11). We have indeed, to leading order in the regime where all the variables x i are small, det (F j (x i )) ≈ det (F j,k-1 )∆ N (x), (A.5) where

F j (x) = k≥0 F j,k x k , F j,k = F (k) j (0) k! , (A.6)
whereas ∆ N (x) is the Vandermonde determinant of the x i (see (2.13)). In the present case, F j (x) = e xyj/t , and so F j,k = (y j /t) k . We are thus left with P (x, y, t) ≈ ∆ N (x)∆ N (y) (2π) N/2 G(N + 1)

t N 2 /2 exp   - 1 2t j y 2 j   . (A.7)
The first quantity of interest is the survival probability S N (x, t), i.e., the probability that no two trajectories of the N walkers have intersected up to time t. The behavior of this quantity in the long-time regime is obtained by integrating (A.7) over the allowed range of final positions (-∞ < y 1 < . . . < y N < +∞). The result is proportional to a Mehta integral of the form (B.3). We thus obtain S N (x, t) ≈ σ N ∆ N (x) G(N + 2) t N (N -1)/4 , (A.8)

where σ N is given by (B.4). The decay exponent N (N -1)/4 of the survival probability can be found in [START_REF] Fisher | Walks, walls, wetting, and melting[END_REF][START_REF] Huse | Commensurate melting, domain walls, and dislocations[END_REF][START_REF] Grabiner | Brownian motion in a Weyl chamber, non-colliding particles, and random matrices[END_REF][START_REF] Katori | Noncolliding Brownian motion and determinantal processes[END_REF]. Reference [START_REF] Grabiner | Brownian motion in a Weyl chamber, non-colliding particles, and random matrices[END_REF] also contains the expression of the prefactor.

for Brownian particles, and p w (x, y, t) = e -t (I y-x (t) -I y+x (t)) (A.16) for particles executing continuous-time random walks on the lattice. In the continuum limit, the discrete expression (A.16) again becomes the continuous one (A.15). For a long time and for fixed (i.e., bounded) initial positions x i , and arbitrary final positions y j , the expression (A.1) simplifies to (A.17)

Furthermore, the determinant can be evaluated along the lines of the derivation of (3.9). In the case where all the functions F j (x) are odd, (A. and x 2 = (x 2 1 , . . . , x 2 N ). In the present situation, we have F j (x) = sinh(xy j /t), and so F j,k = (y j /t) 2k+1 . Using (C.10), we obtain

P w (x, y, t) ≈ 2 N N ! π N G(2N + 2) i (x i y i )∆ N (x 2 )∆ N (y 2 ) t N (2N +1)/2 × exp   - 1 2t j y 2 j   . (A.20)
The survival probability S w N (x, t) is now the probability that no trajectory has either crossed the origin or intersected another one up to time t. The behavior of this quantity in the long-time regime is obtained by integrating (A.20) over the allowed range of final positions (0 < y 1 < . . . < y N < +∞). The result is proportional to a Mehta integral of the form (B.7). We thus obtain

S w N (x, t) ≈ 2 N G(N + 2) i x i ∆ N (x 2 ) π N N !G(2N + 2) t N 2 /2 . (A.21)
The return probability R w N (x, t) is now the probability (or probability density) that the walkers return at (or close to) their initial positions and that no trajectory has either crossed the origin or intersected another one up to time t. This quantity can be directly read off from (A.20):

R w N (x, t) ≈ 2 N N ! π N G(2N + 2) i x 2 i ∆ 2 N (x 2 ) t N (2N +1)/2 . (A.22)
The return probability conditioned on survival scales as R w N (x, t) ≈ i x i ∆ N (x 2 ) G(N + 1) t N (N +1)/2 .

(A.23)

. 9 )

 9 ‡ Throughout this work, unless specified explicitly, determinants are of size N × N , with indices in the range 1 ≤ m, n ≤ N .

  .36) § The normalization A N (0) = 1 for all N provides a useful check of the formalism.

  p are evaluated in Appendix C. Their explicit expressions (C.11) and (C.12) yield

  5) becomes det (F j (x i )) ≈ det (F j,k-1 ) i x i ∆ N (x 2 ), (A.18)withF j (x) = k≥0 F j,k x 2k+1 , F j,k = F

= M 00 M 11 -M 10 M 01 , and so

	with
	c (even) p	= det (M 2k,2l ) 0≤k,l≤p-1 ,	(2.23)
	c (odd) p	= det (M 2k-1,2l-1 ) 1≤k,l≤p .	(2.24)
	Let us illustrate this in the case of two fermions. For an arbitrary potential, we
	have C 2 R 2 (t) ≈	|M 00 M 11 -M 10 M 01 | 4 4π 2 t 4	.	(2.25)

Acknowledgments

We are grateful to Jean-Marie Stéphan for very interesting discussions.

Appendix A. Classical analogue: non-colliding random walkers

A classical analogue of the problem considered in the body of this work is a collection of N independent random walkers on the line, or on the half-line, conditioned to never collide. This system has been studied by various approaches [START_REF] Karlin | Coincidence probabilities Pacific[END_REF][START_REF] Fisher | Walks, walls, wetting, and melting[END_REF][START_REF] Huse | Commensurate melting, domain walls, and dislocations[END_REF][START_REF] Grabiner | Brownian motion in a Weyl chamber, non-colliding particles, and random matrices[END_REF][START_REF] Katori | Noncolliding Brownian motion and determinantal processes[END_REF]. The goal of this appendix is to derive in a self-consistent way many results on the survival and return probabilities, some of which are already known but scattered in the literature. The analogies and the differences between the classical and the quantum situations and between the geometries of the line and of the half-line are briefly summarized in section 4. Hereafter we follow the approach initiated by Karlin and McGregor [START_REF] Karlin | Coincidence probabilities Pacific[END_REF], and pursued by Lindström [START_REF] Lindström | On the vector representation of induced matroids Bull[END_REF] and Gessel and Viennot [START_REF] Gessel | Binomial determinants, paths and hook length formulae[END_REF], yielding to determinantal formulas such as (A.1).

Appendix A.1. Walkers on the line

Consider N independent random walkers on the line starting at time t = 0 from the positions x = (x 1 , . . . , x N ), with x 1 < . . . < x N . At any subsequent time t, the probability (or probability density) P (x, y, t) that the walkers are at the positions y = (y 1 , . . . , y N ), with y 1 < . . . < y N , and that their trajectories have not intersected, is given by the Karlin-McGregor formula [START_REF] Karlin | Coincidence probabilities Pacific[END_REF] P (x, y, t) = det p(x i , y j , t), (A.1)

The second quantity of interest is the return probability R N (x, t), i.e., the probability (or probability density) that the walkers return at (or close to) their initial positions and that no two trajectories of the N walkers have intersected up to time t. This quantity can be directly read off from (A.7):

(A.9)

The classical analogue of the quantum return probability studied in the body of this work is the return probability conditioned on survival, i.e.,

whose decay is predicted to be

The above predictions for both return probabilities seem to be novel.

In the case of continuous-time lattice walks, if the walkers are launched from any N consecutive sites, we have

and so the above expressions read

where σ N is given by (B.4). Finally, when the number of walkers becomes large, the above expressions can be further simplified by means of the expansions (C.4) and (C.5). Keeping only leading terms in N 2 in the exponentials, we obtain

. Walkers on the half-line

Consider now N independent random walkers on the half-line (x > 0), starting at time t = 0 from the positions x = (x 1 , . . . , x N ), with 0 < x 1 < . . . < x N . The probability (or probability density) P w (x, y, t) that the walkers are at the positions y = (y 1 , . . . , y N ) at time t, with 0 < y 1 < . . . < y N , and that their trajectories have neither intersected nor gone through the origin, is again given by (A.1), albeit with

In the case of continuous-time lattice walks, if the walkers are launched from the first N sites of the half-infinite chain (x i = i), we have

The second equality is equivalent to (C.12). The above expressions become

Finally, when the number of walkers becomes large, the above expressions can be further simplified by means of the expansions (C.4) and (C.5). Keeping only leading terms in N 2 in the exponentials, we obtain

Appendix B. Mehta integrals

In this appendix we give the expressions of two of the well-known Mehta integrals [START_REF] Mehta | Random Matrices[END_REF]. These multiple integrals, which can be derived as limiting values of the Selberg integral and play a central part in random matrix theory, are used in the present work at several places. Reference [START_REF] Forrester | The importance of the Selberg integral Bull[END_REF] provides a comprehensive historical overview of the Selberg and related integrals.

• First Mehta integral [22, Eq. (17.6.7)]:

We have in particular

for γ = 1, where G denotes the Barnes G-function (see Appendix C), and

• Second Mehta integral [22, Eq. (17.6.6)]:

We have in particular

for β = 3/2 and γ = 1, and

for β = 1 and γ = 1/2.

Appendix C. The Barnes G-function and related identities

Let us begin with a brief summary of the main properties of the Barnes G-function (see [START_REF] Whittaker | A Course in Modern Analysis[END_REF]). The Euler Γ-function and the Barnes G-function are meromorphic functions of the complex variable z obeying the recursion relations

When z is a positive integer, the Γ-function becomes the usual factorial:

whereas the G-function becomes the 'superfactorial':

The Γ and G-functions have the following asymptotic expansions as z → +∞: The following products can be expressed in terms of values of the G-function:

The above identities allow us to evaluate in closed form the determinants d