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Author: Grafiati

Published: 11 March 2023

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1

Boccato, Chiara, Christian Brennecke, Serena Cenatiempo, and Benjamin Schlein. "Complete Bose–Einstein Condensation in the Gross–Pitaevskii Regime." Communications in Mathematical Physics 359, no. 3 (November 9, 2017): 975–1026. http://dx.doi.org/10.1007/s00220-017-3016-5.

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2

Basti, Giulia, Serena Cenatiempo, Alessandro Olgiati, Giulio Pasqualetti, and Benjamin Schlein. "Ground state energy of a Bose gas in the Gross–Pitaevskii regime." Journal of Mathematical Physics 63, no. 4 (April 1, 2022): 041101. http://dx.doi.org/10.1063/5.0087116.

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We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere.

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3

Cenatiempo, Serena. "Bogoliubov theory for dilute Bose gases: The Gross-Pitaevskii regime." Journal of Mathematical Physics 60, no. 8 (August 2019): 081901. http://dx.doi.org/10.1063/1.5096288.

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4

Béthuel, Fabrice, Raphaël Danchin, and Didier Smets. "On the linear wave regime of the Gross-Pitaevskii equation." Journal d'Analyse Mathématique 110, no. 1 (January 2010): 297–338. http://dx.doi.org/10.1007/s11854-010-0008-1.

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5

Michelangeli, Alessandro, Phan Thành Nam, and Alessandro Olgiati. "Ground state energy of mixture of Bose gases." Reviews in Mathematical Physics 31, no. 02 (February 27, 2019): 1950005. http://dx.doi.org/10.1142/s0129055x19500053.

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We consider the asymptotic behavior of a system of multi-component trapped bosons, when the total particle number [Formula: see text] becomes large. In the dilute regime, when the interaction potentials have the length scale of order [Formula: see text], we show that the leading order of the ground state energy is captured correctly by the Gross–Pitaevskii energy functional and that the many-body ground state fully condensates on the Gross–Pitaevskii minimizers. In the mean-field regime, when the interaction length scale is [Formula: see text], we are able to verify Bogoliubov’s approximation and obtain the second order expansion of the ground state energy. While such asymptotic results have several precursors in the literature on one-component condensates, the adaptation to the multi-component setting is non-trivial in various respects and the analysis will be presented in detail.

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6

Zloshchastiev, Konstantin G. "Sound Propagation in Cigar-Shaped Bose Liquids in the Thomas-Fermi Approximation: A Comparative Study between Gross-Pitaevskii and Logarithmic Models." Fluids 7, no. 11 (November 19, 2022): 358. http://dx.doi.org/10.3390/fluids7110358.

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A comparative study is conducted of the propagation of sound pulses in elongated Bose liquids and Bose-Einstein condensates in Gross-Pitaevskii and logarithmic models, by means of the Thomas-Fermi approximation. It is demonstrated that in the linear regime the propagation of small density fluctuations is essentially one-dimensional in both models, in the direction perpendicular to the cross section of a liquid’s lump. Under these approximations, it is demonstrated that the speed of sound scales as a square root of particle density in the case of the Gross-Pitaevskii liquid/condensate, but it is constant in a case of the homogeneous logarithmic liquid.

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7

Boccato, Chiara, Christian Brennecke, Serena Cenatiempo, and Benjamin Schlein. "Optimal Rate for Bose–Einstein Condensation in the Gross–Pitaevskii Regime." Communications in Mathematical Physics 376, no. 2 (September 13, 2019): 1311–95. http://dx.doi.org/10.1007/s00220-019-03555-9.

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8

Ma, Li, and Jing Wang. "Sharp Threshold of the Gross-Pitaevskii Equation with Trapped Dipolar Quantum Gases." Canadian Mathematical Bulletin 56, no. 2 (June 1, 2013): 378–87. http://dx.doi.org/10.4153/cmb-2011-181-2.

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AbstractIn this paper, we consider the Gross-Pitaevskii equation for the trapped dipolar quantum gases. We obtain the sharp criterion for the global existence and finite time blow-up in the unstable regime by constructing a variational problem and the so-called invariant manifold of the evolution flow.

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9

Brennecke, Christian. "The low energy spectrum of trapped bosons in the Gross–Pitaevskii regime." Journal of Mathematical Physics 63, no. 5 (May 1, 2022): 051101. http://dx.doi.org/10.1063/5.0089630.

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Bogoliubov theory {N. N. Bogoliubov, Izv. Akad. Nauk Ser. Fiz. 11, 77 (1947) [J. Phys. (USSR) 11, 23 (1947) (in English)]} provides important predictions for the low energy properties of the weakly interacting Bose gas. Recently, Bogoliubov’s predictions were justified rigorously by Boccato et al. [Acta Math. 222(2), 219–335 (2019)] for translation invariant systems in the Gross–Pitaveskii regime, where N bosons in [Formula: see text] interact through a potential whose scattering length is of size N−1. In this article, we review recent results from the work of Brennecke et al. [Ann. Henri Poincaré 23, 1583–1658 (2022)], a joint work with Schlein and Schraven, which extends the analysis for translation invariant systems to systems of bosons in [Formula: see text] that are trapped by an external potential.

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10

Nam, Phan Thành, Marcin Napiórkowski, Julien Ricaud, and Arnaud Triay. "Optimal rate of condensation for trapped bosons in the Gross–Pitaevskii regime." Analysis & PDE 15, no. 6 (November 10, 2022): 1585–616. http://dx.doi.org/10.2140/apde.2022.15.1585.

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11

Bellazzini, Jacopo, and David Ruiz. "Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime." American Journal of Mathematics 145, no. 1 (February 2023): 109–49. http://dx.doi.org/10.1353/ajm.2023.0002.

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12

Basti, Giulia. "A second order upper bound on the ground state energy of a Bose gas beyond the Gross–Pitaevskii regime." Journal of Mathematical Physics 63, no. 7 (July 1, 2022): 071902. http://dx.doi.org/10.1063/5.0089790.

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We consider a system of N bosons in a unitary box in the grand-canonical setting interacting through a potential with the scattering length scaling as N−1+ κ, κ ∈ (0, 2/3). This regimes interpolate between the Gross–Pitaevskii regime ( κ = 0) and the thermodynamic limit ( κ = 2/3). In the work of Basti et al. [Forum Math., Sigma 9, E74 (2021)], as an intermediate step to prove an upper bound in agreement with the Lee–Huang–Yang formula in the thermodynamic limit, a second order upper bound on the ground state energy for κ < 5/9 was obtained. In this paper, thanks to a more careful analysis of the error terms, we extend the mentioned result to κ < 7/12.

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13

Gil-Londoño, J., G. Marı́n-Alvarado, and K. Rodrı́guez-Ramı́rez. "Numerical implementation of a Mach-Zehnder interferometer for Bose-Einstein condensates." Suplemento de la Revista Mexicana de Física 1, no. 3 (August 22, 2020): 31–35. http://dx.doi.org/10.31349/suplrevmexfis.1.3.31.

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We numerically implement a Mach-Zehnder interferometer, where the coherence and oscillatory properties of Bose-Einstein condensates are explored and the system is modeled by the Gross-Pitaevskii equation. Several time-dependent external trapping potentials were engineered seeking the adiabatic regime which is quantified using fidelity measurements with respect to the actual ground-state of the trap. The dynamics of both conjugate variables, namely density and phase of the matter-wave function, are shown. Moreover, the density and fidelity profiles of the system are presented when the phase-shifter is switching-on and -off, being found in the presented profiles that the system exhibits three different regimes during the recombination stage among them even an orthogonal BEC to the original one is obtained. We achieve the numerical solution through an adequate implementation of the finite-difference method for the spatial discretization and a Runge-Kutta method for the time evolution.

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14

Herr, Sebastian, and Vedran Sohinger. "Unconditional uniqueness results for the nonlinear Schrödinger equation." Communications in Contemporary Mathematics 21, no. 07 (October 10, 2019): 1850058. http://dx.doi.org/10.1142/s021919971850058x.

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We study the problem of unconditional uniqueness of solutions to the cubic nonlinear Schrödinger equation (NLS). We introduce a new strategy to approach this problem on bounded domains, in particular on rectangular tori. It is a known fact that solutions to the cubic NLS give rise to solutions of the Gross–Pitaevskii (GP) hierarchy, which is an infinite-dimensional system of linear equations. By using the uniqueness analysis of the GP hierarchy, we obtain new unconditional uniqueness results for the cubic NLS on rectangular tori, which cover the full scaling-subcritical regime in high dimensions. In fact, we prove a more general result which is conditional on the domain. In addition, we observe that well-posedness of the cubic NLS in Fourier–Lebesgue spaces implies unconditional uniqueness.

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15

STEVENSON, P. M. "HYDRODYNAMICS OF THE VACUUM." International Journal of Modern Physics A 21, no. 13n14 (June 10, 2006): 2877–903. http://dx.doi.org/10.1142/s0217751x06028527.

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Hydrodynamics is the appropriate "effective theory" for describing any fluid medium at sufficiently long length scales. This paper treats the vacuum as such a medium and derives the corresponding hydrodynamic equations. Unlike a normal medium the vacuum has no linear sound-wave regime; disturbances always "propagate" nonlinearly. For an "empty vacuum" the hydrodynamic equations are familiar ones (shallow water-wave equations) and they describe an experimentally observed phenomenon — the spreading of a clump of zero-temperature atoms into empty space. The "Higgs vacuum" case is much stranger; pressure and energy density, and hence time and space, exchange roles. The speed of sound is formally infinite, rather than zero as in the empty vacuum. Higher-derivative corrections to the vacuum hydrodynamic equations are also considered. In the empty-vacuum case the corrections are of quantum origin and the post-hydrodynamic description corresponds to the Gross–Pitaevskii equation. We conjecture the form of the post-hydrodynamic corrections in the Higgs case. In the (1+1)-dimensional case the equations possess remarkable "soliton" solutions and appear to constitute a new exactly integrable system.

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16

Yan, D., P. G. Kevrekidis, and D. J. Frantzeskakis. "Dark solitons in a Gross–Pitaevskii equation with a power-law nonlinearity: application to ultracold Fermi gases near the Bose–Einstein condensation regime." Journal of Physics A: Mathematical and Theoretical 44, no. 41 (September 20, 2011): 415202. http://dx.doi.org/10.1088/1751-8113/44/41/415202.

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17

Li, Ye. "An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems." Communications in Computational Physics 19, no. 2 (February 2016): 442–72. http://dx.doi.org/10.4208/cicp.021114.140715a.

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AbstractIn this paper, we propose an uniformly convergent adaptive finite element method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation (BEC) and quantum chemistry. We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations for the problem are reviewed to confirm the asymptotic behaviors of the solutions in the boundary/interior layer regions. By using the normalized gradient flow, we propose an adaptive finite element with hybrid basis to solve the singularly perturbed nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively to the small parameter ε. Extensive numerical results are reported to show the uniform convergence property of our method. We also apply the AFEM-HB to compute the ground and excited states of BEC with box/harmonic/optical lattice potential in the semiclassical regime (0 <ε≪C 1). In addition, we give a detailed error analysis of our AFEM-HB to a simpler singularly perturbed two point boundary value problem, show that our method has a minimum uniform convergence order

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18

Деменев, А. А., Н. А. Гиппиус, and В. Д. Кулаковский. "Динамика спинорной экситон-поляритонной системы в латерально сжатых GaAs микрорезонаторах при резонансном фотовозбуждении." Физика твердого тела 60, no. 8 (2018): 1567. http://dx.doi.org/10.21883/ftt.2018.08.46245.06gr.

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AbstractThe evolution of the spatial coherence and the polarization has been studied in a freely decaying polariton condensate that is resonantly excited by linearly polarized picosecond laser pulses at the lower and upper sublevels of the lower polariton branch in a high-Q GaAs-based microcavity with a reduced lateral symmetry without excitation of the exciton reservoir. It is found that the condensate inherits the coherence of the exciting laser pulse at both sublevels in a wide range of excitation densities and retains it for several dozen picoseconds. The linear polarization of the photoexcited condensate is retained only in the condensate at the lower sublevel. The linearly polarized condensate excited at the upper sublevel loses its stability at the excitation densities higher a threshold value: it enters a regime of internal Josephson oscillations with strongly oscillating circular and diagonal linear degrees of polarization. The polariton–polariton interaction leads to the nonlinear Josephson effects at high condensate densities. All the effects are well described in terms of the spinor Gross–Pitaevskii equations. The cause of the polarization instability of the condensate is shown to be the spin anisotropy of the polariton–polariton interaction.

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19

KASAMATSU, KENICHI, MAKOTO TSUBOTA, and MASAHITO UEDA. "VORTICES IN MULTICOMPONENT BOSE–EINSTEIN CONDENSATES." International Journal of Modern Physics B 19, no. 11 (April 30, 2005): 1835–904. http://dx.doi.org/10.1142/s0217979205029602.

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We review the topic of quantized vortices in multicomponent Bose–Einstein condensates of dilute atomic gases, with an emphasis on the two-component condensates. First, we review the fundamental structure, stability and dynamics of a single vortex state in a slowly rotating two-component condensates. To understand recent experimental results, we use the coupled Gross–Pitaevskii equations and the generalized nonlinear sigma model. An axisymmetric vortex state, which was observed by the JILA group, can be regarded as a topologically trivial skyrmion in the pseudospin representation. The internal, coherent coupling between the two components breaks the axisymmetry of the vortex state, resulting in a stable vortex molecule (a meron pair). We also mention unconventional vortex states and monopole excitations in a spin-1 Bose–Einstein condensate. Next, we discuss a rich variety of vortex states realized in rapidly rotating two-component Bose–Einstein condensates. We introduce a phase diagram with axes of rotation frequency and the intercomponent coupling strength. This phase diagram reveals unconventional vortex states such as a square lattice, a double-core lattice, vortex stripes and vortex sheets, all of which are in an experimentally accessible parameter regime. The coherent coupling leads to an effective attractive interaction between two components, providing not only a promising candidate to tune the intercomponent interaction to study the rich vortex phases but also a new regime to explore vortex states consisting of vortex molecules characterized by anisotropic vorticity. A recent experiment by the JILA group vindicated the formation of a square vortex lattice in this system.

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20

Schobesberger, Sonja O., Tanja Rindler-Daller, and Paul R. Shapiro. "Angular momentum and the absence of vortices in the cores of fuzzy dark matter haloes." Monthly Notices of the Royal Astronomical Society 505, no. 1 (May 11, 2021): 802–29. http://dx.doi.org/10.1093/mnras/stab1153.

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ABSTRACT Scalar field dark matter (SFDM), comprised of ultralight (≳ 10−22 eV) bosons, is distinguished from massive (≳GeV), collisionless cold dark matter (CDM) by its novel structure-formation dynamics as Bose–Einstein condensate (BEC) and quantum superfluid with wave-like properties, described by the Gross-Pitaevskii and Poisson (GPP) equations. In the free-field (‘fuzzy’) limit of SFDM (FDM), structure is inhibited below the de Broglie wavelength λdeB, but resembles CDM on larger scales. Virialized haloes have ‘solitonic’ cores of radius ∼λdeB that follow the ground-state attractor solution of GPP, surrounded by CDM-like envelopes. As superfluid, SFDM is irrotational (vorticity-free) but can be unstable to vortex formation. We previously showed this can happen in halo cores, from angular momentum arising during structure formation, when repulsive self-interaction (SI) is present to support them out to a second length scale λSI with λSI > λdeB (the Thomas–Fermi regime), but only if SI is strong enough. This suggested FDM cores ($ {\rm without}$ SI) would not form vortices. FDM simulations later found vortices, but only outside halo cores, consistent with our previous suggestion based upon TF-regime analysis. We extend that analysis now to FDM, to show explicitly that vortices should not arise in solitonic cores from angular momentum, modelling them as either Gaussian spheres, or ( n = 2)-polytropic, irrotational Riemann-S ellipsoids. We find that, for typical halo spin parameters, angular momentum per particle is below ℏ, the minimum required even for one singly-quantized vortex in the centre. Even for higher angular momentum, however, vortex formation is not energetically favoured.

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21

Kinjo, Kayo, Eriko Kaminishi, Takashi Mori, Jun Sato, Rina Kanamoto, and Tetsuo Deguchi. "Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons." Universe 8, no. 1 (December 21, 2021): 2. http://dx.doi.org/10.3390/universe8010002.

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We study quantum double dark-solitons, which give pairs of notches in the density profiles, by constructing corresponding quantum states in the Lieb–Liniger model for the one-dimensional Bose gas. Here, we expect that the Gross–Pitaevskii (GP) equation should play a central role in the long distance mean-field behavior of the 1D Bose gas. We first introduce novel quantum states of a single dark soliton with a nonzero winding number. We show them by exactly evaluating not only the density profile but also the profiles of the square amplitude and phase of the matrix element of the field operator between the N-particle and (N−1)-particle states. For elliptic double dark-solitons, the density and phase profiles of the corresponding states almost perfectly agree with those of the classical solutions, respectively, in the weak coupling regime. We then show that the scheme of the mean-field product state is quite effective for the quantum states of double dark solitons. Assigning the ideal Gaussian weights to a sum of the excited states with two particle-hole excitations, we obtain double dark-solitons of distinct narrow notches with different depths. We suggest that the mean-field product state should be well approximated by the ideal Gaussian weighted sum of the low excited states with a pair of particle-hole excitations. The results of double dark-solitons should be fundamental and useful for constructing quantum multiple dark-solitons.

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22

Serhan, M. "Bose-Einstein Condensation of Confined Atomic Gases at Ultra Low Temperatures." Applied Physics Research 9, no. 5 (September 23, 2017): 96. http://dx.doi.org/10.5539/apr.v9n5p96.

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In this work I solve the Gross-Pitaevskii equation describing an atomic gas confined in an isotropic harmonic trap by introducing a variational wavefunction of Gaussian type. The chemical potential of the system is calculated and the solutions are discussed in the weakly and strongly interacting regimes. For the attractive system with negative scattering length the maximum number of atoms that can be put in the condensate without collapse begins is calculated.

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23

Richberg, Roham, and Andrew Martin. "The Influence of s-Wave Interactions on Focussing of Atoms." Atoms 9, no. 3 (June 25, 2021): 37. http://dx.doi.org/10.3390/atoms9030037.

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The focusing of a rubidium Bose–Einstein condensate via an optical lattice potential is numerically investigated. The results are compared with a classical trajectory model which underestimates the width of the focused beam. Via the inclusion of the effects of interactions into the classical trajectories model, we show that it is possible to obtain reliable estimates for the width of the focused beam when compared to numerical integration of the Gross–Pitaevskii equation. Finally, we investigate the optimal regimes for focusing and find that for a strongly interacting Bose–Einstein condensate focusing of order 20 nm may be possible.

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24

Zhang, Jian Wei, Hai Jun Chen, Sheng Jun Wang, and Yuan Ren. "Variational Solution of Steady-Structure in Exciton-Polariton Condensates with a Modified Lagrangian Approach." Key Engineering Materials 787 (November 2018): 113–22. http://dx.doi.org/10.4028/www.scientific.net/kem.787.113.

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Exciton-polariton condensate is a new kind of system exhibiting spontaneous coherence, which is a new quantum dissipation system. Numerical simulation and analytical methods can be used to study the static and dynamical properties of exciton-polariton condensate. In this paper, A modified Lagrangian method is developed for exciton-polariton system to find the steady-state structure and regimes among the parameters of the system, and two new forms of trial wave function are proposed. The modified Lagrangian method is successfully applied to the exciton-polariton system described by the open-dissipative Gross-Pitaevskii equation for the first time. Furthermore, static version of the modified Lagrangian method provides stationary shape of the steady-state structure, while the time-dependent version can be used to study small amplitude oscillations around stationary states. On the one hand, comparison of the profiles for steady-state structure, predicted by the modified Lagrangian and those found from numerical solution of the open-dissipative Gross-Pitaevskii(dGP) equation shows good agreement, thereby proving the accuracy of the trial wave function and validating the proposed approach. Particularly, this new method promotes the deeper cognition and understanding for the dissipative exciton-polariton system and is helpful to explore the mechanism of the gain and dissipation effect on the steady-state structure of the system.

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25

Adhikari, Arka, Christian Brennecke, and Benjamin Schlein. "Bose–Einstein Condensation Beyond the Gross–Pitaevskii Regime." Annales Henri Poincaré, December 26, 2020. http://dx.doi.org/10.1007/s00023-020-01004-1.

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AbstractWe consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order $$N^{-1+\kappa }$$ N - 1 + κ , for $$\kappa >0$$ κ > 0 . Assuming that $$\kappa \in (0;1/43)$$ κ ∈ ( 0 ; 1 / 43 ) , we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.

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26

Brennecke, Christian, Marco Caporaletti, and Benjamin Schlein. "Excitation Spectrum of Bose Gases beyond the Gross–Pitaevskii regime." Reviews in Mathematical Physics, June 4, 2022. http://dx.doi.org/10.1142/s0129055x22500271.

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27

Brennecke, Christian, Benjamin Schlein, and Severin Schraven. "Bogoliubov Theory for Trapped Bosons in the Gross–Pitaevskii Regime." Annales Henri Poincaré, February 25, 2022. http://dx.doi.org/10.1007/s00023-021-01151-z.

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28

Lieb, Elliott H., Robert Seiringer, and Jakob Yngvason. "Yrast line of a rapidly rotating Bose gas: Gross-Pitaevskii regime." Physical Review A 79, no. 6 (June 24, 2009). http://dx.doi.org/10.1103/physreva.79.063626.

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29

Caraci, Cristina, Serena Cenatiempo, and Benjamin Schlein. "Bose–Einstein Condensation for Two Dimensional Bosons in the Gross–Pitaevskii Regime." Journal of Statistical Physics 183, no. 3 (May 20, 2021). http://dx.doi.org/10.1007/s10955-021-02766-6.

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AbstractWe consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose–Einstein condensation, with almost optimal bounds on the number of orthogonal excitations.

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30

Boccato, Chiara. "The excitation spectrum of the Bose gas in the Gross–Pitaevskii regime." Reviews in Mathematical Physics, April 9, 2020, 2060006. http://dx.doi.org/10.1142/s0129055x20600065.

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We consider a gas of interacting bosons trapped in a box of side length one in the Gross–Pitaevskii limit. We review the proof of the validity of Bogoliubov’s prediction for the ground state energy and the low-energy excitation spectrum. This note is based on joint work with C. Brennecke, S. Cenatiempo and B. Schlein.

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31

Hainzl, Christian, Benjamin Schlein, and Arnaud Triay. "Bogoliubov theory in the Gross-Pitaevskii limit: a simplified approach." Forum of Mathematics, Sigma 10 (2022). http://dx.doi.org/10.1017/fms.2022.78.

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Abstract We show that Bogoliubov theory correctly predicts the low-energy spectral properties of Bose gases in the Gross-Pitaevskii regime. We recover recent results from [6, 7]. While our main strategy is similar to the one developed in [6, 7], we combine it with new ideas, taken in part from [15, 25]; this makes our proof substantially simpler and shorter. As an important step towards the proof of Bogoliubov theory, we show that low-energy states exhibit complete Bose-Einstein condensation with optimal control over the number of orthogonal excitations.

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32

Enciso, Alberto, and Daniel Peralta-Salas. "Approximation Theorems for the Schrödinger Equation and Quantum Vortex Reconnection." Communications in Mathematical Physics, July 28, 2021. http://dx.doi.org/10.1007/s00220-021-04177-w.

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AbstractWe prove the existence of smooth solutions to the Gross–Pitaevskii equation on $$\mathbb {R}^3$$ R 3 that feature arbitrarily complex quantum vortex reconnections. We can track the evolution of the vortices during the whole process. This permits to describe the reconnection events in detail and verify that this scenario exhibits the properties observed in experiments and numerics, such as the $$t^{1/2}$$ t 1 / 2 and change of parity laws. We are mostly interested in solutions tending to 1 at infinity, which have finite Ginzburg–Landau energy and physically correspond to the presence of a background chemical potential, but we also consider the cases of Schwartz initial data and of the Gross–Pitaevskii equation on the torus. In the proof, the Gross–Pitaevskii equation operates in a nearly linear regime, so the result applies to a wide range of nonlinear Schrödinger equations. Indeed, an essential ingredient in the proofs is the development of novel global approximation theorems for the Schrödinger equation on $$\mathbb {R}^n$$ R n . Specifically, we prove a qualitative approximation result that applies for solutions defined on very general spacetime sets and also a quantitative result for solutions on product sets in spacetime $$D\times \mathbb {R}$$ D × R . This hinges on frequency-dependent estimates for the Helmholtz–Yukawa equation that are of independent interest.

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33

Brennecke, Christian, Benjamin Schlein, and Severin Schraven. "Bose–Einstein Condensation with Optimal Rate for Trapped Bosons in the Gross–Pitaevskii Regime." Mathematical Physics, Analysis and Geometry 25, no. 2 (April 12, 2022). http://dx.doi.org/10.1007/s11040-022-09424-7.

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34

徐红萍. "Interaction-modulated tunneling dynamics of in a mixture of Bose-Fermi superfluid." Acta Physica Sinica, 2022, 0. http://dx.doi.org/10.7498/aps.71.20212168.

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In this paper, we study the interaction-modulated tunneling dynamics of a Bose-Fermi superfluid mixture, where a BEC with weak repulsive interaction are confined in a symmetric deep double-well potential and an equally populated two-component Fermi gas in a harmonic potential symmetrically positioned at the center of the double-well potential. The tunneling between the two wells is modulated by fermions trapped in a harmonic potential. When the temperature is adequately low and the bosonic particle number is adequately large, we can employ the mean-field theories to describe the evolution of the BEC in the double-well potential with the time-dependent Gross–Pitaevskii equation. For the Fermi gas in the harmonic potential trap, we consider the case where the inter-fermion interaction is tuned on the deep Bose-Einstein condensate of the inter-fermion Feshbach resonance, where two fermions of spin-up and spin-down form a two-body bound state. Within the regime, the Fermi gas is well described by a condensate of these fermionic dimers, hence can be simulated as well by a Gross–Pitaevskii equation of dimers. The inter-species interactions couple the dynamics of the two species, which result in interesting features in the tunneling oscillations. The dynamic equations of the BEC in the double-well potential is described by a two-mode approximation. Coupling it with time-dependent Gross-Pitaevskii equation of the harmonically potential trapped molecular BEC, we numerically investigate the dynamical evolution of the Boson-Fermi hybrid system under different initial conditions. It is found that the interaction of fermions in a harmonic potential leads to strong non-linearity in the oscillations of the bosons in the double-well potential and enriches the tunneling dynamics of the bosons. Especially, it strengthens macroscopic quantum self-trapping. And the macroscopic quantum self-trapping can be manifested in three forms: the phase tends to be negative and monotonously decrease with time, the phase evolves bounded with time, and the phase tends to be positive and monotonously increase with time. This means that it is possible the tunneling dynamics of the BEC in double-well potential is adjustable. Our results can be verified experimentally in Bose–Fermi superfluid mixture by varying different interactions parameters via Feshbach resonance and confinement-induced resonance.

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35

Basti, Giulia, Serena Cenatiempo, Alessandro Olgiati, Giulio Pasqualetti, and Benjamin Schlein. "A Second Order Upper Bound for the Ground State Energy of a Hard-Sphere Gas in the Gross–Pitaevskii Regime." Communications in Mathematical Physics, December 5, 2022. http://dx.doi.org/10.1007/s00220-022-04547-y.

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AbstractWe prove an upper bound for the ground state energy of a Bose gas consisting of N hard spheres with radius $$\mathfrak {a}/N$$ a / N , moving in the three-dimensional unit torus $$\Lambda $$ Λ . Our estimate captures the correct asymptotics of the ground state energy, up to errors that vanish in the limit $$N \rightarrow \infty $$ N → ∞ . The proof is based on the construction of an appropriate trial state, given by the product of a Jastrow factor (describing two-particle correlations on short scales) and of a wave function constructed through a (generalized) Bogoliubov transformation, generating orthogonal excitations of the Bose–Einstein condensate and describing correlations on large scales.

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36

Zin, Paweł, Maciej Pylak, Zbigniew Idziaszek, and Mariusz Franciszek Gajda. "Self-consistent Description of Bose-Bose Droplets: Modified Gapless Hartree-Fock-Bogoliubov Method." New Journal of Physics, November 9, 2022. http://dx.doi.org/10.1088/1367-2630/aca175.

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Abstract We define a formalism of a self-consistent description of the ground state of a weakly interacting Bose system, accounting for higher order terms in expansion of energy in the diluteness parameter. The approach is designed to be applied to a Bose-Bose mixture in a regime of weak collapse where quantum fluctuations lead to stabilization of the system and formation of quantum liquid droplets. The approach is based on the Generalized Gross -- Pitaevskii equation accounting for quantum depletion and renormalized anomalous density terms. The equation is self-consistently coupled to modified Bogoliubov equations. We derive well defined procedure to calculate the zero temperature renormalized anomalous density - the quantity needed to correctly describe the formation of quantum liquid droplet. We pay particular attention to the case of droplets harmonically confined in some directions. The method allows to determine the Lee-Huang-Yang-type contribution to the chemical potential of inhomogeneous droplets when the local density approximation fails.

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37

Kati, Yagmur, Xiaoquan Yu, and Sergej Flach. "Density resolved wave packet spreading in disordered Gross-Pitaevskii lattices." SciPost Physics Core 3, no. 2 (October 8, 2020). http://dx.doi.org/10.21468/scipostphyscore.3.2.006.

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We perform novel energy and norm density resolved wave packet spreading studies in the disordered Gross-Pitaevskii (GP) lattice to confine energy density fluctuations. We map the locations of GP regimes of weak and strong chaos subdiffusive spreading in the 2D density control parameter space and observe strong chaos spreading over several decades. We obtain a renormalization of the ground state due to disorder, which allows for a new disorder-induced phase of disconnected insulating puddles of matter due to Lifshits tails. Inside this Lifshits phase, the wave packet spreading is substantially slowed down.

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