scholarly journals Quantum Dark Solitons in the 1D Bose Gas: From Single to Double Dark-Solitons

Universe ◽  
2021 ◽  
Vol 8 (1) ◽  
pp. 2
Author(s):  
Kayo Kinjo ◽  
Eriko Kaminishi ◽  
Takashi Mori ◽  
Jun Sato ◽  
Rina Kanamoto ◽  
...  

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.

Author(s):  
Phan Thành Nam ◽  
Marcin Napiórkowski

AbstractWe consider the homogeneous Bose gas on a unit torus in the mean-field regime when the interaction strength is proportional to the inverse of the particle number. In the limit when the number of particles becomes large, we derive a two-term expansion of the one-body density matrix of the ground state. The proof is based on a cubic correction to Bogoliubov’s approximation of the ground state energy and the ground state.


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Pierre Roux ◽  
Delphine Salort

<p style='text-indent:20px;'>The Nonlinear Noisy Leaky Integrate and Fire (NNLIF) model is widely used to describe the dynamics of neural networks after a diffusive approximation of the mean-field limit of a stochastic differential equation. In previous works, many qualitative results were obtained: global existence in the inhibitory case, finite-time blow-up in the excitatory case, convergence towards stationary states in the weak connectivity regime. In this article, we refine some of these results in order to foster the understanding of the model. We prove with deterministic tools that blow-up is systematic in highly connected excitatory networks. Then, we show that a relatively weak control on the firing rate suffices to obtain global-in-time existence of classical solutions.</p>


2004 ◽  
Vol 18 (09) ◽  
pp. 1339-1349 ◽  
Author(s):  
YAN XU ◽  
DUO-JE JIA ◽  
ZHAO-YANG CHEN ◽  
YUAN GAO ◽  
FA-SHEN LI

The deviation effect of spinor mode from the single-mode for a spin-1 Bose gas of trapped atoms is studied beyond the mean field theory. Based on the effective Hamiltonian with nondegenerated level of the collective spin states, the splitting level of the system energy due to the deviation effect has been calculated. For the large condensates of 87 Rb and 23 Na with atom number N>105, the splitting fraction of the energy, arising from the magnetization exhibited by the trapped Bose gas, is found to have a typical order of (10-4~10-8), decreasing as N-2 for 87 Rb and increasing as -N-2 for 23 Na , respectively.


1969 ◽  
Vol 188 (1) ◽  
pp. 522-525 ◽  
Author(s):  
M. Schick ◽  
P. R. Zilsel

2002 ◽  
Vol 293 (5-6) ◽  
pp. 287-292 ◽  
Author(s):  
A.Yu. Cherny ◽  
A.A. Shanenko

2005 ◽  
Vol 31 (8) ◽  
pp. S1367-S1375
Author(s):  
M Bender ◽  
P-H Heenen
Keyword(s):  

2020 ◽  
Vol 2 (2) ◽  
Author(s):  
Michał Kowalski ◽  
Rafał Ołdziejewski ◽  
Kazimierz Rzążewski

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