Transition of a mesoscopic bosonic gas into a Bose-Einstein condensate

The condensate number distribution during the transition of a dilute, weakly interacting gas of N=200 bosonic atoms into a Bose-Einstein condensate is modeled within number conserving master equation theory of Bose-Einstein condensation. Initial strong quantum fluctuations occuring during the exponential cycle of condensate growth reduce in a subsequent saturation stage, before the Bose gas finally relaxes towards the Gibbs-Boltzmann equilibrium.Comment: 5 pages, 3 figure

Monte Carlo Simulation for the Frequency Comb Spectrum of an Atom Laser

A theoretical particle-number conserving quantum field theory based on the concept of imaginary time is presented and applied to the scenario of a coherent atomic laser field at ultra-cold temperatures. The proposed theoretical model describes the analytical derivation of the frequency comb spectrum for an atomic laser realized from modeling a coherent atomic beam of condensate and non-condensate quantum field components released from a trapped Bose–Einstein condensate at a given repetition phase and frequency. The condensate part of the atomic vapor is assumed to be subjected to thermal noise induced by the temperature of the surrounding thermal atomic cloud. This new quantum approach uses time periodicity and an orthogonal decomposition of the quantum field in a complex-valued quantum field representation to derive and model the quantum field's forward- and backward-propagating components as a standing wave field in the same unique time and temperature domain without quantitative singularities at finite temperatures. The complex-valued atom laser field, the resulting frequency comb, and the repetition frequency distribution with the varying shape of envelopes are numerically monitored within a Monte Carlo sampling method, as a function of temperature and trap frequency of the external confinement.Quanta 2023; 12: 171–179

Josephson Oscillations of Two Weakly Coupled Bose-Einstein Condensates

A numerical experiment based on a particle number-conserving quantum field theory is performed for two initially independent Bose–Einstein condensates that are coherently coupled at two temperatures. The present model illustrates ab initio that the initial phase of each of the two condensates does not remain random at the Boltzmann equilibrium, but is distributed around integer multiple values of 2π from the interference and thermalization of forward and backward propagating matter waves. The thermalization inside the atomic vapors can be understood as an intrinsic measurement process that defines a temperature for the two condensates and projects the quantum states to an average wave field with zero (relative) phases. Following this approach, focus is put on the original thought experiment of Anderson on whether a Josephson current between two initially separated Bose–Einstein condensates occurs in a deterministic way or not, depending on the initial phase distribution.Quanta 2024; 13: 28–37

Two-dimensional representation of time for a quantum particle in a Bose-Einstein condensate

A quantitative quantum field approach for interacting particles using a complex two dimensional representation of time is presented. The representation of a two-dimensional complex valued time arises from the constraint of particle number conservation and can be used to account simultaneously for both oscillations of the quantum state of a particle in a Bose-Einstein condensate as well as coherence times of the particle in the atomic cloud below the critical temperature. It is illustrated that, in contrast to so far established theories of purely imaginary complex time, two dimensional complex time has a preferred direction in positive direction of the imaginary axis below the critical temperature in agreement with the observation of a spontaneously broken phase gauge symmetry of the underlying fugacity spectrum. The results reduce to the standard scheme of purely imaginary time above the critical temperature.Comment: 5 pages, 4 figure

Spontaneously broken gauge symmetry in a Bose gas with constant particle number

The interplay between spontaneously broken gauge symmetries and Bose-Einstein condensation has long been controversially discussed in science, since the equation of motions are invariant under phase transformations. Within the present model it is illustrated that spontaneous symmetry breaking appears as a non-local process in position space, but within disjoint subspaces of the underlying Hilbert space. Numerical simulations show that it is the symmetry of the relative phase distribution between condensate and non-condensate quantum fields which is spontaneously broken when passing the critical temperature for Bose-Einstein condensation. Since the total number of gas particles remains constant over time, the global U(1)-gauge symmetry of the system is preserved.Comment: 4 pages, 2 figures, final versio

Number-conserving master equation theory for a dilute Bose-Einstein condensate

We describe the transition of $N$ weakly interacting atoms into a Bose-Einstein condensate within a number-conserving quantum master equation theory. Based on the separation of time scales for condensate formation and non-condensate thermalization, we derive a master equation for the condensate subsystem in the presence of the non-condensate environment under the inclusion of all two body interaction processes. We numerically monitor the condensate particle number distribution during condensate formation, and derive a condition under which the unique equilibrium steady state of a dilute, weakly interacting Bose-Einstein condensate is given by a Gibbs-Boltzmann thermal state of $N$ non-interacting atoms

Environment-induced dynamics in a dilute Bose-Einstein condensate

We directly model the quantum many particle dynamics during the transition of a gas of N indistinguishable bosons into a Bose-Einstein condensate. To this end, we develop a quantitative quantum master equation theory, which takes into account two body interaction processes, and in particular describes the particle number fluctuations characteristic for the Bose-Einstein phase transition. Within the Markovian dynamics assumption, we analytically prove and numerically verify the Boltzmann ergodicity conjecture for a dilute, weakly interacting Bose-Einstein condensate. The new physical bottom line of our theory is the direct microscopic monitoring of the Bose-Einstein distribution during condensate formation in real-time, after a sudden quench of the non-condensate atomic density above the critical density for Bose-Einstein condensation

Environment-induced dynamics in a dilute Bose-Einstein condensate

We directly model the quantum many particle dynamics during the transition of a gas of N indistinguishable bosons into a Bose-Einstein condensate. To this end, we develop a quantitative quantum master equation theory, which takes into account two body interaction processes, and in particular describes the particle number fluctuations characteristic for the Bose-Einstein phase transition. Within the Markovian dynamics assumption, we analytically prove and numerically verify the Boltzmann ergodicity conjecture for a dilute, weakly interacting Bose-Einstein condensate. The new physical bottom line of our theory is the direct microscopic monitoring of the Bose-Einstein distribution during condensate formation in real-time, after a sudden quench of the non-condensate atomic density above the critical density for Bose-Einstein condensation

Umgebungsinduzierte Dynamik in einem Bose-Einstein Kondensat

We directly model the quantum many particle dynamics during the transition of a gas of N indistinguishable bosons into a Bose-Einstein condensate. To this end, we develop a quantitative quantum master equation theory, which takes into account two body interaction processes, and in particular describes the particle number fluctuations characteristic for the Bose-Einstein phase transition. Within the Markovian dynamics assumption, we analytically prove and numerically verify the Boltzmann ergodicity conjecture for a dilute, weakly interacting Bose-Einstein condensate. The physical bottom line of our theory is the direct microscopic monitoring of the Bose-Einstein distribution during condensate formation in real-time, after a sudden quench of the non-condensate atomic density above the critical density for Bose-Einstein condensation.Wir beschreiben die Vielteilchen-Quantendynamik eines Gases von N ununterscheidbaren Teilchen während des Übergangs in ein Bose-Einstein Kondensat. Hierfür entwickeln wir eine quantitative Mastergleichungstheorie, welche den Phasenübergang des Gases in die kondensierte Phase realistisch beschreibt -- unter Einschluss von Zweiteilchenwechselwirkungen und unter der Berücksichtigung von Teilchenfluktuationen. Im Rahmen unseres Ansatzes gelingt ein analytischer Beweis der Boltzmannschen Ergodizitätshypothese für schwach wechselwirkende Quantengase unter der Annahme Markovscher Dynamik, in Übereinstimmung mit numerischen Simulationsergebnissen. Das übergreifende physikalische Ergebnis unserer Theorie ist die direkte ikrokopische Echtzeitbeschreibung der Bose-Einstein Verteilungsfunktion während der Kondensation, nach einer instantanen Änderung der atomaren ichtkondensatsdichte oberhalb der kritischen Dichte für die Bose-Einstein Kondensation

Sentiment Analysis of Tesla Tweets: Leveraging XGBoost for Social Media Insights

This study conducts an extensive sentiment analysis of 7,357 English Tesla-related tweets using an XGBoost classifier, addressing the critical need to understand public perception of innovative companies in the electric vehicle (EV) sector (Jain et al., 2019). The methodology involves advanced preprocessing with tweet-preprocessor and NLTK, feature engineering using TF-IDF (2,000 features) and weighted VADER sentiment scores, and model optimization via GridSearchCV with SMOTE balancing (Chawla et al., 2002). The model achieved an accuracy of 71.67% and a macro F1-score of 67.73% ± 5.97%, with a sentiment distribution of 37.31% negative, 30.58% neutral, and 32.11% positive. Theoretical assumptions explore the impact of social media on EV sentiment (Thelwall et al., 2010), while results and discussions highlight model performance and Tesla-specific insights (Chen & Guestrin, 2016). The study concludes with implications for EV marketing and future research directions in NLP