![]() ![]() The Bose–Hubbard model (BHM) is an archetypal quantum lattice system exhibiting a quantum phase transition between its superfluid (SF) and Mott-insulator (MI) phase. In applying all the above models to physical situations, the need for an exact analysis of small-scale systems is established due to large quantum fluctuations which render mean-field approaches inaccurate. Finally, we discuss the model of Bardeen, Cooper and Schrieffer in this framework, which is appropriate for systems of ultracold fermionic atomic gases, as well as being applicable for the description of superconducting correlations in metallic grains with nanoscale dimensions. Additionally, these same two models are relevant to studies in quantum optics. The theory is also applicable to models of atomic-molecular Bose-Einstein condensates, with two examples given and analysed. ![]() It can be used not only for the study of tunnelling between condensates of atomic gases, but for solid state Josephson junctions and coupled Cooper pair boxes. The first model we introduce describes Josephson tunnelling between two coupled Bose-Einstein condensates. As examples we apply the theory to several models of current interest in the study of Bose-Einstein condensates, which have been successfully created using ultracold dilute atomic gases. In this review we demonstrate how the algebraic Bethe ansatz is used for the calculation of the energy spectra and form factors (operator matrix elements in the basis of Hamiltonian eigenstates) in exactly solvable quantum systems. A brief quantum analysis shows that quantum fluctuations can put serious limitations on the applicability of the mean-field results. The validity of the phase diagram obtained from the time-average calculation is discussed by using the orbit tracking method, and the difference in contrast with the maximum absolute deviation analysis is shown as well. The stability diagram for the range of modulation amplitudes and periods that stabilize the dynamics is given. To control and stabilize the system, periodic modulation is applied that suddenly shifts the relative phase between the atomic and the molecular modes and limits their further interconversion. The system is initialized to an unstable equilibrium state corresponding to a saddle point in the classical phase space, where subsequent free evolution gives rise to atom-molecule conversion. Steam achievements are given for the successful completion of each level.We numerically demonstrate the dynamic stabilization of a strongly interacting many-body bosonic system which can be realized by coupled ultracold atom-molecule gases. ![]() Players may also choose to simply fall off the platform or run into a wall to end the level and move on to the next one. Players may compete for a top spot on the global leaderboards by continuing to progress through the level and filling up the meter even more, however, this is not necessary. Once the player has filled the completion bar for a given level, it does not end, but rather speeds up. Additionally, some levels have fewer than the standard eight or ten platform lanes to bounce between, reaching all the way down to two lanes. Certain levels will also include red platforms that begin to fall once the player lands their character on them. Some levels, however, include not only platforms of various heights and gaps between them, but a rotating beam that must be dodged or walls that prevent the player from moving or seeing through that section. Each level has a different set of patterns and speed at which those patterns approach to challenge the player with, as is expected of any runner. Levels vary significantly in pattern, speed, additional hazards, and shape. Within each group, the same six particles are found: Geon, Acceleron, Radion, Graviton, Y Boson, and X Boson. There are the standard particles, Dark particles, and Anti-particles, each group being progressively more difficult. ![]() Levels are broken down into three groups of six. Character choice does not affect gameplay in any way. Players may choose their protagonist, given the option of a male scientist, a female scientist, and a robot. Reaching the blue spaces are easier said than done, each level is randomly generated and platforms constantly fall from the sky, meaning that it is impossible to view too much of the course up ahead.Players can jump in three directions (Forward, left or right) jumping left or right causes the level to rotate however it can only rotate once per jump. Like in that game, the player is running inside a 360 loop with platforms on all side, the goal of each level is to discover the element within each level, the player accomplishes this by running on certain blue platforms spread across each level until a meter reaches 100% or above. Boson X is an endless runner game which is similar in concept to the special stages of Chaotix. ![]()
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