Final Activity Report Summary - BECANDDECFROMSEQ (Bose-Einstein condensation and quantum decoherence as scaling limits of the Schrodinger equation)
The main purpose of our project was the rigorous study of the quantum decoherence. It firstly has to be stressed that, for a quantum mechanical particle, some states which are impossible for a classical particle are allowed. Such states are called superposition or Schroedinger cat-like states and are represented by a wave function with two spatially separate components. Therefore, when lying in a state like these, a particle is neither here nor there, but in both places at the same time. This is difficult to conceive and impossible to experience directly. However, the existence of a superposition state can be put in evidence letting the two components of the wave function come closer and eventually overlap, since interference fringes appear in the overlap region. Nowadays, such fringes are commonly observed and testified to the existence of a superposition state.
As a matter of fact, the interaction with an external environment tends to destroy interference. This process is usually called decoherence and is commonly invoked to provide an explanation to the transition from the quantum to the classical world. It is certainly a mechanism that turns a quantum state to a classical, however probabilistic, one. Because of both decoherence and the small scale of the quantum world, superposition states are fragile and difficult to preserve. Nevertheless, a new light on them is expected to be shed by the technology of Bose-Einstein condensates, which are ultracold gases of identical bosons, in which all particles lie in the same quantum state. Despite the fact that they consist of a large, i.e. greater than 100, number of particles, they behave like a unique giant quantum particle and can have access to superposition states. Because of their macroscopic length scale, they offer the possibility to directly observe the interference fringes that reveal superpositions, as well as their destruction due to decoherence.
We studied the two regimes relevant for such an experiment, namely the Bose-Einstein condensates and the small mass ratio ones. We planned to unify the two treatments in the future, after the project completion. Mathematically, the problem was translated into the study of systems made by numerous interacting quantum particles. As for classical physics, quantum mechanical systems of more than two particles were not exactly tractable, but in the mentioned regime some simplifying assumption could be made. The main results we achieved were the following:
1. rigorous derivation of the effective Gross-Pitaevskii equation for cigar-shaped condensates from the first principles of quantum mechanics, namely from the many-body Schroedinger equation.
2. construction of the first many-particle model that allowed for a rigorous understanding and computation of the decoherence effect.
From these results some new research directions arose and, by the time of the project completion, we were constructing rigorous models for decoherence that were induced by more realistic models of environment, like quantised fields.
As a matter of fact, the interaction with an external environment tends to destroy interference. This process is usually called decoherence and is commonly invoked to provide an explanation to the transition from the quantum to the classical world. It is certainly a mechanism that turns a quantum state to a classical, however probabilistic, one. Because of both decoherence and the small scale of the quantum world, superposition states are fragile and difficult to preserve. Nevertheless, a new light on them is expected to be shed by the technology of Bose-Einstein condensates, which are ultracold gases of identical bosons, in which all particles lie in the same quantum state. Despite the fact that they consist of a large, i.e. greater than 100, number of particles, they behave like a unique giant quantum particle and can have access to superposition states. Because of their macroscopic length scale, they offer the possibility to directly observe the interference fringes that reveal superpositions, as well as their destruction due to decoherence.
We studied the two regimes relevant for such an experiment, namely the Bose-Einstein condensates and the small mass ratio ones. We planned to unify the two treatments in the future, after the project completion. Mathematically, the problem was translated into the study of systems made by numerous interacting quantum particles. As for classical physics, quantum mechanical systems of more than two particles were not exactly tractable, but in the mentioned regime some simplifying assumption could be made. The main results we achieved were the following:
1. rigorous derivation of the effective Gross-Pitaevskii equation for cigar-shaped condensates from the first principles of quantum mechanics, namely from the many-body Schroedinger equation.
2. construction of the first many-particle model that allowed for a rigorous understanding and computation of the decoherence effect.
From these results some new research directions arose and, by the time of the project completion, we were constructing rigorous models for decoherence that were induced by more realistic models of environment, like quantised fields.