By Daniel Braun
Dissipative Quantum Chaos and Decoherence offers an over- view of the state-of-the-art of study during this interesting box. the most emphasis is at the improvement of a semiclassical formalism that permits one to include the impact of dissipation and decoherence in an actual, but tractable manner into the quantum mechanics of classically chaotic platforms. The formalism is hired to bare how the spectrum of the quantum mechanical propagator of a density matrix will depend on the spectrum of the corresponding classical propagator of part area density. easy quantum--classical hybrid formulae for experimentally correct correlation features and time-dependent expectation values of observables are derived. the matter of decoherence is taken care of intimately, and hugely unforeseen circumstances of very gradual decoherence are printed, with very important results for the long-debated realizability of Schrödinger cat states in addition to for the development of quantum computers.
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Extra resources for Dissipative Quantum Chaos and Decoherence
The quantum number j is just this number of quanta. It will turn out that js of the order of 5–10 can already lead to rather classical behavior; and 5–10 is still far away from the angular momenta which we encounter in the classical mechanics of everyday life, which have values of j of the order of 1034 . The surface of the unit sphere limj→∞ (J /j)2 = 1 becomes the phase space in the classical limit. It is two-dimensional, or, in other words, we have but a single degree of freedom. Besides j, it is convenient to introduce also J = j + 1/2 since this parameter simpliﬁes many formulae.
Owing to the periodicity of H(t), the evolution operator for one period is always the same, so that U (nT ) = U (T )n = F n . The matrix F is called the Floquet matrix . It contains all the information about the stroboscopic dynamics. Typical situations where the Hamiltonian is a periodic function of time are the interaction of a laser with atoms, electron spin resonance, nuclear magnetic resonance (see P. H¨anggi in ), or driven chemical reactions . Many of the classical maps that have played an important role in understanding classical chaos have been quantized.
These authors have shown that the quantum mechanical entanglement between two coupled kicked tops increases linearly in time with a rate proportional to the sum of the positive Lyapunov exponents. Miller and Sarkar also generalized the classical concept of entropy production as a criterion for chaoticity to the quantum world : the von Neumann entropy tr ρ ln ρ increases much more rapidly for chaotic systems than for integrable ones. Deﬁnitely the most popular criterion for chaoticity in the quantum world is the based on the so-called random-matrix conjecture.