Conditions for the Quantum to Classical Transition: Trajectories vs. Phase Space Distributions

We contrast two sets of conditions that govern the transition in which classical dynamics emerges from the evolution of a quantum system. The first was derived by considering the trajectories seen by an observer (dubbed the ``strong'' transition) [Bhattacharya, et al., Phys. Rev. Lett. 85: 4852 (2000)], and the second by considering phase-space densities (the ``weak'' transition) [Greenbaum, et al., Chaos 15, 033302 (2005)]. On the face of it these conditions appear rather different. We show, however, that in the semiclassical regime, in which the action of the system is large compared to $\hbar$, and the measurement noise is small, they both offer an essentially equivalent local picture. Within this regime, the weak conditions dominate while in the opposite regime where the action is not much larger than Planck's constant, the strong conditions dominate.Comment: 8 pages, 2 eps figure

Parameter scaling in the decoherent quantum-classical transition for chaotic systems

The quantum to classical transition has been shown to depend on a number of parameters. Key among these are a scale length for the action, $\hbar$, a measure of the coupling between a system and its environment, $D$, and, for chaotic systems, the classical Lyapunov exponent, $\lambda$. We propose computing a measure, reflecting the proximity of quantum and classical evolutions, as a multivariate function of $(\hbar,\lambda,D)$ and searching for transformations that collapse this hyper-surface into a function of a composite parameter $\zeta = \hbar^{\alpha}\lambda^{\beta}D^{\gamma}$. We report results for the quantum Cat Map, showing extremely accurate scaling behavior over a wide range of parameters and suggest that, in general, the technique may be effective in constructing universality classes in this transition.Comment: Submitte

Note and Comment

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4. The School Develops

Between 1947 and 1953, when M.P. Catherwood left the deanship to become New York’s industrial commissioner, the ILR School developed into a full fledged enterprise. These pages attempt to capture some of the excitement of this period of the school’s history, which was characterized by vigor, growth, and innovation. Includes: Alumni Recall Their Lives as Students; The Faculty Were Giants; Alice Cook: Lifelong Scholar, Consummate Teacher; Frances Perkins; Visits and Visitors; Tenth Anniversary: Reflection and Change; The Emergence of Departments at ILR; Development of International Programs and Outreach