The Born Rule in Quantum and Classical Mechanics

Considerable effort has been devoted to deriving the Born rule (e.g. that $|\psi(x)|^2 dx$ is the probability of finding a system, described by $\psi$, between $x$ and $x + dx$) in quantum mechanics. Here we show that the Born rule is not solely quantum mechanical; rather, it arises naturally in the Hilbert space formulation of {\it classical} mechanics as well. These results provide new insights into the nature of the Born rule, and impact on its understanding in the framework of quantum mechanics.Comment: 5 pages, no figures, to appear in Phys. Rev.

An efficient approach to the quantum dynamics and rates of processes induced by natural incoherent light

In many important cases, the rate of excitation of a system embedded in an environment is significantly smaller than the internal system relaxation rates. An important example is that of light-induced processes under natural conditions, in which the system is excited by weak, incoherent (e.g., solar) radiation. Simulating the dynamics on the time scale of the excitation source can thus be computationally intractable. Here we describe a method for obtaining the dynamics of quantum systems without directly solving the master equation. We present an algorithm for the numerical implementation of this method, and, as an example, use it to reconstruct the internal conversion dynamics of pyrazine excited by sunlight. Significantly, this approach also allows us to assess the role of quantum coherence on biological time scales, which is a topic of ongoing interest

Uniform Semiclassical Wavepacket Propagation and Eigenstate Extraction in a Smooth Chaotic System

A uniform semiclassical propagator is used to time evolve a wavepacket in a smooth Hamiltonian system at energies for which the underlying classical motion is chaotic. The propagated wavepacket is Fourier transformed to yield a scarred eigenstate.Comment: Postscript file is tar-compressed and uuencoded (342K); postscript file produced is 1216

Chaos and Lyapunov exponents in classical and quantal distribution dynamics

We analytically establish the role of a spectrum of Lyapunov exponents in the evolution of phase-space distributions $\rho(p,q)$. Of particular interest is $\lambda_2$, an exponent which quantifies the rate at which chaotically evolving distributions acquire structure at increasingly smaller scales and which is generally larger than the maximal Lyapunov exponent $\lambda$ for trajectories. The approach is trajectory-independent and is therefore applicable to both classical and quantum mechanics. In the latter case we show that the $\hbar\to 0$ limit yields the classical, fully chaotic, result for the quantum cat map.Comment: 5 RevTeX pages + 2 ps figs. Phys. Rev. E (to appear,'97

Exponentially rapid decoherence of quantum chaotic systems

We use a recent result to show that the rate of loss of coherence of a quantum system increases with increasing system phase space structure and that a chaotic quantal system in the semiclassical limit decoheres exponentially with rate $2 \lambda_2$, where $\lambda_2$ is a generalized Lyapunov exponent. As a result, for example, the dephasing time for classically chaotic systems goes to infinity logarithmically with the temperature, in accord with recent experimental results.Comment: 4 ReVTeX pages + 1 postscript figure. PRL (to appear

Overlapping Resonances in the Resistance of Superposition States to Decoherence

Overlapping resonances are shown to provide new insights into the extent of decoherence experienced by a system superposition state in the regime of strong system- environment coupling. As an example of this general approach, a generic system comprising spin-half particles interacting with a thermalized oscillator environment is considered. We find that (a) amongst the collection of parametrized Hamiltonians, the larger the overlapping resonances contribution, the greater the maximum possible purity, and (b) for a fixed Hamiltonian, the larger the overlapping resonances contribution, the larger the range of possible values of the purity as one varies the phases in the system superposition states. Systems displaying decoherence free subspaces show that largest overlapping resonances contribution.Comment: 5 figures, submitte

Long-lived oscillatory incoherent electron dynamics in molecules: trans-polyacetylene oligomers

We identify an intriguing feature of the electron-vibrational dynamics of molecular systems via a computational examination of \emph{trans}-polyacetylene oligomers. Here, via the vibronic interactions, the decay of an electron in the conduction band resonantly excites an electron in the valence band, and vice versa, leading to oscillatory exchange of electronic population between two distinct electronic states that lives for up to tens of picoseconds. The oscillatory structure is reminiscent of beating patterns between quantum states and is strongly suggestive of the presence of long-lived molecular electronic coherence. Significantly, however, a detailed analysis of the electronic coherence properties shows that the oscillatory structure arises from a purely incoherent process. These results were obtained by propagating the coupled dynamics of electronic and vibrational degrees of freedom in a mixed quantum-classical study of the Su-Schrieffer-Heeger Hamiltonian for polyacetylene. The incoherent process is shown to occur between degenerate electronic states with distinct electronic configurations that are indirectly coupled via a third auxiliary state by the vibronic interactions. A discussion of how to construct electronic superposition states in molecules that are truly robust to decoherence is also presented