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Register a new userSignatures of quantum stability in a classically chaotic system
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We experimentally and numerically investigate the quantum accelerator mode dynamics of an atom optical realization of the quantum delta-kicked accelerator, whose classical dynamics are chaotic. Using a Ramsey-type experiment, we observe interference, demonstrating that quantum accelerator modes are formed coherently. We construct a link between the behavior of the evolutions fidelity and the phase space structure of a recently proposed pseudoclassical map, and thus account for the observed interference visibilities.
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The Loschmidt echo (LE) is a measure of the sensitivity of quantum mechanics to perturbations in the evolution operator. It is defined as the overlap of two wave functions evolved from the same initial state but with slightly different Hamiltonians. Thus, it also serves as a quantification of irreversibility in quantum mechanics. In this thesis the LE is studied in systems that have a classical counterpart with dynamical instability, that is, classically chaotic. An analytical treatment that makes use of the semiclassical approximation is presented. It is shown that, under certain regime of the parameters, the LE decays exponentially. Furthermore, for strong enough perturbations, the decay rate is given by the Lyapunov exponent of the classical system. Some particularly interesting examples are given. The analytical results are supported by thorough numerical studies. In addition, some regimes not accessible to the theory are explored, showing that the LE and its Lyapunov regime present the same form of universality ascribed to classical chaos. In a sense, this is evidence that the LE is a robust temporal signature of chaos in the quantum realm. Finally, the relation between the LE and the quantum to classical transition is explored, in particular with the theory of decoherence. Using two different approaches, a semiclassical approximation to Wigner functions and a master equation for the LE, it is shown that the decoherence rate and the decay rate of the LE are equal. The relationship between these quantities results mutually beneficial, in terms of the broader resources of decoherence theory and of the possible experimental realization of the LE.
The interest in the properties of quantum systems, whose classical dynamics are chaotic, derives from their abundance in nature. The spectrum of such systems can be related, in the semiclassical approximation (SCA), to the unstable classical periodic orbits, through Gutzwillers trace formula. The class of systems studied in this work, tiling billiards on the pseudo-sphere, is special in this correspondence being exact, via Selbergs trace formula. In this work, an exact expression for Greens function (GF) and the eigenfunctions (EF) of tiling billiards on the pseudo-sphere, whose classical dynamics are chaotic, is derived. GF is shown to be equal to the quotient of two infinite sums over periodic orbits, where the denominator is the spectral determinant. Such a result is known to be true for typical chaotic systems, in the leading SCA. From the exact expression for GF, individual EF can be identified. In order to obtain a SCA by finite series for the infinite sums encountered, resummation by analytic continuation in $hbar$ was performed. The result is similar to known results for EF of typical chaotic systems. The lowest EF of the Hamiltonian were calculated with the help of the resulting formulae, and compared with exact numerical results. A search for scars with the help of analytical and numerical methods failed to find evidence for their existence.
We show that strongly localized wave functions occur around classical bifurcations. Near a saddle node bifurcation the scaling of the inverse participation ratio on Plancks constant and the dependence on the parameter is governed by an Airy function. Analytical estimates are supported by numerical calculations for the quantum kicked rotor.
Gutzwillers trace formula and Bogomolnys formula are applied to a non--specific, non--scalable Hamiltonian system, a two--dimensional anharmonic oscillator. These semiclassical theories reproduce well the exact quantal results over a large spatial and energy range.
We present experimental and theoretical results for the fluctuation properties in the incomplete spectra of quantum systems with symplectic symmetry and a chaotic dynamics in the classical limit. To obtain theoretical predictions, we extend the random-matrix theory (RMT) approach introduced in [O. Bohigas and M. P. Pato, Phys. Rev. E 74, 036212 (2006)] for incomplete spectra of quantum systems with orthogonal symmetry. We validate these RMT predictions by randomly extracting a fraction of levels from complete sequences obtained numerically for quantum graphs and experimentally for microwave networks with symplectic symmetry and then apply them to incomplete experimental spectra to demonstrate their applicability. Independently of their symmetry class quantum graphs exhibit nongeneric features which originate from nonuniversal contributions. Part of the associated eigenfrequencies can be identified in the level dynamics of parameter-dependent quantum graphs and extracted, thereby yielding spectra with systematically missing eigenfrequencies. We demonstrate that, even though the RMT approach relies on the assumption that levels are missing at random, it is possible to determine the fraction of missing levels and assign the appropriate symmetry class by comparison of their fluctuation properties with the RMT predictions.
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