Bala Sundaram

Experimental evidence for non-exponential decay in quantum tunnelling

An exponential decay law is the universal hallmark of unstable systems and is observed in all fields of science. This law is not, however, fully consistent with quantum mechanics and deviations from exponential decay have been predicted for short as well as long times. Such deviations have not hitherto been observed experimentally. Here we present experimental evidence for short-time deviation from exponential decay in a quantum tunnelling experiment. Our system consists of ultra-cold sodium atoms that are trapped in an accelerating periodic optical potential created by a standing wave of light. Atoms can escape the wells by quantum tunnelling, and the number that remain can be measured as a function of interaction time for a fixed value of the well depth and acceleration. We observe that for short times the survival probability is initially constant before developing the characteristics of exponential decay. The conceptual simplicity of the experiment enables a detailed comparison with theoretical predictions.

Quantum-Classical Transition in Nonlinear Dynamical Systems

Viewed as approximations to quantum mechanics, classical evolutions can violate the positive semidefiniteness of the density matrix. The nature of the violation suggests a classification of dynamical systems based on classical-quantum correspondence; we show that this can be used to identify when environmental interaction (decoherence) will be unsuccessful in inducing the quantum-classical transition. In particular, the late-time Wigner function can become positive without any corresponding approach to classical dynamics. In the light of these results, we emphasize key issues relevant for experiments studying the quantum-classical transition.

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

The quantum to classical transition for a system depends on many parameters, including a scale length for its action, variant Plancks over 2 pi, a measure of its coupling to the environment, D, and, for chaotic systems, the classical Lyapunov exponent, lambda. We propose measuring the proximity of quantum and classical evolutions as a multivariate function of (Plancks over 2 pi,lambda,D) and searching for transformations that collapse this hypersurface into a function of a composite parameter zeta= Plancks over 2 pi alpha)lambda beta D gamma. We report results for the quantum Cat Map and Duffing oscillator, showing accurate scaling behavior over a wide parameter range, indicating that this may be used to construct universality classes for this transition.

Wave analysis of ray chaos in underwater acoustics

The dispersion of a wave packet in an acoustic medium is considered in the paraxial wave approximation, where the effective potential, due to variation of the speed of propagation, varies both with depth and propagation distance. The analysis of the resulting parabolic equation, similar to the Schrodinger equation, clearly demonstrates the role of ray chaos in enhancing the dispersion of the initial packet. However, wave coherence effects are also seen that suppress the effects of the ray chaos in a manner analogous to the effects of quantum chaos. (c) 1999 American Institute of Physics.

Perturbation theory for the Stark effect in a double δ quantum well

We study the Stark effect in a symmetric double ? quantum well, for which there are two kinds of resonances: the familiar resonances stemming from the bound states, and a doubly infinite family of resonances stemming from the zero-field continuum threshold. We derive explicit expressions for the Borel-summable Rayleigh?Schr?dinger perturbation series for the resonances stemming from the bound states, for the imaginary part of these same resonances and for all the resonances stemming from the zero-field continuum threshold. The techniques used in this paper are directly applicable to realistic models of quantum square well potentials with or without barriers.

Chaos and Quantum Mechanics

Abstract: The relationship between chaos and quantum mechanics has been somewhat uneasy—even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes—such as chaos—that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time‐series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next‐generation experiments.

The semiclassical regime of the chaotic quantum-classical transition.

An analysis of the semiclassical regime of the quantum-classical transition is given for open, bounded, one-dimensional chaotic dynamical systems. Environmental fluctuations-characteristic of all realistic dynamical systems-suppress the development of a fine structure in classical phase space and damp nonlocal contributions to the semiclassical Wigner function, which would otherwise invalidate the approximation. This dual regularization of the singular nature of the semiclassical limit is demonstrated by a numerical investigation of the chaotic Duffing oscillator.

Comment on "nonmonotonicity in the quantum-classical transition: chaos induced by quantum effects".

In a recent Letter [PRL 101, 074101 (2008)], Kapulkin and Pattanayak presented evidence that a quantum Duffing oscillator, sufficiently damped so that it is not classically chaotic, becomes chaotic in the transition region between quantum and classical motion. If true, this would be a striking result. However, Kapulkin and Pattanayak did not calculate the Lyapunov exponent for the system, usually regarded as the litmus-test of chaos. Here we perform this calculation, which throws considerable doubt upon their conclusions.

Chaotic dynamics of the relativistic kicked rotor

The relativistic periodically driven classical and quantum rotor problems are studied. Kinetical properties of the relativistic standard map is discussed. Quantum rotor is treated by solving the Dirac equation in the presence of the periodic δ-function potential. The relativistic quantum mapping which describes the evolution of the wave function is derived. The time-dependence of the energy are calculated.