We present a numerically feasible semiclassical (SC) method to evaluate quantum fidelity decay (Loschmidt echo) in a classically chaotic system. It was thought that such evaluation would be intractable, but instead we show that a uniform SC expression not only is tractable but it also gives remarkably accurate numerical results for the standard map in both the Fermi-golden-rule and Lyapunov regimes. Because it allows Monte Carlo evaluation, the uniform expression is accurate at times when there are 1070 semiclassical contributions. Remarkably, it also explicitly contains the “building blocks” of analytical theories of recent literature, and thus permits a direct test of the approximations made by other authors in these regimes, rather than an a posteriori comparison with numerical results. We explain in more detail the extended validity of the classical perturbation approximation and show that within this approximation, the so-called “diagonal approximation” is automatic and does not require ensemble averaging.
A general semiclassical approach to quantum systems with system-bath interactions is developed. We study system decoherence in detail using a coherent-state semiclassical wave-packet method which avoids singularity issues arising in the usual Green’s function approach. We discuss the general conditions under which it is approximately correct to discuss quantum decoherence in terms of a “dephasing” picture and we derive semiclassical expressions for the phase and phase distribution. Remarkably, an effective system wavefunction emerges whose norm measures the decoherence and is equivalent to a density-matrix formulation.
In this article, we employ a recently discovered criterion for selecting important contributions to the semiclassical coherent state propagator [T. Van Voorhis and E. J. Heller, Phys. Rev. A 66, 050501 (2002)] to study the dynamics of many dimensional problems. We show that the dynamics are governed by a similarity transformed version of the standard classical Hamiltonian. In this light, our selection criterion amounts to using trajectories generated with the untransformed Hamiltonian as approximate initial conditions for the transformed boundary value problem. We apply the new selection scheme to some multidimensional Henon–Heiles problems and compare our results to those obtained with the more sophisticated Herman–Kluk approach. We find that the present technique gives near-quantitative agreement with the the standard results, but that the amount of computational effort is less than Herman–Kluk requires even when sophisticated integral smoothing techniques are employed in the latter.
We find a uniform semiclassical (SC) wave function describing coherent branched flow through a two-dimensional electron gas (2DEG), a phenomenon recently discovered by direct imaging of the current using scanned probed microscopy [M.A. Topinka, B.J. LeRoy, S.E.J. Shaw, E.J. Heller, R.M. Westervelt, K.D. Maranowski, and A.C. Gossard, Science 289, 2323 (2000)]. The formation of branches has been explained by classical arguments [M.A. Topinka, B.J. LeRoy, R.M. Westervelt, S.E.J. Shaw, R. Fleischmann, E.J. Heller, K.D. Maranowski, and A.C. Gossard, Nature (London) 410, 183 (2001)], but the SC simulations necessary to account for the coherence are made difficult by the proliferation of catastrophes in the phase space. In this paper, expansion in terms of “replacement manifolds” is used to find a uniform SC wave function for a cusp singularity. The method is then generalized and applied to calculate uniform wave functions for a quantum-map model of coherent flow through a 2DEG. Finally, the quantum-map approximation is dropped and the method is shown to work for a continuous-time model as well.
Recent experimental work in the Westervelt laboratory at Harvard has succeeded in directly imaging electron flow in two degree of freedom electron gasses formed in semiconductor microstructures. Here, we give a brief account of the unexpected high resolution of the resulting images, the surprising branching of the flow which was observed, and the survival of quantum fringing beyond where it was thought to have been obliterated by thermal effects.
We introduce a very general approximation to the quantum propagator that is based on the assumption that the most important contributions to the complex semiclassical propagator evolve from real classical trajectories that almost satisfy the desired boundary conditions. Our results for two systems — the autocorrelation function for the quartic anharmonic oscillator and the photodissociation spectrum of CO2 — show that these nearly real contributions yield an excellent approximation to the quantum propagator for quite long times. The approach taken here is applicable to problems with many (e.g., several hundred) degrees of freedom, and hence promises to provide an accurate and useful representation of the quantum dynamics for a wide variety of physically interesting systems.
This paper presents a perturbative model for the vibrational predissociation dynamics of inert gas hydrogen halide (RgHX) complexes. The predissociation is modeled as a Fermi Golden Rule (FGR) process from a bound state residing on a two-dimensional potential energy surface (PES) obtained by averaging over the HX vibrational state of interest to a series of one-dimensional exit channels obtained by averaging over the HX rovibrational state of interest. This model is applied to ArHF, for which a high-quality ab initio interaction potential is available. In particular, we focus on the v → v − 1 transition for the bound states (1000), (2000), (2110), (3000), and (3110). We confirm the experimental observation that the product HF tends to come off at the highest accessible j state, which is j = 13 for this system. This results from a strong angular anisotropy in the ArHF interaction potential that couples low-j and high-j HF states. The basic mechanism for this high-j preference is determined to be the suppression of the low-j exit channels arising from highly oscillatory low-j outgoing wave functions. We also observed that the tails of the bound-state wave functions, in the inner wall region of the interaction potential, gave the main contribution to the predissociation rate, indicating that the vibrational predissociation process is due to tunneling and is therefore a purely quantum effect. The calculations also confirm the strong v-dependence of the predissociation rates, as well as the stabilization of the complex that occurs when energy is placed into the HF bending mode. For the (1000) and (2110) states, we obtain rates well below 1600 s-1, which is consistent with the observation by Miller that the vibrational predissociation rates are too slow to be measured. The (2000) state does give a measurable rate, with a computed decay into the j = 13 exit channel of 14 000 s-1. The (3000) state gives a corresponding rate of 200 000 s-1, in good agreement with the overall dissociation rate of 250 000 s-1. The (3110) rate is slower, with a value of 12 000 s-1. Though this rate seems somewhat small, given that the lifetime of the (3110) state was measured to only be twice as long as that of the (3000) state, this rate and all our rates are within an order of magnitude of the measured rates.
We provide a general and nonperturbative theoretical basis for quantum reflection of an ultracold atom incident on a cold or warm surface. Sticking is identified with the formation of a long-lived resonance, from which it emerges that the physical reason for not sticking is that the many internal degrees of freedom of the target serve to decohere the incident one body wave function, thereby upsetting the delicate interference process necessary to form a resonance. We then explore the transition to the post-threshold behavior, when sticking prevails at higher incident energies. Studying the WKB wave functions of the atom provides a quick understanding of our results even in the ultracold regime where WKB is not applicable. Explicit examples are examined in detail and we predict the temperatures required to reach the various regimes.
We consider a classically chaotic system that is described by a Hamiltonian H(Q,P;x), where (Q,P) describes a particle moving inside a cavity, and x controls a deformation of the boundary. The quantum eigenstates of the system are \|n(x)>. We describe how the parametric kernel P(n\|m)=\|<n(x)\|m(x0)>\|2, also known as the local density of states, evolves as a function of δx=x-x0. We illuminate the nonunitary nature of this parametric evolution, the emergence of nonperturbative features, the final nonuniversal saturation, and the limitations of random-wave considerations. The parametric evolution is demonstrated numerically for two distinct representative deformation processes.
Quantum tunnelling breaks the rules of classical physics — and leads to ghost-like transfer of matter through barriers. Demonstrations of a new type of quantum tunnelling have the ghosts taking new liberties.
We consider the response of a chaotic cavity in d dimensions to periodic driving. We are motivated by older studies of one-body dissipation in nuclei, and also by anticipated mesoscopic applications. For calculating the rate of energy absorption due to time-dependent deformation of the confining potential, we introduce an improved version of the wall formula. Our formulation takes into account that a special class of deformations causes no heating in the zero-frequency limit. We also derive a mesoscopic version of the Drude formula, and explain that it can be regarded as a special example of our calculations. Specifically we consider a quantum dot driven by an electro-motive force which is induced by a time-dependent homogeneous magnetic field.
We present a semiclassical technique that relies on replacing complicated classical manifold structure with simpler manifolds, which are then evaluated by the usual semiclassical rules. Under circumstances where the original manifold structure gives poor or useless results semiclassically the replacement manifolds can yield remarkable accuracy. We give several working examples to illustrate the theory presented here.
We explain the origin of the Kondo mirage seen in recent quantum corral Scanning Tunneling Microscope (STM) experiments with a scattering theory of electrons on the surfaces of metals. Our theory combined with experimental data provides the first direct observation of a single Kondo atom phase shift. The Kondo mirage observed at the empty focus of an elliptical quantum corral is shown to arise from multiple electron bounces off the corral wall adatoms in a manner analagous to the formation of a real image in optics. We demonstrate our theory with direct quantitive comparision to experimental data. *This research was supported by the National Science Foundation under Grant No. CHE9610501 and by ITAMP.
This paper presents a PDE-based, gradient-descent approach (GDA) to the EBK quantization of nearly separable Hamiltonians in the quasi-integrable regime. The method does this by finding an optimal semiclassical basis of invariant tori which minimizes the angular dependence of the Hamiltonian. This representation of the Hamiltonian is termed an intrinsic resonance representation (IRR), and it gives the smallest possible basis obtainable from classical mechanics. Because our method is PDE-based, we believe it to be significantly faster than previous IRR algorithms, making it possible to EBK quantize systems of higher degrees of freedom than previously studied. In this paper we demonstrate our method by reproducing results from a two-degree-of-freedom system used to demonstrate the previous Carioli, Heller, and Moller (CHM) implementation of the IRR approach. We then go on to show that our method can be applied to higher dimensional Hamiltonians than previously studied by using it to EBK quantize a four- and a six-degree-of-freedom system.
Presents a study to observe electron flow through a narrow constriction in a semiconductor nanostructure. Methods; Results; Conclusion that a branching of current flux is due to focusing of the electron paths by ripples in the background potential.
Scanning a charged tip above the two-dimensional electron gas inside a gallium arsenide/aluminum gallium arsenide nanostructure allows the coherent electron flow from the lowest quantized modes of a quantum point contact at liquid helium temperatures to be imaged. As the width of the quantum point contact is increased, its electrical conductance increases in quantized steps of 2e2/h, where e is the electron charge and h is Planck's constant. The angular dependence of the electron flow on each step agrees with theory, and fringes separated by half the electron wavelength are observed. Placing the tip so that it interrupts the flow from particular modes of the quantum point contact causes a reduction in the conductance of those particular conduction channels below 2e2/h without affecting other channels.
We consider chaotic billiards in d dimensions, and study the matrix elements Mnm corresponding to general deformations of the boundary. We analyze the dependence of |Mnm|2 on ω=(En−Em)/ħ using semiclassical considerations. This relates to an estimate of the energy dissipation rate when the deformation is periodic at frequency ω. We show that, for dilations and translations of the boundary, |Mnm|2 vanishes like ω4 as ω→0, for rotations such as ω2, whereas for generic deformations it goes to a constant. Such special cases lead to quasiorthogonality of the eigenstates on the boundary.
We consider a classically chaotic system that is described by a Hamiltonian H(Q,P;x), where x is a constant parameter. Specifically, we discuss a gas particle inside a cavity, where x controls a deformation of the boundary or the position of a "piston." The quantum eigenstates of the system are |n(x)>. We describe how the parametric kernel P(nmid R:m) = |<n(x)mid R:m(x(0))>|(2) evolves as a function of deltax = x-x(0). We explore both the perturbative and the nonperturbative regimes, and discuss the capabilities and the limitations of semiclassical as well as random waves and random-matrix-theory considerations.
Heller, E.J.Semiclassical wave packets. The Physics and Chemistry of Wave Packets; Yeazel, J.; Uzer, T. eds, Wiley: New York (1999).