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1.
Philos Trans A Math Phys Eng Sci ; 374(2069)2016 Jun 13.
Article in English | MEDLINE | ID: mdl-27140977

ABSTRACT

Echoes are ubiquitous phenomena in several branches of physics, ranging from acoustics, optics, condensed matter and cold atoms to geophysics. They are at the base of a number of very useful experimental techniques, such as nuclear magnetic resonance, photon echo and time-reversal mirrors. Particularly interesting physical effects are obtained when the echo studies are performed on complex systems, either classically chaotic, disordered or many-body. Consequently, the term Loschmidt echo has been coined to designate and quantify the revival occurring when an imperfect time-reversal procedure is applied to a complex quantum system, or equivalently to characterize the stability of quantum evolution in the presence of perturbations. Here, we present the articles which discuss the work that has shaped the field in the past few years.

2.
Article in English | MEDLINE | ID: mdl-25353553

ABSTRACT

We address the time decay of the Loschmidt echo, measuring the sensitivity of quantum dynamics to small Hamiltonian perturbations, in one-dimensional integrable systems. Using a semiclassical analysis, we show that the Loschmidt echo may exhibit a well-pronounced regime of exponential decay, similar to the one typically observed in quantum systems whose dynamics is chaotic in the classical limit. We derive an explicit formula for the exponential decay rate in terms of the spectral properties of the unperturbed and perturbed Hamilton operators and the initial state. In particular, we show that the decay rate, unlike in the case of the chaotic dynamics, is directly proportional to the strength of the Hamiltonian perturbation. Finally, we compare our analytical predictions against the results of a numerical computation of the Loschmidt echo for a quantum particle moving inside a one-dimensional box with Dirichlet-Robin boundary conditions, and find the two in good agreement.

3.
Proc Math Phys Eng Sci ; 469(2160): 20130308, 2013 Dec 08.
Article in English | MEDLINE | ID: mdl-24353468

ABSTRACT

We develop a systematic asymptotic description for domain wall motion in one-dimensional magnetic nanowires under the influence of small applied magnetic fields and currents and small material anisotropy. The magnetization dynamics, as governed by the Landau-Lifshitz-Gilbert equation, is investigated via a perturbation expansion. We compute leading-order behaviour, propagation velocities and first-order corrections of both travelling waves and oscillatory solutions, and find bifurcations between these two types of solutions. This treatment provides a sound mathematical foundation for numerous results in the literature obtained through more ad hoc arguments.

4.
Article in English | MEDLINE | ID: mdl-23767609

ABSTRACT

We consider the problem of quantum scattering of a localized wave packet by a weak Gaussian potential in two spatial dimensions. We show that, under certain conditions, this problem bears close analogy with that of focusing (or defocusing) of light rays by a thin optical lens: Quantum interference between straight paths yields the same lens equation as for refracted rays in classical optics.


Subject(s)
Lenses , Models, Theoretical , Nonlinear Dynamics , Quantum Theory , Scattering, Radiation , Computer Simulation , Light
5.
Phys Rev E Stat Nonlin Soft Matter Phys ; 83(5 Pt 2): 056210, 2011 May.
Article in English | MEDLINE | ID: mdl-21728633

ABSTRACT

We study the short-time stability of quantum dynamics in quasi-one-dimensional systems with respect to small localized perturbations of the potential. To this end, we analytically and numerically address the decay of the Loschmidt echo (LE) during times that are short compared to the Ehrenfest time. We find that the LE is generally a nonmonotonic function of time and exhibits strongly pronounced minima and maxima at the instants when the corresponding classical particle traverses the perturbation region. We also show that, under general conditions, the envelope decay of the LE is well approximated by a Gaussian, and we derive explicit analytical formulas for the corresponding decay time. Finally, we demonstrate that the observed nonmonotonic nature of the LE decay is only pertinent to one-dimensional (and, more generally, quasi-one-dimensional) systems, and that the short-time decay of the LE can be monotonic in a higher number of dimensions.

6.
Phys Rev E Stat Nonlin Soft Matter Phys ; 83(1 Pt 2): 016214, 2011 Jan.
Article in English | MEDLINE | ID: mdl-21405767

ABSTRACT

We study fidelity decay in classically chaotic microwave billiards for a local, pistonlike boundary perturbation. We experimentally verify a predicted nonmonotonic crossover from the Fermi golden rule to the escape-rate regime of the Loschmidt echo decay with increasing local boundary perturbation. In particular, we observe pronounced oscillations of the decay rate as a function of the piston position which quantitatively agree with corresponding theoretical results based on a refined semiclassical approach for local boundary perturbations.

7.
Phys Rev Lett ; 104(14): 147202, 2010 Apr 09.
Article in English | MEDLINE | ID: mdl-20481956

ABSTRACT

We address the dynamics of magnetic domain walls in ferromagnetic nanowires under the influence of external time-dependent magnetic fields. We report a new exact spatiotemporal solution of the Landau-Lifshitz-Gilbert equation for the case of soft ferromagnetic wires and nanostructures with uniaxial anisotropy. The solution holds for applied fields with arbitrary strength and time dependence. We further extend this solution to applied fields slowly varying in space and to multiple domain walls.

8.
J Chem Phys ; 133(24): 244113, 2010 Dec 28.
Article in English | MEDLINE | ID: mdl-21197982

ABSTRACT

The flux-flux correlation function formalism is a standard and widely used approach for the computation of reaction rates. In this paper we introduce a method to compute the classical and quantum flux-flux correlation functions for anharmonic barriers essentially analytically through the use of the classical and quantum normal forms. In the quantum case we show that for a general f degree-of-freedom system having an index one saddle the quantum normal form reduces the computation of the flux-flux correlation function to that of an effective one-dimensional anharmonic barrier. The example of the computation of the quantum flux-flux correlation function for a fourth order anharmonic barrier is worked out in detail, and we present an analytical expression for the quantum mechanical microcanonical flux-flux correlation function. We then give a discussion of the short-time and harmonic limits.

9.
J Chem Phys ; 131(14): 144103, 2009 Oct 14.
Article in English | MEDLINE | ID: mdl-19831429

ABSTRACT

The quantum normal form approach to quantum transition state theory is used to compute the cumulative reaction probability for collinear exchange reactions. It is shown that for heavy-atom systems such as the nitrogen-exchange reaction, the quantum normal form approach gives excellent results and has major computational benefits over full reactive scattering approaches. For light atom systems such as the hydrogen-exchange reaction however, the quantum normal approach is shown to give only poor results. This failure is attributed to the importance of tunneling trajectories in light atom reactions that are not captured by the quantum normal form as indicated by the only very slow convergence of the quantum normal form for such systems.

10.
Phys Rev E Stat Nonlin Soft Matter Phys ; 79(4 Pt 2): 046211, 2009 Apr.
Article in English | MEDLINE | ID: mdl-19518316

ABSTRACT

The Loschmidt echo (LE) (or fidelity) quantifies the sensitivity of the time evolution of a quantum system with respect to a perturbation of the Hamiltonian. In a typical chaotic system the LE has been previously argued to exhibit a long-time saturation at a value inversely proportional to the effective size of the Hilbert space of the system. However, until now no quantitative results have been known and, in particular, no explicit expression for the proportionality constant has been proposed. In this paper we perform a quantitative analysis of the phenomenon of the LE saturation and provide the analytical expression for its long-time saturation value for a semiclassical particle in a two-dimensional chaotic billiard. We further perform extensive (fully quantum mechanical) numerical calculations of the LE saturation value and find the numerical results to support the semiclassical theory.

11.
Phys Rev Lett ; 101(17): 174101, 2008 Oct 24.
Article in English | MEDLINE | ID: mdl-18999749

ABSTRACT

We address the decay in open chaotic quantum systems and calculate semiclassical corrections to the classical exponential decay. We confirm random matrix predictions and, going beyond, calculate Ehrenfest time effects. To support our results we perform extensive numerical simulations. Within our approach we show that certain (previously unnoticed) pairs of interfering, correlated classical trajectories are of vital importance. They also provide the dynamical mechanism for related phenomena such as photoionization and photodissociation, for which we compute cross-section correlations. Moreover, these orbits allow us to establish a semiclassical version of the continuity equation.

12.
Phys Rev E Stat Nonlin Soft Matter Phys ; 75(1 Pt 2): 015201, 2007 Jan.
Article in English | MEDLINE | ID: mdl-17358212

ABSTRACT

We investigate the sensitivity of the time evolution of semiclassical wave packets in two-dimensional chaotic billiards with respect to local perturbations of their boundaries. For this purpose, we address, analytically and numerically, the time decay of the Loschmidt echo (LE). We find the LE to decay exponentially in time, with the rate equal to the classical escape rate from an open billiard obtained from the original one by removing the perturbation-affected region of its boundary. Finally, we propose a principal scheme for the experimental observation of the LE decay.

13.
Phys Rev E Stat Nonlin Soft Matter Phys ; 74(1 Pt 2): 016204, 2006 Jul.
Article in English | MEDLINE | ID: mdl-16907174

ABSTRACT

We consider the time evolution of a wave packet representing a quantum particle moving in a geometrically open billiard that consists of a number of fixed hard-disk or hard-sphere scatterers. Using the technique of multiple collision expansions we provide a first-principle analytical calculation of the time-dependent autocorrelation function for the wave packet in the high-energy diffraction regime, in which the particle's de Broglie wavelength, while being small compared to the size of the scatterers, is large enough to prevent the formation of geometric shadow over distances of the order of the particle's free flight path. The hard-disk or hard-sphere scattering system must be sufficiently dilute in order for this high-energy diffraction regime to be achievable. Apart from the overall exponential decay, the autocorrelation function exhibits a generally complicated sequence of relatively strong peaks corresponding to partial revivals of the wave packet. Both the exponential decay (or escape) rate and the revival peak structure are predominantly determined by the underlying classical dynamics. A relation between the escape rate, and the Lyapunov exponents and Kolmogorov-Sinai entropy of the counterpart classical system, previously known for hard-disk billiards, is strengthened by generalization to three spatial dimensions. The results of the quantum mechanical calculation of the time-dependent autocorrelation function agree with predictions of the semiclassical periodic orbit theory.

14.
Phys Rev E Stat Nonlin Soft Matter Phys ; 71(2 Pt 2): 026225, 2005 Feb.
Article in English | MEDLINE | ID: mdl-15783405

ABSTRACT

We consider the quantum-mechanical propagator for a particle moving in a d -dimensional Lorentz gas, with fixed, hard-sphere scatterers. To evaluate this propagator in the semiclassical region, and for times less than the Ehrenfest time, we express its effect on an initial Gaussian wave packet in terms of quantities analogous to those used to describe the exponential separation of trajectories in the classical version of this system. This result relates the spread of the wave packet to the rate of separation of classical trajectories, characterized by positive Lyapunov exponents. We consider applications of these results, first to illustrate the behavior of the wave-packet autocorrelation functions for wave packets on periodic orbits. The autocorrelation function can be related to the fidelity, or Loschmidt echo, for the special case that the perturbation is a small change in the mass of the particle. An exact expression for the fidelity, appropriate for this perturbation, leads to an analytical result valid over very long time intervals, inversely proportional to the size of the mass perturbation. For such perturbations, we then calculate the long-time echo for semiclassical wave packets on periodic orbits.

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