Table of contents

The 395th Wilhelm and Else Heraeus Seminar: `Time-dependent phenomena in Quantum Mechanics' took place at the Heinrich Fabri Institute in Blaubeuren, Germany, 12–16 September 2007.

The conference covered a wide range of topics connected with time-dependent phenomena in quantum mechanical systems. The 20 invited talks and 15 short talks with posters at the workshop covered the historical debate between Schrödinger, Dirac and Pauli about the role of time in Quantum Mechanics (the debate was carried out sometimes in footnotes) up to the almost direct observation of electron dynamics on the attosecond time-scale. Semiclassical methods, time-delay, monodromy, variational principles and quasi-resonances are just some of the themes which are discussed in more detail in the papers. Time-dependent methods also shed new light on energy-dependent systems, where the detour of studying the time-evolution of a quantum states allows one to solve previously intractable problems.

Additional information is available at the conference webpage http://www.quantumdynamics.de The organizer would like to thank all speakers, contributors, session chairs and referees for their efforts in making the conference a success. We also gratefully acknowledge the generous financial support from the Wilhelm and Else Heraeus Foundation for the conference and the production of this special volume of Journal of Physics: Conference Series.

Manfred Kleber Physik Department T30, Technische Universität München, 85747 Garching, Germany mkleber@ph.tum.de

Tobias Kramer Institut I: Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany tobias.kramer@physik.uni-regensburg.de Guest Editors

Front row (from left): W Schleich, E J Heller, J B Delos, H Friedrich, K Richter, M Kleber, P Kramer, M Man'ko, A del Campo, V Man'ko, M Efremov, A Ruiz, M O Scully Middle row: A Zamora, R Aganoglu, T Kramer, J Eiglsperger, H Cruz, P Raab, I Cirac, G Muga, J Larson, V Dodonov, W Becker Back row: A Eckardt, A Siddiki, K Vafayi, M Holthaus, E Räsänen, M Rodriguez, O Kullie, D Milošević, J Briggs, A Ribeiro, (not in the picture W Zwerger)

In stationary quantum mechanical problems time is absent. However time comes back if an experiment is performed on a stationary wave function. This property is demonstrated for a real experiment (photodetachment). The wave function is shown to set the stage for mutually exclusive experimental results, each of them having its own history of Feynman paths.

Some of the difficulties that have occurred in the past associated with the derivation of a time-energy uncertainty relation are discussed. Then it is shown that, beginning with a time-independent quantum composite of system and environment, a time-energy uncertainty relation for the system alone can be derived in the limit that the environment becomes classical, to provide a classical clock with which to monitor the quantum system.

We study the transient dynamics that arise during the formation of an atom laser beam in a tight waveguide. The time dependent density profile develops a series of wiggles which are related to the diffraction in time phenomenon. The apodization of matter waves, which relies on the use of smooth aperture functions, allows to suppress such oscillations in a time interval, after which there is a revival of the diffraction in time. The revival time scale is directly related to the inverse of the harmonic trap frequency for the atom reservoir.

We have solved the in-plane momentum-dependent effective-mass nonlinear Schrödinger equation for a spin-polarized electron wave packet in a InAs double quantum well system with an interlayer voltage. Considering a time-dependent Hartree potential, we have calculated the spin-polarized nonlinear electron dynamics between both quantum wells at different in-plane momentum values and applied bias. The spin-splitting caused by the Rashba effect is combined with the level matching between the spin dependent resonant tunneling levels making possible the observed local spin density oscillations which depend on the applied bias value. The filtering efficiency has been studied using time-dependent calculations.

The word 'monodromy' means 'once around a course', and it refers to changes that might occur when a system goes around some closed loop [1]. We discuss here a phenomenon whose proper name is 'nontrivial monodromy of action and angle variables in a Hamiltonian system'; for the obvious reason, we just call it 'Hamiltonian monodromy'. In this paper we describe two manifestations of Hamiltonian monodromy: a manifestation in a time-dependent classical system, and a manifestation in a stationary quantum system. Then we give a brief description of the mathematical theory, and finally close with a short survey of previous work on this subject.

This is a brief review of the recent progress in the theory of dynamical (non-stationary) Casimir effect in non-ideal cavities and of existing proposals to observe this effect in a laboratory, with an emphasis on the experiment which is under preparation in the University of Padua. The main idea of this experiment is to simulate periodical displacements of the cavity wall by periodical strong changes of conductivity of a thin semiconductor slab illuminated by picosecond laser pulses. In this connection the theory of quantum damped oscillator with arbitrary time dependence of the frequency and damping coefficient has been developed in order to take into account intrinsic losses in the semiconductor slab due to a finite conductivity during the intermediate part of the excitation-recombination cycle. The influence of different parameters, such as the diffusion and mobility coefficients of carriers, surface recombination velocity, absorption coefficient of laser radiation, thickness of the slab and geometry of the cavity, is analysed. Analytical and numerical evaluations show that under realistic experimental conditions, several thousand of quanta of EM field ('Casimir photons') could be produced from the initial vacuum state in a high-quality cavity at the lowest resonance frequency 2.5 GHz.

We suggest to view ultracold atoms in a time-periodically shifted optical lattice as a 'dressed matter wave', analogous to a dressed atom in an electromagnetic field. A possible effect lending support to this concept is a transition of ultracold bosonic atoms from a superfluid to a Mott-insulating state in response to appropriate "dressing" achieved through time-periodic lattice modulation. In order to observe this effect in a laboratory experiment, one has to identify conditions allowing for effectively adiabatic motion of a many-body Floquet state.

The time delay of an almost monochromatic wave packet can be defined unambiguously via the energy derivative of the phase of the amplitude defining the superposition of stationary states making up the wave packet. We illustrate how this concept, originally formulated by Eisenbud and Wigner in the context of scattering theory, can be applied to quantum reflection by the nonclassical region of a long-ranged attractive potential tail. We pay special attention to the case of a wave packet incident from the near side of the nonclassical region and interpret the results with the help of regularized step potentials.

The time-dependent variational principle associates to a Hamiltonian quantum system a set of trajectories running on a classical phase space with canonical equations of motion. When the quantum observables generate a Lie group G and the states are taken as functions on an appropriate homogeneous quotient space space := G/G₀ under under G, they can be equipped with a classical Poisson bracket which reproduces the commutators of the Lie group. The quantum system maps into a classical system whose equations of motion are governed by the expectation value of the quantum Hamiltonian H. We consider examples of this construction and show that the analysis of generalized classical systems provides insight into quantum many-body dynamics like chemical reactions.

Time-dependent quantum mechanics provides an intuitive picture of particle propagation in external fields. Semiclassical methods link the classical trajectories of particles with their quantum mechanical propagation. Many analytical results and a variety of numerical methods have been developed to solve the time-dependent Schrödinger equation. The time-dependent methods work for nearly arbitrarily shaped potentials, including sources and sinks via complex-valued potentials. Many quantities are measured at fixed energy, which is seemingly not well suited for a time-dependent formulation. Very few methods exist to obtain the energy-dependent Green function for complicated potentials without resorting to ensemble averages or using certain lead-in arrangements. Here, we demonstrate in detail a time-dependent approach, which can accurately and effectively construct the energy-dependent Green function for very general potentials. The applications of the method are numerous, including chemical, mesoscopic, and atomic physics.

The Jaynes-Cummings model, with and without the rotating wave approximation, is expressed in the conjugate variable representation and solved numerically by wave packet propagation. Both cases are then cast into systems of two coupled harmonic oscillators, reminiscent of coupled bound electronic potential curves of diatomic molecules. Using the knowledge of such models, this approach of the problem gives new insight of the dynamics. The effect of the rotating wave approximation is discussed. The collapse-revival phenomenon is especially analyzed in a non-standard manner. Extensions of the method is briefly mentioned in terms of a three-level atom and the Dicke model.

Review of the paraxial approximation to the problem of light beams propagating in optical fibers is presented. The analogy of time-dependent integrals of motion for the quantum system and space-dependent invariants for the radiation in inhomogeneous optical waveguides is elucidated. A specific tomographic probability distribution associated with the light-pulse profile is considered analogously to tomographic probabilities in quantum mechanics. The new inequality is presented for the tomographic probability.

Time-dependent integrals of motion for systems with both time-independent and time-dependent Hamiltonians are studied and expressed in terms of the evolution operator. The probability representation in which the system quantum states are described by tomographic probabilities (tomograms), is reviewed. Examples of systems of parametric oscillators and the entanglement phenomena induced by time-dependent coupling of the oscillators are discussed. Transition probabilities between the energy levels of parametric oscillators expressed in terms of special functions like Laguerre and Legendre polynomials are given in terms of tomographic probabilities.

First we discuss briefly the importance of time and time keeping, explaining the basic functioning of clocks in general and of atomic clocks based on Ramsey interferometry in particular. The usefulness of cold atoms is discussed, as well as their limits if Bose-Einstein condensates are used. We study as an alternative a different cold-atom regime: the Tonks-Girardeau (TG) gas of tightly confined and strongly interacting bosons. The TG gas is reviewed and then generalized for two-level atoms. Finally, we explore the combination of Ramsey interferometry and TG gases.

The near-threshold behaviour of the phase loss due to reflection ("the reflection phase") at the outer turning point in an attractive, homogeneous potential tail is obtained analytically up to linear order in energy by adapting the effective-range formalism of scattering theory to the bound state regime. A generalization of the Bohr-Sommerfeld quantization rule based on these results accurately reproduces the eigenenergies close to the threshold. We derive an expression for the reflection phase at arbitrary energy by matching the near-threshold expansion to known formulas for large negative energies. This model, which includes only one free parameter is a significant improvement over the eigenenergies obtained by other approximate methods. Also, the scattering length is completely determined by the knowledge of one of the highest bound energy levels (not necessarily by the highest one) and the asymptotic behavior of the potential.

A uniform approximation for the coherent state propagator, valid in the vicinity of phase space caustics, was recently obtained using the Maslov method combined with a dual representation for coherent states. In this paper we review the derivation of this formula and apply it to two model systems: the one-dimensional quartic oscillator and a two-dimensional chaotic system.

We analyze the dynamical melting of two-component atomic Mott-Insulator states in a completely-connected optical lattice within the adiabatic approximation. We examine in detail the effect of the dynamical phase acquired by the state during the adiabatic melting of the lattice potential. We show how for certain limits an on-site superposition state with two particles per site melts into a macroscopic superposition state, while an on-site superposition state with only one particle per site melts into a coherent state.

The only global autonomous system in which quasiresonant processes have been previously described is the seminal example of one atom colliding with a diatom molecule. In this work we describe classical quasiresonances in a different context, the grazing angle atom-surface collisions. While in the first system the actions related to the process correspond to the internal vibro-rotational degrees of freedom of the molecule, in this new example they turn to be the two components of the momentum of the atom parallel to the surface. We discuss the range of initial actions where quasiresonant processes arise, and suggest a new method to localize quasiresonance regions using the dwell time of the incoming atom in the vicinity of the surface.

The time dependent Schrodinger equation is frequently 'derived' by postulating the energy E → iℏ(∂/∂t) and momentum → (ℏ/i) operator relations. In the present paper we review the quantum field theoretic route to the Schrodinger wave equation which treats time and space as parameters, not operators. Furthermore, we recall that a classical (nonlinear) wave equation can be derived from the classical action via Hamiltonian-Jacobi theory. By requiring the wave equation to be linear we again arrive at the Schrodinger equation, without postulating operator relations. The underlying philosophy is operational: namely 'a particle is what a particle detector detects.' This leads us to a useful physical picture combining the wave (field) and particle paradigms which points the way to the time-dependent Schrodinger equation.

We present the essential findings of the screening theory of the integer quantum Hall effect (IQHE) considering a quantum point contact (QPC). Our approach is to solve the Poisson and the Schrödinger equations self-consistently, taking into account electron interactions, within a Hartree type approximation for a two dimensional electron gas (2DEG) subject to high perpendicular magnetic fields. The Coulomb interaction between the electrons separates 2DEG into two co-existing regions, namely quasi-metallic compressible and quasi-insulating incompressible regions, which exhibit peculiar screening and transport properties. In the presence of an external current, we show that this current is confined into the incompressible regions where the drift velocity is finite. In particular, we investigate the distribution of these incompressible strips and their relation with the quantum Hall plateaus considering a quasi 1D constriction, i.e. a QPC.