arXiv subjects

Gerhard C. Hegerfeldt

Publications and source records attributed to Gerhard C. Hegerfeldt.

At least 19 recordsLinked to original sources

arXiv2023

Time-Optimal Transport of a Harmonic Oscillator: Analytic Solution

Motivated by the experimental transport of a trap with a quantum mechanical system modeled as a harmonic oscillator (h.o.) the corresponding classical problem is investigated. Protocols for the fastest possible transport of a classical h.o. in a wagon over a distance d are derived where both initially and finally the wagon is at rest and the h.o. is in its equilibrium position and also at rest. The acceleration of the wagon is assumed to be bounded. For fixed oscillator frequency Ωit is shown that there are in general three switches in the acceleration and for special values of Ωonly one switch. In the latter case the optimal transport time is T_abs , that of a wagon without oscillator. The optimal transport time and the switch times are determined. It is shown that in some cases it is advantageous to go backwards for a while. In addition a time-dependent Ω(t), bounded by Ω_ and Ω+ , is allowed. In this case the behavior depends sensitively on Ω_+ and is spelled out in detail. In particular, depending on Ω_+ , T_abs may be obtained in continuously many ways.

quant-ph↗

arXiv2021

On the ocean beach -- why elliptic pebbles do not become spherical

Among pebbles strewn across a sandy ocean beach one can find relatively many with a nearly perfect elliptical (ellipsoidal) shape, and one wonders how this shape was attained and whether, during abrasion, the pebbles would remain elliptical or eventually become spherical. Mainly the latter question was addressed in a previous publication which identified frictional sliding and rotation of an elliptic pebble as main abrasion processes in the surf waves. In particular, it was predicted that the ellipticity $ε$ converges to a common equilibrium value for elliptic-like pebbles. Unfortunately, the derivation was based on an invalid force expression and a dimensionally unsuitable curvature. In this paper, not only force and curvature but also the contact duration with the sand surface during rotations is taken into account by fairly simple physical arguments, and it is shown that elliptic pebbles neither approach the same ellipticity and nor become more spherical nor more disk-like but rather that the ellipticity $ε$ increases.

physics.class-ph↗

arXiv2014

High-Speed Driving of a Two-Level System

A remarkably simple result is found for the optimal protocol of drivings for a general two-level Hamiltonian which transports a given initial state to a given final state in minimal time. If one of the three possible drivings is unconstrained in strength the problem is analytically completely solvable. A surprise arises for a class of states when one driving is bounded by a constant $c$ and the other drivings are constant. Then, for large $c$, the optimal driving is of type bang-off-bang and for increasing $c$ one recovers the unconstrained result. However, for smaller $c$ the optimal driving can suddenly switch to bang-bang type. It is also shown that for general states one may have a multistep protocol. The present paper explicitly proves and considerably extends the author's results contained in Phys. Rev. Lett. {\bf 111}, 260501 (2013).

quant-ph↗

arXiv2013

Driving at the quantum speed limit: Optimal control of a two-level system

A remarkably simple result is derived for the minimal time $T_{\rm min}$ required to drive a general initial state to a final target state by a Landau-Zener type Hamiltonian or, equivalently, by time-dependent laser driving. The associated protocol is also derived. A surprise arises for some states when the interaction strength is assumed to be bounded by a constant $c$. Then, for large $c$, the optimal driving is of type bang-off-bang and for increasing $c$ one recovers the unconstrained result. However, for smaller $c$ the optimal driving can suddenly switch to bang-bang type. We discuss the notion of quantum speed limit time.

quant-ph↗

arXiv2007

How to measure the wave-function absolute squared of a moving particle by using mirrors

We consider a slow particle with wave function $ψ_t(\vec{x})$, moving freely in some direction. A mirror is briefly switched on around a time $T$ and its position is scanned. It is shown that the measured reflection probability then allows the determination of $|ψ_T(\vec{x})|^2$. Experimentally available atomic mirrors should make this method applicable to the center-of-mass wave function of atoms with velocities in the cm/s range.

quant-ph↗

arXiv2006

Passage-time distributions from a spin-boson detector model

The passage-time distribution for a spread-out quantum particle to traverse a specific region is calculated using a detailed quantum model for the detector involved. That model, developed and investigated in earlier works, is based on the detected particle's enhancement of the coupling between a collection of spins (in a metastable state) and their environment. We treat the continuum limit of the model, under the assumption of the Markov property, and calculate the particle state immediately after the first detection. An explicit example with 15 boson modes shows excellent agreement between the discrete model and the continuum limit. Analytical expressions for the passage-time distribution as well as numerical examples are presented. The precision of the measurement scheme is estimated and its optimization discussed. For slow particles, the precision goes like $E^{-3/4}$, which improves previous $E^{-1}$ estimates, obtained with a quantum clock model.

quant-ph↗

arXiv2006

Investigation of a quantum mechanical detector model for moving, spread-out particles

We investigate a fully quantum mechanical spin model for the detection of a moving particle. This model, developed in earlier work, is based on a collection of spins at fixed locations and in a metastable state, with the particle locally enhancing the coupling of the spins to an environment of bosons. Appearance of bosons from particular spins signals the presence of the particle at the spin location, and the first boson indicates its arrival. The original model used discrete boson modes. Here we treat the continuum limit, under the assumption of the Markov property, and calculate the arrival-time distribution for a particle to reach a specific region.

quant-ph↗

arXiv2006

Simplified approach to double jumps for fluorescing dipole-dipole interacting atoms

A simplified scheme for the investigation of cooperative effects in the quantum jump statistics of small numbers of fluorescing atoms and ions in a trap is presented. It allows the analytic treatment of three dipole-dipole interacting four-level systems which model the relevant level scheme of Ba+ ions. For the latter, a huge rate of double and triple jumps was reported in a former experiment and the huge rate was attributed to the dipole-dipole interaction. Our theoretical results show that the effect of the dipole-dipole interaction on these rates is at most 5% and that for the parameter values of the experiment there is practically no effect. Consequently it seems that the dipole-dipole interaction can be ruled out as a possible explanation for the huge rates reported in the experiment.

quant-ph↗

arXiv2004

Matter diffraction at oblique incidence: Higher resolution and the Helium Trimer Efimov state

We study the diffraction of atoms and weakly-bound three-atomic molecules from a transmission grating at non-normal incidence. Due to the thickness of the grating bars the slits are partially shadowed. Therefore, the projected slit width decreases more strongly with the angle of incidence than the projected period, increasing, in principle, the experimental resolution. The shadowing, however, requires a revision of the theory of atom diffraction. We derive an expression in the style of the Kirchhoff integral of optics and show that the diffraction pattern exhibits a characteristic asymmetry which must be accounted for when comparing with experimental data. We then analyze the diffraction of weakly bound trimers and show that their finite size manifests itself in a further reduction of the slit width by (3/4) where is the average bond length. The improved resolution at non-normal incidence may in particular allow to discern, by means of their bond lengths, between the small ground state of the helium trimer ( =1 nm, Barletta and Kievsky, Phys. Rev. A 64, 042514 (2001)) and its predicted Efimov-type excited state ( =8 nm, ibid.), and in this way to experimentally prove the existence of this long-sought Efimov state.

quant-ph↗

arXiv2004

Cooperative quantum jumps for three dipole-interacting atoms

We investigate the effect of the dipole-dipole interaction on the quantum jump statistics of three atoms. This is done for three-level systems in a V configuration and in what may be called a D configuration. The transition rates between the four different intensity periods are calculated in closed form. Cooperative effects are shown to increase by a factor of 2 compared to two of either three-level systems. This results in transition rates that are, for distances of about one wavelength of the strong transition, up to 100% higher than for independent systems. In addition the double and triple jump rates are calculated from the transition rates. In this case cooperative effects of up to 170% for distances of about one wavelength and still up to 15% around 10 wavelengths are found. Nevertheless, for the parameters of an experiment with Hg+ ions the effects are negligible, in agreement with the experimental data. For three Ba+ ions this seems to indicate that the large cooperative effects observed experimentally cannot be explained by the dipole-dipole interaction.

quant-ph↗

arXiv2003

Cooperative fluorescence effects for dipole-dipole interacting systems with experimentally relevant level configurations

The mutual dipole-dipole interaction of atoms in a trap can affect their fluorescence. Extremely large effects were reported for double jumps between different intensity periods in experiments with two and three Ba^+ ions for distances in the range of about ten wave lengths of the strong transition while no effects were observed for Hg^+ at 15 wave lengths. In this theoretical paper we study this question for configurations with three and four levels which model those of Hg^+ and Ba^+, respectively. For two systems in the Hg^+ configuration we find cooperative effects of up to 30% for distances around one or two wave lengths, about 5% around ten wave lengths, and, for larger distances in agreement with experiments, practically none. This is similar for two V systems. However, for two four-level configurations, which model two Ba^+ ions, cooperative effects are practically absent, and this latter result is at odds with the experimental findings for Ba^+.

quant-ph↗

arXiv2003

Blinking molecules: Determination of Photo-Physical Parameters from the Intensity Correlation Function

An explicit expression is given for the correlation function of blinking systems, i.e. systems exhibiting light and dark periods in their fluorescence. We show through the example of terrylene in a crystalline host that it is possible to determine by means of this explicit expression photo-physical parameters, like Einstein coefficients and the mean light and dark periods by a simple fit. In addition we obtain further parameters like the frequency of the various intensity periods and the probability density of photons scattered off the host crystal. It turns out that this approach is simpler and allows greater accuracy than previous procedures.

quant-ph↗

arXiv2003

Diffraction of Weakly Bound Clusters: Spectroscopy and Size Effects

Exciting experiments in the field of atom and molecule optics have lately drawn much attention to the effects involved in the coherent diffraction of particle beams. We review the influence of the finite size of the particles and of their energy level spectrum on the diffraction pattern. In turn, we demonstrate how experimental diffraction measurements allow to determine these quantities of weakly bound molecules by considering the diffraction of the dimer and trimer of helium, and the deuterium molecule dimer.

quant-ph↗

arXiv2002

The intensity correlation function of "blinking" quantum systems

Explicit expressions are determined for the photon correlation function of ``blinking'' quantum systems, i.e. systems with different types of fluorescent periods. These expressions can be used for a fit to experimental data and for obtaining system parameters therefrom. For two dipole-dipole interacting $V$ systems the dependence on the dipole coupling constant is explicitly given and shown to be particularly pronounced if the strong driving is reduced. We propose to use this for an experimental verification of the dipole-dipole interaction.

quant-ph↗

arXiv2001

Quantum Effects in Matter-Wave Diffraction

Advances in micro-technology of the last years have made it possible to carry optics textbooks experiments over to atomic and molecular beams, such as diffraction by a double slit or transmission grating. The usual wave-optical approach gives a good qualitative description of these experiments. However, small deviations therefrom and sophisticated quantum mechanics yield new surprising insights on the size of particles and on their interaction with surfaces.

quant-ph↗

arXiv2001

Particle localization and the Notion of Einstein Causality

The notion of Einstein causality, i.e. the limiting role of the velocity of light in the transmission of signals, is discussed. It is pointed out that Nimtz and coworkers use the notion of signal velocity in a different sense from Einstein and that their experimental results are in full agreement with Einstein causality in its ordinary sense. We also show that under quite general assumptions instantaneous spreading of particle localization occurs in quantum theory, relativistic or not, with fields or without. We discuss if this affects Einstein causality.

quant-ph↗

arXiv2000

Towards the elusive Efimov state of the ^4He_3 molecule through a new atom-optical state-selection technique

Excited states and excitation energies of weakly bound systems, e.g. atomic few-body systems and clusters, are difficult to study experimentally. For this purpose we propose a new and very general atom-optical method which is based on inelastic diffraction from transmission gratings. The technique is applicable to the recently found helium trimer molecule ^4He_3, allowing for the first time an investigation of the possible existence of an excited trimer state and determination of its excitation energy. This would be of fundamental importance for the famous Efimov effect.

quant-ph↗

arXiv1999

Theoretical challenges in atom optics: Atomic and molecular diffraction by transmission gratings

A few years ago, diffraction of atoms by double slits and gratings was achieved for the first time, and standard optical wave-theory provided an excellent description of the experiments. More recently, diffraction of weakly bound molecules and even clusters has been observed. Due to their size and to possible breakup processes optical wave-theory is no longer adequate and a more sophisticated approach is needed. Moreover, surface effects, which can modify the diffraction pattern of atoms and molecules, give rise to further complications. In this article a fully quantum mechanical approach to these questions is discussed.

quant-ph↗