Fakultät für Physik - Digitale Hochschulschriften der LMU - Teil 01/05

Gemäß den aktuellen kosmologischen Modellen besteht der Großteil der Masse im Universum aus dunkler Materie. Aus früheren Studien ist es bekannt, daß Galaxien verschiedener Typen auf verschiedene Weise verteilt sind. Die räumliche Verteilung von Galaxien folgt nicht der Verteilung der Masse. Die Relation zwischen der Galaxienverteilung und der Massenverteilung wird »»Bias der Galaxienverteilung¶¶ genannt.

Laut den derzeitigen Modellen für die Bildung von Galaxien entstehen die Galaxien durch das Abkühlen und die Kondensation des baryonischen Gases innerhalb der Potentialtöpfe von virialisierten Klumpen aus dunkler Materie (dunkle Halos). Obwohl die hydrodynamischen Prozesse, die an der Entstehung von Galaxien beteiligt sind, noch wenig verstanden sind, wird angenommen, daß diese Prozesse für die Entstehung einzelner Objekte relevant sind und daß sie möglicherweiser keine bedeutende Rolle bei der gesamten räumlichen Verteilung der Galaxien spielen. Das bedeutet, daß das Problem der Verteilung der Galaxien und des sogenannten Bias der Galaxienverteilung gut untersucht werden kann, indem man die Verteilung von dunklen Halos betrachtet. Diese Annäherung ist sehr praktisch, weil bei der Haloentstehung und der Verteilung die Gravitation der einzige beteiligte physikalische Prozess ist.

In dieser Arbeit beschäftige ich mich mit den Eigenschaften der räumlichen Verteilung von dunklen Halos auf kosmischen Dichtefeldern. Die Analyse wird in zwei Hauptteilen durchgeführt. Im ersten Schwerpunkt studiere ich deterministische Bias- Modelle, die auf einem sphärischen Kollapsmodell, sowie auf einem ellipsoidförmigen Kollapsmodell beruhen. Im zweiten Teil meiner Arbeit konzentriere ich mich auf die stochastische Beschaffenheit des Bias der Halo- und Galaxien-Verteilung unter Verwendung der bedingten Wahrscheinlichkeitsfunktion.

Ich studiere den deterministischen Bias der Haloverteilung mit Hilfe von verschiedenen Modellen für die Bias Relation zwischen dunklen Halos und der darunterliegenden Materie. Mit der Benutzung von N-Körper Simulationen mit hoher Auflösung prüfe ich einige theoretische Modelle für die Streuung und für höherwertige Momente der Halo Verteilung in Modellen mit kalter dunkler Materie (CDM, englisch cold dark matter). Ich habe herausgefunden, daß die theoretischen Modelle des Bias, die auf einem sphärischen Kollapsmodell beruhen, die simulierten counts-in-cells Momente für Halos mit Massen grösser

als M* ziemlich genau beschreiben. M* wird als die Massenskala, auf der die Fluktuation des Dichtefeldes ein rms von ungefähr 1 hat, definiert. Eine bedeutende Verbesserung der theoretischen Beschreibung der simulierten counts-in-cells Momente für unter-M* Halos wird erzielt, wenn ein ellipsoidförmiges Kollapsmodell anstelle eines sphärischen für die Definition von dunklen Halos benutzt wird. Beide Versionen der Modelle sind besonders genau in der Beschreibung der counts-in-cells Momente der Nachkommen von Halos, die

bei hohen Rotverschiebungen ausgewählt worden sind. Deswegen sind diese Bias-Modelle ziemlich nützlich für die Interpretierung der Momente der Galaxienverteilung.

Als eine Anwendung der Bias-Modelle berechne ich die Voraussage der Modelle für die höherwertigen Momente der Verteilung der Lyman break Galaxien und deren Nachkommen. Es wird angenommen, daß die Lyman break Galaxien im Zentrum der massivsten Halos bei der Rotverschiebung z » 3 entstehen. Ich habe festgestellt, daß, obwohl der lineare Bias-Parameter b stark von der angenommenen Kosmologie abhängt, die Werte der höherwertigen Momente praktisch dieselben in beiden LamdaCDM und TafCDM Modelle sind.

Folglich können die höherwertigen Momente der räumlichen Verteilung dieser Objekte die kosmologische Parameter nicht eingrenzen.

Au¼erdem betrachte ich die stochastische Natur der Bias Relation vom Gesichtspunkt der bedingten Wahrscheinlichkeitsfunktion aus. Die stochastische Natur der Verteilung von dunklen Halos in einem kosmischen Dichtefeld zeigt sich in der Vert

Efficient Generation of Entangled Photon-Pairs

The first experiments with correlated photons have been performed in the context of

EPR-Bell experiments on the realistic and local properties of quantum mechanics. The

source used there produced pairs of polarization entangled photons from a 2-photon

decay of Calcium atoms. The technical requirements of these experiments were high

(vacuum systems, stronge dye-lasers, etc.) whereas the efficiency of the source was quite

low.

An important step forward was the introduction of spontaneous parametric down

conversion (SPDC), which has become the most common source in quantum optics for

generating correlated or entangled photon pairs. In this process photons of an intense

pump laser convert to photon pairs in an optical nonlinear crystal. Conservation of energy

and momentum leads to strong correlations between the generated photons. With this

kind of two-photon source it was possible to realize or improve many experiments on the

foundations of quantum mechanics addressing the EPR-Paradoxon and in the new field

of quantum information.

But again, more advanced experiments and applications suffer from the limited ef-

ficiency of the fluorescence process. Many photon pairs are lost by spatial and spectral

filtering, which is necessary to achieve polarization entanglement and long coherence times.

Different techniques have been implemented to increase the number of photon pairs

using two-crystal arrangements, focusing techniques or periodically poled crystals. Most

of these methods have the disadvantage that no entangled photons have been observed.

It is the aim of this work to increase the yield and to improve the mode definition

of entangled photon pairs generated by resonant enhancement of the pump mode and

the fluorescence modes. As a first step a linear cavity for the pump mode was realized.

Since the conversion probability is proportional to the pump power it was possible to

increase the photon pair count rate by factor of 7 over the previous source. Besides the

possibility of further improvement on already established pair correlation experiments,

such an enhancement allows to build a compact source for photon pairs, in which an

expensive argon-ion laser is replaced by a cheap diode laser. Among other applications

such sources are of strong interest for quantum cryptography.

3 In many quantum information experiments optical fibers are use to carry the photons

over long distance. Therefore, light from the parametric down-conversion source has to

be efficiently coupled into fibers. In the second part we report on a new method to

optimize collection efficiency by matching the angular distribution of the parametric

fluorescence to the spatial mode of an optical fiber. By using this technique, we detected

366500 polarization-entangled photon pairs per second in the near-infrared region in

single-mode optical fibers for 465 mW pump power (at 351.1 nm) with a 2 mm BBOcrystal.

The entanglement of the photon pairs was verified by measuring polarization

correlations of more than 96% in a HV-basis and in a ±45◦-basis. To our knowledge,

such enormous count rates of highly entangled photon pairs have not been reached yet

with any other technique.

In the third part of this thesis we investigated the process of parametric downconversion

in a cavity which is resonant to certain longitudinal down-conversion modes

only. The idea of placing the parametric down-conversion source inside a cavity is not

new. Such a device is usually referred to as a single or double resonant optical parametric

oscillator (OPO) and is mainly used to generate squeezed quantum states. In

that kind of application the system is operating close to but still under the threshold of

oscillation. In our application the situation is quite different. The system is operating far

below threshold so that mainly spontaneous emission occurs. In that mode correlations

between single photons can still be observed. But bouncing the light back and forth

inside the cav

A refined theory of line formation for neutral and singly-ionized iron in the atmospheres

of cool stars in presented. It accounts for departures from local thermodynamic

equilibrium –so-called non-LTE effects –an d serves the derivation

of spectroscopic stellar gravities “g” via the ionization equilibrium constraint.

The respective influence of various model assumptions (atomic data sets including

photo-ionization cross-sections and transition probabilities; the efficiency

of collisions with neutral particles and so forth) is investigated by means of extensive

profile analyses of iron lines in the optical spectrum of the Sun. Due to

the large scatter among laboratory transition probabilities the main free parameter

of the model –th e strength of collisions with neutral hydrogen SH which

govern excitation and ionization –can not be determined by inspecting the Sun

alone. Depending on the choice of SH non-LTE effects in neutral iron range from

0.01 dex to 0.10 dex in the logarithmic abundance, while singly-ionized iron follows

LTE.

To calibrate SH a set of local standard stars is employed whose surface gravities

can be inferred from astrometry (the Hipparcos satellite mission) on a given

temperature scale.

With the exception of the metal-rich standard star Procyon for which a mild

temperature correction is needed to fulfil the ionization balance, excellent consistency

is obtained for the metal-poor calibration stars in various evolutionary

stages if the Balmer profile temperatures are combined with an SH of 3. Departures

from LTE are moderate and amount to 0.1 dex in log g at most. The

correct choice of the van-der-Waals damping constants is generally more important

for these stars than non-LTE. The long-standing discrepancy with respect

to an independent spectroscopic gravity scale discussed in the literature can be

fully resolved.

Recent modifications to the formation theory of Balmer lines are implemented,

but do not yield consistent results: the temperature of the Sun is not recovered,

the observed Balmer line profiles of metal-poor stars are not reproduced.

The devised model is applied to two extreme objects, the most metal-poor star

known (100 000 times less metals than the Sun) and a so-called r-process star,

both recently identified in wide-angle sky survey. Here non-LTE effects are much

more pronounced (0.3 dex in log g ). This result emphasizes the importance of

considering departures from LTE for iron, especially when it comes to studying

giant stars in the Galactic halo.

The aim of this dissertation is to study the kinematics of 2-qubit pairs. A 2-qubit

state is specified by the Bloch vectors for each qubit and a 3 × 3 dyadic,

representing the expection values of the joint observables. In what follows the

main results are listed.

• The behaviour of a 2-qubit state under a local transformation is investi-gated,

where we obtain a set of 9 quantities, constructed from the Bloch

vectors and the cross dyadic, which are invariant under local transforma-tion.

We establish a generic form for a 2-qubit state, which helps us decide

if given two 2-qubit states belong to the same family or not. Using this

form, which depends on the characteristic values of the cross dyadic, one

can distinguish between all two-qubit families, where we obtain six classes

of families of locally equivalent states. Those families classify all 2-qubit

states, four of them consist of two subclasses each.

• A simple criterion is obtained to check if a 2-qubit state is separable or

not. This criterion does not depend on the eigenvalues of the partial trans-position

of the state in question but only on three numbers, those are the

coefficients of the eigenvalue equation. These numbers are written as a

function of the Bloch vectors and the cross dyadic. Also using the charac-teristic

values of the cross dyadic, one can decide if the CHSH inequality is

obeyed or violated.

• The properties of Lewenstein-Sanpera decompositions are employed to split

a given state into its optimal separable and pure parts. The degree of

separability is an important part in this decomposition, where a state is

more useful for quantum information purposes, the smaller its degree of

separability. The optimal degree of separability is obtained analytically for

some important cases as a function of the Bloch vectors and the cross dyadic

of the state in question: For the simple Werner state, where our result coincides with that obtained numerically. Also for the self-transposeed

states, the generalized for Werner states and rank-2 states.

• The technique of the rank-2 case is employed to study the case of rank-3

and the full rank, where we obtain all possible decompositions. Among

them, we choose the one which has the largest splitting parameter. In this

situation a numerical procedure could help in obtaining the optimal de-composition.

Given a density matrix, one obtains its eigenvalues and corre-sponding

eigenvectors. Using the rank-2 method to obtain all the possible

decompositions, then choose the one corresponding to the largest degree of

separability. If this decomposition is the optimal one, a technique is used

to check the optimality. If it is not optimal, an optimization procedure is

performed.

• There is an important relation between the degree of separability S and

what is the so called concurrence C which appears in the definition of the

entanglement of formation. In general, the sum of the degree of separabil-ity

and the concurrence is less than one, but equal to one for states with

vanishing Bloch vectors. On the other hand both of them run from 0 to 1.

• A criterion is introduced to decide if a given decomposition is the optimal

one or not. Two inequalities are obeyed if the decomposition in question is

the optimal one and are violated if it is not.

• The effect of the unitary operators and BCNOT operations are described

on the dynamical variables. An alternative presentation of the IBM and

Oxford purification protocols is obtained by using theses variables. One

could introduce the degree of separability as a purification parameter, where

the purified state has a smaller degree of separability than the initial one.

Employing the properties of the characteristic values of the cross dyadic,

makes the Oxford protocol much faster.