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Concept# Phase space

Summary

In dynamical systems theory and control theory, a phase space or state space is a space in which all possible "states" of a dynamical system or a control system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. It is the direct product of direct space and reciprocal space. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs.
Principles
In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system or allowed combination of values of the system's parameters, a point is included in the multidimensional space. The system's evolving state

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The exceptional performance of self-assembled Quantum Dot (QD) materials renders them extremely appealing for their use as optical communications devices. As lasers, they feature reduced and temperature independent threshold current and proper emission wavelength at the fiber telecommunication windows. These characteristics, together with the low linewidth enhancement factor and broad spectrum, make QD materials extremely attractive for application as light emitters or amplifiers. There exist, nevertheless, several unclear issues which prevent QDs from conquering the new generation of optoelectronic devices. Their differential efficiency is lower than expected. The output power of QD lasers is lower than that of their quantum well counterpart. Still, it is their dynamics which has incited the majority of studies. The modulation bandwidth of these devices seems to be limited by the relaxation of carriers from the upper energetic layers to the low levels within the dot. Besides, the electron-hole interaction is widely unknown, the extent of the electron-hole Coulombic attraction is not yet established. Throughout this thesis I present a theoretical and experimental study of the gain and phase dynamics of quantum dot lasers. I explain the appearance of different decay times observed in pump and probe experiments in QD amplifiers as a result of the different electron and hole relaxation times, by means of an electron-hole rate-equation model. The ultrafast hole relaxation first leads to an ultrafast recovery of the gain, which is then followed by electron relaxation and, on the nanosecond timescale, radiative and non-radiative recombinations. The phase dynamics is slower and is affected by thermal redistribution of carriers within the dot. Our results corroborate with spectral measurements of the dephasing and gain in QD amplifiers. Additionally, our work is compared with existing pump and probe results. Exploiting the capacity of QD lasers to emit at two different wavelengths corresponding to the ground state (GS) and excited state (ES), I present a theoretical study of the QD dynamics, based on a linearization of the QD rate-equations. The results predict the existence of single oscillation frequency of GS and ES, meaning that both states are highly coupled. In order to verify our theory, we perform two kinds of experiments. By modulating these lasers at high frequency, we measure separately the dynamics of GS and ES. However, in contradiction to our theory, two different modulation frequencies are found. Additional temporally-resolved measurements of the laser dynamics reveal a surprising effect. By injecting a sub-bandgap pump in an InAs/InGaAs QD laser, the emitted photons are depleted. Through additional transmission and photocurrent measurements, we relate this observation with carrier photoexcitation, which was so far only theoretically addressed. The role of carrier photoexcitation in our experimental laser dynamics is further supported by a rate-equation model. Impelled by this finding, we study the effect of carrier photoexcitation in the static and dynamic characteristics. We find that carrier photoexcitation reduces the efficiency of QD lasers, which is one of the major QD handicaps, and depletes the GS lasing after the ES threshold, as observed experimentally. Moreover, by adding carrier photoexcitation to our linearization of the rate-equations, we find that the theory predicts the appearance of two lasing resonance frequencies, in agreement with our previous experimental results. Additionally, we deal with the improvement of carrier relaxation. In tunnel injection devices, carriers are given an additional path towards the ground state of the dot by growing a quantum well layer close to the QD active plane. Through the quantum-mechanical tunneling effect, carriers relax from the nearby quantum well layer to the QDs, which speeds up relaxation. We aim at the increase of the modulation bandwidth while keeping the good performances quantum dot lasers have exhibited, such as low and temperature insensitive threshold current and proper emission wavelength. In the final part of this work, we present dynamical measurements of 1.5 µm InAs/InP tunnel injection and non-tunnel injection QD lasers, which display remarkable static characteristics. After proving with static measurements that tunnel injection is actually taking place in these structures, we show several dynamic measurements. Pump and probe measurements on QD devices show that the tunnel injection samples exhibit a slightly faster relaxation time than the non-tunnel injection samples used as reference, meaning that relaxation time is improved with tunnel injection. 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Starting from the quantum statistical master equation derived in [1] we show how the connection to the semi-classical Boltzmann equation (SCBE) can be established and how irreversibility is related to the problem of separability of quantum mechanics. Our principle goal is to find a sound theoretical basis for the description of the evolution of an electron gas in the intermediate regime between pure classical behavior and pure quantum behavior. We investigate the evolution of one-particle properties in a weakly interacting N-electron system confined to a finite spatial region in a near-equilibrium situation that is weakly coupled to a statistical environment. The equations for the reduced n-particle density matrices, with n < N are hierarchically coupled through two-particle interactions. In order to elucidate the role of this type of coupling and of the inter-particle correlations generated by the interaction, we examine first the particular situation where energy transfer between the N-electron system and the statistical environment is negligible, but where the system has a finite memory. We then formulate the general master equation that describes the evolution of the coarse grained one-particle density matrix of an interacting confined electron gas including energy transfer with one or more bath subsystems, which is called the quantum Boltzmann equation (QBE). The connection with phase space is established by expressing the one-particle states in terms of the overcomplete basis of coherent states, which are localized in phase space. In this way we obtain the QBE in phase space. After performing an additional coarse-graining procedure in phase space, and assuming that the interaction of the electron gas and the bath subsystems is local in real space, we obtain the semi-classical Boltzmann equation. The validity range of the classical description, which introduces local dynamics in phase space is discussed.

Computer simulations of biomolecules such as molecular dynamics simulations are limited by the time scale of conformational rearrangements. Several sampling techniques are available to search the multi-minima free energy landscape but most efficient, time-dependent methods do generally not produce a canonical ensemble. A sampling algorithm based on a self-regulating ladder of searching copies in the dihedral subspace is developped in this paper. The learning process using short-and long-term memory functions allows an efficient search in phase space while combining a deterministic dynamics and stochastic swaps with the searching copies conserves a canonical limit. The sampling efficiency and accuracy are indicated by comparing the ansatz with conventional molecular dynamics and replica exchange simulations.

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