The subject of this thesis is the study of coherent interaction of light with matter (atoms) to improve the precision measurements and techniques. Special attention is drawn to get the narrow subnatural electromagnetically induced transmission (EIT), electromagnetically induced absorption (EIA) and nonlinear magneto-optic rotation (NMOR) caused by alkali atoms contained in a vapor cell. Subnatural polarization rotation introduces by a strong circularly polarized light in the absence of any external magnetic field was also studied. A detailed theoretical treatment, given in this dissertation, allows to associate each of the features of the spectra with a special physical mechanism. Many quantum phenomena related to interferences, coherences, optical pumping etc. experiments are studied using home-built diode lasers. This thesis also describes laser cooling and trapping of rubidium atoms using two techniques. Deflection of cold atoms horizontally from MOT using pushing beams are discussed in close consideration for the improvements in the precision measurements. This thesis is organized as follows. In Chapter 1, an introduction to the importance of sub-natural narrow resonance and simplified technique in the precision measurement will be discussed. In Chapter 2, an introduction to EIT, EIA and NMOR resonance are discussed. This chapter will provide a basic theoretical background of atom-field interactions, especially for Λ-type and N -type systems and its steady state solution using density matrix analysis and experimental tools. The most important notion of laser cooling of atoms, ions or molecule i.e., exchange of momentum between light and atoms combining with the Doppler effect will be discussed. In Chapter 3, the observation of subnatural EIT and subnatural EIA in closed and open degenerate two-level system using room temperature vapor cell filled with Rb will be presented. Physical mechanisms that contribute to EIT and EIA, and the contrast of our results from the coherent population trapping (CPT)–type resonances are discussed in detail in appendix A.2 according to our experimental results. In Chapter 4, the narrowing of subnatural EIT and subnatural EIA linewidth in closed and open systems again in degenerate two level transition, using the “Laguerre-Gaussian” control beam instead of generally studied Gaussian beam, will be discussed in detail. In Chapter 5, the conversion between subnatural EIT to subnatural EIA in a degenerate Λ system will be discussed. The physical mechanism created by the introduction of a coherent counter propagating control laser to the co-propagating probe and the control laser are studied. The effect of polarization and axial velocity Doppler averaging will also be studied. In Chapter 6, we will discuss the sensitive technique for precise measurement of small magnetic fields using the NMOR, by chopping the resonant laser beam. We will study the sensitivity and the potential application of this technique in the measurement of an atomic electric-dipole moment. In Chapter 7, we will be studying about the induced optical rotation by a circularly polarized control laser on the linearly polarized probe laser. The effect of the intensity of the control laser beam on the higher order optical rotation will also be studied. In Chapter 8, we will be studying about the cooling and trapping of 87Rb in magneto-optic trap. We will be studying two techniques of trapping of atoms in MOT. The cold cloud of atoms from the MOT are deflected horizontally by using different configuration of pushing beam are studied. A brief summary and outlook of my thesis work will be discussed at Chapter 9.

In dieser Arbeit untersuchen wir kollektive Strahlungseffekte in Nanofaser-gekoppelten atomaren Ensembles, die sich über Tausende von optischen Wellenlängen erstrecken. Wir koppeln bis zu 1000 Atome optisch an die geführten Moden einer optischen Nanofaser, die langreichweitige Dipol-Dipol Wechselwirkungen zwischen den Atomen vermittelt. Wir realisieren eine unidirektionale Kopplung und damit ein kaskadiertes Quantensystem, in dem die Dynamik jedes Atoms ausschließlich durch die Dynamik der vorgelagerten Atome bestimmt wird. Wir regen die Atome mit nanofasergeführten optischen Pulsen kohärent an, was uns ermöglicht, den gesamten Parameterbereich von schwacher Anregung bis hin zur voll-ständigen Inversion zu erforschen. Wir stellen fest, dass die kohärente Vorwärtsstreuung, die für die Superradianz im Regime der schwachen Anregung verantwortlich ist, auch nahe voller Inversion eine wichtige Rolle für die Dynamik spielt. Wir beobachten superradiante Puls-Dynamik, die in unserem System trotz des makroskopischen Abstands zwischen den Atomen und einer asymmetrischen Kopplung auftritt. Wir stellen fest, dass die emittierte Spitzenleistung noch schneller mit der Anzahl der Atome skaliert als im Fall der idealen Dicke Superradianz, was auf eine kollektiv erhöhte Sammeleffizienz der nanofasergeführten Mode zurückzuführen ist. Die Analyse der Kohärenz-Eigenschaften des superradianten Pulses erlaubt es uns, zwei Regime der Puls-Dynamik zu identifizieren. Wir entwickeln ein kaskadiertes Wechselwirkungsmodell und zeigen, dass es die kollektive Dynamik unseres Systems über den gesamten in dieser Arbeit untersuchten Parameterbereich akkurat beschreibt. Schließlich untersuchen wir die getriebene Dynamik eines Nanofaser-gekoppelten Ensembles von Drei-Niveau-Atomen. Wir treiben Zwei-Photonen-Rabi-Oszillationen zwischen den beiden Grundzuständen eines $\Lambda$-Systems und beobachten die damit verbundene oszillatorische Raman-Verstärkung und -Absorption. / In this thesis, we study collective radiative effects in nanofiber-coupled atomic ensembles that extend over thousands of optical wavelengths. We optically couple up to 1000 atoms to the guided modes of an optical nanofiber, which mediates long-range dipole-dipole interactions between the atoms. We engineer the coupling to be unidirectional, realizing a cascaded quantum system in which the dynamics of each atom is solely determined by the dynamics of upstream atoms. We coherently excite the atoms using nanofiber-guided optical pulses, allowing us to explore the entire parameter regime from weak excitation to full inversion. We find that coherent forward scattering, which is responsible for superradiance in the weak excitation regime, plays an important role for the dynamics even close to full inversion. We observe superradiant burst dynamics, which occurs in our system despite the macroscopic separation between the atoms and an asymmetric coupling. We find that the peak-emitted power scales even faster with the number of atoms than in the case of ideal Dicke superradiance due to a collectively enhanced channeling efficiency into the nanofiber-guided mode. By analyzing the coherence properties of the superradiant burst, we directly identify two regimes of burst dynamics. In the second regime, there is no initial coherence, and the superradiant burst is seeded by vacuum fluctuations. We introduce a cascaded interaction model and find that it accurately describes the collective dynamics of our system over the entire parameter regime explored in this thesis. Finally, we study the driven dynamics of a nanofiber-coupled ensemble of three-level atoms. We drive two-photon Rabi oscillations between the two ground states of a $\Lambda$ system and observe the associated oscillatory Raman gain and absorption.

The physics of ultracold quantum gases has been the subject of a long-lasting and intense research activity, which started almost a century ago with purely theoretical studies and had a fluorishing experimental development after the implementation of laser and evaporative cooling techniques that led to the first realization of a Bose Einstein condensate (BEC) over 25 years ago. In recent years, a great interest in ultracold atoms has developed for their use as platforms for quantum technologies, given the high degree of control and tunability offered by ultracold atom systems. These features make ultracold atoms an ideal test bench for simulating and studying experimentally, in a controlled environment, physical phenomena analogous to those occurring in other, more complicated, or even inaccessible systems, which is the idea at the heart of quantum simulation. In the rapidly developing field of quantum technologies, it is highly important to acquire an in-depth understanding of the state of the quantum many-body system that is used, and of the processes needed to reach the desired state. The preparation of the system in a given target state often involves the crossing of second order phase transitions, bringing the system strongly out-of-equilibrium. A better understanding of the out-of-equilibrium processes occurring in the vicinity of the transition, and of the relaxation dynamics towards the final equilibrium condition, is crucial in order to produce well-controlled quantum states in an efficient way. In this thesis I present the results of the research activity that I performed during my PhD at the BEC1 laboratory of the BEC center, working on ultracold gases of 23Na atoms in an elongated harmonic trap. This work had two main goals: the accurate determination of the equilibrium properties of a Bose gas at finite temperature, by the measurement of its equation of state, and the investigation of the out-of-equilibrium dynamics occurring when a Bose Einstein condensate is prepared by cooling a thermal cloud at a finite rate across the BEC phase transition.To study the equilibrium physics of a trapped atomic cloud, it is crucial to be able to observe its density distribution in situ. This requires a high optical resolution to accurately obtain the density profile of the atomic distribution, from which thermodynamic quantities can then be extracted. In particular, in a partially condensed atomic cloud at finite temperature, it is challenging to resolve well also the boundaries of the BEC, where the condensate fraction rapidly drops in a narrow spatial region. This required an upgrade of the experimental apparatus in order to obtain a high enough resolution. I designed, tested and implemented in the experimental setup new imaging systems for all main directions of view. Particular attention was paid for the vertical imaging system, which was designed to image the condensates in trap with a resolution below 2 μm, with about a factor 4 improvement compared to the previous setup. The implementation of the new imaging systems involved a partial rebuilding of the experimental apparatus used for cooling the atoms. This created the occasion for an optimization of the whole system to obtain more stable working conditions. Concurrently I also realized and included in the experiment an optical setup for the use of a Digital Micromirror Device (DMD) to project time-dependent arbitrary light patterns on the atoms, creating optical potentials that can be controlled at will. The use of this device opens up exciting future scenarios where it will be possible to locally modify the trapping potential and to create well-controlled barriers moving through the atomic cloud. Another challenge in imaging the density distribution in situ is determined by the fact that the maximum optical density (OD) of the BEC, in the trap center, exceeds the low OD of the thermal tails by several orders of magnitude. In order to obtain an accurate image of the whole density profile, we developed a minimally destructive, multi-shot imaging technique, based on the partial transfer of a fraction of atoms to an auxiliary state, which is then probed. Taking multiple images at different extraction fractions, we are able to reconstruct the whole density profile of the atomic cloud avoiding saturation and maintaining a good signal to noise ratio. This technique, together with the improvements in the imaging resolution, has allowed us to accurately obtain the optical density profile of the Bose gas in trap, from which the 3D density profile was then calculated applying an inverse Abel transform, taking advantage of the symmetry of the trap. From images of the same cloud after a time-of-flight expansion, we measured the temperature of the gas. From these quantities we could find the pressure as a function of the density and temperature, determining the canonical equation of state of the weakly interacting Bose gas in equilibrium at finite temperature. These measurements also allowed us to clearly observe the non-monotonic temperature behavior of the chemical potential near the critical point for the phase transition, a feature that characterizes also other superfluid systems, but that had never been observed before in weakly interacting Bose gases. The second part of this thesis work is devoted to the study of the dynamical processes that occur during the formation of the BEC order parameter within a thermal cloud. The cooling at finite rate across the Bose-Einstein condensation transition brings the system in a strongly out-of-equilibrium state, which is worth investigating, together with the subsequent relaxation towards an equilibrium state. This is of interest also in view of achieving a better understanding of second order phase transitions in general, since such phenomena are ubiquitous in nature and relevant also in other platforms for quantum technologies. A milestone result in the study of second order phase transitions is given by the Kibble-Zurek mechanism, which provides a simple model capturing important aspects of the evolution of a system that crosses a second-order phase transition at finite rate. It is based on the principle that in an extended system the symmetry breaking associated with a continuous phase transition can take place only locally. This causes the formation of causally disconnected domains of the order parameter, at the boundaries of which topological defects can form, whose number and size scale with the rate at which the transition is crossed, following a universal power law. It was originally developed in the context of cosmology, but was later successfully tested in a variety of systems, including superfluid helium, superconductors, trapped ions and ultracold atoms. The BEC phase transition represents in this context a paradigmatic test-bench, given the high degree of control at which this second-order phase transition can be crossed by means of cooling ramps at different rates. Already early experiments investigated the formation of the BEC order parameter within a thermal cloud, after quasi-instantaneous temperature quenches or very slow evaporative cooling. In the framework of directly testing the Kibble-Zurek mechanism, further experiments were performed, both in 2D and 3D systems, focusing on the emergence of coherence and on the statistics of the spontaneously generated topological defects as a function of the cooling rate. The Kibble-Zurek mechanism, however, does not fully describe the out-of-equilibrium dynamics of the system at the transition, nor the post-quench interaction mechanisms between domains that lead to coarse-graining. Most theoretical models are based on a direct linear variation of a single control parameter, e.g. the temperature, across the transition. In real experiments, the cooling process is controlled by the tuning of other experimental parameters and a global temperature might not even be well defined, in a thermodynamic sense, during the whole process. Moreover, the temperature variation is usually accompanied by the variation of other quantities, such as the number of atoms and the collisional rate, making it difficult to accurately describe the system and predict the post-quench properties. Recent works included effects going beyond the Kibble-Zurek mechanism, such as the inhomogeneity introduced by the trapping potential, the role of atom number losses, and the saturation of the number of defects for high cooling rates. These works motivate further studies, in particular of the dynamics taking place at early times, close to the crossing of the critical point. The aim of the work presented in this thesis is to further investigate the timescales associated to the formation and evolution of the BEC order parameter and its spatial fluctuations, as a function of the rate at which the transition point is crossed. We performed experiments producing BECs by means of cooling protocols that are commonly used in cold-atom laboratories, involving evaporative cooling in a magnetic trap. We explored a wide range of cooling rates across the transition and found a universal scaling for the growth of the BEC order parameter with the cooling rate and a finite delay in its formation. The latter was already observed in earlier works, but for a much more limited range of cooling rates. The evolution of the fluctuations of the order parameter was also investigated, with an analysis of the timescale of their decay during the relaxation of the system, from an initial strongly out-of-equilibrium condition to a final equilibrium state. This thesis is structured as follows: The first chapter presents the theoretical background, starting with a brief introduction to the concept of Bose Einstein condensation and a presentation of different models describing the thermodynamics of an equilibrium Bose gas. The second part of this chapter then deals with the out-of-equilibrium dynamics that is inevitably involved in the crossing of a second-order phase transition such as the one for Bose-Einstein condensation. The Kibble-Zurek mechanism is briefly reviewed and beyond KZ effects are pointed out, motivating a more detailed investigation of the timescales involved in the BEC formation. In the second chapter, I describe the experimental apparatus that we use to cool and confine the atoms. Particular detail is dedicated to the parts that have been upgraded during my PhD, such as the imaging system. In the third chapter I show our experimental results on the measurement of the equation of state of the weakly interacting uniform Bose gas at finite temperature. In the fourth chapter I present our results on the out-of-equilibrium dynamics in the formation of the condensate order parameter and its spatial fluctuations, as a function of different cooling rates.