#### A more detailed description of the experiments can be found here.

## Introduction

In current day computer and communication technology information is represented in bits. Computation is performed by applying a series of logical gates to the bits, and in between storing their value in a memory. Quantum computation takes these ideas to the the quantum world. Loosely speaking, using the superposition concept, many computations can be done in parallel thus achieving an exponential speed up. Quantum bits (qubits) replace the classical bit, with the important feature that and they are capable of being in a coherent superposition of the two underlying logical states. Other counterparts of classical computation span from quantum memories to quantum processors.

From a practical point of view, a qubit can be implemented as any system with two energy levels. We shall refer to it in this work as a two-level system (TLS). In most isolated quantum systems there are many discrete levels, but we will restrict transitions and measurements to only two of them, and treat it as an effective TLS. Many physical realizations of TLS exist, including photons, ions, quantum dots, Josephson junction loops and atoms. Each of these systems has advantages and disadvantages, and it seems that ultimately a quantum network will combine several of them to benefit from their very different properties.

Two of the most used physical qubits are photons and neutral atoms. Photons are easy to manipulate and produce. They interact weakly with their environment and therefore can remain coherent for long travel distances. This last advantage is also their disadvantage: interaction between photons is usually very small, making the implementation of an all optical two-qubit gate very difficult. Atoms, on the other hand, are easy to keep in one place, and can interact strongly with both other atoms and external electromagnetic fields. It is therefore sensible to use atoms as “stationary qubits” for storage and manipulation and use photons as “flying qubits” that interchange the information between separated sites.

One of the controlled ways of interaction between atoms and photons which is widely used is the electro-magnetically induced transparency (EIT). In this scheme atoms with a lambda-shape energy structure interact with two light fields called “pump” and “probe”. The pump is usually much stronger than the probe, and is used to control the interaction strength between the probe and the atomic ensemble. Closing the pump light while the probe is propagating in the atomic ensemble leads to a conversion of the photonic excitation into the coherence between the two low lying states of the atoms. This is sometimes called “storage of light”, although only the coherence which was carried by the light is actually stored in the ensemble. The beauty of this conversion process is that it is reversible, which makes the atomic ensemble a true memory.

In order to increase the efficiency of the storage and retrieval processes, it is desired to work with atomic ensembles with high optical depth. This is because the coupling of the atoms to the external electromagnetic field scales as the square root of the number of atoms in a volume where the light intensity is approximately uniform. Working at high optical depth, however, usually implies that the atomic density is high and the rate of inter-particle collisions is large compared to the storage time. One of the goals of this work is to understand how this fact changes the time dynamics of a stored coherence. The question we address is how rapid velocity changing elastic collisions change the decay function of the ensemble coherence, and what is the asymptotic coherence time in such a scenario.

The ensemble I consider in the thesis is that of cold atoms trapped in a conservative potential. In practice, the atoms are cooled and trapped by lasers and are very well isolated from their surrounding. When trapped in an optical dipole potential, the atoms can be kept for many seconds without scattering photons or interacting with other atoms coming from the walls of the vacuum chamber. This, together with the excellent controllability one has over the experimental parameters, makes this system ideal for studying the effect elastic collisions have on the ensemble coherence. Another point which simplifies things in a cold ensemble is the fact that s-wave scattering is the dominant collision process.

From the point of view of a particular atom, other atoms can be regarded as the environment, and collisions with them can be treated as the coupling to a bath. Also, since the phase space density is low enough, the motion of each atom is to very good approximation classical. The theoretical framework I adopt in this thesis, therefore, will be that of an effective two level system in contact with a Lorentzian reservoir. The experimental results confirm that this is indeed a good approximation.

## The experimental setup

In the experiment, ^{87}Rb atoms are trapped in a dipole potential created by a far-off-resonance laser, and can be cooled down to quantum degeneracy (Bose-Einstein condensation).

The two relevant internal states are |1>=|F=1;m_{f}=-1> and |2>=|F=2;m_{f}=1> in the 5^{2}S_{1/2} manifold, which are, to first order, Zeeman insensitive to magnetic fluctuations in the applied magnetic field of 3.2G. Initially ~10^{9} atoms are trapped and cooled in a magneto-optical trap, and further cooled by Sisyphus and Raman sideband techniques. The technique of rapid adiabatic passage with constant RF radiation and a ramped magnetic field is then used to transfer the atoms from state |5^{2}S_{1/2},F=1;m_{f}=1> ending with 80% of the atoms at |1>, and the rest in state |5^{2}S_{1/2},F=1;m_{f}=0>. The atoms are loaded into an optical dipole trap created by two horizontal crossing beams at an angle of 28^{o}, creating an oval trap with an aspect ratio of 1:3.9. The 50um waist laser beams originate from a single frequency Ytterbium fiber laser at 1064nm. Their polarization is parallel to the magnetic field, and frequency differ by 120MHz to eliminate standing waves. The external control is composed of RF radiation at 2.15MHz and microwave radiation at ~6.8GHz, both locked to an atomic standard.

The state populations is measured by recording the fluorescence of a detection beam resonant with a transition to an excited 5^{2}_{P3/2} state. An effect which should be taken into account is the levels shift induced by the MW field. We have carefully measured this shift, which in the maximum MW power reaches to ~50Hz, and it is taken into account when setting the frequency of the external fields. In most of the experiments we carry out evaporative cooling for 2.5sec to a laser power of 0.16W and back to the final laser value which determines the thermodynamic conditions in the experiment. The typical spontaneous scattering rate is less than 1 1/s and the trap lifetime is longer than 5s. The temperature is measured by an absorption imaging after a time of flight.

The experiments

The first question I have addressed is how does the spectrum change due to elastic collisions. A different coupling of the internal states to the external potential induces a static inhomogeneous broadening of the ideally delta-function spectrum of an isolated atom. Intriguingly, time-dependent fluctuations narrow the linewidth and prolong the coherence time – a phenomenon first observed in NMR and aptly named “motional narrowing”. We have performed experiments revealing the analogous effect in optically trapped ^{87}Rb atoms, where there is a prolongation of the coherence time as the density increases owing to velocity-changing elastic collisions. We have further showed theoretically and experimentally that the new dephasing timescale universally depends only on the atomic phase space density. We support our findings with classical molecular dynamics Monte-Carlo simulations.

The second question I have studied is how this effect depends on the physical model of the fluctuations. We considered a fluctuation model in which the ensemble is coupled to an environment which induces random jumps separated by times which have a Poisson distribution. For this model a closed-form formula for the spectrum in terms of the inhomogeneous spectrum and the Poisson rate constant exists. We have shown that even for a normal distribution of frequencies in the ensemble, the spectrum is different for “hard” s-wave scattering collisions compared to soft forward scattering like processes. We also demonstrated experimentally that this model correctly accounts for the spectrum arising from cold elastic collisions in an optically trapped ensemble, without fitting parameters.

Another benefit of the spectral jumps model is that the solution for the spectrum holds for any frequency distribution. In particular, we have found that for a a fat-tail distribution fluctuations have the reverse effect, namely instead of narrowing the spectrum they lead to broadening. Using ideas from the mathematical theory of sums of identical and independent variables, we proved that this behavior arises for frequency distributions which belong to the domain of attraction of a Levy stable law with a characteristic exponent smaller than 1.

Finally, I have addressed the question of whether the coherence time of a collisional-narrowed atomic ensemble can be extended by applying external control fields. Fluctuations at low frequencies can be overcome by a single population inverting pulse – the celebrated coherence echo technique. As the collision rate increases this is no longer possible since the average frequency of each atom before and after the echo pulse is not the same. Dynamical decoupling theories generalize this technique to multi-pulse sequences and enable the suppression of noise at a higher frequency. We have employed these ideas in our apparatus and demonstrated a 20-fold increase of the coherence time when a dynamical decoupling sequence with more than 200 pi-pulses was applied. Using quantum process tomography we have showed that a dense ensemble with an optical depth of 230 can be used as an atomic memory with coherence times exceeding 3 seconds. We have also found that although the ensemble is a non-Gaussian many-body system which is almost decoupled from the environment, the coherence time with the dynamical decoupling sequence is described well by an effective single spin coupled to a Gaussian reservoir.