The quantum phase of a Bose-Einstein condensate has long been a subject fraught with misunderstanding and confusion. In this thesis we provide a consis- tent description of this phenomenon and, in particular, discuss how phase may be defined, created, manipulated, and controlled. We begin by describing how it is possible to set up a reference condensate against which the phase of other condensates can be compared. This allows us to think of relative phases as if they were absolute and gives a clear and precise definition to 'the phase of a condensate'. A relative phase may also be established by coupling condensates and we show how this can be controlled. We then extend this model to explain how the phase along a chain of coupled condensates can lock naturally without the need for any measurements. The second part of the thesis deals primarily with the link between entangle- ment and phase. We show that, in general, the more entangled a state is, the better its phase resolution. This leads us to consider schemes by which maximally entangled states may be able to be created since these should give the best prac- tical advantages over their classical counterparts. We consider two such states: a number correlated pair of condensates and a Schrodinger cat state. Both schemes are shown to be remarkably robust to loss. A comparison of the merits of these two states, as the inputs to an interferom- eter, reveals very different behaviours. In particular, the number correlated state performs significantly better than the cat state in the presence of loss, which means that it might be useful in interferometry and frequency standard schemes where phase resolution is of the utmost importance. Finally, we propose a scheme for concentrating the entanglement between con- densates, which is an important step in quantum communication protocols. This, along with the ability to manipulate phase and entanglement, suggests that the future for condensates holds not only academic interest but great potential for practical applications.
The quantum phase of a Bose-Einstein condensate has long been a subject fraught with misunderstanding and confusion. In this thesis we provide a consis- tent description of this phenomenon and, in particular, discuss how phase may be defined, created, manipulated, and controlled. We begin by describing how it is possible to set up a reference condensate against which the phase of other condensates can be compared. This allows us to think of relative phases as if they were absolute and gives a clear and precise definition to 'the phase of a condensate'. A relative phase may also be established by coupling condensates and we show how this can be controlled. We then extend this model to explain how the phase along a chain of coupled condensates can lock naturally without the need for any measurements. The second part of the thesis deals primarily with the link between entangle- ment and phase. We show that, in general, the more entangled a state is, the better its phase resolution. This leads us to consider schemes by which maximally entangled states may be able to be created since these should give the best prac- tical advantages over their classical counterparts. We consider two such states: a number correlated pair of condensates and a Schrodinger cat state. Both schemes are shown to be remarkably robust to loss. A comparison of the merits of these two states, as the inputs to an interferom- eter, reveals very different behaviours. In particular, the number correlated state performs significantly better than the cat state in the presence of loss, which means that it might be useful in interferometry and frequency standard schemes where phase resolution is of the utmost importance. Finally, we propose a scheme for concentrating the entanglement between con- densates, which is an important step in quantum communication protocols. This, along with the ability to manipulate phase and entanglement, suggests that the future for condensates holds not only academic interest but great potential for practical applications.
This thesis investigates thermalisation, correlations and entanglement in Bose-Einstein condensates. Bose-Einstein condensates are ultra-cold collections of identical bosonic atoms which accumulate in a single quantum state, forming a mesoscopic quantum object. They are clean and controllable quantum many-body systems that permit an unprecedented degree of experimental flexibility compared to other physical systems. Further, a tractable microscopic theory exists which allows a direct and powerful comparison between theory and experiment, propelling the field of quantum atom optics forward at an incredible pace. Here we explore some of the fundamental frontiers of the field, examining how non-classical correlations and entanglement can be created and measured, as well as how non-classical effects can lead to the rapid heating of atom clouds. We first investigate correlations between two weakly coupled condensates, a system analogous to a superconducting Josephson junction. The ground state of this system contains non-classical number correlation arising from the repulsion between the atoms. Such states are of interest because they may lead to more precise measurement devices such as atomic gyroscopes. Unfortunately thermal fluctuations can destroy these correlations, and great care is needed to experimentally observe non-classical effects. We show that adiabatic evolution can drive the isolated quantum system out of thermal equilibrium and decrease thermal noise, in agreement with a recent experiment [Esteve et al. Nature 455, 1216 (2008)]. This technique may be valuable for observing and using quantum correlated states in the future. Next, we analyse the rapid heating that occurs when a condensate is placed in a moving periodic potential. The dynamical instability responsible for the heating was the subject of much uncertainty, which we suggest was due to the inability of the mean-field approximation to account for important spontaneous scattering processes. We show that a model including non-classical spontaneous scattering can describe dynamical instabilities correctly in each of the regimes where they have been observed, and in particular we compare our simulations to an experiment performed at the University of Otago deep inside the spontaneous scattering regime. Finally, we proposed a method to create and detect entangled atomic wave-packets. Entangled atoms are interesting from a fundamental perspective, and may prove useful in future quantum information and precision measurement technologies. Entanglement is generated by interactions, such as atomic collisions in Bose-Einstein condensates. We analyse the type of entanglement generated via atomic collisions and introduce an abstract scheme for detecting entanglement and demonstrating the Einstein-Podolsky-Rosen paradox with ultra-cold atoms. We further this result by proposing an experiment where entangled wave-packets are created and detected. The entanglement is generated by the pairwise scattering that causes the instabilities in moving periodic potentials mentioned above. By careful arrangement, the instability process can be controlled to to produce two well-defined atomic wave-packets. The presence of entanglement can be proven by applying a series of laser pulses to interfere the wave-packets and then measuring the output populations. Realising this experiment is feasible with current technology.
We discuss a scheme for using entangled Bose-Einstein condensates to detect phase differences with a resolution better than the standard quantum limit. To date, schemes have shown that the enhancement in phase resolution gained by entangling condensates is lost when dissipation is present. Here we show how this can be overcome by using number correlated condensates, as have been produced recently in the laboratory. We also outline a scheme for measuring this phase that is not destroyed when the effects of finite detector efficiency are considered.
Three different schemes for producing Schrödinger cat states in Bose-Einstein condensates are outlined and the effects of loss in each of them compared. The first scheme involves coupled interacting condensates and proves to be very fragile to loss. This is improved upon with a second scheme which first evolves a cat state in phase space and then rapidly transforms it to a number cat state. Finally a third scheme is discussed which makes use of number correlated condensates and is remarkably robust to loss. It may prove to be valuable for experimentally creating such states.
In the years following the first Bose-Einstein condensation of dilute atomic gases in 1995, several experiments demonstrated many fundamental properties of two-component BEC. Subsequent to these pioneering experiments, almost a decade passed with little experimental activity, relative to investigations of single-component condensates. In somewhat of a renaissance, the initial observations of two-component BEC have been revisited in recent years, consolidating and clarifying results of the early works, and illuminating new and surprising behaviour of these rich quantum fluids. This thesis presents new experimental results on the relative phase evolution of two-component Bose-Einstein condensates (BECs). We approach our study of this system from the dual perspective of binary superfluidity and trapped atom interferometry. We observe nonequilibrium dynamics of the condensate that are deeply connected to the physics of interpenetrating superfluids. These dynamics have consequences for interferometers whose two paths are different internal spin states of the condensate atoms. Using trapped condensates for precision measurement draws upon the benefits of confined atom interferometry - long interrogation times, precise position control to measure near-field potentials - in addition to the unique coherence properties of BECs. Precision measurementwith trapped particles, however, is obstructed by the inherent inhomogeneities associated with the confinement itself. Spatial variation of the external potentials and atomic density can give rise to dephasing during the free evolution and imperfect splitting and recombination of the interferometric paths. In this thesis, we consider the dephasing of the two-componentmean field order parameter, which tends to diminish the interference signal attained with the condensate. The promise of fuhs for atom interferometry lies within using nonclassical many body states to reduce the fundamental noise limit imposed upon measuring the projection of atomic pseudospin. Before the many body quantum properties of two-component condensates can be used to their full potential, the ideal classical limit must be achieved and investigated. As in optical interferometry, the classical ideality of an interference measurement is contingent upon control of the spatialmodes - thematter wave fields, or single particle wave functions. Whilst much attention is currently focused on preparing nonclassical many body states for entanglement and squeezing of the condensate pseudospin, this thesis focuses on the ancillary yet important issue of spatial inhomogeneity of the condensate mean field. Reversal of the spatially inhomogeneous mean field evolution is demonstrated, employing the techniques of nuclear magnetic resonance. A new method is demonstrated to probe pseudospin -1/2 condensates involving microwave adiabatic passage. The technique permits simultaneous, spatially resolved absorption imaging of the spatial mode of each component. Controlled manipulation of the components into Zeeman states with different magnetic moments allows spatial separation using a Stern-Gerlach separation technique, providing a spatially resolved projection of the local condensate pseudospin from a single absorption image. We propose an interferometric technique which employs this imaging method to directly image the relative phase. The miscibility of the two components depends on the relative magnitude of the interactions between various collision partners. The boundary of this miscibility is found to be less pronounced than expected from the commonly applied results of homogeneous super-fluids.The effects of kinetic energy are shown to be significant at the threshold of miscibility, leading to a breakdown in the Thomas-Fermi approximation. Magnetic dipole transitions amongst alkali ground states are often used to prepare, manipulate, and interrogate pseudospin -1/2 condensates. We present a thorough investigation of the two-photonmagnetic dipole transitions between the |F = 1,mF = -1〉 and |F = 2,mF = +1〉 states, and the |F = 1,mF = +1〉 and |F = 2,mF = -1〉 states of 87Rb. Taking into account all 8 Zeeman levels of the electronic ground state reveals markedly different radiative shifts and effects beyond the rotating wave approximation, compared to the three-level formalism commonly used for these transitions.This has implications for Ramsey interferometry using the two-photon magnetic dipole transitions. The unchartered experimental territory offered by two-component condensates is vast, and future directions, supplementary to the work in this thesis, are elucidated.
We discuss how a consistent phase standard for Bose-Einstein condensates may be defined. We show that it has the properties we would wish for in a phase standard: it is not corrupted by subsequent comparisons. A quantum jump technique is employed to study the time evolution of a three mode condensate system on which we make measurements, which entangle the modes and so establish relative phases between them. By establishing, in turn, the phases of two condensates relative to a reference condensate, we show that the relative phase between them can be predicted accurately. The existence of such a phase standard gives a precise definition to the phase of a condensate.
The ability to make ultra-precise measurements is fundamentally important to science as it allows theories to be tested and refined. Interferometers offer unrivalled measurement precision and therefore form the basis of many metrology schemes. Research has shown that by using quantum states as inputs to interferometers, precisions better than anything possible classically can be achieved. Nevertheless, these states are difficult to produce and fragile to particle losses. Consequently, classical inputs, which are extremely robust, are used in experiments. Here, however, we propose experimentally accessible schemes to make quantum-limited measurements, in particular rotation measurements using Bose-Einstein condensates, that are robust to losses. We begin by describing how, by loading a Bose-Einstein condensate into an optical ring lattice, multiport beam splitters are created through a simple raising and lowering of potential barriers between sites. We then use these ‘splitters’ to create an atomic gyroscope. We demonstrate how to create several quantum states in the gyroscope, all capable of making rotation measurements. Whilst NOON states afford best precision in idealised set-ups, we find they are outperformed by ‘bat’ states for modest loss rates. However, bat states are not ideal as they are outperformed by classical states for large losses. A second gyroscope scheme is therefore developed. Using multiple momentum modes, rather than just two, we show quantum-limited precisions can be reached using states that have similar robustness to classical states. The final section focuses on the precision of linear interferometers. Recent work[1, 2] has calculated the theoretical optimum initial states for two-mode lossy interferometers. Here we present an experimental way to produce initial states that afford similar precisions to this optimum. We also consider lossy multimode interferometry and demonstrate a potential advantage over two-mode systems. It is thought with further investigation other advantages will be found.