Final Report Summary - NUM2BEC (Number Conserving Approaches to Bose-Einstein Condensates)
The Marie Curie project NUM2BEC (No. 300285), entitled 'Number Conserving Approaches to Bose-Einstein Condensates', had the following objectives:
This research project aimed to provide a deeper understanding of the way in which a Bose-Einstein condensate responds when it is driven far from equilibrium and to use this in the development of accurate nonlinear interferometers. This will help to characterize and control non-equilibrium condensates. Specifically, we will need to look at the effects of a temperature field and at the interface of a two-component condensate. The three objectives were
1) Finite-temperature effects.
The first research objective was to develop a consistent number-conserving approach as a microscopic description of a finite-temperature Bose-Einstein condensate.
2) Interfacial dynamics.
The second research objective is to consider a magnetically trapped two-component condensate specifically when the two components have separated and we have a thin interface layer between the two components. More concretely, it is important to characterise the condensate and non-condensate fraction in both components in this interface layer when particular effects such as any rotation or topological defects and any instabilities are taken into account.
3) Analysing and developing an interferometer.
The third research objective builds on the first two objectives and is to consider techniques that can lead to experimentally feasible nonlinear interferometers that will give improved experimental sensitivity.
The main results of our investigations are as follows:
1) Over the course of the fellowship, excellent progress has been made on objective (1). A description of the dynamics of the Bose-Einstein condensate at finite temperatures is a difficult theoretical problem, particularly when considering low-temperature, nonequilibrium systems in which depletion of the condensate occurs dynamically as a result of external driving. We have been able to describe a fully time-dependent numerical implementation of a second-order, number-conserving description of finite-temperature Bose-Einstein condensate dynamics. This description consists of equations of motion describing the coupled dynamics of the condensate and noncondensate fractions in a self-consistent manner, and is ideally suited for the study of low-temperature, nonequilibrium, driven systems. We have demonstrated that the qualitative features of the system dynamics at zero temperature are generally preserved at finite temperatures, and have predicted a quantitative finite-temperature shift of resonance frequencies which would be relevant for, and could be verified by, future experiments. Specifically, objective (1) has resulted in a published piece of work:
T. P. Billam, P. Mason & S. G. Gardiner
Second-order number-conserving description of nonequilibrium dynamics in finite-temperature Bose-Einstein condensates,
Physical Review A 87, 033628, 2013.
2) In relation to objective (2), and as mentioned in the Mid-Term report, this objective has been modified. Previously the aim was to model a two-component condensate. This has now been generalised to an n-component condensate, for which the two-component condensate, which formed the original objective (2), is a special case. This generalisation was found to contain non-tirival analysis, and as such, objective (2) took longer than anticipated. However, the progress towards this (modified) objective (2) has been excellent.
An n-component condensate is a mixture of different states, different isotopes or different atomic species. The general system may contain a number of coherent or incoherent components. Within this setting, we have a developed a generalised framework to account for all the possible combinations of states. This has been accomplished by a partitioning of the condensate field operators into a condensate and noncondensate part. The resulting analysis collects quantum field operators into distinct coherent sets. By the use of a suitable expansion parameters, the Hamiltonian of the system can be expanded to differing orders. We consider all orders of expansion, upto and including the order that gives a self-consistent set of dynamical equations that govern the condensate and noncondensate parts for each of the n components. Examples of the equations are given for a single component, a two-component and a three-component condensate. Specifically, objective (2) has resulted in three publications, as listed below (note that the first of these three is now in press having been recently accepted)
P. Mason & S. A. Gardiner
A Number-Conserving Approach to n-component Bose-Einstein Condensates
Physical Review A, accepted 2014.
P. Mason
Ground states of two-component condensates in a harmonic plus Gaussian trap,
Eur. Phys. J. B 86, 453, 2013.
P. Mason
Calculating the properties of a coreless vortex in a two-component condensate,
Physical Review A 88, 043608, 2013.
At the time of the ending of the fellowship, preliminary work has begun on objective (3). It is anticipated that this final objective will result in publishable material in an international journal.
This research project aimed to provide a deeper understanding of the way in which a Bose-Einstein condensate responds when it is driven far from equilibrium and to use this in the development of accurate nonlinear interferometers. This will help to characterize and control non-equilibrium condensates. Specifically, we will need to look at the effects of a temperature field and at the interface of a two-component condensate. The three objectives were
1) Finite-temperature effects.
The first research objective was to develop a consistent number-conserving approach as a microscopic description of a finite-temperature Bose-Einstein condensate.
2) Interfacial dynamics.
The second research objective is to consider a magnetically trapped two-component condensate specifically when the two components have separated and we have a thin interface layer between the two components. More concretely, it is important to characterise the condensate and non-condensate fraction in both components in this interface layer when particular effects such as any rotation or topological defects and any instabilities are taken into account.
3) Analysing and developing an interferometer.
The third research objective builds on the first two objectives and is to consider techniques that can lead to experimentally feasible nonlinear interferometers that will give improved experimental sensitivity.
The main results of our investigations are as follows:
1) Over the course of the fellowship, excellent progress has been made on objective (1). A description of the dynamics of the Bose-Einstein condensate at finite temperatures is a difficult theoretical problem, particularly when considering low-temperature, nonequilibrium systems in which depletion of the condensate occurs dynamically as a result of external driving. We have been able to describe a fully time-dependent numerical implementation of a second-order, number-conserving description of finite-temperature Bose-Einstein condensate dynamics. This description consists of equations of motion describing the coupled dynamics of the condensate and noncondensate fractions in a self-consistent manner, and is ideally suited for the study of low-temperature, nonequilibrium, driven systems. We have demonstrated that the qualitative features of the system dynamics at zero temperature are generally preserved at finite temperatures, and have predicted a quantitative finite-temperature shift of resonance frequencies which would be relevant for, and could be verified by, future experiments. Specifically, objective (1) has resulted in a published piece of work:
T. P. Billam, P. Mason & S. G. Gardiner
Second-order number-conserving description of nonequilibrium dynamics in finite-temperature Bose-Einstein condensates,
Physical Review A 87, 033628, 2013.
2) In relation to objective (2), and as mentioned in the Mid-Term report, this objective has been modified. Previously the aim was to model a two-component condensate. This has now been generalised to an n-component condensate, for which the two-component condensate, which formed the original objective (2), is a special case. This generalisation was found to contain non-tirival analysis, and as such, objective (2) took longer than anticipated. However, the progress towards this (modified) objective (2) has been excellent.
An n-component condensate is a mixture of different states, different isotopes or different atomic species. The general system may contain a number of coherent or incoherent components. Within this setting, we have a developed a generalised framework to account for all the possible combinations of states. This has been accomplished by a partitioning of the condensate field operators into a condensate and noncondensate part. The resulting analysis collects quantum field operators into distinct coherent sets. By the use of a suitable expansion parameters, the Hamiltonian of the system can be expanded to differing orders. We consider all orders of expansion, upto and including the order that gives a self-consistent set of dynamical equations that govern the condensate and noncondensate parts for each of the n components. Examples of the equations are given for a single component, a two-component and a three-component condensate. Specifically, objective (2) has resulted in three publications, as listed below (note that the first of these three is now in press having been recently accepted)
P. Mason & S. A. Gardiner
A Number-Conserving Approach to n-component Bose-Einstein Condensates
Physical Review A, accepted 2014.
P. Mason
Ground states of two-component condensates in a harmonic plus Gaussian trap,
Eur. Phys. J. B 86, 453, 2013.
P. Mason
Calculating the properties of a coreless vortex in a two-component condensate,
Physical Review A 88, 043608, 2013.
At the time of the ending of the fellowship, preliminary work has begun on objective (3). It is anticipated that this final objective will result in publishable material in an international journal.