Dynamics of spinor Bose-Einstein condensates
My Masters project on geometry of specific quantum mechanical states
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This post provides a high-level summary of my masters project, which was on the Majorana dynamics of spinor Bose-Einstein condensates (BECs). The below is an adapted version of the introduction - the full file can be found here. If you have any questions about this, feel free to get in touch! The GitHub repository also contains some Python and Mathematica files which can be used to explore the Majorana dynamics of spinor BECs.
Quantum mechanics
Over the last century, quantum mechanics has become a frontier of modern physics, describing how light and matter behave with unerring accuracy. Modern-day technology, including lasers, transistors and magnetic resonance imaging in healthcare, relies extensively on quantum phenomena. My project brings together three key developments in modern quantum mechanics: spin, Bose-Einstein condensates and Berry’s phase.
Spin
‘Spin’ as a concept was first proposed by Wolfgang Pauli in 1925 as an intrinsic form of angular momentum (or an "internal degree of freedom") but its experimental origin dates back to 1922, in the Stern-Gerlach experiment. In this experiment, a beam of silver atoms was fired through a magnetic field onto a photographic plate. According to theories at the time, it was expected that the beam would have a continuous spread, showing a continuous line on the plate, but instead, the beam split into 2 distinct points. Stern and Gerlach intended to demonstrate the quantisation of angular momentum according to Bohr’s so-called ‘old‘ quantum theory (now referred to as old as it is accepted to be incorrect). The results matched Bohr’s prediction, but it was realised shortly afterwards that the interpretation of the Stern-Gerlach experiment was incorrect. The experiment had in fact measured the internal spin degree of freedom of the electron - unbeknownst to the duo at the time as the concept did not yet exist! It was not until 1927 that theoreticians realised that Stern and Gerlach’s work had measured the electron’s spin. Soon after, Markus Fierz proved the spin statistics theorem in 1939 showing that particles could be classified as either:
- Fermions, which have half-integer spin, or
- Bosons, which have integer spin
His argument was generalised by Pauli in 1940, who in the same paper also extended his result from 1925 that identical fermions can not occupy the same quantum state, now known as the Pauli exclusion principle.
Bose-Einstein condensates
Pauli’s exclusion principle highlights one of the key differences between fermions and bosons and has huge ramifications: bosons have no restriction regarding occupying the same quantum state. As the theory of spin was being developed, in 1924, Satyendra Nath Bose developed some ideas on the statistics of photons and Albert Einstein built on these in 1925, predicting that a sufficiently large number of bosons could occupy the same quantum state at a sufficiently low temperature. A system in this regime is called a Bose-Einstein condensate (BEC). It was shown that these systems would have extremely interesting properties, for example being linked to the superfluidity of helium. However, the temperatures required to achieve Bose-Einstein condensation are so close to absolute zero that for a long time it was virtually impossible to observe them. Consequently, relatively little was done until 1995, when the first BEC was experimentally realised using rubidium atoms in a single spin state. Not long after, in 1998, influential papers by Tin Lun Ho and Tetsuo Ohmi and Kazushige Machida paved the way for significantly more work in this area, which remains an active area of research today.
Spinors
Spin plays an important role in the theory of BECs. Since BECs require extremely low temperatures, they are incredibly unstable. Most experimental realisations require them to be ‘trapped’. This is mostly done via magnetic and optical traps. In the case of magnetic traps, the spin degrees of freedom of the condensate are frozen. This does not occur in the cases of optical traps. We distinguish between these two cases by referring to the former case as scalar BECs and the latter case as spinor BECs. More broadly, different spins result in characteristically different behaviours. In this paper we will focus on (spin-1 and 2) spinor BECs.
Majorana's stellar representation (warning: technical!)
Mathematically, spin-1/2 states can be described by a 2-dimensional complex vector. A priori, such a quantity has 4 (real) degrees of freedom, but with some work these states can be visualised on a 3-dimensional sphere, the Bloch sphere. This suffices for spin-1/2 systems, but the exact construction cannot be repeated for higher spins. In a relatively understated work, Ettore Majorana outlined a new geometric idea enabling visualisation of arbitrary spin systems on the unit sphere. In particular, his method provided a means of visualing a spin-F state as 2F states (also referred to as points or stars) on the unit sphere. This is called Majorana’s stellar representation (MSR).
Berry phase
We require one more important ingredient. In 1984, Michael Berry, generalising results from the 1950s, found that under certain conditions, a quantum system evolving adiabatically (‘slowly’ in a sense) remains in its initial state, up to a phase. This phase has two contributions - one contribution is related to the time evolution of the state, and the other on the set of parameters the system depends on. The former is the dynamical phase and the latter is the Berry phase. The Berry phase reveals the underlying geometric nature of quantum mechanics and has applications in many areas of physics, including magnetism, Hall effects and the classification of insulators.
This paper brings together all of the aforementioned ideas. In particular, we analyse the dynamics of spinor BECs in the context of the MSR and Berry phase.