| EPSRC Reference: |
EP/H027750/1 |
| Title: |
A classical analogue for non-Hermitian quantum systems: Towards a semiclassical framework |
| Principal Investigator: |
Graefe, Dr E |
| Other Investigators: |
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| Department: |
Mathematics |
| Organisation: |
University of Bristol |
| Scheme: |
Postdoc Research Fellowship |
| Starts: |
01 August 2010 |
Ends: |
31 July 2013 |
Value (£): |
207,819
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| EPSRC Research Topic Classifications: |
| Mathematical Analysis |
Mathematical Physics |
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| EPSRC Industrial Sector Classifications: |
| No relevance to Underpinning Sectors |
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| Panel History: |
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Summary on Grant Application Form |
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My research is about non-Hermitian generalisations of quantum theories. In quantum mechanics physical quantities are described by linear operators, which have to fulfil some conditions to make them interpretable as physical observables. One of these is the requirement that measurement outcomes, given by the eigenvalues of the operators, have to be real numbers. Thus, only operators with real spectrum are allowed as quantum observables. A special class of operators fulfilling this consists of the so-called Hermitian operators. Yet, there are two main motivations to investigate more general non-Hermitian quantum theories. First, the demand that an operator be Hermitian is a stronger restriction than necessary; consequently, non-Hermitian theories have been suggested as a generalisation. The second motivation is that to describe quantum states with finite lifetime one can use complex energies (as in an oscillation where a complex frequency can be used to describe a damping), which can be regarded as eigenvalues of non-Hermitian operators. This was used, e.g., already in 1928 by Gamov to describe the alpha-decay. Thus, non-Hermitian observables were always around in quantum mechanics and proved useful in the description of a variety of phenomena of great experimental relevance. It was long thought, though, that non-Hermitian quantum dynamics is just like Hermitian dynamics with an additional overall decay. However, this is not true! Only recently new experimental areas are opening up, where behaviour drastically differing from that of Hermitian systems is discovered (such as lopsided diffraction in optical structures). We are only beginning to glimpse the variety and richness of new phenomena connected with non-Hermitian quantum dynamics and to seriously investigate it; but many features are still poorly understood. I propose here a new approach by developing a semiclassical framework which proved extremely useful in the Hermitian case. The subtlety of the semiclassical limit in which macroscopic classical mechanics (in the Hamiltonian formulation) arises from the microscopic Hermitian quantum mechanics has been a big issue from the early days of quantum mechanics and was even regarded as a major flaw of the theory by some of its founders. Today, we have powerful mathematical tools resulting from a fruitful interplay of theoretical physics and pure mathematics to understand this semiclassical limit, and a variety of semiclassical methods allows for accurate quantum calculations using classical objects. The main issue for a non-Hermitian generalisation is that so far nobody has found the structure associated with the semiclassical limit (many, who regard non-Hermiticity as an effective description, do not even think that it exists). Yet, I recently identified such a structure in some examples and mathematically derived it for an important special case. It turns out to be very similar to Hamiltonian mechanics, but additionally, it can describe dissipative motion! This will now help us to understand both non-Hermitian quantum and dissipative classical dynamics, and will allow us to build up a semiclassical framework. This will open up a new line of theoretical research which I plan to initiate. Related experiments are likely to provide the basis for the development of new technology in the field of lasers, optical communication and even quantum computers.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
| Summary |
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| Date Materialised |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.bris.ac.uk |