Third AGM meeting
Third AGM Meeting on Geometric Quantum Dynamics 2014-2015
3 October 2014, Brunel University
This one-day meeting will cover various disciplines related to geometric quantum dynamics. Specific topics will include:
-- geometry of quantum states,
-- quantum integrable systems,
-- reduction by symmetry.
Timetable:
10:00 -- 10:30 Coffee
10:35 -- 11:20 Dorje Brody (Brunel)
11:30 -- 12:15 Karol Zyczkowski (Jagiellonian University, Cracow, PL)
12:30 -- 13:30 Lunch
13:35 -- 14:20 Eva-Maria Graefe (Imperial)
14:25 -- 15:10 Lane Hughston (Brunel)
15:10 -- 15:40 Coffee
15:40 -- 16:25 David Meier (Brunel)
16:30 -- 17:15 Alessandro Torrielli (Surrey)
17:20 -- 18:05 Henry Jacobs (Imperial)
The meeting will take place in room 128, John Crank building (more info here)
Titles (abstracts to follow soon)
Dorje Brody
Geometric quantum mechanics
This talk will cover an overview of the development of the geometric formulation of quantum theory, and will point to a number of specific ideas that emerged out of the geometric approach.
David Meier
Quantum Zermelo navigation for states and gates
The solution to the problem of finding a time-optimal control Hamiltonian to generate a given unitary gate, in an environment in which there exists an uncontrollable ambient Hamiltonian (e.g., a background field), is obtained. In the classical context, finding the time-optimal way to steer a ship in the presence of a background wind or current is known as the Zermelo navigation problem, whose solution can be obtained by working out geodesic curves on a space equipped with a Randers metric. The solution to the quantum Zermelo problem, which is shown to take a remarkably simple form, is likewise obtained by finding explicit solutions to the geodesic equations of motion associated with a Randers metric on the space of unitary operators. The result reveals that the optimal control is in a sense to ‘go along with the wind’. (Joint work with Dorje Brody)
[1] Dorje C. Brody, David Meier, Solution to the quantum Zermelo navigation problem, arXiv:1409.3204
Eva-Maria Graefe
Mean-field and many-particle correspondence for a quantum atom-molecule conversion system
There is currently considerable interest in experiments with Bose-Einstein condensates (BECs) of cold atoms that can associate to form multi-atomic molecules and vice versa. A full theoretical description of cold atoms and BECs requires modelling of many-particle quantum dynamics, which quickly goes beyond the scope of computational accessibility for realistic setups. For low densities, large particle numbers, and short times, a BEC can be effectively described by a single macroscopic wave function. This mean-field approximation is closely related to the classical limit of single particle physics. Thus, semiclassical methods can be applied to model many-particle features on top of the mean-field description.
Here we consider the simplest example system of an atom-molecule conversion system where atoms can combine into two-atomic molecules, and only one basis state is considered for atoms and molecules respectively. The many-particle system can be described by a deformed SU(2) algebra, and the mean-field dynamics is confined to a teardrop shape that replaces the Bloch sphere of a conventional two-mode system. We demonstrate that many-particle effects can be accurately recovered from the mean-field approximation via semiclassical quantisation techniques.
Lane Hughston
Quantum heat bath - a geometric perspective
A model for a quantum heat bath is introduced. When the bath molecules have finitely many degrees of freedom, it is shown that the assumption that the molecules are weakly interacting is sufficient to enable one to derive the canonical distribution for the energy of a small system immersed in the bath. While the specific form of the bath temperature, for which we provide an explicit formula, depends (i) on spectral properties of the bath molecules, and (ii) on the choice of probability measure on the state space of the bath, we are in all cases able to establish the existence of a strictly positive lower bound on the temperature of the bath. The results can be used to test the merits of different hypotheses for the equilibrium states of quantum systems. Two examples of physically plausible choices for the probability measure on the state space of a quantum heat bath are considered in detail, and the associated lower bounds on the temperature of the bath are worked out.
Henry Jacobs
Quantum theoretic representations of diffeomorphisms and functions for computational advection schemes
We consider the problem of numerically computing the advection of a probability density by an autonomous dynamical system. A direct spectral discretization, such as the transfer operator or the Koopman operator, exhibit spurious phenomena such as "negative probabilities" and a failure to advect algebraic structures, such as the ring structure of real-valued functions. We present an algorithm which addresses these issues by computing the advection of quantum wave functions in lieu of computing the advection of probability densities and real-valued functions. The existence of dense coordinate charts allows us to transport the machinery of wavelet analysis to a variety of manifolds by paying close attention to the stalks (in the sense of algebraic geometry) at the coordinate singularities. The resulting multi-scale scheme behaves well, even at low resolutions, and yields a unified approach to advection by preserving the positivity of probability densities and the algebraic structures of functions, densities, and vector-fields.
Alessandro Torrielli
Exact quantisation of integrable systems
In this talk we will review the exact quantisation of a prototypical integrable system (the Non Linear Schrödinger Equation) by the method of the Quantum Inverse Scattering. The emphasis will be on symmetries and their interplay with the quantisation procedure.
Karol Zyczkowski
On the geometry of the set of quantum states
We study geometry of the set of mixed quantum states acting on an N--dimensional Hilbert space. For N>2 this convex body of dimension N2-1 is neither a ball nor a polytope. Investigation of its properties for composite dimensions contributes to our understanding of the structure of quantum entanglement.
[1] I. Bengtsson, S. Weis and K. Zyczkowski, Geometry of the set of mixed quantum states: An apophatic approach, in Geometric Methods in Physics Trends in Mathematics, pp 175-197, Springer Basel 2013
[2] Z. Puchala, J. A. Miszczak, P. Gawron, C.F. Dunkl, J. A. Holbrook and K. Zyczkowski, Restricted numerical shadow and geometry of quantum entanglement, J. Phys. A 45 (2012), 415309 (2012)
Contacts: