Many physical problems are controlled by two
very different spatial scales. For example, in microelectronics, we want to
describe both the atomic scale (which determines the band structure and
therefore the semiconductor properties) and the scale of the device itself
(typically >1,000 times larger). When this is the case, direct numerical
simulations become very difficult because the system must be discretized to
sizes smaller than the smallest physical scale. At Irig/PHELIQS-GT, Xavier Waintal is developing proofs of
concept showing that we can solve this type of problem very efficiently using
quantum-inspired tensor networks for academic applications (in this
case, Bose-Einstein condensates).
The technique can be directly generalised to many other applications, e.g. electrostatic, electromagnetism, heat propagation, stochastic control...
Here, we show an example of a simulation of the Gross-Pitaevskii equation (which describes Bose-Einstein condensates in ultra-cold atom experiments) in a quasi-crystalline potential (order 8 symmetry). The presence of two scales (that of the condensate and that of the potential), coupled with the presence of a nonlinear term in the equation, makes the problem very difficult at first glance,
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(in green
the crystalline potential, in red the
non-linear term).
The potential V is shown in the figure opposite. The important point is that this simulation contains 1 000 000 000 000 pixels
and would therefore be completely impossible with conventional techniques.
Here, it is performed on a simple PC.
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The result of the simulation is
shown in the figure below. It shows the propagation of a cold atom condensate
in an immeasurable potential that can be created by crossing several lasers. Poetic, isn't it?
