ELFT HÍRADÓ

[Janos Asboth <asboth.janos AT ttk.bme.hu>]

Meghívó

BME Elméleti Fizika Szeminárium,

okt. 20. péntek 10h15,

1111 Budapest, Budafoki út 8., BME F III. magasföldszint 01.,
Elméleti Fizika Tanszék szemináriumi szoba
and online, in a Microsoft Teams meeting,
https://shorturl.at/kBFL4

Balázs Hetényi (BME Elméleti Fizika Tanszék):
Fluctuations, uncertainty relations, and the geometry of quantum state
manifolds

I will discuss our recent study of the quantum geometry.  For a pedagogic
presentation geometric tensor quantities will be introduced: metric tensor,
Christoffel symbol, and briefly the four index Riemann curvature.  There is
some mystery in the terms "connection" and "curvature", as they appear to
refer to different quantites when used in the context of Riemannian
geometry or in the case of adiabatic cycles in quantum systems. I will try
to demystify this through a detailed derivation. Our main result [1] is
that the fidelity can be used as a cumulant generating function: the first
cumulant generates a Berry connection, the second cumulant generates the
two two-index geometric quantities, one being the quantum metric the other
the Berry curvature.  In the fidelity language this second cumulant is the
fidelity susceptibility.  The series can be continued, the third cumulant
(or skew) corresponding to what one would call the "quantum Christoffel
symbol" (the real part of which corresponds to a true Christoffel symbol of
the parameter space of the given quantum system), the fourth cumulant
(kurtosis) giving a four index "quantum Riemann curvature tensor". The
formalism will be applied to several model systems. For coupled quantum
classical systems moving on a Born-Oppenheimer surface, we show that a
complex Hermitian inverse mass tensor leads to a mixing of the "molecular
electric" and "molecular magnetic" fields.  Requiring the determinant of
the second cumulant to be greater than or equal to zero leads to
uncertainty relations.  In the end I will discuss our calculations for
coherent states (Glauber, SU(2) and SU(1,1)), where we find that the
quantum metric tensor has a determinant of zero for minimum uncertainty
states, meaning that the geometry is trivial, while for generalized
coherent states this is not the case.

[1]: B Hetényi and P Lévay, Phys. Rev. A 108, 032218 (2023).

Minden érdeklődőt szeretettel várunk.
Asbóth János,
 szemináriumi koordinátor