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).