Demos.ConicalInter.GeoPhase

Geometric phase effects

The lower potential sheet of a two-dimensional spin-boson system appears like a "Mexican hat". For strong (linear) Jahn-Teller coupling (κ>>ω), wavepackets placed near the bottom of the (circular) well, encircle the conical intersection without significant non-adiabatic population transfer. Nonetheless, the effect of the conical intersection manifests itself in corresponding wavepacket dynamics through a geometric phase upon complete encirclement, see also work by Schön and Köppel.

Dynamics on two coupled states

We present the results of a solution for two coupled time-dependent Schrödinger equations giving the vibrational quantum dynamics on the lower and upper electronic potential energy surface of the E x e Jahn Teller system. The wavepacket is initially placed at the bottom of the well. Due to its dispersion, it spreads in angular direction until it has completely encircled the locus of the conical intersection. To compensate the double-valuedness of real electronic states upon complete encirclement, these states are multiplied with a complex phase factor which is equivalent to the geometric (Berry) phase. Here the left and right half acquire a geometric phase of exp(iπ/2) and exp(-iπ/2), respectively. The phase difference of exp(iπ) at the encounter is the reason for the destructive interference resulting in a nodal line. As you can see from the expectation values, the second adiabatic state is never occupied and hence indeed does not contribute to the dynamics.

• Animated wavepacket
• Input data file
• Logfile output

Lower state dynamics

For comparison, we carried out a quantum-dynamical simulation on the lower potential energy surface. One might be tempted to think that this should be approximately equal to the two-state calculation presented above, because of the absence of notable non-adiabatic population transfer. However, the neglect of geometric (Berry) phase effects leads to a qualitatively different evolution of the density. In particular, no nodal line is observed at the encounter of the left and right part of the wavepacket.

• Animated wavepacket
• Input data file
• Logfile output

It remains to be seen how this can be connected with 2π-antiperiodic solutions of the planar generalized pendulum problem.