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Dynamics of quantum systems, controlled by external fields

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WavePacket demo example: Generic conical intersections

Once considered to be of merely academic interest, by now conical intersections of potential energy surfaces are known to be of paramount importance for many processes in photophysics, photochemistry, and photobiology. The most elementary, generic case of a conical intersection arises in molecular physics is the socalled case of E x e Jahn Teller coupling. It arises when a molecule with (at least!) threefold rotational symmetry is in a degenerate electronic state (E irrep), coupled to a pair of degenerate nuclear distortions of the same symmetry (e irrep). The result of the Jahn Teller theorem stating the incompatibility of degenerate (electronic) states and symmetric (nuclear) configurations predicts spontaneous symmetry breaking for all non-linear molecules.

Formalism

Diabatic potential energy matrix (Cartesian coordinates)

$V(x,y)=\frac{\omega}{2}\begin{pmatrix}x^2+y^2&0klzzwxh:00000&x^2+y^2\end{pmatrix}+\kappa\begin{pmatrix}x&yklzzwxh:0001y&-x\end{pmatrix}+\frac{g}{2}\begin{pmatrix}x^2-y^2&-2xyklzzwxh:0002-2xy&y^2-x^2\end{pmatrix}$

Adiabatic potentials (using polar coordinates ρ and φ)

$E(\rho,\phi)=\frac{\omega}{2}\rho^2\pm\sqrt{\kappa^2\rho^2+g\kappa\rho^3\cos(3\phi)+g^2\rho^4/4}$

Four conical intersections

One at ρ_X=0 and for g>0 three more at $\rho_X=2\kappa/g,\,\phi_X=\pi/3,\pi,5\pi/3$

Three minima (lower adiabatic potential)

$\rho_M=\frac{\kappa}{\omega-g},\,\phi_M=0,2\pi/3,4\pi/3,\,E_M=-\frac{\kappa^2}{2(\omega-g)}$

Three saddles (lower adiabatic potential)

$\rho_S=\frac{\kappa}{\omega+g},\,\phi_S=\pi/3,\pi,5\pi/3,\,E_S=-\frac{\kappa^2}{2(\omega+g)}$

see e.g. work by Zwanziger and Grant
see e.g. work by Eisfeld and Viel

Linear E x e Jahn-Teller system: Double cone potentials

In linear approximation, i.e., for vanishing quadratic (and higher) terms (κ ≠ 0, ω=g=0), the two adiabatic potential energy surfaces of the corresponding Jahn-Teller system can be represented by a pair of funnels forming a linear conical intersection. A wavepacket touching the catchment area of the intersection is subject to strong nonadiabatic effects. Learn more ...

Spin-Boson system

If harmonic potentials are added to the linear E x e Jahn-Teller system, the prototypical spin-boson Hamiltonian is obtained with quadratic potentials on the diagonals and linear terms on the off-diagonal entries of the (diabatic!) potential matrix (κ ≠ 0, ω ≠ 0, g=0). Due to the multiple passing of high energy wavepackets of the conical intersection, the quantum dynamics is governed by complex interference patterns. Learn more ...

Spin-Boson system in 1D

In a diabatic representation, a one-dimensional spin-boson Hamiltonian can be represented by two harmonic potentials displaced in position and energy and a constant coupling between the two states. In the adiabatic representation, the two potential energy curves exhibit an avoided crossing. The corresponding quantum dynamics becomes incresingly complicated as time goes by: Upon each passage of the avoided crossing, each wavepacket spwns another one, eventually leading to rather complicated interference patterns. Learn more ...

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. Learn more ...

Quadratic E x e Jahn-Teller system: Warped potentials

If quadratic Jahn-Teller coupling (g ≠ 0) is added to the spin boson system, the circular symmetry of the system is broken. The corresponding (adiabatic) potential energy surfaces become deformed ("warped") and the ground state exhibits threefold rotational symmetry with three additional conical intersections. The lower potential surface exhibits an alternating sequence of three minima and three saddles. Learn more ...