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Dynamics of quantum systems, controlled by external fields

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Demos.ConicalInter.OneDim

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Adiabatic.gif (4985 bytes)

Coupling.gif (3342 bytes)

Crossing.gif (2443 bytes)

Diabatic.gif (8432 bytes)

Minima.gif (2915 bytes)

SpinBoson system in 1D

A spin-boson system consists of a discrete N-state "spin" system (usually N=2 for a spin 1/2 particle) coupled to M "bosonic" degrees of freedom (usually harmonic oscillators). Despite of its apparent simplicity, the resulting quantum dynamics becomes fairly involved thus posing the spin-boson system a challenging test for approximate and/or numerical schemes in quantum dynamics.

Formalism

Diabatic potential energy matrix

$V(x)=\frac{\omega^2}{2}\begin{pmatrix}x^2&0klzzwxh:00000&x^2\end{pmatrix}+\kappa\begin{pmatrix}x&0klzzwxh:00010&-x\end{pmatrix}+\gamma\begin{pmatrix}0&1klzzwxh:00021&0\end{pmatrix}+\frac{\delta}{2}\begin{pmatrix}1&0klzzwxh:00030&-1\end{pmatrix}$

Crossing of diabatic potentials

$x_C=-\frac{\delta}{2\kappa}\,,V_C=\frac{\omega^2\delta^2}{8\kappa^2}$

Minima of diabatic potentials

$x_M=\mp\frac{\kappa}{\omega^2}\,,V_M=-\frac{\kappa^2}{2\omega^2}\pm\frac{\delta}{2}$

Adiabatic potential energy curves

$E_{1,2}(x)=\frac{\omega^2}{2}x^2\pm\sqrt{\gamma^2+\left(\frac{\delta}{2}+\kappa x\right)^2}$

Non-adiabatic coupling

$C_{12}(x)=-\frac{2\gamma\kappa}{\left(E_1(x)-E_2(x)\right)^2}$

Note that the Hamiltonian here is a simple one-dimensional cut of a conical intersection occuring, e. g., for a general Jahn-Teller coupling in molecular physics.

Simulation

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 spawns another one, eventually leading to rather complicated interference patterns.

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 increasingly complicated as time goes by: Upon each passage of the avoided crossing, each wavepacket spawns another one, eventually leading to rather complicated interference patterns.

The parameters for this example (ω=0.01, κ=0.1 √5, γ=0.5, δ=14.88) are adapted from the diploma thesis of Monika Hejjas (HU Berlin, Dept. of Physics, 2004). Note the change of the "squeeze parameter" of the phase space packets (tilting of elliptic contours) upon passing the avoided crossing.

Wiki: Demos.ConicalInter.Main
Wiki: Demos.ConicalInter.SpinBoson
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