Demos.OptDoubleWell.BorisSB
Optimal control of quantum dynamics:
Double well with two localized states
In the work by Boris Schäfer-Bung et al., also showcased as one of our double well demo examples, we introduced a one-dimensional, asymmetric quantum double well system which has 6 (5) states localized within the left (right) potential well. In addition to drive populations between left and right well and delocalized states, optimal control fields can also be used to induce ladder climbing within each of the wells. Here we consider open system quantum dynamics including dissipation and decoherence due to interaction with a thermal environment. In order to reduce the computation effort for solving the corresponding Liouville-von Neumann equation, we employ here the balanced truncation method as well as H2 optimal models with the designing optimal control fields in reduced dimensionality.
Optimal control of in full dimensionality
First let us look at results in full dimensionality, i.e. for 441 elements of (vectorized!) density matrix (constructed from the first 21 quantum states). For chosen relaxation rate Γ2→0=0.01 and a time span of T=100, the optimization iteration quickly yields a pulse that increases the initial thermal population of the right well from 26% to 41%.
| Time dependence | Iteration convergence | FROG spectra of u(t) |
|---|---|---|
| Control field u(t) | Target functional | |
| State vector x(t) | Cost functional | |
| Observables y(t) | Difference | |
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Optimal control after balanced truncation
Next we use the balanced truncation method to construct a reduced model, in this case with d=170. The graphs below show that the optimized field found in reduced dimension is practically the same as that found in full dimension. Upon further dimension reduction (d=160) the field changes qualitatively, thus indicating a limit in dimension reduction.
| Time dependence | Iteration convergence | FROG spectra of u(t) |
|---|---|---|
| Control field u(t) | Target functional | |
| State vector x(t) | Cost functional | |
| Observables y(t) | Difference | |
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Optimal control after H2 optimal model reduction
Finally we also use the H2 optimal models to construct a reduced model, again with d=170. The graphs below show that the optimized field found in reduced dimension is practically the same as that found in full dimension. Upon further dimension reduction (d=160) the field changes qualitatively.
Note that the resulting fields, the limit for dimension reduction, and the iteration dependence of the control functionals are very similar!
| Time dependence | Iteration convergence | FROG spectra of u(t) |
|---|---|---|
| Control field u(t) | Target functional | |
| State vector x(t) | Cost functional | |
| Observables y(t) | Difference | |
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Related
Wiki: Demos.DoubleWell.DimReduce_1
Wiki: Demos.OptDoubleWell.Main
