Connections and Questions

The entry below summarizes the 'brainstorming' roundtable from Thursday May 28

The first part of the discussion focussed on

1. The connection between control in NMR and molecules.

In NMR, control in few-level systems is very successful. Optimal control
yields highly complex pulse shapes, but those sometimes can be well understood with the right kind of analysis. Is it possible to use NMR kind of approaches and analysis for other quantum systems?
Is it possible to use analysis for other quantum systems for control in NMR systems?

Possibly 'yes'. A possible route is a connection between
a) A resonantly driven multiple-level system,
b) A lattice with finite number of sites, or a set of quantum wells, with sites (wells) coupled by e.g. nearest neighbour interactions (tunnelling)
c) A molecule where charge transfer /migration is described via coupling of different localized
'orbitals' (excitations in a localized basis)

When a few-level system is subject to near-resonant driving fields, the rotating wave approximation (RWA) maps its Hamilonian onto a system of nearly degenerate levels. Frequency and amplitude modulations of the driving resonant fields are mapped onto modulations of couplings between these levels. The same picture arises in quantum wells /finite lattices. Diffrent levels of a few level system (after RWA) in this case correspond to localized states on different sites, and the couping between the sites is due to e.g. tunnelling. Frequency and amplitude modulation of resonant fields in NMR corresponds to modulating energies on each lattice site and modulating tunnelling between the sites.

In a set of quantum wells (finite lattice) control over energies and tunneling rates between the sites can be achieved even when the initial Hamiltonian H0 is degenerate -- zeroes on the main diagonal, same next-neighbor interactions on the two diagonals just below and just above the main diagonal.

The route to control relies on lifting the degeneracy by introducing 2-color field: w+2w with controlled phase between the two colors. Nonlinear interaction between lattice and external field is effectively described by the susceptibilities \chi(0; +w,+w,-2w) and \chi(0; -w,-w,+2w) -- these are resonant diagrams. Response at zero frequency means that effective DC field is introduced, which lifts
degeneracy of the original system. Finite size of the lattice -- the presence of edge defects --results in site-dependence of the effective DC field. In principle, this is enough to control dynamics, moving a particle from a given site to a given site. From here, we can go back to NMR systems, using the lattice-NMR analogy.

Important things to keep in mind:
- With H0 degenerate, controllability is a big problem. Once defects are introduced, controllability is easier to analyze and achieve (S. Schirmer).
- In general, for a lattice with degenerate H0 on the main diagonal, one needs:
-1) w+2w, which adds effective DC bias and creates a current, with current direction controlled by the relative phase between colors. This lifts degeneracy
-2) extra blocker (controlled defect) (M Shapiro and S Schirmer), which should ideally be moved to block the current at the desired lattice site. Not clear how to introduce such a blocker in general.
In NMR, this is easier -- turn off the resonant field that couples to the desired level

Relevant work is done by M Shapiro and P Brumer on controlling currents in polyacetylenes, using nonresonant w+2w field. This current is robust. However, they have not yet addressed the problem of moving charge from a specific site in a molecule to another specific site
- Decoherence. How robust is control wrt to phonons?
- Can one phase-control site-specific tunnelling rates (locally!) with external fields?

Main point of this discussion: Use lattice - NMR analogy to
a) map NMR fields onto control in lattices -- see if this yields any useful insight
b) map lattice control on NMR -- see if this yields any useful insight
Work to look at:
- work of Ivan Deutsch and Carl Williams on lattices
- controllability in dipole-dipole coupled qubits

The second part of the discussion touched upon the following question:

2. Why reactions are harder to control than finite-level systems?

Possible reasons:
- Structure of H0 -- multiple degeneracies, especially in the continuum, with access only to outgoing scattering states and not to incoming waves
- H0 is not known. Potential energy surfaces (PES) are inaccurate. (Personally, I - Misha Ivanov- do not think this is a problem for control. It is more of a problem for interpreting the control field)
- Technology: in NMR, generating very complex control fields is not a problem. They can desigh any field they want. Not so in reactions, where we are so far limited to very narrow frequency range, not tuned to the molecule of interest.
- Inverse scattering problem is notoriously tough. Is it also not robust?

Related point: data mining, and construction of predictive 'black-box' models. We may never be able to build a full-scale model of how a complex molecule is operating, but we may not need to. If we can treat it as a black box, and build an effective and relatively simple model of that black box in terms of a system of relatively simple coupled diff. equations, based on the full range of available experimental data, then we can design our control experiments using this effective model.

Unfortunately, the problem is that the model could fail as soon as we go outside the range of data it was built upon.

- How do we generalize Bloch sphere to, say, 2-active particle system? Answer ( M. Shapiro): Use hyperspherical coordinates. This also shows that our sphere is not compact -- R can extend to infinity. This non-compactness is, possibly, the biggest problem.

The third part of the discussion focussed on

3. How to analyze complex pulse shapes?

- First, one needs to blow up the 1D signal into 2D space (D. Uskov also mentioned connection to related work in meteorology). 2D spectrograms are very useful for analysis, much more than original 1D signals.
- Suggestion: Design a _set_ of special time-frequency basis sets for the spectrograms, so that each member of the _set_ is trained to recognize a particular control mechanims (such as pi-pulses, adiabatic passage, etc). Example -- recent work of E Shapiro, V Milner, and M Shapiro on using a sequence of short pulses to generate _piece-wise_ adiabatic passage. It is not recognizable as adiabatic passage if the 2D spectrogram is created with short time-gates (short wavelets) but is immediately recognizable as such when frequency-resolved wavelets are used.
- Criticism of the idea: for a mixed control mechanism, its components might be completely obscured even for the specially trained basis sets

- Important related points:
- a) Constraints are very important in designing pulse shapes. Different constraints yield completely different optimal pulses for the same system
-b) How to identify _useless_parts of the spectrum in the control pulse?