Optimal control is based on the ability to optimize properties of the laser pulses such as the phase [76], the amplitude [77] and the polarization [78] which act on a quantum system until a desired product is obtained. The experimental realization the optimal control experiments covers a wide domain, including: control over molecular dissociation and ionization [79,80,81,82] , fragmentation [83,84,85, 80], chemical bond breaking [86] , control over fluorescence of dye molecules [87], shaping of molecular wavefunction [88], quantum information processing [89], control of attosecond dynamics [90, 91], control over izomerization of proteins [92], dynamical processes in the light-harvesting complexes [93], semiconductors [94,95,96,97,98], and many other applications.
One of the challenges and issues emerging, when employing quantum control experiments, is finding the optimal laser pulses, that lead to the desired outcome.
In theory, optimally laser pulse can be calculated by solving the time-dependent Schrödinger equation of the system, and a-priori knowledge of molecular Hamil-tonians is needed. In particular, for a complex system, such as large molecules in the condensed phase, the molecular Hamiltonian is known usually to a limited de-gree, and solving the Schrödinger equation is challenging. However, tailored laser pulses steering the quantum system from its initial state to a desired final state can be found, by using the optimal control introduced by Rabitz and co-workers [71] in 1992. It was proposed to use a feedback from a molecule observable to iteratively optimize the laser pulse characteristics until an optimally shaped laser field is found. In this framework, there is no need to have prior information of the molecular system, the experimental apparatus ’solves’ Schrödinger equation in the laboratory field.
Optimal control can be adapted for the discrimination task of molecular systems even if their spectral properties are very similar. Optimal Dynamic Discrimination (ODD) exploits the dynamics of the molecules, in order to discriminate them. Ul-trafast excitation of the quantum system creates the molecular wavepacket, which is then probed by the second detection laser pulse. The detection pulse projects the molecular wavepacket that has evolved under the influence of the first optimally control field into the detection state. As a result, we observe the dynamical re-sponse of the molecule, that depends on the frequencies of the lasers, polarization and the relative time-delay. The use of a pulse shaping technique combined with a closed-loop approach to control the molecular dynamics differently, is at the basis of ODD.
To introduce the concept of optimal control, let us consider a quantum system described by a wavefunctionΨ(t), for example a molecular wavepacket generated at a potential energy surface. Its evolution under the influence of a control field ε(t)is governed by the time-dependent Schrödinger equation:
i¯hδ
δtΨ(t) =HΨ(t) (2.2)
whereH=H0−µε(t)is the total Hamiltonian comprising molecular Hamilto-nian and the interaction with external light field. The external fieldε(t)influences primarily the phases of molecule. The optimal electric field, which can be gener-ated by pulse-shapers, can guide the quantum system, evolving along the multiple pathways, to the desired state by manipulation over constructive and destructive interferences.
Let us consider example of quantum systems represented by multiple chemi-cal species, that we intend to discriminate. Then for each of them characterized by Ψν0,Ψν1,...,ΨνN−1, there is a detection stateDwhich can describe a final population.
The detection state is associated with the observable such as fluorescence (or fluo-rescence depletion) in the experiment. For each chemical species, the wavepacket is defined by [99] The dynamics of each system is controlled by its Schrödinger equation [99]
i¯h∂
∂tΨ(t) = (H0ν −µνεc(t))Ψ(t) (2.4)
unknown and unexplored. Recent theoretical evidence indi-cates that realistic quantum systems differing even infinitesi-namic evolution when acted upon with an optimally shaped vides a means to transiently resolve the product channels of quantum systems with nearly indistinguishable initial states.
The present work builds upon the one-parameter control of
Figure 2.6: Graphic representation of the ODD mechanism. Left:
Initial state, before the excitation by the laser fieldεc(t).
The state vectorscν are the wave function components.
Right: An optimal control laser field, prepares a statecξ that will be parallel to the detection stateD, and all the other vectors will be orthogonal to the detection state.
Picture is taken from [11]
A graphical representation of the ODD mechanism is depicted in Figure 2.6.
The signal from a detection stateDνcan be used as a feedback in order to optimize the laser field using a closed loop approach.
Implementation of optimal control
Optimal control experiments generally use in a closed-loop learning strategy which involves the quantum system controlled repeatedly. An example of the closed-loop learning algorithm scheme is shown in Figure 2.7. The Genetic Al-gorithm (GA) at the centre of the search strategy starts with an initial random pop-ulation, generated by a pulse shaper, and which is applied to the molecular system.
The observable, for example, fluorescence, is recorded by a suitable detection sys-tem. Step by step, an optimally shape should be found by converging to the best solution found by the GA.
The implementation of ODD opens up numerous applications in fluorescence microscopy, label-free diagnostics, and even remote identification of different bac-teria. It has been already successfully applied for discrimination of biomolecules like free amino acids [10] and flavins (riboflavin and flavin mononucleotide) [11]
Laser pulse shaper
Quantum system
Detector
Learning Algorithm
Objective Trial E(t)
Product
Result modified E(t)
iteration
fitness
Figure 2.7: Scheme of an optimal control experiment.
which are indistinguishable by spectroscopic means. This approach can be ex-tended further toward label free cellular imaging and detection of chemical or bi-ological substances. In the following chapters we will demonstrate the application of ODD in the deep UV for the discrimination of small peptides and serum pro-teins. Moreover, the extension will be done for proteins placed in the mixture, in order retrieve their relative concentration.