Abstract
So far, we have been concerned with the development of novel spectroscopic techniques to investigate the dynamics of complex quantum systems such as molecular aggregates. This final chapter will present novel protocols to obtain nonlinear spectroscopic signals in a very different setup - namely in linear chains of trapped ions. The goal in this chapter is to provide a versatile toolbox to probe and to characterize the emergent complexity in such well-controlled systems. In strong contrast to the spirit of tomographic methods in quantum information, these methods do not intend to characterize the full quantum state of the system, but instead to tailor custom-fit measurement protocols to investigate desired processes or properties of the system. The crucial ingredient in this undertaking is the use of ladder diagrams, which yield the intuition necessary to estimate the information content of the created signals beforehand.
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- 1.
The results presented in this chapter were obtained in collaboration with Manuel Gessner, so a similar presentation thereof can be found in [Gessner15].
- 2.
We trust that these protocols are equally applicable to similar synthetic quantum systems such as, e.g., cold atoms in optical lattices [Bloch08], or Rydberg atoms in optical tweezers [Saffman10, Fuhrmanek11].
- 3.
Details can be found in the dissertation of M. Gessner [Gessner15].
- 4.
The interaction term is clearly invariant under the simultaneous flip of all the spin in the x-direction, which changes \(\sigma ^{(k)}_x \rightarrow - \sigma ^{(k)}_x\).
- 5.
In the present chapter, we only consider signals in the impulsive limit. Hence, to simplify our notation, we dropped the superscript “\(^{(0)}\)” for the time delays in Eq. (2.141).
- 6.
In case a single excitation pulse interacts multiple times with the sample (e.g. when driving two consecutive excitations in the system), it is understood that the intermediate propagators are evaluated at \(t_i = 0\), and hence only yield the identity, \(\mathcal {G} (0) = \mathcal {I}\). Such a scenario will be discussed in the next section.
- 7.
For the Liouville space notation, see Sect. 2.2.
- 8.
Our following discussion may also be applied to systems in thermal equilibrium, for instance. One simply needs to take additional diagrams into account, just like in our discussion of multiple excitations in the previous Sect. 7.3.4.
- 9.
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Schlawin, F. (2017). Trapped Ion Spectroscopy. In: Quantum-Enhanced Nonlinear Spectroscopy. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-44397-3_7
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