Two-Dimensional Spectroscopy for the Study of Ion Coulomb Crystals

Ion Coulomb crystals are currently establishing themselves as a highly controllable test bed for mesoscopic systems of statistical mechanics. The detailed experimental interrogation of the dynamics of these crystals, however, remains an experimental challenge. In this work, we show how to extend the concepts of multidimensional nonlinear spectroscopy to the study of the dynamics of ion Coulomb crystals. The scheme we present can be realized with state-of-the-art technology and gives direct access to the dynamics, revealing nonlinear couplings even in the presence of thermal excitations. We illustrate the advantages of our proposal showing how two-dimensional spectroscopy can be used to detect signatures of a structural phase transition of the ion crystal, as well as resonant energy exchange between modes. Furthermore, we demonstrate in these examples how different decoherence mechanisms can be identified.

Figure 1

Parts (a) and (b) show two example pathways carrying the phase signature ${\varphi }_2-{\varphi }_3-{\varphi }_4$. Starting from a population all pathways have to end in a population in order to be observable. In paths (a) the coherences oscillate with the same frequency during the evolution period $t_3$ as during $t_1$ thus giving rise to diagonal peaks in the spectrum. In paths (b) the oscillation frequency during $t_3$ is shifted by $-{\mathrm{\Omega }}_{\mathrm{SI}}$ with respect to $t_1$ leading to off-diagonal peaks below the main diagonal.

Figure 2

The central plot shows the 2D spectrum $|S({\omega }_1,{\omega }_3)|=|\mathcal{F}(s(t_1,t_3))|$ obtained by a four-pulse sequence with the simulation parameters given in Table 1, including up to fourth-order terms in the Hamiltonian, in the neighborhood of the linear-to-zigzag transition. The diagonal peaks are due to paths of type (a) in Fig. 1. They are blurred because of static dephasing caused by thermal populations of the spectator modes leading to the diagonal line. The dominant off-diagonal [paths (b) in Fig. 1] is shifted by $-{\mathrm{\Omega }}_{\mathrm{SI}}$ along the ${\omega }_3$ axis and can thus be used to infer the self-interaction of the zigzag mode. The small plots along ${\omega }_{1/3}$ show the spectra obtained by integrating along the other frequency direction. This is the result that would be obtained by a 1D experiment with only one free evolution period $|S({\omega }_{1/3})|=|S({\omega }_{1/3},t_{3/1}=0)|$.

Figure 3

2D spectrum due to the resonant third-order terms $H_{\text{res}}^{(3)}$, Eq. (9), in the Coulomb potential. Simulation parameters are given in the main text. A strong peak at ${\omega }_1={\omega }_3=-{\omega }_{\mathrm{ZZ}}$ was removed from the spectrum for clarity. Eigenvalues of $H_{\text{res}}^{(3)}$ for low phonon numbers are identified and the effect of homogeneous broadening is clearly visible as broadening of the peaks in vertical and horizontal directions.