Fig. 4: Valence bond resonances.
From: Probing resonating valence bonds on a programmable germanium quantum simulator
a Probabilities of having horizontal singlet pairs \({P}_{{S}_{12}{S}_{34}}\) and vertical singlet pairs \({P}_{{S}_{23}{S}_{14}}\) as a function of dwell time tD. All the exchange couplings are tuned toward an identical value of Jij/h ≃ 25 MHz. Lines are fits to the data with \({P}_{SS}=1/2\,{{{\mathcal{V}}}}\cos (2\pi {f}_{SS}{t}_{{{{\rm{D}}}}}+\phi )\exp (-{({t}_{{{{\rm{D}}}}}/{T}_{\varphi })}^{2})+{A}_{0}\) giving respectively Tφ = 167 ns and 143 ns for data corresponding to \({P}_{{S}_{12}{S}_{34}}\) and \({P}_{{S}_{23}{S}_{14}}\). The state is initialized as \(\left\vert {S}_{x}\right\rangle\). b Illustration showing valence bond resonances characterized by oscillations between the singlet product states \(\left\vert {S}_{x}\right\rangle\) and \(\left\vert {S}_{y}\right\rangle\). c \({P}_{{S}_{12}{S}_{34}}\) and, d \({P}_{{S}_{23}{S}_{14}}\) as a function of tD and virtual barrier gate voltage variation \(\delta {V}_{x}^{{\prime} }\). The state is initialized as \(\left\vert {S}_{x}\right\rangle\). e The oscillation frequency as a function of \(\delta {V}_{x}^{{\prime} }\). The blue (red) points are extracted from (c, d). The black points are the theoretical predictions \({f}_{SS}=\sqrt{{J}_{x}^{2}+{J}_{y}^{2}-{J}_{x}{J}_{y}}/h\) computed using the exchanges Jx and Jy measured in Fig. 3e. f Visibility \({{{{\mathcal{V}}}}}_{x,y}\) as a function of gate voltage variation \(\delta {V}_{x}^{{\prime} }\). The triangles in blue (red) are extracted from (c, d). The expected values are derived from equations (5) and (7) of Supplementary Note 4 using the measured exchanges. The shaded areas correspond to one standard deviation from the best fit for the experimental data, and for the theoretical data they correspond to the uncertainties on the amplitude and the frequency computed using the uncertainties on the exchange couplings values. g Ratio of the visibilities \({{{{\mathcal{V}}}}}_{y}/({{{{\mathcal{V}}}}}_{x}+{{{{\mathcal{V}}}}}_{y})\) as a function of the gate voltage variation \(\delta {V}_{x}^{{\prime} }\).
