Fig. 4: Tune diagram, (\(\langle {{\varOmega }}\rangle\), \({\sigma }_{{{\varOmega }}}\)) diagram and \((\langle \bar{a}_{x}\rangle,\langle {\bar{a}}_{y} \rangle)\) chart.

All shots were measured for the same resonance excitation and machine tune settings. a, Tune diagram showing the measured average beam tunes along the interval of 3,000 turns. From the total of about 400 shots measured, around 150 shots are on the resonance. The solid line Q_x + 2Q_y = 79 is the resonance excited by our sextupole settings. Q_x and Q_y are the betatron tunes, and 79 is the harmonic number of the resonance. The statistical error of tune evaluation is within the marker size. b, (\(\langle {{\varOmega }}\rangle\), \({\sigma }_{{{\varOmega }}}\)) diagram. For each shot, we compute the turn-by-turn evolution of \(\varOmega\) along the storage period, and the corresponding mean value \(\langle {{\varOmega}}\rangle\) and standard deviation \({\sigma }_{{{\varOmega }}}\). The black solid line represents \({{{\varOmega }}}_{{{{\rm{fl}}}}}^{{{{\rm{drt}}}}}\). The blue area shows the selected shots close to the excited fixed line. Recall that \({\varOmega}={{{\phi}_x}}+2{{{\phi}_y}}\) with ϕ_x and ϕ_y the phase advances of the horizontal and vertical planes. \({{{\varOmega }}}_{{{{\rm{fl}}}}}^{{{{\rm{drt}}}}}\) is the resonance phase advance of the resonance we have excited. The dashed lines show what is expected when \(\varOmega\) is not bounded and evolves randomly. In this case, the average of \({\varOmega}\) is 0.5, and its standard deviation \({\sigma }_{\varOmega }=1/\sqrt{12}\), (in units of 2π). The random error (precision) of \(\langle{{\varOmega }}\rangle\) is \({\sigma }_{(\varOmega)}\) ≲ 6 × 10⁻⁴ and, hence, is within the marker size. c, Mean values \(\langle {\bar{a}}_{x}\rangle\) and \(\langle {\bar{a}}_{y}\rangle\) of the scaled Courant-Snyder forms for the measured shots that are close to the resonance. The quantities \({\bar{a}}_{x}\) and \({\bar{a}}_{y}\) are the scaled Courant–Snyder forms of the particles. The statistical error bar (precision) of the measured quantities \(\langle {\bar{a}}_{x}\rangle\) and \(\langle {\bar{a}}_{y}\rangle\) when the beam is on the fixed line is \({\sigma }_{\langle {\bar{a}}_{x}\rangle }\lesssim 0.5\) mm² and \({\sigma }_{\langle {\bar{a}}_{y}\rangle }\lesssim 1.0\) mm². In case the beam is not centred on a fixed line, the error bars remain nevertheless within the marker size. The quantity D (Methods and equation (9)), shown by the colour of the markers, indicates the degree of shift in the \({\bar{a}}_{x}\) and \({\bar{a}}_{y}\) values and, thus, how stable the machine conditions are during the measurement or how close the beam is to the centre of the resonance structure. Small values of D (yellow band) signify that the beam is close to a fixed line and is stationary. If D is large (blue band), then either the beam is far from the fixed line or there has been a drift of the machine parameters (Methods).