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3.2Array control of two ion speciesAs described above, it was conrmed that even an ion species that dees direct laser cooling can be brought into a quiescent state by utilizing sympathetic cooling. e next issue is to control the arrangement of the ions: the target ion must be positionally controlled for its quantum state to be measured. For example, In+ ion should preferably be positioned as in Fig. 8 (a) in view of optical frequency standard applications. For this purpose, the authors devel-oped a method to maintain specic ion arrangements, whereby the dependency of the collective vibration mode frequency on the arrangement and mass of each ion species is utilized [9]. e calculated in-phase frequencies (all ions vibrate in coordinated phase along the array axis) were ν1 (=100.5 kHz) for array (a), and ν2 (=98.5 kHz) for arrays (b) and (c). Figure 9, from (1) to (4), represents time evolution of the array, in which time (vertical axis) is plot-ted against coordinate values of the array axis (uorescence intensity is color-coded against the background). Spontaneous emission from Ca+ (as shown in (1)) can produce recoil leading to random transpositions among the ions as shown in (a), (b), and (c). is situation can be simulated by controlling the trap potential appropriately. Applying the frequency characteristic to array (b) and (c) (i.e. ν2) to trap potential excites a mode of collective vibra-tion that makes the arrays (b) and (c) unstable as shown in (2). is unstable situation is resolved at the moment the array is recongured to (a). Conversely, applying the frequency of array (a) (i.e. ν1) to modulate the intensity of a 397-nm laser (used for laser cooling of Ca+) makes the array unstable and induces its transition to (b) and (c) to regain stability (see (3)). Figure 9 (4) represents the situa-tion in which array (a) is constantly maintained: at the moment array (b) or (c) arises, ν2 is applied to destabilize the system to induce a return to array (a). is selective destabilization approach enables control and measurement of quantum states with In+ always positioned at the center. e authors consider this approach applicable to systems consisting of a larger number of ions.3.3Sideband coolinge vibrational state of ionic motion, when quantized in an ion trap, can be specied by the number of phonons (a quantum number) existing in the system. e elec-tronic state of the ions is manipulated by the laser, thus the number of phonons in the system can be controlled by adjusting the laser frequency. By removing the phonons from the system one by one through careful adjustment of the laser frequency, the vibration state can be brought into the ground state. is approach is called sideband cooling. e authors conducted research [10], in collaboration with researchers from Osaka University Graduate School of Engineering Science, whose objective was to implement sideband cooling in a two-ion system (an ionic array consisting of a Ca+ and an In+) whereby electronic transi-tion of Ca+ is exploited. When sideband cooling is applied to the system while it is vibrating in an out-of-phase mode (one of the vibration modes in a two-ion system where the ions vibrate antiphase to each other), the average number of phonons was found to be 0.096 [10]. is value indicates FiF9　Control of ion array(2)(3)(4)(1)200sFiF8　Sympathetic cooling of an In+ using two Ca+ ions8(a)(b)(c)754-4 Quantum State Engineering of Trapped Ions