Ultra-cold atoms in an optical lattice, which are highly controllable and quite clean quantum systems, allow us to simulate quantum many-body phenomena found in condensed matters and related fields. As a typical example, a quantum phase transition from superfluid (SF) to Mott insulator (MI) of bosons has been experimentally demonstrated by tuning lattice depths [1]. However, systematic and comprehensive measurements across the two phases, SF and MI, are still lacking. This is because measurement methods used in the previous works are basically standing on the strategy to detect a certain characteristic signal for one of the two phases and then to determine the critical point where such a signal disappears: For example, the time-of-flight imaging has been used to identify SF-MI transitions by detecting the disappearance of the coherent peak that characterizes the existence of SF states [1].

Kyoto Univ. and NTT collaboratively developed high-resolution spectroscopy [2] and a numerical scheme for calculating observed spectra [3], respectively. The developed spectroscopy has been used to comprehensively measure spectra of bosons in a three-dimensional optical lattice across the SF and MI phases [2]. Our numerical scheme takes into account the physical sum rule [3], which allows us to systematically analyze the spectra without any fitting parameters. As shown in Fig. 1, experiments and calculations show good agreements with each other, and the obtained spectra clearly capture the signatures of the existence of both SF and MI states in shallow and deep lattices, respectively [2]. Furthermore, we found that SF and MI states coexist in a middle lattice [2]. A combination of the high-resolution spectroscopy and the numerical scheme can determine quantum phase transitions of bosons in detail, and in addition, it will be applicable to the future quantum simulation on the various systems, e.g., fermions or boson-fermion mixtures confined in optical lattices.

This work was partly supported by CREST (JPMJCR08F5), JST and JSPS KAKENHI JP25287104.

- [1] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys.
**80**, 885 (2008), and references therein. - [2] S. Kato, K. Inaba et al., Nature Commun.
**7**, 11341 (2016). - [3] K. Inaba and M. Yamashita, Phys. Rev. A
**93**, 043608 (2016).

Fig. 1. Spectrum of bosons obtained experimentally (blue circles) and those calculated numerically (orange solid lines) in (a) shallow, (b) middle, and (c) deep optical lattices. The broadened spectra originate from the superfluid states, and the discrete narrow spectra result from the Mott insulators. The both characteristics can be seen in the middle region. The data is extracted from Ref. [2]. |