Quantum metrology is the science that aims to achieve precision measurements by making use of quantum principles. Attribute to the well-developed techniques of manipulating and detecting cold atoms, cold atomic systems provide an excellent platform for implementing precision quantum metrology. In this chapter, we review the general procedures of quantum metrology and some experimental progresses in quantum metrology with cold atoms. Firstly, we give the general framework of quantum metrology and the calculation of quantum Fisher information, which is the core of quantum parameter estimation. Then, we introduce the quantum interferometry with single and multiparticle states. In particular, for some typical multiparticle states, we analyze their ultimate precision limits and show how quantum entanglement could enhance the measurement precision beyond the standard quantum limit. Further, we review some experimental progresses in quantum metrology with cold atomic systems.

Universal dynamics of spontaneous symmetry breaking is central to understanding the universal behavior of spontaneous defect formation in various systems from the early universe, condensed-matter systems to ultracold atomic systems. We explore the universal real-time dynamics in an array of coupled binary atomic Bose-Einstein condensates in optical lattices, which undergo a spontaneous symmetry breaking from the symmetric Rabi oscillation to the brokensymmetry self-trapping. In addition to Goldstone modes, there exist gapped Higgs modes whose excitation gap vanishes at the critical point. In the slow passage through the critical point, we analytically find that the symmetry-breaking dynamics obeys the Kibble-Zurek mechanism. From the scalings of bifurcation delay and domain formation, we numerically extract two Kibble-Zurek exponents, b1 = ν/(1 + νz) and b2 = 1/(1 + νz), which give the static correlation-length critical exponent ν and the dynamic critical exponent z. Our approach provides an efficient way for the simultaneous determination of the critical exponents ν and z for a continuous phase transition.

We investigate the non-equilibrium dynamics across the miscible-immiscible phase separation in a binary mixture of Bose-Einstein condensates. The excitation spectra reveal that the Landau critical velocity vanishes at the critical point, where the superfluidity spontaneously breaks down. We analytically extract the dynamical critical exponent z=2 and static correlation length critical exponent v=1/2 from the Landau critical velocity. Moreover, by simulating the real-time dynamics across the critical point, we find the average domain number and the average bifurcation delay show universal scaling laws with respect to the quench time. We then numerically extract the static correlation length critical exponent v=1/2 and the dynamical critical exponent z=2 according to Kibble-Zurek mechanism. The scaling exponents (v=1/2, z=2) in the phase separation driven by quenching the atom-atom interaction are different from the ones (v=1/2, z=1) in the phase separation driven by quenching the Rabi coupling strength (2009 Phys. Rev. Lett. 102 070401; 2011 Phys. Rev. Lett. 107 230402). Our study explores the connections between the spontaneous superfluidity breakdown and the spontaneous defect formation in the phase separation dynamics.

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According to the famous Kibble-Zurek mechanism (KZM), the universality of spontaneous defect generation in continuous phase transitions (CPTs) can be understood by the critical slowing down. In most CPTs of atomic Bose-Einstein condensates (BECs), the universality of spontaneous defect generations has been explained by the divergent relaxation time associated with the nontrivial gapless Bogoliubov excitations. However, for atomic BECs in synthetic gauge fields, their spontaneous superfluidity breakdown is resulted from the divergent correlation length associated with the zero Landau critical velocity. Here, by considering an atomic BEC ladder subjected to a synthetic magnetic field, we reveal that the spontaneous superfluidity breakdown obeys the KZM. The Kibble-Zurek scalings are derived from the Landau critical velocity which determines the correlation length. In further, the critical exponents are numerically extracted from the critical spatial-temporal dynamics of the bifurcation delay and the spontaneous vortex generation. Our study provides a general way to explore and understand the spontaneous superfluidity breakdown in CPTs from a single-well dispersion to a double-well one, such as, BECs in synthetic gauge fields, spin-orbit coupled BECs, and BECs in shaken optical lattices.Introduction. Engineered synthetic gauge fields for neutral atoms [1-8] provide new opportunities to explore exotic collective quantum phenomena [9][10][11][12][13][14]. The dispersion relation plays an important role in the emergence of many collective quantum phenomena. Through controlling the applied external fields, the dispersion relation can be tuned from a single-well shape into a double-well one, such as, spin-orbit coupled quantum gases [15][16][17][18], ultracold atoms in shaken optical lattices [19][20][21] and Bose ladders within magnetic fields [22][23][24][25]. At the transition point, due to the interplay between the synthetic gauge fields and the atom-atom interactions, the Landau critical velocity vanishes [17,18,20] and thus the superfluid spontaneously breaks down. Such a spontaneous superfluidity breakdown is very different from the conventional Landau instability which requires the superfluid velocity exceeding a nonzero critical velocity [26][27][28]. Although the static phase transitions in synthetic gauge fields have been extensively studied, the underlying dynamics of phase transitions is still unclear.