Bose-Einstein condensation denotes the formation of a collective quantum ground state of identical particles with integer spin or intrinsic angular momentum. In magnetic insulators, the magnetic properties are due to the unpaired shell electrons that have half-integer spin. However, in some such compounds (KCuCl3 and TlCuCl3), two Cu2+ ions are antiferromagnetically coupled to form a dimer in a crystalline network: the dimer ground state is a spin singlet (total spin zero), separated by an energy gap from the excited triplet state (total spin one). In these dimer compounds, Bose-Einstein condensation becomes theoretically possible. At a critical external magnetic field, the energy of one of the Zeeman split triplet components (a type of boson) intersects the ground-state singlet, resulting in long-range magnetic order; this transition represents a quantum critical point at which Bose-Einstein condensation occurs. Here we report an experimental investigation of the excitation spectrum in such a field-induced magnetically ordered state, using inelastic neutron scattering measurements of TlCuCl3 single crystals. We verify unambiguously the theoretically predicted gapless Goldstone mode characteristic of the Bose-Einstein condensation of the triplet states.

A quantum critical point (QCP) is a singularity in the phase diagram arising due to quantum mechanical fluctuations. The exotic properties of some of the most enigmatic physical systems, including unconventional metals and superconductors, quantum magnets, and ultracold atomic condensates, have been related to the importance of the critical quantum and thermal fluctuations near such a point. However, direct and continuous control of these fluctuations has been difficult to realize, and complete thermodynamic and spectroscopic information is required to disentangle the effects of quantum and classical physics around a QCP. Here we achieve this control in a high-pressure, high-resolution neutron scattering experiment on the quantum dimer material TlCuCl 3 . By measuring the magnetic excitation spectrum across the entire quantum critical phase diagram, we illustrate the similarities between quantum and thermal melting of magnetic order. We prove the critical nature of the unconventional longitudinal ("Higgs") mode of the ordered phase by damping it thermally. We demonstrate the development of two types of criticality, quantum and classical, and use their static and dynamic scaling properties to conclude that quantum and thermal fluctuations can behave largely independently near a QCP.In "classical" isotropic antiferromagnets, the excitations of the ordered phase are gapless spin waves emerging on the spontaneous breaking of a continuous symmetry [1]. The classical phase transition, occurring at the critical temperature T N , is driven by thermal fluctuations. In quantum antiferromagnets, quantum fluctuations suppress long-range order, and can destroy it completely even at zero temperature [2]. The ordered and disordered phases are separated by a quantum critical point (QCP), where quantum fluctuations restore the broken symmetry and all excitations become gapped, giving them characteristics fundamentally different from the Goldstone modes on the other side of the QCP (Fig. 1). At finite temperatures around a QCP, the combined effects of quantum and thermal fluctuations bring about a regime where the characteristic energy scale of spin excitations is the temperature itself, and this quantum critical (QC) regime has many special properties [3].Physical systems do not often allow the free tuning of a quantum fluctuation parameter through a QCP. The QC regime has been studied in some detail in heavy-fermion [14], and the green points depict the temperature and pressure values studied. Full details of this panel are presented in Fig. 4(c). The centre (E, T ) panel shows neutron intensity data collected from T = 1.8 K to 12.7 K at p = 1.75 kbar, where TN = 5.8 K. The rightmost (E, T ) panel shows the corresponding data at p = 3.6 kbar, where TN = 9.2 K. The data in both (E, T ) panels display a clear softening of the magnetic excitations at TN (p). The bottom (p, E) panel indicates the softening of the excitations, measured at T = 1.8 K, across the QPT [19]. T1, T2 and L denote the three gapped triplet excitations of th...