Bragg spectroscopy shows the evolution of gapless Goldstone modes and single-particle-like excitations in an atomic Fermi superfluid as it crosses from a Bardeen–Cooper–Schrieffer superfluid to the Bose–Einstein condensate regime. Spontaneous symmetry breaking is a central paradigm of elementary particle physics1, magnetism2, superfluidity3 and superconductivity4. According to Goldstone’s theorem, phase transitions that break continuous symmetries lead to the existence of gapless excitations in the long-wavelength limit5. These Goldstone modes can become the dominant low-energy excitation, showing that symmetry breaking has a profound impact on the physical properties of matter. Here, we present a comprehensive study of the elementary excitations in a homogeneous strongly interacting Fermi gas through the crossover from a Bardeen–Cooper–Schrieffer (BCS) superfluid to a Bose–Einstein condensate (BEC) of molecules using two-photon Bragg spectroscopy. The spectra exhibit a discrete Goldstone mode, associated with the broken-symmetry superfluid phase, as well as pair-breaking single-particle excitations. Our techniques yield a direct determination of the superfluid pairing gap and speed of sound in close agreement with strong-coupling theories.
Significant experimental progress has been made recently for observing long-sought supersolidlike states in Bose-Einstein condensates, where spatial translational symmetry is spontaneously broken by anisotropic interactions to form a stripe order. Meanwhile, the superfluid stripe ground state was also observed by applying a weak optical lattice that forces the symmetry breaking. Despite the similarity of the ground states, here we show that these two symmetry breaking mechanisms can be distinguished by their collective excitation spectra. In contrast to gapless Goldstone modes of the spontaneous stripe state, we propose that the excitation spectra of the forced stripe phase can provide direct experimental evidence for the gapped pseudo-Goldstone modes. We characterize the pseudo-Goldstone mode of such lattice-induced stripe phase through its excitation spectrum and static structure factor. Our work may pave the way for exploring spontaneous and forced or approximate symmetry breaking mechanisms in different physical systems.
We study the decay rate $\mathrm{\ensuremath{\Gamma}}(k)$ of density excitations of two-component Bose-Einstein condensates at zero temperature. Those excitations, where the two components oscillate in phase, include the Goldstone mode resulting from condensation. While within Bogoliubov approximation the density sector and the spin (out-of-phase) sector are independent, they couple at the three-phonon level. For a Bose-Bose mixture we find that the Belyaev decay is slightly modified due to the coupling with the gapless spin mode. At the phase separation point the decay rate changes instead from the standard
k5 to a
k5/2 behavior due to the parabolic nature of the spin mode. If instead a coherent coupling between the two components is present, the spin sector is gapped and, away from the ferromagnetic-like phase transition point, the decay of the density mode is not affected. On the other hand, at the transition point, when the spin fluctuations become critical, the Goldstone mode is not well defined anymore since $\mathrm{\ensuremath{\Gamma}}(k)\ensuremath{\propto}k$. As a consequence, we show that the friction induced by a moving impurity is enhanced---a feature which could be experimentally tested. Our results apply to every nonlinear 2-component quantum hydrodynamic Hamiltonian which is time-reversal invariant and possesses an $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry.