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arXiv:2210.06237 · Physical Review A

Lifetime of excitations in atomic and molecular Bose-Einstein condensates

Matteo Bellitti, Garry Goldstein, Chris R. Laumann

Published Mar 6, 2023• 1 citation
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Abstract

Recent experimental progress produced molecular superfluids (MSF) in thermal equilibrium; this opens the door to a new class of experiments investigating the associated thermodynamic and dynamical responses. We review the theoretical picture of the phase diagram and quasiparticle spectrum in the atomic superfluid (ASF) and MSF phases. We further compute the parametric dependence of the quasiparticle lifetimes at one-loop order. In the MSF phase, the $\text{U}(1)$ particle-number symmetry breaks to ${\mathbb{Z}}_{2}$ and the spectrum exhibits a gapless Goldstone mode in addition to a gapped ${\mathbb{Z}}_{2}$-protected atom-like mode. In the ASF phase, the $\text{U}(1)$ symmetry breaks completely, leaving behind a Goldstone mode and an unprotected gapped mode. In both phases, the Goldstone mode decays with a rate given by the celebrated Belyaev result, as in a single component condensate. In the MSF phase, the gapped mode is sharp up to a critical Cherenkov momentum beyond which it emits phonons. In the ASF phase, the gapped mode decays with a constant rate even at small momenta. These decay rates govern the spectral response in microtrap tunneling experiments and lead to sharp features in the transmission spectrum of atoms fired through molecular clouds.

DOI
10.1103/physreva.107.033304

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Recent experimental progress produced molecular superfluids (MSF) in thermal equilibrium; this opens the door to a new class of experiments investigating the associated thermodynamic and dynamical responses. We review the theoretical picture of the phase diagram and quasiparticle spectrum in the atomic superfluid (ASF) and MSF phases. We further compute the parametric dependence of the quasiparticle lifetimes at one-loop order. In the MSF phase, the U(1)\text{U}(1)U(1) particle-number symmetry breaks to Z2{\mathbb{Z}}_{2}Z2​ and the spectrum exhibits a gapless Goldstone mode in addition to a gapped Z2{\mathbb{Z}}_{2}Z2​-protected atom-like mode. In the ASF phase, the U(1)\text{U}(1)U(1) symmetry breaks completely, leaving behind a Goldstone mode and an unprotected gapped mode. In both phases, the Goldstone mode decays with a rate given by the celebrated Belyaev result, as in a single component condensate. In the MSF phase, the gapped mode is sharp up to a critical Cherenkov momentum beyond which it emits phonons. In the ASF phase, the gapped mode decays with a constant rate even at small momenta. These decay rates govern the spectral response in microtrap tunneling experiments and lead to sharp features in the transmission spectrum of atoms fired through molecular clouds.
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.
Beliaev damping in a superfluid is the decay of a collective excitation into two lower-frequency collective excitations; it represents the only decay mode for a bosonic collective excitation in a superfluid at T=0T=0T=0. The standard treatment for this decay assumes a linear spectrum, which in turn implies that the final-state momenta must be collinear to the initial state. We extend this treatment, showing that the inclusion of a gradient term in the Hamiltonian yields a realistic spectrum for the bosonic excitations; we then derive a formula for the decay rate of such excitations and show that even moderate nonlinearities in the spectrum can yield substantial deviations from the standard result. We apply our result to an attractive Fermi gas in the BCS--Bose-Einstein condensate crossover: Here the low-energy bosonic collective excitations are density oscillations driven by the phase of the pairing order field. These collective excitations, which are gapless modes as a consequence of the Goldstone mechanism, have a spectrum that is well established both theoretically and experimentally and whose linewidth, we show, is determined at low temperatures by the Beliaev decay mechanism.
The paradoxes and disparities in the contemporary microscopic theory of superfluid helium (He–II) are discussed along with possible ways of resolving them by taking pair correlations of He4 atoms into consideration. It is shown that most paradoxes are associated with the commonly accepted initial assumption concerning the dominating role of single-particle Bose condensate (SPBC) in the quantum microstructure of the superfluid component ρs. The existence of intensive SPBC leads to a strong hybridization of the elementary excitation branches and to a common dispersion law for all boson branches, which is identified with the quasiparticle spectrum E(p) observed experimentally from slow neutron scattering in liquid helium. However, the stability of this spectrum during a transition through the λ-point and the large value of the gap in the vicinity of the “rotonic” minimum contradict both the Landau theoretical criterion of superfluidity and the small value of experimentally measured critical velocity. At the same time, a strong interaction between particles in the Bose liquid He4 strongly suppresses the SPBC which amounts to less than 1% of all He4 atoms and hence cannot be the main constituent of the superfluid component, unlike the case of a weakly nonideal Bose gas. Moreover, for a quite strong attraction between particles in a certain region of the momentum space, bound pairs of bosons can be formed in the superfluid Bose liquid, and a coherent pair condensate (CPC) analogous to the Cooper pair condensate in superconductors may appear. Such a strong CPC may completely suppress the weak SPBC. In this case, the one-particle spectrum ε(p) of elementary excitations does not hybridize with the collective (two-particle) spectrum and does not appear in the structure of the dynamic form factor S(p,ε), i.e., does not coincide with the spectrum measured from neutron scattering. The dispersion of one-particle spectrum is defined by the momentum dependence of the pair order parameter Ψ̃(p) and may have a minimum or a point of inflection at p≠0. This peculiarity in the one-particle spectrum of a Bose liquid with CPC but without SPBC vanishes together with Ψ̃(p) at the temperature Tc=Tλ of the phase transition from the superfluid to the normal state (unlike the rotonic minimum in the collective spectrum), while the corresponding critical velocity vc=min[ε(p)/p] vanishes at the λ-point in accordance with the Landau criterion and the experimental data. The assumption that the strong “Cooper-like” CPC is responsible for the quantum structure of the superfluid component ρs is confirmed indirectly by the successful application of the Justrow approximation (based on strong pair correlations) for describing the properties of liquid He4 and quantum liquid mixtures He3–4He on one hand, and by an anomalously large effective mass of He3 impurity atoms in He4, which is approximately equal to total mass of He3 and He4 atoms, thus pointing to the existence of helium atoms in superfluid liquid He–II. The value of the superfluid velocity circulation quantum in the Onsager–Feynman vortices in a Bose liquid with CPC but without SPBC is discussed as well as the critical velocities of superfluid He4 in ultrathin films and channels in which the creation and motion of quantum vortices are ruled out, and the quasiparticle spectrum undergoes dimensional quantization.
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We investigate the equilibrium properties of a hybrid atomic and molecular Bose-Einstein condensate in the context of the Feshbach-resonant quantum many-body model. We analyze fully the mean-field solutions of the system for a general choice of interaction parameters. The quantum fluctuations of the system are obtained from the Bogoliubov theory that characterizes the quasiparticle excitations of the system. In particular we find that the excitation energies have two branches: a Goldstone mode and a second one containing a gap. Both gap energy and phonon velocities are functions of the composition, and our analysis indicates that for a large range of parameter space the system is unstable near a Feshbach resonance.