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Physical Review A

Beliaev damping of gapped excitations in a two-component Bose-Einstein condensate

Rukuan Wu

Published Jun 7, 2024• 1 citation
Publisher

Abstract

We investigate the Beliaev damping of gapped excitations in a homogenous two-component Bose-Einstein condensate with Rabi coupling and find different damping behavior of gapped excitations in two quantum phases. In the paramagnetic phase, due to constraints imposed by the ${\mathbb{Z}}_{2}$ symmetry and energy-momentum conservation, the only damping channel available, that of decaying into one phonon and one new gapped mode, cannot occur unless the group velocity of the gapped modes surpasses the sound speed. Conversely, in the ferromagnetic phase, three distinct decay channels exist. Once the momentum of the excitations falls below a critical threshold, only one of these channels, that of decaying into two gapless modes, is activated, and the damping rate remains finite even at zero momentum.

DOI
10.1103/physreva.109.063314

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Cambridge University Press eBooks•Feb 19, 2009•0 citations
We investigate the Beliaev damping of gapped excitations in a homogenous two-component Bose-Einstein condensate with Rabi coupling and find different damping behavior of gapped excitations in two quantum phases. In the paramagnetic phase, due to constraints imposed by the Z2{\mathbb{Z}}_{2}Z2​ symmetry and energy-momentum conservation, the only damping channel available, that of decaying into one phonon and one new gapped mode, cannot occur unless the group velocity of the gapped modes surpasses the sound speed. Conversely, in the ferromagnetic phase, three distinct decay channels exist. Once the momentum of the excitations falls below a critical threshold, only one of these channels, that of decaying into two gapless modes, is activated, and the damping rate remains finite even at zero momentum.
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{k}^{5}k5 to a k5/2{k}^{5/2}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.
The condensates of cold atoms at zero temperature in the tunable binary Bose-Einstein condensate system are studied with the Rabi transition between atomic hyperfine states where the system can be represented by a coupled two-field model of gapless excitations and gapped excitations. We set up the configuration of the supersonic and subsonic regimes with the acoustic horizon between them in the elongated two-component Bose-Einstein condensates, trying to mimic Hawking radiations, in particular due to the gapped excitations. The simplified steplike sound speed change is adopted for the subsonic-supersonic transition so that the model can be analytically treatable. The effective-energy gap term in the dispersion relation of the gapped excitations introduces the threshold frequency ${\ensuremath{\omega}}_{\mathrm{min}}$ in the subsonic regime, below which the propagating modes do not exist. Thus, the particle spectrum of the Hawking modes significantly deviates from that of the gapless cases near the threshold frequency due to the modified gray-body factor, which vanishes as the mode frequency is below ${\ensuremath{\omega}}_{\mathrm{min}}$. The influence from the gapped excitations to the quantum entanglement of the Hawking mode and its partner of the gapless excitations is also studied according to the Peres-Horodecki-Simon (PHS) criterion. It is found that the presence of the gapped excitations will deteriorate the quantumness of the pair modes of the gapless excitations when the frequency of the pair modes in particular is around $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\omega}}_{\mathrm{min}}$. On top of that, when the coupling constant between the gapless and gapped excitations becomes large enough, the huge particle density of the gapped excitations in the small $\ensuremath{\omega}$ regime will significantly disentangle the pair modes of the gapless excitations. The detailed time-dependent PHS criterion will be discussed.
Beliaev damping provides a fundamental mechanism for dissipation of quasiparticles. Previous research has shown that the two-component internal degrees of freedom has no nontrivial effect on Beliaev damping. Here we provide the first example where the spinor nature of Bose gases can manifest itself in the Beliaev damping by way of spin-obit coupling. We identify novel features of the Beliaev decay rate due to spin-orbit coupling; in particular, it shows an explicit dependence on the spin-density interaction and diverges at the interaction-modified phase boundary between the zero-momentum and plane wave phases. This represents a manifestation of the effect of spin-orbit coupling in the beyond-mean-field regime, which by breaking Galilean invariance couples excitations in the density and spin channels. We further show that the measurement of the Beliaev damping rate is experimentally feasible through the measurement of spin polarizability susceptibility, which has been already achieved in spin-orbit-coupled Bose gases.
We explore the quantum many-body physics of a driven Bose-Einstein condensate in optical lattices. The laser field induces a gap in the generalized Bogoliubov spectrum proportional to the effective Rabi frequency. The lowest-lying modes in a driven condensate are characterized by zero group velocity and nonzero current. Thus, the laser field induces roton modes, which carry interaction in a driven condensate. We show that collective excitations below the energy of the laser-induced gap remain undamped, while above the gap they are characterized by a significantly suppressed Landau damping rate.
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.
With very few exceptions (such as the centre-of-mass dipole mode), collective oscillations in trapped superfluid Bose gases are damped. In the “collisionless” region the damping is second order in the interaction strength. There are three possible components. One is Beliaev damping, which is due to the decay of a single excitation into two excitations; this can occur even at T = 0. In addition, there is Landau damping, which is due to a collective mode scattering from thermally excited excitations. This process only occurs at finite temperatures but quickly becomes the dominant damping mechanism as the temperature increases. Both Landau and Beliaev damping arise naturally from the imaginary part of the Beliaev second-order self-energies, as given in (5.40) in the case of a uniform Bose gas. Finally there is the damping that arises from the C 22 and C 12 collision processes; this is discussed in Chapters 8, 12 and 19. In Chapter 12 we calculated the damping of various condensate modes at finite temperatures using direct numerical simulations of the ZNG equations. These numerical results were generally in very good agreement with the available experimental data. From a theoretical perspective, one advantage of the simulations is that the Landau damping contribution can be isolated simply by setting the C 12 and C 22 collision terms to zero. After providing an introduction to Landau damping in uniform Bose gases in Section 13.1, we present in Section 13.2 a detailed discussion of Landau damping based on a general formula in terms of Bogoliubov–Popov excitations. This discussion makes it clear that the Landau damping of condensate oscillations arises from the interaction with a thermal cloud of excitations.