Reflection phase shifts of bouncing Bogoliubov waves

Abstract

The Bogoliubov-de Gennes equations are solved for an inhomogeneous condensate in the vicinity of a turning point, addressing the full continuous spectrum. A basis change in the space of the two Bogoliubov "particle" and "hole" amplitudes is introduced that decouples them approximately. We find a spatially extended mode that governs mainly excitations in the condensate phase, while another mode is localised to regions with density gradients. An analytical and numerical discussion of the phase shift is provided that incident matter waves suffer upon reflection at the turning point, forming standing waves. As an application, we compute eigenfrequencies in a gravitational trap, without recourse to the local density approximation. The non-condensate density at finite temperature and its quantum depletion are discussed in a companion paper.

Figures (10)

• Figure 1: Kaleidoscope of potentials and condensate densities considered in this paper. From top left to bottom right: hard-wall potential $V = 0$, quarter-pipe potential that connects smoothly a hard wall at $z = 0$ to a flat bottom at $z \ge R$ (dotted line), exponential $V(z) = \mu \exp(- a z)$ and linear $V(z) = \mu - F z$ potential. The function $V_1 = V - \mu + g \phi^2$ (thick blue lines) applies to the elementary excitations of the condensate phase, see Eq. (\ref{['eq:BdG-phase']}). Energies are scaled to the chemical potential $\mu$, except for the linear potential where, since $\mu = 0$ by a suitable choice of coordinates, the natural unit $F \ell$ [see Table \ref{['t:units']}] is used. The turning point is at $z = 0$ in all cases, although the boundary conditions there depend on the type of potential (Dirichlet $\phi(0) = 0$ for the hard-wall potentials in the top row). Position coordinate scaled by the inverse healing length $\kappa$, as defined in Eq. (\ref{['eq:tanh-condensate']}) and Table \ref{['t:units']}.
• Figure 2: Bogoliubov mode functions $\varphi$ (left) and $f$ (right) for the hard-wall potential in the adiabatic representation. For clarity, we have shifted the potentials (thick solid lines) of Eqs. (\ref{['eq:BdG-adiabatic-phi']}, \ref{['eq:BdG-adiabatic-f']}) by the energy $E$ to $U_{\rm ad}(z) + E$ (left) and $V_{\rm ad}(z) + E$ (right); the mode functions are plotted on these “energy levels” $E \approx 1.4\,\mu$ (blue), and $0.41\,\mu$ (orange). The $k$-vectors (see inset legend) are computed from the dispersion relation (\ref{['eq:Bogoliubov-dispersion']}), and $\kappa = \sqrt{\mu/2}$. The black dashed curves illustrate the “source terms” $\hat{L} f$ (left) and $-\hat{L} \, \varphi$ (right) in Eqs. (\ref{['eq:BdG-adiabatic-phi']}, \ref{['eq:BdG-adiabatic-f']}). To enhance visibility, a few curves are multiplied as indicated in the legends. The thick gray curves correspond to the eigenvalues $M_{\varphi}(z)$, $M_{f}(z)$ of Eq. (\ref{['eq:adiabatic-potentials-0']}), the bumps in the coloured curves are due to the “geometric” potential $(\theta'(z)/2)^2$ in Eqs. (\ref{['eq:BdG-adiabatic-phi']}, \ref{['eq:BdG-adiabatic-f']}).
• Figure 3: Comparison of exact (numerical) and adiabatic solutions for the Bogoliubov modes in the density/phase ($\varphi, f$) representation. Exact results in color, adiabatic solutions black dotted. Numerical parameters as in Table \ref{['t:units']}, except $\kappa R = 8$ for the quarter-pipe case (vertical dotted line). The plotted energies are somewhat arbitrary, as they are determined by boundary conditions at the right end of the computational grid. Here, a Neumann condition is applied at $z \approx 20/\kappa\,(32\,\ell)$ for flat-bottom (linear) potentials, respectively.
• Figure 4: Phase shift $\delta(E)$ of the oscillatory Bogoliubov mode $\varphi(z)$, obtained by matching it to the asymptotic form (\ref{['eq:def-phase-1']}) or (\ref{['eq:def-phase-2']}) (for the linear potential). Symbols: numerical diagonalisation with boundary condition at the open right end as indicated, dashed lines (“adiab+”): adiabatic approximation plus the non-adiabatic correction discussed in Sec. \ref{['s:phase-shift-2']}, dotted lines (“adiabatic”): without this correction. Thick solid line (left panel): exact solution Eq. (\ref{['eq:delta-hw']}); pale thick solid line (right panel): simple turning-point construction $\delta = \pi/2 - k z_*$ (see main text). In the linear potential (right panel), the phase shift is extrapolated to infinity (Sec. \ref{['s:linear-phase']}), starting from the position $z \approx 32\,\ell$ (see Table \ref{['t:units']} for the units). Parameters: quarter-pipe potential with $\kappa R = 5.5$, exponential potential with $\kappa / a = 1/\sqrt{2}$. We plot against the root of the mode energy to enlarge the low-energy region. Results for the quarter-pipe potential with other parameters are shown in Fig. \ref{['fig:scan-R']}.
• Figure 5: Extrapolation of phase shift $\delta(z, E)$ to $z = \infty$. The coloured curves give the numerically computed phase shift based on solutions with Dirichlet boundary conditions at $L \approx 32\,\ell$ (terminating at the open dots). The vertical grid lines mark the zeroes of the reference solution $j_E( z )$. The dotted black curves are obtained as explained after Eq. (\ref{['eq:exact-d-delta-dz']}), starting from the open dots (phase shift at $z = L$) to both sides. The phase shift resulting from continuing the dotted curves to infinity is represented by the full dots with $\delta(E) \approx 0$.