Phonon evaporation in freely expanding Bose-Einstein condensates

Phonon like excitations can be imprinted into a trapped Bose-Einstein condensate of cold atoms using light scattering. If the condensate is suddenly let to freely expand, the initial phonons lose their collective character by transferring their energy and momentum to the motion of individual atoms. The basic mechanisms of this evaporation process are investigated by using the Gross-Pitaevskii theory and dynamically rescaled Bogoliubov equations. Different regimes of evaporation are shown to occur depending on the phonon wavelength. Distinctive signatures of the evaporated phonons are visible in the density distribution of the expanded gas, thus providing a new type of spectroscopy of Bogoliubov excitations.

Figure 1

Column (below) and linear (above) density, in arbitrary units, of a condensate subject to a Bragg pulse with $k=0.5a_{\rho }^{-1}$, $\omega =1.1{\omega }_{\rho }$, $V_B=0.2\hslash {\omega }_{\rho }$, and $t_B=4{\omega }_{\rho }^{-1}$ and a subsequent free expansion of $t=25{\omega }_{\rho }^{-1}$. The white ellipse is the shape of the condensate at $t=0$, at the beginning of the expansion. Distances are in units of $a_{\rho }={[\hslash ∕\left(m{\omega }_{\rho }\right)]}^{1∕2}$.

Figure 2

Same as Fig. 1, but for a Bragg pulse with $k=2.5a_{\rho }^{-1}$, $\omega =4.7{\omega }_{\rho }$, $V_B=1\hslash {\omega }_{\rho }$, and $t_B=1{\omega }_{\rho }^{-1}$.

Figure 3

Lower part: the quantity $\mathrm{\Delta }n(\rho ,z)$ at $t=25{\omega }_{\rho }^{-1}$ is plotted, in arbitrary units, for the same simulation of Fig. 1. Black (white) means positive (negative) $\mathrm{\Delta }n(\rho ,z)$. The dashed line is the position of the surface of the expanding condensate. The uniform grey color far outside the condensate corresponds to $\mathrm{\Delta }n=0$. Solid lines are the position of maxima of $\mathrm{\Delta }n(\rho ,z)$ (wave fronts). Dot-dashed lines are the same wave fronts estimated by means of Eq. (49). Upper part: integrated density variation $2\pi \int d\rho \rho \mathrm{\Delta }n$, calculated at $t=0$ (dashed line), $10{\omega }^{-1}$ (dot-dashed line), and $25{\omega }_{\rho }^{-1}$ (solid line).

Figure 4

Lower part: the quantity $\mathrm{\Delta }n(\rho ,z)$ at $t=25{\omega }_{\rho }^{-1}$ is plotted, in arbitrary units, for the same simulation of Fig. 2. The meaning of the grey scale is the same as in the lower part of Fig. 3. Upper part: integrated density variation $2\pi \int d\rho \rho \mathrm{\Delta }n$, calculated at $t=0$, $10{\omega }_{\rho }^{-1}$, and $25{\omega }_{\rho }^{-1}$ from the thin to thick solid line, respectively.

Figure 5

Position $z\left(t\right)$ of a wave front for $k=0.5a_{\rho }^{-1}$ and $\rho =0$, $0.4R$, and $0.8R$ (points $A$, $B$, and $C$ in Fig. 3), where $R$ is the radial size of the expanding condensate. The thin solid line is $z\left(t\right)=ct$, where $c={[gn\left(0\right)∕\left(2m\right)]}^{1∕2}$ is the sound velocity in a cylindrical condensate with the same central density $n\left(0\right)$.

Figure 6

Spectrum of axially symmetric Bogoliubov excitations of a cylindrical condensate with $\eta =9.1$. The frequency ${\omega }_{nk}$, in units of the radial trapping frequency ${\omega }_{\rho }$, is plotted as a function of the axial wave vector $k$, in units of $a_{\rho }^{-1}$. The number of radial nodes is $n=0,1,2,\dots$, starting from the lowest branch. The quasiparticle amplitudes $u$ and $v$ of the modes $a$, $b$, $c$, $d$, $e$, and $f$ are shown in the first column of 7, 8.

Figure 7

Time evolution of the functions $\mid \tilde{u}\left(\rho \right)\mid$ (solid lines) and $\mid \tilde{v}\left(\rho \right)\mid$ (dashed lines) obtained from Eq. (18). At $t=0$ these functions coincide with the Bogoliubov amplitudes $u_{0k}$ and $v_{0k}$, solutions of Eq. (7) for the modes (a), (b), and (c) in the spectrum of Fig. 6. The radial coordinate $\rho$ is given in units of $a_{\rho }$. The Thomas-Fermi radius of the condensate is $R=4.27a_{\rho }$.

Figure 8

Same as in Fig. 7, but for the modes (d), (e), and (f). At $t=0$ the functions $\mid \tilde{u}\mid$ and $\mid \tilde{v}\mid$ coincide with the Bogoliubov amplitudes $u_{1k}$ and $v_{1k}$, solutions of Eq. (7) with one radial node. The position of the $t=0$ node is shown as a black dot on the horizontal axis.

Figure 9

Normalized radial average of ${\mid \tilde{v}\mid }^2$ as a function of $t$, in units of ${\omega }_{\rho }^{-1}$, for different values of $k$, in units of $a_{\rho }^{-1}$. Solid lines are the results obtained from Eq. (18) for the infinite cylinder. Dashed lines are the results obtained from Eq. (23) or, equivalently, from Eq. (48) and correspond to the behavior of ${\mid \tilde{v}\mid }^2$ in a uniform gas whose density is equal to the radial average of the density of the expanding infinite cylinder. Dot-dashed lines correspond to the adiabatic following approximation (45). For larger values of $k$ all curves converge to the analytic result (46) shown as a dotted line in the bottom-right quadrant.

Figure 10

Wave front position vs time for $k=0.5a_{\rho }^{-1}$ and $k=2a_{\rho }^{-1}$, obtained from the maxima of $\delta \tilde{n}$ defined in Eq. (20) with $\tilde{u}$ and $\tilde{v}$ given by Eq. (18). As in Fig. 5, the wave front position is plotted for $\rho =0$ (solid lines), $0.4R$ (dot-dashed lines) and $0.8R$ (dashed lines), where $R$ is the Thomas-Fermi radius of the condensate. The slope of the thin solid and dashed lines is the Bogoliubov sound velocity $c={[gn\left(0\right)∕\left(2m\right)]}^{1∕2}$ and the phase velocity of free particles, $\hslash k∕\left(2m\right)$, respectively. The inset shows the shape of two nearest fronts at $t=0$ (dashed lines) and $t=10{\omega }_{\rho }^{-1}$ (solid lines). Each front is plotted from $\rho =0\text{to}\rho =R=4.27a_{\rho }$. The distance between the two fronts at $t=0$, in units of $a_{\rho }$, is $2\pi ∕k$.

Figure 11

Same as in Fig. 10, but using the local velocity (29) to get the wave front position at each $\rho$ and $t$.

Figure 12

Ratio ${\mid c_{-}\mid }^2∕{\mid c_{+}\mid }^2$ at $t\to \infty$ obtained by solving Eq. (48) for different values of $k$, in units of $a_{\rho }^{-1}$, with the condition $c_{+}=1$ and $c_{-}=0$ at $t=0$.

Figure 13

Wave front position vs time for the elongated condensate of Figs. 1, 3. Points correspond to the crests of the linear density obtained from the GP simulation. Solid lines are the results of the local density model, with local phase velocity given by Eq. (49). Thin dashed lines represent the axial size of the expanding condensate. Length and time are in units of $a_{\rho }$ and ${\omega }_{\rho }^{-1}$, respectively.

Figure 14

Radial density profiles, in arbitrary units, of the released-phonon cloud at $z=\hslash kt∕m$, with $t=25{\omega }_{\rho }^{-1}$, $k=2.5a_{\rho }^{-1}$, and three different values of $\omega$. Points with dashed lines are the results of GP simulations with $V_B=0.3\hslash {\omega }_{\rho }$ and $t_B=4{\omega }_{\rho }^{-1}$. Solid lines are the functions $\mid u_{nk}+v_{nk}\mid$ for an infinite cylinder with $\eta =9.1$, the same $k$ and $n=0$ (bottom), 1 (mid), and 2 (top).