High Sensitivity Phonon Spectroscopy of Bose-Einstein Condensates using Matter-Wave Interference

We study low-momentum excitations of a Bose-Einstein condensate using a novel matter-wave interference technique. In time-of-flight expansion images we observe strong matter-wave fringe patterns. The fringe visibility is a sensitive spectroscopic probe of in-trap phonons and is explained by use of a Bogoliubov excitation projection method applied to the rescaled order parameter of the expanding condensate. Gross-Pitaevskii simulations agree with the experimental data and confirm the validity of the theoretical interpretation. We show that the high sensitivity of this detection scheme gives access to the quantized quasiparticle regime.

Figure 1

Density $n(\rho ,z)$ of the expanding condensate, initially excited with phonons of momentum $q$. In (a) and (b): $q{a}_{\rho }=2.03$. The density in (a) is obtained by computerized tomography of TOF column density absorption images, $n_{\mathrm{c}\mathrm{o}\mathrm{l}}(y,z)$. In this image we measure nearly $2\times {10}^4$ excitations, while (b) corresponds to a full GP simulation. In both cases, note the clearly separated excitation cloud on the right of the condensate. The same quantities are plotted in (c) and (d) for $q{a}_{\rho }=0.31$. Note the strong density modulation, with no significant outcoupled fraction. The number of excitations in the simulation is $9.6\times {10}^2$.

Figure 2

Linear density $2\pi {\int }_0^{\infty }\rho n(\rho ,z)d\rho$ measured from TOF image with very low-momentum excitation $q{a}_{\rho }=0.31$ (solid line). Note the slight asymmetry to right, which is in agreement with the GPE simulation (dashed line).

Figure 3

Fourier transform of the column density $n_{\mathrm{c}\mathrm{o}\mathrm{l}}(y,z)$ for the same image as Fig. 1c. (a) Fourier transform. (b) Integration over central strip of (a). Note the clear separation between the central mode and the sidelobes, despite the low-momentum of the excitation.

Figure 4

Fringe visibility as a function of the Bragg excitation frequency (points). The error bars represent the uncertainty due to four measurements. We observe a clear double-peaked spectrum, which is finite-time broadened. The peaks are found at $\omega =\pm 137\pm 10\mathrm{H}\mathrm{z}$, close to the expected Bogoliubov frequency ($138\pm 5\mathrm{H}\mathrm{z}$). The solid line is a full GPE simulation of the experiment. The dashed line is the result of Eq. (5).

Figure 5

Theoretical fringe visibility after TOF for $q{a}_{\rho }=0.31$ (circles) compared with the standard atom counting technique for $q{a}_{\rho }=2.03$ (boxes), as a function of the number of excitations. The visibility of the atom counting method grows linearly in excitation number with a very weak slope of $1/N$, whereas the fringe visibility is not negligible ($\sim 0.76\%$) even for a singly quantized excitation, and grows as the square root of the number of excitations.