Excitation Modes of Bright Matter-Wave Solitons

We experimentally study the excitation modes of bright matter-wave solitons in a quasi-one-dimensional geometry. The solitons are created by quenching the interactions of a Bose-Einstein condensate of cesium atoms from repulsive to attractive in combination with a rapid reduction of the longitudinal confinement. A deliberate mismatch of quench parameters allows for the excitation of breathing modes of the emerging soliton and for the determination of its breathing frequency as a function of atom number and confinement. In addition, we observe signatures of higher-order solitons and the splitting of the wave packet after the quench. Our experimental results are compared to analytical predictions and to numerical simulations of the one-dimensional Gross-Pitaevskii equation.

Figure 1

Experimental setup and oscillation measurements. (a) Sketch of the experimental setup. Inset: Density profiles for a BEC (solid red line) and for a soliton (dashed blue line). (b) Total energy of a soliton, $a=-5.2a_0$, ${\omega }_r=2\pi \times 95\mathrm{Hz}$, $N=2000$, with an external trap, ${\omega }_z=2\pi \times 5\mathrm{Hz}$ (dashed blue line), and without external trap, ${\omega }_z=0\mathrm{Hz}$ (solid red line). (c) Absorption images after a free-expansion time of 16 ms [from dataset with circles in (d)], integrated density profile for $t=60\mathrm{ms}$ (blue line) and fit (dashed red line). (d) Oscillations of a quantum gas after the quench procedure. Blue diamonds, quench of only ${\omega }_z$ for a BEC ($Q1$); red circles, additional interaction quench to create soliton ($Q3$); green squares, optimized quench parameters to minimize breathing of the soliton ($Q2$). Uncertainty intervals indicate $\pm 1$ standard error.

Figure 2

Breathing frequency ${\omega }_{\mathrm{sol}}$ of the soliton. (a) Atom number dependence ($Q4$). Red circles, experimental data; the uncertainty bars for the atom number indicate the standard deviation of $N$ over the first 100 ms of each frequency measurement. Blue triangles, simulation of the 1D GPE [15]. Red line, analytical approximation [13, 15]. Dashed gray line, oscillation frequency of a noninteracting gas, $2{\omega }_{z,f}$. (b) Dependence of ${\omega }_{\mathrm{sol}}$ on the trap frequency ($Q5$). Red circles, experimental data points for $N\approx 1450$. Blue area, simulation of the 1D GPE for $N=1300–1500$. Red line, analytical approximation. Dashed gray line, $2{\omega }_{z,f}$.

Figure 3

Time evolution after a strong quench of interactions and trap frequency ($Q6$). (a) 1D GPE simulation of the density profiles for a second-order soliton with 1100 atoms, $a_f=-5.3a_0$, and with an oscillation period $T^{(2)}$ of 432 ms. (b) Absorption images at time $t$ after the quench and after 11 ms of free expansion. (c) Time evolution of the measured width $l_z$ of the central wave packet (red circles), sinusoidal fit with period 420(30) ms (dashed red line). The uncertainty intervals indicate $\pm 1$ standard deviation.

Figure 4

Second-order soliton and splitting after the double quench $Q7$. (a) Absorption images at time $t$ after the quench and after 7 ms of free expansion. (b) Time evolution of the measured width $l_z$ of the central wave packet (red circles), sinusoidal fit with period 180(20) ms (dashed red line). The expected period from the 1D GPE simulations is 192 ms. The histogram counts the fraction of images showing a splitting of the wave function (9 repetitions per time step). Inset: Absorption image of a split matter wave for $t=210\mathrm{ms}$.