Slow sound in matter-wave dark soliton gases

We demonstrate the possibility of drastically reducing the velocity of phonons in quasi-one-dimensional Bose-Einstein condensates. Our scheme consists of a dilute dark soliton “gas” that provide the trapping for the impurities that surround the condensate. We tune the interaction between the impurities and the condensate particles in such a way that the dark solitons result in an array of qutrits (three-level structures). We compute the phonon-soliton coupling and investigate the decay rates of these three-level qutrits inside the condensate. As such, we are able to reproduce the phenomenon of acoustic transparency based purely on matter-wave phononics, in analogy with the electromagnetically induced transparency effect in quantum optics. Thanks to the unique properties of transmission and dispersion of dark solitons, we show that the speed of an acoustic pulse can be brought down to ∼5 $\mu \mathrm{m}$/s, ∼${10}^3$ times lower than the condensate sound speed. This is a value that greatly underdoes most of the reported studies for phononic platforms. We believe the present work could pave the stage for a new generation of “stopped-sound”-based quantum information protocols.

Figure 1

Schematic representation of dark soliton qutrit array immersed in a BEC. The impurities (free particles) are trapped inside a dark soliton potential. The wiggly lines represent the quantum fluctuations on top of the condensate (phonons).

Figure 2

(a) The impurity states and the respective transition frequencies ${\omega }_i(i=0,1)$ in the dark soliton potential. (b) The dependence of the transition frequencies on the BEC-impurity coupling $g_{12}$. The vertical dashed lines correspond to the range $4/5\le \nu <9/7$ defined for the qutrit. For definiteness, we have used $m_2=1.56m_1$, corresponding to a ${}^{134}\mathrm{Cs}$ impurity loaded in a ${}^{85}\mathrm{Rb}$ BEC dark soliton.

Figure 3

Dependence of the decay rates ${\gamma }_0$ (blue line) and ${\gamma }_1$ (red line) on the BEC-impurity coupling. The solid lines correspond to the range $\frac{4}{5}\le \nu <\frac{9}{7}$ that defines the qutrit. We have used $m_2=1.56m_1$, as in Fig. 7.

Figure 4

Acoustic susceptibility $\chi$ dependence on the probe detuning ${\mathrm{\Delta }}_p$. (a) The dispersion and (c) absorption spectra calculated for $g_{12}=1.1g_{11}$ (dashed line) and $g_{12}=1.85g_{11}$ (solid line), with ${\mathrm{\Omega }}_c>{\gamma }_1$. (b) The dispersion and (d) absorption for ${\mathrm{\Omega }}_c=0.2{\gamma }_0$ (dashed line) and ${\mathrm{\Omega }}_c=2{\gamma }_0$ (solid line).

Figure 5

(a) The dispersion curves, where the solid line illustrates the solutions of Eq. (12) for $g_{12}=1.85g_{11}$ and the dashed line shows the Bogoliubov energy spectrum. The inset shows the dependence of the absorption (transparency) width on the control Rabi frequency ${\mathrm{\Omega }}_c$. (b) The group velocity $v_g$ of the order of 0.06 of the sound speed $c_s$. In all cases, we have set $m_2=1.56m_1$ and a soliton concentration of $N\xi =0.2$.

Figure 6

Schematic representation of a four-level system obtained for $\nu =10/7$. The highest excited state exists at the border of the potential created by the dark soliton.

Figure 7

Qutrits in a possible experimental situation: numerical profiles of the dark soliton (black lines) and the impurity eigenstates (red lines). From left to right, we depict the ground state ${\phi }_0\left(x\right)$ and the first and second states, ${\phi }_1\left(x\right)$ and ${\phi }_2\left(x\right)$, respectively, of a fermionic ${}^{134}\mathrm{Cs}$ impurity trapped in a ${}^{85}\mathrm{Rb}$ BEC dark soliton. The solid lines are the numerical solutions, while the dashed lines are the analytical expression described in the main text. We have used the following parameters: $m_2=1.56m_1,g_{12}=1.25g_{11}$ (corresponding to $\nu =1.13$). We fix the number of depleted condensate atoms by the dark soliton to be $n_0\xi \simeq 50$.

Figure 8

Interband $g_{l{l}^{\prime }}^k$ ($l\ne l^{\prime }$; solid lines) and intraband $g_{l{l}^{\prime }}^k$ ($l=l^{\prime }$; dashed and dot-dashed lines) coupling amplitudes. Near resonance ($k\sim 0.9{\xi }^{-1}$ for the first transition and $k\sim 0.7{\xi }^{-1}$ for the second transition), the interband terms clearly dominate over the intraband transitions, allowing us to neglect the latter within the rotating wave approximation.