Quantum Flutter: Signatures and Robustness

We investigate the motion of an impurity particle injected with finite velocity into an interacting one-dimensional quantum gas. Using large-scale numerical simulations based on matrix product states, we observe and quantitatively analyze long-lived oscillations of the impurity momentum around a nonzero saturation value, called quantum flutter. We show that the quantum flutter frequency is equal to the energy difference between two branches of collective excitations of the model. We propose an explanation of the finite saturation momentum of the impurity based on the properties of the edge of the excitation spectrum. Our results indicate that quantum flutter exists away from integrability and provide parameter regions in which it could be observed in experiments with ultracold atoms using currently available technology.

Figure 1

Schematic illustration of the system. An impurity atom (large blue sphere) moves with momentum $⟨P_{\downarrow }(t)⟩$ through a background gas of interacting bosonic atoms (small red spheres). The shaded area illustrates an equipotential surface confining the atomic motion to one dimension. Initially the background gas is prepared in its ground state, and the impurity is injected as a plane wave with finite momentum $⟨P_{\downarrow }(0)⟩=Q$. The background particles interact with the impurity and with each other through a repulsive $\delta$-function potential of strengths $\gamma$ and ${\gamma }_{\mathrm{bg}}$, respectively.

Figure 2

Impurity momentum $⟨P_{\downarrow }(t)⟩$ as a function of time. Red solid curve: ${\gamma }_{\mathrm{bg}}=\infty$ the integrable Tonks-Girardeau model studied in Ref. [34]. Blue dashed curve: ${\gamma }_{\mathrm{bg}}=12$, the integrable bosonic Yang-Gaudin model. Black dotted curve: ${\gamma }_{\mathrm{bg}}=4$, a nonintegrable case. The initial momentum is $Q=1.16k_F$ and the impurity-background coupling strength is $\gamma =12$ for all curves. The masses of the impurity and the background particles are equal, $m_{\downarrow }=m_{\uparrow }$. All curves exhibit a rapid drop at short times followed by pronounced slowly decaying oscillations around a finite saturation value of momentum. We call the frequency of these oscillations the quantum flutter frequency ${\omega }_f$.

Figure 3

Quantum flutter frequency for the integrable cases of model (1). Red circles: ${\gamma }_{\mathrm{bg}}=\infty$ TEBD simulations. Black diamonds: ${\gamma }_{\mathrm{bg}}=\infty$ Bethe ansatz (BA) data from Ref. [34]. Blue boxes: ${\gamma }_{\mathrm{bg}}=\gamma$ TEBD simulations. Each data point and its error bar is obtained by taking twice the distance in time between all neighboring extrema of $⟨P_{\downarrow }(t)⟩$ (exemplary curves of which are shown in Fig. 2), converting them into frequencies, and calculating their mean and standard deviation. Solid (dashed) curve is the plasmon-magnon energy difference ${\omega }_{\mathrm{pm}}$ for ${\gamma }_{\mathrm{bg}}=\infty$ (${\gamma }_{\mathrm{bg}}=\gamma$) at momentum $k_F$ obtained from Bethe ansatz.

Figure 4

Excitation spectrum. Plasmons are the lowest energy excitations of the background gas (2). The plasmon dispersion $E_p(k)$ is shown with dotted lines. Magnons are the lowest energy excitations of model (1), i.e., background gas plus impurity. The magnon dispersion $E_m(k)$ is shown with solid lines. Two integrable cases are illustrated: (a) ${\gamma }_{\mathrm{bg}}=\infty$ and (b) ${\gamma }_{\mathrm{bg}}=\gamma$. The curves are obtained from Bethe ansatz.

Figure 5

Quantum flutter for the nonintegrable cases of model (1). Top panels: Flutter frequency ${\omega }_f$. Data are obtained from TEBD simulations and error bars are obtained in the same way as for Fig. 3. The dashed line shows the plasmon energy at the Fermi momentum, $E_p(k_F)$. (a) ${\omega }_f$ as a function of $1/{\gamma }_{\mathrm{bg}}$ for $m_{\downarrow }=m_{\uparrow }$ and $\gamma =12$. (b) ${\omega }_f$ as a function of $m_{\downarrow }/m_{\uparrow }$ for ${\gamma }_{\mathrm{bg}}=\infty$, $\gamma =12$, and $Q=1.16k_F$. The quantum flutter frequency for the integrable Tonks-Girardeau model is indicated by the red diamond and for the integrable Yang-Gaudin model by the blue circle. Bottom panels: Saturated momentum $⟨P_{\downarrow }(\infty )⟩$. Data used for panels (c) and (d) are the same as for (a) and (b). Error bars indicate the standard deviation of $⟨P_{\downarrow }(t)⟩$ after the transient decay, $t>15t_F$.