Characterization of stimulated excitations in a driven Bose-Einstein condensate

We apply the time-dependent Hartree-Fock-Bogoliubov (td-HFB) theory to describe the stimulated excitation driven by periodically modulating the interactions in a Bose-Einstein condensate (BEC). A comparison with the results calculated from the typical Bogoliubov approximation indicates that the additional interaction terms contributed by the excited modes play an important role to explain the dynamics of the stimulating process. The td-HFB model has not only painted a clear picture of the density wave propagation, but also partly explained the generation of the second order harmonic of the excited modes. The theoretical framework can be directly employed to study similar driven processes.

Figure 1

Diagram of the stimulated excitation in a driven BEC. (a) The one-dimensional density distributions. The BEC gets spatially separated after a modulation, indicating that the stimulated emission happens. (b) The time-dependent $s$-wave scattering length with a modulation amplitude of $a_{\text{s,t}}$ and a modulation frequency of $\omega$. The dashed line indicates the initial background $s$-wave scattering length value $a_{\text{s,bg}}$. (c) The distribution of the population in each excited mode. $N\left({\omega }_{\text{e}}\right)$ means the excited atom number in the mode with energy $\hslash {\omega }_{\text{e}}$. The peak locates around the mode with energy $\hslash \omega /2$ with a quasi-Gaussian shape. A Gaussian fit tells us the full width of half maximum (FWHM).

Figure 2

Stimulated excitation in a quasi-one-dimensional BEC. The trap frequency ${\omega }_{\text{x}}={\omega }_{\text{y}}=2\pi \times 500\text{Hz}$, ${\omega }_{\text{z}}=2\pi \times 20\text{Hz}$. The total atom number $N=5000$, and the modulating frequency $\omega =2\pi \times 2000\text{Hz}$. (a) The time dependence of the excited fraction $N_{\text{e}}/N$ with typical Bogoliubov theory (black dashed line) and td-HFB theory (red solid line), respectively. The horizontal dotted dashed line indicates a critical excited fraction where the contributions from the excited atoms become significant. The time is in unit of the modulation period $\tau =2\pi /\omega$. (b) The spatial density distributions of the excited atoms calculated from the typical Bogoliubov theory (black dashed line) and the td-HFB theory (red solid line), respectively, with a modulating time of $8\tau$. Here $a_z=\sqrt{\hslash /m{\omega }_z}$. (c) and (d) The dependencies of the excited fraction on the modulating amplitude and the background scattering length, respectively. In (c), the background $a_{\text{s,bg}}=0$ (black circle), $5a_0$ (red square), and $10a_0$ (green triangle), respectively, while in (d), the amplitude $a_{\text{st}}=30a_0$. (e) The FWHM of the spectra distribution of the excited modes shown in Fig. 1 as a function of the background $a_{\text{s,bg}}$. In (c)–(e), a short modulating time of $2\tau$ is used and the dashed lines are just guides to eyes.

Figure 3

Dynamics of the stimulated excitation in a quasi-one-dimensional BEC. The trap frequency ${\omega }_{\text{x}}={\omega }_{\text{y}}=2\pi \times 500\text{Hz}$, ${\omega }_{\text{z}}=2\pi \times 20\text{Hz}$. The total atom number $N=5000$, and the modulating frequency $\omega =2\pi \times 2000\text{Hz}$ with the amplitude $a_{\text{st}}=30a_0$. For (a)–(f), $a_{\text{s,bg}}=0$. (a)–(c) The time evolutions of the spatial density distributions of the condensate $n_0(z,t)$, the excitations $n_{\text{e}}(z,t)$, and the total $n(z,t)$, respectively. (d)–(f) The density profiles $n_0(z,t)$, $n_{\text{e}}(z,t)$, and $n(z,t)$, respectively, at finite modulating time $t=4\tau$ (red dashed lines) and $t=10\tau$ (black solid lines). (g) The time evolution of the defined density overlap $\chi \left(t\right)$. The green dotted dashed line is calculated from the typically used Bogoliubov theory ($a_{\text{s,bg}}=0$), while the red solid line and black dashed line are calculated with td-HFB theory for $a_{\text{s,bg}}=0$ and $a_{\text{s,bg}}=2a_0$, respectively. (h) The time evolution of the amplitude of the excited pairing field $|m_{\text{e}}|$ which shows a similar behavior with the excited density profile in (b).

Figure 4

Generation of the second order matter-wave harmonic in a driven quasi-one-dimensional BEC. The trap frequency ${\omega }_{\text{x}}={\omega }_{\text{y}}=2\pi \times 500\text{Hz}$, ${\omega }_{\text{z}}=2\pi \times 20\text{Hz}$. The total atom number $N=5000$, and the modulating frequency $\omega =2\pi \times 2000\text{Hz}$. The background scattering length $a_{\text{s,bg}}=5a_0$ and the modulation amplitude $a_{\text{st}}=60a_0$. The modulating time is $10\tau$.