Breathing mode in two-dimensional binary self-bound Bose-gas droplets

In this work, we study the stationary structures and the breathing mode behavior of a two-dimensional self-bound binary Bose droplet. We employ an analytical approach using a variational ansatz with a super-Gaussian trial order parameter and compare it with the numerical solutions of the extended Gross-Pitaevskii equation. We find that the super-Gaussian is superior to the often used Gaussian ansatz in describing the stationary and dynamical properties of the system. For sufficiently large nonrotating droplets the breathing mode is energetically favorable compared to the self-evaporating process. However, for small self-bound systems our results differ based on the ansatz. Inducing angular momentum by imprinting multiply quantized vortices at the droplet center, this preference for the breathing mode persists independent of the norm.

Figure 1

Optimal value for the exponent $m$ of the super-Gaussian ansatz with increasing norm $N$, solved numerically for Eq. (14) (dashed line) and approximated for large $N$ according to Eq. (15) (dotted-dashed line). The inset shows the difference $\mathrm{\Delta }m=m_{\text{exact}}-m_{\text{approx}}$. With increasing $N$ the error $\mathrm{\Delta }m$ approaches zero, suggesting that the analytical approximation is suitable for flat-top droplets.

Figure 2

Mean radius $\sqrt{\langle r^2\rangle }$ of the self-bound droplet as a function of the norm $N$, based on different approaches as indicated by the legend. While the Gaussian trial order parameter ($m=1$) is a good approximation for small $N$, it fails to describe the mean radius accurately for larger norm. The inset shows the relative error $\delta =1-\sqrt{\langle r_{\text{m=1}}^2\rangle /\langle r_{m_{\text{exact}}}^2\rangle }$ in the range of small norms $N\le 10$ (in dimensionaless units as defined in Eq. (3)).

Figure 3

Maximum density $n_m$ of the droplet based on the variational and numerical solutions, as indicated in the legend. As proposed in Refs. [20, 25], $n_m$ approaches $1/\sqrt{e}$ with increasing $N$ for the super-Gaussian trial order parameter in its exact and approximate forms. With increasing norm $N$ the Gaussian ($m=1$) deviates significantly. The numerical solutions of the eGPe, Eq. (4), are in good agreement with the variational results for $m_{\text{exact}}$ and $m_{\text{approx}}$. The inset shows the relative error $\delta =1-n_{m=1}/n_{m_{\text{exact}}}$ for small norm $N\le 10$ (in dimensionaless units as defined in Eq. (3)).

Figure 4

Density distributions $n\left(R\right)$ based on the Gaussian trial order parameter (dashed), the super-Gaussian trial order parameter with $m_{\text{exact}}$ (dotted-dashed), and a numerical solution of the eGPe (full lines) for different $N$. With increasing $N$ the super-Gaussian density distribution approaches the flat-top limit (in dimensionaless units as defined in Eq. (3)).

Figure 5

Chemical potential $\mu$ of the droplet based on the variational and numerical solutions, as specified by the legend. The super-Gaussian ansatz correctly predicts the large $N$ limit, while the Gaussian ansatz clearly deviates from it. For small $N$ the approximate approach overestimates the chemical potential, while the Gaussian offers a good fit. The inset shows the relative error $\delta =1-{\mu }_{m=1}/{\mu }_{m_{\text{exact}}}$ for small norm $N\le 10$ (in dimensionaless units as defined in Eq. (3)).

Figure 6

Breathing frequency ${\omega }_0$ of the nonrotating droplet. For large $N$ the numerical solutions of the eGPe, Eq. (4), coincide with the super-Gaussian ansatz calculation, while for smaller $N$ the assumption of a single breathing frequency in Eq. (19) no longer holds, due to the emergence of a beating pattern. The inset shows the relative error $\delta =1-{\omega }_{0,m=1}/{\omega }_{0,m_{\text{exact}}}$ for small norm $N\le 10$ (in dimensionaless units as defined in Eq. (3)).

Figure 7

${\omega }_0/\left|\mu \right|$ for droplets based on the different approaches, as specified in the legend. All three analytical solutions coincide for the high norm regime, and are energetically favorable over the self-evaporation for $N\gtrsim 36$ (in dimensionaless units as defined in Eq. (3)).

Figure 8

Density distributions for self-bound droplets at $N=850$ with angular momentum $L/N=1.0,2.0,3.0$. The size of the droplets and minimum norm $N$ for stability follow the heuristically derived relation in Ref. [25]. The phase in cases with nonzero angular momentum is shown as small insets in each figure (in dimensionaless units as defined in Eq. (3)).

Figure 9

Frequency spectrum for droplets with angular momentum as specified in the legend. With increasing $L/N$ the breathing frequency ${\omega }_0$ decreases. The onset of each branch is given by the real-time stability of the system. For the multiply quantized systems, the stability requirement for the minimum required norm $N$ is equal to that found in [25] (in dimensionaless units as defined in Eq. (3)).

Figure 10

Availability of the breathing mode for droplets with angular momentum as given in the figure. The breathing mode exists if ${\omega }_0<\left|\mu \right|$ and is energetically favorable for a rotating system as long as the stability requirement is fulfilled and occurs for $L/N=0$ if $N\gtrsim 21$ (in dimensionaless units as defined in Eq. (3)).

Figure 11

(a) Mean radius $\sqrt{\langle r^2\rangle }$ in the first 10000 timesteps after interaction strength modulation for a droplet with $N=1000$. (b) Fourier spectrum for a droplet with $N=1000$ after interaction strength modulation. (c) Mean radius $\sqrt{\langle r^2\rangle }$ in the first $20000$ timesteps after interaction strength modulation for a droplet with $N=25.12$. The beating pattern besides one main frequency is clearly observable. (d) Fourier spectrum for a droplet with $N=25.12$ after interaction strength modulation. Besides the main frequency, we can also observe smaller peaks with lower frequency, which emerge from the beating pattern (in dimensionaless units as defined in Eq. (3)).