Breathing mode of a quantum droplet in a quasi-one-dimensional dipolar Bose gas

We investigate the breathing mode and the stability of a quantum droplet in a tightly trapped one-dimensional dipolar gas of bosonic atoms. When the droplet with a flat-top density profile is formed, the breathing-mode frequency scales as the inverse of the number of atoms in the cloud. This is straightforwardly derived within a phenomenological hydrodynamical approach and confirmed using both a variational method based on a generalized Gross-Pitaevskii action functional and the sum-rule approach. We extend our analysis also to the presence of axial confinement showing the effect of the trap on the density profile and therefore on the breathing-mode frequency scaling. Our analysis confirms the stability of the quantum droplet against the particle emission when the flat-top density profile is observed. Our results can be used as a guide to the experimental investigations of collective modes to detect the formation of quantum droplets in quasi-one-dimensional dipolar gases.

Figure 1

In panel (a) we show the evolution of the density profile as the number of particles in the cloud is increased for the case $a_{1\mathrm{D}}=-7000a_0$, for some selected cases, i.e., $N=50$, 100, 200, and 400. In panel (b), we show, for several scattering lengths $a_{1\mathrm{D}}$, the ratio between the density at the center $n(x=0)$ of the cloud and the equilibrium density reached when the density profile becomes flat as a function of the number of particles $N$ in the droplet. Data for scattering lengths $a_{1\mathrm{D}}/a_0=-6500$, $-7000$, $-7500$, $-8000$, and $-8500$ are shown using orange, dark-red, dark-blue, dark-green, and black solid dots, respectively. All lines are intended to be guides to the eye.

Figure 2

Breathing mode square of the droplet in the absence of trapping potential multiplied by $N^2$ for several values of $a_{1\mathrm{D}}$. Dark-red, blue, dark-green, and black solid dots are estimates of ${({\mathrm{\Omega }}_b/{\omega }_{\perp })}^2$ using the sum-rule approach from Eq. (2) for $-a_{1\mathrm{D}}/a_0=7000$, 7500, 8000, and 8500, using as multiplicative factors $\alpha =1$, 1/3, 1/30, and 1/900, respectively. In the inset we show the phase diagram for the dipolar system from Ref. [8], where the self-bound droplet occurs when the chemical potential $\mu <0$. The dashed lines correspond to the values shown in the main figure with the same color code. All lines are intended to be guides to the eye.

Figure 3

Breathing mode square of the droplet as a function of the number of particles $N$ without the trap for $a_{1\mathrm{D}}=-7000a_0$ (${\omega }_{\perp }$ units). Black, blue, and red solid dots correspond to estimates for ${\mathrm{\Omega }}_b$ using the sum-rule approach, Eq. (2), derivation from the Gross-Pitaevskii equation (27) and the hydrodynamic approach equation (12), respectively. Solid red line is a fitting curve ${\mathrm{\Omega }}_b\left(N\right)=a{(N-N_0)}^{\beta }$, with $\beta =0.49\left(3\right)$. All the lines between points are intended to be guides to the eye.

Figure 4

Ratio of breathing mode, using the sum-rule approach, of the droplet and chemical potential as a function of the number of particles $N$ without the trap for $-a_{1D}/a_0=7000$, 7500, 8000, and 8500 (dark-red, blue, dark-green, and black solid dots, respectively). All lines are intended to be guides to the eye.

Figure 5

In panel (a) we show $\alpha {({\mathrm{\Omega }}_b/{\omega }_{\mathrm{ho}})}^2{N}^2$ for the scattering length $a_{1\mathrm{D}}=-8000a_0$. The dashed black curve is the same quantity computed in the absence of longitudinal trapping, exhibiting a plateau for large $N$. In the lower panels we show the density profiles of the trapped cloud for different numbers of particles in different regions. Density profiles are represented by solid dark-green lines for $N=10$, 50, 100, and 200 [panel (b)]; by red solid lines for $N=400$, 600, 800, and 1000 [panel (c)]; and by blue solid lines for $N=2000$, 4000, and 5000 [panel (d)]. Open dots in panel (a) correspond to results for the same number of particles as in the density profile shown in panels (b)–(d) with identical color code. Lines are intended to be guides to the eye.

Figure 6

${\mathrm{\Omega }}_b^2/{\omega }_{\mathrm{ho}}^2$ as a function of number of particles $N$ with the trap for $a_{1\mathrm{D}}=-8000a_0$ [panel (a)] and $a_{1\mathrm{D}}=-7000a_0$ [panel (b)]. Black, blue, and red solid dots are estimates for ${\mathrm{\Omega }}_b$ using the sum-rule approach, Eq. (2), derivation from the Gross-Pitaevskii equation, Eq. (28), and the hydrodynamic approach, Eq. (12), respectively. All lines are intended to be guides to the eye.