Polarization angle dependence of the breathing mode in confined one-dimensional dipolar bosons

Probing the radial collective oscillation of a trapped quantum system is an accurate experimental tool to investigate interactions and dimensionality effects. We consider a fully polarized quasi-one-dimensional dipolar quantum gas of bosonic dysprosium atoms in a parabolic trap at zero temperature. We model the dipolar gas with an effective quasi-one-dimensional Hamiltonian in the single-mode approximation and derive the equation of state using a variational approximation based on the Lieb-Liniger gas Bethe ansatz wave function or perturbation theory. We calculate the breathing mode frequencies while varying polarization angles by a sum-rule approach and find they are in good agreement with recent experimental findings.

Figure 1

Energy per unit length $\epsilon \left(n\right)$ in units of ${\hslash }^2{n}^3/\left(2m\right)$ within three different approximations: ${\epsilon }_{LL}\left(n\right)$ with $A=3.6$ (dashed lines), perturbation theory using $A=3.6$ (solid lines), and the variational Bethe ansatz (solid dots). The results are shown for three selected scattering lengths, $a_{1\mathrm{D}}=-100a_0,-1000a_0$, and $-5000a_0$, the black, red, and blue data, respectively, and for $\theta =\pi /2$.

Figure 2

Energy per unit length $\epsilon \left(n\right)$ in units of ${\hslash }^2{n}^3/\left(2m\right)$ within three different approximations: using perturbation theory with $A=3.6$ (solid lines) and using $A=0.0$ (dashed lines) compared with those within the variational Bethe ansatz (solid dots). Data are shown for two selected scattering lengths, $a_{1\mathrm{D}}=-1000a_0$ and $-5000a_0$, represented by the red and blue data points, respectively. Results are for $\theta =0$.

Figure 3

Squared breathing mode frequency over trapping frequency squared ${\omega }_b^2/{\omega }_{ho}^2$ as a function of $\mathrm{\Lambda }=N{(a_{1\mathrm{D}}/a_{ho})}^2$, using different approximations. All results are for $N=20$, and lines are only a guide to the eyes. Red, dark-green, and blue solid curves are estimates using the Lieb-Liniger model and the perturbation theory with $A=0$ and with $A=3.6$, respectively. The dashed red curve represents ${\omega }_b^2/{\omega }_{ho}^2$ calculated using Eq. (11) with $A=3.6$. the black solid squares are estimates based on the variational Bethe ansatz equation of state.

Figure 4

Density profile for a system of $N$ dipoles in a trap using two different equations of states at $\mathrm{\Lambda }=434$, namely, the Lieb-Liniger (LL) (red lines) and VBA (black lines). Dashed and solid lines refer to the cases with $N=25$ and $N=20$ particles in the trap, respectively. Results are for $\theta =\pi /2$. In the inset, we show the density at the center of the trap as a function of $\mathrm{\Lambda }$ for $N=25$; the red and black solid dots represent estimates using the LL and VBA equations of state, respectively. Lines joining dots are only a guide to the eye.

Figure 5

Squared breathing mode frequency over trapping frequency squared ${\omega }_b^2/{\omega }_{ho}^2$ as a function of $\mathrm{\Lambda }=N{(a_{1\mathrm{D}}/a_{ho})}^2$, using different approximations. The red solid and dashed lines represent the breathing mode estimates after using the Lieb-Liniger (LL) model without dipolar interaction and using $A=3.6$ in Eq. (11), respectively. Blue solid curves refer to estimates based on the perturbation theory with $A=3.6$, while black solid squares are based on the variational Bethe ansatz (VBA). All results are for $N=20$, and lines are only guides to the eye. In the inset, the density at the center of the trap is shown as a function of $\mathrm{\Lambda }$, with the same legend as in the main panel.

Figure 6

Squared breathing mode frequency over trapping frequency squared ${\omega }_b^2/{\omega }_{ho}^2$ as a function of the polarization angle $\theta$ for selected values of the scattering length, $a_{1\mathrm{D}}/a_0=-100,-2000,-5000,-7000$, represented by red, dark-green, blue, and black solid dots, respectively. The solid lines joining the data are only guides to the eye. The vertical blue dashed line splits the regions with negative $V\left(\theta \right)<0$ (left) and positive $V\left(\theta \right)>0$ (right). Data refer to estimates based on the variational Bethe ansatz equation of state.

Figure 7

Squared breathing mode frequency over trapping frequency squared ${\omega }_b^2/{\omega }_{ho}^2$ as a function of $\mathrm{\Lambda }=N{(a_{1\mathrm{D}}/a_{ho})}^2$, using different approximations, for the attractive case (top panel) and the repulsive one (bottom panel). Dark-red triangles represent the experimental data taken from Refs. [9, 24]. All results are shown for two values of $N, N=50,80$ (top panel) and $N=25,40$ (bottom panel), and lines are only a guide to the eyes. The dashed black and red lines represent estimates using the Lieb-Liniger model. Black and red solid squares are estimates based on the variational Bethe ansatz (VBA) equation of state.

Figure 8

Ratio of breathing mode frequency to trap frequency squared ${\omega }_B^2/{\omega }_{ho}^2$ as a function of the Hartree parameter $\lambda$ for different numbers of particles in the trap, namely, $N=4,10,15,20,25$, and 30, using the generalized Gross-Pitaevskii equation (A2). The red solid line represents the Hartree approximation, independent of the number of particles.