Breathing and quadrupole modes in squeezed Bose-Bose liquid drops

The gist: This study investigates how breathing and quadrupole modes in squeezed Bose-Bose liquid drops change when confined harmonically in one direction, revealing coupling between these modes and showing that squeezing reduces the critical number of particles needed to sustain stable oscillations.

Theoretical Framework

The investigation utilizes a density functional built from first-principles quantum Monte Carlo calculations (QMC) which incorporates the effects of finite range interactions The energy is expressed as a function of the density E[ρ] = ¯h 2/2m ∫dr [∇pρ(r) squared + Z ∫dr (Eint[ρ(r)+Vext[ρ(r)])] The many-body wave function is built as the product of single particle orbitals Ψ(r1, r2,..., rN) = Y N N1 ∫ψ1(ri) Y N(N+1) ψ2(rj) To include finite-range effects, the energy functional is Eint = ρ E/N = αρ + βργ The MF+LHY energy per particle can be written as E/N E0/N = −3ρ/ρ0 + 2ρ 3/ρ0 3/2.

Mode Analysis and Symmetry Breaking

The study focuses on two oscillation modes: the breathing mode and the quadrupole mode which correspond to the oscillations of the monopole moment Q0 and the quadrupole moment Q2, respectively. The discrete definition of the monopole moment is defined as Q0 = Σ N i r 2i, where r 2i is the squared distance of particle i to the center of mass of the system. Similarly, the quadrupole moment is defined as Q2 = Σ N i (3x 2i − r 2i) = Σ N i (2x 2i − y 2i − z 2i), where x is selected as the axis of interest. The choice of the perturbation potential Vi ensures coupling to the desired collective mode. Perturbing with V0 uniformly increases or decreases the radius of the droplet for a small amount (volume mode), while perturbing with V2 elongates the droplet along the chosen axis (surface mode), which excites quadrupole oscillations.

Effect of Squeezing and Dimensional Crossover

When an external harmonic trap is introduced along the z axis, it breaks the SO(3) symmetry, resulting in an SO(2) × Z2 symmetry. Consequently, the l = 0 state becomes m = 0, and the l = 2 state decomposes into the m = 0, m = ±1, and m = ±2 states. In Fig. 1 we observe two modes: l = 0, m = 0⟩ (upper branch) and l = 2, m = 0⟩ (lower branch) states. When increasing the squeezing parameter f, the upper branch asymptotically approaches the breathing mode frequency of a 3D spherical droplet. On the other side of the spectrum, in the lower f range, there is a new mode that needs to be identified and which grows stronger as the squeezing increases.

Stability and Critical Atom Number

The oscillations are stable when their energy is lower than the absolute value of the chemical potential µ. The dependence of µ = ∂E/∂N on the droplet size is determined using the Liquid Drop Model (LDM) E N = EV + ESx + EC x squared. For very large N, surface and curvature contributions to the chemical potential vanish, and µ approaches EV. The study shows that squeezing reduces the minimum number of atoms needed to sustain stable excitation modes. With an increase in the number of atoms, we observe the reduction of the frequency, which generally decreases with the increase in the cluster size.

Comparison of Functionals and Interaction Strengths

The QMC functional predicts higher frequencies than the MF+LHY functional. The difference between the two predictions increases up to 30%. For weaker interactions, excitation mode frequencies are lower and the size of the droplets required to sustain stable oscillations increases. The MF+LHY functional generally predicts lower frequencies (see Fig. 8).

Conclusion

The droplet loses its spherical symmetry when confined harmonically in one direction, resulting in a coupling between the breathing and axial quadrupole modes. Increasing the squeezing, the breathing mode frequency increases and eventually disappears when its energy becomes higher than the chemical potential. The study demonstrates that squeezing reduces the minimum number of atoms needed to sustain stable excitation modes. The findings suggest that stable oscillations could be observed for experimentally achievable droplet sizes, potentially demonstrating the coupling between different modes.

How it works

The researchers used a two-dimensional energy functional, with logarithmic dependence on the density [18] and connected it with a quasi-2D setting via a2D = 2e(-γpπa2ho/2/A exp[-pπ/2aho/a3D]]. The monopole perturbation operator V0 = λr squared is invariant under any rotation, therefore, it is a scalar and couples to the l = 0 mode (breathing mode). The quadrupole perturbation operator V2 = λ(3x squared − r 2) couples to the l = 2 quadrupole mode.

Key Findings Summary

** The coupling between breathing and axial quadrupole modes results from the breaking of spherical symmetry due to the external harmonic trap. **

** Increasing squeezing leads to an increase in the upper branch frequency, which eventually disappears when its energy exceeds the chemical potential. **

** The lower branch mode corresponds to axial quadrupole oscillations with m=0, which transitions to effectively a 2D breathing mode as squeezing is increased. **

** Squeezing reduces the minimum number of atoms needed to sustain stable excitation modes. **

** The 2D breathing frequencies are higher than the quadrupole ones, as happens in spherical droplets. **

** The critical number of atoms needed to support stable oscillations is systematically lower for quadrupole modes across the entire range of squeezings. **

** We have further shown that squeezing reduces the minimum number of atoms needed to sustain stable excitation modes. **

** The timescale of the oscillations is of the order of a few milliseconds, suggesting they could be detected in experiments. **

** We have further shown that squeezing reduces the minimum number of atoms needed to sustain stable excitation modes. **

** The effects observed in the present work are expected to also occur for heteronuclear droplets, which have the same binding mechanism, but importantly can reach lower central densities. **

** Particularly appealing are K-Rb droplets, which have a significantly longer lifetime than 39K droplets, on the order of 100 ms. **

** Our research in progress for these droplets shows the lowering of the critical number of atoms in K-Rb droplets with squeezing. **

** Stable oscillations could be observed for experimentally achievable droplet sizes, potentially demonstrating the coupling between different modes.

1. Improved AI can predict droplet stability by calculating if an excitation energy is below this threshold, determined by the absolute value of the chemical potential mu. This allows for a more accurate assessment of whether a self-bound state will remain stable under different squeezing conditions.

2. Improved AI can model excitation mode coupling by analyzing how the coupling between breathing and quadrupole modes results from breaking spherical symmetry, which is observed in squeezed droplets. This enables the AI to predict the specific frequency spectrum when moving from 3D to quasi-2D settings.

3. Improved AI can determine the optimal confinement parameters by analyzing how the oscillation frequencies change with squeezing in one direction, as well as with the strength of the tunable interaction. This allows for precise experimental design targeting specific collective modes based on interaction strengths and confinement geometry.

4. Improved AI can predict critical atom numbers needed for stable oscillations by showing the reduction of the number of particles needed to sustain stable oscillations when droplets are squeezed in one direction. This directly informs experimental feasibility studies regarding droplet size requirements.

5. Improved AI can distinguish between different theoretical models by comparing results, as The differences between the results obtained with the QMC functional and the extended Gross-Pitaevskii theory are increased with respect to the case of previously studied spherical droplets. This allows for model selection based on experimental data fidelity.

We investigate the lowest excitation modes of binary 39 K ultradilute droplets confined harmonically in one direction, which reduces the critical number of particles needed to sustain a self-bound state. Using the density functional built from first-principles quantum Monte Carlo calculations (QMC), which incorporates the effects of finite range interactions, we study the crossover of the oscillation modes from three to two dimensions. For weak squeezing, we observe the coupling between breathing and quadrupole modes, resulting from the breaking of spherical symmetry. Increasing the squeezing, the higher-frequency mode corresponding to three-dimensional breathing mode becomes unstable. For squeezings stronger and comparable to experiments, we analyze the dependence of the excitation modes on the droplet size for several interactions and show the reduction of the number of particles needed to sustain stable oscillations. The differences between the results obtained with the QMC functional and the extended Gross-Pitaevskii theory are increased with respect to the case of previously studied spherical droplets.