Quantum nature of gravity in self-bound quantum droplets

TL;DR

This work investigates testing quantum gravity in ultracold, self-bound 1D Bose–Bose droplets by embedding a Kempf–Mangano–Mann generalized uncertainty principle (GUP) into a generalized Gross–Pitaevskii framework that includes Lee–Huang–Yang corrections. Using a variational super-Gaussian ansatz and numerical split-step Fourier simulations, the authors derive a dimensionless GGPE with a quartic derivative term $\bar{\beta}\,\partial_x^4$, study how GUP corrections modify binding energies, density profiles, and equilibrium widths, and extract modified minimal lengths. By comparing with recent ${}^{39}$K quantum-droplet experiments, they place upper bounds on the GUP deformation parameter $\beta_0$ and the corresponding minimal length $(\Delta x)_{\min}$, finding $\beta_0 \approx 10^{49}$ for large droplets with $(\Delta x)_{\min} \approx 0.8$ nm and $\beta_0 \approx 10^{53}$ for small droplets with $(\Delta x)_{\min} \approx 21.9$ nm, improving existing constraints. The results indicate that self-bound quantum droplets offer a promising platform to bridge quantum mechanics and gravity, with the potential to extend the approach to higher dimensions.

Abstract

We explore the possibility of testing the quantum nature of the gravitational field with an ultracold self-bound quantum droplet of one-dimensional Bose-Bose mixtures. To this end, we solve variationally and numerically the underlying generalized Gross-Pitaevskii equation which includes the effects of quadratic and cubic nonlinearities. We derive the associated generalized uncertainty principle and its corresponding minimal length. The obtained modified uncertainty relation enables us to search for the quantum gravity signatures in both small and large droplets. We place bounds on the parameter using existing experimental data from recent experiment of dilute droplets of potassium. Improved upper bounds on the generalized uncertainty principle parameters are found from our analysis.

Figures (5)

• Figure 1: (a) Energy functional, $E/N$, of the quantum large droplet versus its width for different values of QG parameter, $\bar{\beta}$. (b) Equilibrium width, $q_0$ as a function of $\bar{\beta}$. Parameters are: $N=5$, $m=4$ and $\alpha=0.001$. (c)-(d) The same as (a) and (b) but for a small droplet with $N=1$, and $m=1$. Here we used a relatively large value of $\bar{\beta}$, ($\bar{\beta}=1$) to highlight the energy shift.
• Figure 2: Droplet-GUP from the super-Gaussian wavepackets for $\bar{\beta}=0.5$ and its corresponding equilibrium width $q=q_0=8.25$ for large droplet and $q=q_0=5.2$ for small droplet. Dotted line is the HUP for comparison. It is important to note that the obtained large droplet-GUP has no minimal length at $(\Delta x)_{\text{min}}= 8.5$. However the small droplet-GUP has no minimal length.
• Figure 3: Droplet density in Fourier space, $|\psi(k)|^2$, with $\bar{\beta}=0.001$, for both small and large self-bound droplets.
• Figure 4: Density profiles $|\psi(x)|^2$ for different values of $\bar{\beta}$, (a) corresponding to a small Gaussian-like droplet and (b) to a large flat-top droplet. Dashed lines represent the variational solutions.
• Figure 5: The droplet width as a function of the norm, $N$, for different values of $\bar{\beta}$.