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arXiv:1705.01814 · Scientific Reports

Equation of state and self-bound droplet in Rabi-coupled Bose mixtures

Alberto Cappellaro, Tommaso Macrì, Giovanni F. Bertacco, Luca Salasnich

Published Oct 11, 2017• 75 citations
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Abstract

Abstract Laser induced transitions between internal states of atoms have been playing a fundamental role to manipulate atomic clouds for many decades. In absence of interactions each atom behaves independently and their coherent quantum dynamics is described by the Rabi model. Since the experimental observation of Bose condensation in dilute gases, static and dynamical properties of multicomponent quantum gases have been extensively investigated. Moreover, at very low temperatures quantum fluctuations crucially affect the equation of state of many-body systems. Here we study the effects of quantum fluctuations on a Rabi-coupled two-component Bose gas of interacting alkali atoms. The divergent zero-point energy of gapless and gapped elementary excitations of the uniform system is properly regularized obtaining a meaningful analytical expression for the beyond-mean-field equation of state. In the case of attractive inter-particle interaction we show that the quantum pressure arising from Gaussian fluctuations can prevent the collapse of the mixture with the creation of a self-bound droplet. We characterize the droplet phase and discover an energetic instability above a critical Rabi frequency provoking the evaporation of the droplet. Finally, we suggest an experiment to observe such quantum droplets using Rabi-coupled internal states of K39 atoms.

DOI
10.1038/s41598-017-13647-y

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Abstract Laser induced transitions between internal states of atoms have been playing a fundamental role to manipulate atomic clouds for many decades. In absence of interactions each atom behaves independently and their coherent quantum dynamics is described by the Rabi model. Since the experimental observation of Bose condensation in dilute gases, static and dynamical properties of multicomponent quantum gases have been extensively investigated. Moreover, at very low temperatures quantum fluctuations crucially affect the equation of state of many-body systems. Here we study the effects of quantum fluctuations on a Rabi-coupled two-component Bose gas of interacting alkali atoms. The divergent zero-point energy of gapless and gapped elementary excitations of the uniform system is properly regularized obtaining a meaningful analytical expression for the beyond-mean-field equation of state. In the case of attractive inter-particle interaction we show that the quantum pressure arising from Gaussian fluctuations can prevent the collapse of the mixture with the creation of a self-bound droplet. We characterize the droplet phase and discover an energetic instability above a critical Rabi frequency provoking the evaporation of the droplet. Finally, we suggest an experiment to observe such quantum droplets using Rabi-coupled internal states of K39 atoms.
We analytically calculate the leading quantum corrections of the ground-state energy of two- and one-dimensional weakly interacting Rabi-coupled Bose-Bose mixtures in the frame of the Bogoliubov approximation. We show that to repulsive intraspecies and attractive interspecies interactions, the effect of quantum fluctuations favors the formation of self-bound droplets. These liquidlike states are crucially affected by the Rabi coupling, leading thus to the appearance of a quantum instability. We derive meaningful formulas to describe the droplet phase in the one-dimensional case.
In the first part of the thesis we investigate properties of multicomponent quantum Bose gas with a Rabi coupling, using theoretical methods based on functional integration. At very low temperatures beyond-mean-field contribution crucially affect the equation of state of many-body systems. We derive the equation of state of the gas including the quantum fluctuations. The divergent zero-point energy of gapless and gapped elementary excitations of the uniform system is properly regularized obtaining a meaningful analytical expression for the beyond-mean-field equation of state in the weak Rabi coupling approximation. In the case of attractive inter-particle interaction we show that the quantum pressure arising from Gaussian fluctuations can prevent the collapse of the mixture with the creation of a self-bound droplet. In the second part we study the effects of Rabi coupling along the BCS-BEC crossover of a Feshbach resonance for a two-spin-component Fermi gas. Results such as the gap equation, the number equation and the critical temperature in the mean-field approximation have been obtained. In the last part we include quantum fluctuations contributions in the previous calculations in order to investigate the phase diagram of the system.
Self-bound quantum droplets form when the mean-field tendency of the gas to collapse is stabilized by the effectively repulsive beyond-mean-field fluctuations. The beyond-mean-field effects depend on Rabi frequency ${\ensuremath{\omega}}_{R}$ and quadratic Zeeman effect qqq for the Rabi-coupled Bose mixtures and the spinor gases, respectively. For a quantum droplet, the effects of varying ${\ensuremath{\omega}}_{R}$ and qqq have recently been examined only for unpolarized Rabi-coupled Bose mixtures and unpolarized spinor gases. In this paper, we theoretically explore the stability of the droplet phase for polarized Rabi-coupled Bose mixtures and polarized spinor gases. We calculate the beyond-mean-field corrections for both gases with polarized order parameters and obtain the phase diagram of the droplets on the parameter space of Rabi frequency ${\ensuremath{\omega}}_{R}$ and detuning $\ensuremath{\delta}$ for Rabi-coupled mixtures and quadratic Zeeman energy qqq and linear Zeeman energy ppp for spinor gases. Finally, we highlight the similarities and differences between the two systems and discuss their experimental feasibility.
We uncover a new mechanism for realizing three-component quantum droplets in ultracold Bose gases, where only one inter-species interaction is attractive. In this scheme, the inter-species attraction leads to a self-bound binary droplet, and the third component joins through Rabi coupling with one component of the binary droplet. We find that a stronger Rabi coupling leads to a larger fraction of the third component, but also destabilizes the entire droplet due to the involvement of more repulsive forces. Such instability can be remedied by a finite detuning between the Rabi-coupled components. We demonstrate these results in realistic Na-Rb mixtures, using both thermodynamic analyses and numerical simulations based on extended Gross-Pitaevskii equations. Our work outlines a general route for stabilizing multi-component droplets by bridging an existing binary droplet with additional components via suitable single-particle fields.
We theoretically investigate the finite-temperature structure and collective excitations of a self-bound ultradilute Bose droplet in a flat space realized in a binary Bose mixture with attractive inter-species interactions on the verge of mean-field collapse. As the droplet formation relies critically on the repulsive force provided by Lee–Huang–Yang quantum fluctuations, which can be easily compensated by thermal fluctuations, we find a significant temperature effect in the density distribution and collective excitation spectrum of the Bose droplet. A finite-temperature phase diagram as a function of the number of particles is determined. We show that the critical number of particles at the droplet-to-gas transition increases dramatically with increasing temperature. Towards the bulk threshold temperature for thermally destabilizing an infinitely large droplet, we find that the excitation-forbidden, self-evaporation region in the excitation spectrum, predicted earlier by Petrov using a zero-temperature theory, shrinks and eventually disappears. All the collective excitations, including both surface modes and compressional bulk modes, become softened at the droplet-to-gas transition. The predicted temperature effects of a self-bound Bose droplet in this work could be difficult to measure experimentally due to the lack of efficient thermometry at low temperatures. However, these effects may already present in the current cold-atom experiments.
According to the mean-field theory a condensed Bose-Bose mixture collapses when the interspecies attraction becomes stronger than the geometrical average of the intraspecies repulsions, g_{12}^{2}>g_{11}g_{22}. We show that instead of collapsing such a mixture gets into a dilute liquidlike droplet state stabilized by quantum fluctuations thus providing a direct manifestation of beyond mean-field effects. We study various properties of the droplet and find, in particular, that in a wide range of parameters its excitation spectrum lies entirely above the particle emission threshold. The droplet thus automatically evaporates itself to zero temperature, the property potentially interesting by itself and from the viewpoint of sympathetic cooling of other systems.