Ultradilute quantum liquid drops
V. Cikojević, K. Dželalija, P. Stipanović, L. Vranješ Markić, J. Boronat
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug Ultradilute quantum liquid drops
V. Cikojević, K. Dželalija, P. Stipanović, L. Vranješ Markić
Faculty of Science, University of Split, Ruđera Boškovića 33, HR-21000 Split, Croatia
J. Boronat
Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain
Using quantum Monte Carlo methods we have studied dilute Bose-Bose mixtures with attractiveinterspecies interaction in the limit of zero temperature. The calculations are exact within somestatistical noise and thus go beyond previous perturbative estimations. By tuning the intensity ofthe attraction, we observe the evolution of an N -particle system from a gas to a self-bound liquiddrop. This observation agrees with recent experimental findings and allows for the study of anultradilute liquid never observed before in Nature. PACS numbers: 02.70.Ss,67.85.-d,03.75.Kk,03.75.Mn,05.30.Jp
The high tunability of interactions in ultracold Boseand Fermi gases is allowing for exploration of regimesand phases difficult to find in other condensed-mattersystems [1]. By adjusting properly the applied mag-netic field, Bose and Fermi gases are driven to Fesh-bach resonances with an increase of interaction practi-cally at will, and with the possibility of turning the sys-tem from repulsive to attractive and vice versa. Thisis obviously not possible in conventional condensed mat-ter where interactions are generally not tunable at thislevel. A significant example of this versatility has beenthe clean experimental realization of the unitary limitfor fermions [2, 3] and the precise characterization of theBCS-BEC crossover [4, 5], which up to that moment, wasonly a theoretical scenario.Recently, it has been possible to explore the forma-tion of liquid/solid patterns in dilute gases by modifyingthe strength of the short-range interatomic interactions.Probably, the most dramatic example of this progresshas been the observation of the Rosensweig instability ina confined system of
Dy atoms with a significant mag-netic dipolar moment [6]. By tuning the short-range in-teraction, Kadau et al. [6] have observed the spontaneousformation of an array of self-bound droplets remember-ing the characteristics of a classical ferrofluid. The ob-servation of solid-like arrangements in dilute gases hasalso been possible working with highly-excited Rydbergatoms [7]. By direct imaging, Schauss et al. [7] have ob-tained ordered excitation patterns with a geometry closeto the well known arrangements observed in few-bodyconfined Coulomb particles.In the line of obtaining other dense systems startingfrom extremely dilute Bose and Fermi gases, it is no-ticeable the mechanism suggested by Petrov relying onBose-Bose mixtures [8]. According to this proposal, it ispossible to stabilize a mixture with attractive interspeciesinteraction in such a way that the resulting system is self-bound, i.e., a liquid. Whereas a mean-field treatment ofthe mixture predicts a collapsed state, the first beyondmean field correction, the Lee-Huang-Yang (LHY) term, is able to stabilize the system by properly selecting theinterspecies s -wave scattering length. Further work hasshown that reducing the dimensionality of the setup totwo or quasi-two dimensions may help to stabilize the liq-uid phase [9]. The LHY correction has also been used toaccount for the formation of dipolar drops [10] and thenconfirmed by full first-principles quantum Monte Carlo(QMC) simulations [11, 12].The exciting idea of producing self-bound liquid dropsby using interspecies attractive interaction acting as glueof the entire Bose-Bose mixture has been put forward byTarruell and collaborators [13]. Results obtained with amixture of K atoms in different hyperfine states haveshown the formation of these drops, that do not re-lease for a significant time when the confining trap is re-moved. Therefore, the theoretical prediction seems con-firmed and thus a new window for exploring matter inunprecedented situations is open. On one side, it provesthe way of forming liquid drops with large density in theworld of cold gases and, on the other, makes possible thestudy of a liquid state of matter with an extremely lowdensity, lower than any other existing liquid.In the present work, we study the formation of liquiddrops in a Bose-Bose mixture using the diffusion MonteCarlo (DMC) method, which solves stochastically the N -body Schrödinger equation in an exact way withinsome statistical uncertainties. The DMC method wasextensively used in the past for determining the struc-ture and energy properties of liquid drops of He [14, 15], He [16, 17], H [18], and spin-polarized tritium [19]. Atdifference with previous perturbative estimates, DMC al-lows for an exact study of the quantum properties of thesystem relying only on its Hamiltonian. Our results con-firm the LHY prediction on the stability of self-boundmixtures and determine quantitatively the conditions un-der which liquid drops are stable and how they evolvewhen the attractive interaction is increased. Within theregime here explored, we do not observe a full collapse ofthe drop but an increase of the density and reduction ofthe size, which is rather progressive. −a /a −5−4−3−2−10123 E / N [ − ħ / ( m a ) ] FIG. 1. (Color online). Energy per particle of the Bose-Bose mixture as a function of the scattering length a /a .Different symbols and lines correspond to DMC calculationswith different number of particles. The Bose-Bose mixture under study is composed of N bosons of mass m and N bosons of mass m withHamiltonian H = − ¯ h m N X i =1 ∇ i − ¯ h m N X j =1 ∇ j + 12 X α,β =1 N α ,N β X i α ,j β =1 V ( α,β ) ( r i α j β ) , (1)with V ( α,β ) ( r ) the interatomic interaction betweenspecies α and β . 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