Rotating binary Bose-Einstein condensates and vortex clusters in quantum droplets

TL;DR

This paper investigates the formation and stability of vortex clusters in quantum droplets derived from Bose-Einstein condensates, highlighting new methods to generate persistent currents in binary condensates under rotation.

Contribution

It introduces the concept of vortex cluster formation in quantum droplets and proposes a novel approach to create stable vortex states in binary condensates using non-monotonic traps.

Findings

Multiple singly-quantized vortices can form in quantum droplets at moderate angular momenta.

Droplets with vortex precursors can remain self-bound after trap removal for certain times.

Persistent currents can be induced in binary condensates using non-monotonic trapping potentials.

Abstract

Quantum droplets may form out of a gaseous Bose-Einstein condensate, stabilized by quantum fluctuations beyond mean field. We show that multiple singly-quantized vortices may form in these droplets at moderate angular momenta in two dimensions. Droplets carrying these precursors of an Abrikosov lattice remain self-bound for certain timescales after switching off an initial harmonic confinement. Furthermore, we examine how these vortex-carrying droplets can be formed in a more pertubation-resistant setting, by starting from a rotating binary Bose-Einstein condensate and inducing a metastable persistent current via a non-monotonic trapping potential.

Equations

$$\displaystyle i\frac{\partial\psi}{\partial t}=\Big{(}-\frac{1}{2}\nabla^{2}+\frac{1}{2}\omega^{2}{r}^{2}+V_{0}e^{-(r/a_{\perp})^{2}}+$$

$$\displaystyle+~{}\left|\psi\right|^{2}\ln\left|\psi\right|^{2}-iL_{3}\left|\psi\right|^{4}-\Omega L_{z}\Big{)}\psi,$$

$$\displaystyle E=\int d^{2}\mathbf{r}\Bigg{[}\frac{1}{2}|\nabla\psi|^{2}$$

$$\displaystyle+\frac{1}{2}|\psi|^{4}\ln{\left(\frac{|\psi|^{2}}{\sqrt{e}}\right)}\Bigg{]}$$

Full text

The formation of self-bound droplets is a well-known macroscopic phenomenon. For an exemplary droplet of water, stability and shape rely on the balance of effective forces between its constituent particles – attractive ones that keep it together, and repulsive ones that prevent it from collapse. Their interplay defines the droplets’ surface tension, stabilizing the system in a metastable state. Such droplets do not only occur at a macroscopic level, but are ubiquitous also in the quantum realm, where nuclei Bohr and Mottelson (1998) and superfluid helium droplets Donelly (1991); Toennies and Vilesov (2004); Ancilotto et al. (2018a) are prominent examples. While these are rather dense and strongly interacting many-body systems, recent experiments with ultra-cold quantum gases of bosonic atoms uncovered a novel type of quantum liquid: Self-bound droplets may form out of a gaseous Bose-Einstein condensate (BEC) of dysprosium Kadau et al. (2016); Schmitt et al. (2016); Ferrier-Barbut et al. (2016a, b, 2018) or erbium Chomaz et al. (2016), atomic species that are known for their strong dipolar interactions Lu et al. (2010, 2011); Aikawa et al. (2012). Similar droplet states have more recently also been realized with binary Bose gases of potassium in different hyperfine states Cabrera et al. (2018); Semeghini et al. (2018), where the inter- and intracomponent interactions are short-ranged. These quantum droplets can be large, containing thousands of atoms. Importantly, they are very dilute – by more than eight orders of magnitude when compared with liquid helium Cabrera et al. (2018). While the discovery with dysprosium Kadau et al. (2016); Schmitt et al. (2016); Ferrier-Barbut et al. (2016a) at first came as a surprise, the binary self-bound droplet states were theoretically predicted a year before Petrov (2015) for a scenario similar to the experiments with potassium, and also in lower dimensions Petrov and Astrakharchik (2016). That higher-order corrections beyond mean field may lead to self-bound states was discussed earlier in a different setting in Refs. Bulgac (2002); Hammer and Son (2004). For the dipolar or binary self-bound bosonic systems of Refs. Kadau et al. (2016); Schmitt et al. (2016); Ferrier-Barbut et al. (2016a); Semeghini et al. (2018); Cabrera et al. (2018) the physical mechanism of droplet formation is based on tuning the interactions in gas such that only a weak effective attraction remains. While in pure mean field this would lead to a collapse of the system, weak first-order corrections to the mean field energy, often referred to as the Lee-Huang-Yang (LHY)-correction Lee et al. (1957), can become comparable in size and may thus stabilize the system.

Bound states that are merely a consequence of quantum corrections beyond mean field have been known since long from alkali-metallic clusters Knight et al. (1984); Nishioka et al. (1990), where the jellium of the ionic charge background cancels the electronic Hartree term, resulting in a liquid-like electronic state bound mainly due to a balance between kinetic and exchange-correlation energy Koskinen et al. (1995). The fact that self-bound bosonic droplets can form with atoms as intrinsically different as lanthanides and alkali metals shows that this phenomenon for ultra-cold gases is a general one, giving evidence for a novel state of quantum matter with new and unexpected properties.

In atomic quantum droplets the effective interactions between the constituent bosonic atoms are relatively weak. This eases their theoretical description, making it largely accessible to lowest-order corrections beyond mean field Petrov (2015). On the theoretical side, progress has been made in the framework of both the extended Gross-Pitaevskii approach Baillie et al. (2016); Wächtler and Santos (2016a, b); Chomaz et al. (2016); Bisset et al. (2016); Baillie et al. (2016, 2017) where the LHY-correction as well as atom losses are added to the non-linear Schrödinger equation in an efficient ad hoc manner, by quantum Monte-Carlo approaches Saito (2016); Macia et al. (2016); Cinti et al. (2017); Cikojević et al. (2018), or by solving the Bogoliubov-de Gennes equations Bisset et al. (2016); Baillie et al. (2017).

As these droplets form out of a BEC there is good reason to assume that they have superfluid properties. One of the signatures of superfluidity are vortices – topologically non-trivial states well known from harmonically trapped BECs (see e.g. Butts and Rokhsar (1999); Matthews et al. (1999); Madison et al. (2000); Chevy et al. (2000); Kavoulakis et al. (2000); Abo-Shaeer et al. (2001); Raman et al. (2001); Madison et al. (2001); Haljan et al. (2001) or the reviews Fetter (2009); Saarikoski et al. (2010)), characterized by a depletion of the density accompanied by a phase shift. So far, however, experimental evidence for vortices in these droplets appears elusive. Only a few theoretical works yet considered the LHY-stabilized quantum droplets’ rotational properties. Very recent work found metastable necklace-like clustered droplets carrying angular momentum Kartashov et al. (2019). Stability of an imprinted singly quantized vortex was reported for a prolate dipolar droplet Cidrim et al. (2018), accompanied by a splitting of the droplet into two smaller droplet fragments. Imprinted vortices at the droplet center with similar fragment formation were also reported for binary droplets in both two Li et al. (2018) and three dimensions, in the latter case carrying up to two units of angular momentum in a region of experimentally accessible parameters Kartashov et al. (2018). A doubly quantized vortex was found to decay into two singly-quantized vortices upon a quadrupolar deformation Ancilotto et al. (2018b). In these studies, the vorticity was imprinted on the droplet by a phase factor of the initial state.

In this Letter we investigate the rotational properties of a trapped binary BEC in relation to self-bound quantum droplets. We find that by starting from the rotational ground state of this binary BEC, metastable droplets containing vortex clusters may form after a sufficiently slow release from a harmonic confinement. We also explore the feasibility of creating vortex droplets in this way by starting from the ground state of a trapped condensate with a non-monotonic trapping potential, where we find the existence of a metastable persistent current.

Let us now consider a species-symmetric binary BEC confined to two dimensions Petrov and Astrakharchik (2016) that is interacting weakly via short-range interactions, where the inter- and intraspecies interactions are assumed to be attractive and repulsive, respectively. For such a binary BEC with equal masses of the atoms in the two components, the coupled Gross-Pitaevskii equations reduce to that of a one-component BEC with an accordingly modified interaction term  Petrov (2015); Petrov and Astrakharchik (2016). We initally confine the gas in a harmonic trap with an added Gaussian at the trap center. The LHY-amended Gross-Pitaevskii equation for such a system in a frame rotating with angular frequency $\Omega$ can then be written as

[TABLE]

where the scaling invariances of the system have been used to bring the equation into this dimensionless form. Here $\omega$ is the harmonic trapping frequency, $V_{0}$ the amplitude of the Gaussian, $a_{\perp}=1/\sqrt{\omega}$ the oscillator length and $L_{3}$ the rate of three-body losses. The order parameter is normalized according to $\int d^{2}\mathbf{r}|\psi|^{2}=N$. The energy in the non-rotating frame is

[TABLE]

and the angular momentum $L=\int d^{2}\mathbf{r}\psi^{\ast}L_{z}\psi$. Equation (Rotating binary Bose-Einstein condensates and vortex clusters in quantum droplets) is solved with the usual split-step Fourier method Chin and Krotscheck (2005) in real and imaginary time. For the imaginary time-propagation we use set of different randomly perturbed initial conditions in order to avoid local minima in the energy landscape.

We first look for the ground state of free droplets in a rotating frame, but before convergence can be reached, the droplets are found to decay to fragments similar to those in the three-dimensional case Kartashov et al. (2018). As a first remedy to these inherently unstable solutions, we add a weak stabilizing harmonic confinement to the system. The dimensionless parameters considered here to illustrate our findings are $N=1000$ and $\omega=0.04$. For these values the corresponding free droplet has the characteristic flat-top shape Li et al. (2018), and the trapping frequency is sufficiently weak to keep the droplet at a density close to its (free) equilibrium value. We first of all consider a purely harmonic trap ($V_{0}=0$) and identify the rotational ground states at distinct rotation frequencies $\Omega$. The density distributions for some ground states at different $\Omega$ are shown in the leftmost column of Fig. 1.

Clearly, with increased rotation, vortices are induced in a way similar to that of one-component condensate, proceeding from the formation of a unit vortex at the trap center, to two- and three-vortex states with the usual two- and three-fold symmetries. Figure 2 shows the ground state energy in the rotating frame $E-\Omega L$ and the corresponding angular momentum as a function of the trap rotation. The first three steps in $L$, corresponding to kinks in the rotational energy, are seen for $L/N\approx 1.0,1.8$ and $2.5$ for the first three vortex states shown in Fig. 1 similarly to scalar BECs Butts and Rokhsar (1999); Kavoulakis et al. (2000).

Instead of minimizing the energy by solving Eq. (Rotating binary Bose-Einstein condensates and vortex clusters in quantum droplets) for fixed $\Omega$, one may instead study the energy at a fixed angular momentum by minimizing $E+C(L-L_{0})^{2}$ Komineas et al. (2005), where $C$ and $L_{0}$ are dimensionless constants. For sufficiently large values of $C$, the energy minimum then occurs at an angular momentum $L=L_{0}$, making it possible to obtain solutions for states that are not rotational ground states, i.e. for arbitrary $L$. The dispersion relation obtained in this way is displayed in Fig. 3 for angular momenta up to and beyond the unit vortex that nucleates at the trap center at $L/N=1.0$. At this value, for $V_{0}=0$, the energy has a kink, which can turn into a energetic minimum in the rotating frame when the energy $E-\Omega L$ is tilted downwards by a constant slope, resulting in the first step in $L$ when solving Eq. (Rotating binary Bose-Einstein condensates and vortex clusters in quantum droplets) for a corresponding value of $\Omega$, see Fig. 2. For higher vortex numbers the mechanism is similar, with kinks in $E$ leading to the plateaus in $L$ for certain values of $\Omega$. In the absence of three-body losses, $L_{3}=0$, Eq. (Rotating binary Bose-Einstein condensates and vortex clusters in quantum droplets) conserves angular momentum. This implies that a condensate in a unit vortex ground state in the rotating frame will remain so even after the rotation ceases by virtue of this conservation law. However, since there is no local minimum in the dispersion relation for the purely harmonic case, this state is susceptible to small pertubations. It will thus slide down in energy to the non-rotating ground state. Let us next consider a non-monotonic trapping potential by adding a Gaussian to the center of the harmonic trap (as has been realized experimentally, see for example Bretin et al. (2004)). Trapping potentials of this kind have been shown to cause local minima in the dispersion relation for scalar BECs Kärkkäinen et al. (2007). The energy as a function of angular momentum for $V_{0}=0.2$ shown in Fig. 3 confirms that this is also the case for this binary system. Such a mexican-hat type of confinement can thus support a metastable persistent current even in the presence of weak pertubations. The energy in the rotating frame and the corresponding angular momentum as a function of rotation frequency with this central Gaussian is shown in Fig. 2.

Since we are interested in rotational properties of self-bound condensates, we now imagine a scenario where the ground state at a particular rotation frequency is maintained even after the trap rotation has stopped (as could be realistic in an experimental setting when there exists a local minimum in the dispersion relation). The condensate is then released from the trap by decreasing $\omega$ linearly in time in order to reduce the radial velocity that results from the expansion. The real time propagation for the purely harmonic case is shown in Fig. 1.

Intriguingly, in this ideal case where conservation laws are intact, the droplets stay stable even after the trap is fully turned off while still carrying angular momentum in the form of vortices and rigid-body rotation. Additionally, the shape of the droplets is deformed according to the number of vortices they contain. For a condensate in a trap with the Gaussian discussed previously, we consider a similar release, but now leaving the Gaussian even after the harmonic trap is turned off. Density contours and phases for the release of zero- and unit-vortex states in such a setup are displayed in Fig. 4, where we have also included a comparison with the corresponding systems including three-body losses with $L_{3}=0.01$. For both the cases with and without a vortex, the droplet is pinned to the remaining Gaussian, and they stay metastable and self-bound even when the symmetry-breaking three-body losses term is present.

In order to relate our results to experimental values, let us consider 39K atoms tightly confined by a harmonic trap in the transversal direction with an oscillator length $\ell_{z}=0.1\mu\mathrm{m}$, and three dimensional s-wave scattering lengths equal to $a_{\uparrow\downarrow}=-50.0a_{0}$ and $a=50.5a_{0}$ for the inter- and intraspecies interactions, respectively ($a_{0}$ is the Bohr radius). Note that these values for the scattering lengths correspond to the stable gas phase in three dimensions Cabrera et al. (2018); the liquefaction is due to the transition to two dimensions Petrov and Astrakharchik (2016). This choice, when transformed from the dimensionless parameters used above, corresponds roughly to $\omega\sim 10\mathrm{Hz}$, $N\sim 10^{5}$ and $n\sim 10^{14}\mathrm{cm}^{-3}$, with units of time and space in $\sim\mathrm{ms}$ and $\sim\mu\mathrm{m}$, respectively Petrov and Shlyapnikov (2001). The value used for the three-body losses in Fig. 4 corresponds approximately to $10^{-27}\mathrm{cm}^{6}/\mathrm{s}$.

In conclusion, binary self-bound bosonic droplets as realized in recent experiments with potassium Semeghini et al. (2018); Cabrera et al. (2018) show the formation of vortices in a way similar to scalar BECs with weak short-range interactions, but with the additon of a deformation to the droplets’ shape. In order to stabilize these droplets and prevent their decay into fragments (such as they were found both in two and three dimensions, see  Li et al. (2018); Kartashov et al. (2018)), we found that it is crucial to first stabilize the droplets by a weak harmonic confinement, chosen such that it barely confines the self-bound droplet. When switching off the trap rotation and slowly releasing the droplet, cusps in the yrast line lead to rotational ground states that can generate rotating droplets containing multiple singly-quantized vortices. To study a more pertubation-resistant system, we considered a binary BEC trapped in a mexican-hat potential, where we found the existence of a metastable persistent current that potentially could be utilized in order to produce droplets carrying angular momentum in the form of vortices. The findings presented here should be in the range of present experiments for binary condensates, and we expect that similarily metastable persistent currents may occur in dipolar quantum droplets. While the present analysis made use of the two-dimensional extended Gross-Pitaevskii approach Petrov and Astrakharchik (2016), it will also be interesting to study the crossover between two and three dimensions  Ilg et al. (2018); Kartashov et al. (2018) from the perspective of vortex formation.