1 Introduction

Liquid droplets are formed through the interplay of attractive and repulsive forces from different parts of the corresponding energy functional, regardless of their composition, such as in water or helium, or even in nuclei. Recently, ultra-cold quantum droplets in degenerate states emerged as an intriguing new state of matter. It results from the same underlying principle that involves a delicate balance between forces originating from the mean-field interaction and beyond-mean-field quantum fluctuations. In a setting without the involvement of quantum fluctuations, under certain circumstances mean-field theory predicts a collapse of the system when the attraction between species reaches a critical point. However, considering beyond-mean-field effects, the system can be stabilized and the collapse be prevented, which may lead to a self-bound state [1,2,3]. In a remarkable experiment [4], the role of beyond-mean-field effects was substantiated, where self-organized and self-bound structures in an atomic dysprosium Bose–Einstein Condensate (BEC) were found. These structures exhibit rich pattern formation similar to what one observes in a classical ferrofluid [5]. Self-bound droplet crystals in magnetic erbium atoms were also observed soon after [6]. Furthermore, it has been pointed out that this dilute magnetic quantum liquid requires a critical atom number below which the liquid will undergo evaporation into an expanding gas [7].

While the discovery with dipolar dysprosium at first came as a surprise, similar self-bound droplet states were theoretically predicted  [2, 3] for two-component (binary) Bose gases just a year earlier, as mentioned above. Experimental evidence of such binary quantum droplets has now also been reported for a mixture of \(^{39}\)K atoms in the presence [8, 9] and absence of a confining potential [10]. The formation of heteronuclear quantum droplets [11] in the mixture of \(^{41}\)K and \(^{87}\)Rb increased the droplet lifetime. By maneuvering the atom number and contact interaction which is determined by the s-wave scattering length, the system may be rendered into a regime where the mean-field interaction cancels completely, and the system is only governed by quantum fluctuations, forming the so-called LHY fluid that was theoretically predicted in  [12] and found its experimental realization in [13].

The transition from a regular superfluid to the droplet regime depends on the particle number and mean-field interaction strength. The corresponding phase diagram (as a function of scattering length and particle number) is more complex for dipolar condensates compared to short-range interacting binary condensates. This isotropic short-range interaction leads to the formation of a singular droplet for a binary gas. The anisotropic long-range interaction of a dipolar condensate, however, allows for multiple droplets to form. This property is also a consequence of the so-called roton mode  [14, 15].

The droplet phase emerging from a superfluid allows for the exciting possibility of connecting the individual droplets, leading to global phase coherence with spatial periodicity. In fact, it took very little time from the discovery of dipolar droplets [4, 16] to realize the so-called “supersolid” state in three breakthrough experiments [17,18,19] involving highly magnetic dysprosium and erbium atoms.

The supersolid state can also occur in spin-orbit [20] and light-coupled BECs [21, 22]. However, the crystals are rigid—in these systems, external light initiates the localization, while supersolidity is usually characterized by the spontaneous breaking of the above mentioned symmetries.

This mini-review is part of the proceedings of the FQMT-22 in Prague and it is of a more tutorial character. It should neither be taken as a review of this by now immense research field, nor is the list of references complete—rather, this paper lifts a few aspects that we found of relevance in the context of our own work presented at the conference, and summarized in the second half of this paper.

2 Quantum crystal formation in ultra-cold atomic gases

We here provide a brief overview of the quantum droplet lattice formation (to which we in the following also refer to as droplet “crystal” formation, despite the apparent differences to usual crystals) in ultra-cold atomic gases. During the time period when in Helium the interpretation of experimental results as possible supersolidity was actively debated [23,24,25], continuous and tremendous progress in trapping and manipulating BEC’s since its inception occurred as an alternative platform for the long-sought supersolid state of matter. Since BEC often goes hand-in-hand with superfluidity in the system, the natural question arose whether a superfluid can possibly develop a spatially modulated structure. The main three candidates in the context of cold atoms are Rydberg gases, light-coupled BEC and dipolar systems, that we briefly discuss below.

2.1 Rydberg gases

Supersolids were experimentally realized with condensates coupled to optical cavities [21, 22]. To develop a spontaneous density modulation, a momentum-dependent interaction is crucial. It is important to note that in Refs. [21, 22], the specific k-vector dependency was imposed externally, resulting in incompressible crystals in such systems. However, the interparticle potential can be tailored to create a momentum-dependent interaction. This tailoring is primarily aimed at flattening the interaction potential below a specific cut-off distance. As a consequence of the constancy of the potential at short distances, a soft-core potential is formed, which makes it energetically favorable for the system to form a crystal [26].

The Rydberg dressing of BEC atoms, which couples them to highly excited Rydberg states, was proposed to result in a three-dimensional supersolid state with a roton–maxon excitation spectrum [27]. Crystal formation in the supersolid regime was subsequently demonstrated for an ensemble interacting via a repulsive dipolar potential softened at short distances. It was also predicted that the supersolid phase would be sandwiched between an insulating droplet crystal at high density and a homogeneous superfluid phase at low density [28]. Monte-Carlo simulations showed that crystal states could also be obtained in a gas of ground-state atoms by weakly admixing excited Rydberg states with laser light [29]. However, note that the dressing mechanism is valid only in the dilute limit, and an increasing particle density leads to a crossover from a two-particle interaction to a collective many-body regime [30]. Exotic crystal structures, including hexagonal, square, and bilayer structures, were discussed in one component [31]. Two-component Rydberg-dressed BECs were also studied [32], reporting the familiar density structures of two-dimensional quantum spin 1/2 systems.

2.2 Light-coupled BEC

Another possible pathway to achieve a supersolid regime involves the interaction of a BEC with an electromagnetic wave. In Ref. [33], the authors investigated a trapped atomic BEC interacting with an electromagnetic field generated by two far off-resonant counter-propagating orthogonally polarized light beams. The atom–light interaction resulted in an emergent optical lattice, trapping the atoms at the intensity maxima. However, the possible appearance of a supersolid state in this system has not yet been thoroughly investigated.

Another setup, described in [34], involves a transversely pumped BEC trapped along the axis of a ring cavity and coherently coupled to a pair of degenerate counter-propagating cavity modes. After reaching a threshold pump strength, the interference of photons scattered into cavity modes results in an emergent super-radiant lattice that spontaneously breaks the translational symmetry towards a periodic atomic pattern. A gravimeter was also proposed based on this supersolid-like phase of BEC confined in the ring cavity [35]. Generally, the resulting phase diagram of a self-ordering crystalline phase in an optical resonator is well understood on a mean-field level. However, it has been showcased that close to the phase transition point, quantum fluctuations and atom-field entanglement can be significant [36]. Furthermore, Ref. [37] has demonstrated the generation of long-range atomic interactions through a single optical feedback loop in a BEC, which has resulted in the emergence of self-bound crystals of quantum droplets with different lattice structures.

It is important to note that the aforementioned works have only considered BECs with short-range contact interactions. However, a more complex setup of long-range interacting BEC and infinite-range light-induced interactions have also been studied. An amorphous phase, referred to as a crystalized phase without any long-range periodic order, was found in a long-range potential BEC system due to the Rydberg dressing [38]. A long-range magnetic dipolar interaction in a similar setup has also been considered, showcasing the bi-roton instability [39].

2.3 Experimental realization of dipolar supersolids

In a simple picture, a dipolar supersolid can be imagined as an array of phase-coherently overlapping droplets that share a common superfluid background, such that atoms can be exchanged between the background and the droplets in a phase-coherent manner, which is crucial for the identification of the state as a supersolid. Three breakthrough experiments [17,18,19] were published almost simultaneously, involving highly magnetic dysprosium and erbium atoms, that reported signs of supersolidity. In fact, the use of an anisotropic trap confinement was imperative for the experimental discovery of the supersolid state. The stage was in principle already set by the first droplet experiments by Kadau et al. [4]. However, in a pancake trap, establishing such supersolid state is a significantly more complex endeavor, due to the softening of both radial and angular rotons  [40,41,42], see also the discussion in the review by Chomaz et al. [43].

In the following, we will briefly also mention some of the more recent experimental efforts that have been undertaken since the dipolar supersolid state of matter was reported. One major focus of these experimental efforts has been on unraveling the low-lying excitation spectra. Bragg spectroscopy, a well-established technique, has been utilized to probe the phonon–maxon–roton spectrum. The distinct excitation spectra associated with the crystal and supersolid transitions were also probed in a subsequent experiment. In situ measurements of density fluctuations and the estimation of frequencies of elementary excitations were demonstrated in Refs. [44, 45]. These effects were also studied in Ref. [40] across the phase transition, revealing the signature of radial and angular rotons in a two-dimensional oblate trap. The axial breathing mode is also known to bifurcate across the superfluid–supersolid transition [46].

Another experimentally investigated property of supersolidity is the re-establishment of phase coherence following an interaction quench from an initial supersolid state to the independent droplet regime, which destroys global phase coherence [47]. The scattering length is then returned to the supersolid regime to study the re-establishment of phase coherence.

The direct transition to the supersolid regime via evaporative cooling was explored in Ref. [48]. It was observed that the system first establishes a periodic density modulation, followed by long-range phase coherence. The experimental findings hinted at the prominent role of thermal fluctuations, which have also been confirmed theoretically. The formation of crystals linked via a background density was realized in an oblate trap. The impact of the in-plane trap aspect ratio and the one-dimensional to two-dimensional crossover was demonstrated in Ref. [49]. The evidence of two-dimensional supersolidity was further demonstrated in a two-dimensional circular trap in the form of a hexagonal density structure [50].

Ref. [51] investigated the character of the superfluid-to-supersolid phase transition via the dimensional crossover from quasi-one to two dimensions (e.g., from linear to planar droplet arrays or supersolid density modulations) as a function of particle number, finding that one-dimensional supersolidity can be both a continuous and discontinuous transition with respect to the particle number, while two-dimensional supersolidity is continuous.

3 A survey on theoretical works on crystal structures in dipolar quantum gases

Even long before the experimental realization of dipolar supersolidity, the presence of the roton minimum has been studied in dipolar condensate [52] (for an early general review on dipolar condensates see Ref. [53]). For a trap shaped like a pancake and highly confined in the axial direction, the in-plane momentum k is significantly smaller than the characteristic length scale along the z-direction. When a fixed magnetic field is applied along the axial direction, dipolar magnetic atoms position themselves perpendicular to the trap. In the plane, particles repel each other, leading to phonon excitations. As the axial confinement weakens and ultimately reaches \(k \gg 1/L\), the excitation energy is minimized as k increases. Therefore, the excitation energy reaches a minimum at k before continuously growing and ultimately transitioning into single-particle excitations. This creates the phonon–maxon–roton spectrum in the dispersion relation. The roton spectrum was suggested to be observable through the static and dynamical structure factors, which can be measured using Bragg spectroscopy [54]. For sufficiently low densities, the roton softening can lead to oscillatory density structures, but a mean-field collapse occurs for large densities [55]. It was also proposed that the roton excitation spectrum could be measured via atom number fluctuations, which emerge as a super-Poissonian peak as the size of the measurement cell varies, with the maximum value occurring where the size is comparable to the roton minimum [56]. The role of non-condensate atoms on the roton modes has also been discussed, as well as their real and momentum space distribution [57] and manipulation by utilizing a perturbing potential [58].

After the remarkable experimental observation of droplet formation, it was argued that the inclusion of the three-body interaction could prevent the mean-field collapse that the roton instability suggests in the high-density regime [59, 60]. A corresponding phase diagram was computed showcasing the droplet crystals and single droplets [61]. The role of quantum fluctuations in stabilizing the trapped droplets was also put forward [62, 63], extending the idea from the homogeneous setup [64, 65], resorting to the local density approximation. Subsequently, the ground-state phase diagram of dipolar condensates, including quantum fluctuations, was developed by means of variational ansatz and full numerical simulation [66]. Collective excitation spectra were also discussed [67], and the transition from the single to multiple crystal droplet ground states has also been demonstrated [68].

For a dipolar condensate confined in a tube the supersolid state showcased two hallmark properties [69]. These include a finite non-classical transitional inertia along the tube axis and the emergence of two gapless modes associated with density and phase fluctuations [70, 71]. Subsequently, these excitation spectra have been calculated in one and two-dimensional trapped systems [72, 73]. The second-order nature of the phase transition to the supersolid regime in the thermodynamic limit has also been demonstrated [74]. Additionally, a detailed investigation of the phase diagram has revealed the existence of both the superfluid and supersolid phase. It has been shown that the supersolid triangular lattice honeycomb-to-stripe pattern can emerge depending upon the particle number and interaction strength [41, 75, 76]. Different crystal structures in box potentials have also been demonstrated, revealing strong depletion in the bulk and accumulation at the box edge.

A promising research direction that has been pursued is exploring the rotational properties of supersolids. Specifically, the transition from the superfluid to the supersolid phase in a two-dimensional rotating trap has been characterized by observing a jump in the moment of inertia [77]. The coexistence of vortices in a honeycomb structure with the high-density crystals in a triangular geometry has also been demonstrated [78]. Furthermore, the dynamic formation of vortices during a quench from the superfluid to supersolid region has been observed [79].

The current study focuses on more complex structures in a mixture of magnetic atoms [80, 81]. As part of this effort, different crystal structures that go beyond the regular triangular structure have also been demonstrated [82,83,84]. Recently, the supersolid state in a mixture of dipolar–nondipolar atoms has been studied, without requiring the beyond-mean-field term for the stabilization of the system [85, 86]. This supersolid state essentially emerges from the softening of the spin-roton excitation spectrum [87].

4 Exemplary results

In the preceding section, we have provided a brief overview of the formation of superfluid quantum droplets in ultra-cold atomic gases, with a focus on systems involving long-range (or momentum-dependent) interactions enabling the supersolid state of matter. In parallel to the rapidly expanding research field of dipolar supersolidity [43], following the seminal work of Petrov [2], the formation of superfluid droplets in short-range contact-interacting BECs has also been extensively studied, contributing to a significant body of literature. While in the following sections, the authors provide a brief summary of some of their own contributions to this field, along with relevant technical details, a comprehensive discussion of the rapidly growing field is beyond the scope of this mini-review paper. Interested readers are referred to review papers on the topic, such as for example Refs. [43, 88, 89].

Regarding the structure of this review, in Sect. 4.1, we introduce the extended Gross–Pitaevskii (eGP) equation (4.1.1), which we use to elucidate the formation of superfluid droplet crystals in dipolar condensates (4.1.2) and to assess whether any classical behavior can be emulated in these systems (4.1.3). We discuss persistent currents in Sect. 4.2 for dipolar supersolids and in Sect. 4.3 for binary quantum droplets. In Sect. 4.4, we discuss the signature of droplet formation and associated rotational properties, focusing on the few-body limit. Section 4.5 outlines a new phase, the so-called “mixed bubble” resulting from the beyond-mean field interaction energy in binary droplets. Finally, in Sect. 4.6, we provide some discussion of the variational principle that can be used to capture the breathing mode in binary quantum droplets (4.6.1) and the properties of the isolated stacked droplet structure in an anti-dipolar condensate (4.6.2).

4.1 Superfluid droplet crystals in the dipolar and anti-dipolar regime

4.1.1 The extended Gross–Pitaevskii equation

We consider a dBEC of atoms with mass M and magnetic dipole moment \(\mu _{\text {m}}\). The dBEC is confined by a three-dimensional external trapping potential that can be either a harmonic potential,

$$\begin{aligned} V_{\textrm{trap}}(\mathbf {{r}}) = M(\omega _x^2 x^2 + \omega _y^2 y^2 + \omega _z^2 z^2)/2 , \end{aligned}$$
(1)

or a toroidal potential,

$$\begin{aligned} V_{\textrm{trap}}({\textbf{r}})= M\omega ^2\left[ (\rho - \rho _0)^2 + \lambda ^2 z^2 \right] /2. \end{aligned}$$
(2)

Here, \(\omega _x\), \(\omega _y\), \(\omega _z\) and \(\omega\) are the conventional trapping frequencies of the system. \(\rho _{0}\) is the radius of the torus, and \(\rho = \sqrt{x^2 + y^2}\) and \(\lambda = \omega _z/\omega\). The system, at zero-temperature and under a fixed magnetic field \(\textbf{B}\), along the z-axis, is governed by the following extended Gross–Pitaevskii (eGP) equation  [6, 62,63,64]

$$\begin{aligned}&i\hbar \frac{\partial \psi ({\textbf {r}},t)}{\partial t} = \bigg [-\frac{\hbar ^2}{2M}\nabla ^2 + V_{\textrm{trap}}({\textbf {r}}) + g \left| \psi ({\textbf {r}},t) \right| ^2 + \frac{3 g_{\textrm{dd}}}{4 \pi } \nonumber \\&\quad \times \int {\text {d}}{} {\textbf {r}}^{\prime } \frac{1 -3\cos ^{2}\theta }{\left| {\textbf {r}} - {\textbf {r}}' \right| ^2}\left| \psi ({\textbf {r}}^{\prime },t) \right| ^2 + \gamma (\epsilon _{\textrm{dd}})\left| \psi ({\textbf {r}},t) \right| ^3 \bigg ] \\ & \quad \psi (\textbf{r},t). \end{aligned}$$
(3)

Here, the short-range repulsive contact interaction, \(g = 4 \pi \hbar ^2a/M\), is determined by the scattering length a. The dipolar interaction coefficient is \(g_{\textrm{dd}} = 4\pi \hbar ^2a_{\textrm{dd}}/M\) with \(a_{\textrm{dd}}= \mu _{0}\mu ^2_{\text {m}}M /12\pi \hbar ^2\) being the so called dipolar length. The final term in Eq. (3) is given by the repulsive Lee–Huang–Yang (LHY) correction with \(\gamma (\epsilon _{\textrm{dd}}) = \frac{32}{3}g \sqrt{\frac{a^3}{\pi }} \left( 1+\frac{3}{2}\epsilon _{\textrm{dd}}^2\right)\) [64, 65], where the dimensionless parameter \(\epsilon _{\textrm{dd}} = a_{\textrm{dd}}/a\) quantifies the relative strength of the DDI in comparison to the contact interaction. The dipoles, for \(a_{\textrm{dd}} > 0\), attempt to minimize energy by adopting a head-to-tail arrangement that strengthens mutual attraction. But, they push away from each other when placed next to each other.

A more general case can be considered where the dipoles are forced to align with a rotating uniform magnetic field of an arbitrary direction \(\textbf{e}(t) = \textbf{B}(t)/|\textbf{B}(t)|)\) with respect to the z-direction [90, 91] and at the frequency \(\Omega\). The time-averaged DDI is given by [92], for a \(\Omega\) which is larger than the trap frequency but smaller than the Larmor frequency:

$$\begin{aligned} U_{\textrm{dd}} ({\textbf {r}},t) = \frac{\mu _0 \mu ^2_{\text {m}}}{4\pi }\left( \frac{3 \cos ^2\phi - 1}{2} \right) \left[ \frac{1-3\cos ^2\theta }{\textbf{r}^3}\right] ~. \end{aligned}$$
(4)

Here, \(\theta\) denotes the angle between \(\textbf{r}\) and the z-axis. The behavior of the system under such a situation can also be described by Eq. (3), provided that the dipolar length \(a_{\textrm{dd}}\) is scaled as \(a_{\textrm{dd}} \rightarrow a_{\textrm{dd}}(3 \cos ^2 \phi -1)/2\). The orientation of the constituent dipoles plays a crucial role in determining the nature of dipole–dipole interaction (DDI). Notably, the interaction completely disappears at the magic angle, denoted by \(\phi _{\text {m}}\approx 54.7^{\circ }\), and the system behaves like a regular contact-interacting BEC. When \(\phi < \phi _{\text {m}}\), the dipoles prefer to orient themselves in the head-to-tail configuration as usual. Interestingly, for \(\phi > \phi _{\text {m}}\), a side-by-side arrangement of dipoles in an anti-dipolar configuration becomes energetically more favorable. In this anti-dipolar regime, an inverted configuration takes place when the attractive (repulsive) component of dipolar interaction increases if the dipoles reside side-by-side (head-to-tail).

Depending on the absolute value of the parameter \(\epsilon _{\textrm{dd}}\), the system can exhibit different phases. In the case where \(|\epsilon _{\textrm{dd}}|\) is small enough, the system is in a superfluid phase. However, as \(|\epsilon _{\textrm{dd}}|\) increases beyond a certain threshold, the supersolid phase becomes more favorable within a specific range of \(|\epsilon _{\textrm{dd}}|\) values. If \(|\epsilon _{\textrm{dd}}|\) goes beyond this range, the system exhibits a transition to the isolated droplet phase.

4.1.2 Different droplet crystal structures

Fig. 1
figure 1

Representative of the one- and two-dimensional droplet arrays (see, respectively, panels a–d) in a dipolar condensate, shown as density plots (left) and three-dimensional isosurfaces (right). The color bar represents the integrated densities in the unit of \(1000 \,\upmu {\mathrm m}^{-1}\) in a and \(1000 \,\upmu {\mathrm m}^{-2}\) in c, respectively. The dipolar length is \(a_{\textrm{dd}}=131a_{0}\). The isolated three-stack droplets are also shown for an anti-dipolar BEC in e, f. See the text for the parameters

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The nature of the droplet crystal structure intriguingly depends upon the geometry of the trapping potential as well as relative strength of dipolar interaction to the contact interaction (more specifically, the parameter \(\epsilon _{\textrm{dd}}\), as defined above). The exemplary density structures showcasing a one-dimensional linear array or crystal of droplets, and two-dimensional hexagonal structures made of dipolar droplets under a fixed magnetic field along z-direction are shown in Fig. 1a–d, respectively. These structures have been generated considering \(N =\) 45,000 and \(N =\) 80,000 atoms in quasi-one- and two- dimensional trapping potentials with frequencies \((\omega _x, \omega _y, \omega _z)/(2 \pi )=(33, 100, 167) \mathrm Hz\) and \((\omega _x, \omega _y, \omega _z)/(2 \pi )=(43, 43, 133) \mathrm Hz\), respectively. Note that the crystal formation occurs in the weakly confined direction (along the x-direction) or x–y-plane as evidenced by the two-dimensional integrated density \(n_{\textrm{2D}}(x, y)=\int {\mathrm d}z \left| \psi (x, y, z) \right| ^2\), see Fig. 1a, c. Additionally, the three-dimensional isosurfaces [\((50\%, 16\%, 2\%)\) of the density maximum] are also demonstrated in Fig. 1b, d, respectively. These structures reveal that the droplets are typically extended along the z-direction, i.e., along the direction of polarization of the magnetic field.

In an anti-dipolar Bose–Einstein condensate (BEC), atoms tend to align themselves in a side-by-side configuration, which can lead to the formation of superfluid droplet crystals. By applying a rotating magnetic field in the x–y plane, it is possible to reach the maximum anti-dipolar regime with a dipole–dipole interaction strength parameter of \(a_{\textrm{dd}} =-65.5a_{0}\). If the atoms are tightly confined in the radial plane, and the contact interaction is weakened, the crystal structure can break along the weakly confined z-direction into multiple superfluid droplets that take the form of two-dimensional disks.

Figure 1 demonstrates such a crystal structure for \(N=\) 70,000 particles. The confinement is a harmonic potential with frequencies \((\omega _x, \omega _y, \omega _z)/(2\pi ) = (100, 100, 50) \mathrm Hz\) and interacting with \(a=98a_{0}\). Increasing the interaction strength, for example, to \(a=102a_{0}\), these isolated stacks can connect with each other, forming a stack supersolid. For a detailed discussion on the generation, dynamics, and collective excitation of these structures, we refer to our work in Ref. [93].

4.1.3 Classicality of the isolated droplets

Fig. 2
figure 2

(a Periodic time evolution of the integrated density \(n_{\textrm{1D}}\) capturing vibrational pattern of a three droplet state. The dynamics is realized, kicking two outermost droplets along opposite directions. b The time evolution of the displacement [\(x_1(t)\)] of the left-most droplet, corresponding to a calculated from the eGP equation and normal mode analysis. c Same as in a but for a supersolid state. See the text for the parameters of the simulation

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One fundamental question is whether the quantum droplet crystals exhibit any classical behavior. Recent research has substantiated the classical liquid-like behavior in binary quantum liquid droplets by colliding them [94]. To assess the classicality of the dipolar quantum crystals, we consider a system of three droplet states, under the influence of a fixed magnetic field, in a trapping potential with frequencies \((\omega _x, \omega _y, \omega _z)/(2\pi )=(19, 53, 81)\mathrm Hz\) for \(N =\) 35,000 particles and analyze their vibrational dynamics. We trigger the vibrational dynamics by kicking the outermost droplets in opposite directions with a velocity of \(v=0.4v_{\textrm{osc}}\), where \(v_{\textrm{osc}}\) is the harmonic oscillator velocity scale. The result is shown in Fig. 2a, where we calculate the integrated one-dimensional density \(n_{\textrm{1D}}=\int \textrm{d}x \textrm{d}y \left| \psi (x,y,z) \right| ^2\). The outermost droplets oscillate periodically while the central droplets remain motionless. This behavior is reminiscent of the vibrational modes of a classical three-particle system connected by a spring.

The trajectories, \(x_{1(3)}\), of the outermost particles, can be calculated using Hooke’s law: \(x_{1}(t)=- (v/(\sqrt{2}\omega _1)\sin (\sqrt{2}\omega _1t)\), \(x_2(t) = 0\), \(x_{3}(t)= (v/(\sqrt{2}\omega _1)\sin (\sqrt{2}\omega _1t)\). The trajectories calculated from the eGP simulation and Hooke’s law are shown in Fig. 2b, and they exhibit remarkable agreement for \(\omega _1/(2 \pi )=24 \mathrm Hz\). We emphasize that the system we consider here has three degrees of freedom and, thus, possesses three fundamental modes of vibration. Depending on the initial kick velocity, a multitude of trajectories resulting from the mixing of different fundamental modes can emerge [95]. At sufficiently low velocities (\(v < v_{\textrm{osc}}\)), where Hooke’s law holds, there is remarkable agreement between the eGP and normal mode analysis. However, the applicability of the normal mode analysis breaks down when droplets collide with each other and exchange particles at larger velocities.

The classical behavior does not apply when the droplets are immersed in a superfluid background, i.e., in the supersolid limit.. To illustrate this point, we consider three-crystal supersolid states prepared at \(a=94a_{0}\) for \(N=\) 35,000 in the same trapping potential. As shown in Fig. 2, a mass flow occurs between the crystals, characterized by multiple compressional modes. This is in contrast to the single compressional mode observed for the \(a=84a_{0}\) case. The deviation from the classical normal-mode-of-vibration in the supersolid state is a generic behavior that occurs for different velocity configurations, as discussed in our previous work, see Ref. [95].

4.2 Persistent current in dipolar supersolids

Fig. 3
figure 3

Persistent currents in a toroidal dipolar supersolid. The left panel shows the energy dispersion as a function of angular momentum for an increasing ratio between dipolar and contact interactions. The insets show the phase at selected points. The density surfaces in the right panel exemplify the transition from superfluid to supersolid, droplet crystal, and single droplet. After Nilsson Tengstrand et al. [96]

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The fact that dipolar droplets mutually repel each other when localizing side-by-side [16] may inhibit the droplet overlap necessary for supersolid phase coherence. We, thus, suggested [96] to confine the dipolar condensate in a toroidal geometry to avoid edge effects as they occur in a simply connected confinement. Such confinement can be created for example by employing a Laguerre–Gaussian light beam (see, e.g.,  [97]).

Figure 3 (left) shows the energy as a function of angular momentum in the rotating frame, for an increasing ratio between dipolar and contact interactions, for a dipolar gas of \(^{164}\)Dy and \(10^4\) atoms in a ring trap (see Eq. 2). In particular, we choose \(\rho _0= 1\mu m\) as the radius of the ring, \(\omega /2\pi = 1\textrm{kHz}\), and the transversal trapping ratio is \(\lambda =1.7\). We note here that the following discussions are rather general and can hold also for potentials with larger radius of the ring (see, for example, the recent work in  [98]).

The insets depict the phase of the order parameter for the angular momenta and relative interaction strengths \(\varepsilon _{\textrm{dd}}\) as indicated by the dashed lines. Here, the system is constrained to have a specific angular momentum (which in a symmetry-breaking mean-field approach may even be non-integer, see  [99, 100]).

Notably, the energy dispersion changes curvature upon increasing the dipolar strength \(\varepsilon _{\textrm{dd}}\). In both the superfluid and supersolid limit, a clear minimum at non-zero angular momentum is established, which is protected by an energy barrier from the non-rotating state. At the minimum, a singly quantized vortex has fully established, with the typical \(2\pi\) phase jump which in the supersolid case is accompanied by a modulation of the density. The plots to the right schematically sketch the evolution from a homogeneous superfluid (top) to the density-modulated supersolid state, to a droplet crystal, and finally to a single-droplet state (bottom) in a toroidal trap (sketched by the gray iso-surface) similar to droplets confined in a linear tube [101]. In the crystallized and single-droplet cases, the rotation has become a merely classical center-of-mass motion.

4.3 Vortex and persistent current in binary droplets

Fig. 4
figure 4

Upper panel: ground-state energy in the rotating frame for a harmonic trap \((V_0=0)\) (light blue) and with a Gaussian added at the trap center \((V_0=0.2)\) (blue). (After Nilsson Tengstrand et al. [102])

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Fig. 5
figure 5

Self-bound persistent current state with unit quantization, indicated by the phase changes shown as a color map between yellow and blue, from 0 to \(2\pi\), in the position plane, upon release from a harmonic trap. The white lines show the density contours. The Gaussian pins the circulating state that is stable also against atom losses (After Nilsson Tengstrand et al. [102], where also the quantitative data are given)

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As discussed above, the formation of quantum droplets is not limited to single-component dipolar Bose gases. Petrov first proposed the formation of quantum droplets in binary Bose gases [2], which was also earlier proposed in a different setting by Bulgac [1]. We already mentioned that in three dimensions, the Lee–Huang–Yang (LHY) correction leads to an effective repulsion that prevents the collapse of weakly attracting bosons to a metastable state. This effect is also present in low-dimensional systems, where quantum fluctuation contributions are enhanced and can have different signs than in three dimensions, but the principle of balancing residual and LHY contributions to form a bound state remains the same [3]. The dimensional crossover has also been discussed [103].

We investigated the superfluidity of binary quantum droplets in quasi-two dimensions and showed that a precursor of an Abrikosov vortex lattice may form in a rotating binary condensate that remained self-bound for certain timescales when released from a trap [102]. The vortices can be detected in experiments employing so-called in situ imaging [104].

In two dimensions, by scaling the parameters accordingly, the LHY-amended Gross–Pitaevskii equations can be written as [3]:

$$\begin{aligned} i{\partial \Psi \over \partial t} = {\left( - {1\over 2} \nabla ^2 + {1\over 2} \omega ^2 r^2 + \mid \Psi \mid ^2 \ln \mid \Psi \mid ^2 \Omega \hat{L}_z\right) \Psi } ~, \end{aligned}$$
(5)

where \(\Omega\) is the rotation frequency of the harmonic trap with a confinement frequency of \(\omega\) and \(\hat{L}_z\) is the angular momentum operator in two dimensions. Figure 4 shows the energy in the rotating frame in the upper panel and the ground-state angular momentum in the lower panel as a function of the relative trap rotation frequency \(\Omega /\omega\). Similar to single-component BECs [105], the angular momentum increases in steps as the rotation increases, indicating the build-up of a regular vortex lattice, as shown in the density plots in the insets. These steps correspond to the distinct kinks in the energy as a function of angular momentum shown in the upper panel. When a Gaussian potential \(V_0\exp (-r^2/a_{\perp })\) with an oscillator length of \(a_{\perp }=1/\sqrt{\omega }\) is added at the trap center, the values of L/N for which vortices nucleate shift the trap rotation frequencies. In a one-dimensional ring, i.e., for a strong Gaussian potential, the vortices would accumulate as multiply-quantized vortices in the trap center.

Pinning a droplet carrying vorticity, intriguingly, a self-bound metastable persistent current forms, that is even relatively stable against losses (see Fig. 5).

4.4 Signatures of droplets in the few-body limit

Fig. 6
figure 6

Ground-state energy and six lowest excitation energies as a function of total angular momentum, L, for a system of \(N = 8\) particles with \(g = 2\), \(g_{AB} = -0.1\) (a) and \(g_{AB} = -1.8\) (b). c, d Pair correlations for \(L = 0\) ground states. The total pair correlations are plotted in yellow/blue color. The pair correlations for the species of the fixed atom are presented as dashed lines. The pair correlations of the opposite species are presented as solid lines. (After Chergui et al. [106])

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Fig. 7
figure 7

Right: Zero angular momentum ground-state pair correlations for \(N=8\), \(g=2.0\) and various \(g_{AB}\) with reference particle position \(\theta ' = 0\). Left: The corresponding interaction parameters represented by colored dots. The background indicates the contrast \(({n_{\text {max}}- n_{\text {min}}})/({n_{\text {max}}+ n_{\text {min}}})\) of the numerical ground-state solution of a corresponding eGP equation at each point in the phase diagram. \(n_{\text {max}}\) and \(n_{\text {min}}\) are the maximum and minimum values of the eGP ground-state density, respectively. The dashed black curve denotes the boundary of the region of energetic and dynamic stability of the homogeneous solution to the eGP equation. Below this curve, the homogeneous solution is unstable. (The black area, \(g+g_{AB}<0\), marks the region where the eGP equation lost its validity). (From Chergui et al. [106])

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Let us now take a look at the few-body case of a one-dimensional ring consisting of a binary bosonic mixture of equal masses and equal particle numbers interacting through contact interactions. We note here that the LHY correction is rather different from its two- or three-dimensional counterparts [3]. In 1D droplets, the intraspecies interaction is repulsive and equal for both species. The interspecies interaction is thus taken to be attractive with interaction parameter \(g_{AB} < 0\). This system can exist in at least two distinct phases: a homogeneous phase that can be considered as a precursor to states supporting persistent currents, and a localized phase [106] similar to the quantum liquid droplets observed in the many-body limit [9, 10]. An increase in interspecies attraction can drive a phase transition from the homogeneous phase to the localized phase for a sufficiently strong fixed intraspecies repulsion. This phase transition is evident from the exact low-lying energy spectra obtained by numerically diagonalizing the system Hamiltonian given by

$$\begin{aligned} \hat{H}= & {} \sum _{\sigma ,m} \frac{m^2}{2} \hat{a}_{\sigma ,m}^\dagger \hat{a}_{\sigma ,m} \nonumber \\{} & {} + \frac{1}{2}\sum _{\begin{array}{c} \sigma ,\sigma '\\ m_1,m_2,k \end{array}}\frac{g_{\sigma \sigma '}}{2\pi } \hat{a}_{\sigma ,m_1 +k}^\dagger \hat{a}_{\sigma ',m_2 -k}^\dagger \hat{a}_{\sigma ',m_1}\hat{a}_{\sigma ,m_2}. \end{aligned}$$
(6)

Here, \(\hat{a}_{\sigma , m}^\dagger (\hat{a}_{\sigma , m})\) creates (annihilates) a boson of species \({\sigma \in \{A,B\}}\) in the single-particle angular momentum eigenstate \({\phi _{\text {m}} (\theta ) = \frac{1}{\sqrt{2\pi }} e^{im\theta }}\), where \(\theta\) is the azimuthal position on the ring, m is the integer one-body angular momentum quantum number, and we have set the particle mass M, the ring radius R, and \(\hbar\) to unity, \((M = R = \hbar = 1)\).

To obtain the Hilbert space in which Eq. (6) is diagonalzied one may implement a one-body angular momentum cut-off \(|m|\le m_{\text {max}}\) and truncate the many-body basis by means of a so-called importance truncated configuration interaction (ITCI) method (see e.g.[107, 108]). We here construct a Hilbert space with \(m_{\text {max}}=60\) and employ the ITCI method according to the procedure and convergence parameters outlined in [106] for each set of interaction parameters considered. In Fig. 6, we present the resulting low-lying energy spectra for a total of \(N=8\) particles with \(g = 2.0\) and \(g_{AB} = -0.1, -1.8\). Due to the circular symmetry of the ring, total angular momentum, L, is conserved, and we may analyze the energy spectra as a function of L.

The periodic boundary condition of the ring together with Bloch’s theorem [109] imply that the energy spectra can be expressed as the sum of a parabolic function of angular momentum \(L^2/2NMR^2\) corresponding to the kinetic energy of rotation of a rigid body of mass NM about the circumference of the ring, and a term accounting for the internal energy of the system, which must be periodic in L with periodicity \(L=N\) [109, 110]. Furthermore, the invariance of the two-body interaction term of Eq. (6) under the transformation \({m\rightarrow m-1}\) of all one-body angular momentum quantum numbers implies that the periodic component of the energy spectra must be symmetric about \(L = N/2\). This is most readily observed in Fig. 6a.

The persistent dissipation-less flow associated with superfluidity on a ring is enabled by the presence of local minima in the ground-state energy at finite angular momentum [109, 111]. In Fig. 6a, we observe a shallow local minimum at \(L=4\), indicating the presence of a few-body precursor of states that may support persistent currents. In Fig. 6b we see that increasing the interspecies attraction for fixed intraspecies repulsion drives a change in the ground-state energy curvature from the superfluid-like curvature present in Fig. 6a to the nearly parabolic ground-state energy curvature indicative of rigid body rotation around the circumference of the ring. The transition from a homogeneous to a localized state is further evidenced by the zero angular momentum ground-state pair correlations.

While any exact eigenstate \(\vert {\Psi }\rangle\) of Eq. (6) must preserve the azimuthal symmetry of the Hamiltonian, the pair correlations

$$\begin{aligned} \rho ^{(2)}_{\sigma \sigma '}(\theta ,\theta ')= & {} \sum _{m,n,k,l}\phi _{\text {m}}^*(\theta )\phi _n^*(\theta ')\phi _{k}(\theta ')\phi _{l}(\theta ) \nonumber \\{} & {} \times \langle \Psi | \hat{a}^\dagger _{\sigma ,m}\hat{a}^\dagger _{\sigma ',n}\hat{a}_{\sigma ',k}\hat{a}_{\sigma ,l}|\Psi \rangle \end{aligned}$$
(7)

allow for the investigation of the internal structure of such a state. Here, \(\theta '\) is the fixed position of a particle of species \(\sigma '\) on the ring and \(\rho ^{(2)}_{\sigma \sigma '}(\theta ,\theta ')\) is proportional to the probability of finding a particle of species \(\sigma\) at position \(\theta\). Note that for symmetric components A and B, we have \({\rho ^{(2)}_{\text {tot}} = \rho ^{(2)}_{AA} + \rho ^{(2)}_{BA} = \rho ^{(2)}_{BB} + \rho ^{(2)}_{AB}}\) with the normalization condition \({\int \rho ^{(2)}_{\sigma \sigma '}(\theta , \theta ')d\theta = N_\sigma - \delta _{\sigma \sigma '}}\). In Fig. 6c, d \(\rho ^{(2)}_{\text {tot}}\) is plotted in color, \(\rho ^{(2)}_{AA}\) is plotted as dashed lines, and \(\rho ^{(2)}_{BA}\) as solid lines. We see in Fig. 6c that the total pair correlation for the weakly attractive case is approximately homogeneous, with a small indent at the location of the fixed particle \(\theta ' = 0\) due to the intraspecies repulsion. In the strongly attractive case of Fig. 6c, we see that \(\rho ^{(2)}_{\text {tot}}\) begins to localize.

In Fig. 7 (right column), we present the zero angular momentum ground-state pair correlations for fixed \(g = 2.0\) and various \(g_{AB}\). The interaction parameters associated with each set of pair correlations are indicated by a dot of the same color in the phase diagram of Fig. 7 (left). For comparison to the phase transition predicted by beyond-mean-field theory, we plot the contrast \((n_{\text {max}}- n_{\text {min}})/(n_{\text {max}}+ n_{\text {min}})\) calculated from the numerical ground-state solution to the eGP equation at each point in the phase diagram. Here, \(n_{\text {max}}\) and \(n_{\text {min}}\) are the maximum and minimum density of the numerical ground-state solution. The dashed black curve in Fig. 7 denotes the condition for the energetic and dynamic stability of the homogeneous solution to the eGP equation. Below the black dashed curve, the homogeneous solution is unstable [106]. The black area is outside the region of validity of the eGP equation.

4.5 Mixed bubble

Fig. 8
figure 8

Graphical abstract of a mixed bubble. Without the inclusion of quantum fluctuations, the miscible–immiscible phase transition is direct and first order. However, including quantum fluctuations, shifts this phase transition depending on the dimensionality of the system and gives rise to a partial miscibility phase. (After http://www.riken.jp/press/2021/20210322-1/index.html. See also Naidon and Petrov [112])

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Fig. 9
figure 9

Density distributions for four different parameters of \(\sqrt{n_+}\delta g/g^{3/2}\) as indicated in the figures, with the orange and black components corresponding to the first and second component respectively. The first distribution shows the system in a miscible state, while the remaining in a mixed bubble state. (After [113])

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Fig. 10
figure 10

a Radial density profiles n(R) of binary bosonic contact-interacting droplets calculated by utilizing the Gaussian trial order parameter (dashed), the super-Gaussian trial order parameter with \(m_{\text {exact}}\) (dotted-dashed) and from the direct numerical simulation the eGP equation (full lines) for different N. With increasing N the super-Gaussian density distribution approaches the flat-top limit. b Chemical potential \(\mu\) of the droplet based on the variational and numerical solutions, as specified by the legend. The super-Gaussian Ansatz correctly predicts the large N limit, while the Gaussian ansatz clearly deviates from it. For small N, the approximate approach overestimates the chemical potential, while the Gaussian offers a good fit. c Breathing frequency \(\omega _0\) of the non-rotating droplet. For large N, the numerical solutions of the eGP equation coincide with the super-Gaussian ansatz calculation; while for smaller N, the assumption of a single breathing frequency no longer holds, due to the emergence of a beating pattern. (After Ref. [114])

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Until now, research on quantum fluctuations has mainly focused on preventing collapse in bosonic systems (at least in the many-particle limit) to form droplets and supersolids. However, what if we investigate the phase transition of a Bose–Bose mixture from the miscible to the immiscible regime? In a homogeneous system, this transition is a first-order process, where the two gases occupy the same space in the miscible regime, while they completely separate in the immiscible regime. By considering the impact of quantum fluctuations on this phase transition, we observe two effects. First, the transition is shifted to the miscible or immiscible regime, depending on the system’s dimensionality. Second, an intermediate regime of partial miscibility, known as a “Mixed Bubble,” arises, see the Fig. 8. In this state, one component is entirely localized but surrounded by a halo of the other component. It is important to note that this effect exists only in a system with an imbalance in intraspecies interactions and particle numbers. Moreover, an imbalance in the masses of the mixture enhances this effect. Although this effect is so far purely theoretical and based on Naidon and Petrov’s proposal [112], we here aim to identify this novel system in a one-dimensional Bose–Bose mixture. The resulting ground states when varying \(g_{12}\) via \(\sqrt{n_+}\delta g/g^{3/2}\) starting from the miscible regime for a one-dimensional system with periodic boundary conditions are illustrated in Fig. 9.

$$\begin{aligned} \mu _\sigma \psi _\sigma =&\Bigg [-\frac{\hbar ^2}{2m}\partial _{\theta \theta }+g_{\sigma \sigma }n_\sigma +g_{\sigma \sigma '}n_{\sigma '}\nonumber \\&-\frac{m^{1/2}g_{\sigma \sigma }}{\pi \hbar }\left( \sum _\sigma g_{\sigma \sigma } n_\sigma \right) ^{1/2}\Bigg ]\psi _\sigma . \end{aligned}$$
(8)

The phase transition between the miscible and bubble phases is of the second order, resulting in the sudden localization of the second component (dark), leaving a dip in the density of the first component (orange). With an increase in \(g_{12}\), the dip grows larger until both components are completely localized, and the system enters the immiscible regime. It is important to note that, excluding quantum fluctuations, negative values of \(\sqrt{n_+}\delta g/g^{3/2}\) correspond to the miscible regime. Interestingly, the system exhibits similarities to that of a repulsive impurity placed in the surrounding component, while also displaying rotational behavior comparable to that of a weak link. Therefore, the mixed bubble on a ring can be considered as a self-forming “atomtronic” superconducting quantum interference device. For a detailed discussion, we refer to Ref. [113].

4.6 Variational principle

4.6.1 Breathing mode of binary quantum droplets

The order parameter of a condensate, which is confined in a trap and has non-negligible interactions, can be modeled variationally using a Gaussian ansatz [115, 116]. This approach has also been employed to model self-bound droplets in a Bose–Bose mixture. However, as the number of particles in the system increases, the ground-state density profile of the droplet takes on a flat-top shape that cannot be accurately captured by a Gaussian profile alone. This limitation necessitates the use of a more sophisticated Ansatz that goes beyond the typically used Gaussian ansatz. One possible solution in this direction is to use a so-called super-Gaussian which replaces the quadratic exponent with a variational parameter denoted as 2m and reads as follows,

$$\begin{aligned} \psi (r,t)= A(t)\exp \left( ib(t)r^2-\frac{1}{2}\left( \frac{r}{R(t)}\right) ^{2m}\right) ~. \end{aligned}$$
(9)

As can be seen from equation (9), the flat-top shape of the droplet density distribution becomes more prominent as the value of m increases, showing that the function is a suitable choice to effectively encapsulate the droplet density distribution. In this study, we employ the super-Gaussian ansatz and compare its effectiveness in predicting various equilibrium properties and the monopole oscillation frequency with that of a Gaussian ansatz and numerical solutions obtained by solving the GP equation. To evaluate the effectiveness of the super-Gaussian ansatz, we first treat the exponent m as a variational parameter in a dimensionless two-dimensional eGP framework. This treatment yields a transcendental equation for \(m_{\textrm{exact}}\):

$$\begin{aligned} 0 = \frac{m_{\text {exact}}}{2}(\ln {2}-m_{\text {exact}})+\frac{N}{4\pi }2^{-1/m_{\text {exact}}}(2\ln {2}-1).{} \end{aligned}$$
(10)

In the limit of large N this gives the approximation \(m_{\textrm{approx}}\)

$$\begin{aligned} m_{\textrm{approx}}\approx \sqrt{\frac{N}{2\pi }(\ln {4}-1)}. \end{aligned}$$
(11)

Utilizing these expressions, we can now model the density distributions, chemical potential, and monopole mode frequency for different norms N. As expected, the super-Gaussian manages to model the density distribution more accurately than the \(m=1\) Gaussian ansatz (here the plot of the density distributions) and qualitatively agrees well with the ground-state solutions obtained numerically from the eGP equation. For large N, the dimensionless chemical potential \(\mu\) is expected to approach \(-0.5/\sqrt{e}\), which is correctly approached by the super-Gaussian ansatz,

$$\begin{aligned} \mu = -\frac{n_{\text {m}}}{2}\left( \frac{m\pi }{N}+2^{-1/m}\right) , \end{aligned}$$
(12)

while the \(m=1\) ansatz only agrees in the soliton-like regime for low N, the super-Gaussian ansatz accurately describes density as well as the chemical potential of the system with increasing norm N, see Fig. 10a, b. However, both Ansätze fail to predict the monopole oscillation frequency (breathing mode frequency) in an intermediate N regime

$$\begin{aligned} \omega ^2_0 = n_{\text {m}}^2\frac{\Gamma _1}{\Gamma }\frac{2\pi }{Nm}2^{-1/m}\, . \end{aligned}$$
(13)

(here, \(\Gamma \left( \frac{1}{m}\right) = \Gamma _1\) and \(\Gamma \left( \frac{2}{m}\right) /\Gamma \left( \frac{1}{m}\right) = \Gamma\)), where the numerical solution obtains a beating pattern. As the variational ansatz only allows for a single oscillation frequency to occur it is natural that it fails to describe the dynamics accurately (Fig. 10c). For a more detailed discussion, we refer to our earlier work in Ref. [114].

4.6.2 Capturing the stacked isolated droplets by the variational principle

Fig. 11
figure 11

a Energy E as a function of the scattering length a for a different number \(\nu\) of isolated droplet crystals calculated from the variational principle. b The ground- state energy as a function of the scattering length a calculated from the variational principle and eGP simulation. Also the c integrated axial profile (along z) and d profile in the y–z-plane for the eGP method and variational principle is shown (see the legends)

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Here, we describe that isolated stack droplets in anti-dipolar condensates can also be captured using a variational principle. While the variational principle has been used to illustrate the droplet crystal structures in regular dipolar condensates [68], to the best of our knowledge, the same has not been done for isolated stack droplets. We describe the wave function for droplet phase using a variational ansatz that takes the form

$$\begin{aligned} \psi (\textbf{x}) = \sum _{i = 1}^\nu \psi _i(\textbf{x}), \end{aligned}$$
(14)

where each of the \(\nu\) droplets is given by a Gaussian profile

$$\begin{aligned} \psi _i(r,z) = \sqrt{\frac{N_i}{(2\pi )^{3/2}\sigma _{r,i}^2\sigma _{z,i}}}\exp \left[ -\frac{r^2}{4\sigma ^2_{r,i}}-\frac{(z-z_i)^2}{4\sigma ^2_{z,i}}\right] , \end{aligned}\\$$
(15)

where \(N_i\), \(\sigma _{z,i}\), \(\sigma _{r,i}\), \(z_{i}\) are the variational parameters. With the definition above, and assuming that the overlap between different droplets is negligible one has that \(\sum N_i=N\) is the total number of particles. In what follows, we express energies in units of \(\hbar \omega _z\) and lengths in units of the harmonic oscillator length \(\sqrt{\hbar /m\omega _z}\). The ground state of the system is found by minimizing the energy with respect to the free parameters. The energy functional can be written straightforwardly as

$$\begin{aligned} E = E_0 + V_{\textrm{con}} + V_{\textrm{LHY}} + E_{\textrm{dd}}, \end{aligned}$$
(16)

where \(E_0\) is the energy of the non interacting system, namely the sum of kinetic and trap energy, and it is equal to

$$\begin{aligned} E_0 = \sum _{i=1}^\nu N_i\left( \frac{1}{4\sigma ^2_{r,i}} + \frac{1}{8\sigma ^2_{z,i}}+\frac{\omega ^2_r}{\omega _z^2}\sigma ^2_{r,i}+\frac{1}{2}(\sigma ^2_{z,i} + z^2_i)\right) . \end{aligned}$$
(17)

The mean-field contribution of the short-range interaction is given by

$$\begin{aligned} V_{\textrm{con}} = \frac{g}{16 \pi ^{3/2}}\sum _{i=1}^\nu \frac{N^2_i}{\sigma _{r,i}^2\sigma _{z,i}}. \end{aligned}$$
(18)

The beyond-mean-field correction to the energy functional in the local density approximation is equal to

$$\begin{aligned} V_{LHY} = \gamma (\epsilon _{\textrm{dd}})\frac{2^{1/4}}{25\sqrt{5}\pi ^{9/4}}\sum _{i=1}^\nu \frac{N_i^{5/2}}{\sigma _{r,i}^3\sigma _{z,i}^{3/2}}. \end{aligned}$$
(19)

Finally, the dipole–dipole interaction is calculated in momentum space:

$$\begin{aligned} E_{\textrm{dd}} = \frac{1}{2}\int \frac{{\text {d}}\textbf{k}}{(2\pi )^3} \tilde{n}(-\textbf{k})\tilde{V}_{\textrm{dd}}(\textbf{k})\tilde{n}(\textbf{k}), \end{aligned}$$
(20)

where \(\tilde{n}\) is the Fourier transform of the density and \(\tilde{V}(\textbf{k}) = C_{\textrm{dd}}(\cos ^2{\vartheta } - 1/3)\) is the dipole potential in momentum space and \(\vartheta\) is the angle between \(\textbf{k}\) and the z axis. Upon replacing Eq. (14) in Eq. (20), we can expand the integral and identify two terms:

$$\begin{aligned} E_{\textrm{dd}} = \sum _{i=1}^\nu E_{i}^{(1)} + \sum _{i<j} E_{i,j}^{(2)}. \end{aligned}$$
(21)

The first term, giving the intra-droplet dipole interaction, has already been calculated in Ref. and takes the form

$$\begin{aligned} E_{\textrm{dd}}^{(1)} = \frac{\sqrt{\pi }C_{\textrm{dd}}N^2_i}{12(2\pi )^2 \sigma _{r,i}^2\sigma _{z,i}}f(\eta _i) \end{aligned}$$
(22)

and the function f depends only on the ratio \(\eta _i=\sigma _{r,i}/\sigma _{z,i}\) and takes the form

$$\begin{aligned} f(\eta ) = \frac{\sqrt{\eta ^2 -1}(2 \eta ^2-1) - 3\eta ^2\tan ^{-1}{\sqrt{\eta ^2 -1}}}{(\eta ^2-1)^{3/2}} . \end{aligned}$$
(23)

For the second term in Eq. (21), which represents the inter-droplet interaction, there is no analytical expression. However, we can carry out the integral in the radial coordinate,

$$\begin{aligned} E_{\textrm{dd}}^{(1)}& = \frac{C_{\textrm{dd}}N_i N_j}{(2\pi )^{\frac{3}{2}}} \int _0^1 {\text {d}}x \left( x^2 - \frac{1}{3}\right)\\ & \quad \times e^{-\frac{\Delta z^2_{ij}x^2}{2h(x)}}\frac{h(x) - \Delta z^2_{ij}x^2}{h^{5/2}(x)}, \end{aligned}$$
(24)

where \(\Delta z_{ij}=z_i-z_j\), and \(h(x) = (\sigma _{r,i}^2+\sigma _{r,j}^2)(1-x^2)+(\sigma _{z,i}^2 + \sigma _{z,j}^2)x^2\). The integral in Eq. (24) is evaluated numerically. Note that unlike the isolated droplet regime in a quasi-2D geometry (see Ref. [68]), where one can treat the droplets as point-like objects to simplify the expression for \(E_{\textrm{dd}}^{(2)}\), one cannot approximate the flat droplets in the anti-dipolar gas to infinitely thin disks, as it would lead to large discrepancies in the energy for the parameter regime considered in this case. The total energy functional obtained by replacing the previous expressions in Eq. (16) is minimized numerically with respect to the variational parameters for different values of \(\nu\), using a standard numerical multidimensional minimization method that keeps the total number of particles constant. Additional constraints are given by the symmetry of the system with respect to reflections along the z-axis, effectively reducing the number of independent parameters.

In Fig. 11, we present the results for \(N = 70000\) and \(a_{\textrm{dd}} = -65.5a_{0}\). We observe that the energies of the four-crystal state and three-crystal state are very close, and a transition to the three-crystal state (\(\nu = 3\)) occurs when \(a < 98.2a_{0}\) (see Fig. 11a). The ground-state energy, calculated using the eGP equation and variational principle, is shown in Fig. 11b. However, the energy obtained from the variational principle overestimates that from the eGP simulation. Furthermore, we display the axial (\(x=0\) and \(y=0\) plane) and radial profiles (at \(z=0\) plane) in Fig. 11c, d, respectively. In the eGP simulation, the central peak is more localized compared to the variational principle. Also, the eGP results in a broader density distribution in the radial direction as the anti-dipoles try to arrange themselves in the radial plane. However, the variational principle underestimates the true profile.

It is important to note that a Gaussian profile is a poor ansatz to capture the localized droplet structure, as demonstrated for the binary droplet. Therefore, using a super-Gaussian ansatz to model the individual crystal might result in better quantitative agreement with the exact solution of the eGP simulation. However, it would make the corresponding integrals, such as the one in Eq. 20, more computationally involved.

5 Future prospects

In the first half of this article, we presented a brief overview of the research activities related to the formation of superfluid droplet crystals in ultra-cold atomic gases, with a focus on long-range interacting systems. In the second half, we then summarized some of the more recent contributions made by the authors.

Several research directions could be pursued in the future. For example, there is a need to gain a detailed understanding of the formation, stability, and dynamics of the supersolid and droplet states of single or multicomponent dipolar BECs at finite temperatures. To this end, one possible approach is to use the bosonic Hartree–Fock–Bogoliubov theory to compute the finite-temperature excitation spectra of these states [117]. Additionally, one could analyze the dynamical growth of the single and multicomponent supersolid and droplet states from a cloud of thermal atoms using the stochastic Gross–Pitaevskii equation for dipolar condensates [118]. During such a quench, nonlinear defects like vortices can be created, which are currently actively being sought in these novel types of quantum matter [119, 120].

So far, the droplet lattice structures in dipolar condensates have been investigated mainly in a clean harmonic potential, where they are highly symmetric. It will, thus, be interesting to analyze how introducing imperfections of the confinement or weak disorder may affect the symmetry of the droplet lattice arrangement, and how it is influenced by geometrical frustration [121]. Introducing disorder could also be a potential route to create turbulence in the system  [122], opening up a novel direction of research. In one-dimensional systems, the dynamics of droplets in binary mixtures has been studied evincing the enhanced role of correlations [123, 124]. It would be interesting to further investigate the few-to-many-body transition across criticality and explore how the precursors of the phase transition can be encoded in the excitation spectra of finite quantal systems that are far from the thermodynamic limit [125,126,127,128]. Some of these works are in progress and will be reported in the near future.