Excitation spectrum of a double supersolid in a trapped dipolar Bose mixture

Supersolids constitute a peculiar phase of matter in which superfluidity coexists with a crystal-like density modulation Boninsegni and Prokof’ev (2012). Although their experimental realization in helium has remained elusive Kim and Chan (2004); Balibar (2010); Kim and Chan (2012), recent experiments in trapped ultracold quantum gases have explored the realization of supersolids in spin-orbit-coupled Bose-Einstein condensates (BECs) Li et al. (2017); Geier et al. (2021), optical cavities Léonard et al. (2017), and in dipolar BECs formed by magnetic atoms Chomaz et al. (2022); Recati and Stringari (2023). In the latter case, dipolar supersolids Tanzi et al. (2019a); Böttcher et al. (2019); Chomaz et al. (2019), formed by ultradilute quantum droplets immersed in a superfluid halo, result from the non-trivial interplay between $s$-wave collisions, dipole-dipole interactions, quantum fluctuations Petrov (2015), and the external confinement.

Recent interest in dipolar supersolids has focused on their elementary excitations as a key tool for understanding their superfluid properties and associated phase transitions Recati and Stringari (2023); Roccuzzo and Ancilotto (2019); Guo et al. (2019); Tanzi et al. (2019b); Natale et al. (2019); Hertkorn et al. (2019). Supersolids break simultaneously translational invariance, associated with the crystal order, and the $U(1)$ symmetry related to superfluidity. As a result, typical experiments performed in quasi-one-dimensional droplet arrays are expected to present two gapless Goldstone modes Watanabe and Brauner (2012); Roccuzzo and Ancilotto (2019). However, crucially, dipolar supersolids are created in the presence of an external confinement, and hence their spectrum is discretized due to finite size. For a trapped dipolar supersolid, the lowest-lying Goldstone modes are characterized, respectively, by an in-phase or out-of-phase oscillation of the condensate and the crystal structure. The in-phase mode is the center-of-mass oscillation of the whole cloud (dipole mode), whereas the out-of-phase mode has been experimentally revealed by the absence of center-of-mass motion Guo et al. (2019). The two-Goldstone mode character of the spectrum has been experimentally revealed as well by the study of compressional modes, which present characteristic two-frequency excitations in the supersolid phase Tanzi et al. (2019b); Natale et al. (2019).

The recent realization of mixtures of two different dipolar components Trautmann et al. (2018); Durastante et al. (2020); Politi et al. (2022) has opened new avenues for studying dipolar supersolid mixtures Bisset et al. (2021); Bland et al. (2022); Li et al. (2022); Arazo et al. (2023); Kirkby et al. (2023); Halder et al. (2023); Kirkby et al. (2024). The crystal-like modulation may occur in the overall density in the case of miscible mixtures Scheiermann et al. (2023), or as the formation of alternating domains of each component (spin modulation) in immiscible ones Bland et al. (2022). In both cases, the system enters a unique double supersolid phase where mutually interacting components remain superfluid, albeit with distinct superfluid fractions.

The double supersolid is expected to present a non-trivial excitation spectrum. Whereas an unmodulated binary condensate breaks a $U(1)\times U(1)$ symmetry, leading to two sound modes, a double supersolid additionally breaks translational symmetry, resulting in an extra Goldstone mode Watanabe and Brauner (2012). The three gapless modes have been recently discussed for the case of an immiscible mixture of a dipolar and a non-dipolar component in the idealized case of an infinite tube geometry, in which the mixture is not axially confined Kirkby et al. (2024). However, as for the case of a single component, in experiments binary mixtures are confined in all spatial directions by a typically harmonic confinement. This raises the challenge of extracting information about the double-supersolid character, the distinct superfluid properties of the components, and the various phase transitions from the discrete excitation spectrum of trapped mixtures.

This paper provides a detailed analysis of the lowest-lying excitation spectrum of a trapped supersolid dipolar mixture, focusing on a miscible case. We show that even a mildly asymmetric mixture, illustrated here by the case of two components with slightly different dipole moments, results in a highly non-trivial spin nature of the dipolar rotons, the Higgs excitation, and the Goldstone modes. Moreover, in the double supersolid phase, the lowest-lying dipole and breathing modes present a characteristic three-mode structure. In particular, compressional excitations, which can be readily excited and probed in experiments Tanzi et al. (2019b); Natale et al. (2019), are characterized by the doubling of the superfluid breathing modes. These breathing excitations may be employed as well to reveal the markedly different superfluid fraction in each component. Finally, we discuss how the spectrum reveals the onset of the incoherent droplet regime in one component, while the other remains superfluid.

The structure of the paper is as follows. Section II introduces the employed model, whereas Sec. III analyzes the different possible ground-state phases of the mixture. Section IV evaluates and characterizes the collective excitations. In Sec. V, we discuss the particular case of a symmetric mixture, whereas in Sec. VI we analyze the general case of an asymmetric one, in which the spectrum provides clear insights about the nature of superfluidity in the mixture. Finally, we conclude in Sec. VII.

We consider a Bose mixture of two dipolar components labeled by $\sigma=1,2$. We focus on magnetic dipoles with moments $\mu_{1,2}$, although similar conclusions could apply to electric ones. The dipoles, which are oriented along the $z$-axis by an external magnetic field, can belong to the same species or to two different ones. In addition, the atoms interact via short-range interactions characterized by the intra- and inter-component scattering lengths $a_{11}$, $a_{22}$, and $a_{12}$. The mixture is well described by the extended Gross-Pitaevskii equations (eGPEs) Smith et al. (2021); Bisset et al. (2021)

with

where $\psi^{\sigma}({\bf r},t)$ is the condensate wave function of component $\sigma$, $n_{\sigma}({\bf r})=|\psi^{\sigma}({\bf r})|^{2}$, and $U_{\sigma\sigma^{\prime}}(\mathbf{r})=g_{\sigma\sigma^{\prime}}\delta(\mathbf{r})+V_{dd}^{\sigma\sigma^{\prime}}({\bf r})$, with $g_{\sigma\sigma^{\prime}}=4\pi\hbar^{2}a_{\sigma\sigma^{\prime}}/m$, $m$ the mass of the bosons (which we assume for simplicity equal for both components), and $V^{\sigma\sigma^{\prime}}_{\rm dd}({\bf r})=\frac{\mu_{0}\mu_{\sigma}\mu_{\sigma^{\prime}}}{4\pi r^{3}}\left(1-3\cos^{2}\theta\right)$ the dipole-dipole interaction potential, where $\theta$ is the angle sustained by the polarization direction ($z$-axis) and ${\bf r}$. The atoms are confined in a harmonic trap defined by the potential $V_{\rm trap}({\bf r})=\frac{1}{2}m(\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2}+\omega_{z}^{2}z^{2})$. Quantum fluctuations are accounted for by the Lee-Huang-Yang (LHY) term $\mu_{{\rm LHY}}^{\sigma}[n_{1,2}({\bf r},t)]=\delta E_{\rm LHY}/\delta n_{\sigma}$, where Bisset et al. (2021)

is the LHY energy density, and

with $\eta_{\sigma\sigma^{\prime}}=g_{\sigma\sigma^{\prime}}+g^{d}_{\sigma\sigma^{\prime}}(3\cos^{2}\theta_{k}-1)$, with $g^{d}_{\sigma\sigma^{\prime}}=\mu_{0}\mu_{\sigma}\mu_{\sigma^{\prime}}/3$ and $\theta_{k}$ the angle sustained by momentum ${\bf k}$ with the dipole moment.

A single component dipolar condensate displays three possible ground-state phases: an unmodulated phase with a Thomas-Fermi density profile for sufficiently strong interactions; a supersolid phase, characterized by a marked density modulation while still preserving superfluidity; and the incoherent droplet (ID) regime, in which the condensate fragments into mutually incoherent droplets.

The interplay between inter- and intra-component interactions results in a rich ground-state phase diagram for miscible mixtures Scheiermann et al. (2023), which we illustrate in Fig. 1 for the particular case of a mixture of two ¹⁶²Dy components with $\mu_{1}=10\,\mu_{B}$ and $\mu_{2}=9\,\mu_{B}$, assuming $a_{11}=a_{22}=100\,a_{0}$, a total number of atoms $N=3\times 10^{4}$, and trap frequencies $\omega_{x,y,z}/2\pi=30$, $110$ and $90$ Hz (we employ below $a_{x}=\sqrt{\hbar/m\omega_{x}}$ as the length unit). Similar phase diagrams should generally characterize other miscible mixtures. As a function of $a_{12}$ and the fraction of atoms in component $2$, $N_{2}/N$, the two components may be: (i) both unmodulated: this phase presents a broken $U(1)\times U(1)$ symmetry similar to that discussed in non-dipolar binary condensates; (ii) both supersolid (double supersolid regime): in which due to density modulation, the system breaks (in an axially infinite system) translational symmetry, in addition to the gauge symmetry; (iii) one component remains supersolid, whereas the other is in the ID regime (ID-supersolid if the second component remains supersolid, or supersolid-ID in the opposite case): the system presents a broken $U(1)$ symmetry associated with the superfluid component, in addition to the broken translational symmetry; (iv) both components are in the ID regime (ID-ID): the system behaves like a crystal of droplets, with only broken translational symmetry.

We determine the phase diagram from the density profile of each component. The contrast of the profile of component $\sigma$ is calculated from the maximal and minimal densities $n^{\rm max}_{\sigma}$ and $n^{\rm min}_{\sigma}$, as $C_{\sigma}=(n_{\sigma}^{\rm max}-n_{\sigma}^{\rm min})/(n_{\sigma}^{\rm max}+n_{\sigma}^{\rm min})$. Figure 1 shows iso-contours (black lines) at $C_{1}=0.99$, and $C_{2}=0.1$ and $0.99$. Those lines may be employed to determine the approximate boundaries of the different regimes.

The different broken symmetries result in markedly different spectral properties for the different phases. In particular, for an axially-untrapped infinitely elongated mixture (infinite tube), the unmodulated phase should present two gapless excitation branches, corresponding to the broken $U(1)\times U(1)$ symmetry, whereas the double supersolid in contrast should present three, due to the additional broken translational symmetry Watanabe and Brauner (2012); Kirkby et al. (2024). The ID-supersolid or supersolid-ID regimes should present two branches, related to the broken translational symmetry and the remaining broken $U(1)$ symmetry associated with the component that remains superfluid. Finally the ID-ID regime should just present one branch of crystal phonons. In this paper, we focus on how these expected properties in idealized infinite tube geometries, may be revealed under typical experimental conditions, characterized by a three-dimensional confinement.

As a preliminary step, we start our analysis with the discussion of the very particular and especially simple case of a symmetric mixture, with two equally populated ($N_{1}=N_{2}=N/2$) components with the same mass, $a_{11}=a_{22}=a$, and $\mu_{1}=\mu_{2}=\mu$. For a miscible mixture (which is the case in all our calculations), $\psi_{0}^{1}(\mathbf{r})=\psi_{0}^{2}(\mathbf{r})=\psi_{0}(\mathbf{r})$, $\hat{B}_{1}=\hat{B}_{2}=\hat{B}$ and $\hat{X}_{11}=\hat{X}_{22}=\hat{X}$. Hence, Eqs. (8) exactly decouple into:

where $f_{+,D}(\mathbf{r})=f_{+,1}(\mathbf{r})+f_{+,2}(\mathbf{r})$ and $f_{+,S}(\mathbf{r})=f_{+,1}(\mathbf{r})-f_{+,2}(\mathbf{r})$, characterize, respectively, the density modulations $\delta n(\mathbf{r})=2\psi_{0}(\mathbf{r})f_{+,D}(\mathbf{r})$ and the spin modulations $\delta s(\mathbf{r})=2\psi_{0}(\mathbf{r})f_{+,S}(\mathbf{r})$. As a result, the modes acquire either a purely density character ($Q=1$) or spin character ($Q=-1$).

The ground-state total density profile $2|\psi_{0}(\mathbf{r})|^{2}$, is that of a single-component condensate with dipole moment $\mu$ and effective scattering length $a_{\mathrm{eff}}=(a+a_{12})/2$. Decreasing $a_{12}$, reduces $a_{\mathrm{eff}}$, hence increasing the relative dipolar strength ($\propto\mu^{2}/a_{\mathrm{eff}}$), driving eventually the unmodulated-to-supersolid transition Scheiermann et al. (2023). The modes across this transition are depicted in Fig. 2, where we consider the same case as in Fig. 1, but assuming $\mu_{1}=\mu_{2}=10\mu_{B}$, and fixing $N_{1}=N_{2}$. For this particular case, the dipolar mixture develops three central droplets at the unmodulated-to-supersolid transition. Whereas the spin modes are mildly affected at the transition, the density modes present the characteristic softening of two degenerate roton modes, as in the case of a single-component condensate. These rotons split at the transition into a Higgs mode that abruptly hardens when entering the supersolid regime, and a Goldstone mode that softens Guo et al. (2019); Hertkorn et al. (2019).

In principle, we can probe the double supersolid regime by monitoring the axial compressional modes after an $x^{2}$ perturbation equally applied to both components. This perturbation, however, couples only to the overall density. Hence, in a symmetric mixture, the applied compressional perturbation is orthogonal to the spin modes, preventing the probing of the double-supersolid nature of the spectrum. Similar to single-component condensates Tanzi et al. (2019b); Natale et al. (2019), the axial breathing density mode hybridizes with the Higgs mode and other higher-frequency modes at the unmodulated-to-supersolid transition. This leads to a hardening of the breathing mode (crystal mode) and a softening of a superfluid mode, se Fig. 2 (a). Note that, as in Ref. Natale et al. (2019), the additional observed even density mode, which results in a three-droplet scenario from the relative breathing of the droplets and the halo, is only weakly affected by the perturbation (this mode is absent in the two-droplet scenario discussed in the next section).

In the special case of a symmetric mixture, density and spin modes decouple, and density excitations behave as those of an effective single-component condensate. The situation becomes substantially more complex in asymmetric mixtures, where the masses and/or interactions between the components are different. In that case $\psi^{1}(\mathbf{r})\neq\psi^{2}(\mathbf{r})$ and, crucially, spin and density features hybridize, resulting in a much richer physics. We consider below the same mixture discussed above, but with different $\mu_{1}=10\mu_{B}$ and $\mu_{2}=9\mu_{B}$, as in Fig. 1. Nevertheless, the results are to a large extend representative of other asymmetric mixtures, although the particular details of the spectrum may differ.

Figure 3 summarizes our results of the low-lying spectrum. As discussed above, we characterize the modes by determining the relative weight of the two components, given by $P$ (panels a and c) and the density versus spin character, given by $Q$ (panels b and d). The upper (lower) panels corresponds to even- (odd-) parity modes. In contrast to the symmetric case discussed above, the mixture transitions into a two-droplet ground-state when entering the double supersolid regime. Similar to the experiment in Ref. Tanzi et al. (2019a), this simple two-droplet scenario significantly simplifies the analysis of the excitation spectrum.

In this paper, we have studied the excitation spectrum of a trapped dipolar Bose mixture, and how this spectrum provides key insights about the nature of the double-supersolid phase, with two coexisting interacting superfluids with a generally different superfluid fraction. While we have focused on the simple, yet experimentally relevant, case of two-droplet supersolids due to numerical complexity, our conclusions are expected to hold more generally. In particular, the doubling of the superfluid breathing modes, and their basically single-component character, which should be relatively straightforward to monitor Tanzi et al. (2019b); Natale et al. (2019), would generally be a clear proof of the double-supersolid nature and the markedly different superfluid fraction of the two components in asymmetric mixtures. Moreover, the modes exhibit a highly nontrivial nature in what concerns their contribution to the modulation of the overall density, of the relative density (the spin), and of the phases of the two components. This is in particular the case of the dipolar rotons, the Higgs excitation, and the lowest-lying Goldstone modes, which present a very different density-spin hybridization. The different superfluid fraction of the two components becomes especially clear from the detailed analysis of the Goldstone modes, which reveal the crossover of one of the components, but not of the other, into the incoherent droplet regime.

We acknowledge the support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 274200144 – SFB 1227 DQ-mat within the projects B01 and A04, and under Germany’s Excellence Strategy – EXC-2123 Quantum-Frontiers – 390837967).