Dipolar Josephson Relation in Quantum Supersolids

• Dipolar Josephson relation describes a family of generalized Josephson equations where dipole–dipole interactions and self-induced density modulations determine phase-controlled supercurrents and vortex positions.
• It extends the standard bosonic Josephson framework by incorporating effects like pair tunneling, long-range interactions, and geometric constraints in rotating droplet lattices.
• Experimental studies in supersolid systems reveal clear signatures such as phase slips, vortex nucleation, and self-trapping regimes, validating the theoretical models.

Searching arXiv for the cited papers to ground the article in current preprints. The dipolar Josephson relation denotes a class of Josephson-type relations in dipolar quantum fluids in which the phase difference between weakly linked condensates controls supercurrent, phase evolution, and, in rotating dipolar supersolids, even vortex position and velocity. In these systems the weak links may be self-induced by the density modulation of the supersolid itself, and dipole–dipole interactions renormalize effective tunneling energies, onsite nonlinearities, and threshold conditions. In the rotating triangular droplet lattice studied in a dipolar supersolid, the relation becomes especially explicit: local Josephson phases between droplets directly encode vortex nucleation, transport, and the microscopic phase-slip processes mediated by vortex and vortex–antivortex dynamics (Alaña et al., 5 Jun 2026).

1. Definition and conceptual scope

In the standard bosonic Josephson picture, two canonically conjugate variables are used: a population imbalance and a relative phase. In dipolar systems, the same structure survives, but the parameters entering the equations are shaped by long-range dipole–dipole interactions, by self-organized density modulation, and, in finite arrays, by geometric constraints. In a self-induced supersolid junction, the generalized Josephson relations for the central pair of droplets take the form

$$\dot z = 2K \frac{\alpha}{\alpha-1}\sqrt{1-z^2}\,\sin(\Delta\theta), \qquad \dot{\Delta\theta} = -U N_{23}\, z,$$

with the current–phase relation obtained by identifying the current with $\dot z$, and with $\dot{\Delta\theta}=-(\mu_3-\mu_2)$ in the discrete AC Josephson form when $\hbar=1$ (Donelli et al., 28 Jan 2025).

The phrase is not tied to a single universal formula. In dipolar atomic Josephson junctions described by an extended two-site Bose–Hubbard model, pair tunneling adds a second harmonic to the current–phase relation,

$$I(\phi) \simeq -\frac{JN}{\hbar}\sin\phi + \frac{PN^2}{\hbar}\sin 2\phi,$$

so the dipolar Josephson relation becomes explicitly non-sinusoidal (Vianello et al., 23 Feb 2026). In electric-field-controlled dipolar condensates, the basic Josephson structure is preserved while the effective charging and tunneling energies become field dependent through Stark-shift-induced renormalization of the effective Gross–Pitaevskii parameters (Galvez-Poblete et al., 12 Aug 2025). At the most general level, for multi-component bosons with conserved internal charges, the Josephson relation becomes a superfluid-density tensor $\rho_s^{\alpha\beta}$ expressed through the inverse Nambu–internal Green’s function, providing a formal setting for internal spin-like or dipolar charges (Zhang, 2017).

This suggests that “dipolar Josephson relation” is best understood as a family of dipolar-modified Josephson relations rather than as a single equation.

2. Rotating dipolar supersolid realization

A concrete realization is provided by a rotating dipolar supersolid of about $N\simeq 1.1\times10^5$ $^{162}\mathrm{Dy}$ atoms at zero temperature in a 3D harmonic trap with $\{\omega_\perp,\omega_z\}=2\pi\times\{60,120\}$ Hz and dipoles oriented along $z$. Dipole–dipole interactions together with Lee–Huang–Yang repulsion stabilize a supersolid phase consisting of a triangular lattice of self-bound droplets. In the geometry analyzed, one central droplet ($\dot z$0) is surrounded by six equivalent droplets ($\dot z$1), producing hexagonal low-density interstitial regions between droplets (Alaña et al., 5 Jun 2026).

The microscopic dynamics are governed by the extended Gross–Pitaevskii equation with contact, dipolar, and LHY terms, supplemented by a rotating six-fold symmetric “egg-box” potential that pins the droplets and transfers angular momentum. Because the droplets are well separated and connected through low-density valleys, the condensate wave function can be expanded as a sum of localized droplet modes,

$\dot z$2

The first phase factor is the geometric rigid-rotation contribution, while $\dot z$3 is the Josephson phase associated with inter-droplet superflow.

Under the imposed six-fold symmetry, the ring droplets remain equivalent. The many-mode dynamics then reduce to two collective variables,

$\dot z$4

The stationary ground-state supersolid has a finite equilibrium imbalance $\dot z$5, quoted as $\dot z$6, because the central well is deeper and more populated. The mapping to weakly linked condensates relies on three approximations: weak mode overlap, slowly varying droplet shapes relative to phase dynamics, and sufficiently slow rotation that the system follows the instantaneous supersolid configuration without melting the droplet lattice (Alaña et al., 5 Jun 2026).

3. Josephson current, AC relation, and dynamical regimes

Within the multimode reduction, the $\dot z$7 dynamics follow generalized Josephson equations of the form

$\dot z$8

where $\dot z$9 is an effective tunneling energy and $\dot{\Delta\theta}=-(\mu_3-\mu_2)$0 contains interaction and asymmetry contributions. The dipolar character enters through the renormalization of both $\dot{\Delta\theta}=-(\mu_3-\mu_2)$1 and the interaction energy, since dipole–dipole interactions reshape the droplet modes, the barrier between them, and the onsite energies (Alaña et al., 5 Jun 2026).

For the link between the central droplet and a ring droplet,

$\dot{\Delta\theta}=-(\mu_3-\mu_2)$2

and therefore

$\dot{\Delta\theta}=-(\mu_3-\mu_2)$3

This is the particle-current version of the dipolar Josephson relation in the rotating supersolid: the current is sinusoidal in the phase difference, but the critical current depends on the instantaneous imbalance and hence on dipolar-modified populations and energies.

Two dynamical regimes appear. In the Josephson regime, $\dot{\Delta\theta}=-(\mu_3-\mu_2)$4 is bounded and $\dot{\Delta\theta}=-(\mu_3-\mu_2)$5 crosses its equilibrium value. In the self-trapping regime, the imbalance never crosses equilibrium and the phase becomes a running phase that increases or decreases almost linearly in time. The self-induced supersolid analysis of a harmonically trapped four-droplet chain shows the same phenomenology in a generalized Josephson-array form, including small-amplitude oscillations and macroscopic quantum self-trapping, with trap inhomogeneity entering through a current-renormalization factor $\dot{\Delta\theta}=-(\mu_3-\mu_2)$6 and an effective coupling $\dot{\Delta\theta}=-(\mu_3-\mu_2)$7 (Donelli et al., 28 Jan 2025).

The conceptual importance is that dipolarity does not remove Josephson phenomenology; it modifies the effective parameters, threshold conditions, and, in lattice geometries, the number of coupled links that must be retained.

4. Vortex coordinate as a Josephson variable

The distinctive feature of the rotating dipolar supersolid is that the local Josephson phases do not merely control current. They determine the coordinates of vortices, which are defined microscopically as zeros of the total wave function. Along low-density valleys between droplets, destructive interference of the relevant localized modes gives explicit vortex-position formulas (Alaña et al., 5 Jun 2026).

Along a nucleation path between two ring droplets, symmetry makes the relevant phase difference purely geometric, and the $\dot{\Delta\theta}=-(\mu_3-\mu_2)$8-th allowed vortex position satisfies

$\dot{\Delta\theta}=-(\mu_3-\mu_2)$9

This gives a ladder of vortex positions set by the rotation frequency $\hbar=1$0 and the inter-droplet spacing $\hbar=1$1. Because vortices enter only within the cloud radius $\hbar=1$2, threshold frequencies follow as

$\hbar=1$3

For the quoted parameters, the first vortex enters around $\hbar=1$4 Hz.

Along a transport path between the central droplet and a neighboring ring droplet, the vortex coordinate depends on the Josephson phase difference itself: $\hbar=1$5 Differentiation yields

$\hbar=1$6

This is the explicit current–phase–position form of the dipolar Josephson relation: the local phase difference is mapped directly onto the vortex coordinate, and its time derivative onto the vortex velocity.

In the Josephson regime, bounded oscillations of $\hbar=1$7 imply oscillatory vortex motion near vertices. A minimum phase excursion is required for a vortex to reach a vertex,

$\hbar=1$8

The quoted values are $\hbar=1$9 at $$I(\phi) \simeq -\frac{JN}{\hbar}\sin\phi + \frac{PN^2}{\hbar}\sin 2\phi,$$0 Hz and $$I(\phi) \simeq -\frac{JN}{\hbar}\sin\phi + \frac{PN^2}{\hbar}\sin 2\phi,$$1 at $$I(\phi) \simeq -\frac{JN}{\hbar}\sin\phi + \frac{PN^2}{\hbar}\sin 2\phi,$$2 Hz. In the self-trapping regime, by contrast, the phase winds by $$I(\phi) \simeq -\frac{JN}{\hbar}\sin\phi + \frac{PN^2}{\hbar}\sin 2\phi,$$3 each self-trapping period $$I(\phi) \simeq -\frac{JN}{\hbar}\sin\phi + \frac{PN^2}{\hbar}\sin 2\phi,$$4, giving

$$I(\phi) \simeq -\frac{JN}{\hbar}\sin\phi + \frac{PN^2}{\hbar}\sin 2\phi,$$5

so the vortex moves at nearly constant speed along each lateral path. The traversal time for one lateral path is

$$I(\phi) \simeq -\frac{JN}{\hbar}\sin\phi + \frac{PN^2}{\hbar}\sin 2\phi,$$6

With $$I(\phi) \simeq -\frac{JN}{\hbar}\sin\phi + \frac{PN^2}{\hbar}\sin 2\phi,$$7 ms, $$I(\phi) \simeq -\frac{JN}{\hbar}\sin\phi + \frac{PN^2}{\hbar}\sin 2\phi,$$8, $$I(\phi) \simeq -\frac{JN}{\hbar}\sin\phi + \frac{PN^2}{\hbar}\sin 2\phi,$$9, and $\rho_s^{\alpha\beta}$0 Hz, this gives $\rho_s^{\alpha\beta}$1 ms, in excellent agreement with the eGPE simulations.

5. Multi-droplet coupling, topology, and phase slips

A central result is that a two-droplet description is insufficient near the vertices of the hexagonal low-density regions. There the vortex is influenced simultaneously by three droplets: the central droplet and two neighboring ring droplets. Near a vertex the condition for a vortex is therefore

$\rho_s^{\alpha\beta}$2

which must be solved with three localized modes rather than two (Alaña et al., 5 Jun 2026).

This three-droplet structure is essential for understanding phase slips. In the self-trapping regime the running phase implies repeated $\rho_s^{\alpha\beta}$3 windings. Whenever the phase difference reaches $\rho_s^{\alpha\beta}$4, the superflow across the junction must reverse. Microscopically, the reversal occurs through a phase slip. In the rotating dipolar supersolid these phase slips are realized by vortex and vortex–antivortex dynamics in the low-density valleys. If a nucleated vortex remains too far from a vertex to provide the necessary slip, the system creates a vortex–antivortex pair at that vertex; the vortex and antivortex then propagate along adjacent lateral paths and annihilate with counterparts arriving from neighboring vertices.

The simulations show these processes as density nodes, $\rho_s^{\alpha\beta}$5-phase jumps in local phase maps, and alternating signs of the droplet phase differences around hexagons. The three-droplet approximation, using Gaussian mode shapes and the actual droplet populations and phases extracted from the full eGPE, reproduces the vortex trajectories around vertices, including oscillations and the positions of vortex–antivortex creation and annihilation. By contrast, the simpler two-droplet formulas are accurate only well inside a valley and away from vertices.

The comparison with eGPE is detailed. For $\rho_s^{\alpha\beta}$6 Hz and $\rho_s^{\alpha\beta}$7, the nucleation position $\rho_s^{\alpha\beta}$8 follows the analytic expression very closely during the ramp. At $\rho_s^{\alpha\beta}$9 Hz, Josephson oscillations reach the vertex threshold and the eGPE trajectories match the three-droplet predictions almost point-by-point. In self-trapping scenarios with $N\simeq 1.1\times10^5$0 or $N\simeq 1.1\times10^5$1, the phase winds monotonically, the imbalance never crosses equilibrium, and vortices travel in one direction around the central droplet with velocities and traversal times consistent with the analytic expressions.

6. Related formulations, control parameters, and broader significance

In the broader dipolar-gas literature, the dipolar Josephson relation appears in several technically distinct forms. In self-induced supersolid junction arrays, the central statement is that a supersolid with interaction-generated weak links still exhibits the standard bosonic Josephson structure, including sinusoidal current–phase behavior and macroscopic quantum self-trapping; the novelty lies in the fact that the junctions are self-organized and the effective parameters are set by dipolar interactions, LHY corrections, superfluid fraction, and trap inhomogeneity (Donelli et al., 28 Jan 2025).

In dipolar atomic Josephson junctions with pair tunneling, the departure from the standard relation is more explicit. The effective energy–phase relation contains both $N\simeq 1.1\times10^5$2 and $N\simeq 1.1\times10^5$3 terms, and the current–phase relation contains both $N\simeq 1.1\times10^5$4 and $N\simeq 1.1\times10^5$5 harmonics. This produces modified quantum phase transitions, altered self-trapping thresholds, and phase-imbalanced oscillations around $N\simeq 1.1\times10^5$6 (Vianello et al., 23 Feb 2026).

External control can also be engineered. In a NaCs-molecule proposal with microwave-coupled internal states, a DC electric field enters through a Stark shift of the dipolar component and thereby renormalizes the effective one-component Josephson problem. In the contact-dominated regime the Josephson frequency decreases monotonically with field, from about $N\simeq 1.1\times10^5$7 Hz at zero field to about $N\simeq 1.1\times10^5$8 Hz at $N\simeq 1.1\times10^5$9. In the dipolar-dominated regime, by contrast, the frequency increases with field until saturation as the population is transferred into the dipolar component (Galvez-Poblete et al., 12 Aug 2025).

At the many-body formal level, the generalized Josephson relation for multi-component bosons replaces the scalar superfluid density by a tensor $^{162}\mathrm{Dy}$0 in the space of conserved charges, expressed through the inverse Green’s function matrix. A plausible implication is that dipolar or spin-like internal degrees of freedom can be incorporated into a unified Josephson framework whenever the relevant conserved generators are identified (Zhang, 2017).

The rotating-supersolid formulation adds a further layer of significance. It makes the topological content of Josephson dynamics directly visible: local phase differences do not merely encode current flow, but also the nucleation, motion, creation, and annihilation of vortices that mediate phase slips. Since droplet populations and phases can be extracted experimentally via in situ imaging and interference, and vortices are directly visible in time-of-flight and in situ images, the dipolar Josephson–vortex correspondence provides concrete experimental predictions for rotating dipolar supersolids (Alaña et al., 5 Jun 2026).