The magnetostatic wormhole requires constructing a tunnel for magnetic fields acting as if was outside the usual 3D space. The first property to be satisfied is to magnetically decouple a given volume from the surrounding 3D space. The volume enclosed by a superconducting shell has this property19; we consider here a spherical superconducting shell (Fig. 1b, center). A second property is that the whole resulting wormhole cannot be magnetically detectable from its exterior. The superconducting spherical shell would distort an applied field and thus be detectable. In Ref. 21, a cylindrical magnetic cloak was made consisting of a superconducting layer surrounded by a magnetic layer; similar bilayer structures have been recently applied to thermal and diffusive cloaks15,26,27. However, the magnetic bilayer was actually 2D - a long cylinder- and was shown to cloak only uniform applied magnetic fields. Realizing an actual magnetic wormhole requires it to be fully 3D and undetectable also for non-uniform fields. To solve this challenge we demonstrate (see Supplementary Information) that a spherical bilayer composed of a superconducting layer of outer radius R 2 surrounded by a ferromagnetic layer of outer radius R 3 exactly cloaks a uniform applied magnetic field if the permeability of the ferromagnetic layer is

Figure 1 (a) The field of a magnetic source (right) is appearing as an isolated magnetic monopole when passing through the magnetostatic wormhole; the whole spherical device is magnetically undetectable. (b) The wormhole is composed of (from left to right) an outer spherical ferromagnetic metasurface, a spherical superconducting layer and an inner spirally wound ferromagnetic sheet. Full size image

where μ 0 is the vacuum permeability. Interestingly, when the applied field is not uniform such a bilayer also cancels the dipolar term of its magnetic response. Furthermore, all the terms of the magnetic response higher than the dipole can be reduced by making R 3 tend to R 2 , that is, by thinning the ferromagnetic layer. This means a very thin ferromagnetic layer with permeability μ 2 surrounding the superconducting shell effectively cancels the global magnetic response. Since the superconducting and ferromagnetic layers will also be magnetically decoupling the region enclosed by them, then the overall effect is changing the topology of space3, as if the interior region had been (magnetically) removed out of the existing 3D space (Fig. 1).

A final requirement for the wormhole is that magnetic fields have to propagate through its interior. Magnetic fields naturally decay with distance, typically as a magnetic dipole (field at a distance d is decreasing as 1/d3). In Ref. 25 a general strategy for transferring magnetic field to long distances was developed. There a magnetic hose consisting of a ferromagnetic tube surrounded by a superconductor was employed. Here we use an alternative option, namely a thin ferromagnetic sheet wound into a spiral (Fig. 1b). This scheme was also theoretically shown to yield a good field transfer at a distance25; here it is experimentally confirmed.

The parts composing the wormhole are shown in Fig. 1b (see also Supplementary Information). The core of the device is the magnetic hose made of a foil of high-permeability mu-metal, folded into a spiral. This is surrounded by the spherical superconducting layer, made of high-temperature superconducting strips glued to a spherical former to provide a tessellation of the sphere. On top of it, there is an outer ferromagnetic layer, ideally made of a homogenous material with permeability μ 2 given by Eq. (1) and very small thickness. Such effective permeability is achieved by a strategy reminiscent of the metasurfaces used for light manipulation28. An array of high-permeability mu-metal plates is specially arranged as to provide the required magnetic response, following an optimization process based on a 3D-numerical simulation of the whole device (see Supplementary Information).

The experimental setup to demonstrate the wormhole properties (see Fig. 2 and Supplementary Information) uses a pair of Helmholtz coils of radius R separated a distance R. They provide a uniform magnetic field in the central zone. There sits the wormhole, oriented with its two ends perpendicular to the applied field. A small coil at one end of the wormhole is fed with a dc current to supply the field to be transferred through it. Two Hall probes are used for the measurements. Probe T, placed at the opposite exit of the wormhole measures the transferred magnetic field. Probe C scans the magnetic field in lines (green lines in Fig. 2b) close to the surface of the wormhole, measuring the distortion of the applied field. Three types of measurements are performed by probe C: (i) only the superconducting layer, without the ferromagnetic outer layer (this measurement requires submerging the superconductor into liquid nitrogen); (ii) only the ferromagnetic layer (actually measuring the whole device at room temperature, with the superconductor deactivated); and (iii) the full structure, at liquid nitrogen temperature, so that both superconducting and ferromagnetic layers are activated. Ideally, one should observe a clear field distortion of the applied magnetic field in cases (i) and (ii) and no distortion for the fully working wormhole of case (iii). Accurate 3D simulations by finite elements of the whole device, considering details like the ferromagnetic metasurface, validate the design (see Supplementary Information).

Figure 2 (a) 3D scheme of the experimental setup. (b) A detailed description of the central plane, including the lines at which probes T (red) and C (green) measure the transferred and cloaked (or distorted) fields, respectively. The uniform applied field is created by the two Helmholtz coils. (d) In this case, the z-component of magnetic field is measured by probe C as a function x and for different distances, z, to the wormhole. (e) Measurements at z = 5 are shown in detail. (c) Analogous measurements are done for a non-uniform applied field, created by exciting only one of the coils and results are shown in (f). Red lines are for only the ferromagnetic layer, green for only the superconducting one and blue for the complete device. Black lines represent the measured applied field for each case. Full size image

We next present the experimental results. Although measured simultaneously, we discuss the transmission and cloaking results separately for clarity. Results of the field distortion for the three cases (i)–(iii) at a distance of 5 mm are shown in Fig. 2e. The complete wormhole of case (iii) shows an excellent cloaking behavior, whereas the superconducting and ferromagnetic parts separately yield clear field distortions. Scans performed at different distances show very little distortion (see Fig. 2d and Supplementary Information). Only at a close distance the effect of the non-uniform ferromagnetic structure can be discerned. We also measure the effect of applying a non-uniform applied field, created by feeding current in only one of the Helmholtz coils (Fig. 2c,f). Even in this case, a very good cancellation of the field distortion is achieved for the full wormhole and not for its components separately. In this way we confirm the magnetic undetectability of the wormhole.

Experiments show a clear transmission of magnetic field through the wormhole as well (Fig. 3). The dipolar-like magnetic field created by the feeding coil at one end of the wormhole is transformed at the opposite end into a monopolar-like field. Actually, the spacial dependence of the exiting field tends to ∼1/d1.5, since close to the opening the field resembles that of a disk of monopoles rather than a single one (∼1/d2) . These monopolar magnetic fields are an alternative to those obtained by exotic spin ices29 and other systems30. Our magnetic wormhole thus creates an illusion of a magnetic field coming out of nowhere.