Inductively guided circuits for ultracold dressed atoms

Recent progress in optics, atomic physics and material science has paved the way to study quantum effects in ultracold atomic alkali gases confined to non-trivial geometries. Multiply connected traps for cold atoms can be prepared by combining inhomogeneous distributions of DC and radio-frequency electromagnetic fields with optical fields that require complex systems for frequency control and stabilization. Here we propose a flexible and robust scheme that creates closed quasi-one-dimensional guides for ultracold atoms through the ‘dressing’ of hyperfine sublevels of the atomic ground state, where the dressing field is spatially modulated by inductive effects over a micro-engineered conducting loop. Remarkably, for commonly used atomic species (for example, 7Li and 87Rb), the guide operation relies entirely on controlling static and low-frequency fields in the regimes of radio-frequency and microwave frequencies. This novel trapping scheme can be implemented with current technology for micro-fabrication and electronic control.


DISTRIBUTION OF INDUCED CURRENTS IN CONDUCTORS OF FINITE CROSS-SECTION
We present details of a numerical procedure to determine the distribution of induced current within conducting rings with uniform cross-section. We focus on the case of magnetic fields oscillating at low frequencies such that the associated wavelength (λ = c/ν) is much larger than the size of the loop, and apply a quasi-static approximation to the Maxwell equations for the electromagnetic field [1,2]. First, we present results for the current distribution in metallic rings, taking parameters corresponding to gold, obtained using the open-source software FEMM.
Second, we detail a procedure to evaluate the current distribution in superconducting rings in the limit of the London equation. In this last case, we consider parameters corresponding to superconducting Niobium and adapt the methods in references [3] and [4] to the present problem.
The current distribution is evaluated using the coordinate systems in Supplementary Fig. 4.
Exploiting the circular symmetry of the ring, the current density is calculated at points r defined by a polar-coordinate system with origin at the ring cross-section centre, as shown in Supplementary Fig. 4(a). The Maxwell equations associated to time-dependent magnetic fields and its sources are coupled to a constitutive relation between the fields and the current in the ring. In our case, such constitutive equations correspond to the Ohm law for metallic conductors, and the London equation for superconducting materials. The resulting set of coupled equations are conveniently expressed in the cylindrical coordinate system with origin at the centre of the ring, as shown in Supplementary Fig. 4(b).

Metallic rings
A time-variation of magnetic flux across a metallic ring induces an electric current whose distribution depends on the properties and geometry of the ring and the frequency of the field.
For a harmonic variation of the magnetic field with angular frequency , the quasi-static Maxwell equation for the vector potential is: where σ is the ring conductivity [2].
We use the open-source software package FEMM [5] to solve Eq. (1), for gold rings of various sizes. We setup an external magnetic field of amplitude B AC = 2 G, oscillating at a frequency  = 2π × 6.8 GHz along the z direction. Supplementary Fig. 1  FEMM also provides the magnetic field distribution, which we use to evaluate the trapping frequencies shown in the main text.
We consider superconducting rings of uniform cross-section, described by the London theory [1], where the supercurrent and the potential vector are related by: with m and e the electron mass and charge, respectively, and n s the density of superconducting electrons. By using this expression, we neglect non-local effects on the current distribution and restrict our calculations to field frequencies smaller than the superconducting gap (typically of the order of a few ∼ 100GHz) [1].
Under quasi-static conditions, the total vector potential corresponding to an external field and a current distribution is: where the integral is taken over the volume of the current-carrying conductors. AC A is the vector potential associated with the applied field which, in the case of a uniform magnetic field along the z axis is (4) where we have used an elementary property of the Dirac delta distribution [4]. It is convenient to separate the integral over the volume of the conductor into an integral over the conductor cross-section and one over its circumference (see Fig. 1(b)): This equation implies that the magnetic flux across the loop C, created by the current distribution, compensates exactly the magnetic flux imposed by the external field. This corresponds to the well-known Meissner effect in superconductors, and implies that the induced current adjusts instantaneously to cancel the total flux of magnetic field across any loop defined within the superconducting ring.
To obtain a solution of Eq. (8), we discretize the conductor cross-section in elements of area ∆A i centred at positions r i , as schematically shown in Supplementary Methods Fig. 1(b). In discrete from, Eq. (8) becomes: where I j = J j ∆A j is the current flowing across the j-th are element ∆A j , and: is the mutual inductance between the i-th and j-th loops, which for i ≠ j becomes: This integral is evaluated following [4].
Following [3], the self-inductance L i,i can be approximated by: We consider a superconducting ring of size a = 100 m, and circular cross-section in the range 1-20 m. For Niobium, the London penetration depth is λ ≈ 100 nm [3]. Supplementary   Fig. 2 presents the current distribution in rings with s = 2.5 m and s = 7 m (a = 100 m), for an applied field B AC = 2 G. In comparison to the case of metallic conductors shown in Supplementary Fig. 1, the current distribution concentrates more strongly near the surface of the conductor. Nevertheless, the impact on the trapping properties of the ring trap discussed in the main text is similar in both cases (see main text - Figure 5).