Spin-flip lifetimes in superconducting atom chips: Bardeen-Cooper-Schrieffer versus Eliashberg theory

We investigate theoretically the magnetic spin-flip transitions of neutral atoms trapped near a superconducting slab. Our calculations are based on a quantum-theoretical treatment of electromagnetic radiation near dielectric and metallic bodies. Specific results are given for rubidium atoms near a niobium superconductor. At the low frequencies typical of atomic transitions, we find that BCS theory greatly overestimates coherence effects, which are much less pronounced when quasiparticle lifetime effects are included through Eliashberg theory. At $4.2\mathrm{K}$, the typical atomic spin lifetime is found to be larger than $1000\mathrm{s}$, even for atom-superconductor distances of one $1\mathrm{\mu }\mathrm{m}$. This constitutes a large enhancement in comparison with normal metals.

Figure 1

(Color online) Schematic geometrical setup. A plane metallic or superconducting slab lies parallel to the $(x,y)$ plane. The atom with magnetic moment $\mathit{\mu }$ indicted by the arrow is located in vacuum at a distance $z$ from the surface. The atom suffers spontaneous or thermally stimulated magnetic spin-flip transitions, as indicated by ${\mathit{G}}_0$ and $\mathit{G}$, thereby becoming more weakly trapped and eventually lost. Johnson current noise $⟨\mathbf{jj}⟩$ within the penetration depth $\lambda$ contributes to magnetic-field fluctuations at the position of the atom.

Figure 2

(Color online) Temperature dependence of (a) real part ${\sigma }^{\prime }\left({\omega }_A\right)$ and (b) imaginary part ${\sigma }^{″}\left({\omega }_A\right)$ of the optical conductivity, normalized to the normal-state conductivity ${\sigma }_0$. ${\omega }_A=2\pi \times 500\mathrm{kHz}$ is the atomic spin-flip frequency, and $T_c=9.2\mathrm{K}$ is the superconductor transition temperature. The different lines correspond to the results for the two-fluid model (dashed line), using ${\delta }_0=16\mathrm{\mu }\mathrm{m}$ and ${\lambda }_L\left(0\right)=35\mathrm{nm}$, BCS theory (solid line), and Eliashberg theory (symbols) for three elastic impurity scattering rates $\gamma$.

Figure 3

(Color online) ${\sigma }^{\prime }\left(\omega \right)$ at $4.2\mathrm{K}$ as a function of frequency for three elastic scattering rates $\gamma$, as computed within the framework of Eliashberg theory. The peak at zero frequency is attributed to the condensate, and the peak at $1\mathrm{THz}$ to the breaking of Cooper pairs. In the inset we show that the condensate peak saturates at low frequencies.

Figure 4

(Color online) Spin-flip lifetime ${\tau }_A$ of a trapped atom near a superconducting slab as a function of (a) temperature (for $10\mathrm{\mu }\mathrm{m}$ atom-surface distance) and (b) distance (at $4.2\mathrm{K}$). The smallest distance in (b) is $1\mathrm{\mu }\mathrm{m}$ (with ${\tau }_A\approx 5000\mathrm{s}$ for $\gamma =1\mathrm{meV}∕\hslash$). For the calculation of ${\tau }_A$ we use Eq. (13) and the optical conductivity computed within Eliashberg theory for different elastic scattering rates $\gamma$. The atomic transition frequency is fixed at $500\mathrm{kHz}$ throughout. The dashed lines correspond to calculations performed with the two-fluid model and the parameters given in Ref. 25 (London length ${\lambda }_L=35\mathrm{nm}$).