Bose-Einstein condensate sub-wavelength confinement via superoscillations

TL;DR

This work tackles sub-wavelength confinement of a Bose-Einstein condensate in optical lattices by exploiting superoscillations in a tri-chromatic, off-resonant optical potential. The authors formulate a quadratic, constrained optimization that minimizes the potential within a localized superoscillatory window, solving a generalized eigenvalue problem to obtain field coefficients that produce sub-diffraction features. Applied to a $^{87}$Rb BEC with $N=2000$, the approach yields a central region containing nine density peaks within about $1.5\,\mu$m, with a central peak spacing of $272.3$ nm (≈ $0.776\,T_{ m opt}$) and a ground-state energy of $\approx 303.8$ nK; the full condensate remains confined within the superoscillating region, despite the presence of larger amplitudes outside it. A bi-chromatic variant retains superoscillations with reduced experimental complexity. Overall, the results establish superoscillations as a viable route to sub-wavelength BEC confinement, with trade-offs in outside-field requirements that future work may mitigate for practical applications in quantum simulations and high-density optical lattices.

Abstract

Optical lattices are essential tools in ultra-cold atomic physics. Here we demonstrate theoretically that sub-wavelength confinement can be achieved in these lattices through superoscillations. This generic wave phenomenon occurs when a local region of the wave oscillates faster than any of the frequencies in its global Fourier decomposition. To illustrate how sub-wavelength confinement can be achieved via superoscillations, we consider a one-dimensional tri-chromatic optical potential confining a spinless Bose-Einstein Condensate of $^{87}$Rb atoms. By numerical optimization of the relative phases and amplitudes of the optical trap's frequency components, it is possible to generate superoscillatory spatial regions. Such regions contain multiple density peaks at sub-wavelength spacing. This work establishes superoscillations as a viable route to sub-wavelength BEC confinement in blue-detuned optical lattices.

Figures (4)

• Figure 1: The six solutions (a - f) to the three-frequency eigenvalue equation shown over the scale of a single period of the shortest wavelength, $\lambda_{min}/2 = 351$ nm, centered about 0. Column (i) demonstrates the potential created by each solution. For illustrative purposes only, we have scaled each potential $\tilde{V}$ such that, within the superoscillating region, the amplitude ranges from 0 to 1. Column (ii) shows the coefficients required to make $\tilde{V}_{\rm scaled}$. Note that the solutions favor exclusively either the sine or cosine amplitudes. We can see in (i) that solutions (a) and (e) basically overlap with the shortest wavelength field $\lambda_{min}=702$ nm, and that (b) and (f) are just a phase shift from (a) and (e). However, in (c) and (d), we see definite evidence of superoscillations both in the potential shape and in the correspondingly large coefficients required to reach the same amplitude as in (a-b) and (e-f).
• Figure 2: A demonstration of the potential created by each individual wavelength field, $V_{\lambda_j}$. We note that the magnitudes of the two far-off-resonance wavelengths are dominant, on the order of $\mu$K, while we must actually move to the nK scale to see the near-resonance wavelength field.
• Figure 3: The bi-chromatic solution given by using only the $E_1$ and $E_2$ fields. We see that the potential is still superoscillating, but with a smoother central well. We also see that the potential does not decrease as fast as with the tri-chromatic solution at the edges of our chosen window, showing that there is some loss due to the exclusion of the third, near-resonance field.
• Figure 4: a) The BEC density within a single superoscillating well. We include two magnified regions: b) The black enclosed region shows the optimal sub-wavelength spacing of BEC peaks. c) The purple enclosed region shows the entire density profile. Note that even though the peak spacings increase as one moves further from the superoscillating region, all nine peaks are contained in a region smaller than 9$T_{\rm opt}$.