Arrays of dark optical traps on a toroidal surface

We have theoretically constructed light fields that can generate optical potentials representing rectangular arrays of dark optical traps on the surface of a torus. The arrays are not diffraction limited, and their period can in principle be deeply subwavelength. We discuss the challenges anticipated in realizing such potentials, which will be of great interest for quantum simulations.

Figure 1

Schematic of the construction of a regular array of dark spots on a torus. The construction is based on fields that contain vortex lines (blue lines) spiralling on the torus, performing $m/n$ full turns of the poloidal angle $\theta$ during each turn of the toroidal angle $\phi$. The geometry of such a vortex line with $m=3$ and $n=2$ is shown schematically in 3D (a) and in a $(\theta ,\phi )$ plot of the torus surface (b). The poloidal angle $\theta$ and the toroidal angle $\phi$ themselves are shown in (c), together with the major and minor radii of the torus, $R^{*}$ and $r^{*}$. Incoherent superposition of two fields containing different vortex lines spiralling on the same torus results in arrays of dark spots on the torus, specifically, at the positions where the vortex lines in the two fields intersect. In the 3D plot shown in (d), the positions of the intersections of vortex lines in two fields, the first with parameters $m_1=3$, $n_1=2$, the second with $m_2=-3$, $n_2=2$, are shown as gray spheres, and in the $(\theta ,\phi )$ plot shown in (e) they are marked as black dots. Throughout, the vortex lines of the first field are shown as solid lines, those of the second field as dashed lines. The graphs were created with the help of the mathematica notebook topologyPlots.nb, which is available in the Supplemental Material [11].

Figure 2

Example of a scalar vortex line (black) spiralling on a distorted torus. The plots show the saturated intensity distribution on the torus, in 3D (a) and in the $(\theta ,\phi )$ coordinate system (b). The field was calculated following Ref. [12] for $m=3$ and $n=2$ and a perturbation amplitude $\varepsilon =0.05$. Black represents intensity 0, white represents intensities $\ge 0.001$. The graphs were created using the mathematica notebook geometryPlots.nb, which is available in the Supplemental Material [11]. The beam coefficients, and the way they were calculated, is outlined in Appendix pp1.

Figure 3

Arrays of dark spots resulting from different parameter choices. (a), (b) Face-centered rectangular array, shown for the case $m_1=5$, $n_1=3$, $m_2=-5$, $n_2=3$; (c, d) skew array ($m_1=5$, $n_1=3$, $m_2=-4$, $n_2=2$); (e), (f) rectangular array ($m_1=5$, $n_1=3$, $m_2=0$, $n_2=3$). The graphs were created with the help of the mathematica notebook topologyPlots.nb, which is available in the Supplemental Material [11].

Figure 4

Contour plots of the intensity in the coherent superposition of ${\mathrm{\Psi }}_{3,2}$ and ${\mathrm{\Psi }}_{-3,2}$ (a) and in incoherent superpositions of the fields ${\mathrm{\Psi }}_{3,2}$ and ${\mathrm{\Psi }}_{-3,2}$ (b) and ${\mathrm{\Psi }}_{5,3}$ and ${\mathrm{\Psi }}_{-5,3}$ (c). The plots show contours with intensity 0.001 in the arbitrary units used in all simulations in this paper. (a) In the coherent superposition of the fields ${\mathrm{\Psi }}_{3,2}$ and ${\mathrm{\Psi }}_{-3,2}$, the dark regions in the beam do not form the desired structure. (b), (c) In the incoherent superpositions, the intensity zeros on the torus are surrounded by approximately ellipsoidal low-intensity surfaces that can clearly be seen to form the desired regular array on a torus. Additional intensity zeros formed by intersections of the threading vortex lines can be seen around the torus axis. The size of the perturbation was chosen to be $\varepsilon =0.05$ in (a) and (b), and $\varepsilon =0.025$ in (c). The plots were created using the mathematica notebook geometryPlots.nb, available in the Supplemental Material [11].

Figure 5

Intensity on the distorted torus in incoherent superpositions of the fields ${\mathrm{\Psi }}_{3,2}$ and ${\mathrm{\Psi }}_{-3,2}$ (a), (b) and ${\mathrm{\Psi }}_{5,3}$ and ${\mathrm{\Psi }}_{-5,3}$ (c), (d). The plots on the left (a), (c) show the distorted torus in 3D, those on the right (b), (d) show the $(\theta ,\phi )$ plane. The dark spots coincide with the positions where vortex lines from the two fields intersect. Figures 4 and 4 show contour plots for the same superpositions, but note that the viewpoints of those plots subtly differ from those used in the 3D plots (a) and (c), resulting in the trap arrays appearing shifted. The intensity range plotted is from 0 to 0.001 in the arbitrary units used throughout this paper. The basic plots were created with the mathematica notebook geometryPlots.nb, which is available in the Supplemental Material online [11].

Figure 6

Intensity contour of the field $|{\vec{\mathrm{\Psi }}}_{5,2}{|}^2+{\left|{\vec{\mathrm{\Psi }}}_{-5,2}\right|}^2$. The field contains intensity minima in the desired configuration. The plots show contours with intensity $5\times {10}^{-4}$. In comparison, the maximum intensity in the beam is approximately 17 (see Fig. 11). The coefficients $a_l$, $b_l$ of the unperturbed beams are listed in Appendix pp1, and the perturbation amplitude is $\varepsilon =2\times {10}^{-2}$.

Figure 7

Ground-state trapping angular frequencies in the toroidal and poloidal directions in units of the angular recoil frequency against the light-field intensity.

Figure 8

Hopping rates in the poloidal direction $J_{\theta }$ and in the toroidal direction $J_{\phi }$, and the on-site collision rate $C$ plotted against the intensity. The graphs were calculated for trapped ${}^{85}\mathrm{Rb}$ atoms and a detuning $\mathrm{\Delta }=100\mathrm{\Gamma }$.

Figure 9

Ratio of ground-state energy $U_g$ and the maximum potential well optical-dipole potential energy $U_b$ plotted against the intensity.

Figure 10

Same as Fig. 6, but with each vector Bessel beam replaced by a numerical approximation of a corresponding vector Bessel-Gauss beam (see Appendix pp3). The intensity minima still possess the desired configuration. The surface shown is the contour with intensity $5\times {10}^{-4}$; the maximum intensity in the beam is $\sim 500$.

Figure 11

Radial intensity distribution of the incoherent superposition of ${\vec{\mathrm{\Psi }}}_{5,2}$ and ${\vec{\mathrm{\Psi }}}_{-5,2}$ along the line $z=0$ and $\phi =0$. The intensity is shown in the same arbitrary units used in Fig. 6.

Figure 12

Optical traps (black dots) whose positions differ only in their poloidal angle, shown in the radial plane in which they are located. The dotted circle is where the torus on which the vortex lines spiral intersects the plane; the $z$ axis is the torus's axis of rotational symmetry, $R^{*}$ and $r^{*}$, are the major and minor radius of the torus, respectively; and $s$ is the separation between neighboring traps.

Figure 13

Fourier-space structure of Bessel-Gauss beams and their approximation as a sum of Bessel beams. The plots in (a, b) show grayscale representations (black = 0, white = 1) of the amplitude distribution as a function of the transverse wave numbers $k_x$ and $k_y$ for a Bessel-Gauss beam (a) and for a superposition of ten Bessel beams used to approximate this Bessel-Gauss beam. (c) Radial cross sections through these distributions. The amplitude of the Bessel-Gauss beam is shown as a solid line, and the amplitudes of the Bessel beams are indicated by black disks. Both are plotted as a function of $k_r=\sqrt{{k_x^2+k_y^2}^{\phantom{0}}}$.