Asymmetrical two-dimensional magnetic lattices for ultracold atoms

A simple method for implementing an asymmetrical two-dimensional magnetic lattice is proposed. The asymmetrical two-dimensional magnetic lattice is created by periodically distributing nonzero magnetic minima across the surface of a magnetic thin film, where the magnetic patterns are formed by milling $n\times n$ square holes on the surface of the film. The method is proposed for trapping and confining quantum degenerate gases, such as Bose-Einstein condensates and ultracold Fermi gases, prepared in low-magnetic-field-seeking states. Analytical expressions and numerical simulation results of the magnetic local minima are shown where we analyze the effect of changing the magnetic lattice parameters, such as the separation of the holes, the hole size, and external bias magnetic fields, to maintain and locate the nonzero local minima at a suitable distance above the film surface to avoid the effect of Majorana spin flips and the Casimir-Polder potential.

Figure 1

(a) Schematic representation of a magnetic lattice. (b) The magnetic lattice parameters are specified by the hole size ${\alpha }_h\times {\alpha }_h$, the separation between the holes ${\alpha }_s$, and the magnetic film thickness $\tau$. (c) Magnetic density plot of the simulated finite magnetic lattice sites in the $z-x$ plane across the center of the lattice. The traps are located at an effective $z$ distance, $d_{\min }$, above the holes. (d,e) Contour plots of the distributed lattice sites across the $x-y$ plane without (d) and (e) with application of bias fields of $B_{x\hspace{0.5em}\mathrm{bias}}=B_{y\hspace{0.5em}\mathrm{bias}}=10$ G. (f) 3D plot of the magnetic field of the distributed sites across the $x-y$ plane at $d_{\min }$. The field is displayed from the center sites to the edge sites. Simulation input parameters: ${\alpha }_s={\alpha }_h=1$ $\mu$m, $M_z=3$ kG, and $\tau =2$ $\mu$m.

Figure 2

(a)–(d) Contour plots of different magnetic-field gradients for different sizes of a single magnetic potential well simulated using different values of hole size ${\alpha }_h$. The simulation is carried out using the parameters $M_z=2.8$ kG, $\tau =2$ $\mu$m, and with external bias fields $B_{x\hspace{0.5em}\mathrm{bias}}=B_{y\hspace{0.5em}\mathrm{bias}}=1.5$ G. ${\alpha }_h={\alpha }_s=$ (a) 1 $\mu$m, (b) 3 $\mu$m, (c) 5 $\mu$m, and (d) 7 $\mu$m.

Figure 3

(a) Biased and unbiased magnetic-field minimum $B_{\min }^z$ along the $z$ axis with $B_{x\hspace{0.5em}\mathrm{bias}}=3$ G. (b) Magnetic-field minima along the $z$ axis at the center and edge lattice sites. (c) Magnetic-field distribution along the $x$ axis for an $11\times 11$ 2D magnetic lattice with ${\alpha }_s={\alpha }_h=3.5$ $\mu$m. (d) Two adjacent magnetic quantum wells at the lattice edge along the $y$ axis separated by the magnetic barrier $\Delta B^y$ and having different values of nonzero local minima through a tilted (magnetic) potential. Simulation input parameters: $M_z=2.8$ kG and $\tau =2$ $\mu$m with no external bias fields applied in (b)–(d).

Figure 4

(a) Effect of changing the size of the holes, ${\alpha }_h$, on the location of the magnetic-field local minima along the $z$ axis at $d_{\min }$ above the holes of the thin film and (b) effect of changing the separation of the holes, ${\alpha }_s$, across the $x-y$ plane. (c) Simulation result of varying the barrier heights $\Delta B_y$ by applying a negative external $B_{z\hspace{0.5em}\mathrm{bias}}$ magnetic field and (d) $B_{x\hspace{0.5em}\mathrm{bias}}$, $B_{y\hspace{0.5em}\mathrm{bias}}$, and $B_{z\hspace{0.5em}\mathrm{bias}}$ effects on the gradient of the magnetic sites near the local minima along the $z$ axis. A film thickness of $\tau =2$ $\mu$m is used in the simulation results shown.

Figure 5

(a)–(d) Effect of applying a $B_{z\hspace{0.5em}\mathrm{bias}}$ field along the negative direction of the $z$ axis on the magnetic barriers $\Delta B_y$ and $\Delta B_z$ and the nonzero magnetic local minima. Enhanced asymmetrical effect and $d_{\min }$ values are realized for different values of $B_{z\hspace{0.5em}\mathrm{bias}}$ field. Simulation input parameters: $n=11$ sites, ${\alpha }_s={\alpha }_h=3.5$ $\mu$m, $M_z=2.8$ kG, and $\tau =2$ $\mu$m.

Figure 6

Applying external magnetic bias fields, $B_{x\hspace{0.5em}\mathrm{bias}}=B_{y\hspace{0.5em}\mathrm{bias}}=5$ G, produces symmetrically distributed magnetic lattices sites with (almost) no offset of each lattice site. Simulation input parameters: $n=9$, ${\alpha }_s={\alpha }_h=3.5$ $\mu$m, $M_z=2.80$ kG, and $\tau =2$ $\mu$m.

Figure 7

(a) Scanning-electron-microscope and (b) atomic-force microscope images of a fabricated 2D magnetic structure. (c) Magnetic field measured using the magnetic-force microscope and (d) simulation result of the in situ biased 2D magnetic lattice. Simulation and experimental input parameters: ${\alpha }_s={\alpha }_h=10$ $\mu$m, $M_z\approx 2.80$ kG, and $\tau =1$ $\mu$m.