Effective many-body parameters for atoms in nonseparable Gaussian optical potentials

We analyze the properties of particles trapped in three-dimensional potentials formed from superimposed Gaussian beams, fully taking into account effects of potential anharmonicity and nonseparability. Although these effects are negligible in more conventional optical lattice experiments, they are essential for emerging ultracold-atom developments. We focus in particular on two potentials utilized in current ultracold-atom experiments: arrays of tightly focused optical tweezers and a one-dimensional optical lattice with transverse Gaussian confinement and highly excited transverse modes. Our main numerical tools are discrete variable representations (DVRs), which combine many favorable features of spectral and grid-based methods, such as the computational advantage of exponential convergence and the convenience of an analytical representation of Hamiltonian matrix elements. Optimizations, such as symmetry adaptations and variational methods built on top of DVR methods, are presented and their convergence properties discussed. We also present a quantitative analysis of the degree of nonseparability of eigenstates, borrowing ideas from the theory of matrix product states, leading to both conceptual and computational gains. Beyond developing numerical methodologies, we present results for construction of optimally localized Wannier functions and tunneling and interaction matrix elements in optical lattices and tweezers relevant for constructing effective models for many-body physics.

Figure 1

Double-well optical tweezer. The double-well optical tweezer potential (4), at $y=0$, with no bias $\left(\mathrm{\Delta }=0\right)$ and spacing $a/w_0=1.23$. Note the different scaling of the $x$ and $z$ coordinates.

Figure 2

Nonseparable optical lattice. The nonseparable optical lattice [Eq. (6)] for $V/V_{\mathrm{const}}=3/22$. A nonzero value of $V_{\mathrm{const}}$ increases the effective transverse $\left(r\right)$ confinement while leaving the axial $\left(z\right)$ motion unchanged, to lowest order. Note the different scalings of the $r$ and $z$ coordinates, as well as the difference in scaling of $z$ coordinates with respect to Fig. 1 due to the additional length scale $a$.

Figure 3

Sinc DVR basis functions. The sinc DVR basis functions $\langle x|{\mathrm{\Delta }}_0\rangle$ (red solid line) and $\langle x|{\mathrm{\Delta }}_1\rangle$ (blue dashed line) are nonzero at their centering grid point and vanish at all other grid points. The black dots on the $x$ axis denote the equally spaced grid points.

Figure 4

Bessel DVR basis functions. The Bessel DVR basis functions $\langle r|{\mathrm{\Delta }}_{3,1}\rangle$ (red solid line) and $\langle r|{\mathrm{\Delta }}_{3,2}\rangle$ (blue dashed line) are nonzero at their centering grid point and vanish at all other grid points. Here, the grid (black points on $x$ axis) is not uniformly spaced.

Figure 5

Convergence of DVR energies with number of DVR grid points and domain size. (a) Convergence in the number of DVR grid points (equivalently, the grid spacing $\mathrm{\Delta }x$) is demonstrated for the first 10 eigenstates of a Gaussian well at fixed $R=3w_0$. The points are differences in energies for neighboring $N$, as described in the main text. Curves from bottom to top are increasing in energy. (b) The analogous plot to (a) at fixed $\mathrm{\Delta }x=0.05w_0$ and varying $R$, the DVR grid domain cutoff. The parameters used are depth $V=96$ kHz, waist $w_0=707$ nm, and the mass of ${}^{87}\mathrm{Rb}$.

Figure 6

Convergence of $s$- and $p$-wave interaction matrix elements in the DVR basis size. The convergence of the one-dimensional analogs of Eqs. (14) and (15) in oscillator units is given by considering the differences with neighboring convergence parameters. These integrals show a similar rate of convergence to the energies in Fig. 5, although the exponential convergence sets in at a larger value of $N$. The trap parameters are the same as those used in Fig. 5.

Figure 7

Convergence of the $m=0$ energies and tunneling amplitudes of the potential (6) with the variational basis size $a_z$. The difference in energies adding another lattice band $\mathrm{\Delta }E\equiv E(a_z+1)-E\left(a_z\right)$ as a function of eigenstate energy are shown for numbers of bands $a_z=2,\cdots ,7$, with lower curves corresponding to larger numbers of bands. The bottom curve is $\mathrm{\Delta }J$ for $a_z=7$, showing that the tunneling amplitudes display similar convergence behavior to the energy. The trap parameters are $w_0=30\mu \mathrm{m},V_{\mathrm{const}}=88E_R$, and $V=12E_R$.

Figure 8

Single-particle Hubbard parameters in an asymmetric double-well optical tweezer. The maximal localization procedure described in the text converts the tunneling doublet whose energy splitting $({\epsilon }_2-{\epsilon }_1)$ shown in magenta into a single-band Hubbard model with a nearly constant tunneling $J$ in red and an effective bias $(E_2-E_1)$ which scales linearly with the applied potential bias.

Figure 9

Tunneling and interaction parameters in a symmetric double-well optical tweezer. The tunneling and $s$-wave interaction energies for the ground $\left(g\right)$ and first $z$ excited $\left(e\right)$ states in a double-well optical tweezer as a function of depth. While the reduction in $U_{ee}$ compared to $U_{gg}$ is an effect of the different spatial extent of the Wannier orbitals in the $g$ and $e$ states that occurs even for separable potentials, the increase of $J_e$ vs $J_g$ is solely a manifestation of nonseparability. Also displayed is $J_g^{1\mathrm{D}}$, the tunneling predicted from a one-dimensional calculation with the potential $V_d\left(x,0,0\right)$, showing significant deviation from the full 3D result.

Figure 10

Nonseparability of states in a symmetric double-well optical tweezer. The von Neumann entropy $S$ of $xy$ and $yz$ bipartitions and the geometric nonseparability $\mathcal{E}$ for the ground state $\left(g\right)$ and the first $z$ excited state $\left(e\right)$. While both states are near separable, the excited state is less separable than the ground state.

Figure 11

Nearest separable states of a double-well optical tweezer and their harmonic approximations. The components of the nearest separable wave function $\tilde{X}(x/w_0)$ (solid red line), $\tilde{Y}(y/w_0)$ (dashed blue line), and $\tilde{Z}(z/z_R)$ (dotted black line) for a symmetric double-well optical tweezer with depth $V=52$ kHz are shown as functions of their respective dimensionless coordinates. The harmonic approximations are also shown for $y$ (blue crosses) and $z$ (black $\times \mathrm{s}$), demonstrating significant anharmonicity in the $z$ degree of freedom.

Figure 12

Tunneling vs energy. The tunneling, in units of the separable lattice tunneling, as a function of Wannier state energy for $V=12E_R$ and $w_0=30\mu \mathrm{m}$. The upper pair of curves corresponds to $V_{\mathrm{const}}=88E_R$, giving a transverse frequency $\hslash \omega \sim 400$ Hz, with the red (green) curves corresponding to azimuthal quantum number $m=0 \left(m=200\right)$. The lower pair of curves has $V_{\mathrm{const}}=388E_R$ ($\hslash \omega \sim 850$ Hz), with blue (magenta) corresponding to $m=0 \left(m=200\right)$. $J$ depends dominantly on the transverse mode energy, as intuitively understood through the lowest-order harmonic expansion of the transverse potential.

Figure 13

Interaction matrix elements in a nonseparable optical lattice The $s$- and $p$-wave interaction integrals (14) and (15), for $m=0$ (top row) and $m=200$ (bottom row). $n$ and $n^{\prime }$ are eigenstate indices, which correlate to transverse mode quantum numbers. $s$-wave ($p$-wave) interactions show a slow decay (growth) with transverse mode energy.

Figure 14

Nonseparability of 1D optical lattice eigenstates. The von Neumann entropy of nonseparability $S$ and the geometric nonseparability $\mathcal{E}$ as a function of Wannier state energy for $V_{\mathrm{const}}=88E_R$. Red and blue (green and magenta) curves correspond to states of azimuthal quantum number $m=0 \left(m=200\right)$. As was the case for the tunneling (Fig. 12), the nonseparability is dominantly a function of transverse mode energy.

Figure 15

Anharmonicity of nonseparable optical lattice states. The overlap of the radial part of the wave function, indexed by $n$, with the approximate harmonic oscillator wave functions, indexed by $n_{\mathrm{ho}}$, shows that the radial wave function is spread across many harmonic oscillator modes in excited states. The values at quasimomentum $q=-0.6\pi /a$ are representative of general quasimomentum away from high-symmetry points.