Emulating Molecular Orbitals and Electronic Dynamics with Ultracold Atoms

In recent years, ultracold atoms in optical lattices have proven their great value as quantum simulators for studying strongly correlated phases and complex phenomena in solid-state systems. Here, we reveal their potential as quantum simulators for molecular physics and propose a technique to image the three-dimensional molecular orbitals with high resolution. The outstanding tunability of ultracold atoms in terms of potential and interaction offer fully adjustable model systems for gaining deep insight into the electronic structure of molecules. We study the orbitals of an artificial benzene molecule and discuss the effect of tunable interactions in its conjugated $\pi$ electron system with special regard to localization and spin order. The dynamical time scales of ultracold atom simulators are on the order of milliseconds, which allows for the time-resolved monitoring of a broad range of dynamical processes. As an example, we compute the hole dynamics in the conjugated $\pi$ system of the artificial benzene molecule.

Popular Summary

Molecules are the constituents of life, and these assemblies of atoms play a dominant role in nature. Even though molecular structure—the positions of atoms within a molecule—is often well known, the molecular orbitals of the electrons have been observed only in a few exceptional cases. Here, we propose a method to simulate molecules using well-controlled atoms at ultracold temperatures. The ultracold atoms mimic the electrons in a molecule, and a laser field (an optical trapping potential) plays the role of the atomic nuclei. Such quantum simulations of molecules make it possible to image molecular orbitals at high resolution.

In the artificial benzene molecules, one can image three-dimensional molecular orbitals in a highly tunable system. High resolution can be obtained by imaging the particles in momentum space after removing the optical trapping potential. The interactions between the ultracold atoms can be tuned via a magnetic field as well, which makes it possible to study the effect of the interactions on the molecular orbitals and the electronic dynamics in detail. A long-standing dream is the direct imaging of the electron dynamics within molecules taking place on femtosecond scales. Since the dynamical time scale of our system is measured in milliseconds, we are able to show that such “artificial molecules” offer deep insights into this dynamical evolution.

Our proposal paves the way for a better understanding of molecular physics and quantum chemistry. We expect that our ideas can also be extended to more complex, nonplanar molecules as well.

Figure 1

Molecular orbitals of artificial benzene. (a) Illustration of a honeycomb optical lattice superposed with a dipole trap. The hexagonal ring and the adjacent sites form a trapping potential for an artificial, benzenelike molecule. (b) Calculated molecular single-particle orbitals of the artificial benzene molecule with low-lying $s$ orbitals (1–6) on the inner “C” ring, hybridized $s{p}^2$ orbitals (7–18, 25–30) including the adjacent “H” atoms, and $p_z$ orbitals (19–24) forming a conjugated $\pi$ system. The orbitals are plotted for the lattice depths $V_0=11E_R$ and $V_{0z}=35E_R$ in units of the recoil energy $E_R$ of the lattice wavelength. The isosurfaces depict the orbital wave functions at $0.4/a\sqrt{b}$ (orange) and $-0.4/a\sqrt{b}$ (green) with the lattice constant $a$ of the honeycomb and $b$ of the orthogonal lattice. (c) Chemical structure of the benzene molecule ${\mathrm{C}}_6{\mathrm{H}}_6$.

Figure 2

Imaging of molecular wave functions via momentum-space mapping. The first row shows the momentum density in two dimensions, the second row the momentum wave function after phase retrieval, the third row the density (the sign of the wave function is indicated), and the fourth row the isosurface of the three-dimensional wave function for $\pm 0.45/a\sqrt{b}$. (a) Theoretical calculation for the lowest orbital ($n=1$) of the artificial benzene molecule. (b) Simulated time-of-flight measurements with 1000 observed momenta in the $x\text{−}y$ plane (randomly distributed) for the orbitals $n=1$ and (c) $n=12$. (d) The momentum density in $y\text{−}z$ direction with 100 random momenta for the lowest $p_z$ orbital ($n=19$), integrated momentum wave function and density in $z$ direction. In combination with the reconstructed wave function in $x\text{−}y$ coordinates [see (b)], this allows one to retrieve the three-dimensional wave function for the lowest $p_z$ orbital [analogous for (b) and (c) using the lowest Wannier orbital in $z$ direction].

Figure 3

Interactions in the conjugated $\pi$ system. (a) Single-particle energy spectrum showing the lowest orbitals ($p_z$ orbitals in red) as a function of the lattice depth $V_0$ (taking the role of the effective nuclei distance $R_0$). See Appendix pp1 for parameters. The energy of the $p_z$ orbitals can be tuned independently by lattice depth $V_{0z}$ (here, $35E_R$). (b) Close-up showing only the $p_z$ orbitals with three bonding and three antibonding orbitals, where the orbitals 2-3 and 4-5 are degenerate. (c) Energy spectrum of the conjugated system incorporating the on-site interaction $U$ for $V_0=6E_R$ (see Appendix pp3). The colored arrows indicate the lowest excitations [labeled in (e)] for vanishing interaction. (d) Logarithmic density plot of the two-dimensional Wannier function for the $p_z$ (or $s$) band. Each contour indicates a density drop by a factor $1/10$. (e) Level scheme depicting orbitals and spin configuration.

Figure 4

Correlated dynamics after particle removal. (a) Quantum revival for the noninteracting system and (b) dynamics in the antiferromagnetically correlated state for $a_s/b=0.02$ and $V_0=6E_R$. The plots display the occupation number of the Wannier state at site $i$ of the ring for spin-up (left) and spin-down (right) as a function of time $t$. The images on the left depict the summed density of both spin states. The three-dimensional isosurface is plotted for $6/a^{2}b$, where the coloring refers to the two-dimensional density.