Spin dynamics of two bosons in an optical lattice site: A role of anharmonicity and anisotropy of the trapping potential

We study the spin dynamics of two magnetic chromium atoms trapped in a single site of a deep optical lattice in a resonant magnetic field. Dipole-dipole interactions couple spin degrees of freedom of two particles to their quantized orbital motion. A trap geometry combined with two-body contact $s$-wave interactions influence a spin dynamics through the energy spectrum of the two-atom system. Anharmonicity and anisotropy of the site results in a “fine” structure of two-body eigenenergies. The structure can be easily resolved by weak magnetic dipole-dipole interactions. As an example we examine the effect of anharmonicity and anisotropy of the binding potential on demagnetization processes. We show that weak dipolar interactions provide a perfect tool for precision spectroscopy of the energy spectrum of the interacting few particle system.

Figure 1

Harmonic trap—dipolar resonances with $\Delta {\mathcal{M}}_z=-1$. Left upper panel: the excitation energy of the two-particle system in the harmonic trap as a function of an external magnetic field. The horizontal line at small $B$ corresponds to the $|\mathrm{ini}\rangle$ state with both atoms fully polarized $|3\rangle |3\rangle$. Only the lowest energy state $|{\mathcal{L}}_1\rangle$, the one which is coupled to the $|\mathrm{ini}\rangle$ state by the $\Delta {\mathcal{M}}_z=-1$ dipole-dipole interaction term is shown. Notice avoided crossing. Right upper panel: a single maximum in the self-energy curve (right upper panel) signifying strong coupling of both involved states at resonant magnetic field. Lower panel: reduced single-particle density matrix component (the diagonal element with magnetization $m=2$) of the state $|{\mathcal{L}}_1\rangle$; left panel—density cut at the $xy$ plane for $z=1/\sqrt{\kappa }$; right panel—density at the $xz$ plane ($y=0$).

Figure 2

Harmonic trap—dipolar resonances with $\Delta {\mathcal{M}}_z=-2$. Left upper panel: excitation energy of the two-particle system in the harmonic trap as a function of an external magnetic field. The horizontal line at small $B$ corresponds to the $|\mathrm{ini}\rangle$ state with both atoms fully polarized, $|3\rangle |3\rangle$. Notice avoided crossing. Right upper panel: a single maximum in the self-energy curve signifying strong coupling of both involved states at the resonant magnetic field. Lower panel: reduced single-particle density (in the $xy$ plane) of the $|{\mathcal{L}}_2\rangle$ state coupled to the $|\mathrm{ini}\rangle$ state.

Figure 3

Anharmonic trap—dipolar resonances with $\Delta {\mathcal{M}}_z=-2$. Upper panel: excitation energy of the two-particle system in the anharmonic trap as a function of an external magnetic field. The horizontal line at small $B$ corresponds to the $|\mathrm{ini}\rangle$ state with both atoms fully polarized, $|3\rangle |3\rangle$. Notice two avoided crossings. Middle panel: two maxima in the self-energy curve signifying strong coupling at the resonant magnetic field. Lower panel: reduced single-particle densities (in the $xy$ plane) of the final states accessible via the $\Delta {\mathcal{M}}_z=-2$ resonant dipolar processes. The densities correspond to the two resonant values of the magnetic field as indicated in the middle panel.

Figure 4

Dipolar matrix elments $D_1=|\langle v_{a_1}|H_D|\mathrm{ini}\rangle |$ and $D_2=|\langle v_{a_2}|H_D|\mathrm{ini}\rangle |$ for an axially symmetric anharmonic potential. With increasing barrier height the coupling to the state with no center-of-mass excitation dominates $D_1\sim \sqrt{V_0}$ and the harmonic limit is recovered.

Figure 5

Anharmonic trap—dipolar resonances with $\Delta {\mathcal{M}}_z=-2$ when the contact interaction coupling the $|2\rangle |2\rangle$ and $|3\rangle |1\rangle$ spin states is included (compare Fig. 3 where this coupling was omitted). Upper panel: excitation energy of the two-particle system in the anharmonic trap as a function of an external magnetic field (upper panel). The horizontal line at small $B$ corresponds to the $|\mathrm{ini}\rangle$ state with both atoms fully polarized $|3\rangle |3\rangle$. Notice four avoided crossings. Middle panel: four maxima in the self-energy curve signifying strong coupling at resonant magnetic field. Lower panel: reduced single-particle densities (in the $xy$ plane) of the final states accessible via $\Delta {\mathcal{M}}_z=-2$ resonant dipolar processes and the spin-dependent contact interactions corresponding to the four resonant values of the magnetic fields as indicated in the middle panel.

Figure 6

Single site of a square lattice—dipolar resonances with $\Delta {\mathcal{M}}_z=-2$. Upper panel: excitation energy of the two-particle system in a single lattice site as a function of an external magnetic field. The horizontal line at small $B$ corresponds to the $|\mathrm{ini}\rangle$ state with both atoms fully polarized, $|3\rangle |3\rangle$. Notice three avoided crossings. Middle panel: three maxima in the self-energy curve signifying strong coupling at the resonant magnetic field. Lower panel: reduced single-particle densities (in the $xy$ plane) of the final states accessible via $\Delta {\mathcal{M}}_z=-2$ resonant dipolar processes corresponding to the three resonant values of the magnetic fields as indicated in the middle panel.

Figure 7

Single site of a square lattice—dipolar resonances with $\Delta {\mathcal{M}}_z=-2$ when the contact interaction coupling the $|2\rangle |2\rangle$ and $|3\rangle |1\rangle$ spin states is included (compare Fig. 6 where this coupling was omitted). Upper panel: the excitation energy of the two-particle system in a single lattice site as a function of external magnetic field. The horizontal line at small $B$ corresponds to the $|\mathrm{ini}\rangle$ state with both atoms fully polarized $|3\rangle |3\rangle$. Notice five avoided crossings. Middle panel: five maxima in the self-energy curve signifying strong coupling at the resonant magnetic field. Lower panel: reduced single-particle densities (in the $xy$ plane) of the final states accessible via $\Delta {\mathcal{M}}_z=-2$ resonant dipolar processes and the spin-dependent contact interactions corresponding to the five resonant values of the magnetic fields as indicated in the middle panel.

Figure 8

Comparison with the experiment: The self-energy as a function of the Larmor frequency for the rectangular lattice of anisotropic sites—black lines. The lattice potential assumed here is separable and in every spatial direction has a form of a $“{\sin }^2”$ function. Quadratic terms in the expansion of the potential correspond to the following harmonic frequencies: ${\omega }_x/2\pi =160$ kHz, ${\omega }_y/2\pi =60$ kHz, and ${\omega }_z/2\pi =90$ kHz. Two orthogonal orientations of the magnetic field in the horizonatal plane, similarly as in the experiment, are considered: $y$ direction, upper panel; $x$ direction, lower panel. Occupation of the initial $m=3$ state as measured experimentally in [11] is shown for comparison—circles. The occupation should exhibit a local minimum at every resonance. The red dashed line corresponds to the self-energy obtained from the harmonic approximation, ${\omega }_x/2\pi =170$ kHz, ${\omega }_y/2\pi =55$ kHz, and ${\omega }_z/2\pi =100$ kHz, without any two-body interactions as used in [11].