Hyperfine molecular Hubbard Hamiltonian

An ultracold gas of heteronuclear alkali-metal dimer molecules with hyperfine structure loaded into a one-dimensional optical lattice is investigated. The hyperfine molecular Hubbard Hamiltonian (HMHH), an effective low-energy lattice Hamiltonian, is derived from first principles. The large permanent electric dipole moment of these molecules gives rise to long-range dipole-dipole forces in a dc electric field and allows for transitions between rotational states in an ac microwave field. Additionally, a strong magnetic field can be used to control the hyperfine degrees of freedom independently of the rotational degrees of freedom. By tuning the angle between the dc electric and magnetic fields and the strength of the ac field, it is possible to control the number of internal states involved in the dynamics as well as the degree of correlation between the spatial and internal degrees of freedom. The HMHH’s unique features have direct experimental consequences such as quantum dephasing, tunable complexity, and the dependence of the phase diagram on the molecular state.

Figure 1

Quantum dephasing in the HMHH. The plot shows the behavior of the total number in state $0$, $\langle {\mathrm{n̂}}_0\rangle \equiv \langle {\sum }_i{\mathrm{n̂}}_{i0}\rangle$, when the system evolves under the Hamiltonian (1) with 12 sites at half filling. Quantum dephasing produces an emergent exponential envelope on the Rabi oscillation pattern between states 0 and 1. Only the number of state 0 is shown for clarity. The dashed red curve is an exponential envelope fit to $N\exp (-t/\tau )$ with $\tau =1441.17$ ms. The nonexponential behavior near $t=200$ is due to the finite size of the lattice.

Figure 2

Tunneling matrix elements in a dc electric field. Tunneling energies (in kHz) of the $N=0$ (solid blue line) and $N=1$ (dashed green line) rotational states and their difference divided by their arithmetic mean, $2(t_1-t_0)/(t_1+t_0)$ (dash-dotted red line), for KRb in a field of 10 kV/cm as a function of the effective isotropic lattice height $\eta \equiv \mathrm{\alpha ̄}\left|{\mathbf{E}}_{\mathrm{opt}}\right|{}^2$ (in recoil energy units). The values of the polarizability tensor are taken from Ref. [20].

Figure 3

Geometry of the HMHH. Counterpropagating laser beams along the $y$ and $z$ directions create an array of one-dimensional (1D) tubes, and an additional pair of laser beams along $x$ creates a lattice potential. A strong dc field orients the dipoles along the direction perpendicular to motion, and a magnetic field orients the nuclear spins. An ac field of circular space-fixed polarization drives transitions between internal levels.

Figure 4

Distribution of dipolar character. The color bar shows the logarithm of the transition dipole moment with the ground state, $\log {}_{10}\langle \mathrm{g}.\mathrm{s}.\left|{\mathrm{d̂}}_1\right|i\rangle$, as a function of the angle between the magnetic and electric fields ${\theta }_B$ and the state index $i$ (ordered by energy). Change of the angle between the electric and magnetic fields breaks the nuclear spin projection selection rule and allows for transition dipole moments between many states. Only dipole moments greater than ${10}^{-7}$ are displayed.

Figure 5

Gigahertz-scale view of the Stark effect for KRb. Introduction of a dc electric field breaks the degeneracy between all states with the same $N$ but different $\left|M_N\right|$. The large electric dipole moment causes gigahertz-scale energy shifts which completely obscure the hyperfine splittings on the scale of this plot. Because the dipole moment is the same for any isotope of KRb, the Stark effect on this scale is the same for all isotopes.

Figure 6

Induced dipoles for KRb in an electric field. The $N=0$ and $N=1$, $M_N=\pm 1$ levels orient along the field, giving rise to positive dipole moments. The $N=1$, $M_N=0$ state anti-aligns with the field for small fields, but aligns in stronger fields. All resonant dipole moments approach the “permanent” value 0.566 D as the field strength increases.

Figure 7

Kilohertz-scale view of the Stark effect for ${}^{40}\mathrm{K}$${}^{87}\mathrm{Rb}$, $N=$ 0. All energies are shown relative to the gigahertz-scale field-dependent average energy for $N=0$ (see Fig. 5). The inset shows the weak-field region where the scalar spin-spin interaction has split the levels according to $I$ (equivalently, $F$), with larger $I$ having lower energy. As the field is increased, the nuclear quadrupole couplings split according to $M_I$, and in large fields $M_{\mathrm{Rb}}$ and $M_{\mathrm{K}}$ also become well defined. See the text for details.

Figure 8

Zeeman effect for ${}^{40}\mathrm{K}$${}^{87}\mathrm{Rb}$, $N=0$. The magnetic field splits the hyperfine levels according to their projections $M_{\mathrm{K}}$ and $M_{\mathrm{Rb}}$ with splittings between adjacent levels of order kHz for the experimentally relevant range $B~550$ G. The lowest- (highest-) energy state corresponds to $m_F=-4+3/2=-5/2$ ($5/2$). The zero-field splitting is set by $c_4$ and is not visible on the scale of this plot.

Figure 9

Zeeman effect for ${}^{40}\mathrm{K}$${}^{87}\mathrm{Rb}$, $N=1$. The zero-field splitting is caused mainly by the nuclear quadrupole intraction and separates the levels into groups of well-defined $F$. The much larger zero-field splitting causes avoided crossings between states with the same $M_F$ to occur at much higher fields than in the $N=0$ case.