Creation on Demand of Higher Orbital States in a Vibrating Optical Lattice

It is shown that the extended Hubbard Hamiltonian describing atoms confined in an optical lattice always contains commonly neglected terms which can significantly change the dynamical properties of the system. Particularly for bosonic systems, they can be exploited for creating orbital states on demand via the parametric resonance phenomenon. This indicates an additional application for optical lattices, namely, the study and emulation of interactions between particles and lattice vibrations.

Figure 1

Parameters of the Hamiltonian (1) as functions of lattice depth for the symmetric case $\mathbf{Q}=(q,q,8)$ and $g=1$. Parameters originating in contact interactions are at least 10 times smaller than the energy gap between the $s$ and $p$ bands.

Figure 2

Creation of the $p$-band states by a vibrating optical lattice with $\mathbf{Q}=\mathbf{(}32+4\sin (\omega t),20,8\mathbf{)}$. Plot (a) presents the transfer efficiency as a function of vibration frequency $\omega$. Two distinct peaks are visible. For these particular frequencies, $p$-band states become highly occupied. The dashed line comes from a generalized model that also takes into account $d$-band orbitals. Additional peaks correspond to resonant frequencies in which one of interacting atoms can be promoted to the $d_x$ or $d_y$ band, respectively. Moreover, resonant frequencies ${\omega }_1$ and ${\omega }_2$ are shifted. Plot (b) presents the occupation of basis states as a function of time for the resonant frequencies. At frequency ${\omega }_1$ (${\omega }_2$) the $p_y$ ($p_x$) orbital is populated. Plot (c) shows a comparison between the situation when tunneling is totally neglected (solid line) and when it is taken into account (filled circles). The inset shows the variance of the on-site number operator as a function of time for $\omega ={\omega }_2$. Detailed explanations are given in the text.

Figure 3

Two experimental scenarios for the creation of a two particle superposition in excited band states. Initially the system is prepared in the ground state of a highly non symmetric lattice ${\mathbf{Q}}_0=(32,20,8)$. Plot (a) In the first scenario, two particles are transferred to the $p_y$ state by applying appropriate vibrations (${\mathbf{P}}_1$) and then lattice depths in both directions are equilibrated ($\mathbf{E}$). If this process is slow enough, then the final state of the system is in the superposition $(|020⟩-|002⟩)/\sqrt{2}$. Plot (b) In the second case, two particles are transferred to the $p_y$ state by applying appropriate vibrations (${\mathbf{P}}_1$). When the initial state is half-depleted the vibration frequency is changed to the other resonance, and the $p_x$ state is filled (${\mathbf{P}}_2$). Then similarly to the previous scenario, lattice depths are equilibrated ($\mathbf{E}$). The final state is in a complex superposition of basis states and the occupation of each basis state varies in time ($\mathbf{F}$). At the moments when the occupation of $p_x$ and $p_y$ orbitals are equal, the system is in one of the vortex states $(|020⟩\pm i|002⟩)/\sqrt{2}$.