Avoided crossings between bound states of ultracold cesium dimers

We present an efficient computational method for calculating the binding energies of the bound states of ultracold alkali-metal dimers in the presence of magnetic fields. The method is based on propagation of coupled differential equations and does not use a basis set for the interatomic distance coordinate. It is much more efficient than the previous method based on a radial basis set and allows many more spin channels to be included. This is particularly important in the vicinity of avoided crossings between bound states. We characterize a number of different avoided crossings in ${\mathrm{Cs}}_2$ and compare our converged calculations with experimental results. Small but significant discrepancies are observed in both crossing strengths and level positions, especially for levels with $l$ symmetry (rotational angular momentum $L=8$). The discrepancies should allow the development of improved potential models in the future.

Figure 1

Molecular potential energy curves $V_0\left(R\right)$ and $V_1\left(R\right)$ for the respective singlet and triplet states of ${\mathrm{Cs}}_2$ correlating with two separated ${}^{2}S_{1∕2}$ ground-state atoms. The inset shows an expanded view of the long-range potentials separating to the two different $f=3$ and 4 hyperfine states of the ${}^{2}S_{1∕2}$ atom with nuclear spin $i=7∕2$ and magnetic field $B=0$. The inset shows the adiabatic potentials obtained from diagonalizing the matrix form of the operator ${\widehat{h}}_1+{\widehat{h}}_2+\widehat{V}\left(R\right)$ at each $R$ for the case of $L=0$, $M_F=+6$. There are five channels, and the $3+4$ and $4+4$ separated-atom limits are doubly degenerate at $B=0$. All five channels have the same long-range variation as $-C_6∕R^6$, with $C_6=6860E_{\mathrm{h}}{a}_0^6$ [12] ($E_{\mathrm{h}}=4.3597\times {10}^{-18}\mathrm{J}$ is the hartree and $a_0=0.0529177\mathrm{nm}$ is the Bohr radius). The level crossings discussed in this paper are for very weakly bound levels that lie within about $E∕h\approx 5\mathrm{MHz}$ of the dissociation limit to two $\left\{f{m}_f\right\}=\{3,+3\}$ atoms in the magnetic-field range from $0\text{to}5\mathrm{mT}$.

Figure 2

(Color online) Bound-state energy $E∕h$ as a function of $B$ for levels of the ${\mathrm{Cs}}_2$ molecules with even $L⩽4$ and $M_{\mathrm{tot}}=+6$. Energies are given relative to the energy of two Cs atoms in their ground Zeeman sublevel ($f=3$, $m_f=+3$). The $FL\left(M_F\right)$ labeling scheme is shown for each level. Off-diagonal coupling between levels with different $FL\left(M_L\right)$ quantum numbers is neglected in this calculation.

Figure 3

(Color online) Example of coupling between different $L\left(M_F\right)$ symmetry blocks with the symmetry of the dipole-dipole interaction of ${\widehat{V}}^{\mathrm{d}}$. Each block represents a Hamiltonian matrix for spin states with the $L\left(M_F\right)$ values indicated. The labels $x$ and $y$ indicate the existence of nonvanishing coupling due to ${\widehat{V}}^{\mathrm{d}}$; a 0 indicates no coupling. The case shown is for a $g\left(6\right)$ and a $g\left(3\right)$ level, which have no direct coupling. The left panel shows the symmetries that give rise to second-order interactions between the two levels and thus contribute to the strength of the avoided crossing between them. The right panel shows a truncated set of interactions through intermediate $d\left(4\right)$ and $d\left(5\right)$ levels.

Figure 4

(Color online) Calculated energy levels $E∕h$ with $M_{\mathrm{tot}}=+6$ as a function of magnetic field $B$ near the crossing of the $4g\left(3\right)$ level with the $6g\left(6\right)$ level near $1.0\mathrm{mT}$. The points and solid line show the propagator calculation with an $sdgi$ basis set. The dashed lines show the crossing levels from two uncoupled DVR calculations with $g\left(3\right)$ or $g\left(6\right)$ basis functions only. The dash-dotted and dotted lines show the crossing levels from DVR calculations with added $g(4,5)$ and $d(4,5)$ functions, respectively. The DVR calculation with $i(4,5)$ basis functions is not shown, but lies near the uncoupled crossing and has a very small splitting, indicating very weak second-order coupling through distant $i$ states.

Figure 5

(Color online) Calculated energy levels $E∕h$ with $M_{\mathrm{tot}}=+6$ as a function of magnetic field $B$ near the crossing of the $4g\left(4\right)$ level with the $6g\left(6\right)$ level near $1.0\mathrm{mT}$. The points and solid line show the propagator calculation with a $dgi$ basis set. The dashed lines shows the crossing levels from two uncoupled DVR calculations with $g\left(4\right)$ or $g\left(6\right)$ basis functions only. The dash-dotted lines show the avoided crossing from a DVR calculation with only the direct coupling in the $g(4,6)$ basis set included. The double-headed arrow at the position of the propagator crossing shows the measured splitting [8]. The actual experimental crossing was observed $0.046\mathrm{mT}$ lower in $B$ value than the propagator crossing.

Figure 6

(Color online) Calculated energy difference $\Delta ∕h$ between the $6g\left(6\right)$ and $4d\left(4\right)$ levels with $M_{\mathrm{tot}}=+6$ as a function of magnetic field $B$. The solid line is from a propagator calculation with the $sdgi$ basis. The diamonds show the experimental results obtained by Ferlaino et al. using their more accurate field modulation method [33]. The dashed line shows the calculated points shifted by $+0.034\mathrm{mT}$. The DVR calculation (not shown) with direct coupling included in the $d\left(4\right)g\left(6\right)$ basis is virtually identical to the dashed line when the DVR results are shifted by $+0.062\mathrm{mT}$.

Figure 7

(Color online) Energy levels $E∕h$ as a function of $B$ for the ${\mathrm{Cs}}_2$ molecule for $L=0$ and 2 only with $M_{\mathrm{tot}}=+6$ ($g$ and $l$ levels are not shown). The dashed lines show the DVR levels calculated with the uncoupled $s\left(6\right)$ and $d\left(M_F\right)$ basis sets, where $M_F=4$, 5, or 6. The diamonds show the results of Mark et al. [8]. The $4d\left(4\right)$ level crosses the $6s\left(6\right)$ level twice, near 0.24 and $4.77\mathrm{mT}$. The closed circles and dotted lines show the levels obtained from a DVR calculation with an $sd$ basis. A propagator calculation with a full $sdg$ basis shows negligible differences for this case.

Figure 8

(Color online) Expanded view of the crossing in Fig. 7 of the $4d\left(4\right)$ and $6s\left(6\right)$ levels near $4.8\mathrm{mT}$. The long-dashed line shows the uncoupled calculation with the $s\left(6\right)$ and $d\left(4\right)$ basis sets. The solid lines show the propagator calculations with an $sdg$ basis. The upper crossing near $5.4\mathrm{mT}$ is due to a $2g\left(2\right)$ level. The open circles show DVR calculations with a full $sd$ basis in a finite box of $5000a_0$. The diamonds show experimental results of Lange et al. [35].

Figure 9

(Color online) Energy levels $E∕h$ as a function of $B$ for the ${\mathrm{Cs}}_2$ molecule for $L=8$ only with $M_{\mathrm{tot}}=+6$. The solid lines show the DVR levels calculated with the uncoupled $l\left(M_F\right)$ basis sets, where $M_F=0,\dots ,6$. The dashed line shows the $6g\left(6\right)$ level for which avoided crossings have been calculated (see Table ) and measured [8].