Fourier-transform spectroscopy and coupled-channels deperturbation treatment of the $A{\hspace{0.5em}}^1{\Sigma }^{+}–b{\hspace{0.5em}}^3\Pi$ complex of KCs

The laser-induced fluorescence (LIF) $A{\hspace{0.5em}}^1{\Sigma }^{+}–b{\hspace{0.5em}}^3\Pi \to X{\hspace{0.5em}}^1{\Sigma }^{+}$ spectra of the KCs molecule were recorded in a near infrared region by a Fourier-transform spectrometer with a resolution of $0.03$ cm${}^{-1}$. Overall more than 200 collisionally enhanced LIF spectra were rotationally assigned to ${}^{39}\mathrm{K}$${}^{133}\mathrm{Cs}$ and ${}^{41}\mathrm{K}$${}^{133}\mathrm{Cs}$ isotopomers yielding more than 3400 rovibronic term values of the strongly mixed singlet $A{\hspace{0.5em}}^1{\Sigma }^{+}$ and triplet $b{\hspace{0.5em}}^3\Pi$ states with the uncertainty of $0.003–0.01$ cm${}^{-1}$. Experimental data massive starts from the lowest vibrational level $v_A=0$ of the singlet and nonuniformly covers the energy range $E\in [\mathrm{10\; 040},\mathrm{13\; 250}]$ cm${}^{-1}$ with rotational quantum numbers $J^{\prime }\in [7,225]$. Besides the dominating regular $A{\hspace{0.5em}}^1{\Sigma }^{+}–b{\hspace{0.5em}}^3{\Pi }_{\Omega =0}$ interactions, the weak local heterogeneous $A{\hspace{0.5em}}^1{\Sigma }^{+}–b{\hspace{0.5em}}^3{\Pi }_{\Omega =1}$ perturbations have been discovered and analyzed. Coupled-channels deperturbation analysis of the experimental ${}^{39}\mathrm{K}$${}^{133}\mathrm{Cs}$ $e$-parity term values of the $A{\hspace{0.5em}}^1{\Sigma }^{+}–b{\hspace{0.5em}}^3{\Pi }_{\Omega =0,1,2}$ complex was accomplished in the framework of the phenomenological $4\times 4$ Hamiltonian accounting implicitly for regular interactions with the remote ${}^1\Pi$ and ${}^3{\Sigma }^{+}$ states. The diabatic potential energy curves of the $A{\hspace{0.5em}}^1{\Sigma }^{+}$ and $b{\hspace{0.5em}}^3\Pi$ states, as well as relevant spin-orbit coupling matrix elements, were defined analytically with the expanded Morse oscillators model. The obtained parameters reproduce $95\%$ of experimental data field of the ${}^{39}\mathrm{K}$${}^{133}\mathrm{Cs}$ isotopomer with a standard deviation of $0.004$ cm${}^{-1}$, which is consistent with the uncertainty of the experiment. Reliability of the derived parameters was confirmed by a good agreement between the predicted and experimental term values of the ${}^{41}\mathrm{K}$${}^{133}\mathrm{Cs}$ isotopomer. The calculated relative intensity distributions in $A-b\to X$ LIF progressions are also consistent with their experimental counterparts. The deperturbation model was applied for simulation of a pump-dump optical cycle $a{\hspace{0.5em}}^3{\Sigma }^{+}\to A{\hspace{0.5em}}^1{\Sigma }^{+}–b{\hspace{0.5em}}^3\Pi \to X{\hspace{0.5em}}^1{\Sigma }^{+}$ proposed for transformation of ultracold KCs molecules to their absolute ground state $v_X=0;J_X=0$.

Figure 1

Scheme of KCs electronic terms based on ab initio nonrelativistic calculation of Ref. [13]. Shaded area marks the studied energy region.

Figure 2

${}^{39}\mathrm{K}$${}^{133}\mathrm{Cs}$ $A-b\to X$ LIF spectrum recorded by InGaAs detector at excitation with 850-nm laser diode (laser frequency 11 608.623 cm${}^{-1}$). The strongest doublet $P_{63},R_{61}$ progression $5⩽v_X⩽78$ originates from the level $J^{\prime }=62$ with energy $E=\mathrm{12\; 164.984}$ cm${}^{-1}$ excited by the transition $(v^{*},62)\leftarrow (6,63)$. A weaker KCs LIF progression excited by $(v^{*},133)\leftarrow (6,134)$ transition is not marked. A mesh of weaker lines around 9500 cm${}^{-1}$ belongs to ${\mathrm{K}}_2$ LIF.

Figure 3

(a) Example of ${}^{39}\mathrm{K}$${}^{133}\mathrm{Cs}$ $A-b\to X$ LIF spectrum recorded at laser frequency 9800.135 cm${}^{-1}$. The following $A-b\leftarrow X$; $(v^{*},J^{\prime })\leftarrow (v_X,J_X)$ excitation transitions have been assigned: $(v^{*},50)\leftarrow (4,51)$; $(v^{*},109)\leftarrow (3,110)$; $(v^{*},57)\leftarrow (7,56)$; $(v^{*},123)\leftarrow (16,122)$; $(v^{*},44)\leftarrow (5,43)$. Weak Cs${}_2$ progressions have been assigned below 8500 cm${}^{-1}$. Hereafter indices for $P,R$ lines denote the respective ground-state rotational levels. (b) The zoomed fragment of the spectrum; doublets originating from the lowest excited vibrational level with $J^{\prime }=50$ and $J^{\prime }=109$ possessing dominant singlet character are marked with black and red vertical bars below the spectrum. The bars above the spectrum mark transitions from the level $(v^{*},J^{\prime }=44)$ with dominant triplet character while numbers at bars denote $v_X$.

Figure 4

(a) Example of ${}^{39}\mathrm{K}$${}^{133}\mathrm{Cs}$ $A-b\to X$ LIF spectrum recorded at laser frequency 9743.083 cm${}^{-1}$. The scattered laser light and LIF at frequencies higher than about 9500 cm${}^{-1}$ are cut by a long-pass edge filter. The following excitation transitions for ${}^{39}\mathrm{K}$${}^{133}\mathrm{Cs}$ have been assigned: $(v^{*},49)\leftarrow (6,50);(v^{*},81)\leftarrow (17,82);(v^{*},25)\leftarrow (13,26);(v^{*},135)\leftarrow (12,134);(v^{*},150)\leftarrow (11,149)$. (b) Zoomed part of the spectrum around $v_X=20$ for the strongest progression with $J^{\prime }=49$. Three groups (II–IV) of satellite $P,R$ branches shifted from the main rotational relaxation pattern (group I) are clearly seen. The inset presents term values of the respective rovibronic levels as dependent of $J^{\prime }(J^{\prime }+1)$; directly excited level $J^{\prime }=49$ is marked.

Figure 5

(a) LIF spectrum recorded at excitation frequency 9826.348 cm${}^{-1}$. The strongest progression originates from the upper level excited in transition $(v^{*},158)\leftarrow (4,157)$. (b) Zoomed part of the spectrum around $v_X=12$; red (upper) and blue (lower) bars denote assigned $P$ and $R$ satellite lines, respectively. Longer bars mark transitions from the upper state levels with dominant triplet $b{\hspace{0.5em}}^3{\Pi }_{\Omega =1}$ character.

Figure 6

Experimental term values of the $A{\hspace{0.5em}}^1{\Sigma }^{+}–b{\hspace{0.5em}}^3\Pi$ complex plotted versus $J^{\prime }(J^{\prime }+1)$. Empty (red) circles relate to the ${}^{41}\mathrm{K}$${}^{133}\mathrm{Cs}$ isotopomer.

Figure 7

(a) Present empirical and available ab initio (ab1-[13]; ab2-[14]; ab3-[15]) PECs of the $A{\hspace{0.5em}}^1{\Sigma }^{+}$ and $b{\hspace{0.5em}}^3{\Pi }_{\Omega =1}$ states corresponding to (a) Hund’s coupling case. Dashed horizontal and vertical lines indicate the experimentally studied region. (b) Present empirical diabatic PECs of the $A{\hspace{0.5em}}^1{\Sigma }^{+},b{\hspace{0.5em}}^3{\Pi }_{\Omega =0,1,2}$ sub-states and the corresponding adiabatic PECs $(2;3)\Omega =0^{+}$ in the (c) Hund’s coupling case states. $r_c$ is the internuclear distance where diabatic $A{\hspace{0.5em}}^1{\Sigma }^{+}$ and $b{\hspace{0.5em}}^3{\Pi }_{\Omega =0}$ PECs intersect.

Figure 8

Comparison of the present empirical first-order SO functions with their ab initio counterparts (ab3) from Ref. [15]. (a) Diagonal empirical $A_{\pm }^{\mathrm{so}}(r)$ and ab initio $A^{\mathrm{so}}(r)$ SO splitting functions. (b) Relevant off-diagonal ${\xi }_{\mathit{Ab}0}^{\mathrm{so}}(r)$ SO coupling functions. $r_c$ is the internuclear distance where diabatic PECs intersect. At large $r$ both SO functions approach asymptotic limit ${\xi }_{\mathrm{Cs}}^{\mathrm{so}}=184.68$ cm${}^{-1}$.

Figure 9

Comparison of the present empirical and ab initio second-order SO functions ($\mathit{ab}$). (a) Empirical $B{\zeta }_{\mathit{Ab}1}(r)$ entering Eq. (3) with ab initio function determined by Eq. (7). (b) Empirical SO splitting functions $A_{\pm }^{\mathrm{so}}(r)$ entering Eq. (4) whith ab initio $A_{\pm }^{\mathrm{so}}(r)$ values calculated by Eq. (8).

Figure 10

(a) Fragment of $J^{\prime }$ dependence of experimental term values as well as their deperturbed diabatic singlet and triplet counterparts. Term values are given by the reduced scale: $E^{\mathrm{red}}=E-0.0222J^{\prime }(J^{\prime }+1)$. Graphs I, II, and III denote experimental data. Almost horizontal solid lines refer to diabatic singlet $A{\hspace{0.5em}}^1{\Sigma }^{+}$ state in the region $v_A=1$–3; dashed and dotted lines refer to diabatic triplet substates $b{\hspace{0.5em}}^3{\Pi }_{\Omega =0}$ ($v_{b0}=18$–19) and $b{\hspace{0.5em}}^3{\Pi }_{\Omega =1}$ ($v_{b1}=17$ and $18$), respectively. For small $J^{\prime }$ the effective vibrational quantum numbers can be attributed as $v_{b0}^{*}$(I) $=20$, $v_{b0}^{*}$(II) $=21$, and $v_A^{*}$(III) $=1$. The inset zooms in the splitting caused by the interaction with $v_{b1}=17$ term. Arrows mark the positions of observed weak local perturbations by $b{\hspace{0.5em}}^3{\Pi }_{\Omega =1}$ substate. Triangles mark the levels which give the LIF progressions analyzed in Figs. 13 and 14. (b) Residuals of the fit $E^{\mathrm{expt}}-E^{\mathrm{CC}}$ for all respective $J^{\prime }$ values. (c) Fractional partition of the relevant nonadiabatic wave functions $P_i$ for the experimental term values presented by graph II. Sharp resonance at $J^{\prime }=113/114$ correspond to the intersection shown by inset in (a). Arrows in (c) mark the predicted influence of the $b{\hspace{0.5em}}^3{\Pi }_{\Omega =2}$ substate.

Figure 11

(a) Experimental term values around local $J^{\prime }=157/158$ perturbation [see Fig. 5b] caused by $v_{b1}=17$ level (solid line) of the diabatic $b{\hspace{0.5em}}^3{\Pi }_{\Omega =1}$ substate. Term values are presented by the reduced scale: $E^{\mathrm{red}}=E-\mathrm{10\; 266}-0.0237J^{\prime }(J^{\prime }+1)$. (b) Residuals of the fit $E^{\mathrm{expt}}-E^{\mathrm{CC}}$ for respective $J^{\prime }$ values. (c) Respective fractional partition functions $P_i$.

Figure 12

(a) Experimental and calculated intensity distribution in the long LIF progression (see Fig. 2) originating from the high rovibronic level $E(J^{\prime }=62)$ of the $A-b$ complex. Hereafter the intensities are normalized to the most intensive transition of the progression. (b) Respective nonadiabatic wave function ${\varphi }_A(r)$ of the singlet $A$ state ($P_A=\langle {\varphi }_A|{\varphi }_A\rangle =0.81$).

Figure 13

Experimental and calculated intensity distributions in LIF progressions originating from close-lying rovibronic levels of the $A-b$ complex [marked by the solid triangles in Fig. 10a]: (a) for the $v_{b0}^{*}=21;J^{\prime }=44$ having an admixture of the triplet $b{\hspace{0.5em}}^3{\Pi }_{\Omega =0}$ component ($P_{b0}=\langle {\varphi }_{b0}|{\varphi }_{b0}\rangle =0.52$); (b) for the $v_A^{*}=1;J^{\prime }=49$ level having a dominant singlet $A{\hspace{0.5em}}^1{\Sigma }^{+}$ character ($P_A=0.74$). (c) Respective nonadiabatic wave functions ${\varphi }_A(r)$ of the singlet $A$ state.

Figure 14

Experimental and calculated intensity distributions in LIF progressions originating from two mutually perturbed levels of the $A-b$ complex with the same $J^{\prime }=158$ value [see Fig. 11a]. (a) Having dominant singlet $A{\hspace{0.5em}}^1{\Sigma }^{+}$ character ($P_A^s=0.58$; $P_{b0}^s=0.14$; $P_{b1}^s=0.28$), marked by “s” in Fig. 11a. (b) Having dominant $b{\hspace{0.5em}}^3{\Pi }_{\Omega =1}$ component ($P_A^t=0.23$; $P_{b0}^t=0.06$; $P_{b1}^t=0.71$), marked by “t” in Fig. 11a. (c) Respective nonadiabatic ${\varphi }_A^{s/t}(r)$ wave functions of the $A$ state.