Nonadiabatic effects in long-pulse mixed-field orientation of a linear polar molecule

We present a theoretical study of the impact of an electrostatic field combined with nonresonant linearly polarized laser pulses on the rotational dynamics of linear molecules. Within the rigid rotor approximation, we solve the time-dependent Schrödinger equation for several field configurations. Using the carbonyl sulfide molecule as the prototype, the field-dressed dynamics is analyzed in detail for experimentally accessible static-field strengths and laser pulses. Results for directional cosines are presented and compared to the predictions of the adiabatic theory. We demonstrate that for the prototypical field configuration used in current mixed-field orientation experiments, the molecular field dynamics is, in general, nonadiabatic, and a time-dependent description of these systems is mandatory. We investigate several field regimes identifying the sources of nonadiabatic effects and provide the field parameters under which the adiabatic dynamics would be achieved.

Figure 1

Laboratory fixed coordinate, Euler angles, schematic field configuration, and the OCS molecule.

Figure 2

Adiabatic results for the (a) energy and expectation values, (b) $\langle {\cos }^2\theta \rangle$, and (c) $\langle \cos \theta \rangle$ as a function $\mathit{I}$ of the adiabatic states ${|0,0,e\rangle }_{\mathrm{p}}^0$ [thick solid (red) line], ${|1,0,e\rangle }_{\mathrm{p}}^0$ [thin solid (gold) line], ${|1,1,e\rangle }_{\mathrm{p}}^0$ [thick short-dashed (orange) line], ${|2,1,e\rangle }_{\mathrm{p}}^0$ [long-dashed (dark-blue) line], ${|2,0,e\rangle }_{\mathrm{p}}^0$ [dot-short-dashed (blue) line], ${|3,0,e\rangle }_{\mathrm{p}}^0$ [dotted (purple) line], ${|2,2,e\rangle }_{\mathrm{p}}^0$ [thin short-dashed (green) line], and ${|3,2,e\rangle }_{\mathrm{p}}^0$ [dot-long-dashed (pink) line]. Insets: Relevant energy and intensity ranges where the formation of the near-degenerate doublets occurs. (d) Polar plots of the square of their wave functions at $I={10}^{12}$ W/cm${}^2$. $E_{\mathrm{s}}=300\mathrm{V}/\mathrm{cm}$ and $\beta =0^{\circ }$ for all data.

Figure 3

(a) For the ground state, time evolution of the expectation value $\langle \cos \theta \rangle$ as a function of $I\left(t\right)$ for Gaussian pulses of $\tau =10$ ns and peak intensities $I_0=2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ [thick solid (red) line], $I_0=5\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ [dashed (orange) line], $I_0={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ [dotted (gold) line], and $I_0=2\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ [dot-dashed (green) line]. The adiabatic results for $\langle \cos \theta \rangle$ (thin solid line) are also included. (b) Squares of the projections of the time-dependent wave functions onto the adiabatic pendular state ${|0,0,e\rangle }_{\mathrm{p}}^0$. (c) Adiabatic criteria $\eta$ as a function of $I\left(t\right)$. The field configuration is $E_{\mathrm{s}}=300\mathrm{V}/\mathrm{cm}$ and $\beta =0^{\circ }$.

Figure 4

Expectation value $\langle \cos \theta \rangle$ at $t=0$ as a function of the peak intensity $I_0$ for the states ${|0,0,e\rangle }^0$ [thick solid (red) line], ${|1,0,e\rangle }^0$ [thin solid (gold) line] ${|1,1,e\rangle }^0$ [thick short-dashed (orange) line], ${|2,1,e\rangle }^0$ [long-dashed (dark-blue) line] ${|2,0,e\rangle }^0$ [dot-short-dashed (blue) line], ${|3,0,e\rangle }^0$ [dotted (purple) line], ${|2,2,e\rangle }^0$ [thin short-dashed (green) line], and ${|3,2,e\rangle }^0$ [dot-long-dashed (pink) line], for $\beta =0^{\circ }$ and (a) $E_{\mathrm{s}}=300\mathrm{V}/\mathrm{cm}$ and (b) $E_{\mathrm{s}}=600\mathrm{V}/\mathrm{cm}$. The FWHM of the laser pulses is 10 ns.

Figure 5

Projections of the time-dependent wave functions onto the corresponding adiabatic states as a function of the peak intensity $I_0$ for the states ${|0,0,e\rangle }^0$ [thick solid (red) line], ${|1,1,e\rangle }^0$ [thick short-dashed (orange) line], ${|2,0,e\rangle }^0$ [dot-short-dashed (blue) line], and ${|2,2,e\rangle }^0$ [thin short-dashed (green) line], for (a) $E_{\mathrm{s}}=300\mathrm{V}/\mathrm{cm}$ and (b) $E_{\mathrm{s}}=600\mathrm{V}/\mathrm{cm}$. We use 10-ns laser pulses and $\beta =0^{\circ }$.

Figure 6

Expectation value $\langle \cos \theta \rangle$ at $t=0$ as a function of $\tau$ for (a) $I_0=2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ and (b) $I_0={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. The fields are parallel and $E_{\mathrm{s}}=300\mathrm{V}/\mathrm{cm}$. The states and their labels are the same as in Fig. 4.

Figure 7

Expectation value $\langle \cos \theta \rangle$ at $t=0$ as a function of $E_{\mathrm{s}}$ for (a) $I_0=2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ and (b) $I_0={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. We use a 10-ns laser pulse and $\beta =0^{\circ }$. The states and their labels are the same as in Fig. 4.

Figure 8

For the ground state, evolution of the expectation values (a) $\langle \cos \theta \rangle$ and (b) $\langle \cos {\theta }_{\mathrm{s}}\rangle$ as a function of $I\left(t\right)$ of a 10-ns Gaussian pulse. The field configurations are $\beta ={30}^{\circ }$ (thick line) and ${45}^{\circ }$ (thin line) with peak intensities $I_0=2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ (dashed line), $5\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ (dot-dashed line), and ${10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ (dotted line). The static electric field is fixed at $E_{\mathrm{s}}=300\mathrm{V}/\mathrm{cm}$. The adiabatic results (solid line) are also included.

Figure 9

(a) For the $|1,1,e\rangle$ state, the expectation value $\langle \cos \theta \rangle$ as a function of $I\left(t\right)$ of a 10-ns Gaussian pulse. The field configurations are $\beta ={30}^{\circ }$ (thick line) and ${45}^{\circ }$ (thin line), with $I_0=2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ (dashed line), $I_0=5\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ (dot-dashed line) and $I_0={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ (dotted line). The adiabatic results (solid line) are also included. For $I_0={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ and $\beta ={30}^{\circ }$, (b) the squares of the projections of the time-dependent wave function on the adiabatic pendular states ${|1,1,e\rangle }_{\mathrm{p}}$ (dot-dashed line), ${|1,0,e\rangle }_{\mathrm{p}}$ (dotted line), ${|0,0,e\rangle }_{\mathrm{p}}$ (long-dashed line), and ${|2,2,e\rangle }_{\mathrm{p}}$ (short-dashed line) and (c) the adiabatic parameter between the pendular states ${|1,1,e\rangle }_{\mathrm{p}}$ and ${|1,0,e\rangle }_{\mathrm{p}}$ (dot-dashed line), ${|1,1,e\rangle }_{\mathrm{p}}$ and ${|0,0,e\rangle }_{\mathrm{p}}$ (dashed line), and ${|1,0,e\rangle }_{\mathrm{p}}$ and ${|2,2,e\rangle }_{\mathrm{p}}$ (dotted line). The dc field is fixed at $E_{\mathrm{s}}=300\mathrm{V}/\mathrm{cm}$.

Figure 10

Expectation value $\langle \cos \theta \rangle$ at $t=0$ as a function of the peak intensity $I_0$ for $|0,0,e\rangle$ [thick solid (red) line], $|1,0,e\rangle$ [thin solid (gold) line], $|1,1,e\rangle$ [thick short-dashed (orange) line], $|2,1,e\rangle$ [long-dashed (dark-blue) line], $|2,0,e\rangle$ [dot-short-dashed (blue) line], $|3,0,e\rangle$ [dotted (purple) line], $|2,2,e\rangle$ [thin short-dashed (green) line], and $|3,2,e\rangle$ [dot-long-dashed (pink) line]. The field configurations are (a), (b) $\beta ={30}^{\circ }$, (c), (d) $\beta ={45}^{\circ }$, and (e), (f) $\beta ={75}^{\circ }$, with $E_{\mathrm{s}}=300$ and 600 V/cm, respectively. The FWHMs of the Gaussian pulses are fixed at 10 ns.

Figure 11

Expectation value $\langle \cos \theta \rangle$ at $t=0$ as a function of $\tau$. The field configurations are (a) $\beta ={30}^{\circ }$ and (b) $\beta ={45}^{\circ }$, with $I_0={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ and $E_{\mathrm{s}}=300\mathrm{V}/\mathrm{cm}$. The states and their labels are the same as in Fig. 10.

Figure 12

Expectation value of $\langle \cos \theta \rangle$ at $t=0$ as a function of $E_{\mathrm{s}}$ for the field configurations $\tau =10$ ns, $I_0={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$, and (a) $\beta ={30}^{\circ }$ and (b) $\beta ={45}^{\circ }$. Labeling of the states is as in Fig. 10.

Figure 13

Expectation values $\langle \cos \theta \rangle$ (thick solid line) and $\langle \cos {\theta }_{\mathrm{s}}\rangle$ (thin solid line) at $t=0$ as a function of $\beta$ for the ground state. The peak intensities are (a) $I_0=2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ and (b) $I_0={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$. The adiabatic results for $\langle \cos \theta \rangle$ (thick dashed line) and $\langle \cos {\theta }_{\mathrm{s}}\rangle$ (thin dashed line) are also presented. The FWHM of the laser pulse is fixed at $\tau =10$ ns, and the dc field at $E_{\mathrm{s}}=300\mathrm{V}/\mathrm{cm}$.

Figure 14

Expectation value $\langle \cos \theta \rangle$ at $t=0$ as a function of $\beta$ for the states (a) $|0,0,e\rangle$ (black line) and $|1,0,e\rangle$ (blue line), (b) $|1,1,e\rangle$ (black line) and $|2,1,e\rangle$ (blue line), (c) $|2,0,e\rangle$ (black line) and $|3,0,e\rangle$ (blue line), and (d) $|2,2,e\rangle$ (black line) and $|3,2,e\rangle$ (blue line). The field configuration is $E_{\mathrm{s}}=300\mathrm{V}/\mathrm{cm}$, with $I_0=2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ (solid lines) and ${10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ (dashed lines). The FWHM of the laser pulse is fixed at $\tau =10$ ns.

Figure 15

Orientation cosines once the peak intensity and dc-field strength are kept constant for $t>0$. For state $|1,1,e\rangle$, (a) the expectation value $\langle \cos \theta \rangle$ with $I_0=2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ and $\beta =0^{\circ }$ (solid line), $\beta ={30}^{\circ }$ (dashed line), and $\beta ={45}^{\circ }$ (dotted line) and (b) the value $\langle \cos {\theta }_{\mathrm{s}}\rangle$ with $\beta ={30}^{\circ }$ and $I_0=2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ (solid line), $I_0=5\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ (dashed line), and $I_0={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ (dotted line). The dc field is $E_{\mathrm{s}}=300\mathrm{V}/\mathrm{cm}$.

Figure 16

For the ground state, (a) expectation value $\langle \cos \theta \rangle$ and (b) weight of the adiabatic ground state on its time-dependent wave function as a function of $E_{\mathrm{s}}\left(t\right)$, for turning-on speeds $v_{\mathrm{s}}={10}^{10}$ Vcm${}^{-1}$s${}^{-1}$ [dashed (orange) line], $v_{\mathrm{s}}={10}^9$ Vcm${}^{-1}$s${}^{-1}$ [solid (red) line], $v_{\mathrm{s}}={10}^8$ Vcm${}^{-1}$s${}^{-1}$ [dotted (blue) line], and $v_{\mathrm{s}}={10}^7$ Vcm${}^{-1}$s${}^{-1}$ [dot-dashed (pink) line], and adiabatic results (thin solid line). The fields are parallel and $I_0=2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$.