Electro-optical trap for polar molecules

A detailed treatment of an electro-optical trap for polar molecules, realized by embedding an optical trap within a uniform electrostatic field, is presented and the trap's properties analyzed and discussed. The electro-optical trap offers significant advantages over an optical trap that include an increased trap depth and conversion of alignment of the trapped molecules to marked orientation. Tilting the polarization plane of the optical field with respect to the electrostatic field diminishes both the trap depth and orientation and lifts the degeneracy of the $\pm M$ states of the trapped molecules. These and other features of the electro-optical trap are examined in terms of the eigenproperties of the polar and polarizable molecules subject to the combined permanent and induced electric dipole interactions at play.

Figure 1

Electrostatic field ${\mathit{\varepsilon }}_1$ and optical field ${\mathit{\varepsilon }}_2$ of a linearly polarized laser acting jointly on the molecular dipole moment $\mu$ and the parallel ${\alpha }_{\parallel }$ and perpendicular ${\alpha }_{\perp }$ components of the molecular polarizability. Also shown are the polar angle $\theta$ between the space-fixed axis $Z$, as defined by the common direction of the field vectors, and the body-fixed (molecular) axis $z$, as well as the projection $M$ of the angular momentum $\mathbf{J}$ on the $Z$ axis together with the uniformly distributed azimuthal angle $\varphi$ of $\mathbf{J}\perp z$ about $Z$. For further details see the text and Appendix pp1.

Figure 2

(a) Induced dipole potential ${\overline{V}}_{\alpha }\left(\theta \right)$ [Eq. (15)] as a function of the polar angle $\theta$. Note that ${\overline{V}}_{\alpha }\left(\theta \right)$ is a double-well potential with equivalent minima $-{\zeta }_{\parallel }$ at $\theta =0^{\circ }$ and ${180}^{\circ }$ and a maximum $-{\zeta }_{\perp }$ at $\theta ={90}^{\circ }$. The potential (in blue) has been drawn and the energy levels calculated for $\mathrm{\Delta }\zeta =10$ and ${\zeta }_{\perp }=2.5$, which implies ${\zeta }_{\parallel }=12.5$. At these values of $\mathrm{\Delta }\zeta$ and ${\zeta }_{\perp }$, only one tunneling doublet comprised of the $|0,0;10\rangle$ and $|1,0;10\rangle$ states is created by the double-well potential. Note that while the lower member $|1,1;10\rangle$ of the next tunneling doublet is bound by the ${\overline{V}}_{\alpha }\left(\theta \right)$ potential, its upper member $|2,1;10\rangle$ is not. Generally, the tunneling splitting decreases for more deeply bound doublets. Also note that the members of a given tunneling doublet have the same $\left|M\right|$ but opposite parity (levels with $p=+1$ are shown in green and levels with $p=-1$ are shown in ochre). The ${\zeta }_{\perp }$ term shifts the potential, as well as the energy levels it binds, uniformly along the energy axis. The calculated energy levels bound by the induced dipole potential are also shown along with the numerical values of their eigenenergies (left) and alignment cosines (right). The blue double-headed arrow indicates that the states created by the induced dipole potential are aligned but not oriented. (b) Combined permanent and induced dipole potential $\overline{V}\left(\theta \right)={\overline{V}}_{\mu }\left(\theta \right)+{\overline{V}}_{\alpha }\left(\theta \right)$ (black) of form A drawn for $\eta =1, \mathrm{\Delta }\zeta =10$, and ${\zeta }_{\perp }=2.5$. The permanent dipole potential ${\overline{V}}_{\mu }\left(\theta \right)$ and the induced dipole potential ${\overline{V}}_{\alpha }\left(\theta \right)$ are shown in red and blue, respectively. The calculated energy levels bound by the combined potential are also shown along with the numerical values of their eigenenergies (left) and orientation cosines (right). For further details see the text.

Figure 3

(a) Dependence of the eigenenergies $E_{\tilde{J},\left|M\right|}/B$ on the induced dipole interaction parameter $\mathrm{\Delta }\zeta$ for the lowest six initial rotational states of a linear polar molecule in the absence of an electrostatic field ($\eta =0$). Even-parity levels are shown in green and odd-parity levels in ochre, with pairs of similarly color-coded vertical arrows indicating the tunneling splitting of the tunneling doublets comprised of the $|\tilde{J}=0,\left|M\right|=0;\mathrm{\Delta }\eta \rangle$ and $|\tilde{J}=1,\left|M\right|=0;\mathrm{\Delta }\eta \rangle$ and the $|\tilde{J}=1,\left|M\right|=1;\mathrm{\Delta }\eta \rangle$ and $|\tilde{J}=2,\left|M\right|=1;\mathrm{\Delta }\eta \rangle$ states [cf. Fig. 2]. The black dotted curve shows the dependence of the ground-state eigenenergy in the low-field limit (see also Table 2 and the text). (b) Dependence of the eigenenergies on $\mathrm{\Delta }\zeta$ in the presence of a superimposed electrostatic field that gives rise to a permanent dipole interaction with $\eta =10$. The red vertical arrows illustrate how the splitting of the tunneling doublets of (a) has been altered or enhanced by the superimposed electrostatic field. The blue arrow shows the position $\mathrm{\Delta }\zeta =25$ of the avoided crossing between the $|2,0;\eta =10,\mathrm{\Delta }\zeta =25\rangle$ and $|1,0;\eta =10,\mathrm{\Delta }\zeta =25\rangle$ states, corresponding to the topological index $k=1$ (see the text for details).

Figure 4

(a) Permanent dipole potential ${\overline{V}}_{\mu }\left(\theta \right)$ drawn for $\eta =10$ along with the calculated energy levels and their numerical values (left) and orientation cosine (right). The red vertical arrow indicates that the states created by the permanent dipole potential are oriented. (b) Combined permanent and induced dipole potential $\overline{V}\left(\theta \right)={\overline{V}}_{\mu }\left(\theta \right)+{\overline{V}}_{\alpha }\left(\theta \right)$ (black) of form B drawn for $\eta =10, \mathrm{\Delta }\zeta =1$, and ${\zeta }_{\perp }=0.25$. The permanent dipole potential ${\overline{V}}_{\mu }\left(\theta \right)$ and the induced dipole potential ${\overline{V}}_{\alpha }\left(\theta \right)$ are shown in red and blue, respectively. The calculated energy levels bound by the combined potential are also shown along with the numerical values of their eigenenergies (left) and orientation cosines (right).

Figure 5

Dependence of the depth of the optical trap for the $|\tilde{J}=0,\left|M\right|=0;\mathrm{\Delta }\zeta \rangle$ state on the induced dipole parameter $\mathrm{\Delta }\zeta$ for different values of the parameter ${\zeta }_{\perp }$. See the text for further details.

Figure 6

Comparison of the trapping potentials for a purely optical trap (blue) and for an electro-optical trap (red) for the same distribution of laser intensity $I(X,Z=0)$ and hence the same distributions of the induced dipole interaction parameters $\mathrm{\Delta }\zeta (X,0)$ and ${\zeta }_{\perp }(X,0)$ along the laser propagation direction $X$, for $\mathrm{\Delta }\zeta (X=0,Z=0)=1, {\zeta }_{\perp }(X=0,Z=0)=\frac{1}{4}$, and $\eta =10$ (cf. Fig. 4). Note that the trapping potential is expressed in terms of the rotational constant $B$ of the trapped molecule and that the length scale of the $X$ coordinate is expressed in terms of the wavelength $\lambda$ of the optical field.

Figure 7

(a) Dependence of the trap depth $U_0$ for the $|\tilde{J}=0,\left|M\right|=0;\mathrm{\Delta }\zeta ,\eta \rangle$ state on the induced dipole interaction parameter $\mathrm{\Delta }\zeta$ for different constant values of the permanent dipole interaction parameter $\eta$. (b) Enhancement of the trap depth compared to that of a purely optical trap as a function of the induced dipole interaction parameter $\mathrm{\Delta }\zeta$ for different constant values of the permanent dipole interaction parameter $\eta$. The dots indicate loci where $\mathrm{\Delta }\zeta =\eta /2$, i.e., form A and form B potentials are to the right and to the left of the dots, respectively. See the text for details.

Figure 8

(a) Dependence of the orientation cosine ${\langle cos\theta \rangle }_{0,0}$ of the $|\tilde{J}=0,\left|M\right|=0;\mathrm{\Delta }\zeta ,\eta \rangle$ state on the permanent dipole interaction parameter $\eta$ for different constant values of the induced dipole interaction parameter $\mathrm{\Delta }\zeta$. (b) Enhancement of the orientation cosine ${\langle cos\theta \rangle }_{0,0}$ of the $|\tilde{J}=0,\left|M\right|=0;\mathrm{\Delta }\zeta ,\eta \rangle$ state compared to that of the $|\tilde{J}=0,\left|M\right|=0;\mathrm{\Delta }\zeta =0,\eta \rangle$ state as a function of the permanent dipole interaction parameter $\eta$ for different constant values of the induced dipole interaction parameter $\mathrm{\Delta }\zeta$. Also shown are polar plots of the squares of the corresponding wave functions for $\eta =5$ and different values of $\mathrm{\Delta }\zeta$ whose color coding is the same as that of the labeled curves. See the text for details.

Figure 9

Trapping potential $U(X,Z)$ of Eq. (34) shown in red together with its harmonic counterpart $U_{\mathrm{H}}(X,Z)$ of Eq. (37) shown in gray, plotted for $Z=0$. Note that the harmonic trapping potential approximates the electro-optical trapping potential faithfully up to about the Rayleigh length $X_{\mathrm{R}}$.

Figure 10

(a) Dependence of the orientation cosine ${\langle cos\theta \rangle }_{0,0}$ of the $|\tilde{J}=0,\tilde{M}=0;\mathrm{\Delta }\zeta ,\eta \rangle$ state on the permanent dipole interaction parameter $\eta$ for different constant values of the induced dipole interaction parameter $\mathrm{\Delta }\zeta$ for perpendicular fields ${\mathit{\varepsilon }}_1$ and ${\mathit{\varepsilon }}_2$ (tilt angle $\beta =\pi /2$). Also shown are polar plots of the squares of the corresponding wave functions for $\eta =5$ and different values of $\mathrm{\Delta }\zeta$ whose color coding is the same as that of the labeled curves. The horizontal bars run through the origin of the polar coordinate system and divide the right-way (upward) and wrong-way (downward) oriented lobes of the wave functions. (b) Dependence of the trap depth $U_0$ for the $|\tilde{J}=0,\tilde{M}=0;\mathrm{\Delta }\zeta ,\eta \rangle$ state in perpendicular fields on the induced dipole interaction parameter $\mathrm{\Delta }\zeta$ for different constant values of the permanent dipole interaction parameter $\eta$. See the text for further details.

Figure 11

Transformation from the space-fixed frame $(X,Y,Z)$ to the body-fixed frame $(x,y,z)$, effected by the inverse direction cosine matrix $\mathrm{\Phi }$. Also shown is the parametrization of the transformation by the Euler angles $\varphi , \theta$, and $\chi$. (Figure has been adapted from Ref. [66].)