# Quantum pendula

Our interest in pendular states in quantum mechanics is mainly motivated by the physical and chemical significance of molecular rotation. While typical measurements in gas or liquid phase involve an averaging over angular distributions, controlling the rotational degrees of freedoms by external fields can narrow these distributions leading to the formation of pendular states. This is of paramount importance, e. g., for molecular imaging techniques as well as for the control of molecular reaction dynamics. Typical approaches to orient molecules are based on the (first-order) interaction of electrostatic fields with molecular dipole moments. Alignment of molecules can be achieved by non-resonant (second-order) interaction of intense optical (laser) fields with molecular (anisotropic) polarizabilities; in many cases a combination thereof appears most promising. Neglecting the influence of all other degrees of freedom, intramolecular rotations can be modeled by a single angular variable (quantum particle on a ring), while the description of external rotation of linear molecules requires two angles (quantum particle on a sphere). Hence, these two cases closely resemble the quantum planar or spherical pendulum, respectively, in between the limiting cases of free rotation and libration (strongly hindered rotation).

See also our 2018, 2022 press releases

from our work in Ref. [69]

## Static alignment and orientation of pendular states

*Burkhard Schmidt with** Marjan Mirahmadi** and Simon Becker*

*Cooperation with B. Friedrich and K. Schatz (Dept. of Molecular Physics, FHI Berlin)*

*Support by Deutsche Forschungsgemeinschaft (DFG), grant SCHM 1202/3-1 (FR 3319/3-1)*

For the case of pure first-order (orienting) or pure second-order (aligning) interaction, the corresponding pendular states can be expressed in terms of Mathieu functions (planar case) or spheroidal wave functions (spherical case). No closed expressions can be given for these functions but certain asymptotic properties are known. Both orienting and aligning interactions lift the degeneracies of the free rotor states; the double-well potential of the latter one gives rise to formation of near-degenerate tunneling pairs.

Combined orienting and aligning interaction (arising e. g. for permanent and induced electric dipole interactions of polar and polarizable molecules with collinear electric fields) leads to a unique topology of the surfaces of eigenenergies and of other observables - such as alignment and orientation cosines - in the plane spanned by the permanent and induced dipole interaction parameters. Given that the crossings only arise for the combined interaction, the dynamics revealed herein is *sui generis*, with no counterpart in the dynamics due to either the orienting or aligning interaction alone.

Interestingly, the loci of the intersections of the surfaces can be traced analytically and the eigenstates as well as the number of their intersections can be characterized by a single topological index. For certain integer values of this index, distinctive for a particular ratio of the interaction parameters, analytic expressions for wavefunctions and certain eigenproperties can be constructed. This is based on the apparatus of supersymmetric quantum mechanics and on the theory of conditionally quasi-exact solvability, see our work on pendular states of planar rotors [69] [76] and spherical rotors [68] [72], as well as spherical tops [79].

from our work in Ref. [53]

## Dynamic alignment and orientation of pendular states

Burkhard Schmidt with Marjan Mirahmadi

Cooperation with B. Friedrich and M. Karra (Dept. of Molecular Physics, FHI Berlin)

In our work on quantum dynamics of pendular states, we try to derive analytical approach as far as possible. This includes the case of ultra-short pulses (sudden limit) as well as instantaneous switch-on and switch-off of orienting and/or aligning interactions, the latter of which leads to pendular analogues of coherent and squeezed states. Full and fractional revivals as well as spatio-temporal structures in the time-evolution of the probability densities (quantum carpets) have been quantitatively analyzed for planar [53], [84] as well as spherical pendula [80]. For the related case of a symmetric double well potential also the question of the tunneling velocities has been addressed [67].

Quantum dynamics of the above scenarios can be characterized as a function of pulse intensity and duration. In the limit of long pulses, the pendular states follow the pulse envelope adiabatically. In the (sudden) limit of short pulses, the induced quantum dynamics is non-adiabatic, exhibiting rich revival patterns [53], [80]. In the intermediate regime, sudden drops of the kinetic energy and the mean orientation imparted by finite length pulses are found [85]. This applies also to the case of combined orienting and aligning interactions exhibiting novel phenomena that cannot be brought forth or explained by either of the two interactions alone; observables such as the orientation cosine reflect the topology of the system's eigensurfaces [84].

Suitable combinations of the two limiting scenarios can be used to achieve spectral selection. While rotational ground state can be transformed into aligned states via rotational heating [55], rotationally excited states can also be anti-aligned via rotational cooling [60]. The revival intensity patterns in transient alignment of molecules in dense gaseous mixtures can also be used to determine timescales of dissipation and/or decoherence thus allowing to characterize underlying collision processes [58]. Note that molecular alignment also occurs as a "side effect" in molecular photodissociation, see our study on centrifugal fragmentation and light-induced conical intersections in the photodissociation of H_{2}^{+} in intense laser fields [59].

from our work in Ref. [47]

## Alignment and orientation of matrix isolated molecules

Burkhard Schmidt with Antonio Accardi and Matthias Berg (Dept. of Chemistry, FU Berlin)

Cooperation with Toni Kiljunen (Dept. of Chemistry, Jyväskylä, Finland)

Molecules embedded in inert clusters or matrices (crystals) are known to undergo librations, i. e., rotational motions which are - to a certain extent - hindered by the "solvent" environment, depending on sizes and interaction strengths [71]. For examples, see our work on diatomic molecules in rare gas matrices [A], [C] [35], [45]. We also showed that the photodissociation yield, and possibly also the further fate of the photofragments, of small molecules embedded in rare gas clusters or matrices can be controlled by appropriate choice of their initial rotational state [C], [27], [28].

These findings have been the main motivation for our work on manipulation of external degrees of freedom of matrix isolated molecules. The central question is whether the techniques successfully established in gas phase alignment and orientation can be transferred to this case as well. Indeed, we could show that even in the case of competitive directions of internal (crystal) and external (laser) fields photo-induced alignment can be achieved [47], [49]. In the time-dependent domain two limits can be distinguished: Relatively (compared with molecular rotational period) long pulses can create alignment adiabatically, while ultrashort pulses can create non-adiabatic alignment with strong signals also in the field-free regime [51], [D].

According to Felix Bloch, Erich Hückel "incited and helped" the students at the University of Zürich to write poems about their great professors.^{[7]} The poem about Erwin Schrödinger went like this:

Gar Manches rechnet Erwin schon

Mit seiner Wellenfunktion.

Nur wissen möcht' man gerne wohl

Was man sich dabei vorstell'n soll.

Erwin with his psi can do

Calculations quite a few.

But one thing has not been seen:

Just what does psi really mean?