Quantum control of molecular rotation

The angular momentum of molecules, or, equivalently, their rotation in three-dimensional space, is ideally suited for quantum control. Molecular angular momentum is naturally quantized, time evolution is governed by a well-known Hamiltonian with only a few accurately known parameters, and transitions between rotational levels can be driven by external fields from various parts of the electromagnetic spectrum. Control over the rotational motion can be exerted in one-, two-, and many-body scenarios, thereby allowing one to probe Anderson localization, target stereoselectivity of bimolecular reactions, or encode quantum information to name just a few examples. The corresponding approaches to quantum control are pursued within separate, and typically disjoint, subfields of physics, including ultrafast science, cold collisions, ultracold gases, quantum information science, and condensed-matter physics. It is the purpose of this review to present the various control phenomena, which all rely on the same underlying physics, within a unified framework. To this end, recall the Hamiltonian for free rotations, assuming the rigid rotor approximation to be valid, and summarize the different ways for a rotor to interact with external electromagnetic fields. These interactions can be exploited for control—from achieving alignment, orientation, or laser cooling in a one-body framework, steering bimolecular collisions, or realizing a quantum computer or quantum simulator in the many-body setting.

Figure 1

Schematic picture of the correspondence between molecules and classical objects. Two examples of rigid asymmetric tops, the chlorobenzene molecule ${\mathrm{C}}_6{\mathrm{H}}_5\mathrm{Cl}$ and a tennis racket, are represented in the top panel. The bottom panel displays a prolate symmetric top, the iodomethane molecule ${\mathrm{CH}}_3\mathrm{I}$ and its classical analog. The body-fixed frame $(\mathbf{x},\mathbf{y},\mathbf{z})$ is defined for each case.

Figure 2

Definition of the Euler angles $(\theta ,\varphi ,\chi )$, which are used to describe the position of the body-fixed frame $(\mathbf{x},\mathbf{y},\mathbf{z})$ in the space-fixed frame $(\mathbf{X},\mathbf{Y},\mathbf{Z})$.

Figure 3

Trajectories of the angular momentum of a rigid body in the body-fixed frame $(\mathbf{x},\mathbf{y},\mathbf{z})$. The blue (dark gray) and red (light gray) lines represent, respectively, the rotating and oscillating trajectories of the angular momentum. The dashed black line is the separatrix.

Figure 4

Energy-momentum diagram of the water molecule. The dashed red line depicts the position of the separatrix and the solid black lines the boundary of the accessible EM. The blue dots indicate the position of the energy levels in the diagram. We use the convention $J=h(j+\frac{1}{2})$, with $h=1$ and $j$ is the quantum number defined in Eq. (6).

Figure 5

Schematic description of the tennis racket effect. Note the $\pi$ flip of the head of the racket when the handle makes a rotation. From [541].

Figure 6

Angular distribution of the ${\mathrm{CO}}_2$ molecule after interaction with two short pulses polarized in orthogonal directions. Note the planar alignment of the molecule in the $(x,y)$ plane. From [204].

Figure 7

Chiral-sensitive three-wave mixing microwave spectroscopy: Right-handed and left-handed enantiomers of chiral molecules with $C_1$ symmetry possess three orthogonal electric dipole moment components, identical in magnitude but differing in sign for one of the components. When rotational transitions are driven using two of the components and fluorescence is collected along the third direction, the different sign accumulated in the three-wave mixing results in a relative phase of $\pi$ between the two enantiomers. From [390].

Figure 8

Experimental observation of shape resonances in Penning ionization reactions. From [188].

Figure 9

Time dependence of the alignment cosine of ${\mathrm{CH}}_3\mathrm{I}$ molecules after a 450 fs laser pulse in helium droplets (red) compared to the gas phase (black). From [396].

Figure 10

The first proposal of a quantum computer based on ultracold molecules in an optical lattice. From [102].

Figure 11

(a) Phase diagram of dipolar bosons on a two-dimensional optical lattice. Supersolid (SS), superfluid (SF), as well as devil’s staircase (DS) are indicated. (b)–(d) Examples of Mott solids with densities of $1/2$, $1/3$, and $1/4$, respectively. From [62].