Physical Sciences
February 18, 2022

Quantum carpets from tumbling molecules: Manipulating molecular rotation by electromagnetic fields

Molecules are comprised of charged particles whose spatial distribution is or could be distorted, lending the molecules a handle by which they can be manipulated via external electromagnetic fields. Dr Burkhard Schmidt at the Freie Universität Berlin and Professor Bretislav Friedrich at the Fritz Haber Institute of the Max Planck Society in Berlin, along with their affiliates Dr Marjan Mirahmadi and Mr Mallikarjun Karra, have been studying the effects of electromagnetic fields on molecular rotation and how these effects could be used to achieve quantum control.

The distribution of electrical charges in molecules is often non-uniform, making such molecules polar, ie, endowed with what is called a permanent electric dipole moment. Even the charge distribution of non-polar molecules can be made non-uniform by action of a strong electromagnetic field such as that of a focused laser beam, which polarises the molecules and lends them an induced electric dipole moment. The dipole moments – permanent and/or induced – can then be used as handles by which the molecules can be manipulated via external electromagnetic fields. Such manipulation can be aimed either at controlling the rotation or the translation of the molecules – or both. While the control of molecular translation encompasses deflection of molecular trajectories and confinement of molecules in traps, the control of molecular rotation entails molecular orientation and alignment. It is the latter that is the subject of the work highlighted herein. Its applications range from reaction dynamics to Stark spectroscopy to high-order harmonic generation and orbital imaging to quantum computing and simulation.

Figure 1. A unipolar electromagnetic pulse. Apart from the larger amplitude of the positive branch than the negative one, the field strength has a different temporal dependence over the two branches, with durations τ+ and τ, respectively. Note that the surface areas under the positive and above the negative branch of the pulse are equal. The positive branch is well approximated by a Gaussian pulse. Friedrich, B (2021) Manipulation of Molecules by Combined Permanent and Induced Dipole Forces. In Sason Shaik, Thijs Stuyver (eds.) The Royal Society of Chemistry, doi.org/10.1039/9781839163043

The ability of electromagnetic fields to interact with the dipoles of molecules has been the focus of a research team in Berlin, Germany. Research leaders Dr Burkhard Schmidt and Professor Bretislav Friedrich, alongside Dr Marjan Mirahmadi and Mr Mallikarjun Karra, have been exploring how polar and polarisable rotors interact with electromagnetic fields in general and how the exact shape and duration of an electromagnetic pulse influence molecular rotation in particular.

By applying an external electromagnetic field, it is possible to manipulate the rotation of a molecule.

Quantum dynamics
The molecular systems Schmidt, Friedrich and colleagues have been focusing on are linear polar molecules (such as NO or OCS) and symmetric top molecules (such as ammonia, NH3, or iodomethane, CH3I). If such a molecule is left unperturbed, it rotates with a period given by factors such as the masses of the constituent atoms and their mutual distance. By applying an external electromagnetic field, it is possible to manipulate the rotation of the molecule.

Figure 2. Quantum carpets, ie, dependences of the wavefunctions squared on time, for delta kicks of a particular nature and strength. The top, middle, and bottom panels correspond, respectively, to cases when the delta kick is orienting, aligning, or both. An orienting delta kick acts on the permanent dipole moment of the molecule. When the delta kick acts on the induced dipole, the interaction is aligning. The quantum carpets reveal complex interference patterns that are related to the space–time structure of the full and fractional revivals of the rotational wavepackets created by the different types of delta kicks. The axes of the figure are the pulse duration τ and the polar angle θ between the molecular axis and the polarisation vector of electromagnetic pulse.

The researchers examined what happens when a polar and polarisable molecule (every molecule is polarisable but not every molecule is polar) is acted upon not by many cycles of an electromagnetic wave, as is usually the case, but only by a half of one cycle (called a half-cycle pulse – HCP). Figure 1 provides an example. In such a case, the electric field of the HCP acts on both the permanent and induced electric dipole moments of the molecule. This gives rise to intriguing dynamics captured by the population quilts and quantum carpets (the latter shown in Figure 2). The electric field’s main effect is the creation of rotational wavepackets – combinations (termed superpositions) of rotor states whose composition changes periodically with time. It is a prerogative of quantum systems to exist in superpositions of different states, which amounts to being in different states at the same time. The quantum carpets reveal that after the passage of the pulse, the rotational wavepackets and the directional properties – orientation and alignment – they encapsulate undergo periodic revivals. This work is an important development in helping us understand what types of electromagnetic pulses are required to control the orientation and alignment of molecules. The team notes that orientation can be visualised by single-headed arrows whereas alignment is visualised by double-headed arrows.

Figure 3. a) Dependence of the electric field strength ε on time t for a rectangular pulse of duration s. b) Rotational kinetic energy imparted to a polar rigid rotor as a function of pulse duration expressed in terms of the rotational period of the molecule. The values of the pulse duration (in terms of the rotational period) for which the imparted rotational energy suddenly drops are indicated. Please note the logarithmic scale of the energy axis. c) Post-pulse orientation for different initial rotational states, J0=0, 1, and 2.
Modified from Karra et al, (2021) Molecular Physics, 119:17-18, doi.org/10.1080/00268976.2021.1966111

In their study, the HCP was approximated either by a temporal delta pulse or by a temporal Gaussian pulse. While the former has no width (this is what ’delta’ means), the latter has a temporal width (or length) that can be varied. For the delta pulse, the researchers were able to solve the Schrödinger equation for the dynamics of its interaction with the permanent and induced dipole moments of the molecule analytically (what is temporally long or short is determined by comparison with the rotational constant of the molecule). This provided a rare insight into the dynamics of the rotational wavepackets that are created by an HCP.

It is a prerogative of quantum systems to exist in superpositions of different states – which amounts to being in different states at the same time.

For the Gaussian pulse, the problem is not analytically solvable, but can be solved numerically with high accuracy. The team’s numerical solution of the Schrödinger equation for the Gaussian pulse of very short duration (approximating a delta pulse) reproduced faithfully the analytic solutions they found for a delta pulse. For longer HCPs, the dynamics deviated from the ’delta-pulse dynamics’. In particular, they found that at certain pulse durations, the rotational energy and orientation imparted by the Gaussian pulse to the polar and polarisable rotor exhibits quasi-periodic drops. At first, they suspected numerical artifacts, but a thorough analysis revealed that the drops were real.

Figure 4. This is what the logarithm of the sinc function of time looks like.

Sudden drops in imparted energy and orientation explained analytically
In a recent publication (Karra, Schmidt and Friedrich, 2021), the authors provided a physical explanation for the quasi-periodic drops. In order to make the problem analytically tractable – and thereby their analysis insightful – they approximated the Gaussian pulse with a rectangular electric pulse (Figure 3). This analysis revealed that the drops result from the oscillating populations of rotational states that make up the wavepacket created by the pulse, and that they are necessarily accompanied by drops in the orientation and by a restoration of the pre-pulse alignment of the rotor. The ’oscillating populations’ arise due to rotational coherences (or phases) driven by the electric pulse whose probability is proportional to a sinc function of pulse duration, sinc τ  = sin τ/τ (Figure 4). By making use of a simplified model comprised of just two rotational states for the interaction of the rectangular pulse with the rotor, the authors were able to recover the sinc probabilities and to predict analytically at which pulse durations the sudden drops will occur. This previously unnoticed effect is an important one: by choosing the wrong pulse duration, the desired effect of the pulse – creation of a particular rotational wavepacket – will not and cannot be accomplished.

Personal Response

 What was the biggest challenge in deriving these new analytical expressions?

To find an approximation – which is what these analytic results are – that is simple enough to provide insight and yet accurate enough to check against key aspects of the numerical calculations.

This feature article was created with the approval of the research team featured. This is a collaborative production, supported by those featured to aid free of charge, global distribution.

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