Quantum Beats in the Rubrene Fluorescence

The basic principles underlying the quantum beats in the fluorescence that arises via geminate fusion of the two triplet excitons created by fission.

When a singlet exciton undergoes fission, it is converted into two quantum states that still have, together, a total spin of 1. Under an applied magnetic field two different states can combine in the right way to give a total spin of 1, but these two states have a slightly different energy and evolve with different frequencies. Triplet exciton fusion can be seen as the projection onto a singlet state of the quantum state that describes the two entangled triplets. Since the triplet-pair quantum states evolve in a way dictated by the two slightly different energies of the two states that contribute to it, its projection onto a singlet state also evolves, giving a time-dependent, oscillating probability of fusion.

This time dependent fusion probability then determines the probability that a photon is emitted at any given time, and it ultimately modulates the fluorescence caused by geminate triplet fusion. This effect has been observed for the first time in 1981 by Chabr and Wild in tetracene crystals. But despite the fact that this observation happend quite a long time ago, tetracene has remained the only crystal where this kind of quantum beats have been studied until 2018.

The rubrene crystal is a close relative to tetracene, even though it has a different molecular arrangement. Under short pulse illumination it shows a strong delayed fluorescence that originates from geminate triplet exciton fusion, but without an applied magnetic field of at least 0.3 Tesla it doesn't show any visible quantum beats.

Under an applied magnetic field (for this figure in the direction of the 2-fold symmetry axis of the molecules in the crystal), the fluorescence induced by a short light pulse oscillates in time in addition to following a relatively complicated decay.

But the situation changes when a magnetic field is applied. Under such conditions there are two states with slightly different energies that combine to create the entangled triplet pair, and therefore the quantum beats have just one characteristic frequency.

This is shown in the figure to the left. The fluorescence decay painted in blue is the one with the applied electric field, and one can easily see slight modulations on top of it. These modulations can then be extracted basically in three ways: Take the ratio between the data measured with and without a magnetic field, subtract the non-oscillating background, or take the ratio between the data and its non-oscillating background. In our group we prefer the latter method because it is the ratio that would correlate better to the fraction of geminate triplet pairs that are spin-coherent and have a sinusoidally modulated fusion probability.

The frequency of the quantum beats depends on the energy difference between the states that contribute to the entangled triplet exciton pair, and depends on the direction of the applied magnetic field. It does not depend on the amplitude of the applied magnetic field for the simple reason that any magnetic-field-induced increase in the energy of one of the triplets in the geminate triplet pair is compensated exactly by the same decrease in the energy of its brother. What matters is the total energy of the pair, and this one is determined by the structure of the molecule at zero applied field.

Dependence of the quantum beat frequency on the direction of the applied magnetic field.

More information:

• Eric A. Wolf, Drew M. Finton, Vincent Zoutenbier, Ivan Biaggio, “Quantum beats of a multiexciton state in rubrene single crystals,” Appl. Phys. Lett. 112, 083301 (2018).

See also:

• Light, Excitons, and transport in Organic Molecular Crystals.
• The rubrene crystal.
• Absorption and luminescence spectroscopy.
• Singlet Exciton Fission.
• Triplet Exciton Fusion.
• Quantum Beats.
• Triplet Exciton Diffusion and Fusion.