## Detuning in Coupled Oscillatory Circuits

We all know them: twins. Either they love and support each other or they are fighting to find out who is the best. The reason is that they are almost equal. In this problem we will find out why they behave as they do understanding coupled oscillatory circuits.

## Problem Statement

Two RLC-circuits are inductively coupled via an inductance $$L_{c}$$. Determine the currents in both circuits. Under which condition can the currents in both circuits become infinite? Find the corresponding frequencies $$\omega_{\pm}$$.

Determine the difference $$\omega_{+}-\omega_{-}$$ if the resonance frequencies of the single circuits are equal. Calculate this so-called detuning for sufficiently small mutual inductance.

Verify that you find the same results using an eigenfrequency analysis assuming vanishing resistances.

## Background: Coupled Systems - Avoided Crossing, Hybridisation and all that

Two coupled systems form a combined system. If some energies of the original systems are comparable, the combined system may have drastically altered characteristics. We may exemplarily consider some coupled harmonic oscillator, see figure on the left: two equal masses attached to some walls with springs of equal modulus. If there was no coupling at all, both masses would oscillate at the same frequency. If, however, both masses are connected with a third spring, there will be two eigenmodes of the system: a symmetric and an antisymmetric oscillation of the two masses. Both oscillations will happen at different frequencies. This detuning strongly depends on the coupling and is rather general. It can be observed in different fields of the natural sciences where we find different names for the phenomenon: avoided crossing for coupled two-level systems in quantum mechanics, hybridisation for interacting orbitals in chemistry, energy splitting in coupled mode theory and so on. There is probably also a name for it in twin studies for supportively and destructively “coupled states”...

However, a more physical state-of-the-art example is “Direct Observation of Controlled Coupling in an Individual Quantum Dot Molecule” published in Physical Review Letters 94 (2005) by Krenner et. al. The authors report the anticrossing for coupled quantum dots and how it can be tuned by an external electric field.