Trimera: Efimov States
Quantum Foundations arc, page 16 of 193. Back to The Pixelated Physics Universe
Who they are
Trimera is three atoms linked in a ring. No two of them can hold together, but all three can. Around her, the rings repeat: every 22.7× bigger, another trio.
Trimera is a character of the Quantum Foundations arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
The physics
In 1970 Vitaly Efimov showed that three identical bosons with a resonant pair attraction can form three-body bound states, even where no pair can bind. In the unitarity limit, where the scattering length |a| is much larger than the range of the force, there is an infinite ladder of these trimers. Each one is eπ/s₀ ≈ 22.7 times larger than the one before, and its binding energy is about 22.7² ≈ 515 times smaller. The first evidence came in 2006 from Innsbruck: a giant three-body loss resonance in an ultracold gas of caesium atoms (Kraemer et al., Nature 440, 315). In 2014 the same lab found the second resonance, with a measured scaling factor of 21.0(1.3), close to the ideal 22.7 (Huang et al., Phys. Rev. Lett. 112, 190401).
$$ \begin{gathered} \frac{E_{n+1}}{E_n} \rightarrow e^{-2\pi/s_0} \approx \frac{1}{515}, \qquad e^{\pi/s_0} \approx 22.7 \\ s_0\cosh\frac{\pi s_0}{2} = \frac{8}{\sqrt{3}}\sinh\frac{\pi s_0}{6}, \qquad s_0 \approx 1.00624 \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $E_n$ | binding energy of the n-th three-body state | J |
| $s_0$ | Efimov's universal constant for three identical bosons | 1 |
| $e^{\pi/s_0}$ | size ratio between successive states | 1 |
Honest caveat: 22.7 is exact only for three identical bosons with zero-range forces at unitarity; other masses and spins give other values. In real atoms |a| and the range of the force are finite, so the ladder stops after a few steps. The 2006 result was a loss signal, not an image of a trimer.Rings drawn not to scale: each real Efimov trio is about 22.7× larger than the last, too big a step to fit on a shirt.
Sources
- Efimov (1970) Energy levels arising from resonant two-body forces in a three-body system, Phys. Lett. B 33, 563
- Braaten & Hammer (2006) Universality in few-body systems with large scattering length, Phys. Rep. 428, 259
- Kraemer et al. (2006) Evidence for Efimov quantum states in an ultracold gas of caesium atoms, Nature 440, 315
- Huang, Sidorenkov, Grimm & Hutson (2014) Observation of the Second Triatomic Resonance in Efimov's Scenario, Phys. Rev. Lett. 112, 190401
Crossovers
Trimera meets The Condensate Queen (Bose–Einstein Condensate). Trimera's Efimov trios are studied in ultracold atomic gases, the Condensate Queen's home ground, where magnetic Feshbach resonances can tune the atoms' scattering length. That is how the predicted three-body states were finally seen.
Trimera meets Trialpha (Hoyle State). Trialpha binds three alpha particles into carbon's Hoyle state. Trimera binds three atoms into Efimov states. Both are three-body quantum problems, but the Hoyle state is a single nuclear resonance, not an Efimov tower.
Wear the lore
Concept keywords: Efimov States, Trimera, Quantum Foundations, physics lore, Pixelated Physics