Atomic Tunneling from an STM/AFM tip: Dissipative Quantum effects from phonons

Don Eigler and colleagues at IBM found that a Xenon atom stuck to an STM tip could spontaneously jump to a surface if brought close enough. They modelled this by assuming thermal hopping in a double well potential with a barrier height Vo, as shown in the figure below:

Thermal Atomic Hopping:

A barrier height of for example Vo = 0.75 eV gives a transfer rate of about one second at room temperature.

Intrigued, we asked ourselves the question: what happens if the atom tunnels?

To answer this, we must first convince ourselves that it does. We model the potential the atom feels by a double well:

Double Well Potential

and find that for small separation (1-2 Angstroms), low barrier heights and light atoms, the tunneling probablility is significant.

But this isn't the whole story! An STM exerts a significant force on a surface, typically of the order of NanoNewtons (that's 0.000000001 Newtons, roughly the scale of weak chemical bonds). The force is different, depending on the position of the atom, and the surface will respond to this difference as shown below:

Response of Surface

In other words, the surface ``measures'' the position of the atom. Quantum Mechanics doesn't allow the strict dichotomy between ``observer'' and ``observed'' we're used to in our macroscopic world. We can't observe or measure a system without affecting it, and thus the surface must affect the tunneling. The question now is how?

First, we rewrite the response of the surface in terms of the lattice vibrations called phonons, and take some hints from the well known problem of an atom tunneling between two positions inside a bulk-material. The effect of the bulk-material is qualitatively accounted for by taking the so-called overlap integral between the state with the atom in position one, and all the phonons relaxed to accomodate that, with the state with the atom in position two, and all phonons relaxed to accommodate that. With the phonons taken into account, the atom still tunnels, but at a lower frequency, a well establised experimental fact. When we try the same approach for the atom tunneling between the STM tip and the surface we find no tunneling at all, a surprizing result! It turns out that the low frequency phonons are the culprits, surpressing all the tunneling by a logarithmic overlap catastrophe.

Finite Tunneling Frequency

But wait, lattice vibrations with frequencies much lower than the tunneling frequency don't have time to relax as we assumed in the calculation of the overlap integral, and we really shouldn't include them in the overlap integral. If we do this self-consistently, we get the same result obtained from more careful calculations: For weak coupling to the surface, the atom still tunnels, but as the coupling gets stronger, there is a transition to no tunneling at all. It turns out that our atom+phonons system is an example of tunneling in the presence of ohmic dissipation , a term coined by Tony Leggett from the University of Illinois at Champaign-Urbana, and colaborators who were working on the problem of Macroscopic Quantum Coherence.

The strength of the dissipation can be characterized by a coupling parameter alpha, and at zero temperature, the atom oscillates between the two postitions for alpha less than one, while for alpha greater than one, there is no tunneling at all. The atom just sits there. At finite temperatures the coherent oscillations slowly break down, and cross over to incoherent hopping. All this is summarized in the figure below:

Temperature-Coupling Diagram

The finite temperature behavior is interesting in and of itself. The rate turns out to be proportional to the temperature to the power (2alpha -1), and thus decreases with increasing temperature if alpha < 1/2, mimicing the so-called Quantum Zeno effect. As the temperature rises, the substrate vibrates more, more phonons are excited, and the combined effect of each phonon ``measuring'' the position of the atom slows down the tunneling.

An interesting aspect of our proposed experiment, is that we can exactly calculate the coupling constant alpha in terms of the elastic constants, the density and the force difference between the positions of the atoms. These can be independently measured, and thus an experiment could carefully test all these theories of how the environment (phonons) affects an embedded quantum system (our tunneling atom).

Finally, you may ask, what does all this have to do with Schroedinger's cat. ? Well, when the atom is coherently tunneling between the tip and the surface, Quantum Mechanics tells us the state of the atom must be described by a ``superposition''. The atom is at the tip and the surface at the same time as it were, much like the cat is in a superposition between dead and alive because it's fate is coupled to the quantum atom decay. However, as our atom+surface system shows, the effect of an environment on a quantum superposition can be drastic, even destroying the superposition, thus hinting that for all practical purposes, the cat's wave function will be ``collapsed'' into either the dead or the alive state.

More detailed info can be obtained by reading our paper. (Phys. Rev. Lett. 74 , 1363 (1995))

This research was paid for by YOUR FEDERAL GOVERNMENT through the National Science Foundation (NSF under Grant No. DMR-19-18065).
Last modified: May 31, 1994

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