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ISIS Report 17/03/04

Quantum Phases and Quantum Coherence

Dr. Mae-Wan Ho investigates the link between quantum phases and quantum coherence, and comes up with the surprising implication that the universe itself may be quantum coherent

A fully illustrated version of this article with sources is posted on ISIS Members’ website. here.

Quantum phases

In classical electromagnetic theory, the vector and scalar potentials in Maxwell’s equations of the electromagnetic field are purely mathematical entities without physical significance, but not so in quantum mechanics.

In 1959, quantum physicists Yakir Aharonov and the late David Bohm at Birbeck College, London University, proposed a thought-experiment in which an electron beam is directed towards a long thin magnetic field trapped inside a tightly-wound coil of electric wire, a solenoid. The beam is split into two so that one passes to the left and the other to the right of the solenoid. Although the magnetic field is zero outside the solenoid, the vector potential associated with the field is not zero. Aharonov and Bohm predicted that, according to quantum theory – described by Schrödinger’s wave function - the two beams, being wave-like, would acquire different phases due to their interaction with the vector potential, even though the field itself was zero, and that the difference between these phases could be detected via interference.

The effect was soon observed in experiments. One of the more remarkable experiments to demonstrate the Aharonov-Bohm (AB) effect was carried out by Akira Tonomura of Hitachi Laboratory in Japan, using microscopic toroidal (doughnut shaped) magnets. In such a magnet, the magnetic field is trapped inside, and no magnetic field exists either in the hole in through the middle or in the space outside the torus. So, electron beams split into components that pass through the middle and the outside would be expected to have a different phase.

In the 1970s, Tonomura succeeded in making a highly coherent electron microscope – with coherence more than two orders of magnitude higher than that of a conventional electron microscope. This was the basis of a technique known as electron holography, an interference electron microscopy that could be used to show up the contour map of the electron phase. This enabled him to pick out perfect magnets that do not leak magnetic fields to the outside.

In 1982, using a holography electron microscope Tonomura and his colleagues measured the phase difference in the form of interference fringes produced by two beams of electrons, one passing through the inside and the other passing through the outside of a doughnut-shaped magnet. They clearly showed that there is a phase difference between the two electrons beams passing through regions of space that have no magnetic field, and that the extent of the phase difference precisely matches the predicted value.

Soon after publishing the results, an objection was raised to this experiment. As the electron beams contacted the magnet, i.e. there were leakage magnetic fields in space, that could have created the phase differential. If so, the experiment did not prove the existence of the AB effect.

To settle this dispute, Tonomura and his colleagues made a doughnut-shaped ferromagnet six micrometers in diameter (Fig. 1(a) and (b)), and covered it with a niobium superconductor to completely confine the magnetic field within the doughnut. With the magnet maintained at a temperature of 5 K, they measured the phase difference from the interference fringes between one electron beam passing though the hole in the doughnut and the other passing on the outside of the doughnut. As can be seen in Fig. 1(a), interference fringes are displaced with just half-fringe spacing inside and outside of the doughnut, proving the existence of the AB effect. In other words, electrons passing through regions free of any electromagnetic field, nevertheless interacted with the vector potentials in the regions. Tonomura received numerous prizes for this work.

Figure 1. Aharonov-Bohm effect caught in the act.

Independently in 1984, Michael Berry at Warwick University, Coventry, UK, showed that quantum systems could acquire what is known as a ‘geometric phase’ under certain conditions. He was studying quantum systems in which the Hamiltonian (energy equation) describing the system is slowly changed so that it eventually returns to its initial form. Berry showed that the ‘adiabatic theorem’, widely used to describe such systems, was incomplete. In particular he found that the system acquired a ‘phase factor’ that depended on the path followed, but not on the rate at which the Hamiltonian was changed. This geometric phase factor, now known as the ‘Berry phase’, was later shown to be a generalization of the AB effect. They arise essentially because of the inherently wave-like character of all quantum systems.

Both the AB effects and the Berry phases have been incorporated into many fields of physics including optics, nuclear physics, fluid physics, chemistry, molecular physics, string theory, gravitational physics, cosmology, solid-state physics, the foundations of quantum mechanics and quantum computing. Aharonov and Berry eventually shared the 1998 Wolf Prize for their work.

Why should quantum phases turn up everywhere, from the microscopic to the macroscopic and even to the cosmological scale? Quantum phases depend ultimately on the coherence of quantum systems; and the fact that they turn up over all scales implies that nature is quantum, and not classical.

Quantum coherence

Indeed, quantum coherence (see Box) has come up in cosmology and quantum computation. According to a quantum cosmological model in which the universe comes into being via a period of ‘cosmic inflation’, the universe will naturally end up in a coherent superposition of classical states; and in quantum computation, people are asking whether anything interesting or useful will result by allowing computers to evolve as coherent superposition of computational states.

What is quantum coherence?

‘Coherence’ is generally understood as ‘wholeness’, a correlation over space and time. Atoms vibrating in phase, teams rowing in synchrony in a boat race, choirs singing harmony, troops dancing in exquisite formations, all conform to our ordinary notion of coherence.

Quantum coherence implies all that and more. Think of a gathering of consummate musicians playing jazz together (‘quantum jazz’) where every single player is freely improvising from moment to moment and yet keeping in tune and in rhythm with the spontaneity of the whole. It is a special kind of wholeness that maximizes both local freedom and global cohesion.

For a succinct technical definition, I like the one offered by cosmologist Andreas Albrecht at University of California at Davis, United States.

"Quantum mechanics is different from classical mechanics in several ways. Firstly, the state of a system is defined most fundamentally by probability amplitudes (the "wavefunction") which must be squared to get the probabilities. Secondly, the space of possible quantum states is quite different from its classical counterpart. Positions and momenta cannot be both specified precisely..

"To the extent that the probabilities are all one needs, I will say one is working with a "classical" probability distribution, regardless of whether the actual space of possible states has quantum mechanical features or not. To the extent that one needs to know the initial probability amplitudes (rather than just the probabilities) in order to do the right calculation, I will say that the system exhibits "quantum coherence".

Albrecht (see Box) is implying that all quantum systems exhibit quantum coherence. And the complex phase of the wave function matters in quantum coherent systems.

But why is it, if nature is fundamentally quantum mechanical, that we see it predominantly as classical in our everyday life?

That is because a quantum system enters into quantum entanglement with the observer. So, how one chooses to observe the system determines what is being observed. There is a "very subjective" element, said Albrecht. We must choose to use an observational basis that reveals the quantum mechanical properties, that preserves quantum coherence, in order to observe them.

But there are practical limitations on the types of measurements that are possible in practice, and most measurements will tend to destroy the quantum coherence of the system in any case.

The reason it took so long to discover quantum coherence is because there is very good agreement (among observers and environments) as to what basis one uses both to measure and describe the world – namely a basis which closely approximates wave packets fairly localized both in position and momentum, and is hence classical.

As heavy macroscopic things tend to have localized position and momentum, "the only quantum coherence effects have a natural interpretation in terms of a classical momentum, as long as one sticks to this "classical" basis". Furthermore, as the environment gets correlated with these macroscopic things in the same basis, the effects of decoherence are unimportant. For these reasons, the effects of quantum coherence are definitely not part of everyday experience, and, according to Albrecht, "it takes clever experimentalists to set up a situation to observe such effects"

One way to observe quantum effects, as we have seen ("Nature is quantum, really", this series), is by "weak measurement", using very sensitive equipment.

Does this not also say something to us about non-invasive, non-destructive and non-violent ways of studying living systems and relating to the world in general?

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