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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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