But there are examples of macroscopic systems that behave like quantum objects: superconductors, superfluids, quantum Hall devices, Bose-Einstein condensates (BEC). Some stars, heavier than the Sun, also behave like quantum systems: white dwarfs, neutron stars .
The quantum systems liable to be used in quantum computing are : photons produced by laser, trapped ions, cold atoms, electrons (e.g. in quantum dots), nuclear spins, (in NMR), Josephson junctions. Their typical size may vary enormously.
atoms 1-3 Å, namely 10 -10 m
nuclei 1-3 Fermis (F) namely 10-15m, i.e. 100,000 times smaller than an atom
electrons 10-18 m , namely 1000 times smaller than a nucleus
photon size of its wavelength,
for visible light it varies between 0.4-0.8 µm, namely 1000 to 10,000 times the size of an atom.
Josephson junction : depends upon the device used; it can be as small as few µm
Each quantum system can be seen from two points of view: as a particle, with mechanical properties, or as a wave, with propagation, diffraction and interference properties. Interferences or diffraction produce maxima and minima of intensity. Max Born (1926) showed that the intensity, at some point x, of the wave attached to a particle, should be interpreted as the particle density probability of being at x.
: characterized by
mass = m
electric charge = q
spin = s
other "quantum numbers" (isotopic spin, baryonnic number, ....)
energy = E
momentum = p = (px, py, p z )
data : characterized
frequency = ν or pulsation ω = 2 π ν
wave vector = k = (kx, ky, kz) |k| = 2 π / λ λ : wavelength
Planck-Einstein-de Broglie relations : wave and mechanical data are related through Planck's constant
h = 6.02 x10-34 Joules.sec
E = hν = (h/2 π ) ω (Planck 1900, Einstein 1905) p = (h/2π ) k (Einstein 1905, de Broglie 1923)
Heisenberg uncertainty inequalities
If particles are wave, the minimal length scale
accessible is the wavelength. Thanks to de Broglies's relation, the
smaller the wavelength the bigger the momentum.
In 1927, Heisenberg remarked that, as a consequence of this remark, there is an inequality relating the accuracy for measurements of position and momentum ofthe form
Δ x . Δ p > h / 2 π
He then realized that such a relation applies to all
pairs of conjugate variables occuring in Hamiltonian Mechanics.
One way to get a mathematical framework liable to provide such an
inequality automatically consists in representing the observables
by noncommuting objects called q-numbers
(Dirac 1925). Heisenberg was the first to realized that in his seminal
paper written in May 1925 ( Heisenberg, W. "Über
quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen
", Z. Phys., 33, (1925) 878--893) . His
proposal was soon interpreted by Max Born in July 1925: what Dirac
later called q-numbers were nothing else than
The spin of a particle, like electron, provides the
most elementary model for a quantum bit of information called the qbit
. It is the quantum version of an angular momentum,
characterizing the rotating motion of the particle on itself
(spinning). However, when this angular momentum becomes very small (of
the order of Planck's constant), thanks to Heisenberg's inequality, it
becomes impossible to measure simultaneously the three
components of this angular momentum. In particular the three spin
components must be matrices
s = (sx , sy , sz ) = (h /2 π )(σ x , σ y ,σ z )
where the σ i 's are the so-called Pauli matrices and are given by
| X = σx =
| Y = σy =
| Z = σz =
Applications of this observation range from the explanation of the photoelectric effect (Einstein 1905) to the electronic microscop in which light is replaced by electrons, going through the structure of atoms, Compton's effect, etc.. Particle accelerators can be seen as an extrapolation of electron microscops.
In the photoelectric effect, the light must
as made of elementary particles called photons with energy
the Planck relation. An energy threshold is observed
the minimum energy the photon must provide to an electron of the
photoelectric cell to be accelerated from one electrode to the other.
The photon energy is transformed into an electric current. Thus only
photon of frequency high enough can be detected by the cell. By
measuring the relation between the light frequency and the electron
energy (electric current), Planck's constant can be measured accurately
(Franck & Hertz).
In electron microscops, or in particle accelerators, there is a relation between energy and momentum: for non relativistic particle it reads
E = p 2 /2m
The electron energy can be varied through applying a potential difference through the equipment: it increases the momentum thus, thanks to the de Broglie relation, it decreases the wavelength. This can be used to produce an optical image as well as a diffraction patterns (transmission electron microscopy T.E.M.). Such technics is currently used in crystallography to analyze the structure of materials, crystals, large molecules like proteines or DNA. By decreasing the wavelength, the optical imaging provides a way of seeing smaller and smaller objects. The cost is the increase of the incoming electron energy. In particle accelerators, the energy is so large that it is possible to see inside the nucleus, or even inside the protons or neutrons.
It is believed that the total energy of the Universe is finite. If true, there is a minimal wavelength below which it is impossible to distinguish between points in space: the Planck scale which of the order of 10-30m ! At this scale the Universe should be quantized: the components of the position-time should become noncommutative with an uncertianty relation.
On the other side of the application are cold atoms. By
slowing down the velocity of an atomic jet, like Cesium (Cs)
or Rubidium (Rb),
the atomic wavelength can become very large if compared to the actual size of the atom. Typically for a velocity of few cm/s, the wavelength can be as big as few µm : that is 10,000 times the size of the original atom ! By exchanging energy between light and cold atoms, the information contained in photons can be stored, transmitted and read. Moreover, such atoms can combine if they behave according to the Bose-Einstein statistics (this is the case for Rb) to condensate into a macroscopic quantum object, the BEC, that may be used eventually in quantum computers.