shot noise in mesoscopic conductors - Low Temperature Laboratory

Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 157
Such entangled states are very important in the "eld of quantum computation. Condensed
matter systems are full of entangled states: there is hardly a system for which the ground state can
be expressed simply in terms of a product of wave functions. The key problem is to "nd ways in
which entangled states can be generated and manipulated in a controlled way. Optical experiments
on noise have reached a sophisticated stage since there exist optical sources of entangled states
(the production of twin photon-pairs through down conversion). It would be highly desirable to
have an electronic equivalent of the optical source and to analyze to what extent such experiments
can be carried out in electrical conductors [352,108]. An example of such source is a p-n junction
which permits the generation of an electron-hole pair and the subsequent separation of the
particles. The disadvantage of such an entangled state is that the electron and hole must be kept
apart at all times. Similarly, a Cooper pair entering a normal conductor, represents an entangled
state. But in the normal conductor it is described as an electron-hole excitation and again we have
particles with di!erent charge. To date most proposals in condensed matter related to quantum
computation consider entangled states in closed systems.
Theoretically, entanglement opens a number of interesting opportunities. One of the questions is:
Provided we were able to prepare entangled states, how do we know the states are really entangled?
Since we deal with two-particle states, it is clear that entanglement can only be measured in the
experiments which are genuinely two-particle. Burkard et al. [353,354] investigate the multiterminal
noise. Indeed, add to the structure two more reservoirs (electron detectors) 2 and 4, and
imagine that there is no re#ection back to the sources (the geometry of the exchange HBT
experiment, Fig. 15b, with the additional `entanglera creating the states (B.1)). The system acts as
a three-terminal device, with an input of entangled electrons and measuring the current}current
correlation at the two detectors.
Since the shot noise is produced by the motion of the electron charge, it is plausible that the noise
measurements are in fact sensitive to the symmetry of the orbital part of the wave function, and
not to the whole wave function. Thus, the noise power seen at a single contact is expected to be
enhanced for the singlet state (symmetric orbital part) and suppressed for the triplet state
(anti-symmetric orbital part). It is easy to quantify these considerations by repeating the calculation
of Section 2 in the basis of entangled states (B.1). Assuming that the system is of "nite size, so that
the set of energies E is discrete, and the incoming stream of entangled electrons is noiseless,
Burkard et al. [353] obtained the following result I "I , S "S "!S , with
S "2e¹(1!¹)(1G ) . (B.2)
Thus, indeed, the shot noise is suppressed for the triplet state and enhanced for the singlet state,
provided the electrons are taken at (exactly) the same energy. For the singlet state this suppression
is an indication of the entanglement, since there are no other singlet states. One can also construct
the triplet states which are not entangled,
"a( t (E )a( t (E )0 , (B.3)
and an analogous state with spins down. These states, as shown by Burkard et al. [353], produce
the same noise as the entangled triplet state. Thus, the noise suppression in this geometry is not
a signature of the entanglement.
Another proposal, due to Loss and Sukhorukov [303], is that the entangled states prepared in
the double quantum dot can be probed by the Aharonov}Bohm transport experiments. The shot