shot noise in mesoscopic conductors - Low Temperature Laboratory

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156 Ya.M. Blanter, M. Bu( ttiker / Physics Reports 336 (2000) 1}166 symmetric chaotic cavities. Eq. (A.9) can be also obtained classically [347], using the generalization of the minimal correlation approach. Muzykantskii and Khmelnitskii [169] investigate the counting statistics for the NS interface and "nd ()" [¹ e#1!¹ ], ¹ ,2¹/(2!¹ ) . (A.10) It is clearly seen from the comparison with Eq. (A.4) that the particles responsible for transport have e!ective charge 2e. Other important developments include the generalization to the multi-terminal [341] and time-dependent [348}350] problems, and numerical investigation of the counting statistics for the non-degenerate ballistic conductors. Thus, the counting statistics certainly reveal more information about the transport properties of conductors than is contained in either the conductance or the second order shot noise. The drawback is that it is not quite clear how these statistics can be measured. A proposal, due to Levitov et al. [349], is to use the spin- galvanometer, precessing in the magnetic "eld created by the transmission current. The idea is to measure the charge transmitted during a certain time interval through the evolution of the spin precession angle. However, the time-dependent transport is a collective phenomenon (Section 3), and thus the theory of such an e!ect must include electron}electron interactions. In addition, this type of experiments is not easy to realize. On the other hand, measurements of photon numbers are routinely performed in quantum optics. In this "eld concern with counting statistics has already a long history. However, typical mesoscopic aspects } disorder, weak localization, chaotic cavities } and e!ects particular to optics, like absorption and ampli"cation, make the counting statistics of photons a promising tool of research. Some of these aspects (relating to disorder and chaos, where the random matrix theory may be applied), have been recently investigated by Beenakker [351]; however, there are still many unsolved problems. Appendix B. Spin e4ects and entanglement A notion which mesoscopic physics recently borrowed from quantum optics is entanglement. States are called entangled, if they cannot be written simply as a product of wave functions. For our purpose, we will adopt the following de"nition. Imagine that we have two leads, 1 and 3, which serve as sources of electrons. The entangled states are de"ned as the following two-particle states described in terms of the creation operators, $"(1/2)(a( s (E )a( t (E )$a( t (E )a( s (E ))0 , (B.1) where a( (E) is the operator which creates an electron with the energy E and the spin projection in the source . The state corresponding to the lower sign in Eq. (B.1) is the spin singlet with the symmetric orbital part of the wave function, while the upper sign describes a triplet (antisymmetric) state.
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
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8. Concluding remarks, future prosp
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text; they are usually extensions o
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such reviews are not readily availa
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The above considerations are, of co
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and the initial state of our two-pa
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2.3.3. Multi-terminal case We consi
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Note that in the absence of time-de
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equilibrium and shot noise. For low
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and S" e A e< dE ¹(E )[1!¹
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arrier, situated in the region !w/2
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we include weak localization e!ects
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Edge channels in the quantum Hall e
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Fig. 19. (a) Geometry of a ring thr
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a manifestation of interference e!e
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Physically, this corresponds to ele
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Our starting point is Eq. (51), whi
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introducing the transmission coe$ci
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investigated experimentally. A self
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and the non-equilibrium resistance
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the sample. These electrons are des
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with D ,¹ (2!¹ ). Performing t
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For completeness, we mention that i
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