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Full counting statistics in mesoscopic fermion and boson systems

Figure 1 Two examples demonstrating the use of the counting statistics concept. In a quantum point contact (a) electrons impinge on a barrier in a constriction, some are reflected and some transmitted in a randommanner. Thenumberof incoming electrons M in a time interval t can be understood in a energy diagram (b). The number is related to the energy window spanned by the applied bias voltage eV and given by M = eV t/h. As the particles are independent the statistics of the total transmitted number is binomial. In a similar way we can discuss the spatial fluctuations of the particle density, e.g. in cloud of ultra cold atoms, depicted symbolically in (c). By considering a cell or a bin defining a certain spatial region, the particle number in it is a stochastic quantity and follows from someprobability distribution. Suchadistribution (d) typically centered around some mean value represents the complete information on the system available by particle number measurements and, hence, this is called the full counting statistics. (Figs. a and b modified from Physik Journal 4, 75– 80 (2005); Figs. c and d modified from [7]). Quantum mechanics is an inherently stochastic description of physical systems. In particular, the outcome of a measurement is probabilistic and many repetitions of the same experiment reveal fluctuations of the observable. It is important to recall that this is the case although the state of the systems is fully determined and described by a deterministic time evolution. Of course, an uncertainty in the state preparation, e.g., due to a large number of degrees of freedom also leads to fluctuations, but these are less fundamental from a quantum perspective (of course they are of importance in all practical systems, which are, e.g., at a finite temperature). The quantum fluctuations, which remain after all sources of uncertainty have been eliminated, are of central interest in the mesoscopic physics of ultracold electrons and atoms. One of the most important quantities in quantum transport processes is the current through a quantum point contact, a small constriction connecting two large fermionic reservoirs with an electric potential differenceV . Assuming the transmission probability to be T , one finds for the average transferred charge 〈Q〉 in a time period t at zero temperature 〈Q〉 = e2V T t/h. Here, e is the electron charge and h denotes Planck’s constant. It is also straightforward