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Book , Online - Google Books. Quantum interference.

I will start by introducing non-abelian quantum Hall states and their quasi-particles. I will then describe the interferometric probes of their quantum statistics and resolve some subtleties associated with the experimental realization of these probes. A cold-atom loop interferometer is proposed as a direct probe of the Berry-Wilkinson geometric phase structure in the energy bands of a laser-generated optical lattice. A cold atom wave packet prepared with a narrow momentum distribution in a weakly forced two-dimensional optical lattice undergoes Bloch oscillations, remaining localized in both position and momentum.

It follows from a semiclassical analysis that when the lattice potential is separable and the forcing is lattice-commensurate and has non-zero components in both lattice directions, the wave packet performs periodic Lissajous motion, while winding the quasi-momentum Brillouin zone torus. When the lattice potential is reflection-symmetric, there is no geometric phase, and the phase accumulated by the wave packet during one period is zero.

Reflection symmetry is broken when a double periodic optical lattice is used. Then the energy band is split, and if the gap is wide enough, the wave packet continues to perform Bloch oscillations in the sub-band, and accumulates the geometric phase associated with its orbit in the Brillouin zone.

This phase can be observed by letting the wave packet impinge on a cold atom beam splitter. The reflected wave packet follows a Brillouin zone orbit parallel to that of the transmitted wave packet, and after one period the transmitted and reflected wave packet return to the beam split- ter and recombine. The geometric phase density can then be mapped by a time-of-flight measurement of the transmission and reflection probabilities of the recombined wave packet as a function of the initial momentum.

P article-wave duality is one of the most essential concepts in quantum theory and provides the impressive illustration of Bohr's complementarity principle [1]. The wave characteristic arises if the particles are not seen in the different possible paths.

By introducing the which-path detector [2], we can determine the actual path taken by the particle necessarily involved coupling to an environment. As a result, the observation by the which-path detector gives rise to the dephasing.

Mesoscopic systems can be often used to study the interplay between interference and dephasing of electrons. Nano-fabrication and low-temperature techniques using semiconductors have enabled us to observe a variety of coherent effects of electrons such as Aharonov-Bohm AB [3,4,5], Fano, and Kondo effects. The AB effect is more intriguing in an AB interferometer containing a double quantum dot DQD [6], providing a refreshing subject for both theoretical and experimental studies.

In particular, the quantum interference effects are sensitive to the inter-dot coherent coupling. U s i n g t h e q u a n t u m d o t QD a s a charge sensor, the shot noise behaves intriguingly because of the correlation effect. Thus, we expect that the coupling with a QD charge sensor leads to the novel dephasing. I n this work, we investigate the effects of observation on quantum interference in a laterally coupled DQD due to a coupling with a QD charge sensor as shown in Fig. We employ the Schwinger-Keldysh nonequilibrium perturbation theory [7].

In particular, we introduce the notion of the coherent indirect coupling, which characterizes the coupling strength of the coherent indirect coupling between two QDs via the reservoirs [6]. Let t. We consider an Aharonov-Bohm AB ring with an tunable asymmetric electron injection, for which electrons incoming from the left lead can be preferentially injected into the lower arm of the ring and electrons incoming from the right lead mostly injected into the upper arm, in a comple- mentary way.

Such configuration reproduces at low magnetic fields the effect of the Lorentz force. By strongly enhancing the degree of asymmetry of the electron injection, the minimum becomes narrower, until the whole pattern becomes a single very narrow dip. In this particular point the electron are totally reflected. An external classical fluctuating field affects the dynamics of electrons traveling in one of the two arms, say the upper one, by randomly shifting its phase.

Coto, Erin T. A search for an analog of the AB effect involving isospin current has also been carried out with neutron interferometry. Gerrits, A. Lita, F. Lee, and G.

The coherent amplitude addition is then destroyed and the response of the device to the presence of the dephasing field changes drastically from total reflection to total transmission. This way, the ring is capable to witness the presence of a source of dephasing. Such a behavior is clearly understood as an incoherent sum of probabilities, with a very strong sensitivity due to the narrow resonance. On the other hand, the coherence of the outgoing signal shows a particular feature. There are several ways to analyze the coherence of the signal.

We choose to study the fraction of coherent signal, given by sum of the squares of the dephased transmission and reflection amplitudes.

In the coherent case the sum is identically one. Alternatively one can study the visibility of the outgoing signal with respect to a signal of known phase. By varying the degree of asymmetry we can inject the electron preferentially in the upper ring, where the dephasing field randomly shifts the phase, or in the lower ring where the propagation is coherent. We see that in the first case the coherence is rapidly destroyed as the strength of the dephasing source is enhanced, whereas in the second case the fraction of coherent signal as first is slightly degraded, for weak dephasing, but then it saturates to one for strong dephasing.

We interpret this behavior in the context of interaction- free measurements, A. Elitzur and L. Vaidman, Found. As an electron is mostly injected in the coherent channel, only a very small fraction of the wave function tests the region subject to dephasing and the outgoing signal is capable to witness the presence of a dephasing source without loosing its coherence. A very well known problem, referred to as Klein tunneling, is the scattering of a relativistic fermion from an infinite potential step, where the energy of the particle is smaller than the height of the step.

Surprisingly, and in sharp contrast to the scattering of a non-relativistic particle, the reflection probability of such a Dirac fermion is not one, and the particle has a non-zero probability to be transmitted [1]. Even more surprising result is found when one considers a flux of massless fermions: the reflection probability vanishes and all particles are transmitted [2, 3].

Kleins work is now 80 years old, but the idea still cannot be tested theoretically in its original context. However, since the discovery of graphene, the scattering of massless fermions from a potential step is no longer a fantasy, since due to the conical intersections the quasi-particles behave as massless fermions. In fact, other interesting phenomena, such as anomalous quantum Hall effect in graphene are also explained through the geometrical phase.

We study deformed honeycomb lattices, and show that at some critical deformation, the Dirac points merge, and the geometrical charges annihilate one another, leaving zero geometrical phase. For deformations beyond the critical one, a gap forms between the bands. Moreover, when the potential step is in the perpendicular direction, the transmission probability vanish at normal incidence.

For increasing angles, it rises rapidly to 1, and then slowly decrease to zero Fig. Note that the transmission peak is not a consequence of a resonant phenomena, since the waves are scattered from a potential step. For deformations greater than the critical one, the Hamiltonian is a slightly more complicated, and the ma jor difference is the 'gap' term.

The behavior of the transmission probability in the gaped phases resembles the effect of a mass, but it is not completely equivalent Fig. Klein, Z. Physicss 53, Ando, T. Nakanishi, and R. Saito, Journal of the Phys. Society of Japan 67, Katsnelson, S. Novoselov, and A. Geim, Nature The integer quantm Hall transition, a fascinating critical phenomenon driven by the interplay between disorder and quantum interference in the presence of Aharonov-Bohm phases due strong magnetic field , still presents a mystery in terms of analytic theoretical description, after almost 30 years since its discovery.

We study critical multifractal wave functions at the transition, as well as the so called point-contact conductances PCC.

The multifractal wave functions provide a window for answering the question of presence of conformal invariance at the transition, as well as testing various existing theoretical proposals. We also argue that the PCC related to correlation functions of the wave functions admit a natural description in terms of the so-called conformal restriction, a notion of stochastic conformal geometry closely related to the by now famous Schramm-Loewner evolution.

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We predict analytically the behavior of the PCC in presence of various boundary conditions. In a quantum dot embedded in an Aharon o v-Bohm AB ring, the F ano- K ondo e f fect has been theoretically predicted [1] and e xperimentally obser v ed [2].

The conductance sh o ws an asymmetric shape with a peak and a dip, as a function of ene r gy l e v el in the quantum dot. This is due to an interplay between the ma n y-body K ondo e f fect and one-body interference e f fect. There is another problem on the K ondo e f fect in this system; whether the K ondo e f fect by itself is influenced by the interference e f fect or not. Simon et al. In the present w ork, we perform the scaling analysis for this K ondo e f fect and sh o w analytical e xpressions for the K ondo temperature and conductance.

Our model is shown in the figure.

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