Talks   @   AB50

Plenary Session 11:30-13:00

The hidden horizon and black hole unitarity

François Englert

Since the theoretical discovery of black hole evaporation through Hawking radiation the question of the unitarity of its evolution and hence of the validity of quantum mechanics has been a major challenge for our understanding of black hole physics and of physics in general.

One seems to be faced with the following alternative :

1. There is, as in the original derivation by Hawking, no information in the thermal state. In that case, unitarity is violated except if the radiation terminates on a infinitely degenerate remnant. This is hard to believe, particularly in view of the theoretical achievements of string theory and AdS-CFT.

2. The original information is contained in the radiation and hence unitarity is preserved. This led to suspect that the original derivation was fundamentally incorrect. This also is very puzzling to me because of the simplicity and the consistency of the derivation.

The present work offers a different perspective. It is shown that the weak value of Aharonov and al. drives the detailed gravity response to quantum matter. Unitarity then follows because generic pure out-states appear inconsistent with a causally non-trivial geometry, but observers unable to test unitarity uncover the conventional geometry with its event horizon. Although the computations of unitary amplitudes requires a detailed theory of quantum gravity, the validity of this complementary scheme does not.

Anomalous dephasing in electronic Mach-Zehnder interferometers

Ivan P. Levkivskyi, and Eugene V. Sukhorukov

University of Geneva

Recently, Aharonov-Bohm (AB) effect in electronic Mach-Zehnder (MZ) interferometers has attracted much attention among experimental and theoretical physicists. These interferometers, for the first time experimentally realized in the group of Heiblum [1], utilize quantum Hall edge states in place of optical beams, and quantum point contacts (QPC) as beam splitters, to partition edge channels. Theoretical attempts to explain experimentally observed puzzling lobe-type behavior of the visibility of AB oscillations as a function of voltage bias [2-5], have led to a number of publications [6-9]. They have focused on the filling factor ν=1 state, and suggested different mechanisms of dephasing, including the resonant interaction with a counter-propagating edge state [6], the dispersion of the Coulomb interaction potential [7], and non-Gaussian noise effects [8,9]. To date, however, all the experiments, reporting multiple side lobes in the visibility function of voltage bias, have been done at filling factor ν=2. We will argue that, in fact, there are two main mechanisms of dephasing in MZ interferometers. One mechanism [10], due to spontaneous emission of edge magneto-plasmons, leads to a size effect, which explains the lobes and many other details of experiments [2-5]. According to the second mechanism [11], dephasing in electronic MZ interferometers is due to an external noise source. Experimentally [2], such a noise is created with the help of an additional QPC with the transparency T that partitions incident edge channels. We predict that a phase transition occurs at T=1/2, where the visibility function of voltage bias sharply changes its behavior. An important role in this phenomenon is played by a non-Gaussianity of noise, which is typically negligible because of a weak coupling. It turns out that MZ interferometers are strongly coupled to noise. They, therefore, can be considered efficient detectors of full counting statistics [12].

[1] Y. Ji et al., Nature (London) 422, 415 (2003).

[2] I. Neder et al., Phys. Rev. Lett. 96, 016804 (2006).

[3] E. Bieri et al., arXiv:cond-mat/0812.2612.

[4] P. Roulleau et al., Phys. Rev. B 76, 161309(R) (2007).

[5] L.V. Litvin et al., Phys. Rev. B 75, 033315 (2007).

[6] E.V. Sukhorukov, and V.V. Cheianov, Phys. Rev. Lett. 99, 156801 (2007).

[7] J.T. Chalker, Y. Gefen, and M.Y. Veillette, Phys. Rev. B 76, 085320 (2007).

[8] S.-C. Youn, H.-W. Lee, and H.-S. Sim, Phys. Rev. Lett. 100, 196807 (2008).

[9] I. Neder and E. Ginossar, Phys. Rev. Lett. 100, 196806 (2008).

[10] I.P. Levkivskyi, and E.V. Sukhorukov, Phys. Rev. B 78, 045322 (2008).

[11] I.P. Levkivskyi, E.V. Sukhorukov, unpublished.

[12] L.S. Levitov, H. Lee, and G.B. Lesovik, J. Math. Phys. 37, 4845 (1996).

Paradoxes in the context of the Aharonov-Bohm and Aharonov-Casher effects

Lev Vaidman

The following two pairs of apparently contradicting statements are all correct:     - There is no particular place where the wave packets of the particle in the Aharonov-Bohm experiment gain their relative phase.
    - It is possible to measure the relative phase of spatially separated wave packets of a charged particle (if it is a boson).
    - The wave packets of the neutron passing around the line of charge in the Aharonov-Casher experiment move exactly in the same way as the wave packets of the electron passing around the solenoid in the Aharonov-Bohm experiment.
    - In the Aharonov-Casher experiment, in contrast with the Aharonov-Bohm experiment, different forces are exerted on two wave packets of the neutron.

Parallel Session A. 15:00-16:30

Random Aharonov-Bohm vortices and some exact families of integrals

Stephane Ouvry

A review of the random magnetic impurity model, introduced in the context of the quantum Hall effect, is presented. It models an electron moving in a plane and coupled to a poissonian random distribution of Aharonov-Bohm vortices carrying a fraction of the flux quantum. Recent results on its perturbative expansion are given and a new family of numbers is introduced.

Counting statistics in multiple path geometries and quantum stirring

Doron Cohen

Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel

The amount Q of particles that are transported via a path of motion is charac- terized by its expectation value <Q> and by its variance Var(Q). We analyze what happens in Aharonov-Bohm geometries, where a particle has two optional paths available to get from one site to another site, and in particular what is Var(Q) for the current which is induced in a closed ring due to a quantum stirring process. A novel interference effect shows up in the counting statistics calculation for subsequent multiple path Landau-Zener crossings.

[1] M. Chuchem and D. Cohen, J. Phys. A 41, 075302 (2008)

[2] M. Chuchem and D. Cohen, Phys. Rev. A 77, 012109 (2008)

[3] http://physics.bgu.ac.il/∼dcohen/ARCHIVE/cnt.pdf

The AB effect and some topological effects in cosmology and nanophysics

Yu. A. Sitenko

Bogolyubov Institute for Theoretical Physics, Kiev, 03680, Ukraine

Since the discovery by Y. Aharonov and D. Bohm of the famous effect named after them, it has become clear that the fundamentals of quantum physics involve topology. Topological phenomena are of great interest and importance owing to their universal nature connected with the properties of space-time as a whole. The ideas and approaches inspired by the seminal work of Y. Aharonov and D. Bohm go far beyond the initial setup dealing with the quantum-mechanical scattering of a charged particle on a tube enclosing the magnetic flux. One of the approaches is concerned with the study of the influence of the enclosed flux on the properties of the matter which is second-quantized outside the tube (see, e.g., [1]). Meanwhile, in various setups the enclosed fluxes might be of nonmagnetic nature as well.

In cosmology, one deals with cosmic strings - linear topological defects appearing as a consequence of phase transitions with spontaneous symmetry breakdown in early universe. Cosmic strings are characterized by fluxes of nonmagnetic nature, since they correspond to gauge fields with masses generated via the Higgs mechanism. A surface which is transverse to the cosmic string axis is isomorphic to a surface of a cone with the deficit angle which is less than . The Aharonov-Bohm-type scattering on a cosmic string has been considered [2].

In nanophysics, one deals with topological defects (disclinations) in graphene - strictly two-dimensional layer of carbon atoms. A disclination warps a sheet of graphene, rolling it into a nanocone with the deficit angle which can take both positive and negative values that are equal to multiples of . Also, disclinations are characterized by pseudomagnetic fluxes rather than the magnetic ones; the value of the pseudomagnetic flux is related to the value of the deficit angle. The influence of disclinations in graphene on its electronic properties has been considered [3].

1.     Yu.A. Sitenko, A.Yu. Babansky. Mod. Phys. Lett. A 13, 379-386 (1998).

2.     Yu.A. Sitenko, A.V. Mishchenko. JETP 81, 831-850 (1995).

3.     Yu.A. Sitenko, N.D. Vlasii. Nucl. Phys. B 787 [FS], 241-259 (2007).

Aharonov-Bohm and the Vacuum State of Gravitation

Pawel O. Mazur

Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208, USA. E-mail: mazur@physics.sc.edu, mazurmeister@gmail.com

I will talk about the gravitational Aharonov-Bohm phenomenona connected with gravitational vortices, which are just fluxons of the so-called gravimagnetic field, and about the quantized fluxes in the gravitational vacuum. It turns out that gravitational fluxes are quantized in units of h/M where is a mass of the hypothetical heavy bosons which condense in the gravitational vacuum state. The physical interpretation of the 'spinning cosmic string' metric is given and the apparent violation of causality (the presence of closed time-like curves) in the vortex core is now understood as the failure of general relativity in the unbroken phase. This interpretation fits nicely with our present understanding of the Onsager-Feynman vortices in superfluids. The cores of vortices are regions where the global phase symmetry is restored.

There is an exact one to one relation between the quantization of gravitational 'gravimagnetic' fluxes in units of h/M and the London-Aharonov-Bohm flux quantization in which the AB effect is periodic (superconducting rings; mesoscopic rings etc.). I consider scattering of slow (non-relativistic) and also relativistic particles on the gravitational vortex. The scattering cross-section I have derived a long time ago is exactly the same as the Aharonov-Bohm result with the proviso that now the magnetic flux is replaced by the gravimagnetix flux or vorticity of a vortex while the electric charged in the original AB paper is now replaced by mass-energy of a particle. This is also why I have originally called the phenomenon I discuss here the gravitational Aharonov-Bohm effect.

I will also talk about the experimental/observational aspects of detecting periodicity in the gravimagnetic flux (gravitational vorticity) in the gravitational vacuum state. I propose the direct observation of the moment of inertia of the rapidly rotating black hole candidates such as the dark supermassive dark object in Sagittarius A* in the center of Milky Way. Ordered macroscopic quantum states such as the Einstein Condensates of Bosons (BECs) are characterized by the non- clasical moments of inertia. Measuring the non-classical moments of ineria of giant dark objects (physical black holes) would be the direct test of quantum coherence on the galactic scales. Perhaps it is appropriate to remark here that non-classical moments of inertia were previously measured for a very low temperature rotating superfluid He II and some vortex isomers (massive nuclei).

I will focus mostly on the most recent work on this subject. This is all about the hypothetical connection between gravitation and superfluidity. If this hypothesis will get verified in the proposed experiments/observations of giant dark objects (black holes) then we shall witness the most wonderful example of macroscopic quantum coherence at work on the scale of the universe.

Parallel Session B. 15:00-16:30

The Aharonov-Bohm effect does not enable superluminal signaling but in a subtle way.

Avshalom Elitzur1 and Shmuel Marcovich2

When the AB effect is combined with the EPR experiment in a special way, a causality violation seems to emerge, the resolution of which necessitates the complementary Aharonov-Casher effect. This way, the latter effect could have been predicted on the grounds of the AB plus causality preservation.

We present a gedankenexperiment in which current carriers, e.g., electrons, are split by a BS before entering a pair of solenoids. After traversing the two solenoids, these superposed electrons interfere, such that they all re-emerge through one arm of a second BS.

An observer, Bob, is located next to solenoid pair. He splits an electron with a Stern-Gerlach magnet aligned along the x direction. The split electron interferes around one of the solenoids, thus performing an AB experiment. In this respect, Bob's electron performs a "which path" measurement on the current carriers within the solenoid, destroying (at least partially) the abovementioned interference of the two solenoid currents.

Suppose, however, that Bob's electron has a definite x spin, thereby not interfering with itself around the solenoid. The current carriers, then, will remain superposed over the two solenoids, unaffected by Bob's electron. Their interference will therefore remain intact.

Causality violation seems to be imminent: Bob's electron may be one out of an EPR pair, the other electron going to Alice. She can choose whether or not to perform an x-spin measurement on her electron, thereby apparently being capable of sending Bob a superluminal signal.

We discuss the paradox's solution in two- and three-dimensional settings.

Quantum probabilities from quantum phase behavior

Alonso Botero

It is shown that Quantum-Mechanical transition probabilities can be obtained from the local behavior of the phase of the transition amplitude when all possible variations of the quantum state are considered. Some interesting connections with geometric phases and the Aharonov-Bohm effect are pointed out.

Duality of the Aharonov-Bohm and Aharonov-Casher effects

Daniel Rohrlich

Physics Department, Ben Gurion University

The Aharonov-Casher (AC) effect [1] is dual to the Aharonov-Bohm (AB) effect. The proof of the duality [2] is surprisingly subtle and even controversial [3]; the AC effect is not dual to the AB effect in general but only to the AB effect for a straight solenoid. What follows is an intuitive proof of these aspects of duality.

Let us begin with a two-dimensional effect. The figure shows an electron moving in a plane, and also a "fluxon", i.e. a small region of magnetic flux from which the electron is excluded. The fluxon is in a superposition of two positions, and the electron diffracts around one of the positions but not the other. Initially, the fluxon and electron are in a product state, but their final state is entangled. Measuring the position of the fluxon and the relative phase of the electron, we discover that the electron acquires a relative phase if and only if the fluxon lies between the two electron paths. Measuring the position of the electron and the relative phase of the fluxon, we discover that the fluxon acquires the relative phaseif and only if the electron passes between the two fluxon wave packets. In two dimensions, these two effects are equivalent, but there are two inequivalent ways to go from two to three dimensions while preserving the topology: either the electron stays a particle and the fluxon becomes a line of flux (a solenoid) - the AB effect - or the fluxon stays a neutral particle with a magnetic moment (say, a neutron) and the electron becomes a line of charge - the AC effect.

The nonrelativistic Lagrangian for a neutral particle of magnetic moment μ interacting with a particle of charge e is , where M, R, V and m, r, v are the mass, position and velocity of the neutral and charged particle, respectively, and A(r - R) =. Note that L is invariant under interchange of the two particles. Thus L is the same whether an electron interacts with a line of magnetic moments (AB effect) or a magnetic moment interacts with a line of electrons (AC effect). However, if we begin with the AC effect and replace the magnetic moment with an electron, and all the electrons with the original magnetic moment, we end up with magnetic moments that all point in the same direction, i.e. with a straight line of magnetic flux. Hence the original line of electrons must have been straight. We see intuitively that the effects are dual only in this case [4].

[1] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319-321 (1984).

[2] Y. Aharonov, P. Pearle and L. Vaidman, Phys. Rev. A37, 4052 (1988).

[3] T. H. Boyer, Phys. Rev. A36, 5083 (1997).

[4] I thank Prof. Aharonov for a conversation on this point.

On geometric interpretation of the Aharonov-Bohm effect and Berry's phase

Mikhail Katanaev

A geometric interpretation of Berry's phase, its Wilczek-Zee non-Abelian generalization, and the Aharonov-Bohm effect is given in terms of connections on principal fiber bundles. It is shown that a principal fiber bundle in all cases can be trivial while the connection and its holonomy group are nontrivial. Therefore the main role is played not by topological effects but geometrical ones. The Berry phase is also shown not to be connected to the adiabatic changes of the Hamiltonian parameters.

Propagators on multiply connected spaces with a nonvanishing gauge field and the noncommutative Bloch analysis

Pavel Štovíček

Department of Mathematics, Faculty of Nuclear Science, Czech Technical

University in Prague, Trojanova 13, 120 00 Praha, Czech Republic

Let be a multiply connected Riemannian manifold with a (in general, non-vanishing) gauge field and a scalar potential. Let be a (not necessarily universal) covering of with a (possibly noncommutative) structure group Γ. The gauge field on lifts to a Γ-invariant gauge field on . Similarly, the scalar potential lifts to a Γ-periodic potential on . To any unitary representation Λ of Γ one relates a quantum Hamiltonian on (for the given gauge field and scalar potential). At the same time, one considers a Γ-periodic Hamiltonian on the covering space . One can express the propagator for in terms of the propagator corresponding to Hamiltonian on the covering space as a weighted sum over the group Γ. This construction is well known in theoretical physics for the case when the gauge field vanishes. Conversely, there exists the (noncommutative) Bloch decomposition of the periodic Hamiltonian into a direct integral over the dual space whose components are the Hamiltonians . We show that the two constructions are mutually inverse. This contribution extends results from the paper [P. Kocabova, P. Stovicek: J. Math. Phys. 49 (2008) art. no. 033518] where the case with vanishing gauge field has been worked out in detail.

Plenary Session 17:00-18:50

Things I Do and Do Not Understand About the Aharonov-Bohm Effect

Murray Peshkin*

Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA

I will discuss some features of simple quantum mechanics that went unnoticed until Aharonov and Bohm forced them upon the attention of physicists. This talk is intended to be accessible to students.

When the AB effect was announced in 1959, most physicists found it mysterious if not incredible because the motion of an ion was shown to be influenced by a non-local electric or magnetic field. That was contrary to everything we knew from classical physics and it was not immediately obvious why quantum mechanics should be different in that respect. However it was soon recognized that at least in the magnetic case, it would have been mysterious if the AB effect did not exist. When an ion orbits a flux line, its kinetic angular momentum , where F is the flux. The allowed values of the canonical L_z are integer multiples of h. Then the allowed values of K depend upon the flux and since the centrifugal barrier in the Hamiltonian depends upon K, both bound state energies and scattering state phase shifts must depend upon F. Moreover, the source of the F dependent shift in the allowed values of K was understandable. The electric field whose source is the ion, multiplied by the external magnetic field, gives an electromagnetic angular momentum just equal to . The canonical angular momentum is equal to the sum of the kinetic and electromagnetic angular momenta, and that total angular momentum is quantized in integer units.

Still, there remained the locality issue, and that had profound consequences for the meaning of the vector potential and for the gauge principle. Those matters have been elucidated by C.N. Yang and I will not pursue them here. Instead, I will discuss the elementary classical physics that underlies how past local interactions between the ion and the electromagnetic field provide a simple understanding of where the electromagnetic angular momentum resides, how it got there, and how a then-local interaction with the electromagnetic field changed the kinetic angular momentum of the ion and consequently its future behavior.

A flux line is an idealization of a long thin solenoid whose return flux is far from the solenoid except near its distant ends. The electromagnetic angular momentum turns out to reside in the return flux region, not in the flux line. An electron brought in from infinity must have traversed that return flux. There it experienced a torque whose time integral along any classical trajectory equals , and there it exerted an opposite torque on the magnetic field, storing angular momentum in the crossed fields. Alternatively, the flux may have been turned on after the electron was between the flux line and the return flux and so no longer needed to traverse the return magnetic field. Then the ion experienced a local induced electric field that had the same effect so the flux dependence of the present motion of the ion can be understood as a consequence of a past local interaction with the electromagnetic field. The Hamiltonian retains the memory of that past local interaction through eA(r), the momentum transferred to the electromagnetic field when an ion the ion was brought to the point r.

The causal connection between past forces and present motion of the ion in the magnetic AB effect is intuitively understandable to me on the level described above, but what if the ion is created after the magnetic flux was established? Suppose two gamma rays collide to produce an electron-positron pair somewhere in the field-free region between the flux line and the return flux. The states available to the newly created electron have integer L_z and shifted values of K, as in the case of an ion that experienced an earlier magnetic force. One can perhaps argue that the gamma rays contained virtual pairs and their motion was earlier influenced by a local field, but that interpretation of the phenomenon is at best no longer simple.

The electric AB effect exhibits causation by past electromagnetic fields in an even more mysterious way. In that case, the ion has never experienced a local electromagnetic field. However, one can prove from the Schroedinger equation that there can be no effect of an electric field from which the ion is shielded unless the ion passes through some region where an electric field previously existed.

*Work supported by the U.S. Dept. of Energy, Office of Nucl. Phys., under Contract DE-AC02-06CHI1357.

The relation between field momentum and the vector potential.

Harry J. Lipkin

In a standard Aharonov-Bohm experiment the electron does not enter the magnetic field. But the coulomb field of the electron does enter the field and there is momentum and angular momentum in the crossed fields. Insight into this effect is seen in a simple example. An electron moving in the z-direction passes through a thin slab in the xy plane containing a uniform magnetic field in the y-direction. The Lorentz force on the electron deflects the momentum of the electron in the x-direction. But the Hamiltonian is invariant under translations in the x-direction
and the x-component of the momentum should not change. Two different methods for resolving this paradox provide insight on the roles of field momentum and the vector potential.

Aharonov-Bohm Effect and Hawking Radiation

Ulf Leonhardt

University of St Andrews, North Haugh, St Andrews KY16 9SS, UK

Is there a connection between Aharonov-Bohm scattering [1] and Hawking radiation [2]? One can view both physical phenomena as implementations of space-time coordinate transformations on waves [3,4]: in the Aharonov-Bohm effect a spatial plane of constant time is mapped onto a helix in space and time, see the figure below.

In the Aharonov-Bohm effect, waves experience space-time as the transformation of a virtual empty space (right) to a helix in space-time (left).

One turn of the helix corresponds to the characteristic phase shift in the Aharonov-Bohm effect. If this shift is not a multiple of 2p the wave corrects for the phase mismatch by scattering, as Aharonov's and Bohm's 1959 scattering solution of the wave equation shows [1]. Normally, coordinate transformations simply transform plane waves into modulated plane waves, without scattering, unless they change the topology.

Another example of a topology-changing coordinate transformation is hidden in the Hawking radiation of the event horizon [2]. Waves near the event horizon effectively experience a 1+1 dimensional space-time geometry, because the extreme wavelength shift normal to the horizon makes all other dimensions of space irrelevant - the waves perceive them as uniform. Two-dimensional space-time geometries are conformally flat; essentially, the waves are just coordinate-transformed. But this coordinate transformation at the horizon is logarithmic with branches in the complex plane of the spatial coordinate normal to the horizon, similar to the branches in the Aharonov-Bohm effect. Such branches also create scattering, the mixing of waves with positive and negative frequencies. This mode mixing is responsible for the creation of particles from the quantum vacuum, Hawking radiation [2]. Hawking radiation thus seems to be related to the Aharonov-Bohm effect in the complex plane.

[1] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).

[2] S. M. Hawking, Nature 248, 30 (1974).

[3] U. Leonhardt and T. G. Philbin, New J. Phys. 8, 247 (2006).

[4] U. Leonhardt and T. G. Philbin, Transformation Optics and the Geometry of Light, Prog. Opt. (in press); arXiv:0805.4778.

A Dynamical Nonlocal Exchange that is Responsible for the Shift of the

Interference Pattern in the AB Effect

Tirzah Kaufherr

The question "when does the AB effect occur?" is of long standing, for in every gauge the relative phase between the two AB wave packets evolves differently. Using the gauge invariant version of modular momentum, i.e. the displacement operator or its Hermitian counterpart , it is found that when the external particle's two wave packets become collinear with the solenoid, an abrupt nonlocal exchange of the otherwise conserved quantity occurs. Using the Heisenberg picture, a dynamical description ensues. Specifically, it follows that this exchange is responsible for the shift of the interference pattern in the AB effect.

In addition, a gedanken experiment is described, which shows that the exchange can, in principle, be tested experimentally. Finally, this exchange gives new insight into the famous two slit quantum interference experiment.

Poster Session 19:00 - 20:30

Day 2 - Monday 12-Oct-09

Plenary Session 09:30-11:00

Pairing Symmetry in High-Temperature Superconductors : Cuprates and Iron Pnictides

C. C. Tsuei

IBM Thomas J. Watson Research Center, Yorktown Heights, NY U. S. A.

Phase-sensitive tests have been developed to definitively establish the pairing symmetry in high-temperature superconductors such as cuprates [1], and more recently in the newly discovered iron pnictides [2]. In these Aharonov-Bohm effect based experiments, the observation of half-integer flux quantization is used as an unambiguous signature of unconventional pairing. In this talk, we will first give a brief overview on the establishment of d-wave pairing symmetry and its experimental consequences in cuprate superconductors. We will then present a recent study of the quantized flux states in a superconducting niobium loop interrupted by a polycrystalline FeAs-based superconductor [2]. Through the observation of integer and half-integer flux-quantum transitions in such composite loops, we establish spin-singlet even-parity pairing state with a sign-reversing s-wave order parameter in the one-layer NdFeAsO_0.88F_{0.12 .} At the end of the talk, we will discuss the implications of the pairing symmetry findings in both the cuprates and the iron pnictides for understanding the origin of high-temperature superconductivity.

[1] C. C. Tsuei and J. R. KiLTRey Rev. Mod. Phys. 72, 969 (2000).

[2] C. -T. Chen et al. to be published (arXiv: 0905.3571)

The AB effect in mesoscopic solid state Physics

Yoseph Imry

Dept of Condensed-Matter Physics, the Weizmann Institute, Rehovot, Israel

The Byers-Yang theorem states the periodicity of all physical properties of a "ring" in the AB flux through its opening, with a period h/q where q is the charge of the relevant charge-carrier. This explained the flux-quantization in superconductors. Bloch later showed that the h/q ratio between frequency and current in a Josephson junction follows from the same general principles. We shall demonstrate how the Laughlin explanation for the integer Quantum Hall effect is also based on the above. Two of the main applications of the AB effect in mesoscopic Physics will then be reviewed:

1. The persistent current that can flow in a finite "ring" in spite of its having a finite resistance (our recent explanation of the magnitude of these currents will be mentioned as well); and

2. The AB oscillations in the conductance of such a ring with contacts. This leads to the important concept of mesoscopic fluctuations, which will be explained if time permits.

Aharonov-Bohm hair for black holes and particles

Georgi Dvali

We show that black holes can carry a long-range quantum-mechanical hair under massive fields of spin-2 (and higher), which is detectable via Aharonov-Bohm effect. We discuss implications of this phenomenon for fundamental gravitational physics and phenomenology.

Plenary Session 11:30-13:00

Time-resolved single electron interference in a semiconductor AB ring

Klaus Ensslin

Ring structures in semiconductors have been used to observe phase coherent electronic transport. Here we show how the B-periodic energy spectrum of a Coulomb blockaded quantum ring can be experimentally determined by transport experiments. By placing a double quantum dot coupled to a charge read-out in a ring geometry, the time-resolved transport of individual electrons can be monitored and the AB-like interference pattern can be established by counting many single electron events while the magnetic flux penetrating the ring is swept. Electron transport through graphene rings is used to determine the phase coherence length and to explore the local potential landscape in this novel material system.

Geometric manipulations, dephasing, and pumping of spin

Alexander Shnirman

University of Karlsruhe

We describe the effect of geometric phases induced by either classical or quantum electric fields acting on single electron spins in quantum dots in the presence of spin‐ orbit coupling. On one hand, applied electric fields can be used to control the geometric phases, which allows performing quantum coherent spin manipulations without using high‐frequency magnetic fields. On the other hand, fluctuating fields induce random geometric phases that lead to spin relaxation and dephasing, thus limiting the use of such spins as qubits. We also present a simple analysis of spin pumping through a single quantum dot with spin‐orbit coupling. We demonstrate that in presence of time‐reversal symmetry no peristaltic spin pumping is allowed. We focus on the analysis of spin pumping in the resonant transport regime and analyze the conditions to have pure spin transport.

Aharonov-Bohm interferometry in non-abelian quantum Hall states

Ady Stern

Weizmann

In this talk I will describe how non-abelian quantum statistics of quasi-particles in the quantum Hall effect may manifest itself in several interferometric measurements. 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.

Parallel Session A. 15:00-16:40

Parallel Session A. 15:00-16:30

Geometric phase interferometry with cold atoms in a two-dimensional optical lattice.

Omri Gat

Racah Institute of Physics, Hebrew University, Jerusalem 91904

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.

Dephasing in an Aharonov-Bohm interferometer containing a laterally coupled double quantum dot due to a coupling with a quantum dot charge sensor

T. Kubo１,*, Y. Tokura１，２, and S. Tarucha１，３

１Quantum Spin Information Project, ICORP, JST, Atsugi-shi, Kanagawa 243-0198, Japan

2NTT Basic Research Laboratories, NTT Corporation, Atsugi-shi, Kanagawa 243-0198, Japan

3Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

* e-mail address: kubo@tarucha.jst.go.jp

Particle-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.

The current fluctuation (shot noise) through the charge sensor is important for dephasing. Using the quantum dot (QD) as 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.

Figure 1: Schematic diagram of a laterally coupled QDQ (QD1, QD2) with a capacitively coupled QD charge sensor. Vinter and VS are inter-dot Coulomb interaction and capacitive coupling energy of the charge sensor.

In 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. 1. 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]. We show that the visibility of AB oscillations in the tunneling current through a DQD decreases linearly for low QD charge sensor source-drain bias voltage (VSDR) and decreases sub-linearly for high VSDR

[1] N. Bohr, Nature 121, 580 (1928).

[2] E. Buks, R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, Nature 391, 871 (1998).

[3] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).

[4] A. Yacoby, M. Heiblum, D. Mahalu, and H. Shtrikman, Phys. Rev. Lett. 74, 4047 (1995).

[5] T. Hatano, M. Stopa, W. Izumida, T. Yamaguchi, T. Ota, and S. Tarucha, Physica E 22, 534 (2004).

[6] T. Kubo, Y. Tokura, T. Hatano, and S. Tarucha, Phys. Rev. B 74 (2006) 205310.

[7] J. Schwinger, J. Math. Phys. 2 (1961) 407; L. V. Keldysh, Sov. Phys. JETP 20 (1965) 1018.

Coherent detection of electron dephasing

E. Strambini,∗ V. Giovannetti, F. Taddei, R. Fazio, V. Piazza, and F. Beltram

NEST, Scuola Normale Superiore and CNR-INFM, Piazza dei Cavalieri, 7, I-56126 Pisa, Italy

L. Chirolli†

Department of Physics, University of Konstanz, D-78457 Konstanz, Germany

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. The system shows characteristic AB oscillation as a function of the magnetic field, with a minimum of zero transmission for Φx = Φ0 /2. 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. 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. C. Elitzur and L. Vaidman, Found. Phys. 23, 987 (1993)), in which the dephasing source plays the role of a bomb in the original proposal, and the degradation of coherence mimics the explosion of the bomb. 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.

∗ Electronic address: e.strambini@sns.it

† Electronic address: luca.chirolli@uni-konstanz.de

Perfect Klein Tunneling with Zero Geometric Phase

Omri Bahat-Treidel, Or Peleg, Mark Grobman, Nadav Shapira, Tamar Pereg-Barnea, and Mordechai Segev

Scattering of relativistic fermions is fundamentally different from that of non-relativistic ones, since the former are described by Dirac equation rather than the Schrö dinger equation. 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. The absence of back-scattering has been explained by the π geometrical phase acquired by surrounding the Dirac point. 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. We derive an effective Hamiltonian for deformed honeycomb lattices, and find it to be hybridization of massless Dirac Hamiltonian with chiral Schrö dinger Hamiltonian, i.e., for the critical deformation we find , where are the momentum, are the Puali matrices and A, B are constants.

We reexamine Klein tunneling using the Hamiltonian above with the potential step is in the y−direction, and find that even though the geometrical phase vanish the transmission probability is one. The angular dependence of the transmission probability, T (θ), is remarkable: it is almost independent of the angle of incidence (Fig.1a). This result is surprising since in non-deformed lattices T(θ) decreases rapidly with the increasing angle. 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.1b). 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.1c).

FIG. 1: (a) T (θ) for a potential step in the y−direction (solid), for comparison T (θ) in non-deformed graphene (dashed). (b) T (θ) for a potential step in the x−direction (solid), for comparison T (θ) in bilayer graphene (dashed). (c) Transmission probability in the gapped phase at normal incidence as a function of the height of the step, V . The step is in the y−direction, and for comparison we also plot T (V ) for massive fermions.

[1] O. Klein, Z. Physicss 53, 157 (1929).

[2] T. Ando, T. Nakanishi, and R. Saito, Journal of the Phys. Society of Japan 67, 2857 (1998). [3] M. I. Katsnelson, S. Novoselov, and A. K. Geim, Nature (2006).

Wave functions and conductances at the integer quantum Hall transition: conformal invariance and possible theories

Ilya Gruzberg

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. We predict analytically the behavior of the PCC in presence of various boundary conditions.

Parallel Session B. 15:00-16:30

Flux-dependent Kondo effect in an Aharonov-Bohm ring with an embedded quantum dot

Ryosuke Yoshii and Mikio Eto

Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

In a quantum dot embedded in an Aharonov-Bohm (AB) ring, the Fano-Kondo effect has been theoretically predicted [1] and experimentally observed [2]. The conductance shows an asymmetric shape with a peak and a dip, as a function of energy level in the quantum dot. This is due to an interplay between the many-body Kondo effect and one-body interference effect. There is another problem on the Kondo effect in this system; whether the Kondo effect by itself is influenced by the interference effect or not. Simon et al. have studied this problem using the slave-boson mean-field theory [3]. In the present work, we perform the scaling analysis for this Kondo effect and show analytical expressions for the Kondo temperature and conductance.

Our model is shown in the figure. A quantum dot with an energy level is connected to two external leads by tunnel couplings, and . Another arm of the AB ring (reference arm) and external leads are represented by a one-dimensional tight-binding model with transfer integral and lattice constant . The reference arm includes a tunnel barrier with transmission probability of with. is the AB phase of magnetic flux penetrating the ring. We consider the Coulomb blockade regime with one electron in the dot: , , where is the Coulomb interaction in the dot (). is the level broadening, where with being the density of states in the leads.

In the small limit of the AB ring (), we apply the two-stage scaling method to our model [4]. First, we renormalize the energy level to , taking account of the charge fluctuation. On the second stage of scaling, we evaluate the Kondo temperature by considering the spin fluctuation. This yields

where is the asymmetric factor of tunnel barriers. This expression indicates that the Kondo temperature is significantly modulated by the magnetic flux. is minimal when and maximal when . The scaling analysis also yields the analytical expression for the conductance in two extreme cases, logarithmic corrections at and unitary limit at [4] .

We extend the scaling analysis to the case of finite size of the ring, . By the first stage of scaling, we find that the renormalized level depends on when . Here, is the Fermi velocity. The renormalized level is almost independent of when . On the second stage, we evaluate the Kondo temperature. In this case the relevant length scale is the size of the Kondo cloud, . depends on the flux significantly when and hardly when , in accordance with the previous study [3].

[1] W. Hofstetter, J. Konig, and H. Schoeller, Phys. Rev. Lett. 87, 156803 (2001).

[2] S. Katsumoto, H. Aikawa, M. Eto, and Y. Iye, phys. stat. sol.(c) 3, 4208 (2006).

[3] P. Simon, O. Entin-Wohlman, and A. Aharony, Phys. Rev. B 72, 245313 (2005).

[4] R. Yoshii and M. Eto, J. Phys. Soc. Jpn. 77, 123714 (2008).

Experimental Observation of Optical Aharonov-Bohm Effect in Stacked Type-II Quantum Dots

Igor L Kuskovsky1,* I. R. Sellers2, V. R. Whiteside2, B. D. McCombe2, and A. O. Govorov3

1Department of Physics, Queens College of CUNY, Flushing NY 11367 USA

2 Department of Physics, University at Buffalo SUNY, Buffalo, NY 14260 USA

3Department of Physics and Astronomy, Ohio University, Athens OH 45701 USA

The Aharonov-Bohm (AB) effect is typically discussed for a quantum charged particle moving along a closed trajectory in a magnetic field. There, however, exists a possibility of the AB effect associated with an overall neutral quasi-particle that possesses a radial electric dipole moment (e.g., an exciton in quantum ring or cylindrical type-II quantum dot (QD)). Type-II excitons have been predicted to be particularly sensitive to the AB effects due to relatively larger spatial separation of charged particles. The AB phase reveals itself in photoluminescence (PL) properties of type-II QDs since, due to the cylindrical symmetry, the exciton ground state initially has a zero orbital angular momentum, which changes to higher values with increasing magnetic field. This transition of the angular momentum to a non-zero value influences the optical properties in two ways: (i) the ground state energy will oscillate as the orbital angular momentum states cross and (ii) the PL intensity will change from strong (bright exciton with zero angular momentum) to weak (dark excitons with non- zero angular momentum) with increasing magnetic field.

We present experimental studies on type-II magneto-excitons in stacked ZnTe/ZnSe type-II QDs. Results show strong AB oscillations in both the energy and intensity of the PL from the same structure (Fig. 1). This is the only system for which the oscillations in both energy and intensity have been reported. In addition, we investigate the temperature dependence of the emission, and show that the AB oscillations are remarkably robust against temperature, with the AB signature visible up to 180 K. We believe this to be the highest temperature at which the AB effect, and therefore quantum coherence, has been demonstrated in semiconductor ring-like structures.

To explain the observations, we first point out that single electron density calculations show that the electron, in the absence of strain, will be located either above or below the dot and, therefore, no AB signature is expected. An observation of the AB effect in such a system is thus possible only due to effects of strain and is inherently dependent on the growth conditions.

In our case, the stacked character of the systems ensures that the electron's wave-function is "pushed" to the side of the dot due to electron-electron interaction, independent of stress, whereas cylindrical geometry nicely defines the ring- like trajectory for an electron. We thus explain the results as a motion of an electron around an entire stack of QDs, one of which is occupied by a hole (inset, Fig. 1).

*http://www.physics.qc.edu/pages/kuskovsky/

Breaking of Phase Symmetry in a "Which-Path?" Interferometer

V. Puller1 and Y. Meir1,2

1Department of Physics, Ben-Gurion University of the Negev, 84105 Beer Sheva, Israel

2The Ilze Katz Center for Meso- and Nano-scale Science and Technology, Ben-Gurion University of the Negev, 84105 Beer Sheva - Israel

Linear response conductance of a two terminal Aharonov-Bohm (AB) interferometer is an even function of magnetic field, as dictated by Onsager-Büttiker relations [1,2]. One of the possibilities to break this phase symmetry is by coupling the interferometer to a non-equilibrium environment [3]. We study this effect in a "Which Path?" detector [4], which consists of an interferometer with quantum dots (QDs) in each of its arms, with one of the QDs being capacitively coupled to a nearby quantum point contact (QPC). AB oscillations asymmetric in magnetic field appear when a finite voltage bias is applied to the QPC. Sweeping in the same time the level in the other QD across the Fermi level allows one to observe smooth change of the phase from 0 to π, rather than the abrupt jump observed in an isolated AB interferometer [2]. In addition, coupling to the biased QPC results in a photo-assisted current, which is non-zero even without source-drain bias voltage applied to the interferometer, and is inherently asymmetric in magnetic field.

The possibility of using such a "Which Path?" detector in order to measure the transmission phase through a QD is discussed.

[1] M. Büttiker, Phys. Rev. Lett., vol. 57, p. 1761 (1986) .

[2] A. Yacoby et al., Phys. Rev. Lett., vol. 74, p. 4047 (1995).

[3] D. Sánchez and K. Kang, Phys. Rev. Lett., vol. 100, p. 036806 (2008).

[4] E. Buks et al., Nature, vol. 391, p. 871 (1998) .

Fluctuation Theorem in a Quantum Dot Aharonov-Bohm Interferometer

Y. Utsumi

Institute of Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan

The fluctuation theorem (FT) is unique, since it is valid even in far from equilibrium regime. It relays only on the microscopic reversibility and relates probabilities for positive and negative entropy productions [1]. Recently, we applied the FT to the quantum transport problem by combining with the theory of the full counting statistics (FCS) [2]. The FT in the frame of FCS leads Onsager's relations in the linear response regime and, moreover, universal relations among nonlinear transport coefficients. Precisely, for a two terminal system, the nth-order nonlinear transport coefficient of m-th cumulant is introduced as where is the source-drain bias voltage and is the temperature. In the present study, we will propose novel relationships beyond the linear response theory in the presence of the magnetic field B, which relate the nonlinear conductance and the linear response of current noise ; [ is the symmetrised/anti-symmetrised transport coefficient].

In order for the theoretical demonstration, we apply the new relationships to a quantum dot (QD) embedded in a two-terminal Aharonov-Bohm interferometer [Fig. (a)] [3]. In this setup, because of a lack of the mirror symmetry along the horizontal axis [dotted line in Fig. (a)] and the Coulomb interaction inside the QD, the asymmetric component of the nonlinear conductance remains finite [4]. Using the path-integral Keldysh generating function, we treat the interaction within the non-equilibrium saddle-point approximation, namely a mean-field level approximation. The approximation properly accounts for the coupling between current and charge fluctuations by the Coulomb interaction. We show that the nonlinear transport coefficients satisfy universal relations imposed by the FT [Fig. (b)].

We also compare our theory with recently proposed universal relations, which do not rely on the micro-reversibility [5]. We point out that the two theories are consistent for the symmetric component. However for the asymmetric component, Ref. [5] provides the necessary condition of the FT. In this sense, Ref. [5] includes our FT. However, it is still possible to distinguish the two theories experimentally.

References

[1] D. J. Evans et al. Phys. Rev. Lett. 71, 2401 (1993); G. Gallavotti Phys. Rev. Lett. 77, 4334 (1996).

[2] K. Saito and Y. Utsumi, Phys. Rev. B 78, 115429 (2008).

[3] Y. Utsumi and K. Saito, arXiv/0810.1113; Phys. Rev. B in press.

[4] D. Sánchez and M. Büttiker, Phys. Rev. Lett. 93, 106802 (2004).

[5] H. Fö rster and M. Büttiker, Phys. Rev. Lett. 101, 136805 (2008); arXiv:0903.1431.

Spin filtering and spin-orbit interaction in Kondo quantum dots

Edson Vernek,1,2 Nancy Sandler,2 and Sergio E. Ulloa2

1 Instituto de Física, Universidade Federal de Uberlândia, Uberlândia, 38400-902, MG - Brazil

2 Department of Physics and Astronomy, and Nanoscale and Quantum Phenomena Institute, Ohio University, Athens, Ohio 45701-2979 - USA

Semiconductor quantum dots (QDs) are a promising platform on which to achieve charge and spin control due to their discrete energy levels, sizable Coulomb interaction and precise level and size manipulation via gate voltages. In these structures, coherent electron propagation at low temperatures and quantum interference may play an essential role in determining their electronic properties. Accurate control of electronic transport is in fact now systematically achieved by exploiting quantum interference in multiple-path geometries, such as those with one or multiple QDs embedded in a ring [1]. In these structures, a few-mT magnetic field through a ring produces drastic changes in transport properties due to the celebrated Aharonov-Bohm (AB) effect. Interestingly enough, the presence of Rashba spin-orbit (SO) interaction [2], which can be modulated by applied gate voltages, provides additional dynamical control of charge and spin transport, although this possibility has received little attention in the literature.

The strong Coulomb interaction in these systems may also result in a Kondo state appearing below a characteristic Kondo temperature TK [3], giving rise to strong antiferromagnetic correlations between localized and itinerant electrons in the leads. This produces an additional transport channel through the system with unique experimental signatures [3], signaling the presence of many-body correlations in these systems. It is in this context that we investigate the competition between SO interaction and AB effect on determining the Kondo correlations in a ring system. The geometry and experimental accessibility allows one us to explore very interesting physics.

Several ring geometries have been proposed as spin polarizers and their behavior has been analyzed in different regimes, although SO interactions have only been treated perturbatively [4]. The basic principle involved in the spin-filtering effect is the modification of the conductance for different spin species as a result of the AB flux and SO effects that introduce spin-dependent dynamical phases for the electrons in the multiply- connected geometry. The correct and complete inclusion of the Kondo physics is crucial in order to provide a proper description of the system, as we describe in this work, especially with respect to its spin transport behavior.

We will present a numerical renormalization group study of this system. The approach is capable of addressing the full spin-dependent character of the coupling to the leads and the p-h asymmetry in a non- perturbative fashion [5]. Our analysis demonstrates that the combination of AB and SO effects can strongly suppress the Kondo state, and in fact eliminate the desired spin filtering effect described previously [4]. Most interestingly, we demonstrate that this suppression can be fully compensated by the application of an in-plane Zeeman field. Under those conditions, the Kondo screening is restored and the spin filtering effect re-established.

Fig. 1. Spin conductance of QD embedded in ring (see inset) in the presence of SO interaction and AB flux. In- plane magnetic field restores Kondo effect and results in perfect spin-filtering.

1. Y. Ji, et al., Science 290, 779 (2000).

2. Y. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6039 (1984).

3. T. Inoshita, Science 281, 526 (1998), and references therein.

4. Q.-F. Sun, J. Wang and H. Guo, Phys. Rev B 71, 165310 (2005).

5. R. Bulla, T. Costi, T. Pruschke, Rev. Mod. Phys. 80, 395 (2008).

Plenary Session 17:00-18:50

Interference between two indistinguishable electrons from independent sources

Moty Heiblum

Weizmann Institute of Science

Very much like the ubiquitous quantum interference of a single particle with itself, quantum interference of two independent, but indistinguishable, particles is also possible. For a single particle, the interference is between the amplitudes of the particle's wave-functions, whereas the interference between two particles is a direct result of quantum exchange statistics. Such interference is observed only in the joint probability of finding the particles in two separated detectors, after they were injected from two spatially separated and independent sources. Evidently, such two-particle correlations are a direct signature of quantum entanglement between the spatial degrees of freedom of the two particles ('orbital entanglement'), even though they do not interact with each other.

The performed experiment is an electronic analogue to the historical Hanbury Brown and Twiss experiment with classical light. It is based on the electronic Mach-Zehnder interferometer that uses edge channels in the quantum Hall effect regime. We partitioned two independent and mutually incoherent electron beams into two trajectories, so that the combined four trajectories enclosed an Aharonov-Bohm flux. Although individual currents and their fluctuations were found to be independent of the Aharonov-Bohm flux, the cross-correlation between current fluctuations at two opposite points across the device exhibited strong Aharonov-Bohm oscillations, suggesting orbital entanglement between the two electron beams.

Interaction-induced dephasing of quantum interference of electrons in quantum wires and Aharonov-Bohm ring

Alexander D. Mirlin

Dephasing of quantum interference by electron-electron interaction in wires and rings is discussed. The systems considered include many-channel and single-channel (Luttinger liquid) wires, as well as Aharonov-Bohm interferometers formed by such wires. At finite temperatures (or else, under non-equilibrium conditions), electron-electron scattering processes induce electron dephasing (which can be thought of as a finite life time of electronic excitations), leading to suppression of quantum coherent phenomena. Two famous quantum interference phenomena are Aharonov Bohm effect and weak localization. Interaction-induced dephasing rates for these two phenomena are analyzed and shown to be parametrically different. We also discuss manifestation of electron dephasing in another related phenomenon - smearing of the zero-bias anomaly in the tunneling density of states

Quantum intereference in an artificial 2D lattice

Jurgen Smet

We demonstrate and discuss quantum interferences unique to 2D artificial crystals. The 2D artificial crystals are implemented by modulating a 2D electron system in two directions. In such a system, electrons are Bragg reflected at the Brillouin zone boundaries of the artificial lattice, where energy gaps appear. In a B-field, electrons move on constant energy contours in k-space, which deviate from the free electron contours due to the gaps at the zone boundaries. Both closed and open electron trajectories emerge. At sufficiently large B-fields, magnetic breakdown allows tunneling across the gaps and enables an entire new network of closed electron paths. These paths enclosing different areas in k-space produce additional 1/B-periodic oscillatory features similar to Shubnikov-de Haas oscillations in the transport quantities. Also quantum interferences of certain pairs of electron paths, which share the same starting and ending point and hence are intimately connected with the Aharonov Bohm effect, are observed as additional 1/B periodic oscillations. The periodicity is proportional to the area enclosed by both paths. These paths are surprisingly complex. To identify the relevant groups of interfering trajectories in k-space a density dependent study has been particularly instrumental. Our experimental observations and interpretations are strongly supported by Monte Carlo simulations.

Quantum transport and Aharonov-Bohm effect in diffusive networks

Gilles Montambaux (with C. Texier and P. Delplace)

Laboratoire de Physique des Solides, Université Paris-Sud, CNRS, 91405-Orsay

We have considered the physics of Aharonov-Bohm oscillations (persistent current, conductance oscillations, ...) for arrays of mesoscopic disordered rings, in the presence of an external magnetic field. If the case of the isolated ring is well known, we show that for connected rings, the winding of the Brownian trajectories around each ring is modified, leading to a new harmonics content of the quantum oscillations. This analysis is based on the calculation of the spectral determinant of the diffusion equation for which we find a simple expression on any network. We study especially the three cases: a single ring connected to an arbitrary network, a linear array of rings connected with long or short wires, and the square network.

Poster Session 19:00 - 20:30

Day 3 - Tuesday 13-Oct-09

Plenary Session 09:30-11:00

Variations on a theme of A & B

Michael Berry, Physics Department, University of Bristol, UK

In the original AB effect, the waves are quantum but the flux is classical. This is one of four possibilities, the others being quantum waves/quantum flux (Tonomura experiment), classical waves/quantum flux (Steinberg proposal) and classical waves/classical flux. The last was demonstrated with ripples traversing a bathtub vortex, with the advantage that the detailed structure of the near field can be observed, including the phase singularities with their peculiar behaviour for half-integer flux. Analogous phenomena occur in optical interferometers and spiral wave plates. The AB wavefunction has a surprising interpretation in terms of the Cornu spiral.

1. Berry, M. V.,1980, Exact Aharonov-Bohm wave function obtained by applying Dirac's magnetic phase factor Eur. J. Phys. 1, 240-244. [97]
2. Berry, M. V., Chambers, R. G., Large, M. D., Upstill, C. & Walmsley, J. C.,1980, Wavefront dislocations in the Aharonov-Bohm effect and its water-wave analogue Eur. J. Phys. 1, 154-162. [96]
3. Berry, M. V.,1986, The Aharonov-Bohm effect is real physics not ideal physics in Fundamental aspects of quantum theory eds. Gorini, V. & Frigerio, A. (Plenum, Vol. 144, pp. 319-320. [157]
4. Berry, M. V.,2007, Wave dislocations threading interferometers Proc. R. Soc. A 463, 1697-1711.[397]
5. Berry, M. V.,1999, Aharonov-Bohm beam deflection: Shelankov's formula, exact solution, asymptotics and an optical analogue J. Phys.A. 32, 5627-5641. [309]
6. Berry, M. V. & Shelankov, A.,1999, The Aharonov-Bohm wave and the Cornu spiral J. Phys. A. 32, L447-L455. [310]
7. Berry, M. V.,2004, Optical vortices evolving from helicoidal integer and fractional phase steps J.Optics. A 6, 259-268. [359]

The Aharonov-Bohm effect for neutral particles

Edouard Sonin

The Aharonov-Bohm (AB) effect was discovered as a quantum-mechanical effect for charged particles, but it has its counterpart in classical wave mechanics. The AB interference arises at scattering of a sound wave by a vortex in classical and quantum hydrodynamics. This interference leads to a transverse force between quasiparticles and vortices in superfluid and superconductors, which is important for the Hall effect in superconductors. The AB effect was also generalized to neutral particles with magnetic or electric dipole momenta. The AB effect for charge particles and its modification for magnetic momenta (the Aharonov-Casher effect) have already been experimentally observed, the AB effect for electrically polarized neutral particles is still waiting on its experimental detection. A possible system for this detection is a Bose-condensate of excitons in double quantum wells. Observation of the AB effect in this system would provide direct evidence of Bose-Einstein condensation.

Weak Measurement, Weak Values, and the Aharonov-Bohm effect

Yuval Gefen^* and Alessandro Romito^❡

Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel 

^❡ Institut für Theoretische Festkörperphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany

The measurement of any observable in quantum mechanics is a probabilistic process described by the projection postulate. Each eigenvalue of the observable happens to be an outcome of the measurement process with a given probability, and the original state of the system collapses into the

corresponding eigenstate. Weakly measuring an observable (i.e., measuring it while weakly disturbing the system), provides only partial information on the state of the system. It has been proposed^1,2 that a two-step procedure -weak measurement followed by a strong one, where the outcome of the first measurement is kept provided a second post-selected outcome occurs- leads to a weak value. The value of the latter may lie well beyond the range of projective measurements. Weak values allow us into an arena of correlations of non-commuting variables that has not been manifestly accessible before. Proposals to measure weak values within the framework of solid state physics did not exist till our recent works^3,4. We will review these works, giving special emphasis to the incorporation of the AB effect within a weak value protocol.

¹ Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988).

² Y. Aharonov and L. Vaidman, Phys. Rev. A, 41, 11 (1990).

³ Alessandro Romito, Yuval Gefen, and Yaroslav M. Balnter, Phys . Rev. Lett. 100, 056801 (2008).

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Parallel Session A. 11:30-13:00

Influence of Coulomb interaction on periodicities of quantum Hall interferometes

Bernd Rosenow

Quantum Hall devices are supposed to be an ideal laboratory for the study of interference effects, because within a conductance plateau, the bulk of a sample is insulating and the current is confined to conducting edge states. Closed interference paths can be defined with the help of two narrow constrictions, which mediate tunneling from one edge to the other. Quantum interference should then manifest itself in flux- and gate-voltage-dependent conductance oscillations. In integer quantum Hall systems, the Coulomb coupling between fully occupied lower Landau levels and the highest partially occupied level gives rise to flux subperiods smaller than one flux quantum. This scenario is generalized to fractional quantum Hall systems, and theoretical predictions are compared to experiments.

Multiple Aharonov-Bohm periodicities in fractional quantum Hall interferometers

Ivan P. Levkivskyi1,2 , Jurg Fröhlich3 and Eugene V. Sukhorukov1

1 Département de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland

2 Physics Department, Kyiv National University, 03022 Kyiv, Ukraine and

3 Institute of Theoretical Physics, ETH Hönggerberg, CH-8093 Zurich, Switzerland