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Franco Nori

RIKEN, Wako, Japan, Center for Emergent Matter Science, Chief Scientist and Group Director

University of Michigan, Physics, Assistant, Associate, full Professor and Research Scientist

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Alberto Pimpinelli

Université Clermont Auvergne

Gordon N Fleming

Pennsylvania State University

University of Illinois at Urbana-Champaign

Pedro de Andres

CSIC (Consejo Superior de Investigaciones Científicas-Spanish National Research Council)

Jonathan D Maltz

Stanford University

University of Regensburg

Thomas Curtright

University of Miami

University of York

California Institute of Technology

Jean-Yves Fortin

Centre National de la Recherche Scientifique / French National Centre for Scientific Research

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Papers by Franco Nori

Scalable quantum computation via local control of only two qubits

by Martin Plenio, Simone Montangero, T. Calarco, and Franco Nori

Physical Review A, 2010

We apply quantum control techniques to control a large spin chain by only acting on two qubits at... more We apply quantum control techniques to control a large spin chain by only acting on two qubits at one of its ends, thereby implementing universal quantum computation by a combination of quantum gates on the latter and swap operations across the chain. It is shown that the control sequences can be computed and implemented efficiently. We discuss the application of these ideas to physical systems such as superconducting qubits in which full control of long chains is challenging. 02.30.Yy Controlling quantum systems at will has been an aspiration for physicists for a long time. Achieving quantum control not only clears the path towards a thorough understanding of quantum mechanics, but it also allows the exploration of novel devices whose functions are based on exotic quantum mechanical effects. Among others, the future success of quantum information processing depends largely on our ability to tame many-body quantum systems that are highly fragile. Although the progress of technology allows us to manipulate a small number of quanta quite well, controlling larger systems still represents a considerable challenge. Unless we overcome difficulties towards the control over large many-body systems, the benefits we can enjoy with the 'quantumness' will be severely limited.

Extraordinary momentum and spin in evanescent waves

by Konstantin Bliokh and Franco Nori

Momentum and spin represent fundamental dynamic properties of quantum particles and fields. In pa... more Momentum and spin represent fundamental dynamic properties of quantum particles and fields. In particular, propagating optical waves (photons) carry momentum and longitudinal spin determined by the wave vector and circular polarization, respectively. Here we show that exactly the opposite can be the case for evanescent optical waves. A single evanescent wave possesses a spin component, which is independent of the polarization and is orthogonal to the wave vector. Furthermore, such a wave carries a momentum component, which is determined by the circular polarization and is also orthogonal to the wave vector. We show that these extraordinary properties reveal a fundamental Belinfante&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#39;s spin momentum, known in field theory and unobservable in propagating fields. We demonstrate that the transverse momentum and spin push and twist a probe Mie particle in an evanescent field. This allows the observation of &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#39;impossible&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#39; properties of light and of a fundamental field-theory quantity, which was previously considered as &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#39;virtual&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#39;.

Relativistic Electron Vortex Beams: Angular Momentum and Spin-Orbit Interaction

by Konstantin Bliokh and Franco Nori

Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM)... more Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM), we construct exact Bessel-beam solutions of the Dirac equation. They describe relativistic and nonparaxial corrections to the scalar electron beams. We describe the spin and orbital AM of the electron with Berry-phase corrections and predict the intrinsic spin-orbit coupling in free space. This can be observed as a spin-dependent probability distribution of the focused electron vortex beams. Moreover, the magnetic moment is calculated, which shows different g-factors for spin and orbital AM and also contains the Berry-phase correction.

Characterizing optical chirality

by Konstantin Bliokh and Franco Nori

We examine the recently introduced measure of chirality of a monochromatic optical field [Y. Tang... more We examine the recently introduced measure of chirality of a monochromatic optical field [Y. Tang and A. E. Cohen, Phys. Rev. Lett. 104, 163901 (2010)] using the momentum (plane-wave) representation and helicity basis. Our analysis clarifies the physical meaning of the measure of chirality and unveils its close relation to the polarization helicity, spin angular momentum, energy density, and Poynting energy flow. We derive the operators of the optical chirality and of the corresponding chiral momentum, which acquire remarkably simple forms in the helicity representation.

Magnetoelectric Effects in Local Light-Matter Interactions

by Konstantin Bliokh and Franco Nori

Physical Review Letters, 2014

We study the generic interaction of a monochromatic electromagnetic field with bi-isotropic nanop... more We study the generic interaction of a monochromatic electromagnetic field with bi-isotropic nanoparticles. Such an interaction is described by dipole-coupling terms associated with the breaking of dual, P-and T-symmetries, including the chirality and the nonreciprocal magnetoelectric effect. We calculate absorption rates, radiation forces, and radiation torques for the nanoparticles and introduce novel characteristics of the field quantifying the transfer of energy, momentum, and angular-momentum in these interactions. In particular, we put forward the concept of 'magnetoelectric energy density', quantifying the local PT-symmetry of the field. Akin to the 'super-chiral' light suggested recently for sensitive local probing of molecular chirality [Phys. Rev. Lett. 104, 163901 (2010); Science 332, 333 (2011)], here we describe a complex field for sensitive probing of the nonreciprocal magnetoelectric effect in nanoparticles or molecules.

Transverse spin of a surface polariton

by Konstantin Bliokh and Franco Nori

ABSTRACT

Dual electromagnetism: Helicity, spin, momentum, and angular momentum

by Konstantin Bliokh and Franco Nori

The dual symmetry between electric and magnetic fields is an important intrinsic property of Maxw... more The dual symmetry between electric and magnetic fields is an important intrinsic property of Maxwell equations in free space. This symmetry underlies the conservation of optical helicity and, as we show here, is closely related to the separation of spin and orbital degrees of freedom of light (the helicity flux coincides with the spin angular momentum). However, in the standard field-theory formulation of electromagnetism, the field Lagrangian is not dual symmetric. This leads to problematic dual-asymmetric forms of the canonical energy-momentum, spin and orbital angular-momentum tensors. Moreover, we show that the components of these tensors conflict with the helicity and energy conservation laws. To resolve this discrepancy between the symmetries of the Lagrangian and Maxwell equations, we put forward a dualsymmetric Lagrangian formulation of classical electromagnetism. This dual electromagnetism preserves the form of Maxwell equations, yields meaningful canonical energy-momentum and angular-momentum tensors, and ensures a self-consistent separation of the spin and orbital degrees of freedom. This provides a rigorous derivation of the results suggested in other recent approaches.

Colloquium: Unusual resonators: Plasmonics, metamaterials, and random media

by Konstantin Bliokh, Franco Nori, and Sergey Savel'ev

Super-resolution, extraordinary transmission, total absorption, and localization of electromagnet... more Super-resolution, extraordinary transmission, total absorption, and localization of electromagnetic waves are currently attracting growing attention. These phenomena are related to different physical systems and are usually studied within the context of different, sometimes rather sophisticated, approaches. Remarkably, all these seemingly unrelated phenomena owe their origin to the same underlying physical mechanism, namely, wave interaction with an open resonator. Here we show that it is possible to describe all of these effects in a unified way, mapping each system onto a simple resonator model. Such description provides a thorough understanding of the phenomena, explains all the main features of their complex behavior, and enables one to control the system via the resonator parameters: eigenfrequencies, Q factors, and coupling coefficients.

Semiclassical dynamics of electron wave packet states with phase vortices

by Konstantin Bliokh and Franco Nori

Physical review letters, 2007

We consider semiclassical higher-order wave packet solutions of the Schrödinger equation with pha... more We consider semiclassical higher-order wave packet solutions of the Schrödinger equation with phase vortices. The vortex line is aligned with the propagation direction, and the wave packet carries a welldefined orbital angular momentum (OAM) @l (l is the vortex strength) along its main linear momentum. The probability current coils around the momentum in such OAM states of electrons. In an electric field, these states evolve like massless particles with spin l. The magnetic-monopole Berry curvature appears in momentum space, which results in a spin-orbit-type interaction and a Berry/Magnus transverse force acting on the wave packet. This brings about the OAM Hall effect. In a magnetic field, there is a Zeeman interaction, which, can lead to more complicated dynamics.

Anderson Localization of Light in Layered Dielectric Structures

by Konstantin Bliokh and Franco Nori

Electron vortex beams in a magnetic field: A new twist on Landau levels and Aharonov-Bohm states

by Konstantin Bliokh and Franco Nori

We examine the propagation of the recently discovered electron vortex beams in a longitudinal mag... more We examine the propagation of the recently discovered electron vortex beams in a longitudinal magnetic field. We consider both the Aharonov-Bohm configuration with a single flux line and the Landau case of a uniform magnetic field. While stationary Aharonov-Bohm modes represent Bessel beams with flux-and vortex-dependent probability distributions, stationary Landau states manifest themselves as nondiffracting Laguerre-Gaussian beams. Furthermore, the Landau-state beams possess field-and vortex-dependent phases: (i) the Zeeman phase from coupling the quantized angular momentum to the magnetic field and (ii) the Gouy phase, known from optical Laguerre-Gaussian beams. Remarkably, together these phases determine the structure of Landau energy levels. This unified Zeeman-Landau-Gouy phase manifests itself in a nontrivial evolution of images formed by various superpositions of modes. We demonstrate that, depending on the chosen superposition, the image can rotate in a magnetic field with either (i) Larmor, (ii) cyclotron (double-Larmor), or (iii) zero frequency. At the same time, its centroid always follows the classical cyclotron trajectory, in agreement with the Ehrenfest theorem. Interestingly, the nonrotating superpositions reproduce stable multivortex configurations that appear in rotating superfluids. Our results open an avenue for the direct electron-microscopy observation of fundamental properties of free quantum-electron states in magnetic fields.

Does a Surface Polariton Have Spin?

by Konstantin Bliokh and Franco Nori

Relativistic electron vortex beams: angular momentum and spin-orbit interaction

by Konstantin Bliokh and Franco Nori

Physical review letters, Jan 21, 2011

Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM)... more Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM), we construct exact Bessel-beam solutions of the Dirac equation. They describe relativistic and nonparaxial corrections to the scalar electron beams. We describe the spin and orbital AM of the electron with Berry-phase corrections and predict the intrinsic spin-orbit coupling in free space. This can be observed as a spin-dependent probability distribution of the focused electron vortex beams. Moreover, the magnetic moment is calculated, which shows different g factors for spin and orbital AM and also contains the Berry-phase correction.

Classical Field Approach to Quantum Weak Measurements

by Konstantin Bliokh and Franco Nori

By generalizing the quantum weak measurement protocol to the case of quantum fields, we show that... more By generalizing the quantum weak measurement protocol to the case of quantum fields, we show that weak measurements probe an effective classical background field that describes the average field configuration in the spacetime region between pre-and postselection boundary conditions. The classical field is itself a weak value of the corresponding quantum field operator and satisfies equations of motion that extremize an effective action. Weak measurements perturb this effective action, producing measurable changes to the classical field dynamics. As such, weakly measured effects always correspond to an effective classical field. This general result explains why these effects appear to be robust for pre-and postselected ensembles, and why they can also be measured using classical field techniques that are not weak for individual excitations of the field.

Observation of the Larmor and Gouy Rotations with Electron Vortex Beams

by Konstantin Bliokh and Franco Nori

Physical Review Letters, Feb 25, 2013

Electron vortex beams carrying intrinsic orbital angular momentum (OAM) are produced in electron ... more Electron vortex beams carrying intrinsic orbital angular momentum (OAM) are produced in electron microscopes where they are controlled and focused by using magnetic lenses. We observe various rotational phenomena arising from the interaction between the OAM and magnetic lenses. First, the Zeeman coupling, proportional to the OAM and magnetic field strength, produces an OAM-independent Larmor rotation of a mode superposition inside the lens. Second, when passing through the focal plane, the electron beam acquires an ...

Photon trajectories, anomalous velocities and weak measurements: a classical interpretation

by Konstantin Bliokh, Franco Nori, and A. Kofman

reported the observation of 'average trajectories of single photons' in a two-slit interference e... more reported the observation of 'average trajectories of single photons' in a two-slit interference experiment. This was possible by using the quantum weak-measurement method, which implies averaging over many events, i.e. in fact, a multi-photon limit of classical linear optics. We give a classical-optics interpretation of this experiment and other related problems. It appears that weak measurements of the local momentum of photons made by Kocsis et al represent measurements of the Poynting vector in an optical field. We consider both the real and imaginary parts of the local momentum and show that their measurements have been realized in classical optics using small-probe particles. We also examine the appearance of 'anomalous' values of the local momentum: either negative (backflow) or exceeding the wavenumber (superluminal propagation). These features appear to be closely related to vortices and evanescent waves. Finally, we revisit a number of older works and find examples of photon trajectories and anomalousmomentum measurements in various optical experiments. Gesellschaft particle [2]. This is because a strong measurement of one quantity destroys the information about its complementary quantity. However, quantum weak measurements, introduced by Aharonov et al, imply a weak coupling between the measuring and the measured sub-systems and do not significantly perturb the measured part . This approach allows the simultaneous determination of complementary quantities, albeit averaged over many events. Furthermore, the weak values of the measured quantity can be 'anomalous', i.e. beyond the range of the spectrum of the corresponding Hermitian operator [3] and even complex . Weak values are closely related to complex conditional probabilities for the system to be found in a certain state under the condition that it is subsequently found in another state . Soon after the introduction of the theoretical concept of weak measurements, this method was applied in observing tiny polarization-dependent wave shifts in classical optics . In these experiments, two weakly coupled degrees of freedom of light (the polarization and spatial distribution) played the roles of the measured and measuring ('meter') sub-systems. For instance, the spin-Hall effect of light (a spin-dependent shift of optical beams) [9, 10] was treated as the shift of the 'meter' which 'measured' the photon spin (helicity) . As a result, the shift was significantly amplified owing to the anomalous weak value of the helicity, which was complex and far outside its normal [−1, 1] range (h = 1 units are used throughout the paper). Despite the quantum weak-measurement formalism, experiments clearly dealt with classical optical systems and all results, including the spin-Hall effect of photons, allowed fully classical descriptions . This is because weak measurements require averaging over many one-particle events, so that in linear photonic systems this approach implies the classical-optics limit. Kocsis et al emphasized that their experiment was realized in the quantum one-photon regime. Still, the measurements represented an average over many events, and the same results would appear in the continuous-wave regime [2, 12] 6 . Nonetheless, the results of [1] have not 6 According to Dirac: 'Each photon interferes only with itself. Interference between two different photons never occurs'. Therefore, the interference pictures obtained from averaging many one-photon events and from the multiphoton classical field coincide with each other. For experimental demonstrations, see .

Relativistic Hall Effect

by Konstantin Bliokh and Franco Nori

We consider the relativistic deformation of quantum waves and mechanical bodies carrying intrinsi... more We consider the relativistic deformation of quantum waves and mechanical bodies carrying intrinsic angular momentum (AM). When observed in a moving reference frame, the centroid of the object undergoes an AM-dependent transverse shift. This is the relativistic analogue of the spin-Hall effect, which occurs in free space without any external fields. Remarkably, the shifts of the geometric and energy centroids differ by a factor of 2, and both centroids are crucial for the Lorentz transformations of the AM tensor. We examine manifestations of the relativistic Hall effect in quantum vortices, and mechanical flywheels, and also discuss various fundamental aspects of this phenomenon. The perfect agreement of quantum and relativistic approaches allows applications at strikingly different scales: from elementary spinning particles, through classical light, to rotating black-holes. PACS numbers: 42.50.Tx, 03.65.Pm, 03.30.+p

Spatiotemporal vortex beams and angular momentum

by Konstantin Bliokh and Franco Nori

We present a space-time generalization of the known spatial (monochromatic) wave vortex beams car... more We present a space-time generalization of the known spatial (monochromatic) wave vortex beams carrying intrinsic orbital angular momentum (OAM) along the propagation direction. Generic spatio-temporal vortex beams are polychromatic and can carry intrinsic OAM at an arbitrary angle to the mean momentum. Applying either (i) a transverse wave-vector shift or (ii) a Lorentz boost to a monochromatic Bessel beam, we construct a family of either (i) time-diffracting or (ii) non-diffracting spatio-temporal Bessel beams, which are exact solutions of the Klein-Gordon wave equations. The proposed spatio-temporal OAM states are able to describe either photon or electron vortex states (both relativistic and nonrelativistic), and can find applications in particle collisions, optics of moving media, quantum communications, and astrophysics.

Transverse relativistic effects in paraxial wave interference

by Konstantin Bliokh, Franco Nori, and Y. Izdebskaya

We consider relativistic deformations of interfering paraxial waves moving in the transverse dire... more We consider relativistic deformations of interfering paraxial waves moving in the transverse direction. Owing to superluminal transverse phase velocities, noticeable deformations of the interference patterns arise when the waves move with respect to each other with non-relativistic velocities. Similar distortions also appear on a mutual tilt of the interfering waves, which causes a phase delay analogous to the relativistic time delay. We illustrate these observations by the interference between a vortex wave beam and a plane wave, which exhibits a pronounced deformation of the radial fringes into a fork-like pattern (relativistic Hall effect). Furthermore, we describe an additional relativistic motion of the interference fringes (a counter-rotation in the vortex case), which becomes noticeable at the same non-relativistic velocities.

Transverse spin and momentum in two-wave interference

by Konstantin Bliokh and Franco Nori

It is well-known, since the seminal works by J.H. Poynting , that light carries momentum and angu... more It is well-known, since the seminal works by J.H. Poynting , that light carries momentum and angular momentum (AM) [2,3]. Typical plane-wave or Gaussian-beam states exhibit longitudinal momentum associated with the wave vector k and also longitudinal spin AM associated with the degree of circular polarization (helicity) σ . Locally, optical momentum and angular-momentum densities can demonstrate unusual features which have recently attracted considerable attention: "super-momentum" with values higher than k per photon [4-8], transverse helicity-independent spin AM [9-12], and transverse helicity-dependent momentum [10,13,14]. So far, such abnormal momentum and spin properties have appeared only in special field configurations, namely, evanescent waves and optical vortices. Here we find that the simplest propagating nonsingular field -two interfering plane waves -also exhibits a variety of extraordinary spin and momentum properties. Despite the seemingly planar and thoroughly-studied character of the two-wave system, we discover that such field possesses a transverse (out-of-plane) helicity-independent spin AM, and also a transverse polarization-dependent momentum with unusual physical properties.

Scalable quantum computation via local control of only two qubits

by Martin Plenio, Simone Montangero, T. Calarco, and Franco Nori

Physical Review A, 2010

We apply quantum control techniques to control a large spin chain by only acting on two qubits at... more We apply quantum control techniques to control a large spin chain by only acting on two qubits at one of its ends, thereby implementing universal quantum computation by a combination of quantum gates on the latter and swap operations across the chain. It is shown that the control sequences can be computed and implemented efficiently. We discuss the application of these ideas to physical systems such as superconducting qubits in which full control of long chains is challenging. 02.30.Yy Controlling quantum systems at will has been an aspiration for physicists for a long time. Achieving quantum control not only clears the path towards a thorough understanding of quantum mechanics, but it also allows the exploration of novel devices whose functions are based on exotic quantum mechanical effects. Among others, the future success of quantum information processing depends largely on our ability to tame many-body quantum systems that are highly fragile. Although the progress of technology allows us to manipulate a small number of quanta quite well, controlling larger systems still represents a considerable challenge. Unless we overcome difficulties towards the control over large many-body systems, the benefits we can enjoy with the 'quantumness' will be severely limited.

Extraordinary momentum and spin in evanescent waves

by Konstantin Bliokh and Franco Nori

Momentum and spin represent fundamental dynamic properties of quantum particles and fields. In pa... more Momentum and spin represent fundamental dynamic properties of quantum particles and fields. In particular, propagating optical waves (photons) carry momentum and longitudinal spin determined by the wave vector and circular polarization, respectively. Here we show that exactly the opposite can be the case for evanescent optical waves. A single evanescent wave possesses a spin component, which is independent of the polarization and is orthogonal to the wave vector. Furthermore, such a wave carries a momentum component, which is determined by the circular polarization and is also orthogonal to the wave vector. We show that these extraordinary properties reveal a fundamental Belinfante&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#39;s spin momentum, known in field theory and unobservable in propagating fields. We demonstrate that the transverse momentum and spin push and twist a probe Mie particle in an evanescent field. This allows the observation of &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#39;impossible&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#39; properties of light and of a fundamental field-theory quantity, which was previously considered as &amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#39;virtual&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#39;.

Relativistic Electron Vortex Beams: Angular Momentum and Spin-Orbit Interaction

by Konstantin Bliokh and Franco Nori

Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM)... more Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM), we construct exact Bessel-beam solutions of the Dirac equation. They describe relativistic and nonparaxial corrections to the scalar electron beams. We describe the spin and orbital AM of the electron with Berry-phase corrections and predict the intrinsic spin-orbit coupling in free space. This can be observed as a spin-dependent probability distribution of the focused electron vortex beams. Moreover, the magnetic moment is calculated, which shows different g-factors for spin and orbital AM and also contains the Berry-phase correction.

Characterizing optical chirality

by Konstantin Bliokh and Franco Nori

We examine the recently introduced measure of chirality of a monochromatic optical field [Y. Tang... more We examine the recently introduced measure of chirality of a monochromatic optical field [Y. Tang and A. E. Cohen, Phys. Rev. Lett. 104, 163901 (2010)] using the momentum (plane-wave) representation and helicity basis. Our analysis clarifies the physical meaning of the measure of chirality and unveils its close relation to the polarization helicity, spin angular momentum, energy density, and Poynting energy flow. We derive the operators of the optical chirality and of the corresponding chiral momentum, which acquire remarkably simple forms in the helicity representation.

Magnetoelectric Effects in Local Light-Matter Interactions

by Konstantin Bliokh and Franco Nori

Physical Review Letters, 2014

We study the generic interaction of a monochromatic electromagnetic field with bi-isotropic nanop... more We study the generic interaction of a monochromatic electromagnetic field with bi-isotropic nanoparticles. Such an interaction is described by dipole-coupling terms associated with the breaking of dual, P-and T-symmetries, including the chirality and the nonreciprocal magnetoelectric effect. We calculate absorption rates, radiation forces, and radiation torques for the nanoparticles and introduce novel characteristics of the field quantifying the transfer of energy, momentum, and angular-momentum in these interactions. In particular, we put forward the concept of 'magnetoelectric energy density', quantifying the local PT-symmetry of the field. Akin to the 'super-chiral' light suggested recently for sensitive local probing of molecular chirality [Phys. Rev. Lett. 104, 163901 (2010); Science 332, 333 (2011)], here we describe a complex field for sensitive probing of the nonreciprocal magnetoelectric effect in nanoparticles or molecules.

Transverse spin of a surface polariton

by Konstantin Bliokh and Franco Nori

ABSTRACT

Dual electromagnetism: Helicity, spin, momentum, and angular momentum

by Konstantin Bliokh and Franco Nori

The dual symmetry between electric and magnetic fields is an important intrinsic property of Maxw... more The dual symmetry between electric and magnetic fields is an important intrinsic property of Maxwell equations in free space. This symmetry underlies the conservation of optical helicity and, as we show here, is closely related to the separation of spin and orbital degrees of freedom of light (the helicity flux coincides with the spin angular momentum). However, in the standard field-theory formulation of electromagnetism, the field Lagrangian is not dual symmetric. This leads to problematic dual-asymmetric forms of the canonical energy-momentum, spin and orbital angular-momentum tensors. Moreover, we show that the components of these tensors conflict with the helicity and energy conservation laws. To resolve this discrepancy between the symmetries of the Lagrangian and Maxwell equations, we put forward a dualsymmetric Lagrangian formulation of classical electromagnetism. This dual electromagnetism preserves the form of Maxwell equations, yields meaningful canonical energy-momentum and angular-momentum tensors, and ensures a self-consistent separation of the spin and orbital degrees of freedom. This provides a rigorous derivation of the results suggested in other recent approaches.

Colloquium: Unusual resonators: Plasmonics, metamaterials, and random media

by Konstantin Bliokh, Franco Nori, and Sergey Savel'ev

Super-resolution, extraordinary transmission, total absorption, and localization of electromagnet... more Super-resolution, extraordinary transmission, total absorption, and localization of electromagnetic waves are currently attracting growing attention. These phenomena are related to different physical systems and are usually studied within the context of different, sometimes rather sophisticated, approaches. Remarkably, all these seemingly unrelated phenomena owe their origin to the same underlying physical mechanism, namely, wave interaction with an open resonator. Here we show that it is possible to describe all of these effects in a unified way, mapping each system onto a simple resonator model. Such description provides a thorough understanding of the phenomena, explains all the main features of their complex behavior, and enables one to control the system via the resonator parameters: eigenfrequencies, Q factors, and coupling coefficients.

Semiclassical dynamics of electron wave packet states with phase vortices

by Konstantin Bliokh and Franco Nori

Physical review letters, 2007

We consider semiclassical higher-order wave packet solutions of the Schrödinger equation with pha... more We consider semiclassical higher-order wave packet solutions of the Schrödinger equation with phase vortices. The vortex line is aligned with the propagation direction, and the wave packet carries a welldefined orbital angular momentum (OAM) @l (l is the vortex strength) along its main linear momentum. The probability current coils around the momentum in such OAM states of electrons. In an electric field, these states evolve like massless particles with spin l. The magnetic-monopole Berry curvature appears in momentum space, which results in a spin-orbit-type interaction and a Berry/Magnus transverse force acting on the wave packet. This brings about the OAM Hall effect. In a magnetic field, there is a Zeeman interaction, which, can lead to more complicated dynamics.

Anderson Localization of Light in Layered Dielectric Structures

by Konstantin Bliokh and Franco Nori

Electron vortex beams in a magnetic field: A new twist on Landau levels and Aharonov-Bohm states

by Konstantin Bliokh and Franco Nori

We examine the propagation of the recently discovered electron vortex beams in a longitudinal mag... more We examine the propagation of the recently discovered electron vortex beams in a longitudinal magnetic field. We consider both the Aharonov-Bohm configuration with a single flux line and the Landau case of a uniform magnetic field. While stationary Aharonov-Bohm modes represent Bessel beams with flux-and vortex-dependent probability distributions, stationary Landau states manifest themselves as nondiffracting Laguerre-Gaussian beams. Furthermore, the Landau-state beams possess field-and vortex-dependent phases: (i) the Zeeman phase from coupling the quantized angular momentum to the magnetic field and (ii) the Gouy phase, known from optical Laguerre-Gaussian beams. Remarkably, together these phases determine the structure of Landau energy levels. This unified Zeeman-Landau-Gouy phase manifests itself in a nontrivial evolution of images formed by various superpositions of modes. We demonstrate that, depending on the chosen superposition, the image can rotate in a magnetic field with either (i) Larmor, (ii) cyclotron (double-Larmor), or (iii) zero frequency. At the same time, its centroid always follows the classical cyclotron trajectory, in agreement with the Ehrenfest theorem. Interestingly, the nonrotating superpositions reproduce stable multivortex configurations that appear in rotating superfluids. Our results open an avenue for the direct electron-microscopy observation of fundamental properties of free quantum-electron states in magnetic fields.

Does a Surface Polariton Have Spin?

by Konstantin Bliokh and Franco Nori

Relativistic electron vortex beams: angular momentum and spin-orbit interaction

by Konstantin Bliokh and Franco Nori

Physical review letters, Jan 21, 2011

Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM)... more Motivated by the recent discovery of electron vortex beams carrying orbital angular momentum (AM), we construct exact Bessel-beam solutions of the Dirac equation. They describe relativistic and nonparaxial corrections to the scalar electron beams. We describe the spin and orbital AM of the electron with Berry-phase corrections and predict the intrinsic spin-orbit coupling in free space. This can be observed as a spin-dependent probability distribution of the focused electron vortex beams. Moreover, the magnetic moment is calculated, which shows different g factors for spin and orbital AM and also contains the Berry-phase correction.

Classical Field Approach to Quantum Weak Measurements

by Konstantin Bliokh and Franco Nori

By generalizing the quantum weak measurement protocol to the case of quantum fields, we show that... more By generalizing the quantum weak measurement protocol to the case of quantum fields, we show that weak measurements probe an effective classical background field that describes the average field configuration in the spacetime region between pre-and postselection boundary conditions. The classical field is itself a weak value of the corresponding quantum field operator and satisfies equations of motion that extremize an effective action. Weak measurements perturb this effective action, producing measurable changes to the classical field dynamics. As such, weakly measured effects always correspond to an effective classical field. This general result explains why these effects appear to be robust for pre-and postselected ensembles, and why they can also be measured using classical field techniques that are not weak for individual excitations of the field.

Observation of the Larmor and Gouy Rotations with Electron Vortex Beams

by Konstantin Bliokh and Franco Nori

Physical Review Letters, Feb 25, 2013

Electron vortex beams carrying intrinsic orbital angular momentum (OAM) are produced in electron ... more Electron vortex beams carrying intrinsic orbital angular momentum (OAM) are produced in electron microscopes where they are controlled and focused by using magnetic lenses. We observe various rotational phenomena arising from the interaction between the OAM and magnetic lenses. First, the Zeeman coupling, proportional to the OAM and magnetic field strength, produces an OAM-independent Larmor rotation of a mode superposition inside the lens. Second, when passing through the focal plane, the electron beam acquires an ...

Photon trajectories, anomalous velocities and weak measurements: a classical interpretation

by Konstantin Bliokh, Franco Nori, and A. Kofman

reported the observation of 'average trajectories of single photons' in a two-slit interference e... more reported the observation of 'average trajectories of single photons' in a two-slit interference experiment. This was possible by using the quantum weak-measurement method, which implies averaging over many events, i.e. in fact, a multi-photon limit of classical linear optics. We give a classical-optics interpretation of this experiment and other related problems. It appears that weak measurements of the local momentum of photons made by Kocsis et al represent measurements of the Poynting vector in an optical field. We consider both the real and imaginary parts of the local momentum and show that their measurements have been realized in classical optics using small-probe particles. We also examine the appearance of 'anomalous' values of the local momentum: either negative (backflow) or exceeding the wavenumber (superluminal propagation). These features appear to be closely related to vortices and evanescent waves. Finally, we revisit a number of older works and find examples of photon trajectories and anomalousmomentum measurements in various optical experiments. Gesellschaft particle [2]. This is because a strong measurement of one quantity destroys the information about its complementary quantity. However, quantum weak measurements, introduced by Aharonov et al, imply a weak coupling between the measuring and the measured sub-systems and do not significantly perturb the measured part . This approach allows the simultaneous determination of complementary quantities, albeit averaged over many events. Furthermore, the weak values of the measured quantity can be 'anomalous', i.e. beyond the range of the spectrum of the corresponding Hermitian operator [3] and even complex . Weak values are closely related to complex conditional probabilities for the system to be found in a certain state under the condition that it is subsequently found in another state . Soon after the introduction of the theoretical concept of weak measurements, this method was applied in observing tiny polarization-dependent wave shifts in classical optics . In these experiments, two weakly coupled degrees of freedom of light (the polarization and spatial distribution) played the roles of the measured and measuring ('meter') sub-systems. For instance, the spin-Hall effect of light (a spin-dependent shift of optical beams) [9, 10] was treated as the shift of the 'meter' which 'measured' the photon spin (helicity) . As a result, the shift was significantly amplified owing to the anomalous weak value of the helicity, which was complex and far outside its normal [−1, 1] range (h = 1 units are used throughout the paper). Despite the quantum weak-measurement formalism, experiments clearly dealt with classical optical systems and all results, including the spin-Hall effect of photons, allowed fully classical descriptions . This is because weak measurements require averaging over many one-particle events, so that in linear photonic systems this approach implies the classical-optics limit. Kocsis et al emphasized that their experiment was realized in the quantum one-photon regime. Still, the measurements represented an average over many events, and the same results would appear in the continuous-wave regime [2, 12] 6 . Nonetheless, the results of [1] have not 6 According to Dirac: 'Each photon interferes only with itself. Interference between two different photons never occurs'. Therefore, the interference pictures obtained from averaging many one-photon events and from the multiphoton classical field coincide with each other. For experimental demonstrations, see .

Relativistic Hall Effect

by Konstantin Bliokh and Franco Nori

We consider the relativistic deformation of quantum waves and mechanical bodies carrying intrinsi... more We consider the relativistic deformation of quantum waves and mechanical bodies carrying intrinsic angular momentum (AM). When observed in a moving reference frame, the centroid of the object undergoes an AM-dependent transverse shift. This is the relativistic analogue of the spin-Hall effect, which occurs in free space without any external fields. Remarkably, the shifts of the geometric and energy centroids differ by a factor of 2, and both centroids are crucial for the Lorentz transformations of the AM tensor. We examine manifestations of the relativistic Hall effect in quantum vortices, and mechanical flywheels, and also discuss various fundamental aspects of this phenomenon. The perfect agreement of quantum and relativistic approaches allows applications at strikingly different scales: from elementary spinning particles, through classical light, to rotating black-holes. PACS numbers: 42.50.Tx, 03.65.Pm, 03.30.+p

Spatiotemporal vortex beams and angular momentum

by Konstantin Bliokh and Franco Nori

We present a space-time generalization of the known spatial (monochromatic) wave vortex beams car... more We present a space-time generalization of the known spatial (monochromatic) wave vortex beams carrying intrinsic orbital angular momentum (OAM) along the propagation direction. Generic spatio-temporal vortex beams are polychromatic and can carry intrinsic OAM at an arbitrary angle to the mean momentum. Applying either (i) a transverse wave-vector shift or (ii) a Lorentz boost to a monochromatic Bessel beam, we construct a family of either (i) time-diffracting or (ii) non-diffracting spatio-temporal Bessel beams, which are exact solutions of the Klein-Gordon wave equations. The proposed spatio-temporal OAM states are able to describe either photon or electron vortex states (both relativistic and nonrelativistic), and can find applications in particle collisions, optics of moving media, quantum communications, and astrophysics.

Transverse relativistic effects in paraxial wave interference

by Konstantin Bliokh, Franco Nori, and Y. Izdebskaya

We consider relativistic deformations of interfering paraxial waves moving in the transverse dire... more We consider relativistic deformations of interfering paraxial waves moving in the transverse direction. Owing to superluminal transverse phase velocities, noticeable deformations of the interference patterns arise when the waves move with respect to each other with non-relativistic velocities. Similar distortions also appear on a mutual tilt of the interfering waves, which causes a phase delay analogous to the relativistic time delay. We illustrate these observations by the interference between a vortex wave beam and a plane wave, which exhibits a pronounced deformation of the radial fringes into a fork-like pattern (relativistic Hall effect). Furthermore, we describe an additional relativistic motion of the interference fringes (a counter-rotation in the vortex case), which becomes noticeable at the same non-relativistic velocities.

Transverse spin and momentum in two-wave interference

by Konstantin Bliokh and Franco Nori

It is well-known, since the seminal works by J.H. Poynting , that light carries momentum and angu... more It is well-known, since the seminal works by J.H. Poynting , that light carries momentum and angular momentum (AM) [2,3]. Typical plane-wave or Gaussian-beam states exhibit longitudinal momentum associated with the wave vector k and also longitudinal spin AM associated with the degree of circular polarization (helicity) σ . Locally, optical momentum and angular-momentum densities can demonstrate unusual features which have recently attracted considerable attention: "super-momentum" with values higher than k per photon [4-8], transverse helicity-independent spin AM [9-12], and transverse helicity-dependent momentum [10,13,14]. So far, such abnormal momentum and spin properties have appeared only in special field configurations, namely, evanescent waves and optical vortices. Here we find that the simplest propagating nonsingular field -two interfering plane waves -also exhibits a variety of extraordinary spin and momentum properties. Despite the seemingly planar and thoroughly-studied character of the two-wave system, we discover that such field possesses a transverse (out-of-plane) helicity-independent spin AM, and also a transverse polarization-dependent momentum with unusual physical properties.