Pier A. Mello and Narendra Kumar
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials
This book presents a statistical theory of complex wave scattering and quantum transport in a class of physical systems of current interest having chaotic classical dynamics (e.g., microwave cavities ...
More
This book presents a statistical theory of complex wave scattering and quantum transport in a class of physical systems of current interest having chaotic classical dynamics (e.g., microwave cavities and quantum dots) or possessing quenched randomness (e.g., disordered conductors). The emphasis here is on mesoscopic fluctuations of the sample-specific transport. The universal character of the statistical behaviour of these phenomena is revealed in a natural way through a novel maximum-entropy approach (MEA). The latter leads to the most probable distribution for the set of random matrices that describe the ensemble of disordered/chaotic samples, which are macroscopically identical but differ in microscopic details. Here, the Shannon information entropy associated with these random matrices is maximized subject to the symmetries and the constraints which are physically relevant. This non-perturbative information-theoretic approach is reminiscent of, but distinct from, the standard random-matrix theory, and indeed forms the most distinctive feature of the book.Less
This book presents a statistical theory of complex wave scattering and quantum transport in a class of physical systems of current interest having chaotic classical dynamics (e.g., microwave cavities and quantum dots) or possessing quenched randomness (e.g., disordered conductors). The emphasis here is on mesoscopic fluctuations of the sample-specific transport. The universal character of the statistical behaviour of these phenomena is revealed in a natural way through a novel maximum-entropy approach (MEA). The latter leads to the most probable distribution for the set of random matrices that describe the ensemble of disordered/chaotic samples, which are macroscopically identical but differ in microscopic details. Here, the Shannon information entropy associated with these random matrices is maximized subject to the symmetries and the constraints which are physically relevant. This non-perturbative information-theoretic approach is reminiscent of, but distinct from, the standard random-matrix theory, and indeed forms the most distinctive feature of the book.
S. R Cloude
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199569731
- eISBN:
- 9780191721908
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199569731.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Geophysics, Atmospheric and Environmental Physics
This chapter considers a basic description of the generation, propagation and scattering of polarised electromagnetic waves. It assumes a starting familiarity with the basic form of Maxwell's ...
More
This chapter considers a basic description of the generation, propagation and scattering of polarised electromagnetic waves. It assumes a starting familiarity with the basic form of Maxwell's equations, and then uses them together with formal matrix methods to develop several key ideas, including the importance of special unitary matrices, the concept of matrix decomposition via the use of the Pauli spin matrices in classical wave problems, and a basic definition of the scattering amplitude matrix. It also includes coverage of the important geometrical concepts of polarimetry such as the Jones vector and related propagation calculus, Stokes parameters, and the Poincaré sphere, as well as considering coordinate issues that arise when using microwave antenna coordinates in radar studies. It concludes with an introduction to the important concept of a scattering vector that forms the basis for studies in later chapters.Less
This chapter considers a basic description of the generation, propagation and scattering of polarised electromagnetic waves. It assumes a starting familiarity with the basic form of Maxwell's equations, and then uses them together with formal matrix methods to develop several key ideas, including the importance of special unitary matrices, the concept of matrix decomposition via the use of the Pauli spin matrices in classical wave problems, and a basic definition of the scattering amplitude matrix. It also includes coverage of the important geometrical concepts of polarimetry such as the Jones vector and related propagation calculus, Stokes parameters, and the Poincaré sphere, as well as considering coordinate issues that arise when using microwave antenna coordinates in radar studies. It concludes with an introduction to the important concept of a scattering vector that forms the basis for studies in later chapters.
Pier A. Mello and Narendra Kumar
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter is devoted to basic potential scattering theory, focusing on the case of a one-dimensional conductor and an open cavity with a one-channel lead connected to it. The contents of this ...
More
This chapter is devoted to basic potential scattering theory, focusing on the case of a one-dimensional conductor and an open cavity with a one-channel lead connected to it. The contents of this chapter include potential scattering in infinite one-dimensional space; Lippmann-Schwinger equation; free Green function; reflection and transmission amplitudes; transfer matrix; T matrix; S matrix and its analytic structure; phase shifts and resonances from the analytic structure of S matrix in complex momentum and complex energy planes; parametrization of the matrices; combination of the S matrices for two scatterers in series; and invariant-imbedding approach for a one-dimensional disordered conductor.Less
This chapter is devoted to basic potential scattering theory, focusing on the case of a one-dimensional conductor and an open cavity with a one-channel lead connected to it. The contents of this chapter include potential scattering in infinite one-dimensional space; Lippmann-Schwinger equation; free Green function; reflection and transmission amplitudes; transfer matrix; T matrix; S matrix and its analytic structure; phase shifts and resonances from the analytic structure of S matrix in complex momentum and complex energy planes; parametrization of the matrices; combination of the S matrices for two scatterers in series; and invariant-imbedding approach for a one-dimensional disordered conductor.
Pier A. Mello and Narendra Kumar
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter extends the potential scattering theory developed in Chapter 2 to a relatively advanced level with emphasis on quasi-one-dimensional (multi-channel) systems, the associated scattering ...
More
This chapter extends the potential scattering theory developed in Chapter 2 to a relatively advanced level with emphasis on quasi-one-dimensional (multi-channel) systems, the associated scattering and transfer matrices, and on how to combine them serially. Both the closed and the open channels are discussed. Scattering by a cavity with an arbitrary number of waveguides (the leads) attached to it is introduced. The Wigner R-matrix theory of two-dimensional scattering is treated in some detail with attention to boundary conditions. A non-trivial exactly soluble example for the two-channel scattering problem is also presented.Less
This chapter extends the potential scattering theory developed in Chapter 2 to a relatively advanced level with emphasis on quasi-one-dimensional (multi-channel) systems, the associated scattering and transfer matrices, and on how to combine them serially. Both the closed and the open channels are discussed. Scattering by a cavity with an arbitrary number of waveguides (the leads) attached to it is introduced. The Wigner R-matrix theory of two-dimensional scattering is treated in some detail with attention to boundary conditions. A non-trivial exactly soluble example for the two-channel scattering problem is also presented.
J. C. Garrison and R. Y. Chiao
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198508861
- eISBN:
- 9780191708640
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508861.003.0009
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This chapter shows that the interaction of photons with a passive, linear device can be described by the scattering matrix of classical optics. Combining this with the paraxial approximation leads to ...
More
This chapter shows that the interaction of photons with a passive, linear device can be described by the scattering matrix of classical optics. Combining this with the paraxial approximation leads to a quantum description of lenses, mirrors, beam splitters, optical isolators, Y-junctions, optical circulators, and stops. In this way, each device is described by means of scattering channels together with input and output ports. Studying quantum noise in the transmitted and reflected signals from a beam splitter or a stop leads to the idea of partition noise, which is ascribed to vacuum fluctuations entering through a classically unused port. This effect is avoided in an optical circulator by arranging for destructive interference of vacuum fluctuation waves traveling in opposite senses of circulation around a ferrite pill containing a static magnetic field.Less
This chapter shows that the interaction of photons with a passive, linear device can be described by the scattering matrix of classical optics. Combining this with the paraxial approximation leads to a quantum description of lenses, mirrors, beam splitters, optical isolators, Y-junctions, optical circulators, and stops. In this way, each device is described by means of scattering channels together with input and output ports. Studying quantum noise in the transmitted and reflected signals from a beam splitter or a stop leads to the idea of partition noise, which is ascribed to vacuum fluctuations entering through a classically unused port. This effect is avoided in an optical circulator by arranging for destructive interference of vacuum fluctuation waves traveling in opposite senses of circulation around a ferrite pill containing a static magnetic field.
Pier A. Mello and Narendra Kumar
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198525820
- eISBN:
- 9780191712234
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525820.003.0006
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter focuses on scattering from classically chaotic cavities. A maximum-entropy approach is employed to derive the probability distribution for the S matrix, with and without the presence of ...
More
This chapter focuses on scattering from classically chaotic cavities. A maximum-entropy approach is employed to derive the probability distribution for the S matrix, with and without the presence of direct processes. The latter (with direct processes) is included as a constraint through the imposition of a given average S matrix, in addition to its analytic properties. Theoretical calculations are compared with the results of computer simulations and experimental data. The contents of this chapter include statistical ensembles of S matrices; invariant measure; distribution of the conductance in the two-equal-lead case; numerical calculations and comparison with theory; dephasing effects; and comparison with experimental data and physical experiments.Less
This chapter focuses on scattering from classically chaotic cavities. A maximum-entropy approach is employed to derive the probability distribution for the S matrix, with and without the presence of direct processes. The latter (with direct processes) is included as a constraint through the imposition of a given average S matrix, in addition to its analytic properties. Theoretical calculations are compared with the results of computer simulations and experimental data. The contents of this chapter include statistical ensembles of S matrices; invariant measure; distribution of the conductance in the two-equal-lead case; numerical calculations and comparison with theory; dephasing effects; and comparison with experimental data and physical experiments.
Jean Zinn-Justin
- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0009
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
This chapter shows how scattering problems are formulated in the framework of path integrals. In quantum mechanics, the state of an isolated system evolves under the action of a unitary operator, as ...
More
This chapter shows how scattering problems are formulated in the framework of path integrals. In quantum mechanics, the state of an isolated system evolves under the action of a unitary operator, as a consequence of the conservation of probabilities and, thus, of the norm of vectors in Hilbert space. Quantum evolution (that is, in real time) is introduced, after which a path integral representation of the scattering matrix is constructed. From this S matrix, the standard perturbative expansion in powers of the potential is recovered. Even the evolution of a free quantum particle is slightly non-trivial; in general, one observes a spreading of wave packets. Scattering is then characterized by the asymptotic deviations at infinite time from this free evolution and this leads to the definition of a scattering or S-matrix. An S-matrix is defined in the example of bosons and fermions. Various other semi-classical approximation schemes are then discussed.Less
This chapter shows how scattering problems are formulated in the framework of path integrals. In quantum mechanics, the state of an isolated system evolves under the action of a unitary operator, as a consequence of the conservation of probabilities and, thus, of the norm of vectors in Hilbert space. Quantum evolution (that is, in real time) is introduced, after which a path integral representation of the scattering matrix is constructed. From this S matrix, the standard perturbative expansion in powers of the potential is recovered. Even the evolution of a free quantum particle is slightly non-trivial; in general, one observes a spreading of wave packets. Scattering is then characterized by the asymptotic deviations at infinite time from this free evolution and this leads to the definition of a scattering or S-matrix. An S-matrix is defined in the example of bosons and fermions. Various other semi-classical approximation schemes are then discussed.
JOHN C. H. SPENCE
- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199552757
- eISBN:
- 9780191708664
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199552757.003.0005
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This chapter introduces many-beam high-resolution imaging by first analysing in detail the case of two or three beams, and showing how aberrations may be arranged to cancel by using off-axis ...
More
This chapter introduces many-beam high-resolution imaging by first analysing in detail the case of two or three beams, and showing how aberrations may be arranged to cancel by using off-axis illumination (as later related to Ptychography). The phenomenon of Fourier or Talbot imaging is reviewed, and the images are shown to be periodic in some aberration coefficients. The manner in which a finite incident beam divergence limits depth of field is explained. The main results of kinematic, two-beam, and the charge-density approximations are given. The full multiple scattering theory is then given, symmetry reduction of the dispersion matrix reviewed, and discussions are provided of partial coherence with multiple scattering, absorption effects due to inelastic scattering with multiple scattering, dynamically forbidden reflections, and the relationship between the various formulations (and sign conventions) of high energy electron scattering theory. A case study of imaging in germanium is given, showing the masking effect of Fourier imaging on defects. The chapter ends with case studies from materials science — imaging dislocation kinks, complex oxides, minerals, quasicrystals, interfaces.Less
This chapter introduces many-beam high-resolution imaging by first analysing in detail the case of two or three beams, and showing how aberrations may be arranged to cancel by using off-axis illumination (as later related to Ptychography). The phenomenon of Fourier or Talbot imaging is reviewed, and the images are shown to be periodic in some aberration coefficients. The manner in which a finite incident beam divergence limits depth of field is explained. The main results of kinematic, two-beam, and the charge-density approximations are given. The full multiple scattering theory is then given, symmetry reduction of the dispersion matrix reviewed, and discussions are provided of partial coherence with multiple scattering, absorption effects due to inelastic scattering with multiple scattering, dynamically forbidden reflections, and the relationship between the various formulations (and sign conventions) of high energy electron scattering theory. A case study of imaging in germanium is given, showing the masking effect of Fourier imaging on defects. The chapter ends with case studies from materials science — imaging dislocation kinks, complex oxides, minerals, quasicrystals, interfaces.
Klaus Morawetz
- Published in print:
- 2017
- Published Online:
- February 2018
- ISBN:
- 9780198797241
- eISBN:
- 9780191838743
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198797241.003.0012
- Subject:
- Physics, Condensed Matter Physics / Materials
The Bose–Einstein condensation and appearance of superfluidity and superconductivity are introduced from basic phenomena. A systematic theory based on the asymmetric expansion of chapter 11 is shown ...
More
The Bose–Einstein condensation and appearance of superfluidity and superconductivity are introduced from basic phenomena. A systematic theory based on the asymmetric expansion of chapter 11 is shown to correct the T-matrix from unphysical multiple-scattering events. The resulting generalised Soven scheme provides the Beliaev equations for Boson’s and the Nambu–Gorkov equations for fermions without the usage of anomalous and non-conserving propagators. This systematic theory allows calculating the fluctuations above and below the critical parameters. Gap equations and Bogoliubov–DeGennes equations are derived from this theory. Interacting Bose systems with finite temperatures are discussed with successively better approximations ranging from Bogoliubov and Popov up to corrected T-matrices. For superconductivity, the asymmetric theory leading to the corrected T-matrix allows for establishing the stability of the condensate and decides correctly about the pair-breaking mechanisms in contrast to conventional approaches. The relation between the correlated density from nonlocal kinetic theory and the density of Cooper pairs is shown.Less
The Bose–Einstein condensation and appearance of superfluidity and superconductivity are introduced from basic phenomena. A systematic theory based on the asymmetric expansion of chapter 11 is shown to correct the T-matrix from unphysical multiple-scattering events. The resulting generalised Soven scheme provides the Beliaev equations for Boson’s and the Nambu–Gorkov equations for fermions without the usage of anomalous and non-conserving propagators. This systematic theory allows calculating the fluctuations above and below the critical parameters. Gap equations and Bogoliubov–DeGennes equations are derived from this theory. Interacting Bose systems with finite temperatures are discussed with successively better approximations ranging from Bogoliubov and Popov up to corrected T-matrices. For superconductivity, the asymmetric theory leading to the corrected T-matrix allows for establishing the stability of the condensate and decides correctly about the pair-breaking mechanisms in contrast to conventional approaches. The relation between the correlated density from nonlocal kinetic theory and the density of Cooper pairs is shown.
Laurent Baulieu, John Iliopoulos, and Roland Sénéor
- Published in print:
- 2017
- Published Online:
- May 2017
- ISBN:
- 9780198788393
- eISBN:
- 9780191830310
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198788393.003.0009
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
Time-ordered products and connection with the path integral. Applications in quantum mechanical problems. T-products and the scattering matrix. The perturbation expansion. The scalar field.
Time-ordered products and connection with the path integral. Applications in quantum mechanical problems. T-products and the scattering matrix. The perturbation expansion. The scalar field.
