Photons in Fock Space and Beyond

The following sections are included:

• Preface and Overview II
• Contents

It took quite a while that the so-called “squeezed states” of light had acquired a broad interest in Quantum Optics. Now they are still in the focus of experimental and technical developments since they increase the accuracy of certain measurement devices and improve the transmission rate and efficiency in optical communication networks. From the foundational viewpoint, they are interesting as a manifestation of the non-classicality of light. For the theoretical implementation of squeezing procedures one commonly uses Hamiltonians which are quadratic in the annihilation and creation operators…

In order to illustrate the difficulties in realizing mathematically infinite-mode squeezing transformations, we stick in the present chapter to a fixed orthonormal basis in the test function space, which diagonalizes the relevant one-particle operators…

The theory of squeezing evolved from rather simple Bogoliubov transformations, concerning the test function space E = ℂe₀ spanned by a single selected transversal photon mode e₀ ∈ ℋ^Τ. We denote the annihilation and creation operators by b = a_F (e₀) and in the Fock space F₊(ℂ). There, the transformed annihilation operator is of the form

Physically we consider the same experimental setup for a black body radiation as in the classical case, described in Chapter 15. In fact, the measurement devices are quite generally in Quantum Theory the same as in classical theory, sometimes refined. But, reversely to the historical process during which the quantum effects in the black body frequency distribution were inductively detected, we start in the present chapter from the full developed quantum mechanical formalism, using the algebraic theory of quantized Boson fields…

The following sections are included:

• On the Fluctuations in Thermal Radiation
• Thermal Quantum Field Theory for Black Body Radiation
  • Thermal GNS Representation for a Local Domain Λ
  • Thermal GNS Representation for Arbitrary Domain Λ
  • Thermal Dynamics and Hamiltonian
• Ergodic and Filter Properties of Photon Fields
  • Ergodic Properties of the Thermal and Vacuum Photons
  • Projected Photon Fields

A mesoscopic many-body system is naturally described in terms of a (non-relativistic) quantum field theory. Especially the quantum field of the electrons, which arises from the one-electron theory via the procedure of “second quantization”, is basic for formulating the interaction with light. The corresponding field algebra is a special CAR algebra, generated by the field polynomials, for which the algebraic relations are based on the canonical anti-commutation relations. For mesoscopic many-body physics it is important not to anticipate the Hilbert space representation of the CAR algebra, especially, not to confine a Fermionic model to the Fock space. The particle densities for mesoscopic systems do not decay for “large” volumina, and so a finite particle density prevails in the limit of increasing volumina (thermodynamic limit). Already this situation requires a non-Fock representation of the CAR algebra, and if macroscopic parameters as the particle density or temperature are varied, one needs various, mutually inequivalent, representations…

In contrast to usual relativistic QED, where one deals with a few real (and many virtual) electrons, we are interested here mostly in mesoscopic systems with so many radiating electrons, that collective phenomena arise. Our main examples are the outer shell electrons of an atomic gas, the conduction electrons of superconductors and SQUID's, and the itinerant electrons and electron holes of semiconductors. In all of these cases it is essential to form the thermodynamic limit as the theoretical tool to include collective variables into the microscopic quantum mechanical description…

We develop in this Chapter a C*-algebraic quantum theory which extends the microscopic, abstract Fermionic theory (𝒜(𝔥), 𝒮), 𝒮 denoting all states on 𝒜(𝔥), by certain classical observables and by the pertinent (microscopically extended) states. The additional central observables are abstracted from representation dependent extension procedures. Our “abstract” extensions are based on so-called f-weak converging nets of local observables. The described extension procedure is considered paradigmatic for deriving the mesoscopic stage from the microscopic theory, without loosing the latter…

The following sections are included:

• Electron Formulation
  • Multi-band Hilbert Space and CAR Algebra
  • Symmetries and Dynamics
  • Quasi-free and Finite Temperature States
  • Generators in Thermal Representations
  • Potentials in the Low-Temperature Limit
  • Ground State Discussion
• Particle–Hole Formulation
  • Particle–Hole CAR Algebra
  • Transformations in Particle–Hole Language
  • Particle-Hole States

The following sections are included:

• General Weakly Inhomogeneous Cluster Interactions
  • Local Interactions
  • Homogeneous Limiting Dynamics
  • Collective Poisson Manifold
  • Inhomogeneous Limiting Dynamics
  • Remarks on Equilibrium Representations
• Couplings to External Currents
  • Current Coupled States
  • States of a Semiconductor Resonator
  • J-Coupled States in Quasi-Spin Formulation

The following sections are included:

• Semiconductor–Photon Couplings
  • Coupling Expressions for a Semiconductor
  • The Coupling Function in the Weak Coupling Limit
  • First Steps to the Dynamics of the Coupled System
    • Free Photons Dynamics for a Semiconductor Coupling
    • Material Dynamics for the Semiconductor Electrons
    • Hamiltonians and Unitaries of the Combined System
  • Weakly Inhomogeneous BCS Models
    • Inhomogeneous BCS Model with Total Electron Algebra
    • BCS Limiting Dynamics in Equilibrium Representations
  • Josephson Junction and SQUID
    • The Josephson Junction
    • Macroscopic Quantum Phenomena with the SQUID
  • Fields Coupled to the Josephson Junction
    • Topology of the Gauged Field Domain
    • The Total Quantized Electromagnetic Field
    • Interaction Between the Junction and the Field
    • Dynamics of the Cohomological and Transversal Quantized Field Parts
    • Cohomological Mesoscopic Quantum Currents
    • Formal Coupling to Classical Vector Potentials
    • Gauge Bundle and Magnetic Flux Quantization
    • Wave Functions from an Associated Line Bundle
    • Current Related to an Associated Line Bundle
    • Microscopic and Macroscopic Quantum Phases

We have described in Sec. 21.6, for arbitrary Coulomb clusters, and in Chapter 37, for semiconductor electron-hole pairs and Cooper pairs, how these electronic configurations interact with the quantized electromagnetic field. In the present chapter, we study — at first for arbitrary finite-electron clusters — the resulting dynamics in terms of a perturbation expansion (which we anticipated for the semiconductor in a sketchy manner already in Sec. 37.1). We emphasize here the algebraic formulation of the Heisenberg dynamics, which we approach in the most simple local implementations. (The details of general unitary implementations are deferred to Chapter 51.) We demonstrate that the perturbation theory in terms of Dyson expansions exists in all orders and in the summed up version…

We start the discussion from a single neutral cluster in free space possessing M electrons. We select two eigenfunctions of its Hamiltonian with level splitting energy ε > 0 between the upper eigenlevel ψ_↑ and the lower eigenlevel ψ_↓. Neglecting the radiation coupling of ψ_↑ and ψ_↓ with the other eigenlevels of the cluster and with the continuous spectral part, the system restricts to our “two-level atom”. As an essential point, the transversal transition function occurring in the creation and annihilation operators of the p · A-interaction is calculated in terms of the two M-electron eigenfunctions…

We elaborate a model for a semiconductor in interaction with photons, as we have it prepared in Sec. 36.2. We restrict the electromagnetic field to the transversal part and neglect, on the material side, the spin of the electrons and holes, as is common in semiconductor physics. The exposition in Sec. 37.1 demonstrates, that transition functions with spin, their averaged form being connected with several resonator frequencies, could in principle be included into the present model discussion. We reduce the material Hamiltonians to the quasi-spin formalism of Sec. 36.2 and repeat shortly the pertinent notions for that kind of a weakly inhomogeneous cluster model…

We have treated already the Josephson contact as a closed system in Sec. 37.3 on page 1188 and in contact with classical and quantized electromagnetic fields in Sec. 37.4 on page 1199. In the present chapter, we elaborate its interaction with only the transverse quantized electromagnetic field but formulate its coupled dynamics and asymptotic radiation in more detail, making use of our general insights on quantum Hertz oscillators in Sec. 38.6 on page 1328. For this model ansatz, there exist many previous discussions and we want first to give a short overview on foregoing works…

In all of our evaluations of mesoscopic radiation models in terms of algebraic QED, we met in both, electronic and photonic systems, an intimate entanglement of microscopic and macroscopic (respectively mesoscopic) features, where not only the microscopic observables exhibited non-commutative behavior. We want to discuss in the present section some general criteria for characterizing true quantum aspects combining them with some further prospects of the light-plus-matter theory.

The following sections are included:

• Bibliography
• Index