Quantum Electrodynamical Bloch Theory with Homogeneous Magnetic Fields

Abstract

We propose a solution to the problem of Bloch electrons in a homogeneous magnetic field by including the quantum fluctuations of the photon field. A generalized quantum electrodynamical (QED) Bloch theory from first principles is presented. In the limit of vanishing quantum fluctuations we recover the standard results of solid-state physics, for instance, the fractal spectrum of the Hofstadter butterfly. As a further application we show how the well known Landau physics is modified by the photon field and that Landau polaritons emerge. This shows that our QED-Bloch theory does not only allow to capture the physics of solid-state systems in homogeneous magnetic fields, but also novel features that appear at the interface of condensed matter physics and quantum optics.Cavity QED materials is a growing research field bridging quantum optics [1, 2], polaritonic chemistry [3-7], and materials science, such as light-induced new states of matter achieved with classical laser fields [8,9]. Photonmatter interactions have recently been suggested to modify electronic properties of solids, such as superconductivity and electron-phonon coupling [10][11][12][13][14]. On the other hand, materials in classical magnetic fields are known to give rise to a variety of novel phenomena such as the Landau levels [15], the integer [16,17] and the fractional quantum Hall effect [18], and the quantum fractal of the Hofstadter butterfly [19] which can be now accessed experimentally with high resolution [20][21][22]. One of the open questions in this field is whether Bloch theory is applicable for solids in the presence of a homogeneous magnetic field. The homogeneous magnetic field breaks explicitly translational symmetry. This issue was solved to some extent by introducing the magnetic translation group. However, the magnetic translation group puts fundamental limitations on the possible directions and values of the strength of the magnetic field [17,23,24].In this Letter, by combining QED with solid-state physics, we provide a consistent and comprehensive theory for solids interacting with homogeneous electromagnetic fields, both classical and quantum. Our main findings are as follows: (i) The quantum fluctuations of the electromagnetic field allow us to restore translational symmetry that is broken due to an external homogeneous magnetic field (see Fig. 1). (ii) We generalize Bloch theory and provide a Bloch central equation for electrons in a solid in the presence of a homogeneous magnetic field and its quantum fluctuations. (iii) Applying our framework to the case of a 2D solid in a perpendicular homogeneous magnetic field, in the limit of no quantum fluctuations, we recover the fractal spectrum of the Hofstadter butterfly (see Fig. 2). (iv) In the case of a 2D * vasil.rokaj@mpsd.mpg.de † angel.rubio@mpsd.mpg.de A A A A A u y x v w = 1 A = 2 A = 3 A = 4 A = 5 A = 1 = 2 = 3 = 4 = 5 a y √ 2 ωc ay A t o t = c o n s t ext tot tot tot tot tot ext ext ext ext FIG.

Reduced Density-Matrix Approach to Strong Matter-Photon Interaction

“…We call them dressed or polaritonic particles, because they depend on electronic and photonic coordinates. (Note that the use of a dressed particle picture allows to also describe Landau polaritons, as shown recently in ref (79). Thus, also such systems can in principle be considered with the presented approach.)…”

Relevance of the Quadratic Diamagnetic and Self-Polarization Terms in Cavity Quantum Electrodynamics

“…Under certain assumptions the diamagnetic term can indeed be absorbed by a redefinition of the frequencies and polarizations of the field modes [15,75,93,94]. These redefinitions depend on the matter subsystem (more specifically the number of charged particles) and lead to the diamagnetic shift of the photon field which can be observed experimentally [98,127,128].…”

“…A). This quadratic term provides the diamagnetic shift [93] of the bare modes and introduces a lowest allowed frequency [94] which then removes the infrared divergence [73]. It is therefore not a drawback to have this term [74].…”

New Class of Landau Levels and Hall Phases in a 2D Electron Gas Subject to an Inhomogeneous Magnetic Field: An Analytic Solution

“…[7][8][9][10] The study of Landau levels and topological edge states is still very actively pursued and is currently even considered in quantum optics and cavity quantum electrodynamics (QED), where ultra-strong coupling of the Landau levels to the quantum vacuum fluctuations and the control of conduction properties have been achieved experimentally in a cavity, [11][12][13][14] with several theoretical studies and proposals accompanying these developments. [15][16][17][18][19] In parallel to the fundamental investigation of quantum systems exposed to magentic fields, the study of impurity models has a long lasting tradition in solid state physics. Some of the fundamental theoretical concepts go back to work of Friedel for charge impurity induced oscillations 20 , whereas Anderson localisation 21 or Kondo effect 22 may emerge due to lattice induced magnetic impurities.…”