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arXiv:2104.00745 · Physical Review A

Dimensional reduction of cavities with axial symmetry: A complete analysis of when an optical fiber is approximately one dimensional

Daniel Grimmer, Richard Lopp, Eduardo Martín-Martínez

Published Jul 28, 2021• 1 citation
arXivHTMLPDF

Abstract

Intuition dictates that a very long, very thin cavity (e.g., a fiber optic cable) could perhaps be modeled as an approximately one-dimensional system. In this paper we rigorously explore the validity of such intuition from the perspective of a localized probe coupling to a quantum field inside a cavity (e.g., an atom or an Unruh-DeWitt particle detector in a fiber optic cable). To do so, we introduce the notion of subfield decomposition in which a ($D+1$)-dimensional quantum field in an axially symmetric cavity can be reduced to an infinite collection of uncoupled, massive ($1+1$)-dimensional fields. We show that the ability to approximate a higher-dimensional scenario by a ($1+1$)-dimensional model is equivalent to making a certain change of the probe's shape in the higher-dimensional space. The approximation is justified whenever this change of shape is ``small enough.'' In this light, we identify the dynamically relevant norm by which the magnitude of these changes in probe shape ought to be judged. Finally, we explore this approximation in particular setups corresponding to quantum optics and superconducting circuits.

DOI
10.1103/physreva.104.013723

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We propose to turn two resonant distant cavities effectively into one by coupling them via an optical fiber which is coated with two-level atoms [J. D. Franson et al., Phys. Rev. A 70, 062302 (2004)]. The purpose of the atoms is to destructively measure the evanescent electric field of the fiber on a time scale which is long compared to the time it takes a photon to travel from one cavity to the other. Moreover, the boundary conditions imposed by the setup should support a small range of standing waves inside the fiber, including one at the frequency of the cavities. In this way, the fiber provides an additional decay channel for one common cavity field mode but not for the other. If the corresponding decay rate is sufficiently large, this mode decouples effectively from the system dynamics. A single nonlocal resonator mode is created.
Abstract We analyze the physics of accelerated particle detectors (such as atoms) crossing optical cavities. In particular we focus on the detector response as well as on the energy signature that the detectors imprint in the cavities. In doing so, we examine to what extent the usual approximations made in quantum optics in cavities (such as the single-mode approximation, or the dimensional reduction of 3 + 1D cavities to simplified 1 + 1D setups) are acceptable when the atoms move in relativistic trajectories. We also study the dependence of these approximations on the state of the atoms and the relativistic nature of the trajectory. We find that, on very general grounds, and already in the weak coupling limit, single- and few-mode approximations, as well as 1 + 1D dimensional reductions, yield incorrect results when relativistic scenarios are considered.
In this thesis, we study the interaction of a first-quantized atomic system with the quantum electromagnetic field within the context of relativistic quantum information (RQI). To that end, we examine common classes of Hamiltonians -- in particular, the dipolar coupling model as well as scalar field analogues (Unruh-DeWitt models) -- for their applicability in RQI setups. Firstly, we investigate how quantum randomness generation based on unbiased measurements on an atom can get compromised by an adversary that has access to the electromagnetic field. We show that preparing the atom in the ground state in the presence of no field excitations is, in general, not the best choice to generate randomness. Secondly, at the study of light-matter interactions inside optical cavities, we show that frequently employed approximations, such as the single-mode approximation and dimensional reduction, fail for relativistic regimes but can be already ill-behaved for non-relativistic scenarios. In particular, we show how approximating a very long and thin cavity by a one-dimensional system can be understood by recasting the D+1 dimensional quantum field inside the cavity as an infinite sum of massive 1+1 dimensional fields. The dimensional reduction approximation can subsequently be identified with ignoring all but one of these subfields or, equivalently, with a change of the atomic localization. Up to this point, we have treated the atomic center of mass classically -- a feature that is shared by usual Hamiltonians in RQI and quantum optics. We therefore revisit the interaction of atoms and light by considering all atomic degrees of freedom to be quantum. Further, we discuss subtleties with respect to the gauge nature of light and the effect that multipole approximations have. This allows us to connect the multipolar Hamiltonian with the common effective models of quantum optics and relativistic quantum information. In particular, we discuss the influence of atomic center-of-mass delocalization and the presence of the so-called Roentgen term. Significantly, Unruh-DeWitt models fail to account for the entangling interaction between all atomic and field degrees of freedom, and we present then a scalar analogue of the Roentgen term. Finally, we demonstrate how the usual dipole model preserves covariance when considering atoms on relativistic trajectories and how this model can be used as a qualitative means to study RQI scenarios.
The quantum theory of a one-dimensional optical cavity is developed with emphasis on the presence of output coupling. First the resonant mode is defined and calculated classically. Then the field is decomposed into a sequence of modes by introducing an imaginary boundary at a large distance. The mode functions are proved to be orthogonal with respect to an integration with the dielectric constant as a weighting factor. The Hamiltonian of the field is shown to be equivalent to that of a collection of independent harmonic oscillators, the mass of which is a function of the frequency of oscillation. This equivalent of mass appears on normalizing the mode functions and proves to carry information on the structure of the cavity. Quantization of the field is carried out, and the commutation relation for the electric fields inside and outside the cavity is derived. The commutator is composed of an infinite set of derivatives of $\ensuremath{\delta}$ functions, which discloses the effect of the presence of output coupling and that of the small size of the cavity on the radiation field. Also, the commutator is shown to have some properties common with the classical resonant mode, i.e., the property of exponential decay and its rate.
A simple but powerful particle detector model consists of a two-level system coupled to a field, where the detected particles are the field excitations. This is known as the Unruh-DeWitt detector. Research using this model has often focused on either a completely classical description of the external degrees of freedom of the detector, or a full field-theoretic treatment, where the detector itself is described as a field. Recently there has been much interest in quantum aspects of the detector's center of mass---either described as moving in superposition along semiclassical trajectories, or dynamically evolving under a nonrelativistic Hamiltonian. However, the processes of interest---the absorption or emission of field particles---necessarily change the detector's rest mass by the amount of energy of the absorbed or emitted field quanta. Neither of the above models can capture such effects. Here we incorporate the quantization of the detector's mass-energy into the Unruh-DeWitt model. We show that internal energy changes due to emission or absorption are relevant even in the lowest energy limit. Specifically, corrections to transition rates due to the detector's mass changing cannot be ignored unless the entire center of mass dynamics is also ignored. Our results imply that one cannot have a consistent model of the Unruh-DeWitt detector as a massive particle without including the mass-energy equivalence.