One-dimensional quantum optical models usually rest on the intuition of large-scale separation or frozen dynamics associated with the different spatial dimensions, for example when studying quasi one-dimensional atomic dynamics, potentially resulting in the violation of (3+1) -dimensional Maxwell's theory. Here, we provide a rigorous foundation for this approximation by means of the light-matter interaction. We show how the quantized electromagnetic field can be decomposed—exactly—into an infinite number of living on a lower-dimensional subspace and containing the entirety of the spectrum when studying axially symmetric setups, such as with an optical fiber, a laser beam, or a waveguide. The approximation then corresponds to a truncation in the number of such subfields that in turn, when considering the interaction with for instance an atom, corresponds to a modification to the atomic spatial profile. We explore under what conditions the standard approach is justified and when corrections are necessary in order to account for the dynamics due to the neglected spatial dimensions. In particular we examine what role vacuum fluctuations and structured laser modes play in the validity of the approximation. Published by the American Physical Society 2024
One-dimensional quantum optical models usually rest on the intuition of large-scale separation or frozen dynamics associated with the different spatial dimensions, for example when studying quasi one-dimensional atomic dynamics, potentially resulting in the violation of (3+1) -dimensional Maxwell's theory. Here, we provide a rigorous foundation for this approximation by means of the light-matter interaction. We show how the quantized electromagnetic field can be decomposed—exactly—into an infinite number of living on a lower-dimensional subspace and containing the entirety of the spectrum when studying axially symmetric setups, such as with an optical fiber, a laser beam, or a waveguide. The approximation then corresponds to a truncation in the number of such subfields that in turn, when considering the interaction with for instance an atom, corresponds to a modification to the atomic spatial profile. We explore under what conditions the standard approach is justified and when corrections are necessary in order to account for the dynamics due to the neglected spatial dimensions. In particular we examine what role vacuum fluctuations and structured laser modes play in the validity of the approximation. Published by the American Physical Society 2024
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.
A rigorous description of the light–matter interaction requires the use of quantized Maxwell equations to describe the light field and the Schrödinger equation to describe the material medium. This book does not attempt to deal with quantum optical problems that necessitate a field quantization. Instead we focus on properties that are generally grouped under the umbrella of nonlinear optics . Although many definitions of this expression exist, we use it in the sense that the properties we deal with do not depend in a crucial way on the field quantization. For practical purposes, this means in general that the average photon number is large and that we can neglect spontaneous emission as a dynamical process. Rather, we include the consequence of spontaneous emission, that is, the instability of the atomic levels, in a purely phenomenological way. When this is done, we call the material equations the Bloch equations instead of the Schrödinger equation. References [1] through [4] are the classical textbooks that contain a tentative justification for the transition from the Schrödinger to the Bloch equations. None of these attempts is satisfactory from a fundamental viewpoint because the real difficulty is to incorporate the finite lifetime (or natural linewidth) of the atomic energy levels in the Schrödinger equation. This remains an open problem as of now. The Maxwell–Schrödinger equations We consider atoms interacting with an intense electromagnetic field. By intense we mean a field for which the quantization is not necessary.
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.
In a realistic treatment of a three-dimensional nonlinear optical experiment, the complete Maxwell equations in (3 + 1) space-time dimensions should be employed. It is then necessary to utilize a multi-mode Hamiltonian that correctly describes the propagating modes. There is an important difference between these experiments and traditional particle scattering. Quantum field dynamics in nonlinear media is dominated by multiple scattering, which is the reason why perturbation theory is less useful. Nevertheless, it is interesting to make a link to conventional perturbation theory. Accordingly, we start by considering a perturbative theory of propagation in a one-dimensional nonlinear optical system in a χ (3) medium. While this calculation cannot treat long interaction times, it does give a qualitative understanding of the important features. This problem is the ‘hydrogen atom’ of quantum field theory: it has fully interacting fields with exact solutions for their energy levels. This is because a photon in a waveguide is an elementary boson in one dimension. The interactions between these bosons are mediated by the Kerr effect, which in quantum field theory is a quartic potential, equivalent to a delta-function interaction. The quantum field theory involved is the simplest model of a quantum field that has an exact solution. This elementary model is still nontrivial in terms of its dynamics, as the calculation of quantum dynamics using standard eigenfunction techniques would require exponentially complex sums over multi-dimensional overlap integrals. This is not practicable, and accordingly we use other methods including quantum phase-space representations to solve this problem.
This thesis presents investigations on the interplay of coherence and many-body effects in the quasi one-dimensional transport of Bose-Einstein condensates (BEC) through scattering potentials. Such configurations can be realized with guided atom lasers that provide a coherent atomic beam. An exact theoretical description of the dynamics is out of reach due to the presence atom-atom interactions. Different levels of approximations are nevertheless possible with their strengths and weaknesses. The mean-field approximation, where the dynamics of the BEC is governed by the Gross-Pitaevskii equation, is most commonly used in the field of ultracold atoms. In this thesis the truncated Wigner method is used to go beyond the standard Gross-Pitaevskii description. This method is adapted in order to study the scattering of Bose-Einstein condensates in one-dimensional waveguides where atom-atom interactions and external potentials are nonvanishing only in a finite region of space. In this case, the truncated Wigner method is combined with the smooth exterior complex scaling method and incorporates quantum noise that originate from the vacuum fluctuations in the waveguide. Inelastic scattering is shown to play a major role in the resonant transport of BEC through a symmetric double potential barrier effectively forming an atomic quantum dot. Indeed, fully resonant transmission is prohibited and incoherent atoms as well as collective oscillations are detected in the transmitted beam. It is also shown that inelastic scattering destroys Anderson localization in the case of transport through disordered potentials. The classical (incoherent) ohmic transmission is recovered for finite atom-atom interactions. The validity of the truncated Wigner method is then assessed using the semiclassical van Vleck-Gutzwiller propagator in the Fock space of the many-body system. It is shown that the truncated Wigner method corresponds to the so-called diagonal approximation, and it is possible to identify the leading correction to the truncated Wigner results, which is provided by the so-called coherent backscattering (CBS) contribution. Coherent basckattering in Fock space is a genuine quantum many-body effect that lies beyond the reach of any mean-field approach. For the case of closed Bose-Hubbard models, the relevance of CBS is confirmed by numerically comparing the (classical) truncated Wigner evolution probabilities to the exact quantum probabilities in Bose-Hubbard models: While a CBS-induced enhancement of the return probability to the initial state is clearly seen in the exact quantum simulations of the bosonic many-body system, this enhancement is absent in the classical calculations. The magnitude and dependence of the CBS contribution on gauge fields, which break time-reversal invariance, is numerically confirmed. For the case of disordered open systems, it can be shown that this contribution as well as next-to leading order contributions vanish thereby confirming the validity of the truncated Wigner method.
We present the results of the numerical investigations of light-matter interactions in one and two dimensions from the quantum mechanical perspective.We investigate the dynamics of two-level systems coupled to quantized electromagnetic fields.We construct a quantum mechanical model to demonstrate how light interacts with classical objects such as mirrors and beam-splitters made from group of atoms, where each atom is modelled as a two-level system.We have been able to simulate behaviour of a singlephoton being reflected and transmitted as a process of absorption and re-emission by the atoms.i