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arXiv:2408.03580 · Physical Review A

Atom-field-medium interactions: Graded influence actions for N harmonic atoms in a dielectric-altered quantum field

Jen-Tsung Hsiang, B. L. Hu

Published Dec 9, 2024• 3 citations
arXivHTMLPDF

Abstract

This paper and planned subsequent ones have two broad aims. First, we construct a theory for multipartite open quantum systems comprising several layers of structure with self-consistent backactions. We develop the graded influence action formalism [Behunin and Hu, Phys. Rev. A 82, 022507 (2010); Phys. Rev. A 84, 012902 (2011)] to account for the influences of successive sublayers on the dynamics of the variables of interest. Second, we apply these methods to the study of atom-field-medium interactions and highlight their merits over conventional methods. We consider a system of $N$ harmonic oscillators, modeling the internal degrees of freedom (IDFs) of $N$ neutral atoms (A), interacting with a quantum field (F), scalar here, for simplicity, altered by the presence of a dielectric medium (M). In this paper we use the coarse-grained and stochastic effective actions in the influence functional formalism to derive the stochastic equations for the reduced density matrices of the dynamical variables in the successive layers of structure. The term ``graded'' refers to the specific ordering of the coarse-graining procedures. Three layers of coarse graining are performed. First, integrating over the common bath of the dielectric oscillators results not only in the appearance of necessary dissipative properties of the dielectric but also in essential nuanced features such as non-Markovian spatial correlations in the dielectric. Second, integrating over the medium variables as a whole results in a dielectric-modified quantum field, the influence of the medium on the quantum field manifesting through a frequency-dependent permittivity function. Finally, integrating over this dielectric-altered quantum field which interacts with the IDFs of the atoms allows us to derive an equation of motion which describes the nonequilibrium stochastic dynamics of the IDFs of the atoms interacting with a dielectric medium-modified quantum field. In the future we plan to calculate the nonequilibrium covariant matrix elements of the correlation functions of the IDFS of $N$ atoms in a dielectric-altered quantum field, which are useful for probing many basic quantum information issues, such as the entanglement dynamics in AFM interactions.

DOI
10.1103/physreva.110.062807

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This paper and planned subsequent ones have two broad aims. First, we construct a theory for multipartite open quantum systems comprising several layers of structure with self-consistent backactions. We develop the graded influence action formalism [Behunin and Hu, Phys. Rev. A 82, 022507 (2010); Phys. Rev. A 84, 012902 (2011)] to account for the influences of successive sublayers on the dynamics of the variables of interest. Second, we apply these methods to the study of atom-field-medium interactions and highlight their merits over conventional methods. We consider a system of NNN harmonic oscillators, modeling the internal degrees of freedom (IDFs) of NNN neutral atoms (A), interacting with a quantum field (F), scalar here, for simplicity, altered by the presence of a dielectric medium (M). In this paper we use the coarse-grained and stochastic effective actions in the influence functional formalism to derive the stochastic equations for the reduced density matrices of the dynamical variables in the successive layers of structure. The term ``graded'' refers to the specific ordering of the coarse-graining procedures. Three layers of coarse graining are performed. First, integrating over the common bath of the dielectric oscillators results not only in the appearance of necessary dissipative properties of the dielectric but also in essential nuanced features such as non-Markovian spatial correlations in the dielectric. Second, integrating over the medium variables as a whole results in a dielectric-modified quantum field, the influence of the medium on the quantum field manifesting through a frequency-dependent permittivity function. Finally, integrating over this dielectric-altered quantum field which interacts with the IDFs of the atoms allows us to derive an equation of motion which describes the nonequilibrium stochastic dynamics of the IDFs of the atoms interacting with a dielectric medium-modified quantum field. In the future we plan to calculate the nonequilibrium covariant matrix elements of the correlation functions of the IDFS of NNN atoms in a dielectric-altered quantum field, which are useful for probing many basic quantum information issues, such as the entanglement dynamics in AFM interactions.
This dissertation treats quantum open system dynamics, focusing on the coherent evolution of a two-level atom (as the system) interacting with an electromagnetic field (as the bath), for purposes relevant to quantum computing. In order to maintain the quantum correlations that develop between the system and bath throughout the evolution path integral formalisms such as the influence functional and closed time path formalisms are used. Predictions of effects due to the quantum correlations in the composite interacting system are computed. Conventional treatments using Schr\"odinger-master equation and Heisenberg-Langevin approaches usually ignore system+bath quantum correlations as a technical simplification. It is argued that although neglect of system+bath correlations is generally a good approximation when the bath has a large continuous set of degrees of freedom, a residual coherence effect remains due to the non-zero bath correlation time. Though small, these effects are becoming more relevant as, with the advent of ultra cold atom sources, atom optics experiments are reaching levels at which such residual effects are becoming measurable. Three specific problems are investigated in this thesis: First is a self-dressing rederivation of the Casimir-Polder retardation force. The well known stationary atom result is reproduced and a result for a slowly moving atom is obtained which is up to twice the stationary atom correction. Second is the entangled evolution of a qubit with an initially thermal low temperature bath. The diagonal matrix elements are found to thermalize and the off-diagonal elements to decohere as expected, however they do so non-exponentially due to the quantum correlations that develop between the qubit and bath. Third is a calculation of qubit dynamics in the presence of quantized atomic motion as well as zero point fluctuations of the electromagnetic field. The decoherence rate of the qubit is found to increase slightly in that case due to the additional degree of freedom.
Mesoscopic physics deals with three fundamental issues: quantum coherence, fluctuations and correlations. Here we analyze these issues for atom optics, using a simplified model of an assembly of atoms (or detectors, which are particles with some internal degree of freedom) moving in arbitrary trajectories in a quantum field. Employing the influence functional formalism, we study the self-consistent effect of the field on the atoms, and their mutual interactions via coupling to the field. We derive the coupled Langevin equations for the atom assemblage and analyze the relation of dissipative dynamics of the atoms with the correlation and fluctuations of the quantum field. This provides a useful theoretical framework for analysing the coherent properties of atom-field systems.
The dynamics of a collection of resonant atoms embedded inside an inhomogeneous nondispersive and lossless dielectric is described with a dipole Hamiltonian that is based on a canonical quantization theory. The dielectric is described macroscopically by a position-dependent dielectric function and the atoms as microscopic harmonic oscillators. We identify and discuss the role of several types of Green tensors that describe the spatio-temporal propagation of field operators. After integrating out the atomic degrees of freedom, a multiple-scattering formalism emerges in which an exact Lippmann-Schwinger equation for the electric field operator plays a central role. The equation describes atoms as point sources and point scatterers for light. First, single-atom properties are calculated such as position-dependent spontaneous-emission rates as well as differential cross sections for elastic scattering and for resonance fluorescence. Secondly, multiatom processes are studied. It is shown that the medium modifies both the resonant and the static parts of the dipole-dipole interactions. These interatomic interactions may cause the atoms to scatter and emit light cooperatively. Unlike in free space, differences in position-dependent emission rates and radiative line shifts influence cooperative decay in the dielectric. As a generic example, it is shown that near a partially reflecting plane there is a sharp transition from two-atom superradiance to single-atom emission as the atomic positions are varied.
We present a numerical framework that integrates a modified Langevin noise formalism into multimode Jaynes- and Tavis-Cummings models for first-principles, non-Markovian analysis of atom-field interactions in dissipative electromagnetic environments under weak coupling. This formalism enables the quantization of electric field operators in terms of boundary- and medium-assisted modes with real-valued eigenfrequencies, accounting for both radiative and absorptive losses. To efficiently handle the continuum nature of the boundary- and medium-assisted mode spectrum, we apply adaptive frequency refinement and mode-space coarse-graining based on finite-element simulations to obtain a tractable discrete set of field modes. These modes are then used to construct matrix representations of the Schr\"odinger equation, enabling direct computation of atomic observables and field quantities---such as single-photon amplitudes and higher-order correlation functions---beyond the Markovian approximation. Importantly, this framework does not rely on any phenomenologically designed memory kernel. Instead, the spectrum of boundary- and medium-assisted modes interacting with atoms inherently and rigorously captures memory effects arising from the open and dissipative electromagnetic environment. We present four numerical examples to validate the proposed framework and demonstrate its ability to model non-Markovian atomic dynamics in open and dissipative electromagnetic environments: (i) a two-level system near the surface of a perfect electric conductor, (ii) dissipative cavity quantum electrodynamics for a single two-level system, (iii) super- and subradiance in two-level system arrays, and (iv) entanglement sudden death between two two-level systems. This framework provides a versatile simulation tool for the design of quantum photonic devices, including nitrogen-vacancy center single-photon sources coupled to plasmonic structures, entangled photon-pair generation near lossy media, and decoherence modeling of solid-state quantum memories based on rare-earth ions and color centers. It is particularly well suited to dissipative quantum electrodynamics in photonic integrated circuits, where accurate modeling of scattering and diffraction is essential.
An open quantum system is a quantum system that interacts with some environment whose degrees of freedom have been coarse grained away. This model describes non-equilibrium processes more general than scattering-matrix formulations. Furthermore, the microscopically-derived environment provides a model of noise, dissipation and decoherence far more general than Markovian (white noise) models. The latter are fully characterized by Lindblad equations and can be motivated phenomenologically. Non-Markovian processes consistently account for backreaction with the environment and can incorporate effects such as finite temperature and spatial correlations. We consider linear systems with bilinear coupling to the environment, or quantum Brownian motion, and nonlinear systems with weak coupling to the environment. For linear systems we provide exact solutions with analytical results for a variety of spectral densities. Furthermore, we point out an important mathematical subtlety which led to incorrect master-equation coefficients in earlier derivations, given nonlocal dissipation. For nonlinear systems we provide perturbative solutions by translating the formalism of canonical perturbation theory into the context of master equations. It is shown that unavoidable degeneracy causes an unfortunate reduction in accuracy between perturbative master equations and their solutions. We also extend the famous theorem of Lindblad, Gorini, Kossakowski and Sudarshan on completely positivity to non-Markovian master equations. Our application is primarily to model atoms interacting via a common electromagnetic field. The electromagnetic field contains correlations in both space and time, which are related to its relativistic (photon-mediated) nature. As such, atoms residing in the same field experience different environmental effects depending upon their relative position and orientation. Our more accurate solutions were necessary to assess sudden death of entanglement at zero temperature. In contrast to previous claims, we found that all initial states of two-level atoms undergo finite-time disentanglement. We were also able to access regimes which cannot be described by Lindblad equations and other simpler methods, such as near resonance. Finally we revisit the infamous Abraham-Lorentz force, wherein a single particle in motion experiences backreaction from the electromagnetic field. This leads to a number of well-known problems including pre-acceleration and runaway solutions. We found a more a more-suitable open-system treatment of the nonrelativistic particle to be perfectly causal and dissipative without any extraneous requirements for finite size of the particle, weak coupling to the field, etc..
This thesis investigates two thematic lines of research, both underpinned by non-Markovian system-reservoir interactions in quantum optics. The overarching focus is on modelling the open system dynamics in a non-perturbative fashion, broadly on - though not restricted to - instances when the environment is structured. A theory is developed by means of enlarging the open system over environmental degrees of freedom to include memory effects in its dynamics. This is achieved using an established technique that involves mapping a bosonic environment onto a 1D chain of harmonic oscillators. Within this setting, we apply a Heisenberg equation-of-motion approach to derive an exact set coupled differential equations for the open system and a single auxiliary oscillator of the chain. The combined equations are shown to have their interpretation rooted in a quantum Markov stochastic process. Including the auxiliary chain oscillator as part of the original system then enables us to obtain an exact master equation for the enlarged system, avoiding any need for the Born-Markov approximations. Our method is valid for a dissipative two-state system, with cases of multiple excitations and added driving discussed. Separately, we apply the framework of quantum Darwinism to an atom-cavity system, and, subsequently, to a more general multiple-environment model. In both cases, the time-dependent spread of correlations between the open system and fractions of the environment is analysed during the course of the decoherence process. The degree to which information is redundant across different fractions is checked to infer the emergence of classicality. In the second case, we go further and present a decomposition of information in terms of its quantum and classical correlations. A quantitative measure of redundancy is also studied with regard to its ability to witness non-Markovian behaviour. Besides fundamental interest, our results have application to quantum information processing and quantum technologies, keeping in mind the potential beneficial use of non-Markovian effects in reservoir engineering.