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arXiv:2212.01935 · Physical Review A

Matrix product states and numerical mode decomposition for the analysis of gauge-invariant cavity quantum electrodynamics

Christopher Jayun Ryu, Dong-Yeop Na, Weng Cho Chew

Published Jun 20, 2023• 9 citations
arXivHTMLPDF

Abstract

There has been a problem of gauge ambiguities with the Rabi Hamiltonian due to the fact that it can be derived from two formally different but physically equivalent fundamental Hamiltonians. This problem has recently been resolved for models with a single quantized electromagnetic mode. In this paper, we mathematically and numerically verify this for multimode models. With this established, we combine the numerical methods, matrix product states (MPS) and numerical mode decomposition (NMD), for analyzing cavity QED systems. The MPS method is used to efficiently represent and time evolve a quantum state. However, since the coupling structure of the Rabi Hamiltonian is incompatible with MPS, it is numerically transformed into an equivalent Hamiltonian that has a chain coupling structure, which allows efficient application of MPS. The technique of NMD is used to extract the numerical electromagnetic modes of an arbitrary environment. As a proof of concept, this combined approach is demonstrated by analyzing one-dimensional cavity QED systems in various settings.

DOI
10.1103/physreva.107.063707

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There has been a problem of gauge ambiguities with the Rabi Hamiltonian due to the fact that it can be derived from two formally different but physically equivalent fundamental Hamiltonians. This problem has recently been resolved for models with a single quantized electromagnetic mode. In this paper, we mathematically and numerically verify this for multimode models. With this established, we combine the numerical methods, matrix product states (MPS) and numerical mode decomposition (NMD), for analyzing cavity QED systems. The MPS method is used to efficiently represent and time evolve a quantum state. However, since the coupling structure of the Rabi Hamiltonian is incompatible with MPS, it is numerically transformed into an equivalent Hamiltonian that has a chain coupling structure, which allows efficient application of MPS. The technique of NMD is used to extract the numerical electromagnetic modes of an arbitrary environment. As a proof of concept, this combined approach is demonstrated by analyzing one-dimensional cavity QED systems in various settings.
Numerical analysis of few-atom cavity quantum electrodynamics (QED) is challenging due to the exponentially scaling size of the Hilbert space when multiple quantized electromagnetic modes are considered. To overcome this, we propose an approach based on coupling matrix transformations that allows adjustment of the coupling structures of QED Hamiltonians so that they are more compatible with tensor network algorithms such as matrix product states. We present the relevant Hamiltonians along with the coupling matrix transformation scheme and present few-atom cavity QED simulations in the time domain that is able to characterize the interactions between atoms efficiently.
This work provides the fundamental theoretical framework for few-mode cavity quantum electrodynamics by resolving the gauge ambiguities between the Coulomb gauge and the dipole gauge Hamiltonians under the photonic mode truncation. We first propose a general framework to resolve ambiguities for an arbitrary truncation in a given gauge. Then, we specifically consider the case of mode truncation, deriving gauge invariant expressions for both the Coulomb and dipole gauge Hamiltonians that naturally reduce to the commonly used single-mode Hamiltonians when considering a single-mode truncation. We finally provide the analytical and numerical results of both atomic and molecular model systems coupled to the cavity to demonstrate the validity of our theory.
We present a matrix product state (MPS) method for including decoherence processes in calculations involving waveguide quantum electrodynamics (waveguide QED) using density matrices. The approach is based on collision quantum optics, where the many-body state of the waveguide is represented using discrete time bins, which are then efficiently encoded using an MPS chain. Our method is a generalization of previous MPS methods, and we demonstrate how one can efficiently extend the method to density matrices, allowing for the inclusion of various loss processes in the form of Lindblad terms in the Liouvillian superoperator responsible for the relevant dissipation dynamics. As an application of the theory, we study various waveguide QED systems and the influence of emitter pure dephasing (which is one of the most important processes in real systems) on the light-matter interactions, including a two-level system (TLS) in a semi-infinite waveguide with time-delayed feedback, two spatially separated TLSs with finite delays, and finally the scattering of few-photon Fock pulses on a TLS. In addition to emitter pure dephasing, we also show how to include off-chip radiative decay, and show how it differs qualitatively from pure dephasing.
This work provides the fundamental theoretical framework for molecular cavity quantum electrodynamics by resolving the gauge ambiguities between the Coulomb gauge and the dipole gauge Hamiltonians under the electronic state truncation. We conjecture that such ambiguity arises because not all operators are consistently constrained in the same truncated electronic subspace for both gauges. We resolve this ambiguity by constructing a unitary transformation operator that properly constrains all light-matter interaction terms in the same subspace. We further derive an equivalent and yet convenient expression for the Coulomb gauge Hamiltonian under the truncated subspace. We finally provide the analytical and numerical results of a model molecular system coupled to the cavity to demonstrate the validity of our theory.
The quantum Rabi model is a widespread description of the coupling between a two-level system and a quantized single mode of an electromagnetic resonator. Issues about this model's gauge invariance have been raised. These issues become evident when the light-matter interaction reaches the so-called ultrastrong coupling regime. Recently, a modified quantum Rabi model able to provide gauge-invariant physical results (e.g., energy levels, expectation values of observables, quantum probabilities) in any interaction regime was introduced [O. Di Stefano, A. Settineri, V. Macr\`{\i}, L. Garziano, R. Stassi, S. Savasta, and F. Nori, Nat. Phys. 15, 803 (2019)]. Here we provide an alternative derivation of this result, based on the implementation in two-state systems of the gauge principle, which is the principle from which all the fundamental interactions in quantum field theory are derived. The adopted procedure can be regarded as the two-site version of the general method used to implement the gauge principle in lattice gauge theories. Applying this method, we also obtain the gauge-invariant quantum Rabi model for asymmetric two-state systems, and the multimode gauge-invariant quantum Rabi model beyond the dipole approximation.
Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems. The matrix product states (MPS) are one particular case of TNS and are used for the simulation of 1+1 dimensional systems. In [1] we considered the MPS formalism for the simulation of the Hamiltonian lattice gauge formulation of 1+1 dimensional one flavor quantum electrodynamics, also known as the massive Schwinger model. We deduced the ground state and lowest lying excitations. Furthermore, we performed a full quantum real-time simulation for a quench with a uniform background electric field. In this proceeding we continue our work on the Schwinger model. We demonstrate the advantage of working with gauge invariant MPS by comparing with MPS simulations on the full Hilbert space, that includes numerous non-physical gauge variant states. Furthermore, we compute the chiral condensate and recover the predicted UV-divergent behavior.