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Digest - IEEE Antennas and Propagation Society. International Symposium

Efficient Numerical Analysis for Few-Atom Cavity Quantum Electrodynamics Using the Coupling Matrix Approach

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

Published Jul 23, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.1109/usnc-ursi52151.2023.10238011

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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.
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.
Quantum light-matter systems at strong coupling are notoriously challenging to analyze due to the need to include states with many excitations in every coupled mode. We propose a nonperturbative approach to analyze light-matter correlations at all interaction strengths. The key element of our approach is a unitary transformation that achieves asymptotic decoupling of light and matter degrees of freedom in the limit where light-matter interaction becomes the dominant energy scale. In the transformed frame, truncation of the matter or photon Hilbert space is increasingly well justified at larger coupling, enabling one to systematically derive low-energy effective models, such as tight-binding Hamiltonians. We demonstrate the versatility of our approach by applying it to concrete models relevant to electrons in crystal potential and electric dipoles interacting with a cavity mode. A generalization to the case of spatially varying electromagnetic modes is also discussed.
With the aid of supercomputer models we numerically investigate the exact solutions for a small number of atoms (typically 1–10) interacting with a single quantized mode of an electromagnetic cavity that is driven by an external coherent field. Effects of spontaneous emission and a nonzero cavity decay rate are included. We investigate the strong coupling regime where the atom-field coupling constant is much greater than the dissipation rates to the external environment. In this regime, the usual system size expansions of quantum statistical physics are not applicable and a full numerical solution is required. We model the recently reported experimental observations of Rabi oscillations in the second order coherence function1 g(2)(τ) and compare and contrast the results with an effective single atom-cavity model with an appropriately scaled coupling constant. In addition, the steady state field distribution for the few-atom-cavity system is explored. A multipeaked phase distribution is found that generalizes the bimodal phase distribution found in the single-atom-cavity case.2 Spectra for the few-atom-cavity system are investigated in the weak and strong driving field limits.
It is shown that a broad class of cavity quantum electrodynamics (QED) problems---which consider the resonant propagation of a single photon interacting with quantum emitters (QEs), such as atoms, quantum dots, or vacancy centers---can be solved directly without application of the second quantization formalism. In the developed approach, the Hamiltonian is expressed through the ket-bra products of collective (photon+cavities+QEs\mathrm{photon}+\mathrm{cavities}+\mathrm{QEs}photon+cavities+QEs) states. Consequently, the SSS matrix of input-output problems is determined exactly by the Mahaux-Weidenm\"uller formula, which dramatically simplifies the analysis of complex cavity QED systems. First, this approach is illustrated for the problem of propagation of a photon resonantly interacting with NNN two-level QEs arbitrary distributed inside the optical cavity. Solution of this problem manifests the effect of cumulative action of QEs previously known for special cases. Can a similar cumulative action of QEs enhance the inelastic resonant transmission of a single photon? We solve this problem for the case of an optical cavity having two modes resonantly coupled to electronic transitions of NNN three-level QEs. It is shown that the described structure is the simplest realistic structure which enables the down-conversion of the single-photon frequency with the amplitude approaching unity in the absence of the external driving field and sufficiently small cavity losses and QE dissipation. Overall, the simplicity and generality of the developed approach suggest a practical way to identify and describe new phenomena in cavity QED.
Light-matter coupled Hamiltonians are central to cavity materials engineering and polaritonic chemistry, but are challenging to simulate with classical hardware due to the scaling of the Hilbert space with the number of quantum photon modes and matter complexity. Leveraging the fact that quantum computers naturally represent photonic modes efficiently, we present a novel approach to simulate quantum-electrodynamical (QED) systems on near-term quantum hardware. After developing the bosonic and mixed operators in the Qiskit Nature framework, we employ them to simulate a first-order Trotterized Hamiltonian for a spontaneous-emission problem of a two-level system in an optical cavity. We find that using a standing-waves photonic basis approach leads to fidelity issues due to hardware connectivity constraints and two-qubits gates errors. Hence, we propose using a localized photonic basis approach that enforces nearest-neighbor couplings, thanks to which we can map the Hamiltonian as a 1D qubit chain. We significantly reduce the noise and, by applying the zero-noise extrapolation error mitigation technique, we recover the accurate quantum dynamics. Finally, we also show that this approach is resilient when relaxing the 1D qubit chain approximation.
Cavity qed is the domain of quantum optics which studies the behaviour of single atoms and photons coupled together in an electromagnetic resonator. During the last ten years, various spectacular effects have been observed on such systems, either on the atom’s or on the field’s properties. The presence of a cavity around an atom can produce an alteration of the atomic radiative rates which has been observed in several experiments. Ideally, cavity qed deals with two-level atoms coupled resonantly or nearly resonantly to a single cavity mode. This is of course a simplification of real situations, in which atoms have many levels and the cavity sustains more than one mode. The amplitudes associated to the upper and lower atomic states oscillate as a function of time, revealing the ‘vacuum Rabi oscillation’ effect. The amplitudes associated to the upper and lower atomic states oscillate as a function of time, revealing the ‘vacuum Rabi oscillation’ effect.