Circuit Modularization of Quantum Optical Systems

Yaakov Lumer and Nader Engheta

Phys. Rev. Applied 14, 054034 – Published 16 November 2020

Abstract

Quantum optics and nanophotonics are the basis of future communications and computation technologies. Thus, there is an ever-increasing need for analysis and design tools that are suitable for large, complex, and multielement systems. However, full quantum analysis of systems requires writing the full quantum Hamiltonian, which is usually possible only in simple cases that are easy to analyze analytically, and especially difficult when the system contains dispersive and dissipative elements. Here we use circuit theory to modularize the modeling of the quantum Hamiltonian, and therefore the entire quantum system. We use the notion of optical metatronics—circuit analog for nanophotonic structures—to model our photonic system as an electronic circuit, and show how each coupling term in the Hamiltonian can be written using classical equivalent circuits. Thus, the quantum-interaction term between any two elements can be analyzed, designed, and optimized using the vast library of circuit engineering tools, which by their very nature enable the design of complex, multielement realistic systems. Our work generalizes the current theoretical treatment of quantized nanophotonic systems, paving the way for a systemic, modularized modeling of large-scale systems.

Received 4 September 2019
Revised 1 September 2020
Accepted 14 September 2020

DOI:https://doi.org/10.1103/PhysRevApplied.14.054034

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Integrated optics Metamaterials Nanophotonics Optical quantum information processing Plasmonics Quantum description of light-matter interaction

Devices

Atomic, Molecular & Optical

Authors & Affiliations

Yaakov Lumer^†,‡ and Nader Engheta ^*

Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

^*engheta@ee.upenn.edu
^†lumer@campus.technion.ac.il
^‡Present address: Department of Physics, Technion—Israel Institute of Technology, Haifa, Israel.

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Vol. 14, Iss. 5 — November 2020

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Figure 1
General principle of modular construction of quantum Hamiltonians. A system of K quantum emitters (marked as two-level systems), connected to each other through a network of transmission lines (shaded blue lines) and to N dissipative loads (colored clouds). Dark green circles represent filters, which can be dissipative or nondissipative. Dashed lines and clouds represent additional elements, which can be arbitrarily many. (Insets) The full quantum Hamiltonian can be written as a sum of $K \cdot N$ equivalent circuits, each circuit describes the connection of a specific quantum emitter (modeled as a source) to a specific dissipative load. The equivalent circuit is the Thevenin or Norton equivalent circuit, as seen from the load and including only the source in question. The rest of the network (transmission lines, nondissipative and dissipative loads) is lumped together in the effective Norton (or Thevenin) sources and internal impedances.
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Figure 2
A quantum emitter connected via a transmission line or waveguide to a dissipative load. (a) A quantum emitter, with transition dipole moment d coupled to a semi-infinite transmission line (blue) that ends in a dissipative load (orange). (b) In order to optimize the coupling of the quantum emitter to the load, we design and place a nanofilter (blue and green) between them. (c) Classical circuit model of the system presented in (a). The quantum emitter is modeled by a classical current source. The load is modeled by a RLC circuit (orange) (generalization to arbitrary dispersions is possible). The filter, assumed to be spatially small, is modeled as a shunt circuit (blue and green). Bottom panel: Norton equivalents of the circuit, as seen from both the load (left) and the semi-infinite end (“environment”). The number of equivalent circuits is the number of sources (one in our case) times the number of loads (two in our case). The full quantum Hamiltonian is then written as a sum of the contribution from each equivalent circuit.
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Figure 3
Full-wave numerical simulation of the coupling of an emitter to a load. For simplicity, the simulations are two dimensional. (a) Sketch of the system. The transmission line is assumed to be made of two conducting parallel plates, with distance $a = 1 (μ m)$ and is filled with air. The transmission line extends to infinitely on the right, and is terminated by a lossy cavity on the left. The cavity’s shape is a circle (in our 2D simulation “a cylinder”) with radius $R = 1 (μ m)$ , and is filled with a dielectric with relative permittivity $ϵ = 4 + i 0.02$ . The emitter is designated by a white star, and by a current source symbol in the circuit diagram. Bottom panel: the circuit model describing our transmission line. (b) The resonant cavity’s impedance versus frequency as calculated in a separate simulation. Two resonances are in our chosen frequency range. We focus on the resonance at $f = 84 (THz)$ . (c),(d) Dissipated power into the load (in units of W/m) versus frequency, with and without a filter designed to optimize the coupling. In (c) the filter is a simple dielectric slab, modeled as a capacitive shunt element. In (d), the filter is a small plasmonic sphere (in our 2D simulation “cylinder”), modeled as a series LC circuit. In both cases, the filter’s parameters and placement are found using circuit-theory tools, and then designed using the optical metatronics concept of lumped electromagnetic elements in optical frequencies. Solid blue line, simulated dissipated power in the absence of the filter; circles, simulated dissipated power with the optimal filter present; solid redline, dissipated power, as calculated using circuit theory where the load impedance and the filter impedance are extracted in separate simulations.
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Figure 4
Temporal dynamics. Temporal dynamics for the spontaneous decay of a QE in the system presented in Fig. 3. (a) Temporal dynamics for the case without a filter, for various values of the resonant frequency of the QE, $Ω$ . Fast decay times relate to either strong coupling to the load, or reflection from the load with constructive interference. The slow decay times correspond to cases with high reflection from the load under the condition of destructive interference. (b) Adding a filter to optimize coupling at frequency of $Ω / 2 π = 84$ (THz). It can be seen that at the chosen frequency the decay time is the shortest. (c) Plots of specific cases from (a) and (b). The magenta line and green line correspond to the respective cases in (b) and (c). The blue curve corresponds to the case where the filter is designed in a nonoptimized way [in our case, shifted in position by 400 (nm)].
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Physical Review Applied