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International Journal of Circuit Theory and Applications

Equivalent circuits for two‐state quantum systems

Pier Paolo Civalleri, Marco Gilli, Michele Bonnin

Published Apr 12, 2007• 9 citations
Publisher

Abstract

Abstract The development of nanocircuits designed to ultimately interact with classical electromagnetic environments demands a description of their performance in circuit terms, i.e. in terms of voltages and currents for incident, reflected and transmitted waves. A class of quantum systems which are particularly promising for applications (e.g. in spintronics) and at the same time easy to analyse and to understand physically are the two‐state systems. Here, we present equivalent circuits for a generic two‐state quantum system imbedded in a one‐port or two‐port structure guiding a linearly polarized electromagnetic wave. For the sake of concreteness we explicitly refer to a system of spin ½ particles whose magnetic moments are aligned with the axis of the guiding structure by a strong constant magnetic field; but the formalism can be easily extended to particles endowed with electric dipole moments or any other system with two degrees of freedom. Copyright © 2007 John Wiley & Sons, Ltd.

DOI
10.1002/cta.408

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Nanocircuits are composed of nanoelements interacting with external electromagnetic fields which are ultimately described in terms of voltages and currents or incident and reflected waves. Hence the necessity of describing their quantum performance in terms of circuit quantities. We present equivalent circuits for the small signal performance of a population of non-interacting spin 1/2 particles about passive and active equilibria induced by external pumping between the ground and the excited state, that describe respectively e.g. nuclear magnetic resonance and reflection amplification.
Abstract A quantum system composed of a large number of non‐interacting spin ½ particles, biased by a large constant magnetic field along the z‐axis and excited by a small circularly polarized electromagnetic wave in the xy‐plane is considered. Its small signal reflectance about the equilibrium state resulting from the biasing field and a pumping activity is calculated in the complex frequency plane. It is suggested that the equilibrium state can be locally passive or active depending on the pumping sign and level. Equivalent circuits containing imaginary resistors, i.e. frequency independent reactances, are presented for both passive and active cases. Copyright © 2006 John Wiley & Sons, Ltd.
A quantum system composed of a large number of non-interacting spin ½ particles, biased by a large constant magnetic field along the z-axis and excited by a small circularly polarized electromagnetic wave in the xy-plane is considered. Its small signal reflectance about the equilibrium state resulting from the biasing field and a pumping activity is calculated in the complex frequency plane. It is suggested that the equilibrium state can be locally passive or active depending on the pumping sign and level. Equivalent circuits containing imaginary resistors, i.e. frequency independent reactances, are presented for both passive and active cases. Copyright © 2006 John Wiley & Sons, Ltd.
Summary In this paper, we introduce small signal equivalent circuit models for quantum dots at equilibrium driven by classical electromagnetic fields. We investigate both an isolated quantum dot and a pair of interacting quantum dots coupled by the electric dipole–dipole interaction. In particular, we derive the impulse responses of the equivalent circuit elements from the density operator of the quantum dots. The use of equivalent circuit elements for complex systems of quantum dots may greatly simplify the analysis and the design of light‐matter interaction processes at the nanoscale. Copyright © 2017 John Wiley & Sons, Ltd.
Abstract Physics is not scale invariant, and today the scale of atoms and molecules challenges designers of machines in which quantum effects have dominant sway. What role could circuit theory play in designing machines described by quantum‐classical models? Classical equivalent circuits do exist for systems composed of metal contacted and wired devices, such as resonant tunneling diodes, single electron transistors, metal–insulator–metal diodes, etc. circuits, but not for quantum‐entangled networks, such as multi‐quantum‐state atoms. If devices were not contacted and wired by macroscopic metals, i.e. devices were classically field coupled, then generalized circuit models can be introduced. Case studies have been presented on the role of circuit models in quantum‐classical systems. However, there are no ideal circuit elements capable of capturing the port properties of quantum‐mechanical and/or quantum‐optical subsystems and their coupling to classical waveguides or cavities. Copyright © 2007 John Wiley & Sons, Ltd.
A systematic description of a spin one-half system endowed with magnetic moment or any other two-level system (qubit) interacting with the quantized electromagnetic field is developed. This description exploits a close analogy between a two-level system and the Dirac electron that comes to light when the two-level system is described within the formalism of second quantization in terms of fermionic creation and annihilation operators. The analogy enables one to introduce all the powerful tools of relativistic QED (albeit in a greatly simplified form). The Feynman diagrams and the propagators turn out to be very useful. In particular, the QED concept of the vacuum polarization finds its close counterpart in the photon scattering off a two level system leading via the linear response theory to the general formulas for the atomic polarizability and the dynamic single spin susceptibility. To illustrate the usefulness of these methods, we calculate the polarizability and susceptibility up to the fourth order of perturbation theory. These ab initio calculations resolve some ambiguities concerning the sign prescription and the optical damping that arise in the phenomenological treatment. We also show that the methods used to study two-level systems (qubits) can be extended to many-level systems (qudits). As an example, we describe the interaction with the quantized electromagnetic field of an atom with four relevant states: one SSS state and three degenerate PPP states.