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Physical Review A

Correspondence rules in macroscopic quantum electrodynamics

Lev I. Plimak

Published Sep 1, 1994• 24 citations
Publisher

Abstract

It is shown that provided two devices are distinguishable, their electromagnetic interactions are in essence classical. This results in the existence of simple correspondence rules connecting q- and c-number statistical descriptions of macroscopic optical arrangements. These rules allow formulas found by c-number techniques to be used as precise quantum relations. For the vast majority of real devices, including all linear devices and real light detectors regardless of their internal system, their c-number macroscopic description can be directly employed in QED. The problem of locality of QED is discussed and a rigorous theory of electrical-feedback arrangements in optics is presented.

DOI
10.1103/physreva.50.2120

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It is shown that provided two devices are distinguishable, their electromagnetic interactions are in essence classical. This results in the existence of simple correspondence rules connecting q- and c-number statistical descriptions of macroscopic optical arrangements. These rules allow formulas found by c-number techniques to be used as precise quantum relations. For the vast majority of real devices, including all linear devices and real light detectors regardless of their internal system, their c-number macroscopic description can be directly employed in QED. The problem of locality of QED is discussed and a rigorous theory of electrical-feedback arrangements in optics is presented.
A rigorous QED theory of electrical-feedback arrangements in optics, based on the Feynman path-integral approach, is presented. It is shown that a feedback arrangement by itself is not capable of producing quantum states of light. For this state of light to appear in the operating loop, the latter should contain at least one device emitting light in a quantum state without feedback. A QED discussion of the Yamamoto-type arrangement with a source of light in a quantum state shows that the conventional corpuscular picture of light fails to describe the properties of the field emitted under the influence of feedback. A possibility of an experimental verification of this conclusion is discussed.
Thus far, we have developed the properties of the electromagnetic field at optical frequencies, based on Maxwell’s equations. These equations further give a classical macroscopic perspective on the coupling of the propagation media to the field, as presented in Chapter 2. The macroscopic properties of a medium are based on averaged microscopic properties. The microscopic energy structure of matter was presented in Chapter 3, covering gases, solids, and liquids by employing mostly quantum models. We now proceed to the next level of development, the dynamic description of the interaction between the optical field and the propagation medium as a function of the field frequency and propagation media variables (e.g., energy structure, temperature, and pressure). In this chapter, the classical electromagnetic field is coupled to discrete frequency oscillators via Newton’s equation of motion. This approach leads to the popular classical oscillator model, often presented in introductory books on lasers. The classical oscillator model is an incomplete theory and can be only a semiempirical model. In the next chapter, a more detailed and comprehensive approach, which also includes statistical and quantum mechanics, is used leading to robust semiclassical and quantum oscillator models. This chapter and the next are the basis for the applied models presented in Part II of this book. Classical electrodynamics is based on Maxwell’s equations, as given in Chapter 2, and the Lorentz force relation, as given below: . . . F = q[E + (v × B)]. (4.1) . . . These equations cover the classical description of the interaction of light and matter. The first term in Eq. 4.1 represents coupling of the electric field to the medium. As discussed in Chapter 2 (Section 2.2), the leading mechanism for this is the electric dipole moment. To see that this is the coupling mechanism in the first term, consider the potential function driving this force, F = −∇V(r) = −∇(−qr · E). The above expression contains the dipole moment, as defined in Chapter 2. The second term in Eq. 4.1 represents coupling of the magnetic field to the medium.
We extend previous work on macroscopic canonical quantization leading to a multipolar Hamiltonian appropriate for application to quantum optics and cavity QED situations involving classical optical devices. In particular, we show that the electric displacement is the negative of the conjugate momentum field and that the Coulomb and polarization energies are equal to the sum of intra-atomic Coulomb and polarization energies and interatomic contact energies. The quantum Hamiltonian is now in a form in which the theory is manifestly gauge invariant.
Devices built using superconducting circuits are a popular approach being pursued to develop quantum information processing hardware. These systems involve the quantized interactions of electromagnetic fields with large coherent collections of charges in superconductors, and so are typically referred to as macroscopic circuit quantum electrodynamics (QED) devices or simply circuit QED devices. Although significant progress has been made recently, substantial engineering efforts are required to improve performance and scale these devices so they can address problems of practical interest. Performing these engineering efforts is difficult due to the absence of rigorous numerical modeling approaches. This work begins to address this by providing a new mathematical description of a commonly used circuit QED system, a transmon qubit coupled to microwave transmission lines. Expressed in terms of three-dimensional vector fields, our model is better suited to developing rigorous numerical solvers than approximate lumped element circuit descriptions used in the literature. We present details on the quantization of our model, and derive equations of motion for the coupled field-transmon system. These results can be used in developing full-wave solvers in the future. To make this work more accessible, we assume a limited amount of training in quantum physics and provide many background details.
The Born rule provides a fundamental connection between theory and observation in quantum mechanics, yet its origin remains a mystery. We consider this problem within the context of quantum optics using only classical physics and the assumption of a quantum electrodynamic vacuum that is real rather than virtual. The connection to observation is made via classical intensity threshold detectors that are used as a simple, deterministic model of photon detection. By following standard experimental conventions of data analysis on discrete detection events, we show that this model is capable of reproducing several observed phenomena thought to be uniquely quantum in nature, thus providing greater elucidation of the quantum-classical boundary.
The macroscopic quantum theory of the electromagnetic field in a dielectric medium interacting with a dense collection of embedded two-level atoms fails to reproduce a result that is obtained from an application of the classical Lorentz local-field condition. Specifically, macroscopic quantum electrodynamics predicts that the Lorentz redshift of the resonance frequency of the atoms will be enhanced by a factor of the refractive index nnn of the host medium. However, an enhancement factor of (n2+2)∕3({n}^{2}+2)∕3(n2+2)∕3 is derived using the Bloembergen procedure in which the classical Lorentz local-field condition is applied to the optical Bloch equations. Both derivations are short and uncomplicated and are based on well-established physical theories, yet lead to contradictory results. Microscopic quantum electrodynamics confirms the classical local-field-based results. Then the application of macroscopic quantum electrodynamic theory to embedded atoms is proved false by a specific example in which both the correspondence principle and microscopic theory of quantum electrodynamics are violated.