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Physical Review A

Space-time descriptions of quantum fields interacting with optical cavities

Julio R. Gea-Banacloche

Published Feb 22, 2013• 20 citations
Publisher

Abstract

The purpose of this paper is to compare two unitary, deterministic approaches to the study of the interaction of quantized fields with atomic systems in optical cavities. In particular it is shown that the ``modes of the universe'' approach, in which each mode has both an intracavity and an outside part, is formally equivalent, in an appropriate limit, to the better known ``input-output'' formalism in its unitary (i.e., Hermitian-Hamiltonian based) form; the latter may be called a ``quasimode'' theory, since it treats the field inside the cavity as a separate degree of freedom. Differences that arise between the two approaches for numerical calculations involving a finite set of discrete modes are also pointed out. The formalism is illustrated with an in-depth discussion of the explicit solution for a single-photon pulse, of arbitrary shape and detuning, incident on a cavity containing a two-level atom.

DOI
10.1103/physreva.87.023832

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The purpose of this paper is to compare two unitary, deterministic approaches to the study of the interaction of quantized fields with atomic systems in optical cavities. In particular it is shown that the modes of the universe'' approach, in which each mode has both an intracavity and an outside part, is formally equivalent, in an appropriate limit, to the better known input-output'' formalism in its unitary (i.e., Hermitian-Hamiltonian based) form; the latter may be called a ``quasimode'' theory, since it treats the field inside the cavity as a separate degree of freedom. Differences that arise between the two approaches for numerical calculations involving a finite set of discrete modes are also pointed out. The formalism is illustrated with an in-depth discussion of the explicit solution for a single-photon pulse, of arbitrary shape and detuning, incident on a cavity containing a two-level atom.
In this paper, we present a very simple approach to input-output relations in optical cavities, limiting ourselves to one- and two-photon states of the field. After field quantization, we derive the nonunitary transformation between inside and output annihilation and creation operators. Then we express the most general two-photon state generated by inside creation operators, through base states generated by output creation operators. After renormalization of coefficients of the inside two-photon state, we calculate the outside photon-number probability distribution in a general case; then we treat with some detail the single-mode and symmetrical cavity case. We found that the statistics of emitted photons may exhibit either quantum or classical behavior, depending both on source properties and on cavity characteristics.
This paper deals with non-Markovian behavior in atomic systems coupled to a structured reservoir of quantum electromagnetic field modes, with particular relevance to atoms interacting with the field in high-Q cavities or photonic band-gap materials. In cases such as the former, we show that the pseudomode theory for single-quantum reservoir excitations can be obtained by applying the Fano diagonalization method to a system in which the atomic transitions are coupled to a discrete set of (cavity) quasimodes, which in turn are coupled to a continuum set of (external) quasimodes with slowly varying coupling constants and continuum mode density. Each pseudomode can be identified with a discrete quasimode, which gives structure to the actual reservoir of true modes via the expressions for the equivalent atom-true mode coupling constants. The quasimode theory enables cases of multiple excitation of the reservoir to now be treated via Markovian master equations for the atom-discrete quasimode system. Applications of the theory to one, two, and many discrete quasimodes are made. For a simple photonic band-gap model, where the reservoir structure is associated with the true mode density rather than the coupling constants, the single quantum excitation case appears to be equivalent to a case with two discrete quasimodes.
We develop quantum-optical input-output theory for resonators with arbitrary coupling strength, and for input fields whose spectrum can be wider than the cavity free-spectral range, while ensuring that the field-operator commutator relations in space-time variables are correct. The cavity-field commutator exhibits a series of space-time ``echoes,'' representing causal connections of certain space-time points by light propagation. We apply the theory to two-photon wave-packet shaping by cavity reflection, which displays a remarkable illustration of dispersion cancellation. We also show that the theory is amenable to inclusion of intracavity absorbing and emitting atoms, allowing, for example, dissipative losses within the cavity to be incorporated in a quantum mechanically correct way.
This paper presents a general master equation formalism for the interaction between traveling pulses of quantum radiation and localized quantum systems. Traveling fields populate a continuum of free-space radiation modes and the Jaynes-Cummings model, valid for a discrete eigenmode of a cavity, does not apply. We develop a complete input-output theory to describe the driving of quantum systems by arbitrary incident pulses of radiation and the quantum state of the field emitted into any desired outgoing temporal mode. Our theory is applicable to the transformation and interaction of pulses of radiation by their coupling to a wide class of material quantum systems. We discuss the most essential differences between quantum interactions with pulses and with discrete radiative eigenmodes and present examples relevant to quantum information protocols with optical, microwave, and acoustic waves.
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Non-Markovian behaviour in atomic systems coupled to a structured reservoir of quantum EM field modes, such as in high Q cavities, is treated using a quasimode description, and the pseudo mode theory for single quantum reservoir excitations is obtained via Fano diagonalisation. The atomic transitions are coupled to a discrete set of (cavity) quasimodes, which are also coupled to a continuum set of (external) quasimodes with slowly varying coupling constants. Each pseudomode corresponds to a cavity quasimode, and the original reservoir structure is obtained in expressions for the equivalent atom-true mode coupling constants. Cases of multiple excitation of the reservoir are now treatable via Markovian master equations for the atom-discrete quasimode system.