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Physical Review A

Quantum electrodynamics of source field detection and quantum beats

I. C. Khoo, Joseph H. Eberly

Published Nov 1, 1978• 6 citations
Publisher

Abstract

The quantum electrodynamics of two atoms in free space, separated by many wavelengths and interacting with one another via the coupling of their dipole moments to the electromagnetic field, is studied using the Heisenberg-operator approach. The problem is formulated in such a way that one of the atoms, which has two levels, plays the role of a detector; the other atom, whose levels are arbitrary but do not involve cascadelike transitions, plays the role of the source. Several aspects of the interaction are considered, including: a review of photodetection theory, the theory of an alternative detection method that uses an excited-state detector, and a study of quantum beats. These dynamical studies of emission and detection provide new insights into the observation of quantum beats in the transition rate of a detector. We show that ground-state beats are not observed by a detector atom making an upward transition, but that ground-state beats are, in principle, observable if the detector atom makes a downward transition.

DOI
10.1103/physreva.18.2184

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Rapid advances in quantum technology have made possible the control of quantum states of elementary material quantum systems, such as atoms or molecules, and of the electromagnetic radiation field resulting from spontaneous photon emission of their unstable excited states to such a level of precision that subtle quantum electrodynamical phenomena have become observable experimentally. Recent developments in the area of quantum information processing demonstrate that characteristic quantum electrodynamical effects can even be exploited for practical purposes provided the relevant electromagnetic field modes are controlled by appropriate cavities. A central problem in this context is the realization of an ideal transfer of quantum information between a state of a material quantum system and a quantum state of the electromagnetic radiation field which contains a small number or even a single photon only. Despite much recent work in cavity quantum electrodynamics, which has explored this problem successfully in cases in which the number of accessible electromagnetic field modes is strongly restricted by a cavity, currently still much less is known about ideal quantum state transfer between matter and few-photon quantum states of the electromagnetic field in the extreme opposite case of free space. In this contribution we review recent work which explores characteristic quantum electrodynamical phenomena governing the interaction of a material quantum system with the electromagnetic field in extreme many-mode cases which are close to cavity-free situations. For this purpose the simplest possible quantum electrodynamical situation, namely the interaction of a single material two-level system with the wave packet of a single optical photon, is investigated in free space and in situations in which a large cavity modifies the mode structure of the electromagnetic field modes. It is demonstrated that perfect excitation of an atom by a properly shaped single-photon wave packet is possible even in free space. In particular, it is demonstrated that perfect excitation of an atom can be achieved by a single optical photon which is prepared initially in an appropriate superposition of plane-wave modes, provided this photon is focused subsequently onto the absorbing atom by a parabolic cavity.

•Jan 1, 2010•4 citations
The quantum electrodynamics of two atoms in free space, separated by many wavelengths and interacting with one another via the coupling of their dipole moments to the electromagnetic field, is studied using the Heisenberg-operator approach. The problem is formulated in such a way that one of the atoms, which has two levels, plays the role of a detector; the other atom, whose levels are arbitrary but do not involve cascadelike transitions, plays the role of the source. Several aspects of the interaction are considered, including: a review of photodetection theory, the theory of an alternative detection method that uses an excited-state detector, and a study of quantum beats. These dynamical studies of emission and detection provide new insights into the observation of quantum beats in the transition rate of a detector. We show that ground-state beats are not observed by a detector atom making an upward transition, but that ground-state beats are, in principle, observable if the detector atom makes a downward transition.
Using the Heisenberg operator approach, we have calculated the emission rate of a multilevel atom whose transitions involve both upper- and lower-state splittings. The emission rate is first calculated by a frequently used power-flow expression and is shown to exhibit lower-state beats. This power-flow expression requires care in its interpretation because it does not account for the measurement process. We then derive an alternative, detection-oriented emission rate and show that lower-state beats are not observable in a photodetection experiment.
The cooperative spontaneous emission from three identical, initially excited two-level atoms separated by distances much smaller than the resonance wavelength is studied taking into account the dipole-dipole near-field interaction. Approximate expressions are obtained for the time evolution of the atomic system, the radiated intensity and the one-time intensity correlation function. As a new result the authors find that the near-field interaction may lead to the occurrence of beats in the intensity. Moreover, the destructive effect of the near-field interaction on the coherence properties of the emitted radiation is discussed.
Quantum beats in the radiation field of "three-level" atoms are analyzed by means of a previously developed boson-second-quantization (BSQ) formalism. Two types of atoms are considered: type I, in which beats are due to transitions from each of two closely spaced upper levels to a single lower level, and type II, in which beats are due to transitions between a single upper level and each of two closely spaced lower levels. This classification is motivated by the claim found in the literature that semiclassical theory and quantum theory give qualitatively different results concerning the existence of beats in the case of type-II atoms. The terminology associated with beats is made precise by distinguishing between amplitude modulation at the beat frequency of the transition frequency field EEE, and the existence of a spectral component at the beat frequency in E2{E}^{2}E2. With respect to the former, no qualitative difference is found in the predictions from semiclassical theory and quantum theory; with respect to the latter, results that appear paradoxical from an intuitive viewpoint are discussed. It is shown that for the case in which the atoms are all in the same coherent superposition of energy states, with all levels occupied, semiclassical and quantum-mechanical results become equivalent as the number of atoms increases. Reference is made to the disturbance caused by the measurement processes in the case of a single atom. The essential difference in spontaneous emission beats between type-I and type-II atoms [with the ground state(s) unoccupied] is explained by the fact that the phase of the beat amplitude may be determined by preparation of the initial state in type-I atoms but may not be determined in type-II atoms.
Quantum detection processes in quantum field theory (QFT) must play a key role in the description of quantum-field correlations, such as the appearance of entanglement, and of causal effects. We consider the detection in the case of a simple QFT model with a suitable interaction to exact treatment, consisting of a quantum scalar field coupled linearly to a classical scalar source. We then evaluate the response function to the field quanta of two-level pointlike quantum model detectors, and analyze the effects of the approximation adopted in standard detection theory. We show that the use of the RWA, which characterizes the Glauber detection model, leads in the detector response to nonlocal terms corresponding to an instantaneously spreading of source effects over the whole space. Other detector models, obtained with nonstandard or no application of RWA, give instead local responses to field quanta, apart from source-independent vacuum contribution linked to preexisting correlations of zero-point field.
A pair of nonidentical two-level atoms, separated by a fixed distance R, interact through photon exchange. The system is described by a state vector which is assumed to be a superposition of four "essential states": (1) the first atom is excited, the second one is in the ground state, and no photon is present, (2) the first atom is in its ground state, the second one is excited, and no photon is present, (3) both atoms are in their ground states and a photon is present, and (4) both atoms are excited and a photon is also present. The system is initially in state (1). The probabilities of each atom being excited are calculated for both the minimally-coupled interaction and the multipolar interaction in the electric dipole approximation. For the minimally-coupled interaction Hamiltonian, the second atom has a probability of being instantaneously excited, so the interaction is not retarded. For the multipolar interaction Hamiltonian, the second atom is not excited before the retardation time, which agrees with special relativity. For the minimally-coupled interaction the nonphysical result occurs because the unperturbed Hamiltonian is not the energy operator in the Coulomb gauge. For the multipolar Hamiltonian in the electric dipole approximation the unperturbed Hamiltonian is the energy operator. An active view of unitary transformations in nonrelativistic quantum electrodynamics is used to derive transformation laws for the potentials of the electromagnetic field and the static Coulomb potential. For a specific choice of unitary transformation the transformation laws for the potentials are used in the minimally-coupled second-quantized Hamiltonian to obtain the multipolar Hamiltonian, which is expressed in terms of the quantized electric and magnetic fields.
Rapid advances in quantum technology have made possible the control of quantum states of elementary material quantum systems, such as atoms or molecules, and of the electromagnetic radiation field resulting from spontaneous photon emission of their unstable excited states to such a level of precision that subtle quantum electrodynamical phenomena have become observable experimentally. Recent developments in the area of quantum information processing demonstrate that characteristic quantum electrodynamical effects can even be exploited for practical purposes provided the relevant electromagnetic field modes are controlled by appropriate cavities. A central problem in this context is the realization of an ideal transfer of quantum information between a state of a material quantum system and a quantum state of the electromagnetic radiation field which contains a small number or even a single photon only. Despite much recent work in cavity quantum electrodynamics, which has explored this problem successfully in cases in which the number of accessible electromagnetic field modes is strongly restricted by a cavity, currently still much less is known about ideal quantum state transfer between matter and few-photon quantum states of the electromagnetic field in the extreme opposite case of free space. In this contribution we review recent work which explores characteristic quantum electrodynamical phenomena governing the interaction of a material quantum system with the electromagnetic field in extreme many-mode cases which are close to cavity-free situations. For this purpose the simplest possible quantum electrodynamical situation, namely the interaction of a single material two-level system with the wave packet of a single optical photon, is investigated in free space and in situations in which a large cavity modifies the mode structure of the electromagnetic field modes. It is demonstrated that perfect excitation of an atom by a properly shaped single-photon wave packet is possible even in free space. In particular, it is demonstrated that perfect excitation of an atom can be achieved by a single optical photon which is prepared initially in an appropriate superposition of plane-wave modes, provided this photon is focused subsequently onto the absorbing atom by a parabolic cavity.