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Physical Review A

Unifying the p⋅Aand d⋅Einteractions in photodetector theory

P. D. Drummond

Published May 1, 1987• 68 citations
Publisher

Abstract

A new technique is introduced in which a canonical transformation is used to transform the atom-radiation Hamiltonian to one without counter-rotating terms. As well as allowing an exact treatment of a one-mode dipole-coupled photodetection model, the theory predicts the existence of squeezing induced by the detector coupling. A feature of the canonical transformation used here is that the resulting interaction Hamiltonian has a symmetric combination of the p\ensuremath{\cdot}A and d\ensuremath{\cdot}E interactions. This allows the resolution of long-standing questions about the correct interaction Hamiltonian to use in photodetector theory. The new theory is able to handle dipole-coupled radiative transitions over ultrafast time scales of the order of a single oscillation period, where the rotating-wave approximation is invalid. The coupling of the detector to the radiation field is predicted to vary as ${f}^{\mathrm{\ensuremath{-}}1/2}$ at high frequencies, and as ${f}^{+1/2}$ at low frequencies.

DOI
10.1103/physreva.35.4253

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A new technique is introduced in which a canonical transformation is used to transform the atom-radiation Hamiltonian to one without counter-rotating terms. As well as allowing an exact treatment of a one-mode dipole-coupled photodetection model, the theory predicts the existence of squeezing induced by the detector coupling. A feature of the canonical transformation used here is that the resulting interaction Hamiltonian has a symmetric combination of the p\ensuremath{\cdot}A and d\ensuremath{\cdot}E interactions. This allows the resolution of long-standing questions about the correct interaction Hamiltonian to use in photodetector theory. The new theory is able to handle dipole-coupled radiative transitions over ultrafast time scales of the order of a single oscillation period, where the rotating-wave approximation is invalid. The coupling of the detector to the radiation field is predicted to vary as f\ensuremath−1/2{f}^{\mathrm{\ensuremath{-}}1/2}f\ensuremath−1/2 at high frequencies, and as f+1/2{f}^{+1/2}f+1/2 at low frequencies.
In this article, by using the perturbation theory, we analytically solve the eigenvalue problem for the Hamiltonian describing the interaction of a Λ -type three-level atom with a single-mode radiation field without the rotating wave approximation (RWA). For this purpose, the atom–field interaction Hamiltonian, which contains the counter-rotating terms (CRTs), is transformed to an analytically solvable Hamiltonian by applying three successive unitary transformations. According to our calculations, the contribution of CRTs within the transformed Hamiltonian is in fact replaced by transforming the 'constant detuning' with the 'intensity-dependent detuning' in the first order, and the 'constant atom–field coupling' with the intensity-dependent coupling in the second order of the perturbation parameters. Then, by solving the eigenvalue problem for the transformed Hamiltonian, the eigenvector of the considered atom–field Hamiltonian is obtained analytically. Finally, after achieving the state vector of the atom–field system at an arbitrary time, a few nonclassical properties of the system state are investigated numerically. Meanwhile, we compare our results with the presence of RWA, from which the role of CRTs will be established.
The interaction of an atomic system with a single mode of a quantized electromagnetic field is investigated in the dipole approximation. Two approximations that are almost always made in this analysis are the neglect of the diamagnetic term in the Hamiltonian and the dropping of the rotating-wave approximation (RWA) terms that correspond to processes that do not conserve energy. Both approximations involve the neglect of terms that are of the same order of magnitude so that consistency demands that they be considered together. It is demonstrated that, in the dipole approximation, a minor modification of the usual application of quantum electrodynamics to any atomic system can exactly incorporate the effects of the diamagnetic term into a new frequency of the electromagnetic cavity mode; this frequency is slightly different from that of the empty cavity. This result is then applied to the specific case of a charged harmonic oscillator interacting with a single mode. For this simple model an analytic solution can then be found that includes the effects of the diamagnetic term and does not use the RWA. It is found that, for off resonance, the normal modes of the coupled system oscillate at shifted frequencies which are in agreement with the second-order perturbation-theory calculation of the Lamb shift. When the cavity mode is tuned to the atomic oscillator frequency, the perturbation-theory Lamb-shift calculation is no longer valid, and the exact solution derived in this paper predicts a resonant Lamb shift that is significantly larger than the off-resonant single-mode Lamb shift. Furthermore, the frequency spectrum is qualitatively changed from the single frequency of the free atomic system to a doublet spectrum. An analytic solution of this harmonic-oscillator problem can also be found when the RWA terms are dropped from the Hamiltonian. Comparison of the two solutions reveals that the RWA terms contribute a frequency shift of magnitude independent of the field intensity in the simple case of a charged harmonic oscillator.When trajectories of the exact and RWA solutions are compared in phase space, it is found that one effect of the RWA terms is to lift a twofold degeneracy of the phase-space orbits that occurs when the RWA terms are neglected. The harmonic-oscillator problem has several features in common with the interaction of a two-level atom with a single quantized field mode. Both Hamiltonians exhibit inversion symmetry and hence have parity as a good quantum number. Both Hamiltonians contain RWA terms. When these RWA terms are neglected, both Hamiltonians have a new constant of motion that can be identified with the total number of excitations in the combined system.
A theory of single-photon ionization, detachment, and dissociation of quantum systems (atom, molecule, ion, etc) is presented. A unified treatment of excitation from the ground and excited states is performed, via a consideration of the rotating- and counter-rotating-wave terms on equal footing. The familiar pole approximation (PA) is extended to obtain higher-order corrections, which allows one to establish the PA validity conditions and provide a unified description of the cases of above- and below-threshold excitation. As a specific example, photodetachment of negative ions is considered. An exact long-time solution is obtained for the general case. A familiar model of photodetachment is used to validate the present PA formalism. The latter has allowed us to derive the PA validity condition, which holds for all values of the parameters, involving quite different situations. The applicability of the existing versions of the rotating-wave approximation has been checked.