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Journal of Physics B Atomic and Molecular Physics

The rotating-wave approximation in bound-free transitions

Juha Javanainen

Published Apr 28, 1983• 26 citations
Publisher

Abstract

The author solves the Schrodinger equation in terms of continued fractions for a system exhibiting light-induced bound-free transitions over an energy gap, without making the rotating-wave approximation. He describes the evolution of the bound state and the photoelectron spectrum by investigating the analytic properties of the Fourier transforms of the probability amplitudes. The main consequence of non-RWA transitions is a Bloch-Siegert shift of the continuum threshold, which may be comparable to RWA dynamic level shifts. At high intensities the counter-rotating wave also influences the evolution of the bound state, and gives rise to multiphoton transitions competing with RWA multiphoton transitions with an odd number of photons, but these effects are probably hard to isolate experimentally.

DOI
10.1088/0022-3700/16/8/011

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The author solves the Schrodinger equation in terms of continued fractions for a system exhibiting light-induced bound-free transitions over an energy gap, without making the rotating-wave approximation. He describes the evolution of the bound state and the photoelectron spectrum by investigating the analytic properties of the Fourier transforms of the probability amplitudes. The main consequence of non-RWA transitions is a Bloch-Siegert shift of the continuum threshold, which may be comparable to RWA dynamic level shifts. At high intensities the counter-rotating wave also influences the evolution of the bound state, and gives rise to multiphoton transitions competing with RWA multiphoton transitions with an odd number of photons, but these effects are probably hard to isolate experimentally.
Abstract We apply a simple analytical method based on a unitary transformation to calculate the ground state, its excitation spectrum and quantum dynamic evolution of physical quantities for the double-photon quantum Rabi Hamiltonian over the wide coupling-strength range. The concise analytical method possesses the same mathematical simplicity as the approach of the rotating wave approximation (RWA). By quantitative comparison with the numerically exact result obtained by matrix diagonalization, we confirm that our calculated results obtained by transformed rotating-wave method are not only accurate in the weak coupling regime but also correct in intermediate strong-coupling case. In the intermediate ultrastrong-coupling regime, the calculated values of the ground state and lower lying excited states are nearly the same as the exact ones. It turns out that our calculation for the energy spectrum is beyond the ordinary-RWA. Meanwhile, we demonstrate the signatures resulting from the counter-rotating wave terms by monitoring the population, the coherence, the squeezing of the photon under the ultra-strong conditions. In particular, we find that when the frequency of the photon is much larger than the transition frequency of the system, the lineshape of the time evolution becomes complicated with the increase of the coupling strength, which may be verified experimentally.
The interaction between light and matter is one of the oldest research areas of quantum mechanics, and a field that just keeps on delivering new insights and applications. With the arrival of cavity and circuit quantum electrodynamics we can now achieve strong light-matter couplings which form the basis of most implementations of quantum technology. But quantum information processing also has high demands requiring total error rates of fractions of percentage in order to be scalable (fault-tolerant) to useful applications. Since errors can also arise from modelling, this has brought into center stage one of the key approximations of quantum theory, the Rotating Wave Approximation (RWA) of the quantum Rabi model, leading to the Jaynes-Cummings Hamiltonian. While the RWA is often very good and incredibly useful to understand light-matter interactions, there is also growing experimental evidence of regimes where it is a bad approximation. Here, we ask and answer a harder question: for which experimental parameters is the RWA, although perhaps qualitatively adequate, already not good enough to match the demands of scalable quantum technology? For example, when is the error at least, and when at most, 1%? To answer this, we develop rigorous non-perturbative bounds taming the RWA. We find that these bounds not only depend, as expected, on the ratio of the coupling strength and the oscillator frequency, but also on the average number of photons in the initial state. This confirms recent experiments on photon-dressed Bloch-Siegert shifts. We argue that with experiments reporting controllable cavity states with hundreds of photons and with quantum error correcting codes exploring more and more of Fock space, this state-dependency of the RWA is increasingly relevant for the field of quantum computation, and our results pave the way towards a better understanding of those experiments.
We consider the excitation of atomic hydrogen in its ground state by an intense radiation field tuned close to the ionization threshold. Results are presented for the ac Stark shift of the 1s and 2s states, the width of the 1s state, and the photoelectron energy spectrum. In the first instance, our nonperturbative approach includes the 1s state, the p series of excited states, and the p continuum: we demonstrate that the p Rydberg states generate a rich structure in the photoelectron energy spectrum through the action of the counter-rotating wave dynamics when the laser frequency is tuned below the ionization threshold. We then go on to analyze in detail how the interaction of the p states with the other Rydberg series and continua increases the width of the p states; it is shown, however, that at the field intensities considered here (\ensuremath{\approxeq}1013{10}^{13}1013 W ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}2}$), this effect is not sufficient to smear out the structure in the photoelectron energy spectrum.
Recalling that the rotating wave approximation (RWA) is only valid in the weak coupling regimes, the purpose of this paper is to study the Hamiltonian dynamics describing the full quantum mechanical approach of the interaction between various configurations of three-level atoms Lambda, V and ladder distinctly with a two-mode radiation field, while the RWA is not considered; the counter-rotating terms (CRTs) are taken into account. Generally, the presence of CRTs in the Hamiltonian prevents one to achieve an analytical solution. Moreover, as we will show in the present work, using the perturbation theory, analytical solvable Hamiltonians can be successfully obtained. According to our calculations, the contribution of CRTs within the ordinary Hamiltonian is equivalent to the replacement of the constant detuning with a specific intensity dependent detuning in the first order, and the constant atom field coupling with a particular intensity dependent (f deformed) coupling in the second order of the associated perturbation parameter. Moreover, noticing that according to the initial expression of the Hamiltonian, each mode of the field interacts only with a specific pair of the allowed transitions of each type of the three-level atom, it is surprisingly observed that via applying the mentioned approach, the obtained intensity dependent coupling functions depend on both modes of the field, f(na,nb). In this way, it is seen that the CRTs are removed with the price of arriving at some intermixed intensity dependent atom field coupling functions of the two modes of the field. In this way, the obtained final Hamiltonians are analytically solvable. At last, by determining the time evolution of the atom field wave function, we study the effects of CRTs on a few nonclassical properties of the state of the system, including the atomic population inversion and photon statistics.