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arXiv:1001.3955 · Physical Review A

Separation of transitions with two quantum jumps from cascades

Ulrich D. Jentschura

Published Jan 20, 2010• 8 citations
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Abstract

We consider the general scenario of an excited level $|i\ensuremath{\rangle}$ of a quantum system that can decay via two channels: (i) via a single-quantum jump to an intermediate, resonant level $|\overline{m}\ensuremath{\rangle}$, followed by a second single-quantum jump to a final level $|f\ensuremath{\rangle}$, and (ii) via a two-quantum transition to a final level $|f\ensuremath{\rangle}$. Cascade processes $|i\ensuremath{\rangle}\ensuremath{\rightarrow}|\overline{m}\ensuremath{\rangle}\ensuremath{\rightarrow}|f\ensuremath{\rangle}$ and two-quantum transitions $|i\ensuremath{\rangle}\ensuremath{\rightarrow}|m\ensuremath{\rangle}\ensuremath{\rightarrow}|f\ensuremath{\rangle}$ compete (in the latter case, $|m\ensuremath{\rangle}$ can be both a nonresonant as well as a resonant level). General expressions are derived within second-order time-dependent perturbation theory, and the cascade contribution is identified. When the one-quantum decay rates of the virtual states are included into the complex resonance energies that enter the propagator denominator, it is found that the second-order decay rate contains the one-quantum decay rate of the initial state as a lower-order term. For atomic transitions, this implies that the differential-in-energy two-photon transition rate with complex resonance energies in the propagator denominators can be used to good accuracy even in the vicinity of resonance poles.

DOI
10.1103/physreva.81.012112

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We consider the general scenario of an excited level ∣i\ensuremath⟩|i\ensuremath{\rangle}∣i\ensuremath⟩ of a quantum system that can decay via two channels: (i) via a single-quantum jump to an intermediate, resonant level ∣m‾\ensuremath⟩|\overline{m}\ensuremath{\rangle}∣m\ensuremath⟩, followed by a second single-quantum jump to a final level ∣f\ensuremath⟩|f\ensuremath{\rangle}∣f\ensuremath⟩, and (ii) via a two-quantum transition to a final level ∣f\ensuremath⟩|f\ensuremath{\rangle}∣f\ensuremath⟩. Cascade processes ∣i\ensuremath⟩\ensuremath→∣m‾\ensuremath⟩\ensuremath→∣f\ensuremath⟩|i\ensuremath{\rangle}\ensuremath{\rightarrow}|\overline{m}\ensuremath{\rangle}\ensuremath{\rightarrow}|f\ensuremath{\rangle}∣i\ensuremath⟩\ensuremath→∣m\ensuremath⟩\ensuremath→∣f\ensuremath⟩ and two-quantum transitions ∣i\ensuremath⟩\ensuremath→∣m\ensuremath⟩\ensuremath→∣f\ensuremath⟩|i\ensuremath{\rangle}\ensuremath{\rightarrow}|m\ensuremath{\rangle}\ensuremath{\rightarrow}|f\ensuremath{\rangle}∣i\ensuremath⟩\ensuremath→∣m\ensuremath⟩\ensuremath→∣f\ensuremath⟩ compete (in the latter case, ∣m\ensuremath⟩|m\ensuremath{\rangle}∣m\ensuremath⟩ can be both a nonresonant as well as a resonant level). General expressions are derived within second-order time-dependent perturbation theory, and the cascade contribution is identified. When the one-quantum decay rates of the virtual states are included into the complex resonance energies that enter the propagator denominator, it is found that the second-order decay rate contains the one-quantum decay rate of the initial state as a lower-order term. For atomic transitions, this implies that the differential-in-energy two-photon transition rate with complex resonance energies in the propagator denominators can be used to good accuracy even in the vicinity of resonance poles.
The coherent interaction of two smoothly varying, near-resonant, two-photon pulses with a three-level system can be described by "two-photon damped Bloch equations" which are analogous to those for a one-photon transition in a two-level system except for the presence of a two-photon coupling and a frequency shift. These equations are solved for the cases ${\ensuremath{\gamma}}_{1}$, ${\ensuremath{\gamma}}_{2}\ensuremath{\ll}\ensuremath{\Omega}$, ${\ensuremath{\gamma}}_{1}={\ensuremath{\gamma}}_{2}$, and $\frac{{\ensuremath{\gamma}}_{2}{k}^{2}{\ensuremath{\epsilon}}^{4}}{{\ensuremath{\Omega}}^{2}}$, ${\ensuremath{\gamma}}_{1}\ensuremath{\ll}\ensuremath{\Omega}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase relaxation widths, respectively, and $\ensuremath{\Omega}$ is the Rabi frequency. The leading contribution to the refractive index is intensity dependent, caused by the level shifts inherent in multiphoton processes; it includes a relaxation dependent part which is important at times shorter than ${\ensuremath{\gamma}}_{1}^{\ensuremath{-}1}$. The second-order contributions depend on the square of the intensity and the time-integrated square of the intensity. The latter contribution, which is relaxation dependent, causes line asymmetry at the long-wavelength wing; it consists of a term proportional to ${\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$ and only important at early times and a term proportional to $2{\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$.
Using second-order, time-dependent perturbation theory, we calculate the intermediate and final state amplitudes for transitions from an initial to a final state via an intermediate resonance. When our complex energies are replaced by real energies our results are identical to the real energy final state amplitudes obtained by other authors, thus justifying the usual phenomenological replacement of real energies with complex ones when considering resonance behavior. We then derive the Fermi Golden Rule No. 1 with complex energies.
The theory of multiphoton cascade transitions with two-photon links is considered within two different approaches: quantum electrodynamical (QED) and phenomenological quantum mechanical (QM). A problem of regularization of the cascade contributions is investigated in detail. It is argued that the correct regularization should include both initial and intermediate level widths in the singular energy denominators. This result follows both from the QED and from the QM approach. Particular transitions nl → 1s + 2γ with nl = 3s, 4s, 3d, 4d and nl → 1s + 3γ with nl = 3p, 4p are considered as examples. The importance of the proper cascade regularization is also demonstrated.
Four-level atom dynamics is studied in a ladder system in the nine parameter space consisting of driving field strengths, detunings and decay constants, ${{\ensuremath{\Omega}}_{1},{\ensuremath{\Omega}}_{2},{\ensuremath{\Omega}}_{3},{\ensuremath{\Delta}}_{1},{\ensuremath{\Delta}}_{2},{\ensuremath{\Delta}}_{3},{\ensuremath{\Gamma}}_{2},{\ensuremath{\Gamma}}_{3},{\ensuremath{\Gamma}}_{4}}$. One can selectively excite or induce two-level behavior between particular levels of ones choice by appropriately tuning the driving field strengths at three-photon resonance. The dynamics may be classified into two main regions of interest (i) small ${\ensuremath{\Omega}}_{2}$ coupling the \ensuremath{\mid}2⟩-\ensuremath{\mid}3⟩ transition and (ii) large ${\ensuremath{\Omega}}_{2}$. In case (i) one sees two-level behavior consisting of adjacent levels and in a particular region in the parameter space, there is an intermittent shelving of the electrons in one of the two subsystems. In case (ii) the levels consist of the ground state and the upper most level. Emission statistics is studied using the delay function approach in both the cases. In case (i), the behavior of the second order correlation function g2(t){g}^{2}(t)g2(t), is similar to that of two-level emission for low ${\ensuremath{\Omega}}_{1}$ coupling the \ensuremath{\mid}1⟩-\ensuremath{\mid}2⟩ transition, and the correlation increases with ${\ensuremath{\Omega}}_{1}$ for smaller time delays. While, in case (ii) when, in addition, ${\ensuremath{\Omega}}_{3}$ coupling the \ensuremath{\mid}3⟩-\ensuremath{\mid}4⟩ transitionis kept low, g2(t){g}^{2}(t)g2(t) shows superpoissonian distribution, which may be attributed to three-photon processes.
A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.
The multiple-linear-time-scale method is used to construct a perturbation theory for NNN-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}\ensuremath{\ll}\ensuremath{\hbar}\overline{\ensuremath{\omega}}$, where $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}$ represents the inter-action strength and $\overline{\ensuremath{\omega}}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\ensuremath{\sim}{10}^{14}$ W/cm2{\mathrm{cm}}^{2}cm2) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\ensuremath{\sim}{10}^{10}$ W/cm2{\mathrm{cm}}^{2}cm2), the approximate solution is valid for many thousands of Rabi cycles.