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Physical Review A

Nonperturbative approach to the theory of transition probabilities by the stroboscopic method

Attila Aşkar

Published Dec 1, 1974• 12 citations
Publisher

Abstract

The exact differential equations with periodic coefficients characterizing the transition in a two-level quantum system are solved by the stroboscopic method which is a nonperturbative approach. This method is based on an averaging over the fluctuations in the system. The solution is unitary and is valid for large times also, as opposed to the short-time validity of the perturbative solution. It should also be noted that this method gives Rabi's well-known solution to the two-level system at resonance by the so-called transformation to the rotating coordinate frame.

DOI
10.1103/physreva.10.2395

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The exact differential equations with periodic coefficients characterizing the transition in a two-level quantum system are solved by the stroboscopic method which is a nonperturbative approach. This method is based on an averaging over the fluctuations in the system. The solution is unitary and is valid for large times also, as opposed to the short-time validity of the perturbative solution. It should also be noted that this method gives Rabi's well-known solution to the two-level system at resonance by the so-called transformation to the rotating coordinate frame.
A nonperturbative approach to the theory of transition probabilities is presented. The results of this theory are employed to investigate the validity of the assumptions inherent in the perturbative approach. For this purpose, the simple examples of absorption of radiation (first-order process) and Raman scattering (second-order process) are used. The method proposed, which is based upon finite-difference techniques, represents an accurate solution of the time-dependent Schr\"odinger equation in a given representation and has many advantages over methods based upon the perturbation series.
A previous exact formal and practical solution for the time-dependent Schrödinger equation for a two-level non-degenerate system in an arbitrarily strong sinusoidal field is transformed for an N-level system to a Floquet form. Using the Floquet form of the solution computationally convenient expressions are obtained for the phase and the phase-long time-averaged induced transition probabilities. The method exploits the time periodicity of the Schrödinger equation and the solution over the initial period of the hamiltonian provides all the information needed to obtain both the unaveraged and the averaged transition probabilities for the system. The method is illustrated by specific examples which are used to discuss some of the temporal features of the multi-photon spectra of two-level systems.
The object of the research are the quantum transitions of a many-level system between its stationary states under the influence of an alternating external electric field. The transition amplitudes and probabilities are calculated by the method of perturbation theory in the approximation of a classical external field. It is illustrated that in the case when the external field frequency is equal to the frequency of any quantum transition, then the probability of this transition fluctuate nearly the average value which is oscillating with Rabi frequency. Probabilities of other quantum transitions are very small and change chaotically with time.
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.
The interaction of a quantum system with an oscillating field is studied in a formalism which replaces the semiclassical time-dependent Hamiltonian with a time-independent Hamiltonian represented by an infinite matrix. The formalism is developed as a mathematical equivalent to the semiclassical treatment, and interpreted as a classical approximation to the quantum treatment of the field. Combined with a perturbation theory for two nearly degenerate states, the formalism provides a convenient method for determining resonance transition probabilities including frequency shifts and multiple quantum transitions. The theory is illustrated by a detailed study of the simple case of a two-state system excited by a strong oscillating field.