Adiabatic Approximation Applied to N-State System at One-Photon Resonance
We consider N bound states in a molecule or atom. They are connected by a chain of N-1 laser-driven transitions. Transfer of population from one end of the chain to the other can be done by applying N-1 laser pulses in the obvious (intuitive) order. The counterintuitive pulse order can be used if the first and last laser pulses overlap. Its advantages have been demonstrated1, and its generalization to any odd value of N is known2. For the case of N=4, counterintuitive pulse sequences have been treated in refs. 3 and 4, where non-zero laser detunings are considered to be essential. Here, we consider the special case of laser frequencies that are not detuned from the Bohr frequencies. We consider both intuitive and counterintuitive pulse orders, and both odd N and even N. We use a classical description of each laser pulse, and assume that it drives only its own transition. The rotating-wave approximation is used, and all optical frequencies can then be eliminated from the Hamiltonian and the probability amplitudes. The resulting Hamiltonian matrix is a symmetric tridiagonal matrix. Each off-diagonal matrix element is minus half of a Rabi frequency. The maximum Rabi frequency is ΩM for each laser-driven transition. We use N-1 Gaussian laser pulses having the same width, and a uniform spacing between their peaks. Each Rabi frequency has the form ΩM exp[−(t−tp)2] where t is the time and tp is a linear function of position in the matrix. Since all laser detunings are zero, diagonal elements in the Hamiltonian matrix are zero. The initial condition is that only state one is occupied at early times, so that all probability amplitudes but the first vanishast →−∞. Values of the probability amplitudes can he found by numerical integration of the coupled differential equations, and the occupation probabilities can be computed for any time t. If we use a counterintuitive pulse sequence to transfer most of the occupation probability from one end of the chain to the other, the occupation probabilities of states 3 to N-3 remain somewhat low at all times, in agreement with refs. 1 and 5.
KeywordsRabi Frequency Probability Amplitude Adiabatic Approximation Uniform Spacing Hamiltonian Matrix
- 6.L.I. Schiff: Quantum Mechanics, McGraw Hill, NY (1955), Sec. 31.Google Scholar