Rabi cycle
Rabi oscillations occur in a quantum mechanical two-level system (e.g., two states of an atom ) to which an external periodic force (such as a laser - light field in laser spectroscopy, or an oscillating magnetic field in magnetic resonance spectroscopy ) having interacts (circular ) frequency. The excitation frequency is near the resonant frequency of the two states, as the occupation of the states oscillates with a frequency which is also called Rabi frequency. It is named after the American physicist Isidor Isaac Rabi.
Rabi oscillations are especially important for the description of the interaction of coherent light with atoms. Under certain simplifying assumptions, two electronic states of the atom can be approximated as a two -level system, which is disturbed by the (weak) light field. Thus, the properties of the system can be calculated as part of its perturbative analysis. The rise of the occupation probability of the second ( higher energy ) state then corresponds to the absorption of light. Rabi oscillations are experimentally measurable. In many cases, however, damping or Dephasierungsprozesse play an important role, allowing the oscillations die out quickly ( if at all) can only be observed for very short times.
Resonant interaction
The two states 1 and 2 of the system have the energies. The corresponding frequency of the transition is then. If the fault is resonant, ie, as is the Rabi frequency in the case of atom-light interaction:
Here, the strength of the radiated light field ( electric field component which predominates here ), the transition dipole moment of the transition and the reduced Planck constant. In the case of nuclear magnetic resonance spectroscopy (NMR) interacts with an oscillating magnetic field strength produced by the magnetic dipole moment of the nuclear spin of an atom. The Rabi frequency of this system is given by the gyromagnetic ratio:
The occupation probabilities and the states oscillate with the Rabi frequency according to the following formula:
It was assumed that at the time of only the ground state 1 was filled.
Disgruntled interaction
If now instead of a resonant interference field, an irradiated to detuned interference field, so also the Rabi frequency will change to:
The Rabi frequency of the resonant system is disturbed. A complete theoretical treatment is possible with the help of time-dependent perturbation theory. For the occupation probability of the excited state to the following time course this results then (at the time was again only the ground state 1 occupied ):
And the occupation probability of the state 1. The cast is shown in the figure for different moods:
By the external disturbance of the state 2 is filled to a maximum and then reduced again. This behavior continues periodically. It is worth noting in this case that a full implementation is only possible if the detuning is zero, the frequency of the disturbance, the transition frequency so true resonant. It should also be noted that the Rabi oscillations continuously in this ( simplified ) model for all times and therefore can not be a stationary reshuffle of states with active fault.
The amplitude of the Rabi oscillations describes the cross section for excitation of the two- level transition. It reads the above formula for:
This corresponds to a Lorentz curve, as is typical for other resonance phenomena.
In order to derive
The Hamiltonian of the system is divided into two parts:
Where the unperturbed two-level system and the time-dependent perturbation describe. The above -mentioned conditions are now solutions undisturbed Hamiltonian:
These states can be the solution of the time -dependent Schrödinger equation
Start as a linear combination:
The solution of this approach results in:
Resulting in the above occupation probabilities as a result and.
Swell
- PW Milonni, JH Eberly, Lasers, Wiley, 1988, ISBN 0-471-62731-3
- HJ Metcalf, P. van der Straten, Laser Cooling and Trapping, Springer, 1999, ISBN 0-387-98747-9, pp. 4-7
- PW Atkins, RS Friedman, Molecular Quantum Mechanics, 4th edition, Oxford University Press, Oxford, 2004, ISBN 0199274983