Two-state quantum system

A two-state system or two-level system in quantum mechanics is a simple but important model system that can be used to describe many situations. The system can be in only one of two possible states or appointed, or in a superposition of these two states are (Bra - Ket notation). These two states have case usually different energies E1 and E2. One example is a bound electron to an atom that can occupy one of two levels of the atomic spectrum ( ground state, excited state, see figure at right ). Often the model system of a quantum spin -1 /2 ( angular momentum ) is used, which can be in only two settings. There is a transition between the levels (for example, an optical transition which can be excited by visible light). If the system is once again in one of two states, it remains there forever, at least as long as one does not interfere with the system. If a fault is turned on in the system, it can be observed that the states can merge into each other:

If, for example, an electron in the state (which lies energetically below ), so it can be absorbed by a resonant incident laser pulse in the state. An electron in the state can fall back by emitting a photon that carries the energy difference between the states in the state. The figure shows schematically.

If the fault is a long time, so to oscillate the probability of finding the atom in one of the states. After half an oscillation period, the probability is high, to be found the atom in the excited state, after a whole life, it is probably going in the ground state, etc. This phenomenon corresponds to the Rabi oscillations.

Mathematical description in the context of quantum mechanics

Static treatment

For given system has a Hamiltonian. The states are eigenstates of this Hamiltonian to the eigenvalues ​​:

Is also switched to a Hermitian fault, are no longer the eigenstates of the new Hamiltonian. The new eigenstates are denoted by and the new self- energies. Obtained in the base following representation for:

• Is, then move only the energy eigenvalues ​​; the eigenstates remain the same. We then have:

• In case we neglect here the diagonal elements (ie: ) and thus obtain:

Time development

If the system is prepared in an eigenstate of the time, so it remains for all time in this state. But in case the fault is switched ( with non-vanishing off-diagonal elements), then the probability of finding the system at time t in the state, not 0, which is essentially due to the fact that the states and not eigenstates of the system are more. From the somewhat extensive account we obtain:

These oscillations between the states, as shown in the accompanying illustration are also referred to as Rabi oscillations or as Rabiflops.