Two-state quantum system

Two-state quantum system

In quantum mechanics, a two-state system (a.k.a. TLS=Two Level System) is a system which has two possible states. More formally, the Hilbert space of a two state system has two degrees of freedom, so a complete basis spanning the space must consist of two independent states.

The physics of a quantum mechanical two-state system is trivial if both states are degenerate, that is, if the states have the same energy. However, if there is an energy difference between the two states, then nontrivial dynamics can occur.

Two-state dynamics

If the time-independent Hamiltonian is H, and we label the two levels as left.|a ight angle and left.|b ight angle with corresponding orthonormal energy eigenvalues E_a and E_b, then the dynamics of the system can be specified as follows:

At some time t_0, let the system be in an arbitrary (and completely general) state,left.|psi(t_0) ight angle=c_aleft.|a ight angle+c_bleft.|b ight anglethen after evolving under H, at time t,the state will be
left.|psi(t) ight angle = c_a e^{-ifrac{E_a(t-t_0)}{hbarleft|a ight angle+c_b e^{-ifrac{E_b(t-t_0)}{hbarleft|b ight angle

The physics of two state systems can be usefully applied to multi-state systems where the system is known to have only enough energy available to excite the lowest two states, thus effectively creating a two state system. In fact, in nature, it is difficult to identify any true two-state systems; merely systems where the energetics of the circumstances isolate two particular states.

The set of all states in a two-level system can be mapped onto a representation known as the Bloch sphere.

Examples of two-state quantum systems

* Spin-1/2 particles are two-state quantum systems when only the spin degree of freedom is considered.
* The "inverting" degree of freedom in an ammonia molecule; the nitrogen at the vertex of an ammonia molecule exhibits two molecular states - "up" and "down", corresponding to opposite positions with respect to the plane of the three hydrogen atoms. In an electric field, these two states are non-degenerate.
* Two-level systems are important in the field of quantum computing as are used to implement qubits.

ee also

* Rabi cycle
* doublet

Links

* http://www.itp.tu-berlin.de/zweiniveau.html A Java applet that visualizes various effects of two-state systems (laser driven). English user interface, but German documentation.
* http://www.itp.tu-berlin.de/6854.html extended version of the applet. Includes electron phonon interaction.


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