Rabi cycle
In physics, the Rabi cycle (or Rabi flop) is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field.
A great variety of physical processes belonging to the areas of quantum computing, condensed matter, atomic and molecular physics, and nuclear and particle physics can be conveniently studied in terms of two-level quantum mechanical systems, and exhibit Rabi flopping when coupled to an optical driving field.
The effect is important in quantum optics, magnetic resonance, and quantum computing, and is named after Isidor Isaac Rabi.
When an atom (or some other two-level system) is illuminated by a coherent beam of photons, it will cyclically absorb photons and emit them by stimulated emission.
One example of Rabi flopping is the spin flipping within a quantum system containing a spin-1/2 particle and an oscillating magnetic field.
are the strengths of the environment and the oscillating fields respectively, and
We can then write a Hamiltonian describing this field, yielding
, because this formula assume that the Hamiltonian is constant with respect to time.
This can be done by shifting the reference frame that we work in to match the rotating magnetic field.
Therefore, in the rotating reference frame, both the magnetic field and the Hamiltonian are constant with respect to time.
The time dependent Schrödinger equation in the stationary reference frame is
Expanding this using the matrix forms of the Hamiltonian and the state yields
Applying the matrix and separating the components of the vector allows us to write two coupled differential equations as follows
To transform this into the rotating reference frame, we may use the fact that
In some sense, this is a transformed Schrödinger equation in the rotating reference frame.
Crucially, the Hamiltonian does not vary with respect to time, meaning in this reference frame, we can use the familiar solution to Schrödinger time evolution:
This transformed problem is equivalent to that of Larmor precession of a spin state, so we have solved the essence of Rabi flopping.
In other words, the frequency of the rotating magnetic field
must match the environmental field's Larmor frequency
in order for the spin to fully flip; they must achieve resonance.
Within the rotating reference frame, when resonance is achieved, it is as if there is no environmental magnetic field, and the oscillating magnetic field looks constant.
Thus both mathematically (as we have derived) and physically, the problem reduces to the precession of a spin state under a constant magnetic field (Larmor precession).
To transform the solved state back to the stationary reference frame, we reuse the rotation operator with the opposite angle
Rabi flopping may be used to describe a two-level atom with an excited state and a ground state in an electromagnetic field with frequency tuned to the excitation energy.
Using the spin-flipping formula but applying it to this system yields where
Any two-state quantum system can be used to model a qubit.
Rabi flopping provides a physical way to allow for spin flips in a qubit system.
The equations are essentially identical in the case of a two level atom in the field of a laser when the generally well satisfied rotating wave approximation is made, where
is proportional to the product of the transition electric dipole moment of atom
On a quantum computer, these oscillations are obtained by exposing qubits to periodic electric or magnetic fields during suitably adjusted time intervals.
