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.

A two-level system is one that has a Hilbert space spanned by two possible orthogonal states. These states may have two possible energy levels. One level can be a ground state with lower energy, and the other can be an excited state with higher energy. If the energy levels are not degenerate (i.e. don't have equal energies), the system can absorb or emit a quantum of energy and transition from the ground state to the excited state or vice versa. 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 such cycle is called a Rabi cycle, and the inverse of its duration is the Rabi frequency of the system. The effect can be modeled using the Jaynes–Cummings model and the Bloch vector formalism.

More generally, the introduction of an oscillatory driving field to a two-level system can be viewed as a perturbation which couples the previously uncoupled states. The time evolution under this perturbed Hamiltonian therefore results in oscillations, called Rabi cycles, between the two unperturbed states. In the strong coupling limit, a complete oscillation between the unperturbed states is observed.

Mathematical description of spin flipping

is the gyromagnetic ratio are the spin operatorsis known as the Rabi frequency, but this formula cannot directly be used for time dependent Hamiltonians.

The main strategy in solving this problem is to transform the Hamiltonian so that the time dependence is gone, solve the problem in this transformed frame, and then transform the results back to normal. This can be done by shifting the reference frame that we work in to match the rotating magnetic field. If we rotate along with the magnetic field, then from our point of view, the magnetic field is not rotating and appears constant. Therefore, in the rotating reference frame, both the magnetic field and the Hamiltonian are constant with respect to time.

direction respectively, and the state is normalized. We can transform this state to the rotating reference frame by using a rotation operatorwhich rotates the state counterclockwise around the positive z-axis in state space, which may be visualized as a Bloch sphereThe time dependent Schrödinger equationThis transformed problem is equivalent to that of Larmor precession is the generalized Rabi Frequency in order for the spin to fully flip; they must achieve resonance.

, thus yielding a full solution to the problem.

Applications

The Rabi effect is important in quantum optics, magnetic resonance and quantum computing.

Quantum optics

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

is the Rabi frequency.

Quantum computing

Any two-state quantum system can be used to model a qubitThe 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 the frequency of laser wave and Rabi frequency is proportional to the product of the transition electric dipole moment and electric field. On a quantum computer, these oscillations are obtained by exposing qubits to periodic electric or magnetic fields during suitably adjusted time intervals.

See also

• Atomic coherence

• Bloch sphere

• Laser pumping

• Optical pumping

• Rabi problem

• Vacuum Rabi oscillation

• Neutral particle oscillation

References

• Quantum Mechanics Volume 1 by C. Cohen-Tannoudji, Bernard Diu, Frank Laloe,

• A Short Introduction to Quantum Information and Quantum Computation by Michel Le Bellac,

• The Feynman Lectures on Physics, Volume III

• Modern Approach To Quantum Mechanics by John S Townsend,