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 two possible energy levels. These two levels are a ground state with lower energy and an excited state with higher energy. If the energy levels are not degenerate (i.e. not having equal energies), the system can absorb a quantum of energy and transition from the ground state to the "excited" state. When an atom (or some other two-level system) is illuminated by a coherent beam of photons, it will cyclically absorb photons and re-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.

Mathematical description

A detailed mathematical description of the effect can be found on the page for the Rabi problem. For example, for a two-state atom (an atom in which an electron can either be in the excited or ground state) in an electromagnetic field with frequency tuned to the excitation energy, the probability of finding the atom in the excited state is found from the Bloch equations to be where \omega is the Rabi frequency. More generally, one can consider a system where the two levels under consideration are not energy eigenstates. Therefore, if the system is initialized in one of these levels, time evolution will make the population of each of the levels oscillate with some characteristic frequency, whose angular frequency is also known as the Rabi frequency. The state of a two-state quantum system can be represented as vectors of a two-dimensional complex Hilbert space, which means that every state vector is represented by complex coordinates: where c_1 and c_2 are the coordinates. If the vectors are normalized, c_1 and c_2 are related by. The basis vectors will be represented as and. All observable physical quantities associated with this systems are 2 × 2 Hermitian matrices, which means that the Hamiltonian of the system is also a similar matrix.

Derivations

One can construct an oscillation experiment through the following steps: If |1\rangle is an eigenstate of H, P(t)=1 and there will be no oscillations. Also if the two states |0\rangle and |1\rangle are degenerate, every state including |1\rangle is an eigenstate of H. As a result, there will be no oscillations. On the other hand, if H has no degenerate eigenstates, and the initial state is not an eigenstate, then there will be oscillations. The most general form of the Hamiltonian of a two-state system is given here, and a_3 are real numbers. This matrix can be decomposed as, The matrix \sigma_0 is the 2 \times 2 identity matrix and the matrices are the Pauli matrices. This decomposition simplifies the analysis of the system especially in the time-independent case where the values of a_0,a_1,a_2 and a_3are constants. Consider the case of a spin-1/2 particle in a magnetic field. The interaction Hamiltonian for this system is where \mu is the magnitude of the particle's magnetic moment, \gamma is the Gyromagnetic ratio and is the vector of Pauli matrices. Here the eigenstates of Hamiltonian are eigenstates of \sigma_3, that is |0\rangle and |1\rangle, with corresponding eigenvalues of. The probability that a system in the state |\psi\rang can be found in the arbitrary state is given by. Let the system be prepared in state at time t=0. Note that is an eigenstate of \sigma_1: Here the Hamiltonian is time independent. Thus by solving the stationary Schrödinger equation, the state after time t is given by with total energy of the system E. So the state after time t is given by: Now suppose the spin is measured in x-direction at time t. The probability of finding spin-up is given by:where \omega is a characteristic angular frequency given by, where it has been assumed that. So in this case the probability of finding spin-up in x-direction is oscillatory in time t when the system's spin is initially in the direction. Similarly, if we measure the spin in the -direction, the probability of measuring spin as of the system is. In the degenerate case where E_+ = E_-, the characteristic frequency is 0 and there is no oscillation. Notice that if a system is in an eigenstate of a given Hamiltonian, the system remains in that state. This is true even for time dependent Hamiltonians. Taking for example ; if the system's initial spin state is, then the probability that a measurement of the spin in the y-direction results in at time t is. !Example of Rabi oscillation between two states in ionized hydrogen molecule. This oscillation of the electron between the two states corresponds to an oscillation of the mean value of the electric dipole moment of the molecule. Thus when the molecule is not in a stationary state, an oscillating electric dipole moment can appear. Such an oscillating dipole moment can exchange energy with an electromagnetic wave of same frequency. Consequently, this frequency must appear in the absorption and emission spectrum of the ionized hydrogen molecule.

By Pauli matrices

Consider a Hamiltonian of the formThe eigenvalues of this matrix are given bywhere and, so we can take. Now, eigenvectors for E_+ can be found from equationSoApplying the normalization condition on the eigenvectors,. SoLet and. So. So we get. That is, using the identity. The phase of a relative to b should be -\phi. Choosing a to be real, the eigenvector for the eigenvalue E_+ is given bySimilarly, the eigenvector for eigenenergy E_- isFrom these two equations, we can writeSuppose the system starts in state |0\rang at time t = 0; that is,For a time-independent Hamiltonian, after time t, the state evolves asIf the system is in one of the eigenstates |E_+\rang or |E_-\rang, it will remain the same state. However, for a time-dependent Hamiltonian and a general initial state as shown above, the time evolution is non trivial. The resulting formula for the Rabi oscillation is valid because the state of the spin may be viewed in a reference frame that rotates along with the field. The probability amplitude of finding the system at time t in the state |1\rang is given by. Now the probability that a system in the state will be found to be in the state |1\rang is given byThis can be simplified to This shows that there is a finite probability of finding the system in state |1\rang when the system is originally in the state |0\rang. The probability is oscillatory with angular frequency, which is simply unique Bohr frequency of the system and also called Rabi frequency. The formula is known as Rabi formula. Now after time t the probability that the system in state |0\rang is given by, which is also oscillatory. These types of oscillations of two-level systems are called Rabi oscillations, which arise in many problems such as Neutrino oscillation, the ionized Hydrogen molecule, Quantum computing, Ammonia maser, etc.

Applications

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

Quantum optics

Quantum computing

Any two-state quantum system can be used to model a qubit. Consider a spin- system with magnetic moment placed in a classical magnetic field. Let \gamma be the gyromagnetic ratio for the system. The magnetic moment is thus. The Hamiltonian of this system is then given by where and. One can find the eigenvalues and eigenvectors of this Hamiltonian by the above-mentioned procedure. Now, let the qubit be in state |0\rang at time t = 0. Then, at time t, the probability of it being found in state |1\rang is given by where. This phenomenon is called Rabi oscillation. Thus, the qubit oscillates between the |0\rang and |1\rang states. The maximum amplitude for oscillation is achieved at, which is the condition for resonance. At resonance, the transition probability is given by. To go from state |0\rang to state |1\rang it is sufficient to adjust the time t during which the rotating field acts such that or. This is called a \pi pulse. If a time intermediate between 0 and is chosen, we obtain a superposition of |0\rang and |1\rang. In particular for, we have a pulse, which acts as:. This operation has crucial importance in quantum computing. 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. Then is the energy difference between the two atomic levels, \omega is the frequency of laser wave and Rabi frequency \omega_1 is proportional to the product of the transition electric dipole moment of atom \vec{d} and electric field \vec{E} of the laser wave that is. In summary, Rabi oscillations are the basic process used to manipulate qubits. These oscillations are obtained by exposing qubits to periodic electric or magnetic fields during suitably adjusted time intervals.