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 photon beam. The effect can be modeled using the Jaynes–Cummings model and the Bloch vector formalism.
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 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 good complex coordinates:
One can construct an oscillation experiment through the following steps:
Prepare the system in a fixed state; for example,
Let the state evolve freely, under a HamiltonianH for time t
Find the probability , that the state is in
If is an eigenstate of H, and there will be no oscillations. Also if the two states and are degenerate, every state including 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 are real numbers. This matrix can be decomposed as,
The matrix is the 2 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 and are constants. Consider the case of a spin-1/2 particle in a magnetic field . The interaction Hamiltonian for this system is
where is the magnitude of the particle's magnetic moment, is the Gyromagnetic ratio and is the vector of Pauli matrices. Here the eigenstates of Hamiltonian are eigenstates of , that is and , with corresponding eigenvalues of . The probability that a system in the state can be found in the arbitrary state is given by .
Let the system be prepared in state at time . Note that is an eigenstate of :
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 . 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 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 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 , 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 is .
Example of Rabi oscillation between two states in ionized hydrogen molecule.
An ionized hydrogen molecule is composed of two protons and , and one electron. Because of their large masses, the two protons can be considered to be fixed. Let R be the distance between them and the and states where the electron is localised around or . Assume, at a certain time, the electron is localised about proton . According to the results from the previous section, we know that the electron will oscillate between the two protons with a frequency equal to the Bohr frequency associated with the two stationary states and of the 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.
Derivation of Rabi formula in a nonperturbative procedure by means of the Pauli matrices
Consider a Hamiltonian of the form
The eigenvalues of this matrix are given by
where and , so we can take .
Now, eigenvectors for can be found from equation
Applying the normalization condition on the eigenvectors, . So
Let and . So .
So we get . That is , using the identity . Taking arbitrary phase angle , we can write . Similarly .
So the eigenvector for the eigenvalue is given by
As overall phase is immaterial, we can write
Similarly, the eigenvector for eigenenergy
From these two equations, we can write
Suppose the system starts in state at time ; that is,
For a time-independent Hamiltonian, after time t, the state evolves as
If the system is in one of the eigenstates or , 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 is given by .
Now the probability that a system in the state will be found to be in the arbitrary state
is given by
This can be simplified to
This shows that there is a finite probability of finding the system in state when the system is originally in the state . The probability is oscillatory with angular frequency , which is simply unique Bohr frequency of the system and also called Rabi frequency. The formula(1) is known as Rabi formula. Now after time t the probability that the system in state is given by, which is also oscillatory.
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 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 at time . Then, at time , the probability of it being found in state is given by where . This phenomenon is called Rabi oscillation. Thus, the qubit oscillates between the and 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 to state it is sufficient to adjust the time during which the rotating field acts such that or . This is called a pulse. If a time intermediate between 0 and is chosen, we obtain a superposition of and . 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, is the frequency of laser wave and Rabi frequency is proportional to the product of the transition electric dipole moment of atom and electric field 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.