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.

Rabi oscillations, showing the probability of a two-level system initially in to end up in at different detunings Δ.

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

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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[1] 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   and   are the coordinates.[2]

If the vectors are normalized,   and   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

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One can construct an oscillation experiment through the following steps:[3]

  1. Prepare the system in a fixed state; for example,  
  2. Let the state evolve freely, under a Hamiltonian H for time t
  3. 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  .[4] 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  .[5]

By Pauli matrices

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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 So Applying the normalization condition on the eigenvectors,  . So Let   and  . So  .

So we get  . That is  , using the identity  .

The phase of   relative to   should be  .

Choosing   to be real, the eigenvector for the eigenvalue   is given by Similarly, the eigenvector for eigenenergy   is 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.[6]

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 state   is given by This can be simplified to

  (1)

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.

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

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

Quantum optics

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Quantum computing

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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.[7]

See also

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References

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  1. ^ Rabi oscillations, Rabi frequency, stimulated emission. Encyclopedia of Laser Physics and Technology.
  2. ^ Griffiths, David (2005). Introduction to Quantum Mechanics (2nd ed.). p. 341.
  3. ^ Sourendu Gupta (27 August 2013). "The physics of 2-state systems" (PDF). Tata Institute of Fundamental Research.
  4. ^ Griffiths, David (2012). Introduction to Quantum Mechanics (2nd ed.) p. 191.
  5. ^ Griffiths, David (2012). Introduction to Quantum Mechanics (2nd ed.) p. 196 ISBN 978-8177582307
  6. ^ Merlin, R. (2021). "Rabi oscillations, Floquet states, Fermi's golden rule, and all that: Insights from an exactly solvable two-level model". American Journal of Physics. 89 (1): 26–34. Bibcode:2021AmJPh..89...26M. doi:10.1119/10.0001897. S2CID 234321681.
  7. ^ A Short Introduction to Quantum Information and Quantum Computation by Michel Le Bellac, ISBN 978-0521860567