Rabi oscillations, showing the probability of a two-level system initially in $|1\rangle$ to end up in $|2\rangle$ 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 photon beam. The effect can be modeled using the Jaynes–Cummings model and the Bloch vector formalism.
Mathematical treatmentEdit
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
$|c_{b}(t)|^{2}\propto \sin ^{2}(\omega t/2),$
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^{[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$|\psi \rangle$ is represented by good complex coordinates:
where $c_{1}$ and $c_{2}$ are the coordinates.^{[2]}
If the vectors are normalized, $c_{1}$ and $c_{2}$ are related by $|c_{1}|^{2}+|c_{2}|^{2}=1$. The basis vectors will be represented as $|0\rangle ={\begin{pmatrix}1\\0\end{pmatrix}}$ and $|1\rangle ={\begin{pmatrix}0\\1\end{pmatrix}}$.
One can construct an oscillation experiment through the following steps:^{[3]}
Prepare the system in a fixed state; for example, $|1\rangle$
Let the state evolve freely, under a HamiltonianH for time t
Find the probability $P(t)$, that the state is in $|1\rangle$
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
The matrix $\sigma _{0}$ is the 2 $\times$ 2 identity matrix and the matrices $\sigma _{k}\;(k=1,2,3)$ 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 $a_{3}$are constants. Consider the case of a spin-1/2 particle in a magnetic field $\mathbf {B} =B\mathbf {\hat {z}}$. The interaction Hamiltonian for this system is
where $\mu$ is the magnitude of the particle's magnetic moment, $\gamma$ is the Gyromagnetic ratio and ${\boldsymbol {\sigma }}$ 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 $E_{+}={\frac {\hbar }{2}}\gamma B\ ,\ E_{-}=-{\frac {\hbar }{2}}\gamma B$. The probability that a system in the state $|\psi \rangle$ can be found in the arbitrary state $|\phi \rangle$ is given by ${|\langle \phi |\psi \rangle |}^{2}$.
Let the system be prepared in state $\left|+X\right\rangle$ at time $t=0$. Note that $\left|+X\right\rangle$ is an eigenstate of $\sigma _{1}$:
where $\omega$ is a characteristic angular frequency given by $\omega ={\frac {E_{+}-E_{-}}{\hbar }}=\gamma B$, where it has been assumed that $E_{-}\leq E_{+}$.^{[4]} 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 $\left|+X\right\rangle$ direction. Similarly, if we measure the spin in the $\left|+Z\right\rangle$-direction, the probability of measuring spin as ${\tfrac {\hbar }{2}}$ of the system is ${\tfrac {1}{2}}$. 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 ${\textstyle {\hat {H}}=-\gamma \ S_{z}B\sin(\omega t)}$; if the system's initial spin state is $\left|+Y\right\rangle$, then the probability that a measurement of the spin in the y-direction results in $+{\tfrac {\hbar }{2}}$ at time $t$ is ${\textstyle {\left|\left\langle \,+Y|\psi (t)\right\rangle \right|}^{2}\,=\cos ^{2}\left({\frac {\gamma B}{2\omega }}\cos \left({\omega t}\right)\right)}$.^{[5]}
Example of Rabi oscillation between two states in ionized hydrogen molecule.
An ionized hydrogen molecule is composed of two protons $P_{1}$ and $P_{2}$, 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 $|1\rangle$ and $|2\rangle$ states where the electron is localised around $P_{1}$ or $P_{2}$. Assume, at a certain time, the electron is localised about proton $P_{1}$. 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 $|E_{+}\rangle$ and $|E_{-}\rangle$ 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 using nonperturbative procedure by means of the Pauli matricesEdit
where $\mathbf {W} =W_{1}+\imath W_{2}$ and ${\left\vert W\right\vert }^{2}={W_{1}}^{2}+{W_{2}}^{2}=WW^{*}$, so we can take $\mathbf {W} ={\left\vert W\right\vert }e^{i\phi }$.
Now, eigenvectors for $E_{+}$ can be found from equation
If the system is in one of the eigenstates $|E_{+}\rangle$ or $|E_{-}\rangle$, 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 $|1\rangle$ is given by ${\textstyle \left\langle \ 1|\psi (t)\right\rangle =e^{i\phi }\sin \left({\tfrac {\theta }{2}}\right)\cos \left({\tfrac {\theta }{2}}\right)\left(e^{\frac {-iE_{+}t}{\hbar }}-e^{\frac {-iE_{-}t}{\hbar }}\right)}$.
Now the probability that a system in the state $|\psi (t)\rangle$ will be found to be in the arbitrary state
This shows that there is a finite probability of finding the system in state $|1\rangle$ when the system is originally in the state $|0\rangle$. The probability is oscillatory with angular frequency $\omega ={\frac {E_{+}-E_{-}}{2\hbar }}={\frac {\sqrt {{\Delta }^{2}+{\left\vert W\right\vert }^{2}}}{\hbar }}$, 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 $|0\rangle$ is given by ${|\langle \ 0|\psi (t)\rangle |}^{2}=1-\sin ^{2}(\theta )\sin ^{2}\left({\frac {(E_{+}-E_{-})t}{2\hbar }}\right)$, which is also oscillatory.
Any two-state quantum system can be used to model a qubit. Consider a spin-${\tfrac {1}{2}}$ system with magnetic moment ${\boldsymbol {\mu }}$ placed in a classical magnetic field ${\boldsymbol {B}}=B_{0}\ {\hat {z}}+B_{1}\left(\cos {(\omega t)}\ {\hat {x}}-\sin {(\omega t)}\ {\hat {y}}\right)$. Let $\gamma$ be the gyromagnetic ratio for the system. The magnetic moment is thus ${\boldsymbol {\mu }}={\frac {\hbar }{2}}\gamma {\boldsymbol {\sigma }}$. The Hamiltonian of this system is then given by $\mathbf {H} =-{\boldsymbol {\mu }}\cdot \mathbf {B} =-{\frac {\hbar }{2}}\omega _{0}\sigma _{z}-{\frac {\hbar }{2}}\omega _{1}(\sigma _{x}\cos \omega t-\sigma _{y}\sin \omega t)$ where $\omega _{0}=\gamma B_{0}$ and $\omega _{1}=\gamma B_{1}$. One can find the eigenvalues and eigenvectors of this Hamiltonian by the above-mentioned procedure. Now, let the qubit be in state $|0\rangle$ at time $t=0$. Then, at time $t$, the probability of it being found in state $|1\rangle$ is given by $P_{0\to 1}(t)=\left({\frac {\omega _{1}}{\Omega }}\right)^{2}\sin ^{2}\left({\frac {\Omega t}{2}}\right)$ where $\Omega ={\sqrt {(\omega -\omega _{0})^{2}+\omega _{1}^{2}}}$. This phenomenon is called Rabi oscillation. Thus, the qubit oscillates between the $|0\rangle$ and $|1\rangle$ states. The maximum amplitude for oscillation is achieved at $\omega =\omega _{0}$, which is the condition for resonance. At resonance, the transition probability is given by $P_{0\to 1}(t)=\sin ^{2}\left({\frac {\omega _{1}t}{2}}\right)$. To go from state $|0\rangle$ to state $|1\rangle$ it is sufficient to adjust the time $t$ during which the rotating field acts such that ${\frac {\omega _{1}t}{2}}={\frac {\pi }{2}}$ or $t={\frac {\pi }{\omega _{1}}}$. This is called a $\pi$ pulse. If a time intermediate between 0 and ${\frac {\pi }{\omega _{1}}}$ is chosen, we obtain a superposition of $|0\rangle$ and $|1\rangle$. In particular for $t={\frac {\pi }{2\omega _{1}}}$, we have a ${\frac {\pi }{2}}$ pulse, which acts as: $|0\rangle \to {\frac {|0\rangle +i|1\rangle }{\sqrt {2}}}$. 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 $\hbar \omega _{0}$ 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 $\omega _{1}\propto \hbar \ {\vec {d}}\cdot {\vec {E}}$. 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]}
^Rabi oscillations, Rabi frequency, stimulated emission. Encyclopedia of Laser Physics and Technology.
^Griffiths, David (2005). Introduction to Quantum Mechanics (2nd ed.). p. 341.
^Sourendu Gupta (27 August 2013). "The physics of 2-state systems" (PDF). Tata Institute of Fundamental Research.
^Griffiths, David (2012). Introduction to Quantum Mechanics (2nd ed.) p. 191.
^Griffiths, David (2012). Introduction to Quantum Mechanics (2nd ed.) p. 196 ISBN 978-8177582307
^Merlin, R. "Rabi oscillations, Floquet states, Fermi's golden rule, and all that: Insights from an exactly solvable two-level model". American Journal of Physics. 89: 26–34.
^A Short Introduction to Quantum Information and Quantum Computation by Michel Le Bellac, ISBN 978-0521860567
Quantum Mechanics Volume 1 by C. Cohen-Tannoudji, Bernard Diu, Frank Laloe, ISBN 9780471164333
A Short Introduction to Quantum Information and Quantum Computation by Michel Le Bellac, ISBN 978-0521860567
The Feynman Lectures on Physics Vol 3 by Richard P. Feynman & R.B. Leighton, ISBN 978-8185015842
Modern Approach To Quantum Mechanics by John S Townsend, ISBN 9788130913148