User:Rittenuov/Rabi Oscillations

Rabi oscillation or Rabi flopping (named after Isidor Isaac Rabi) is the characteristic oscillatory behaviour shown by a two-level system when driven by an oscillatory electromagnetic field. The phenomenon was first observed by Rabi while studying nuclear magnetic resonance. A spin when placed in a constant magnetic field, will in general undergo Larmor precession about the field, unless it is aligned parallel or anti-parallel to the field. In the Rabi model, an additional time-varying transverse magnetic field is applied which causes the spin to periodically flip between its "up" and "down" states. This "flopping" behaviour can be seen in any two-level system interacting with electromagnetic radiation and is of significant interest in quantum optics in controlling atom-photon interactions.

== Semi-Classical Treatment ==

The semi-classical solution to the Rabi problem considers the two-level system to be quantized whereas the electromagnetic field is treated classically. This setion explains the solution in terms of spins interacting with magnetic field. The results are identical for the case of atoms interacting with light.

The general model is as follows:

A spin-1/2 particle is placed in a strong, constant magnetic field ${\vec {B}}_{0}$ applied along any direction which we choose to be our $z$-axis, i.e., ${\vec {B}}_{0}=B_{\parallel }{\vec {z}}$. This strong field creates splitting between the $|\uparrow _{z}\rangle$ ("spin up") and $|\downarrow _{z}\rangle$ ("spin down") states. The Hamiltonian for the interaction of this field with the spin is:

${\hat {H}}_{0}=-{\hat {\vec {\mu }}}\cdot {\vec {B}}$ (1)

${\vec {\mu }}$ is the spin magnetic moment, and can be written in terms of the Pauli operators as ${\vec {\mu }}=\gamma {\frac {\hbar }{2}}{\hat {\vec {\sigma }}}$ where $\gamma$ is the gyromagnetic ratio.
For the given magnetic field,

${\hat {H}}_{0}=-{\frac {\hbar \omega _{0}}{2}}{\hat {\sigma }}_{z}$ (2)

where $\omega _{0}=\gamma B_{\parallel }$ is the frequency of precession or Larmor frequency. The energy splitting between the two spin-states is $\hbar \omega _{0}$.

An additional oscillatory field ${\vec {B}}_{int}(t)$ is applied transverse to ${\vec {B}}_{0}$, i.e., in the $x-y$ plane. Firstly, we consider a field of constant magnitude rotating with a frequency $\omega$,

${\vec {B}}_{int}(t)=B_{\perp }\left(\cos(\omega t+\phi ){\vec {x}}+\sin(\omega t+\phi ){\vec {y}}\right)$ (3)

The corresponding interaction term in the Hamiltonian is

${\hat {H}}_{int}(t)={\frac {\gamma B_{\perp }}{2}}\left(\cos(\omega t+\phi ){\hat {\sigma }}_{x}+\sin(\omega t+\phi ){\hat {\sigma }}_{y}\right)$ (4)

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We need to solve the time-dependent Schrödinger equation

${\frac {\partial }{\partial t}}|\psi (t)\rangle =i\hbar \left({\hat {H}}_{0}+{\hat {H}}_{int}(t)\right)|\psi (t)\rangle$ (5)

Rotating Frame

The frame rotates at the frequency $\omega$ of the rotating magnetic field about the $z$-axis, i.e., in time $t$ it is rotated through an angle $\omega t$ with the laboratory frame. The unitary operator associated with this rotation is
${\hat {U}}_{RF}(t)=e^{-i(\omega t/2){\hat {\sigma }}_{z}}$
Quantum mechanically, going to the rotating frame involves changing our operators as:

${\hat {O}}_{RF}(t)={\hat {U}}_{RF}^{\dagger }(t){\hat {O}}(t){\hat {U}}_{RF}(t)$ (6)

and the states as:

$|\psi _{RF}(t)\rangle ={\hat {U}}_{RF}^{\dagger }(t)|\psi (t)\rangle$ (7)

The subscript _RF refers to objects in the rotating frame whereas the subscript-free objects are understood to be in the laboratory frame or Schrödinger picture. Taking the time derivative of $|\psi _{RF}\rangle$,

${\begin{aligned}{\frac {\partial }{\partial t}}|\psi _{RF}\rangle &={\frac {\partial }{\partial t}}\left[{\hat {U}}_{RF}^{\dagger }(t)|\psi (t)\rangle \right]\\&=-{\frac {i}{\hbar }}\left[{\hat {U}}_{RF}^{\dagger }(t){\hat {H}}(t)+i\hbar {\frac {\partial {\hat {U}}_{RF}^{\dagger }}{\partial t}}\right]|\psi (t)\rangle \\&=-{\frac {i}{\hbar }}\left[{\hat {U}}_{RF}^{\dagger }{\hat {H}}(t){\hat {U}}_{RF}-\hbar \omega {\hat {\sigma }}_{z}\right]\end{aligned}}$ (8)

The states in the rotating frame therefore evolve according to a modified Hamiltonian,

${\hat {H}}_{RF}={\hat {U}}_{RF}^{\dagger }{\hat {H}}(t){\hat {U}}_{RF}-\hbar \omega {\hat {\sigma }}_{z}$ (9)

By expressing ${\hat {\sigma }}_{x}$ and ${\hat {\sigma }}_{y}$ in terms of the Pauli raising and lowering operators, ${\hat {\sigma }}_{\pm }$, we can rewrite the Schrödinger picture Hamiltonian as

${\hat {H}}={\frac {\hbar \omega _{0}}{2}}{\hat {\sigma }}_{z}+\hbar \gamma B_{\perp }\left(e^{-i(\omega t+\phi )}{\hat {\sigma }}_{+}+e^{i(\omega t+\phi )}{\hat {\sigma }}_{-}\right)$ (10)

The rotating-frame Hamiltonian then reduces to the following time-independent form:

${\hat {H}}_{RF}={\frac {\hbar \Delta }{2}}{\hat {\sigma }}_{z}+{\frac {\hbar \Omega _{\perp }}{2}}\left(e^{-i\phi }{\hat {\sigma }}_{+}+e^{i\phi }{\hat {\sigma }}_{-}\right)$ (11)

$\Delta =\omega -\omega _{0}$ is called the "detuning", a measure of how much the driving frequency is off-resonance, i.e., its difference with the characteristic frequency $\omega _{0}$ associated with the level-splitting. $\Omega _{\perp }=\gamma B_{\perp }$ will be shown to be the frequency of flopping in the resonant case and is known as the Rabi frequency or the bare Rabi frequency.
This Hamiltonian is of the form

${\hat {H}}_{RF}={\frac {\hbar {\vec {\Omega }}_{tot}}{2}}\cdot {\hat {\vec {\sigma }}}$ (12)

and the corresponding unitary evolution operator in the rotating frame is

${\hat {U}}_{Rabi}=e^{-i{\frac {{\vec {\Omega }}_{tot}}{2}}\cdot {\hat {\vec {\sigma }}}t}$ (13)

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This corresponds to an SU(2) rotation about an axis

${\vec {n}}={\frac {{\vec {\Omega }}_{tot}}{\Omega _{tot}}}=-{\frac {\Delta }{\Omega _{tot}}}{\vec {z}}+{\frac {\Omega _{\perp }}{\Omega _{tot}}}\left(\cos {\phi }{\vec {x}}+\sin {\phi }{\vec {y}}\right)$ (14)

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The frequency of the rotation $\Omega _{tot}$ is known as the "Generalized Rabi frequency".
${\hat {U}}_{Rabi}$ can be rewritten in the useful form:

${\hat {U}}_{Rabi}=\cos \left({\frac {\Omega _{tot}t}{2}}\right){\hat {I}}-i\sin \left({\frac {\Omega _{tot}t}{2}}\right){\vec {n}}\cdot {\hat {\vec {\sigma }}}$ (15)

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For illustration, we look at the behaviour of a single spin, initialized in the state $|\downarrow _{z}\rangle$. Note that ${\hat {U}}_{RF}^{\dagger }$ has no effect on this state except changing an overall phase which is of no consequence.

Bloch Sphere Picture

These rotations can be visualized conveniently using the Bloch sphere representation.

On Resonance

On resonance, the frequency of the driving field ($\omega$) matches the frequency associated with the level-separation ($\omega _{0}$), i.e., the detuning, $\Delta$ is 0. The rotating-frame Hamiltoninan simplifies to

${\hat {H}}_{RF}={\frac {\hbar \Omega _{\perp }}{2}}\left(\cos {\phi }{\hat {\sigma }}_{x}+\sin {\phi }{\hat {\sigma }}_{y}\right)$ (16)

i.e., the axis of rotation lies completely in the x-y plane.
The time-evolved state in the rotating frame is

$|\psi _{RF}(t)\rangle =\cos \left({\frac {\Omega _{\perp }t}{2}}\right)|\downarrow _{z}\rangle -ie^{i\phi }\sin \left({\frac {\Omega _{\perp }t}{2}}\right)|\uparrow _{z}\rangle$ (17)

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The probability of the spin getting flipped to the up state is thus The time-evolved state in the rotating frame is

$P_{\uparrow _{z}}(t)=|\langle \uparrow _{z}|\psi _{RF}(t)\rangle |^{2}={\frac {1+\cos(\Omega _{\perp }t)}{2}}$ (18)

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Therefore, on resonance the spin oscillates between the up and down states with a frequency equal to the bare Rabi frequency. The evolution is coherent, meaning that the system is in a quantum superposition of the $|\uparrow _{z}\rangle$ and $|\downarrow _{z}\rangle$ states throughout.

Rabi oscillation on Bloch sphere on resonance

Off Resonance

The driving field is detuned from resonance by an amount $\Delta$. The axis of rotation is no longer in the x-y plane. Starting with $|\downarrow _{z}\rangle$, the rotating-frame state after a time $t$ is

$|\psi _{RF}(t)\rangle =\left[\cos \left({\frac {\Omega _{tot}t}{2}}\right)+i{\frac {\Delta }{\Omega _{tot}}}\sin {\frac {\Omega _{tot}t}{2}}\right]|\downarrow _{z}\rangle -ie^{i\phi }{\frac {\Omega _{\perp }}{\Omega _{tot}}}\sin \left({\frac {\Omega _{tot}t}{2}}\right)|\uparrow _{z}\rangle$ (19)

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The probability of finding the spin in the $|\uparrow _{z}\rangle$ state is now

$P_{\uparrow _{z}}(t)=|\langle \uparrow _{z}|\psi _{RF}(t)\rangle |^{2}={\frac {\Omega _{\perp }^{2}}{\Omega _{\perp }^{2}+\Delta ^{2}}}{\frac {1+\cos(\Omega _{tot}t)}{2}}$ (20)

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With non-zero detuning, this probability never reaches 1. Thus, off resonance, it is impossible to have the state completely flipped to the $|\uparrow _{z}\rangle$ state.

Rabi oscillation on Bloch sphere off resonance

Plots of probability to be in the "up" state over time t both on and off-resonance. The off-resonance probability never reaches 1.

Rotating Wave Approximation

If instead of a rotating driving field, we have a linearly oscillating one, we use the Rotating wave approximation. The axis of oscillation is taken to be along the x-axis,

${\vec {B}}_{int}=B\cos(\omega t){\vec {x}}$ (21)

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This can be re-written as the sum of two circulating fields that rotate in opposite directions; one of these co-rotate with the the precession about ${\vec {B}}_{0}$ while the other is counter-rotating. Under the near-resonance conditions we have considered, the counter rotating term oscillates with a very high frequency in the rotating frame. Over longer time scales, its effect on the spin averages out to be nearly zero and can be neglected. This can be seen clearly from the interaction Hamiltonian. In the Schrödinger picture, this is

${\begin{aligned}{\hat {H}}^{(int)}&={\frac {\hbar \gamma B}{2}}\cos(\omega t){\hat {\sigma }}_{x}\\&={\frac {\hbar \gamma B}{4}}\left(e^{-i\omega t}+e^{i\omega t}\right)({\hat {\sigma }}_{+}+{\hat {\sigma }}_{-})\end{aligned}}$ (22)

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Transforming to the rotating frame as before gives us the following Hamiltonian

${\hat {H}}_{RF}^{(int)}={\frac {\hbar \gamma B}{4}}({\hat {\sigma }}_{+}+{\hat {\sigma }}_{-})+{\frac {\hbar \gamma B}{4}}\left(e^{-2i\omega t}{\hat {\sigma }}_{+}+e^{2i\omega t}{\hat {\sigma }}_{-}|\right)$ (23)

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The first term is stationary in the rotating frame whereas the second term oscillates with a frequency $2\omega$. The dynamics that are of interest to us occur at rates determined by $\Delta$, $\Omega _{\perp }$ which are much slower than $\omega$. The effective rotating-frame Hamiltonian is thus

${\hat {H}}_{RF}^{(int)}={\frac {\hbar \Omega _{\perp }}{2}}({\hat {\sigma }}_{+}+{\hat {\sigma }}_{-})$ (24)

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where the Rabi frequency is now $\Omega _{\perp }={\frac {\gamma B}{2}}$.

Interaction of Light with Two-level Atom

The analysis carried out above is general to any two-level system interacting with an electromagnetic field. In this section, we look at Rabi flopping in light-matter interaction. When driven by an electric field close to resonance, atoms periodically absorb and emit light, oscillating between excited ($|e\rangle$) and ground ($|g\rangle$) states. The atomic Hamiltonian can be written as

${\hat {H}}^{(A)}={\frac {\delta E}{2}}(|g\rangle \langle g|+|e\rangle \langle e|)={\frac {\hbar \omega _{eg}}{2}}{\hat {\sigma }}_{z}$ (25)

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This interacts with a monochromatic electric field produced by a laser,

${\vec {E}}(t)=Re\left({\vec {E}}_{0}e^{i(\phi -\omega _{L}t)}\right)$ (26)

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In the electric dipole approximation, the interaction Hamiltonian is

${\hat {H}}^{(AL)}=-{\frac {1}{2}}\left[{\hat {\vec {d}}}\cdot {\vec {\epsilon }}E_{0}e^{i\phi }e^{-\omega _{L}t}+{\hat {\vec {d}}}\cdot {\vec {\epsilon }}^{*}E_{0}e^{-i\phi }e^{\omega _{L}t}\right]$ (27)

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For light with linear polarization along the quantization axis, the dipole transition term $d_{eg}=\langle e|{\hat {\vec {d}}}\cdot {\vec {\epsilon }}|g\rangle$ can be chosen to be real and the interaction Hamiltonian takes the form:

${\hat {H}}^{(AL)}=-{\frac {d_{eg}E_{0}}{2}}({\hat {\sigma }}_{+}+{\hat {\sigma }}_{-})(e^{i\phi }e^{-i\omega _{L}t}+e^{-i\phi }e^{i\omega _{L}t})$ (28)

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which is analogous to the expression for the Hamiltonian for the linearly oscillating magnetic field we had previously. The Rabi frequency here is given by $\Omega =-{\frac {d_{eg}E_{0}}{\hbar }}$. Using the rotating wave approximation as before, the total Hamiltonian is expressed in the familiar form

${\hat {H}}_{RF}^{(int)}={\frac {\hbar \gamma B}{4}}({\hat {\sigma }}_{+}+{\hat {\sigma }}_{-})+{\frac {\hbar \gamma B}{4}}\left(e^{-2i\omega t}{\hat {\sigma }}_{+}+e^{2i\omega t}{\hat {\sigma }}_{-}|\right)$ (29)

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where $\Delta =\omega _{L}-\omega _{eg}$. Thus, with a laser driving the two-level atom close to resonance, we have Rabi oscillations between the excited and ground states.

Fully Quantum Approach

See also Jaynes–Cummings model

For a full quantum treatment of light-atom interaction, the electromagnetic field has to be quantized. The dynamics of a two-level atom interacting with light in an Electromagnetic cavity is studied using the Jaynes–Cummings model . The JCM Hamiltonian for an atom interacting with a single mode of an electromagnetic field, under the dipole and rotating wave approximations, is given by

${\hat {H}}={\frac {\hbar \omega _{eg}}{2}}{\hat {\sigma }}_{z}+\hbar \omega _{c}{\hat {a}}^{\dagger }{\hat {a}}+\hbar g\left({\hat {a}}{\hat {\sigma }}_{+}+{\hat {a}}^{\dagger }{\hat {\sigma }}_{-}\right)$ (29)

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where ${\hat {a}}$ and ${\hat {a}}^{\dagger }$ are creation and annihilation operators for the field. The eigenstates of the combined atom-photon system (in terms of the atom states $\{|g\rangle ,|e\rangle \}$ and the number states of the field $\{|n\rangle |n=0,1,2\cdots \}$ ) are: $\{|g\rangle \otimes |n\rangle ,|e\rangle \otimes |n\rangle |n=0,1,2,...\}$ The total number of excitations in the combined system, given by ${\hat {N}}_{T}={\hat {a}}^{\dagger }{\hat {a}}+{\hat {\sigma }}_{+}{\hat {\sigma }}_{-}$ is conserved, which implies that the JCM Hamiltonian is block-diagonal within the 2x2 subspaces defined by $\{|g\rangle \otimes |n\rangle ,|e\rangle \otimes |n-1\rangle \}$ for $N_{T}=n$.

The effective Hamiltonian within the subspace $\{|g\rangle \otimes |n\rangle ,|e\rangle \otimes |n-1\rangle \}$ is

${\hat {H}}_{n}=\hbar \omega _{c}\left(n-{\frac {1}{2}}\right){\hat {I}}-{\frac {\hbar \Delta }{2}}{\hat {\sigma }}_{3}+\hbar {\sqrt {n}}g{\hat {\sigma }}_{1}$ (30)

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The first term sets the reference for the energy levels and does not play any role in the dynamics. The remaining part of of ${\hat {H}}_{n}$ is identical to the Rabi Hamiltonian with a Rabi frequency $\Omega _{n}=2{\sqrt {n}}g$, driving oscillations between $|\uparrow \rangle =|g,n\rangle$ and $|\downarrow >=|e,n-1\rangle$.

Vacuum Rabi Oscillations

Vacuum Rabi Oscillation is a unique feature of the fully quantum model whereby we can see Rabi flopping even when there is initially no light in the cavity. The atom is initialized in the state $|e\rangle$ with the cavity at resonance ($\Delta =0$).
The time-evolved state is then

$|\psi (t)\rangle =\cos(2gt)|e,0\rangle -i\sin(2gt)|g,1\rangle$ (31)

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Therefore, it periodically flips between the two allowed states. The atom periodically emits a photon, exciting a cavity mode and then reabsorbs it. The entire process follows unitary dynamics, and is thus coherent.