en>JRSpriggs: italicize variables; standardize appearance of formulas; some rewording; link "pairing function"

2010-08-01T11:04:18Z

italicize variables; standardize appearance of formulas; some rewording; link "pairing function"

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The '''Rabi problem''' concerns the response of an [[atom]] to an applied [[normal mode|harmonic]] [[electric field]], with an applied [[frequency]] very close to the atom's [[natural frequency]]. It provides a simple and generally solvable example of light-atom interactions.

== Classical Rabi Problem ==

In the classical approach, the Rabi problem can be represented by the solution to the [[harmonic oscillator|driven, damped harmonic oscillator]] with the electric part of the [[Lorentz force]] as the driving term:

:<math>\ddot{x}_a + \frac{2}{\tau_0}\dot{x}_a + \omega_a^2 x_a = \frac{e}{m} E(t,\mathbf{r}_a)</math>,

where it has been assumed that the atom can be treated as a charged particle (of charge ''e'') oscillating about its equilibrium position around a neutral atom. Here, ''xa'' is its instantaneous magnitude of oscillation, <math>\omega_a</math> its natural oscillation frequency, and <math>\tau_0</math> its [[natural lifetime]]:

:<math>\frac{2}{\tau_0} = \frac{2 e^2 \omega_a^2}{3 m c^3}</math>,

which has been calculated based on the [[dipole]] oscillator's energy loss from electromagnetic radiation.

To apply this to the Rabi problem, one assumes that the electric field ''E'' is oscillatory in time and constant in space:

:<math>E = E_0[e^{i\omega t} + e^{-i\omega t}]</math>

and ''xa'' is decomposed into a part ''ua'' that is in-phase with the driving ''E'' field (corresponding to dispersion), and a part ''va'' that is out of phase (corresponding to absorption):

:<math>x_a = x_0 (u_a \cos \omega t + v_a \sin \omega t)</math>

Here, ''x0'' is assumed to be constant, but ''ua'' and ''va''are allowed to vary in time. However, if we assume we are very close to resonance (<math>\omega \approx \omega_a</math>), then these values will be slowly varying in time, and we can make the assumption that <math>\dot{u}_a \ll \omega u_a</math>, <math>\dot{v}_a \ll \omega v_a</math> and <math>\ddot{u}_a \ll \omega^2 u_a</math>, <math>\ddot{v}_a \ll \omega^2 v_a</math>.

With these assumptions, the Lorentz force equations for the in-phase and out-of-phase parts can be rewritten as,

:<math>\dot{u} = -\delta v - \frac{u}{T}</math>
:<math>\dot{v} = \delta u - \frac{v}{T} + \kappa E_0</math>

where we have replaced the natural lifetime <math>\tau_0</math> with a more general ''effective'' lifetime ''T'' (which could include other interactions such as collisions), and have dropped the subscript ''a'' in favor of the newly defined [[laser detuning|detuning]] <math>\delta = \omega - \omega_a</math>, which serves equally well to distinguish atoms of different resonant frequencies. Finally, the constant <math>\kappa</math> has been defined:

:<math>\kappa \ \stackrel{\mathrm{def}}{=}\ \frac{e}{m \omega x_0}</math>

These equations can be solved as follows:

:<math>u(t;\delta) = [u_0 \cos \delta t - v_0 \sin \delta t]e^{-t/T} + \kappa E_0 \int_0^t dt' \sin \delta(t-t')e^{-(t-t')/T}</math>
:<math>v(t;\delta) = [u_0 \cos \delta t + v_0 \sin \delta t]e^{-t/T} - \kappa E_0 \int_0^t dt' \cos \delta(t-t')e^{-(t-t')/T}</math>

After all [[Transient state|transients]] have died away, the steady state solution takes the simple form,

:<math>x_a(t) = \frac{e}{m} E_0 \left(\frac{e^{i\omega t}}{\omega_a^2 - \omega^2 + 2i\omega/T} + \mathrm{c.c.}\right)</math>

where "c.c." stands for the [[complex conjugate]] of the opposing term.

== Two-level atom ==

===Semiclassical approach===
:See also [[optical Bloch equations]]
The classical Rabi problem gives some basic results and a simple to understand picture of the issue, but in order to understand phenomena such as [[Population inversion|inversion]], [[spontaneous emission]], and the [[Bloch-Siegert shift]], a fully [[quantum mechanics|quantum mechanical]] treatment is necessary.

The simplest approach is through the [[two-state quantum system|two-level atom]] approximation, in which one only treats two energy levels of the atom in question. No atom with only two energy levels exists in reality, but a transition between, for example, two [[hyperfine splitting|hyperfine states]] in an atom can be treated, to first approximation, as if only those two levels existed, assuming the drive is not too far off resonance.

The convenience of the two-level atom is that any two-level system evolves in essentially the same way as a [[spin-1/2]] system, in accordance to the [[Bloch equations]], which define the dynamics of the [[pseudo-spin vector]] in an electric field:

:<math>\dot{u} = -\delta v</math>
:<math>\dot{v} = \delta u + \kappa E w</math>
:<math>\dot{w} = -\kappa E v</math>

where we have made the [[rotating wave approximation]] in throwing out terms with high angular velocity (and thus small effect on the total spin dynamics over long time periods), and [[Coordinate rotation|transformed]] into a set of coordinates rotating at a frequency <math>\omega</math>.

There is a clear analogy here between these equations and those that defined the evolution of the in-phase and out-of-phase components of oscillation in the classical case. Now, however, there is a third term ''w'' which can be interpreted as the population difference between the excited and ground state (varying from -1 to represent completely in the ground state to +1, completely in the excited state). Keep in mind that for the classical case, there was a continuous energy spectra that the atomic oscillator could occupy, while for the quantum case (as we've assumed) there are only two possible (eigen)states of the problem.

These equations can be also be stated in matrix form:

:<math>\frac{d}{dt} \begin{bmatrix}
u \\
v \\
w \\
\end{bmatrix} =
\begin{bmatrix}
0 & -\delta & 0 \\
\delta & 0 & \kappa E \\
0 & -\kappa E & 0
\end{bmatrix}
\begin{bmatrix}
u \\
v \\
w \\
\end{bmatrix}
</math>

It is noteworthy that these equations can be written as a vector precession equation:

:<math>\frac{d\mathbf{\rho}}{dt} = \mathbf{\Omega}\times\mathbf{\rho}</math>

where <math>\mathbf{\rho}=(u,v,w)</math> is the pseudo-spin vector and <math>\mathbf{\Omega} = (-\kappa E, 0, \delta)</math> acts as an effective torque.

As before, the Rabi problem is solved by assuming the electric field ''E'' is oscillatory with constant magnitude ''E0'': <math>E = E_0 (e^{i\omega t} + \mathrm{c.c.})</math>. In this case, the solution can be found by applying two successive rotations to the matrix equation above, of the form

:<math>\begin{bmatrix}
u \\ v \\ w \end{bmatrix} =
\begin{bmatrix} \cos \chi & 0 & \sin\chi \\
0 & 1 & 0 \\
-\sin\chi & 0 & \cos\chi
\end{bmatrix}
\begin{bmatrix}
u' \\ v' \\ w'
\end{bmatrix}</math>

and

:<math>\begin{bmatrix}
u' \\ v' \\ w' \end{bmatrix} =
\begin{bmatrix}
1 & 0 & 0 \\
0 & \cos \Omega t & \sin\Omega t \\
0 & -\sin\Omega t & \cos\Omega t
\end{bmatrix}
\begin{bmatrix}
u'' \\ v'' \\ w''
\end{bmatrix}</math>

where

:<math>\tan \chi = \frac{\delta}{\kappa E_0}</math>
:<math>\Omega(\delta) = \sqrt{\delta^2 + (\kappa E_0)^2}</math>

Here, the frequency <math>\Omega(\delta)</math> is known as the [[generalized Rabi frequency]], which gives the rate of [[precession]] of the pseudo-spin vector about the transformed ''u' ''-axis (given by the first coordinate transformation above). As an example, if the electric field (or [[laser]]) is exactly on resonance (such that <math>\delta = 0</math>), then the psedo-spin vector will precess about the ''u'' axis at a rate of <math>\kappa E_0</math>. If this (on-resonance) pulse is shone on a collection of atoms originally all in their ground state (''w = -1'') for a time <math>\Delta t = \pi/\kappa E_0</math>, then after the pulse, the atoms will now all be in their ''excited'' state (''w = 1'') because of the <math>\pi</math> (or 180 degree) rotation about the ''u'' axis. This is known as a <math>\pi</math>-pulse, and has the result of a complete inversion.

The general result is given by,

:<math>\begin{bmatrix}
u\\v\\w
\end{bmatrix} =
\begin{bmatrix}
\frac{(\kappa E_0)^2 + \delta^2 \cos \Omega t}{\Omega^2} & -\frac{\delta}{\Omega} \sin{\Omega t} & -\frac{\delta \kappa E_0}{\Omega^2} (1-\cos \Omega t) \\
\frac{\delta}{\Omega}\sin\Omega t & \cos \Omega t & \frac{\kappa E_0}{\Omega}\sin \Omega t \\
\frac{\delta \kappa E_0}{\Omega^2} (1-\cos \Omega t) & -\frac{\kappa E_0}{\Omega} \sin{\Omega t} & \frac{\delta^2 + (\kappa E_0)^2 \cos \Omega t}{\Omega^2}
\end{bmatrix}
\begin{bmatrix}
u_0 \\ v_0 \\ w_0
\end{bmatrix}</math>

The expression for the inversion ''w'' can be greatly simplifed if the atom is assumed to be initially in its ground state (''w0 = -1'') with ''u0 = v0 = 0'', in which case,

:<math>w(t;\delta) = -1 + \frac{2(\kappa E_0)^2}{(\kappa E_0)^2 + \delta^2} \sin^2 \left(\frac{\Omega t}{2}\right)</math>



==== Multimedia ====
A Java applet that visualizes Rabi Cycles of two-state systems (laser driven):
* http://www.itp.tu-berlin.de/menue/lehre/owl/quantenmechanik/zweiniveau/parameter/en/

===Quantum field theory approach===
In Bloch's approach, the field is not quantumized, and neither the resulting coherence nor the resonnance is well explained.

Need work for the QFT approach, mainly Jaynes-Cummings model.

== References ==

* L. Allen and J. H. Eberly, ''Optical Resonance and Two-Level Atoms'', (Dover: New York, 1987).

==See also==
* [[Rabi cycle]]
* [[Rabi frequency]]
* [[Vacuum Rabi oscillation]]
* [[Jaynes-Cummings model]]

[[Category:Atomic physics]]

Code (set theory) - Revision history

en>JRSpriggs: italicize variables; standardize appearance of formulas; some rewording; link "pairing function"