Rabi problem

Rabi problem

The Rabi problem concerns the response of an atom to an applied 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 driven, damped harmonic oscillator with the electric part of the Lorentz force as the driving term:

:ddot{x}_a + frac{2}{ au_0}dot{x}_a + omega_a^2 x_a = frac{e}{m} E(t,mathbf{r}_a),

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, omega_a its natural oscillation frequency, and au_0 its natural lifetime:

:frac{2}{ au_0} = frac{2 e^2 omega_a^2}{3 m c^3},

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:

:E = E_0 [e^{iomega t} + e^{-iomega t}]

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):

:x_a = x_0 (u_a cos omega t + v_a sin omega t)

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 (omega approx omega_a), then these values will be slowly varying in time, and we can make the assumption that dot{u}_a ll omega u_a, dot{v}_a ll omega v_a and ddot{u}_a ll omega^2 u_a, ddot{v}_a ll omega^2 v_a.

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

:dot{u} = -delta v - frac{u}{T}:dot{v} = delta u - frac{v}{T} + kappa E_0

where we have replaced the natural lifetime au_0 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 detuning delta = omega - omega_a, which serves equally well to distinguish atoms of different resonant frequencies. Finally, the constant kappa has been defined:

:kappa stackrel{mathrm{def{=} frac{e}{m omega x_0}

These equations can be solved as follows:

: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}: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}

After all transients have died away, the steady state solution takes the simple form,

:x_a(t) = frac{e}{m} E_0 left(frac{e^{iomega t{omega_a^2 - omega^2 + 2iomega/T} + mathrm{c.c.} ight)

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

Two-level atom

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 inversion, spontaneous emission, and the Bloch-Siegert shift, a fully quantum mechanical treatment is necessary.

The simplest approach is through the 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 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 optical Bloch equations, which define the dynamics of the pseudo-spin vector in an electric field:

:dot{u} = -delta v:dot{v} = delta u + kappa E w:dot{w} = -kappa E v

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 transformed into a set of coordinates rotating at a frequency omega.

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:

:frac{d}{dt} egin{bmatrix}u \v \w \end{bmatrix} = egin{bmatrix}0 & -delta & 0 \delta & 0 & kappa E \0 & -kappa E & 0end{bmatrix}egin{bmatrix}u \v \w \end{bmatrix}

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

:frac{dmathbf{ ho{dt} = mathbf{Omega} imesmathbf{ ho}

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

As before, the Rabi problem is solved by assuming the electric field "E" is oscillatory with constant magnitude "E0": E = E_0 (e^{iomega t} + mathrm{c.c.}). In this case, the solution can be found by applying two successive rotations to the matrix equation above, of the form

:egin{bmatrix}u \ v \ w end{bmatrix} = egin{bmatrix} cos chi & 0 & sinchi \0 & 1 & 0 \-sinchi & 0 & coschiend{bmatrix}egin{bmatrix}u' \ v' \ w'end{bmatrix}

and

:egin{bmatrix}u' \ v' \ w' end{bmatrix} = egin{bmatrix}1 & 0 & 0 \0 & cos Omega t & sinOmega t \0 & -sinOmega t & cosOmega tend{bmatrix}egin{bmatrix}u" \ v" \ w"end{bmatrix}

where

: an chi = frac{delta}{kappa E_0}:Omega(delta) = sqrt{delta^2 + (kappa E_0)^2}

Here, the frequency Omega(delta) 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 delta = 0), then the psedo-spin vector will precess about the "u" axis at a rate of kappa E_0. If this (on-resonance) pulse is shone on a collection of atoms originally all in their ground state ("w = -1") for a time Delta t = pi/kappa E_0, then after the pulse, the atoms will now all be in their "excited" state ("w = 1") because of the pi (or 180 degree) rotation about the "u" axis. This is known as a pi-pulse, and has the result of a complete inversion.

The general result is given by,

:egin{bmatrix}u\v\wend{bmatrix} =egin{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}sinOmega 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}egin{bmatrix}u_0 \ v_0 \ w_0end{bmatrix}

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,

:w(t;delta) = -1 + frac{2(kappa E_0)^2}{(kappa E_0)^2 + delta^2} sin^2 left(frac{Omega t}{2} ight)

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/

References

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

ee also

* Rabi cycle
* Rabi frequency
* Vacuum Rabi oscillation
* Jaynes-Cummings model


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Rabi — may refer to: * Rabi crop, spring harvest in India * Rabi cycle, in physics is the cyclic behaviour of a two state quantum system in the presence of an oscillatory driving field. See: Isidor Isaac Rabi * Rabi Island, volcanic island in northern… …   Wikipedia

  • Rabi cycle — In physics, the Rabi cycle is the cyclic behaviour of a two state quantum system in the presence of an oscillatory driving field. A two state system has two possible states, and if they are not degenerate energy levels the system can become… …   Wikipedia

  • problém — a m (ẹ̑) 1. kar je v zvezi z določenim dejstvom nejasno, neznano in je potrebno pojasniti ali rešiti, vprašanje: odkriti, opisati, razčleniti, rešiti problem; poglobiti se v problem; družbeni, filozofski problemi; lahek, nerešljiv, zapleten… …   Slovar slovenskega knjižnega jezika

  • Problem set — A problem set is a teaching tool used by many universities. Most courses in physics, math, engineering, chemistry, and computer science will give problem sets on a regular basis. [cite book last = Curzan first = Anne coauthors = Lisa Damour title …   Wikipedia

  • Isidor Isaac Rabi — Infobox Scientist name = Isidor Rabi imagesize = 180px caption = Isidor Isaac Rabi birth date = birth date|1898|7|29|df=y birth place = Rymanów, Galicia, Austria Hungary nationality = United States death date = death date and… …   Wikipedia

  • Vacuum Rabi oscillation — A vacuum Rabi oscillation is a damped oscillation of an initially excited atom coupled to an electromagnetic resonator or cavity in which the atom alternately emits photon(s) into a single mode electromagnetic cavity and reabsorbs them. The atom… …   Wikipedia

  • Частота Раби — Поведение во времени населенности возбужденного состояния двухуровневого атома для разных ситуаций: без учета (красная линия) и с учетом (синяя линия) «оттока» населенности на другие, третьи уровни. Населенность уровней в обоих случаях осциллируе …   Википедия

  • Jaynes-Cummings model — The Jaynes Cummings model (JCM) is a theoretical model in quantum optics. It describes the system of a two level atom interacting with a quantized mode of an optical cavity, with or without the presence of light. The JCM is of great interest in… …   Wikipedia

  • J. Robert Oppenheimer — J. Robert Oppenheimer, c. 1944 Born …   Wikipedia

  • Arnold Sommerfeld — Infobox Scientist box width = 300px name = Arnold Sommerfeld |250px image width = 250px caption = Arnold Johannes Wilhelm Sommerfeld (1868 1951) birth date = birth date|df=yes|1868|12|5 birth place = Königsberg, Province of Prussia death date =… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”