The Rabi problem concerns the response of an
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
to an applied
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
, with an applied
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
very close to the atom's
natural frequency
Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force.
The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all pa ...
. It provides a simple and generally solvable example of light–atom interactions and is named after
Isidor Isaac Rabi
Isidor Isaac Rabi (; born Israel Isaac Rabi, July 29, 1898 – January 11, 1988) was an American physicist who won the Nobel Prize in Physics in 1944 for his discovery of nuclear magnetic resonance, which is used in magnetic resonance ima ...
.
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
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
as the driving term:
:
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 ''x
a'' is its instantaneous magnitude of oscillation,
its natural oscillation frequency, and
its
natural lifetime:
:
which has been calculated based on the
dipole
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
*An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system i ...
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:
:
and ''x
a'' is decomposed into a part ''u
a'' that is in-phase with the driving ''E'' field (corresponding to dispersion) and a part ''v
a'' that is out of phase (corresponding to absorption):
:
Here ''x
0'' is assumed to be constant, but ''u
a'' and ''v
a'' are allowed to vary in time. However, if the system is very close to resonance (
), then these values will be slowly varying in time, and we can make the assumption that
,
and
,
.
With these assumptions, the Lorentz force equations for the in-phase and out-of-phase parts can be rewritten as
:
:
where we have replaced the natural lifetime
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 , which serves equally well to distinguish atoms of different resonant frequencies. Finally, the constant
:
has been defined.
These equations can be solved as follows:
:
:
After all
transients
Transience or transient may refer to:
Music
* ''Transient'' (album), a 2004 album by Gaelle
* ''Transience'' (Steven Wilson album), 2015
* Transience (Wreckless Eric album)
Science and engineering
* Transient state, when a process variable or ...
have died away, the steady-state solution takes the simple form
:
where "c.c." stands for the
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of the opposing term.
Two-level atom
Semiclassical approach
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
Inversion or inversions may refer to:
Arts
* , a French gay magazine (1924/1925)
* ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas
* Inversion (music), a term with various meanings in music theory and musical set theory
* ...
,
spontaneous emission, and the
Bloch–Siegert shift
The Bloch–Siegert shift is a phenomenon in quantum physics that becomes important for driven two-level systems when the driving gets strong (e.g. atoms driven by a strong laser drive or nuclear spins in NMR, driven by a strong oscillating magne ...
, 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
In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of . The spin number describes how many symmetrical facets a particle has in one full ...
system, in accordance to the
Bloch equations, which define the dynamics of the
pseudo-spin vector in an electric field:
:
:
:
where we have made the
rotating wave approximation
The rotating-wave approximation is an approximation used in atom optics and magnetic resonance. In this approximation, terms in a Hamiltonian that oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic rad ...
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
.
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 spectrum 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 also be stated in matrix form:
:
It is noteworthy that these equations can be written as a vector
precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
equation:
:
where
is the pseudo-spin vector, and
acts as an effective torque.
As before, the Rabi problem is solved by assuming that the electric field ''E'' is oscillatory with constant magnitude ''E
0'':
. In this case, the solution can be found by applying two successive rotations to the matrix equation above, of the form
:
and
:
where
:
:
Here the frequency
is known as the
generalized Rabi frequency, which gives the rate of
precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
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
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
) is exactly on resonance (such that
), then the pseudo-spin vector will precess about the ''u'' axis at a rate of
. If this (on-resonance) pulse is shone on a collection of atoms originally all in their ground state (''w'' = −1) for a time
, then after the pulse, the atoms will now all be in their ''excited'' state (''w'' = +1) because of the
(or 180°) rotation about the ''u'' axis. This is known as a
-pulse and has the result of a complete inversion.
The general result is given by
:
The expression for the inversion ''w'' can be greatly simplified if the atom is assumed to be initially in its ground state (''w''
0 = −1) with ''u''
0 = ''v''
0 = 0, in which case
:
Rabi Problem in time-dependent perturbation theory
In the quantum approach, the periodic driving force can be considered as periodic perturbation and, therefore, the problem can be solved using time-dependent perturbation theory, with
:
where
is the time-independent Hamiltonian that gives the original eigenstates, and
is the time-dependent perturbation. Assume at time
, we can expand the state as
:
where
represents the eigenstates of the unperturbed states. For an unperturbed system,
is a constant.
Now, let's calculate
under a periodic perturbation
. Applying operator
on both sides of the previous equation, we can get
:
and then multiply both sides of the equation by
:
:
When the excitation frequency is at resonance between two states
and
, i.e.
, it becomes a normal-mode problem of a two-level system, and it is easy to find that
:
where
The possibility of being in the state ''m'' at time ''t'' is
:
The value of
depends on the initial condition of the system.
An exact solution of spin-1/2 system in an oscillating magnetic field is solved by Rabi (1937). From their work, it is clear that the Rabi oscillation frequency is proportional to the magnitude of oscillation magnetic field.
Quantum field theory approach
In Bloch's approach, the field is not quantized, and neither the resulting coherence nor the resonance is well explained.
for the QFT approach, mainly
Jaynes–Cummings model
The Jaynes–Cummings model (sometimes abbreviated 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 (or a bosonic field), with or without the prese ...
.
See also
*
Rabi cycle
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, ...
*
Rabi frequency The Rabi frequency is the frequency at which the probability amplitudes of two atomic energy levels fluctuate in an oscillating electromagnetic field. It is proportional to the Transition Dipole Moment of the two levels and to the amplitude (''not ...
*
Vacuum Rabi oscillation
References
*
* {{cite journal , last=Rabi , first=I. I. , title=Space Quantization in a Gyrating Magnetic Field , journal=Physical Review , publisher=American Physical Society (APS) , volume=51 , issue=8 , date=1937-04-15 , issn=0031-899X , doi=10.1103/physrev.51.652 , pages=652–654
Atomic physics
Spintronics