<s>
The	O
Rabi	B-General_Concept
problem	I-General_Concept
concerns	O
the	O
response	O
of	O
an	O
atom	B-Language
to	O
an	O
applied	O
harmonic	O
electric	O
field	O
,	O
with	O
an	O
applied	O
frequency	O
very	O
close	O
to	O
the	O
atom	B-Language
's	O
natural	O
frequency	O
.	O
</s>
<s>
It	O
provides	O
a	O
simple	O
and	O
generally	O
solvable	O
example	O
of	O
light	O
–	O
atom	B-Language
interactions	O
and	O
is	O
named	O
after	O
Isidor	O
Isaac	O
Rabi	O
.	O
</s>
<s>
In	O
the	O
classical	O
approach	O
,	O
the	O
Rabi	B-General_Concept
problem	I-General_Concept
can	O
be	O
represented	O
by	O
the	O
solution	O
to	O
the	O
driven	O
damped	O
harmonic	O
oscillator	O
with	O
the	O
electric	O
part	O
of	O
the	O
Lorentz	O
force	O
as	O
the	O
driving	O
term	O
:	O
</s>
<s>
where	O
it	O
has	O
been	O
assumed	O
that	O
the	O
atom	B-Language
can	O
be	O
treated	O
as	O
a	O
charged	O
particle	O
(	O
of	O
charge	O
e	O
)	O
oscillating	O
about	O
its	O
equilibrium	O
position	O
around	O
a	O
neutral	O
atom	B-Language
.	O
</s>
<s>
To	O
apply	O
this	O
to	O
the	O
Rabi	B-General_Concept
problem	I-General_Concept
,	O
one	O
assumes	O
that	O
the	O
electric	O
field	O
E	O
is	O
oscillatory	O
in	O
time	O
and	O
constant	O
in	O
space	O
:	O
</s>
<s>
The	O
classical	O
Rabi	B-General_Concept
problem	I-General_Concept
gives	O
some	O
basic	O
results	O
and	O
a	O
simple	O
to	O
understand	O
picture	O
of	O
the	O
issue	O
,	O
but	O
in	O
order	O
to	O
understand	O
phenomena	O
such	O
as	O
inversion	O
,	O
spontaneous	O
emission	O
,	O
and	O
the	O
Bloch	O
–	O
Siegert	O
shift	O
,	O
a	O
fully	O
quantum-mechanical	O
treatment	O
is	O
necessary	O
.	O
</s>
<s>
The	O
simplest	O
approach	O
is	O
through	O
the	O
two-level	O
atom	B-Language
approximation	O
,	O
in	O
which	O
one	O
only	O
treats	O
two	O
energy	O
levels	O
of	O
the	O
atom	B-Language
in	O
question	O
.	O
</s>
<s>
No	O
atom	B-Language
with	O
only	O
two	O
energy	O
levels	O
exists	O
in	O
reality	O
,	O
but	O
a	O
transition	O
between	O
,	O
for	O
example	O
,	O
two	O
hyperfine	O
states	O
in	O
an	O
atom	B-Language
can	O
be	O
treated	O
,	O
to	O
first	O
approximation	O
,	O
as	O
if	O
only	O
those	O
two	O
levels	O
existed	O
,	O
assuming	O
the	O
drive	O
is	O
not	O
too	O
far	O
off	O
resonance	O
.	O
</s>
<s>
The	O
convenience	O
of	O
the	O
two-level	O
atom	B-Language
is	O
that	O
any	O
two-level	O
system	O
evolves	O
in	O
essentially	O
the	O
same	O
way	O
as	O
a	O
spin-1/2	O
system	O
,	O
in	O
accordance	O
to	O
the	O
Bloch	B-Algorithm
equations	I-Algorithm
,	O
which	O
define	O
the	O
dynamics	O
of	O
the	O
pseudo-spin	O
vector	O
in	O
an	O
electric	O
field	O
:	O
</s>
<s>
where	O
we	O
have	O
made	O
the	O
rotating	O
wave	O
approximation	O
in	O
throwing	O
out	O
terms	O
with	O
high	O
angular	O
velocity	O
(	O
and	O
thus	O
small	O
effect	O
on	O
the	O
total	O
spin	O
dynamics	O
over	O
long	O
time	O
periods	O
)	O
and	O
transformed	B-General_Concept
into	O
a	O
set	O
of	O
coordinates	O
rotating	O
at	O
a	O
frequency	O
.	O
</s>
<s>
As	O
before	O
,	O
the	O
Rabi	B-General_Concept
problem	I-General_Concept
is	O
solved	O
by	O
assuming	O
that	O
the	O
electric	O
field	O
E	O
is	O
oscillatory	O
with	O
constant	O
magnitude	O
E0	O
:	O
.	O
</s>
<s>
Here	O
the	O
frequency	O
is	O
known	O
as	O
the	O
generalized	O
Rabi	O
frequency	O
,	O
which	O
gives	O
the	O
rate	O
of	O
precession	O
of	O
the	O
pseudo-spin	O
vector	O
about	O
the	O
transformed	B-General_Concept
u	O
axis	O
(	O
given	O
by	O
the	O
first	O
coordinate	O
transformation	O
above	O
)	O
.	O
</s>
