P. Piot, PHYS 630 – Fall 2008
• Consider an atom located in an optical cavity and consider two of itsenergy levels to be E1 and E2 (assume E1<E2 )
• The cavity support an optical mode with frequency ν0
• Chose v0 such that
the photon energy matches the energy-level difference
• Three types of mechanism are possible:– Spontaneous emission– Absorption– Stimulated emission
Interaction of photon with atoms
!
h" = E2# E
1
P. Piot, PHYS 630 – Fall 2008
Spontaneous emission• Atom is initially in “excited” state E2• Atom decays spontaneously and add the energy hv to the optical
mode• The process is independent of the number of photon already in the
optical mode• The probability density is
!
psp =c
V" (#)
[s-1] Volume of cavity
Transition crosssection [m2]
P. Piot, PHYS 630 – Fall 2008
Absorption• Atom is initially in state E1
• Process is induced by a photon: the photon is annihilated and theatom go into excited state E2
• The process is governed by same law as in spontaneous emission
• However is there are n photons in the optical mode the probability isincreased by a factor n so
this is the probability of absorption of one photon from a mode with nphotons
!
pab =c
V"(#)
!
Pab
=nc
V"(#)
P. Piot, PHYS 630 – Fall 2008
Stimulated emission
• Atom is initially in “excited” state E2 the optical mode contain aphoton
• Atom may be induced to emit another photon into the same mode• This is the inverse of the absorption process• The presence of a photon in the
mode stimulates the emission of a “clone” photon
P. Piot, PHYS 630 – Fall 2008
Stimulated emission
• The probability density of stimulated emission is same law thatgoverns spontaneous emission and absorption
• If the mode has originally n photon, then the probability to stimulateemission of one photon is n times larger
• Note that total probability for an atom to emit a photon is
• Since Pab=Pst there are usually written as Wi the probability density ofstimulated emission and absorption
!
pst =c
V" (#)
!
Pst
=nc
V" (#)
!
Pst + psp =(n +1)c
V"(#)
P. Piot, PHYS 630 – Fall 2008
• The transition cross section characterizes the interaction of the atomwith the radiation its area
is called transition (or oscillator) strength and its shape gives thedependence of the magnitude of the interaction on frequencies
• The line-shape function g(ν) is
It is normalized to unity, centered around the resonance frequencyhas units of Hz-1
its width is ~ the inverse of resonance bandwidth
Lineshape function
!
S = "(#)0
$
% d#
!
g(") =# (")
S
P. Piot, PHYS 630 – Fall 2008
Spontaneous Emission• The equation
gives the probability density for spontaneous emission into one modeof frequency n
• The density of mode for a 3d resonator is
• An atom may emit one photon in any of these modes• The overall spontaneous-emission probability is
!
psp =c
V" (#)
!
M(") =8#$ 2
c3
!
Psp = [M(")V ]0
#
$c
V% (" )d" = c M(")
0
#
$ % (" )d"
Number of mode of frequencyn per unit of volume of the cavity per unit of bandwidth
P. Piot, PHYS 630 – Fall 2008
Spontaneous Emission• The overall spontaneous-emission probability is
• A “spontaneous lifetime” can be defined as
and therefore
• One can infer the transition strength from a measurement of thespontaneous lifetime
• The average cross section is related the the lineshape function via
!
Psp = c M(")0
#
$ % (" )d"
& cM("0) % (" )0
#
$ d" ' cM("0)S =
8(S
)2
σ(ν) is sharply peaked at ν=ν0, and M(ν) is slowly varying
!
tsp =1
Psp
!
1
tsp
=8"S
#2
!
" (#) =$2
8%tspg(#)
P. Piot, PHYS 630 – Fall 2008
Stimulated emission & absorption• Consider the interaction of a single-mode light with an atom• Light characterized by its mean photon flux (photon/m2/s)
• The photon flux is
• So
• And the probability of stimulated emission is
so σ is a coefficient of proportionality between the probability of aninduced transition and the photon flux
!
" =I
h#
!
" =#A
!
" = nc
V
!
Wi="# ($ )
A
V c
P. Piot, PHYS 630 – Fall 2008
Transition induced by broad light I• Consider the interaction of a polychromatic light with an atom• The light spectral energy density (energy per unit volume per unit
banwidth) is
• So average photon in [ν,ν+dν] is
• Each photon has the probability of initiating a transition
• So the overall probability
!
"(#)
!
c
V"(#)
!
"(#)Vd#
h#
!
Wi=
"(#)V
h#
c
V$ (# )
%
& ' (
) * 0
+
, d#
-"(#
0)
h#0
c $(#)0
+
, d# ="(#
0)
h#0
cS
P. Piot, PHYS 630 – Fall 2008
Transition induced by broad light II• So
• Defining the mean number of photon per mode to be
• We have
• Einstein A and B coefficients
!
Wi ="3
8#htsp$(%
0)
!
n ="3
8#h$(%
0)
!
Wi =n
tsp
!
Psp " A
Wi " B#($ 0)
P. Piot, PHYS 630 – Fall 2008
Line broadening• We did not yet specified the lineshape function
• It usually follows a Lorentzian distribution of the form
• Origin of bandwidth:– Collision broadening– Doppler effects!
g(") =#" /2$
(" %"0)2
+ (#" /2)2
P. Piot, PHYS 630 – Fall 2008
Example: application of Doppler broadening
• Laser cooling of atom is based on Doppler broadening• Laser tuned to a frequency slightly below transition frequency
– Only atom with velocity matching the Doppler shifted frequencywill be excited (counter propagating atoms)
– Spontaneous emission velocity component along the laserdirection decreases
– Use high intensity laser to induce significant damping
http://focus.aps.org/story/v21/st11
P. Piot, PHYS 630 – Fall 2008
Photoluminescence
• System excited to higher energy level by absorption and decay
Single-photon photoluminescence
Multiple-photon photoluminescence
Two-photon Three-photon Upconversion