interaction of photon with atoms - northern illinois universitypiot/phys_630/lesson17.pdf ·...

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


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]


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"(#)


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


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"(#)


• 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


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


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(#)


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


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


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)


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


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


Photoluminescence

• System excited to higher energy level by absorption and decay

Single-photon photoluminescence

Multiple-photon photoluminescence

Two-photon Three-photon Upconversion

interaction of photon with atoms - northern illinois universitypiot/phys_630/lesson17.pdf ·...

Documents