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Interaction of photon with

• Consider an located in an optical cavity and consider two of its

levels to be E1 and E2 (assume E1

• The cavity support an optical mode with ν0

• Chose v0 such that

h" = E 2 # E1 the matches the energy-level difference

• Three type!s of mechanism are possible: – – Absorption –

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 c • The probability density is p = "(#) sp V Transition cross [s-1] Volume of section [m2] cavity

!

P. Piot, PHYS 630 – Fall 2008 Absorption

• Atom is initially in state E1 • Process is induced by a photon: the photon is annihilated and the

atom go into E2 • The process is governed by same law as in spontaneous emission c p = "(#) ab V

• However is there are n photons in the optical mode the probability is increased by a factor n so ! nc P = "(#) ab V

this is the probability of absorption of one photon from a mode with n photons ! P. Piot, PHYS 630 – Fall 2008 Stimulated emission

• Atom is initially in “excited” state E2 the optical mode contain a photon • 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 that governs spontaneous emission and absorption c p = "(#) st V • If the mode has originally n photon, then the probability to stimulate emission of one photon is n times larger nc ! Pst = "(#) V • Note that total probability for an atom to emit a photon is (n +1)c P + p = "(#) st sp V !

• Since Pab=Pst there are usually written as Wi the probability density of stimulated emission and absorption ! P. Piot, PHYS 630 – Fall 2008 Lineshape function

• The transition cross section characterizes the interaction of the atom with the its area $ S = % "(#)d# 0 is called transition (or oscillator) strength and its shape gives the dependence of the magnitude of the interaction on

• The line-sh!ap e function g(ν) is #(") g(") = S It is normalized to unity, centered around the resonance frequency has units of Hz-1 its width is ~ the inverse of resonance bandwidth !

P. Piot, PHYS 630 – Fall 2008 Spontaneous Emission

• The equation c p = "(#) sp V gives the probability density for spontaneous emission into one mode of frequency n • The density of mode for a 3d resonator is ! 8#$ 2 Number of mode of frequency M(") = n per unit of volume of the c 3 cavity per unit of bandwidth • An atom may emit one photon in any of these modes • The overall spontaneous-emission probability is

! # c # Psp = $ [M(")V] % (")d" = c $ M(")% (")d" 0 V 0

! P. Piot, PHYS 630 – Fall 2008 Spontaneous Emission

• The overall spontaneous-emission probability is # ( ) is sharply peaked at = , P = c M(")% (")d" σ ν ν ν0 sp $ and M(ν) is slowly varying 0 # 8(S cM( ) ( )d cM( )S & " 0 $ % " " ' " 0 = 2 0 ) • A “spontaneous lifetime” can be defined as 1 1 8"S t = and therefore ! sp = 2 Psp tsp # • One can infer the transition strength from a measurement of the spontaneous lifetime • The average cross section is related the the lineshape function via ! ! $2 " (#) = g(#) 8%tsp

P. Piot, PHYS 630 – Fall 2008

! Stimulated emission & absorption

• Consider the interaction of a single-mode with an atom • Light characterized by its mean photon (photon/m2/s) I " = h# A • The photon flux is " = #A c c V • So " = n V! • And the! pr obability of stimulated emission is

Wi = "#($) ! so σ is a coefficient of proportionality between the probability of an induced transition and the photon flux !

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 "(#) d# • So average photon in [ν,ν+dν] is "(#)V h# c • E!ac h photon has the probability " ( # ) of initiating a transition V • So the overall probabi!lity + "(#)V % c ( Wi = , ' $(#)*d # 0! h# &V ) "(# ) + "(# ) - 0 c , $(#)d# = 0 cS h# 0 0 h# 0

P. Piot, PHYS 630 – Fall 2008 ! Transition induced by broad light II "3 • So Wi = $(% 0 ) 8#htsp • Defining the mean number of photon per mode to be "3 n = $(% ) ! 8#h 0 • We have n W = i t ! sp • Einstein A and B coefficients

P " A ! sp 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

#" /2$ g(") = 2 2 (" %" 0) + (#" /2)

• Origin of bandwidth: – Collision broadening – Do!p pler effects

P. Piot, PHYS 630 – Fall 2008 Example: application of Doppler broadening

• 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 frequency will be excited (counter propagating atoms) – Spontaneous emission velocity component along the laser direction decreases – Use high laser to induce significant damping

http://focus.aps.org/story/v21/st11 P. Piot, PHYS 630 – Fall 2008

• System excited to higher by absorption and decay

Single-photon photoluminescence

Two-photon Three-

Multiple-photon photoluminescence

P. Piot, PHYS 630 – Fall 2008