Time-dependent Perturbation TheoryPeriodic Perturbation:

The first order approximation to the coefficient ck(t) is

cn(0) = 0, n ! m c

n(0) = 1, n = m

!H (t) = !H cos("t)

ck

1(t) = !

i

!d "t #

k

0 "H ( "t ) #m

0e!i

!Em0 !Ek

0( ) "t

0

t

$

Initial conditions:

! (t) = e"i

!H (t )t

! (0) = cn(t)e

"i

!En

0t

!n

0

n

#

H (t) = H0+ !H (t)

Time evolution:

where the sinusoidal perturbation can be expressed as

!H (t) = !H cos("t) = !H ei" t

+ !H†e# i" t( ) 2

Time-dependent Perturbation TheoryPeriodic Perturbation:

The first order approximation to the coefficient ck(t):

cn(0) = 0, n ! m c

n(0) = 1, n = m

!H (t) = !H cos("t)

ck

1(t) = !

i

!d "t #

k

0 "H ( "t ) #m

0e!i

!Em0 !Ek

0( ) "t

0

t

$

Initial conditions:

H (t) = H0+ !H (t)

!H (t) = !H ei" t

+ !H†e# i" t( ) 2

ck

1(t) = !

i

2!"

k

0 #H "m

01! exp i E

k

0 ! Em

0+ !$( )t !%& '(

Ek

0 ! Em

0+ !$

!i

2!"

m

0 #H † "k

01! exp i E

k

0 ! Em

0 ! !$( )t !%& '(Ek

0 ! Em

0 ! !$

Time-dependent Perturbation TheoryPeriodic Perturbation:

Transition probability:

cn(0) = 0, n ! m c

n(0) = 1, n = m

!H (t) = !H cos("t)

Initial conditions:

H (t) = H0+ !H (t)

Pkm(t) =

!i

2!"

k

0 #H "m

01! exp i E

k

0 ! Em

0+ !$( )t !%& '(

Ek

0 ! Em

0+ !$

!i

2!"

m

0 #H † "k

01! exp i E

k

0 ! Em

0 ! !$( )t !%& '(Ek

0 ! Em

0 ! !$

2

The probability of transition to state k is large only if the denominatorin one of the two terms is small.

Time-dependent Perturbation TheoryPeriodic Perturbation:

The second term in the expression for the transition probabilitydominates when

cn(0) = 0, n ! m c

n(0) = 1, n = m

!H (t) = !H cos("t)

Initial conditions:

H (t) = H0+ !H (t)

Pkm(t) =

1

!2

!k

0 "H !m

02

F(t,# )

Ek

0! E

m

0( ) ! !" # 0

!km

=Ek

0" E

m

0

!

F(t,! ) =sin

2(!

km"! )t 2[ ]

!km"!( )

2

F

ωkm −ω

Resonance : Pkm is max if (absorbed energy). !

km"! = 0# E

k

0" E

m

0= !!

Time-dependent Perturbation TheoryEmission and Absorption of Radiation

Consider and atom interacting with an electromagnetic field polarized inthe z direction:

!E = E

0z cos(!t)

The perturbation Hamiltonian is then

!H (t) = !H cos("t) !H = "eE0zwith

Therefore, using our previous analysis of periodic perturbations,

Pkm(t) =

1

!2

!k

0 "H !m

02

F(t,# ) =2e

2u

$0!2

!k

0z !

m

02 sin

2(#

km%# )t 2[ ]

#km%#( )

2

u =!0

2E0

2 is the energy density in the E-M wave

Time-dependent Perturbation TheoryEmission and Absorption of RadiationConsider and atom interacting with an electromagnetic field polarized inthe z direction:

Pkm(t) =

1

!2

!k

0 "H !m

02

F(t,# ) =2e

2u

$0!2

!k

0z !

m

02 sin

2(#

km%# )t 2[ ]

#km%#( )

2

- This describes absorption of radiation, resulting in periodictransitions.- Note that Pkm(t) = Pmk(t). This describes transition from level k tolower energy level m with stimulated emission of a photon.- The atom can also spontaneously go from the k to m level without thedriving field resulting in spontaneous emission of a photon

Absorption Stimulated emission Spontaneous emission

Time-dependent Perturbation TheoryEmission and Absorption of Radiation

Consider and atom interacting with an electromagnetic field polarized inthe z direction:

If the electromagnetic radiation is made up of a range of frequencies withenergy density ρ(ω)dω in the frequency range dω, then

Pkm(t) =

1

!2

!k

0 "H !m

02

F(t,# ) =eE

0

!

$%&

'()2

!k

0z !

m

02 sin

2(#

km*# )t 2[ ]

#km*#( )

2

Pkm(t) ==

2e2

!0!2

"k

0z "

m

02 sin

2(#

km$# )t 2[ ]

#km$#( )

2

0

%

& '(# )d#

Time-dependent Perturbation TheoryEmission and Absorption of Radiation

Consider and atom interacting with an electromagnetic field polarized inthe z direction:

Since F(t,ω) is sharply peaked at ωkm and if ρ(ω) is broad, then

Pkm(t) ==

2e2

!0!2

"k

0z "

m

02 sin

2(#

km$# )t 2[ ]

#km$#( )

2

0

%

& '(# )d#

Pkm(t) !

2e2

"0!2

#k

0z #

m

02

$(%km)sin

2(%

km&% )t 2[ ]

%km&%( )

2

0

'

( d% )*e2

"0!2

#k

0z #

m

02

$(%km)t

Wkm

=d

dtPkm(t) =

!e2

"0!2

#k

0z #

m

02

$(%km) (Fermi’s rule)

Time-dependent Perturbation TheoryEmission and Absorption of Radiation

Consider and atom interacting with an electromagnetic field polarized inthe z direction:

Wkm

=d

dtPkm(t) =

!e2

"0!2

#k

0z #

m

02

$(%km) (Fermi’s rule)

If the radiation is coming from all directions and assuming all possiblepolarizations, then we must average over all directions and polarizations,which results in

Wkm

=d

dtPkm(t) = B

km!("

km)

Bkm

=!e2

3"0!2

#k

0 "r #

m

02

The absorption coefficient Bkm = stimulated emission coefficient Bmk

Time-dependent Perturbation TheoryEmission and Absorption of RadiationConsider N atoms with Nk of them in the energy state k and Nm of them inthe energy state m. Then we can write a rate equation:

dNk

dt= !N

kA ! N

kBmk"(#

km) + N

mBkm"(#

km)

If the atoms are in thermal equilibrium at temperature T, then dNk

dt= 0

and the ratio of atoms in each level follows the Boltzman distribution:

Nm

Nk

= e

!!km

kBT

The first term on the RHS corresponds to spontaneous emission, thesecond term to stimulated emission and the last to absorption.

Time-dependent Perturbation TheoryEmission and Absorption of Radiation

dNk

dt= !N

kA ! N

kBmk"(#

mk) + N

mBkm"(#

km)

dNk

dt= 0

Nm

Nk

= e

!!km

kBT

!("km) =

A

Bmke

!"km

kBT # Bkm

Consider N atoms with Nk of them in the energy state k and Nm of them inthe energy state m.

Time-dependent Perturbation TheoryEmission and Absorption of Radiation

!("km) =

A

Bmke

!"km

kBT # Bkm

Comparing to Planck’s blackbody radiation formula,

!("km) =!" 3 # 2

c3

e

!"km

kBT $1

Bmk

= Bkm

Confirms that absorption rate =stimulated emission rate. This is calledthe Einstein B coefficient

Consider N atoms with Nk of them in the energy state k and Nm of them inthe energy state m.

Time-dependent Perturbation TheoryEmission and Absorption of Radiation

!("mk) =

A

Bmke

!"km

kBT # Bkm

Comparing to Planck’s blackbody radiation formula,

!("mk) =!" 3 # 2

c3

e

!"km

kBT $1

A =!!

km

3

"2c3Bkm

The Einstein A coefficient gives us thespontaneous emission rate

Consider N atoms with Nk of them in the energy state k and Nm of them inthe energy state m.

Time-dependent Perturbation TheoryEmission and Absorption of RadiationConsider N atoms with Nk of them in the energy state k and Nm of them inthe energy state m.

A =!!

km

3

"2c3Bkm

Recall from our past analysis

Bkm

=!e2

3"0!2

#k

0 "r #

m

02

A =!

km

3e2

3"#0!c

3$

k

0 "r $

m

02

Time-dependent Perturbation TheoryEmission and Absorption of Radiation

If there is no driving field then the rate equation becomes

with solutionN

k(t) = N

k(0)e

!At

dNk

dt= !N

kA

The time taken for Nk to reach 1/e of its original value is called thelifetime τ of the state k:

! =1

A

If the state k can make transitions to several states with different Acoefficients, then

! =1

A1+ A

2+ A

3+ ...