lecture 19 radiative transitions - tcm groupbds10/aqp/lec19_compressed.pdf · lecture 19 radiative...

Lecture 19

Radiative transitions

Radiative transitions: background

Previously, we have formulated a quantum theory of atoms (matter)coupling to a classical time-independent electromagnetic field, cf.Zeeman and Stark effects.

To develop a fully quantum theory of light-matter systems, we haveto address both the quantum theory of the electromagnetic fieldand formulate a theory of the coupling of light to matter.

In the following we will address each of these components in turn,starting with light-matter coupling.

Our motivation for developing such a consistent theory is that it:

(a) provides a platform to study radiative tranistions in atoms(which will address)

(b) forms the basis of quantum optics (which will not address but which is well-represented in subsequent courses).

Radiative transitions: outline

Coupling of matter to electromagnetic field

Spontaneous emission, absorption and stimulated emission

Einsteins A and B coefficients

Selection rules

Theory of the laser and coherent states

Coupling of matter to the electromagnetic field

For a single-electron atom in a time-dependent external EM field,the Hamiltonian takes the form,

Hatom =1

2m(p + eA(r, t))2 e(r, t) + V (r)

(with a straightforward generalization to multi-electron atoms).

Previously, we have seen that it is profitable to expand Hamiltonian

in A, Hatom = H0 + Hpara + Hdia., where H0 =p2

2m+ V (r) e(t)

Hpara(t) =e

mA(t) p

the paramagnetic term describes coupling of the atom to the EMfield, and Hdia = (eA)2/2m represents diamagnetic term.

Since we will be interested in absorption and emission of singlephotons, influence of diamagnetic term is (as usual) negligible.


When quantized, EM field is described by the photon Hamiltonian,

Hrad =

k,=1,2

~k(

akak +1

2

), k = c |k|

where ak/ak create/annihilate photons with polarization .

These operators obey (bosonic) commutation relations,

[ak, ak ] = k,k, , [ak, ak ] = [a

k, a

k ] = 0

and act on photon number states, |nk = 1nk! (ak)

nk |, as

ak|nk =

nk|nk 1, ak|nk =

nk + 1|nk + 1

With these definitions, the vector potential is given by

A(r) =

k=1,2

~

20kV

[ekake

ikr + ekakeikr

]

Heisenberg representation

Hrad =k

~k(

akak +1

2

)

To determine radiative transition rates, we will exploit FermisGolden rule. To prepare for this, it is convenient to transfertime-dependence to operators (Heisenberg representation).

As with any operator, the field operators obey equations of motion,

ak =i

~[H, ak] = ik[a

kak, ak] = ikak

i.e. ak(t) = ak(0)eikt , ak(t) = a

k(0)e

ikt and

A(r, t) =k

~

20kV

[ekake

i(krkt) + ekakei(krkt)

]


Putting together all of these components, the total Hamiltonian isthen given by H = Hatom + Hpara + Hrad where (with = 0)

Hatom =p2

2m+ V (r), Hrad =

k

~k(

akak +1

2

)and

Hpara(t) =e

mA(r, t) p with

A(r, t) =k

~

20kV

[ekake


]In the following, we will apply this Hamiltonian to the problem ofradiative transitions in single electron atoms.

Spontaneous emission

Consider probability for an atom, initially instate |i to make transition to |f withemission of a photon of wavevector k andpolarization spontaneous emission.

If radiation field initially prepared in vacuum state, |, then finalstate involves one photon, ak|.Therefore, making use of Fermis Golden rule, with the perturbation

Hpara(t) =e

mA(r, t) p =

k

~

20kV

[ekake


] em

p

transition probability given by,

if(t) =2

~2|f| k|Hpara|i ||2(if )

if,k =2

~

f| |ak em

~20kV

ekakeikr p|i |

2

(Ei Ef ~k)

Spontaneous emission

if,k =2

~

f| em

~20kV

eikrek p|i

2

(Ei Ef ~k)

To determine transition rate, we must analyse matrix elements ofthe form f|eikrek p|i. For typical state, ek p p Zmc.But what about exponential factor? With r ~/p ' ~/mZc, andk = c |k| p

2

2m (for electronic transitions), we have

k r ' kc

~p' ~p

mc' Z

i.e. for Z 1, we can expand exponential as power series in k rwith lowest terms dominant.

Taking zeroth order term, and using p = im~ [H0, r] (cf. Ehrenfest)

f|ek p|i = mek f|i

~[H0, r]|i = im

Ef Ei~

ek f|r|i

f|ek p|i = imkf|ek r|i

Spontaneous emission: electric dipole approximation

f|ek p|i = imkf|ek r|i

This result, which emerges from leading approximation in Z, isknown as electric dipole approximation: Effectively, we have set

Hpara =e

mA(r, t) p ' eE(r, t) r = E(r, t) d

translating to the potential energy of a dipole, with momentd = er, in an oscillating electric field.

Stimulated absorption and emission

Consider now absorption of a photon. If weassume that, in the initial state, there are nkphotons in mode (k) then, after thetransition, there will be nk 1 photons.

Then, if initial state of the atom is |i and final state is |f,

f| (nk, 1)|Hpara|i |nk

= f| (nk, 1)|e

m

~

20kVekake

ikr p|i |nk

Then, using the relation ak|nk =

nk|(nk 1),

f| (nk, 1)|Hpara|i |nk = f|e

m

~nk

20kVe ikrek p|i


Consider now absorption of a photon. If weassume that, in the initial state, there are nkphotons in mode (k) then, after thetransition, there will be nk 1 photons.

As a result, using Fermis Golden rule,

if(t) =2

~2|f| (nk 1)|Hpara|i |nk|2(fi )

we obtain the transition amplitude,

if,k =2

~

f| em

~nk20kV

e ikrek p|i

2

(Ef Ei ~k)

In particular, we find that the absorption rate increases linearlywith photon number, nk.


Similarly, if we consider emission process inwhich there are already nk photons in initialstate,

using the relation ak|nk =

nk + 1|(nk + 1), we have revisedtransition rate,

if,k =2

~

f| em

~(nk + 1)20kV

eikrek p|i

2

(Ef Ei ~k)

Enhancement of transition rate by photon occupancy known asstimulated emission.

Radiative transitions: summary

Altogether, in dipole approximation f|ek p|i ' imkf|ek r|i

if,k 'k0V|f|ek d|i|2

{nk (Ef Ei ~k) absorption

(nk + 1) (Ei Ef ~k) emission

where d = er is electric dipole operator.

If there are no photons present initially, if,k reduces to result forspontaneous emission.

The coincidence of nk-independent coefficients for absorption andemission coincide is known as detailed balance.

Absorption and stimulated emission




Integrated transition rate associated with a small solid angle d inthe direction k given by

dRif, =

kd

if,k = d V

k2dk

(2)3if,k

If we assume that the photon number, nk is isotropic, independentof angle , using the dispersion relation k = ck , we obtain

dRif,d

=V

c3

2d

(2)3

0Vf|ek d|i|2

{n() (Ef Ei ~)(n() + 1) (Ei Ef ~)

dRif,d

=3

820~c3|f|ek d|i|2

{n()n() + 1

where ~ = |Ef Ei|.From this expression, we can obtain the power loss as P = ~R.


In fact, frequency dependence of spontaneous emission rate can beinferred using ingenious argument due to Einstein who showed thatstimulated and spontaneous transitions must be related.

Consider ensemble of atoms exposed to black-body radiation attemperature T . Let us consider transitions between states |j and|k, with Ek Ej = ~.

If number of atoms in two states given by nj and nk , transition ratesper atom given by:

absorption j k Bjku()stimulated emission k j Bkju()

spontaneous emission k j Akj()

where u() denotes energy density of radiation.

A and B are known as Einsteins A and B coefficients, and, as wehave seen, are properties of atomic states.


absorption j k Bjku()stimulated emission k j Bkju()

spontaneous emission k j Akj()

In thermodynamic equilibrium the rates must balance, so that

nk [Akj() + Bkju()] = njBjku()

At the same time, relative populations of two states given byBoltzmann factor,

njnk

=eEj/kBT

eEk/kBT= e~/kBT

Thus we have:

Akj() =[Bjke

~/kBT Bkj]u()


Akj() =[Bjke

~/kBT Bkj]u()

For black-body, energy density u() set by Planck formula,

u() =~3

2c3n() =

~3

2c31

e~/kBT 1

Akj() =[Bjke

~/kBT Bkj] ~32c3

1

e~/kBT 1

Since Akj is intrinsic (independent of temperature), T mustcancel on right hand side, i.e.

Bkj = Bjk and Akj() = Bkj~3

2c3

So, A and B coefficients are related, and if we can calculate Bcoefficient for stimulated emission from Fermis Golden rule, we caninfer A, and vice versa.

Selection rules: parity




Formulae for rates if,k show that radiative transitions will notoccur between states |i and |f unless at least one component ofthe dipole matrix element f|d|i is non-zero.

If matrix elements are zero for certain pairs, they are disallowed (atleast in the electric dipole approximation) leading to selection rules.

Since dipole operator d = er changes sign under parity (r r),matrix element f|d|i will vanish if |f and |i have same parity.

1 The parity of the wavefunction must change in an electric dipoletransition.

Selection rules: orbital angular momentum




From the operator identity, [Li , rj ] = i~ijk rk , it follows that

[Lz , z ] = 0, [Lz , x iy ] = (x iy)~

We therefore obtain,

`,m|[Lz , z ]|`,m = (m m)~`,m|z |`,m = 0

Similarly, since `,m|[Lz , x iy ]|`,m = ~`,m|x iy |`,m,

(m m 1)`,m|x iy |`,m = 0

3 Therefore, to get non-zero component of dipole matrix element,require m` = 0,1.

Selection rules: orbital angular momentum

Using operator identity [L2, [L2, r]] = 2~2(rL2 + L2r), we have

`,m|[L2, [L2, r]]|`,m = [`(` + 1) `(`+ 1)]2`,m|r|`,m= 2[`(` + 1) + `(`+ 1)]`,m|r|`,m

i.e. (`+ `)(`+ ` + 2)[(` `)2 1]`,m|r|`,m = 0. Since`, ` 0, dipole matrix element non-vanishing only if ` = ` 1.

4 To effect an electric dipole transition, we must have ` = 1.

One may summarize the selection rules for ` and m` is by sayingthat the photon carries off (or brings in, in an absorption transition)one unit of angular momentum.

N.B. it is possible, though much less likely in the case of an atom,for EM field to interact with magnetic dipole or electric quadrupolemoment with different selection rules.

Selection rules: polarization

if,k, =k0V|f|ek d|i|2

{nk, (Ef Ei ~k) absorption

(nk, + 1) (Ei Ef ~k) emission

For transitions with m` = 0, the dipole matrix element f|d|i ez and there is no component of polarization along z-direction.

Similarly, for electric dipole transitions with m = m 1,`,m|x iy |`,m = 0 = `,m|z |`,m, and f|d|i (1,i , 0).

(a) If the wavevector of photon lies along z , the emitted light iscircularly polarized with a polarization which depends on helicity.

(b) If the wavevector lies in xy place, the emitted light is linearlypolarized, while in general it is elliptically polarized.

Selection rules: LS coupling

In the presence of spin-orbit coupling, stationary states labelled byquantum numbers J,mJ , `, s where J = L + S.

The selection rules in this case can be inferred by looking for theconditions for non-zero matrix elements J ,mJ , `, s |r|J,mJ , `, s.

By expanding states |J,mJ , `, s in basis states |`,m` |s,ms, onemay uncover the following set of selection rules:

5 For dipole transitions to take place, we require that

mj = 0,1j = 0,1 not 0 0

N.B. These conclusions are consistent with photon carrying on unitof angular momentum.

Radiative transitions: recap

When coupled to a quantized electromagnetic field, the totalHamiltonian for atomic system given by H = Hatom + Hpara + Hradwhere

Hatom =p2

2m+ V (r), Hrad =

k

~k(

akak +1

2

)denotes the Hamiltonian of the isolated atomic and radiation field,and

Hpara(t) =e

mA(r, t) p

denotes the coupling with

A(r, t) =k

~

20kV

[ekake


]


In the electric dipole approximation, the transition rate is given by




where d = er is electric dipole operator.

The coincidence of nk-independent coefficients for absorption andemission coincide is known as detailed balance.

From these results, we turn now to consider the principle of theoperation of an atomic laser.

Theory of laser

Principle of stimulated emissionprovides basis of laser operation:light amplification by stimulatedemission of radiation.

However, laser not only amplifieslight, but provides source of

monochromatic (single mode),

coherent (spatial/temporal),

directional and

intense radiation.

In atomic laser, the gain mediumprovided by a gas of atomsconfined to a cavity and bound byhighly reflective mirrors.

Theory of laser: rate equations

k |f|ek d|i|2{

nk abs.(nk + 1) emiss.

Consider gas of atoms in a cavity subject to an EM field of intensityI n() and angular frequency tuned to energy differencebetween two discrete energy levels of the atoms, i.e. ~ = E2 E1.Taking into account stimulated absorption, atoms are transferredfrom level 1 to level 2 at a rate

12 = WN1n()

where N1 atoms in level 1 and W includes matrix elements.

From spontaneous and stimulated emission processes, the rate oftransfer of atoms from level 2 to level 1 is given by

21 = WN2(n() + 1)

Theory of laser: photon equations

k |f|ek d|i|2{


Since transfer of particles from level 2 to 1 leads to creation ofphotons in cavity while from 1 to 2 they involve absorption, the rateof change of photon number is given by n = W (N2(n + 1) N1n).

However, to make use of cavity as a photon source, we have toallow photons to leak from the cavity through imperfect mirrors.Taking into account this and other loss processes, we have

n = DWn + N2W n

ph

where D = N2 N1 denotes population imbalance and 1/ph is thetotal loss rate.

Theory of laser: matter equations

k |f|ek d|i|2{


Without further external processes, photons would escape fromcavity and the system would relax into ground state To create asteady-state photon population, energy must be pumped into thesystem in the form of excitations.

Achieved by transferring atoms between 1 and 2 via level 3 bynon-resonant optical pump. If lifetime of 3 is short, occupancy iseffectively zero, rate of transfer of particles from 2 to 1,

N2 ' w21N2 + w12N1 (N2 N1)Wn ' N1

where we have dropped small contribution from spontaneousemission, and w12,w21 denote net non-resonant transition rates.

Theory of laser: stationary equations

N2 ' w21N2 + w12N1 (N2 N1)Wn ' N1

Without cavity photons (n = 0), since N1 + N2 ' N, in steady state,

D(0) = N(0)2 N

(0)1 = N

w12 w21w12 + w21

denotes unsaturated inversion.

Restoring the cavity photons, we have

D = N2 N1 =D(0) D

T 2DWn

where 1/T = w12 + w21 represents typical relaxation rate.

Theory of laser: stationary equations

n = DWn nph

, D =D(0) D

T 2DWn

In steady-state operation, n = D = 0, population imbalance

D N2 N1 =D(0)

1 + 2TWn

From this result, we find the steady state photon number

n =D(0)W 1/ph

2TW /ph

This result shows that the system will only start lasing when theunsaturated inversion exceeds a threshold, D(0) > 1/phW .

Theory of laser: coherence

Although the analysis above addressed the threshold conditions forthe laser, it does not provide any insight into the coherenceproperties of the radiation field.

In fact, one may show that the radiation field generated by the lasercavity forms a coherent or Glauber state.

The proof of this statement and the coherence properties thatfollow would take us on a considerable detour see Part IIIquantum optics.

However, we can gain some insight into the properties and physicalmanifestations of coherent states by looking at a toy example; butfirst we must define what we mean by a coherent state.

Coherent states

A coherent state is defined as an eigenstate of the annihilationoperator,

a| = |

Since a is not Hermitian, can take complex eigenvalues.

The eigenstates are constructed from the harmonic oscillator groundstate the by action of the unitary operator,

| = U()|0, U() = eaa, U()U() = I

The proof follows from the identity (problem set I),

aU() = U()(a + ), i.e. aU()|0 = U()|0

i.e. U is a translation operator, U()aU() = a + .

Coherent states

| = U()|0, U() = eaa

Since U() = eaa = e||

2/2eae

a and ea|0 = |0, we

can write

| = e||2/2ea

|0

With |n = 1n!

(a)n|0, we can write

| =

n

e||2/2

n

n!|n

showing that the probability of observing n excitations

Pn = |n||2 = e||2 ||2n

n!

is a Poisson distribution with average occupation, |aa| = ||2.

Driven quantum harmonic oscillator

But how can we prepare a system in acoherent state?

Consider a single two-level atom resonantly coupled to a singlecavity mode the quantum Hamiltonian of the coupled system,

H =1

2~z + ~

(aa +

1

2

)+ ~g(a + +a)

When excitations of two level system are driven by an externalpump, it can behave as a classical dipole source for the cavity modeleading to the driven harmonic oscillator Hamiltonian,

H ' Hrad + V (t) = ~(

aa +1

2

)+ i~

(f (t)a f (t)a

)where f (t) = f0e

it


H = ~(

aa +1

2

)+ i~

(f (t)a f (t)a

)f (t) = f0e

it

If photon system is prepared in ground state, |0, the perturbationdrives the system into a coherent state.

To understand how, let us turn to the interaction representation,

i~t |(t)I = VI|(t)I where |(t)I = e i H0t/~|(t)S. Withe ita

aaeitaa = eita,

VI(t) = ei H0t/~i~

(f (t)a f (t)a

)ei H0t/~ = i~

(f0a f 0 a

)Since VI(t) is time-independent, the time-evolution operator,defined by the equation i~tUI(t) = VIUI(t), is given simply by

UI(t) = exp[(f0a

f 0 a)t]


UI(t) = exp[(f0a

f 0 a)t]

Therefore, if the system was prepared in the ground state |0 att = 0, at later times,

|(t)I = exp[(f0a f 0 a)t]|0 = e|f0|2t2/2ef0a

t |0

Reexpressed in the Schrodinger representation,

|(t)S = ei H0t/~|(t)I = e|f0|2t2/2ef0e

itat |0

A classical oscillatory force drives a system prepared in the vacuumstate into a coherent state with an excitation number which climbsas |f0|2t2.

Synopsis: Lectures 16-19

12 Field theory: from phonons to photons:

From particles to fields: classical field theory of harmonic atomicchain; quantization of atomic chain; phonons. Classical theory ofthe EM field; waveguide; quantization of the EM field and photons.

13 Time-dependent perturbation theory:

Rabi oscillations in two level systems; perturbation series; suddenapproximation; harmonic perturbations and Fermis Golden rule.

14 Radiative transitions:

Light-matter interaction; spontaneous emission; absorption andstimulated emission; Einsteins A and B coefficents; dipoleapproximation; selection rules; lasers.

Synopsis: Lectures 20-24

15 Scattering theory

Elastic scattering; cross section; method of particle waves; Bornapproximation; scattering of identical particles.

16 Relativistic quantum mechanics:

Klein-Gordon equation; Dirac equation; relativistic covariance andspin; free relativistic particles and the Klein paradox; antiparticles;coupling to EM field: minimal coupling and the connection tonon-relativistic quantum mechanics; field quantization.

lecture 19 radiative transitions - tcm groupbds10/aqp/lec19_compressed.pdf · lecture 19 radiative...

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