3, Atom-field interaction, semi-classical and quantum theo ries

1. Semiclassical theory

2. Jaynes-Cummings Hamiltonian

3. Multi-mode squeezing

4. Rabi Oscillation

5. Superradiance

Ref:

Ch. 5, 6 in ”Quantum Optics,” by M. Scully and M. Zubairy.Ch. 5, 6 in ”Mesoscopic Quantum Optics,” by Y. Yamamoto and A. Imamoglu.Ch. 5 in ”The Quantum Theory of Light,” by R. Loudon.Ch. 10 in ”Quantum Optics,” by D. Wall and G. Milburn.Ch. 13 in ”Elements of Quantum Optics,” by P. Meystre and M. Sargent III.

IPT5340, Fall ’06 – p. 1/70

Einstein on Radiation

"On the Quantum Theory of Radiation"

D(ω) =A/B

e~ω/kBT − 1

A

B=

~ω3

π2c3

A. Einstein, Phys. Z. 18, 121 (1917).D. Kleppner, "Rereading Einstein on Radiation," Physics Today 58, 30 (Feb. 2005).

IPT5340, Fall ’06 – p. 2/70

Quantization of the Electromagnetic Field

Like simple harmonic oscillator, H = p2

2m+ 1

2kx2, where [x, p] = i~,

For EM field, H = 12

P

j [mjω2mq

2j +

p2jmj

], , where [qi, pj ] = i~δij ,

the Hamiltonian for EM fields becomes: H =P

j ~ωj(a†j aj + 1

2),

the electric and magnetic fields become,

Ex(z, t) =X

j

(~ωj

ǫ0V)1/2[aje

−iωjt + a†jeiωjt] sin(kjz),

Hy(z, t) = −iǫ0cX

j

(~ωj

ǫ0V)1/2[aje

−iωjt − a†jeiωjt] cos(kjz),

energy level for quantized field, En = (n+ 12)~ω.

IPT5340, Fall ’06 – p. 3/70

Planck’s Law

In the thermal equilibrium at temperature T, the probability Pn that the modeoscillator is thermally excited to the n-th excited state is given by the Boltzman factor,

Pn =exp[−En/kBT ]

P

n exp[−En/kBt],

the mean number n of photons is,

n =X

n

nPn =U

1− U =1

exp(~ω/kBT )− 1,

where U ≡ exp(−~ω/kBT ) andP∞n=0 U

n = 1/(1− U).

energy density of the radiation:

D(ω)dω = n~ωdω = n~ωρωdω,

= n~ω3dω/π2c3 =~ω3

π2c3dω

exp[~ω/kBT ]− 1.

total electromagnetic energy density:R ∞0 D(ω)dω = 1/2V

R

cavity ǫ0|E(r, t)|2dV .

IPT5340, Fall ’06 – p. 4/70

Fluctuations in Photon Number

the ergodic theorem of statistical mechanics: time averages are equivalent toaverages taken over a large number of exactly similar systems, each maintained ina fixed state (ensemble).

the probability of finding n photons,

Pn =exp[−En/kBT ]

P

n exp[−En/kBt]= (1− U)Un =

nn

(1 + n)1+n,

which is a thermal distribution or the geometric distribution.

the root-mean-square deviation:

∆n2 =X

n

(n− n2)Pn = n2 + n,

then

∆n ≈ n+1

2, for n≫ 1.

IPT5340, Fall ’06 – p. 5/70

Probability distribution for n = 1

IPT5340, Fall ’06 – p. 6/70

Einstein’s A and B coefficients

For a two-level atom, the rates of changes of N1 and N2 are,

dN1

dt= −dN2

dt= N2A21 −N1B12D(ω) +N2B21D(ω),

A21 is the probability of photon in state 2 spontaneously fall into the lower state 1,i.e. spontaneous emission;

B12 is the probability of photon absorption in state 1 into state 2, i.e. absorption;

B21 is the probability of photon emission from state 2 into state 1, i.e. stimulatedemission;

in thermal equilibrium, dN1

dt = − dN2

dt = 0,

D(ω) =A21

(N1/N2)B12 −B21,

where the populations N1 and N2 are related by Boltzmann’s law,

N1/N2 = (g1/g2)exp[~ω/kBT ],

IPT5340, Fall ’06 – p. 7/70

Einstein’s A and B coefficients

the density distribution of EM fields in a two-level atom,

D(ω) =A21

(g1/g2)exp[~ω/kBT ]B12 −B21,

where g1 and g2 are the level degenerate parameters.

compare it in free space,

D(ω) =~ω3

π2c31

exp[~ω/kBT ]− 1,

at all temperatures T , we have

(g1/g2)B12 = B21,

(~ω3/π2c3)B21 = A21,

the consistency between the Einstein theory and Planck’s law could not have beenachieved without the introduction of the stimulated emission process.

IPT5340, Fall ’06 – p. 8/70

Einstein’s A and B coefficients

for nondegenerate two-level atom, g1 = g2 = 1 and N1 +N2 = N ,

dN1

dt= −dN2

dt= N2A+ (N2 −N1)BD(ω),

the solution for N1 is,

N1 = [N01 −

N(A+BD(ω)

A+ 2BD(ω)]exp[−(A+ 2BD(ω))t] +

N [A+BD(ω)]

A+ 2BD(ω)

where N01 is the initial value of N1 at t = 0,

if N02 = 0, all atoms are in the ground state at t = 0,

N2 =NBD(ω)

A+ 2BD(ω)[1− exp[−(A+ 2BD(ω))t],

in the steady-state,

N2 =NBD(ω)

A+ 2BD(ω)≈ 0.5, if BD(ω)≫ A,

IPT5340, Fall ’06 – p. 9/70

Macroscopic theory of Absorption

for the excited state,dN2

dt= −N2A,

with the solution N2 = N02 exp[−At], where A ≡ 1/τR the radiative lifetime of the

excited states.

in macroscopic, the polarization P by an applied electric field E is related withP = ǫ0χE, where the susceptibility χ = χ1 + iχ2,

the relation between frequency and the wavevector,kc/ω = 1 + χ = n2 = (η + iκ)2, where η2 − κ2 = 1 + χ1 and 2ηκ = χ2,

the traveling-wave solution propagated in the z−direction becomes,

exp[i(kz − ωt)] = exp[iω(ηz

c− t)− ωκz

c],

the averaged Poynting vector, I = 〈E× B/µ0〉 = 12ǫ0cη|E(r, t)|2, where

I(z) = I0exp[−2ωκz/c],

where 2ωκ/c is called the absorption coefficient.

IPT5340, Fall ’06 – p. 10/70

Microscopic theory of Absorption

total electromagnetic energy density:R ∞0 D(ω)dω = 1/2V

R

cavity ǫ0|E(r, t)|2dV .

for a lossy dielectric medium,R ∞0 D(ω)dω = 1/2V

R

cavity ǫ0η2|E(r, t)|2dV .

in steady-state condition, − dN2

dt = N2A+ (N2 −N1)BD(ω)/η2 = 0, with anadditional factor η2 for the energy density,

the attenuation energy within a small section of dz, cross-section A is,

∂

∂tD(ω)dωAdz = −(N1 −N2)F (ω)dωBD(ω)/η2

~ω(Adz/V ),

for the absorption, − ∂∂tD(ω)dωAdz = − ∂

∂zIdωAdz, or ∂

∂tD(ω) = ∂

∂zI,

for I = 12ǫ0cη|E(r, t)|2, we have cD(ω) = ηI, then,

∂

∂zI = −(N1 −N2)F (ω)(B~ω/V cη)~I,

where F (ω) is the distribution of atomic transition frequencies.

IPT5340, Fall ’06 – p. 11/70

Microscopic theory of Absorption

if N02 = 0, all atoms are in the ground state at t = 0,

N2 =NBD(ω)

A+ 2BD(ω)/η2[1− exp[−(A+ 2BD(ω))t] ≈ NBD(ω)

A+ 2BD(ω)/η2,

and we have,

N1 −N2 =NA

A+ 2BD(ω)/η2=

NA

A+ 2BI/cη,

the equation for the average beam intensity becomes,

1

I(1 +

2BI

Acη)∂

∂zI = −NB~ωF (ω)

V cη

for all ordinary light beams, 2BIAcη≪ 1, then we have,

I(z) = I0exp[−NB~ωF (ω)z/V cη],

= I0exp[−Kz],

where the absorption coefficient, K = 2ωκ/c.IPT5340, Fall ’06 – p. 12/70

Microscopic theory of Absorption

A dielectric with one single resonance may be modeled as a distribution of "+" and"−" charges, the + charges immobile and the − charges tied to the + charges bya spring constant k,

m(d2

d t2+ 2β

d

d t+ ω2

0)d = − e

mE,

for the incident field E = E0exp[−i(ωt− kz)] and the dipoled = aexp[−i(ωt− kz)], we have

a =−(e/m)E0

ω2 − ω20 + 2iβω

,

the polarization P = Np = NP

j edj = Nα(ω)E0e−i(ωt−kz), where

α(ω) =−e2/m

ω2−ω2

0+2iβω

.

Ch. 2, 3, 7, 8 in ”Lasers,” by P. Milonni and J. Eberly.

IPT5340, Fall ’06 – p. 13/70

Microscopic theory of Absorption

the dispersion relation,

k2 =ω2

c2[1 +

Nα(ω)

ǫ0] =

ω2

c2n2(ω2),

the real index of refraction,

nR(ω) = 1 +Ne2

mǫ0

ω20 − ω2

(ω20 − ω2)2 + 4β2ω2

,

the absorption coefficient or extinction coefficient,

a(ω) = 2nI(ω)ω/c =2Ne2

mǫ0c

βω2

(ω20 − ω2)2 + 4β2ω2

,

which has the lineshape of the Lorentzian function,

a(ω) =Ne2

2mǫ0c

δω0

(ω0 − ω)2 + δω20

,

where δω = β

IPT5340, Fall ’06 – p. 14/70

Population Inversion: the Laser

for a three level atom, N1 +N2 +N3 = N , the rate equations are:

dN2

dt= −N2A21 −N2A23 +DpB23(N3 −N2)−D(ω)B21(N2 −N1),

dN1

dt= N2A21 −N1A13 +D(ω)B21(N2 −N1),

dN3

dt= −N2A23 +N1A13 −DpB23(N3 −N2),

the pumping rate γ = DpB23(N3 −N2)/N ,

in steady-state,

N2[A21 +B21D(ω)] = N1[A12 +B21D(ω)],

N2A23 +N1A13 = Nγ,

for A21 < A13, we have N2 > N1.

IPT5340, Fall ’06 – p. 15/70

Purcell effect : Cavity-QED (Quantum ElectroDynamics)

E. M. Purcell, Phys. Rev. 69 (1946).

Nobel laureate Edward Mills Purcell (shared the prize with Felix Bloch) in 1952,

for their contribution to nuclear magnetic precision measurements.

from: K. J. Vahala, Nature 424, 839 (2003).

IPT5340, Fall ’06 – p. 16/70

The motion of a free electron

the motion of a free electron is described by the Schr odinger equation,

−~2

2m∇2Ψ = i~

∂Ψ

∂t,

the probability density of finding an electron at position r and time t is

P (r, t) = |Ψ(r, t)|2,

is Ψ(r, t) is a solution os the Schrödinger equation so is

Ψ1(r, t) = Ψ(r, t)exp[iχ],

where χ is an arbitrary constant phase,

the probability density P (r, t) would remain unaffected by an arbitrary choice of χ,

the choice of the phase of the wave function Ψ(r, t) is completely arbitrary,

two functions differing only by a constant phase factor represent the same physicalstate,

IPT5340, Fall ’06 – p. 17/70

Local gauge (phase) invariance

the motion of a free electron is described by the Schr odinger equation,

−~2

2m∇2Ψ = i~

∂Ψ

∂t,

if the phase of the wave function is allowed to vary locally, i.e.

Ψ1(r, t)→ Ψ(r, t)exp[iχ(r, t)],

the probability P (r, t) remains unaffected but the Schrödinger equation is nolonger satisfied,

to satisfy local gauge (phase) invariance, then the Schrödinger equation must bemodified by adding new terms,

−~2

2m[∇− i e

~A(r, t)]2 + eU(r, t)Ψ = i~

∂Ψ

∂t,

where A(r, t) and U(r, t) are the vector and scalar potentials of the external field,respectively,

IPT5340, Fall ’06 – p. 18/70

Minimal-coupling Hamiltonian

to satisfy local gauge (phase) invariance, then the Schrödinger equation must bemodified by adding new terms,

−~2

2m[∇− i e

~A(r, t)]2 + eU(r, t)Ψ = i~

∂Ψ

∂t,

and

A(r, t) → A(r, t) +~

e∇χ(r, t),

U(r, t) → U(r, t)− ~

e

∂χ(r, t)

∂t,

where A(r, t) and U(r, t) are the vector and scalar potentials of the external field,respectively,

A(r, t) and U(r, t) are the gauge-dependent potentials,

the gauge-independent quantities are the electric and magnetic fields,

E = −∇U− ∂A∂t,

B = ∇A,

IPT5340, Fall ’06 – p. 19/70

Minimal-coupling Hamiltonian

an electron of charge e and mass m interacting with an external EM field isdescribed by the minimal-coupling Hamiltonian,

H =1

2m[p − eA(r, t)]2 + eU(r, t),

where p = −i~∇ is the canonical momentum operator, A(r, t) and U(r, t) are thevector and scalar potentials of the external field, respectively,

the electrons are described by the wave function Ψ(r, t),

the field is described by the vector and scalar potentials A and U,

in this way, the photon has been ’derived’ from the Schrödinger equation plus thelocal gauge invariance arguments,

the gauge field theory leads to the unification of the weak and the electromagneticinteractions,

IPT5340, Fall ’06 – p. 20/70

Dipole approximation and r · E Hamiltonian

if the entire atom is immersed in a plane EM wave,

A(r0 + r, t) = A(t)exp[ik · (r0 + r)] ≈ A(t)exp(ik · r0),

where r0 is the location of the electron,

in this way, the dipole approximation, A(r, t) ≈ A(r0, t),

and the minimal-coupling Hamiltonian becomes,

H =1

2m[p − eA(r0, t)]

2 + eU(r, t) + V (r),

where V (r) is the atomic binding potential,

in the radiation gauge, R-gauge,

U(r, t) = 0, and ∇ · A(r, t) = 0,

the minimal-coupling Hamiltonian becomes,

H =p2

2m+ V (r) + er · ∂A(r0, t)

∂t,

IPT5340, Fall ’06 – p. 21/70

Dipole approximation and r · E Hamiltonian

in the dipole approximation the minimal-coupling Hamiltonian becomes,

H =1

2m[p − eA(r0, t)]

2 + eU(r, t) + V (r),

the wave function with a local phase,

Ψ(r, t) = Φ(r, t)exp[ie

~A(r0, t) · r],

then

i~[ie

~r · ∂A(r0, t)

∂tψ(r, t) +

∂ψ(r, t)

∂t]exp[

ie

~A · r] = [

p2

2m+ V (r)]exp[

ie

~A · r],

in terms of the gauge-independent field E, the Hamiltonian for Ψ(r, t) is,

H =p2

2m+ V (r) + er · ∂A(r0, t)

∂t,

=p2

2m+ V (r)− er · E(r0, t) = H0 + H1,

IPT5340, Fall ’06 – p. 22/70

Dipole approximation and r · E Hamiltonian

in the dipole approximation the minimal-coupling Hamiltonian becomes,

H =p2

2m+ V (r) + er · ∂A(r0, t)

∂t,

=p2

2m+ V (r)− er · E(r0, t) = H0 + H1,

in terms of the gauge-independent field E and where

H0 =p2

2m+ V (r),

H1 = −er · E(r0, t),

this Hamiltonian is for the atom-field interaction,

IPT5340, Fall ’06 – p. 23/70

p · A Hamiltonian

in the radiation gauge, R-gauge,

U(r, t) = 0, and ∇ · A(r, t) = 0,

the latter one implies [p,A] = 0, then

and the minimal-coupling Hamiltonian becomes,

H =1

2m[p − eA(r0, t)]

2 + V (r) = H0 + H2,

where

H0 =p2

2m+ V (r),

H2 = − e

mp · A(r0, t) +

e2

2mA2(r0, t) ≈ −

e

mp · A(r0, t),

IPT5340, Fall ’06 – p. 24/70

Differences in r · E and p · A Hamiltonian

in r · EH1 = −er · E(r0, t),

in p · A Hamiltonian

H2 = − e

mp · A(r0, t),

these two different Hamiltonian H1 and H2 give different physical results,

for example, consider a linearly polarized monochromatic plane-wave field,

E(r0 = 0, t) = E0 cosωt, and A(r0 = 0, t) = − 1

ωE0 sinωt,

the ratio of the matrix elements for the Hamiltonian H1 and H2 is

| 〈f |H2|i〉〈f |H1|i〉

| = | − (e/mω)〈f |p|i〉 · E0

e〈f |r|i〉 · E0| = ωfi

ω,

As was first pointed out by Lamb, this makes a difference in measurable quantitieslike transition rates,

IPT5340, Fall ’06 – p. 25/70

Interaction of a single two-level atom with a single-mode fie ld

consider the interaction of a single-mode radiation field of frequency ν,

and a two-level atom with upper and lower level states |a〉 and |b〉,

the unperturbed part of the Hamiltonian H0 has the eigenvalues ~ωa and ~ωb forthe atom,

the wave function of a two-level atom can be written in the form,

|Ψt〉 = Ca(t)|a〉+ Cb(t)|b〉,

the corresponding Schrödinger equation is

i~∂Ψ(t)

∂t= (H0 + H1)Ψ(t),

where

H0 = |a〉〈a|+ |b〉〈b|)H0|a〉〈a|+ |b〉〈b|) = ~ωa|a〉〈a|+ ~ωb|b〉〈b|,H1 = −er · E(t) = −e(|a〉〈a|+ |b〉〈b|)r(|a〉〈a|+ |b〉〈b|)E,

= −(pab|a〉〈b|+ pba|a〉〈b|)E(t),

IPT5340, Fall ’06 – p. 26/70

Probability amplitude method

in the dipole approximation,

H0 = ~ωa|a〉〈a|+ ~ωb|b〉〈b|,H1 = −(pab|a〉〈b|+ pba|a〉〈b|)E(t),

where pab = p∗ba = e〈a|r|b〉,

for a single-mode field,

E(t) = E0 cos νt,

the equation of motion for the probability amplitude are

ddtCa = −iωaCa + iΩR cos(νt)e−iφCb,

ddtCb = −iωbCb + iΩR cos(νt)e+iφCa,

where ΩR =|pab|E0

~is the Rabi frequency which is proportional to the amplitude

of the classical field,

and φ is the phase of the dipole matrix element pab = |pab|exp(iφ),

IPT5340, Fall ’06 – p. 27/70

Probability amplitude method

the equation of motion for the probability amplitude are

ddtCa = −iωaCa + iΩR cos(νt)e−iφCb,

ddtCb = −iωbCb + iΩR cos(νt)e+iφCa,

define the slowly varying amplitudes,

ca = Caeiωat, and cb = Cbe

iωbt,

then

ddtca = i

ΩR

2e−iφ[ei(ω−ν)t + ei(ω+ν)t]cb ≈ i

ΩR

2e−iφei(ω−ν)tcb,

ddtcb = i

ΩR

2eiφ[e−i(ω−ν)t + e−i(ω+ν)t]ca ≈ i

ΩR

2eiφe−i(ω−ν)tca,

where ω = ωa − ωb is the atomic transition frequency,

we also apply the rotating-wave approximation by neglecting terms withexp[±i(ω + ν)t],

IPT5340, Fall ’06 – p. 28/70

Probability amplitude method

the equation of motion for the probability amplitude are

ddtca = i

ΩR

2e−iφei(ω−ν)tcb,

ddtcb = i

ΩR

2eiφe−i(ω−ν)tca,

the solutions are

ca(t) = [cos(Ωt

2)− i∆

Ωsin(

Ωt

2)]ca(0) + i

ΩR

Ωsin(

Ωt

2)e−iφcb(0)ei∆t/2,

cb(t) = [cos(Ωt

2) + i

∆

Ωsin(

Ωt

2)]cb(0) + i

ΩR

Ωsin(

Ωt

2)eiφcb(0)e−i∆t/2,

where

∆ = ω − ν, frequency detuning,

Ω =q

Ω2R + ∆2,

IPT5340, Fall ’06 – p. 29/70

Rabi oscillation

it is easy to verify that

|ca(t)|2 + |cb(t)|2 = 1

assume that the atom is initially in the excited state |a〉, i.e ca(0) = 1 andcb(0) = 0, then the population inversion is

W (t) = |ca(t)|2 − |cb(t)|2 =∆2 − ΩR2

Ω2sin2(

Ω

2t) + cos2(

Ω

2t)

the population oscillates with the frequency Ω =q

Ω2R + ∆2,

when the atom is at resonance with the incident field ∆ = 0, we get Ω = ΩR, and

W (t) = cos(ΩRt),

the inversion oscillates between −1 and +1 at a frequency ΩR,

IPT5340, Fall ’06 – p. 30/70

Rabi oscillation

ΩR = 1.0, ∆ = 0.0 ΩR = 3.0, ∆ = 0.0

2.5 5 7.5 10 12.5 15 17.5 20

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

2.5 5 7.5 10 12.5 15 17.5 20

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

2.5 5 7.5 10 12.5 15 17.5 20

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

2.5 5 7.5 10 12.5 15 17.5 20

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

ΩR = 1.0, ∆ = 1.0 ΩR = 3.0, ∆ = 5.0

IPT5340, Fall ’06 – p. 31/70

Interaction picture

Consider a system described by |Ψ(t)〉 evolving under the action of a hamiltonianH(t) decomposable as,

H(t) = H0 + H1(t),

where H0 is time-independent.

Define

|ΨI(t)〉 = exp(iH0t/~)|Ψ(t)〉,

then |ΨI(t)〉 evolves accords to

i~ddt|ΨI(t)〉 = HI(t)|ΨI(t)〉,

where

HI (t) = exp(iH0t/~)H1(t)exp(−iH0t/~).

The evolution is in the interaction picture generated by H0.

IPT5340, Fall ’06 – p. 32/70

Interaction picture

in the dipole approximation,

H0 = ~ωa|a〉〈a|+ ~ωb|b〉〈b|,H1 = −(pab|a〉〈b|+ pba|a〉〈b|)E(t) = −~ΩR(e−iφ|a〉〈b|+ eiφ|a〉〈b|) cos νt,

where pab = p∗ba = e〈a|r|b〉 and ΩR =

|pab|E0

~,

the interaction picture Hamiltonian is

HI(t) = exp(iH0t/~)H1(t)exp(−iH0t/~),

= −~

2ΩR[e−iφ|a〉〈b|ei(ω−ν)t + eiφ|b〉〈a|e−i(ω−ν)t

+ e−iφ|a〉〈b|ei(ω+ν)t + eiφ|b〉〈a|e−i(ω+ν)t],

in the rotating-wave approximation,

HI(t) = −~

2ΩR[e−iφ|a〉〈b|ei(ω−ν)t + eiφ|b〉〈a|e−i(ω−ν)t],

IPT5340, Fall ’06 – p. 33/70

Interaction picture

on resonance ω − ν = 0,

HI(t) = −~

2ΩR[e−iφ|a〉〈b|+ eiφ|b〉〈a|],

the time-evolution operator in the interaction picture UI(t) is

UI (t) =←−T exp[− i

~

Z t

t0

dτHI(τ)],

= cos(ΩRt

2)(|a〉〈a|+ |b〉〈b|) + i sin(

ΩRt

2)(e−iφ|a〉〈b|+ eiφ|b〉〈a|),

if the atom is initially in the excited state |Ψ(t = 0)〉 = |a〉, then

|Ψ(t)〉 = UI (t)|a〉,

= cos(ΩRt

2)|a〉+ i sin(

ΩRt

2)eiφ|b〉,

IPT5340, Fall ’06 – p. 34/70

Density Operator

for the quantum mechanical description, if we know that the system is in state |ψ〉,then an operator O has the expectation value,

〈O〉qm = 〈ψ|O|ψ〉,

but we typically do not know that we are in state |ψ〉, then an ensemble averagemust be performed,

〈〈O〉qm〉ensemble =X

ψ

Pψ〈ψ|O|ψ〉,

where the Pψ is the probability of being in the state |ψ〉 and we introduce a densityoperator,

ρ =X

ψ

Pψ |ψ〉〈ψ|,

the expectation value of any operator O is given by,

〈O)〉qm = Tr[ρO],

where Tr stands for trace.

IPT5340, Fall ’06 – p. 35/70

Equation of motion for the density matrix

density operator is defined as,

ρ =X

ψ

Pψ |ψ〉〈ψ|,

in the Schrödiner picture,

i~∂

∂t|Ψ〉 = H|Ψ〉,

then we have

i~∂

∂tρ = Hρ− ρH = [H, ρ],

which is called the Liouville or Von Neumann equation of motion for the densitymatrix,

using density operator instead of a specific state vector can give statistical as wellas quantum mechanical information,

compared to the Heisenberg equation, i~ ddt A(t) = [A, H(t)]

IPT5340, Fall ’06 – p. 36/70

Decay processes in the density matrix

equation of motion for the density matrix,

i~∂

∂tρ = [H, ρ],

the excited atomic levels can also decay due to spontaneous emission or collisionsand other phenomena,

the decay rates can be incorporated by a relaxation matrix Γ,

〈n|Γ|m〉 = γnδnm,

then the density matrix equation of motion becomes,

∂

∂tρ = − i

~[H, ρ]− 1

2Γ, ρ,

where Γ, ρ = Γρ+ ρΓ,

the ijth matrix element is,

∂

∂tρij = − i

~

X

k

(Hikρkj − ρikHkj)−1

2

X

k

(Γikρkj + ρikΓkj),

IPT5340, Fall ’06 – p. 37/70

Two-level atom

a two-level atom with upper and lower level states |a〉 and |b〉,

|Ψt〉 = Ca(t)|a〉+ Cb(t)|b〉,

the density matrix operator is

ρ = |Ψ〉〈Ψ| = |Ca|2|a〉〈a|+ CaC∗b |a〉〈b|+ CbC

∗a |b〉〈a|+ |Cb|2|b〉〈b|,

= ρaa|a〉〈a|+ ρab|a〉〈b|+ ρba|b〉〈a|+ ρbb|b〉〈b|,

diagonal elements, ρaa and ρbb, are the probabilities in the upper and lower states,

off-diagonal elements, ρab and ρba, are the atomic polarizations,

from the equation of motion for the two-level atom ∂∂tρ = − i

~[H, ρ]− 1

2Γ, ρ, we

have

∂

∂tρaa =

i

~[pabEρba − c.c]− γaρaa,

∂

∂tρbb = − i

~[pabEρba − c.c]− γbρbb,

∂

∂tρab = − i

~pabE(ρaa − ρbb)− (iω +

γa + γb

2)ρab,

IPT5340, Fall ’06 – p. 38/70

Inclusion of elastic collisions between atoms

the physical interpretation of the elements of the density matrix allows us toinclude terms associated wither certain processes,

for example, one can have elastic collision between atoms in a gas,

during an atom-atom collision the energy levels experience random Stark shifts,

∂

∂tρab = −i[iω + iδω(t) + γab]ρab,

after integration,

ρab = exp[−(iω + γab)t− iZ t

0dt′δω(t′)]ρab(0),

for a zero-mean random process, 〈δω(t)〉 = 0,

the variations in δω(t) are usually rapid compared to other changes which occur intimes like γph,

〈δω(t)δω(t′)〉 = 2γphδ(t− t′),

IPT5340, Fall ’06 – p. 39/70

Inclusion of elastic collisions between atoms

assume that δω(t) is described by a Gaussian random process, then

〈exp[−iZ t

0dt′δω(t′)]〉 = exp[−γpht],

which gives for the average of ρab,

ρab = exp[−(iω + γab − γph)t]ρab(0),

for the process of atom-atom collisions,

∂

∂tρab = −i[iω + γ]ρab −

i

~pabE(ρaa − ρbb),

where γ = γab + γph is the new decay rate,

IPT5340, Fall ’06 – p. 40/70

Population matrix

for a single two-level atom, its density operator at time t and position z is

ρ(z, t, t0) =X

α,β

ραβ(z, t, t0)|α〉〈β|,

where α, β = a, b and the atom starts interacting with the field at an initial time t0,

for a medium consists of two-level homogeneously broadened atoms,

the effect of all atoms which are pumped at the rate ra(z, t0) atoms per secondper unit volume is the population matrix,

ρ(z, t) =

Z t

−∞dt0ra(z, t0)ρ(z, t, t0) =

X

α,β

Z t

−∞dt0ra(z, t0ραβ(z, t, t0)|α〉〈β|,

where the excitation ra(z, t0) generally varies slowly and can be taken to be aconstant, i.e.

ρ(z, t) =X

α,β

ραβ(z, t)|α〉〈β|,

IPT5340, Fall ’06 – p. 41/70

Population matrix

the macroscopic polarization of the medium, P (z, t) is the ensemble of atoms thatarrive at z at time t, regardless of their time of excitation,

P(z, t) = Tr[p · ρ(z, t)] =X

α,β

ραβ(z, t)pβα,

for a two-level atom, pab = pba = p,

P(z, t) = p[ρab(z, t) + ρba(z, t)] = p[ρab(z, t) + c.c],

the off-diagonal elements of the population matrix determine the macroscopicpolarization,

∂

∂tρaa =

i

~[pabEρba − c.c]− γaρaa,

∂

∂tρbb = − i

~[pabEρba − c.c]− γbρbb,

∂

∂tρab = − i

~pabE(ρaa − ρbb)− (iω +

γa + γb

2)ρab,

IPT5340, Fall ’06 – p. 42/70

Maxwell-Schrödinger equations

the equations for the two-level atomic medium coupled to the field E are

∂

∂tρaa =

i

~[pabEρba − c.c]− γaρaa,

∂

∂tρbb = − i

~[pabEρba − c.c]− γbρbb,

∂

∂tρab = − i

~pabE(ρaa − ρbb)− (iω +

γa + γb

2)ρab,

the condition of self-consistency requires that the equation of motion for the field E isdriven by the atomic population matrix elements,

the field is described by the Maxwell’s equation,

∇ · D = 0, ∇× E = −∂B∂t,

∇ · B = 0, ∇× H = J +∂D∂t,

IPT5340, Fall ’06 – p. 43/70

Maxwell-Schrödinger equations

the field is described by the Maxwell’s equation,

∇× (∇× E) + µ0σ∂E∂t

+ µ0ǫ0∂2E∂t2

= −µ0∂2P∂t2

,

for a running wave polarized along x-direction,

E(r, t) = x1

2E(z, t)exp[−i(νt− kz + φ)] + c.c,

the response of the medium is assumed

P(r, t) = x1

2P (z, t)exp[−i(νt− kz + φ)] + c.c,

where E(z, t), φ(z, t), and P (z, t) are all slowly varying function of position andtime, i.e.

∂E

∂t≪ νE,

∂E

∂z≪ kE,

∂

∂t≪ ν,

∂

∂z≪ k,

∂P

∂t≪ νP,

∂P

∂z≪ kP,

IPT5340, Fall ’06 – p. 44/70

Maxwell-Schrödinger equations

the response of the medium is assumed

P(r, t) = x1

2P (z, t)exp[−i(νt− kz + φ)] + c.c,

in terms of the population matrix,

P (z, t) = 2Pρabexp[i(νt− kz + φ)],

the Maxwell’s equation for the slowly varying envelope function is,

(∂

∂z+

1

c

∂

∂t)(− ∂

∂z+

1

c

∂

∂t)E = −µ0σ

∂E

∂t− µ0

∂2P

∂t2,

along with the equations of motion for the two-level atom,

∂

∂tρaa =

i

~[pabEρba − c.c]− γaρaa,

∂

∂tρbb = − i

~[pabEρba − c.c]− γbρbb,

∂

∂tρab = − i

~pabE(ρaa − ρbb)− (iω +

γa + γb

2)ρab,

IPT5340, Fall ’06 – p. 45/70

Jaynes-Cummings Hamiltonian

in the dipole approximation, the semi-classical Hamiltonian is

H0 = ~ωa|a〉〈a|+ ~ωb|b〉〈b|,H1 = −(pab|a〉〈b|+ pba|a〉〈b|)E(t),

to include the quantized field,

H = HA + HF − er · E,

=X

i

~ωiσii +X

k

~νk(a†kak +

1

2)−

X

i,j

Pij σijX

k

Ek(ak + a†k),

= ~ωiσii +X

k

~νk(a†kak +

1

2) + ~

X

i,j

X

k

gijk σij(ak + a†k),

where

gijk = −Pij · Ek~

is the coupling constant,

IPT5340, Fall ’06 – p. 46/70

Jaynes-Cummings Hamiltonian

to include the quantized field,

H = ~ωiσii +X

k

~νk(a†kak +

1

2) +

X

i,j

X

k

gijk σij(ak + a†k),

for a two-level atom, Pab = Pba, we have gk = gabk = gbak , then

H = ~ωaσaa + ~ωbσbb +X

k

~νk(a†kak + ~

1

2) + ~

X

k

gk(σab + σba)(ak + a†k),

define new operators,

σz = σaa − σbb = |a〉〈a| − |b〉〈b|,σ+ = σab = |a〉〈b|,σ− = σba = |b〉〈a|,

and the new energy level

~ωaσaa + ~ωbσbb =1

2~ωσz +

1

2(ωa + ωb),

where ω = ωa − ωb,IPT5340, Fall ’06 – p. 47/70

Jaynes-Cummings Hamiltonian

the Hamiltonian for a two-level atom interaction with quantized fields becomes

H =1

2~ωσz +

X

k

~νk(a†kak +

1

2) + ~

X

k

gk(σ+ + σ−)(ak + a†k),

where the atomic operators satisfy the spin-1/2 algebra of the Pauli matrices, i.e.

[σ−, σ+] = −σz , and [σ−, σz] = 2σ−,

in the rotating-wave approximation, we drop terms akσ− and a†kσ+, then we haveJaynes-Cummings Hamiltonian

H =1

2~ωσz +

X

k

~νk(a†kak +

1

2) + ~

X

k

gk(σ+ak + a†kσ−),

IPT5340, Fall ’06 – p. 48/70

Interaction of a single two-level atom with a single-mode fie ld

the Jaynes-Cummings Hamiltonian,

H =1

2~ωσz + ~νa†a+ ~g(σ+a+ a†σ−),

the interaction Hamiltonian is,

V = exp[iH0t/~]H1exp[−iH0t/~],

= ~g(σ+aei∆t + a†σ−e

−i∆t),

where ∆ = ω − ν,

the equation of motion for the state |Ψ is

i~∂

∂t|Ψ〉 = V |Ψ〉,

where the state |Ψ〉 is the superposition of

|Ψ(t)〉 =X

n

[ca,n(t)|a, n〉+ ba,n(t)|b, n〉],

IPT5340, Fall ’06 – p. 49/70

Interaction of a single two-level atom with a single-mode fie ld

the interaction Hamiltonian is,

V = ~g(σ+aei∆t + a†σ−e

−i∆t),

which only cause transitions between the states |a, n〉 and |b, n+ 1〉, and

ddtca,n = −ig

√n+ 1ei∆tcb,n+1,

ddtcb,n+1 = −ig

√n+ 1e−i∆tca,n,

compared to the semi-classical equations,

ddtca = i

ΩR

2e−iφei(ω−ν)tcb,

d

dtcb = i

ΩR

2eiφe−i(ω−ν)tca,

IPT5340, Fall ’06 – p. 50/70

Interaction of a single two-level atom with a single-mode fie ld

for the initially excited state, ca,n(0) = cn(0) and cb,n+1(0) = 0, and here cn(0) isthe probability amplitude for the field along,

the solutions are

ca,n(t) = cn(0)[cos(Ωnt

2)− i∆

Ωnsin(

Ωnt

2)]ei∆t/2,

cb,n+1(t) = −cn(0)2ig√n+ 1

Ωnsin(

Ωnt

2)ei∆t/2,

the Rabi frequency is Ωn = ∆2 + 4g2(n+ 1), which is proportional to the photonnumber of the field,

the probability p(n) that there are n photons in the field at time t is,

p(n) = |ca,n(t)|2 + |cb,n(t)|2,

= |cn(0)|2[cos2(Ωnt

2) + (

∆

Ωn)2 sin2(

Ωnt

2)] + |cn−1(0)|2( 4g2n

Ω2n−1

) sin2(Ωn−1t

2)]

IPT5340, Fall ’06 – p. 51/70

Interaction of a single two-level atom with a single-mode fie ld

for n photons in the field at time t = 0 with a coherent state, |cn(0)|2 =〈n〉ne−〈n〉

n!,

∆ = 0, 〈n〉 = 25, gt = 0 gt = 3.0

10 20 30 40 50

0.02

0.04

0.06

0.08

10 20 30 40 50

0.02

0.04

0.06

0.08

0.1

10 20 30 40 50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

10 20 30 40 50

0.02

0.04

0.06

0.08

0.1

0.12

∆ = 0, 〈n〉 = 25, gt = 10 gt = 100

IPT5340, Fall ’06 – p. 52/70

Interaction of a single two-level atom with a single-mode fie ld

the population inversion,

W (t) =X

n

|ca,n(t)|2 − |cb,n(t)|2 =∞

X

0

|cn(0)|2[∆2

Ω2n

+4g2(n+ 1)

Ω2n

cos(Ωnt)],

20 40 60 80 100

-0.5

-0.25

0.25

0.5

0.75

1

∆ = 0, 〈n〉 = 25,

IPT5340, Fall ’06 – p. 53/70

Revival and Collapse of the population inversion

the population inversion,

W (t) =X

n

|ca,n(t)|2 − |cb,n(t)|2 =∞

X

0

|cn(0)|2[∆2

Ω2n

+4g2(n+ 1)

Ω2n

cos(Ωnt)],

each term in the summation represents Rabi oscillation for a definite value of n,

at the initial time t = 0, the atom is prepared in a definite state and therefore all theterms in the summation are correlated,

as times increases, the Rabi oscillations associated with different frequentexcitations have different frequencies and there fore become uncorrelated, leading toa collapse of inversion,

as time is further increased, the correlation is restored and revival occurs,

in the semi-classical theory, the population inversion evolves with sinusoidal Rabioscillations, and collapses to zero when on resonance,

for the quantized fields, the collapse and revival of inversion is repeated withincreasing time, but the amplitude of Rabi oscillations decreasing and the timeduration in which revival takes place increasing,

IPT5340, Fall ’06 – p. 54/70

Vacuum Rabi Oscillation

the revivals occur only because of the quantized photon distribution,

for a continuous photon distribution, like a classical random field, there is only acollapse but no revivals,

compared to Fourier transform and Discrete Fourier transform,

even for initial vacuum field, |cn(0)|2 = δn0, the inversion is

W (t) =1

∆2 + 4g2[∆2 + 4g2 cos(

p

∆2 + 4g2t)],

the Rabi oscillation take place due to the vacuum state,

the transition from the upper level to the lower level in the vacuum becomes possibledue to spontaneous emission,

IPT5340, Fall ’06 – p. 55/70

Collective angular momentum operators

for a two-level atom, one can use Pauli spin operator to describe,

s =1

2~σ,

where

σz = |a〉〈a| − |b〉〈b|, σ+ = |a〉〈b|, σ− = |b〉〈a|,σx = |a〉〈b|+ |b〉〈a|, and σy = −i(|a〉〈b| − |b〉〈a|),

for an assembly of N two-level atoms, the corresponding Hilbert space is spannedby the set of 2N product states,

|Φ〉 =N

Y

n=1

|Ψn〉,

we can define the collective angular momentum operators,

Jµ =1

2σnµ, (µ = x, y, z),

IPT5340, Fall ’06 – p. 56/70

Analogs between J and a, a†

the analogies between the free-field quantization, a and a†, and the free atomquantization,

[Jx, Jy ] = iJz ↔ [q, p] = i~,

J− = Jx − iJy ↔ a =1√2~ω

(ωq + ip),

J+ = Jx + iJy ↔ a† =1√2~ω

(ωq − ip),

Jz =1

2(J+J−J−J+) ↔ n = a†a,

and the commutation relations,

[J−, J+] = −2Jz ↔ [a, a†] = 1,

[J−, Jz ] = J− ↔ [a, n] = a,

[J+, Jz ] = −J+ ↔ [a†, n] = −a†,

when all the atoms are in the ground state, the eigenvalue of Jz is −J = −N2

, the

commutation relation is reduced to a bosonlike one, [J−, J+] = N ↔ [a, a†] = 1,

IPT5340, Fall ’06 – p. 57/70

Angular momentum eigenstates (Dicke states)

the Dicke states are defined as the simultaneous eigenstates of the Hermitianoperators Jz and J2, i.e.

Jz |M,J〉 = M |M,J〉, and , J2|M,J〉 = J(J + 1)|M,J〉,

where (M = −J,−J + 1, . . . , J − 1, J) and

J+|M,J〉 =p

J(J + 1)−M(M + 1)|M + 1, J〉 ↔ a†|n〉 =√n+ 1|n+ 1〉,

J−|M,J〉 =p

J(J + 1)−M(M − 1)|M − 1, J〉 ↔ a|n〉 =√n|n− 1〉,

J−| − J, J〉 = 0 ↔ a|0〉 = 0,

|M,J〉 = 1

(M + J)!

0

@

2J

M + J

1

A

−1/2

J(M+J)+ | − J, J〉 ↔ |n〉 = 1√

n!(a†)n|0〉,

the Dicke states is the counterpart of the Fock state, the state |M,J〉 denotes anatomic ensemble where exactly J +M atoms are in the excited state out ofN = 2J atoms,

IPT5340, Fall ’06 – p. 58/70

Interaction between N two-level atoms and a single-mode field

the Dicke states | − J, J〉 corresponds to the case in which all the atoms are in theground state, J = N/2,

the Dicke states | − J + 1, J〉 corresponds to the case in which only one atom is inthe excited state,

the Dicke states |J, J〉 corresponds to the case in which all the atom are in theexcited state,

the total Hamiltonian for N two-level atoms with a single-mode field is,

H =1

2~ωJz + ~ν(a†a+

1

2) + ~g(J+a+ a†J−),

collective Rabi oscillation

IPT5340, Fall ’06 – p. 59/70

Spontaneous emission of a two-level atom

the interaction hamiltonian, in the rotating-wave approximation, for a two-levelatom is,

V = ~

X

k

(gk(r0)∗σ+akei(ω−νk)t + gk(r0)a†kσ−e

−i(ω−νk)t),

where gk(r0) = gkexp(−ik · r0) is the spatial dependent coupling coefficient,

assume at t = 0 the atom is in the excited state |a〉 and the field modes are in thevacuum state |0〉,

|Ψ(t)〉 = ca(t)|a, 0〉+X

k

cb,k|b, 1k〉,

with ca(0) = 1 and cb,k(0) = 0,

in the interaction picture, |Ψ(t)〉 = − i~|Ψ(t)〉, we have

ca(t) = −iX

k

g∗k(r0)ei(ω−νk)tcb,k(t),

cb(t) = −igk(r0)e−i(ω−νk)tca(t),

IPT5340, Fall ’06 – p. 60/70

Weisskopf-Wigner theory of spontaneous emission

in the interaction picture, |Ψ(t)〉 = − i~|Ψ(t)〉, we have

ca(t) = −iX

k

g∗k(r0)ei(ω−νk)tcb,k(t),

cb(t) = −igk(r0)e−i(ω−νk)tca(t),

the exact solutions are

cb(t) = −igk(r0)

Z t

0dt′e−i(ω−νk)t′ca(t

′),

ca(t) = −X

k

|gk(r0)|2Z t

0dt′ei(ω−νk)(t−t′)ca(t

′),

assuming that the filed modes are closely spaced in frequency,

X

k

→ 2V

(2π)3

Z 2π

0dφ

Z π

0dθ sin θ

Z ∞

0dkk2,

where V is the quantization volume,

IPT5340, Fall ’06 – p. 61/70

Weisskopf-Wigner theory of spontaneous emission

the exact solutions are

ca(t) = −X

k

|gk(r0)|2Z t

0dt′ei(ω−νk)(t−t′)ca(t

′),

the coupling coefficient,

|gk(r0)|2 = |P · Ek~|2 =

νk

2~ǫ0VP2ab cos2 θ,

where θ is the angle between the atomic dipole moment Pab and the electric fieldpolarization vector ǫk , i.e. Ek(r, t) = ǫk(

~νk

ǫ0V)1/2[ak + a†k],

the equation for ca(t) becomes

ca(t) = − 4P2ab

(2π)26~ǫ0c3

Z ∞

0dνk

Z t

0dt′ν3

kei(ω−νk)(t−t′)ca(t

′),

where we have use k = νk/c,

IPT5340, Fall ’06 – p. 62/70

Weisskopf-Wigner theory of spontaneous emission

the equation for ca(t) becomes

ca(t) = − 4P2ab

(2π)26~ǫ0c3

Z ∞

0dνk

Z t

0dt′ν3

kei(ω−νk)(t−t′)ca(t

′),

for most of the optical problems, νk varies little around the atomic transitionfrequency ω,

we can safely replace ν3k by ω3 and the lower limit in the νk integration by −∞,

then

ca(t) = − 4P2abω

3

(2π)26~ǫ0c3

Z ∞

−∞dνk

Z t

0dt′ei(ω−νk)(t−t′)ca(t

′),

= − 4P2abω

3

(2π)26~ǫ0c3

Z t

0dt′2πδ(t− t′)ca(t′),

≡ −Γ

2ca(t),

where Γ =4P2

abω3

12π2~ǫ0c3is the decay rate of the excited state,

IPT5340, Fall ’06 – p. 63/70

Photonic Bandgap Crystals: two(high)-dimension

-2 -1 0 1 2-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5a = 0.452r = 0.15d = 0.45l = 1.0nc = 3.5

Normalized Frequency ωa/2πcT

ransm

itta

nce

[dB

]0.2 0.3 0.4 0.5 0.6 0.7

-50

-40

-30

-20

-10

0

10

# of layers= 2

345

IPT5340, Fall ’06 – p. 64/70

Band diagram and Density of States

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

M Γ X M

Fre

qu

en

cy (

ωd

/2πc)

0 0.5 1 1.5 2 2.5 3

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

DOS (A.U.)

IPT5340, Fall ’06 – p. 65/70

Modeling DOS of PBCs

0

0.5

1

1.5

2

2.5

3x 10

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

DO

S

Frequency

anisotropic model: ωk = ωc + A|k − ki0|

2

D(ω) =√

ω−ωcA3 Θ(ω − ωc)

S. Y. Zhu, et al., Phys. Rev. Lett. 84, 2136 (2000).

IPT5340, Fall ’06 – p. 66/70

Remarks:

1. coupling constant:

gk ≡ gk(d,−→r 0) = |d|ωa

√

1

2~ǫ0ωkVd · E∗k(

−→r0 )

2. memory functions:

G(τ) ≡∑

k

|gk|2ei∆ktΘ(τ)

Gc(τ) ≡∑

k

|gk|2e−i∆ktΘ(τ)

3. Markovian approximation:

G(t) = Gc(t) = Γδ(t)

IPT5340, Fall ’06 – p. 67/70

photon-atom bound state

|1>

|2>

ωa

1 2 3 4 5time

0.3

0.4

0.5

0.6

upper level population

S. John and H. Wang, Phys. Rev. Lett. 64, 2418 (1990).

IPT5340, Fall ’06 – p. 68/70

Hamiltonian of our system: Jaynes-Cummings model

H =~

2ωaσz + ~

∑

k

ωka†kak +

Ω

2~(σ−e

iωLt + σ+e−iωLt)

+ ~

∑

k

(gkσ+ak + g∗ka†kσ−)

And we want to solve the generalized Bloch equations:

σ−(t) = iΩ

2σz(t)e

−i∆t +

∫ t

−∞d t′G(t − t′)σz(t)σ−(t′) + n−(t)

σ+(t) = −iΩ

2σz(t)e

i∆t +

∫ t

−∞d t′Gc(t − t′)σ+(t′)σz(t) + n+(t)

σz(t) = iΩ(σ−(t)ei∆t − σ+(t)e−i∆t) + nz(t)

− 2

∫ t

−∞d t′[G(t − t′)σ+(t)σ−(t′) + Gc(t − t′)σ+(t′)σ−(t)]

IPT5340, Fall ’06 – p. 69/70

Fluorescence quadrature spectra near the band-edge

R.-K. Lee and Y. Lai, J. Opt. B, 6, S715 (Special Issue 2004).

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