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Photon physics: Fotonen bestaan niet

Peter van der Straten (Atom Optics) Photon physics June 5, 2007 1 / 41

Books

1 R. Loudon, The Quantum Theory of Light, Oxford, 2nd Ed., 1973.2 H. Metcalf and P. van der Straten, Laser cooling and trapping,

Springer, 1999.3 H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and

Two-Electron Atoms, Springer-Verlag, Berlin, 1957.4 C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics,

Wiley, New York (1977).5 Bransden and Joachain, The physics of Atoms and Molecules,

Longman, New York, 1983.6 C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Photons and

atoms, Wiley, New York, 1989.7 L. Allen and J.H. Eberly, Optical Resonance and Two-Level Atoms,

Dover, 1975.8 P. Meystre, Atom Optics, Springer, 2001.9 P. Meystre and M. Sargent III, Elements of Quantum Optics,

Springer, 1990.Peter van der Straten (Atom Optics) Photon physics June 5, 2007 2 / 41

Outline

1 What is a photon?BasicsQuotes

2 Driving atomsLorentz oscillatorElectric dipole interactionRabi oscillationsTwo level atomMCWF-method

3 Spontaneous emissionA conversation between father and sonEinstein relationsClassical decay rateVacuum fluctuations


What is a photon?

Outline





What is a photon? Basics

What is a photon?

Property Relation

frequency ν

wavevector k, |k| = 2π/λenergy E = hν

momentum p = kmass 0(< 8 × 10−49 g)

spin S = εhelicity ms = ±1

Special properties:

1 Number is not conservative

2 Motion is relativistic

3 No operator for the position of a photon in Quantum Mechanics


What is a photon? Quotes

The “photon” in the press

We know that light is made of particles because we can take avery sensitive instrument that makes clicks when light shines onit, and if the light gets dimmer, the clicks remain just as loud -there are just fewer of them. Thus light is something likeraindrops - each little lump of light is called a photon - and if thelight is all one color, all the “raindrops” are the same size.Richard Feynman

What the laser does is to produce vast numbers of particles ofexactly the energy and wavelength. With no other stable particlebut the photon is such a feat possible. The laser beam’sremarkable macroscopic properties arise that its constituentparticles are precisely identical. Whether the laser could havebeen invented without quantum mechanics is an interestingquestion. G. Feinberg ‘Lasers and Light’



The “photon” in the press

All these fifty years of conscious brooding have brought me nocloser to the answer to the question, “What are light quanta?”Of course, today, every rascal thinks he knows the answer, but heis mistaken. Albert Einstein

The fuzzy ball picture of a photon often leads to unnessaryconfusion. M. Sculley and M. Sargent

The photon interferes only with itself. P. Dirac

The optical duality is then a relic of the 1905-1927 interregnum,a remains serving mainly to mislead students into believing thatlight is at the same time undulatory and non-undulatory. M.Bunge



The “photon” in the press (cont.)

It should be apparent from the title of this article that the authordoes not like the use of the word “photon”, which dates from1926. In view, there is no such thing as a photon. Only a comedyof errors and historical accidents led to its popularity amongphysicists and optical scientists. W.E. Lamb, Jr. ‘Anti-photon’

The words “table” and “light”, as we ordinarily use them, arewhat we might call “classical words”. They refer to things thatwe imagine to have an objective reality between events ofobservation. We know from our experience with quantummechanics that this imagined reality is only an approximation,usually a very good one. The word “photon”, however, is a“quantum word” that describes a phenomenon for which noclassical word is adequate.” John H. Marburger, III



Anti-photon



Anti-photon (cont.)



Optics & Photonics News (2003)

About the cover

Artist’s rendition of a Wigner function

for six photons (see Mack and Schleich,

p. 28). This issue of OPN Trends was

conceived to bring together different

views regarding a question asked

over the course of centuries: What is

the nature of light? Despite significant

progress in our understanding,

it remains an open question.

Sponsored by

NSG



Optics & Photonics News (2003)

The concept of the photon—revisited

Ashok Muthukrishnan,1 Marlan O. Scully,1,2 and M. Suhail Zubairy1,3

1Institute for Quantum Studies and Department of Physics, Texas A&M University, College Station, TX 778432Departments of Chemistry and Aerospace and Mechanical Engineering, Princeton University, Princeton, NJ 08544

3Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan

The photon concept is one of the most debated issues in the history of physical science. Some thirty years ago, we published anarticle in Physics Today entitled “The Concept of the Photon,”1 in which we described the “photon” as a classical electromagneticfield plus the fluctuations associated with the vacuum. However, subsequent developments required us to envision the photon asan intrinsically quantum mechanical entity, whose basic physics is much deeper than can be explained by the simple ‘classicalwave plus vacuum fluctuations’ picture. These ideas and the extensions of our conceptual understanding are discussed in detailin our recent quantum optics book.2 In this article we revisit the photon concept based on examples from these sources and more.© 2003 Optical Society of America

OCIS codes: 270.0270, 260.0260.


Driving atoms

Outline





Driving atoms Lorentz oscillator

Classical atom-field interaction

−ω0

+

m

Lorentz equation:

x + γx + ω02x =

−eE0

mcos ωt

Solution:x(t) = ae iωt + a∗e−iωt

Substitution:

−ω2a + iγωa + ω02a = −eE0

mor a =

−eE0/2m

(ω02 − ω2) + iγω

Approximation (ω0 ≈ ω, δ = ω − ω0 ω):

a ≈ eE0/2mω0

2δ − iγ≡ 1/2x0e

iφ with x0 =eE0/mω0√γ2 + 4δ2

and φ = arctan( γ

2δ

)Final result:

x = x0 cos(ωt + φ)Peter van der Straten (Atom Optics) Photon physics June 5, 2007 14 / 41

Driving atoms Lorentz oscillator

Classical atom-field interaction

Power absorbed:

P = F x = −eE x = eωx0E0

[cos ωt sin ωt cos φ + cos2 ωt sin φ

]Time average:

〈P〉 =e2γE0

2

2m(γ2 + 4δ2)

Define:

I = 12ε0cE0

2 and Is =ε0mcγ2

ω

2e2or s0 ≡ I

Is=

e2E02

mγ2ω

Scattering rate R:

R =〈P〉ω

=s0γ/2

1 + (2δ/γ)2OBE


Driving atoms Electric dipole interaction

Time-Dependent Perturbation Theory

The time-dependent Schrodinger equation

HΨ(r , t) = i∂Ψ(r , t)

∂t

Field-free Hamiltonian H0, eigenvalues En ≡ ωn, eigenfunctions φn(r)Total Hamiltonian:

H(t) = H0 + H′(t)

Solution Ψ(r , t):

Ψ(r , t) =∑k

ck(t)φk(r)e−iωk t

Substitution:

H(t)Ψ(r , t) = [H0 + H′(t)]∑k


= (i)

(∂

∂t

)∑k




Time-Dependent Perturbation Theory

Result:

idcj(t)

dt=∑k

ck(t)H′jk(t)e iωjk t ,

H′jk(t) ≡ 〈φj |H′(t)|φk〉

andωjk ≡ (ωj − ωk)



Electric-dipole approximation

Hamiltonian:

H =∑α

1

2mα

[pα − qα

A(r , t)]2

+ V (r) +∑α

qαU(r , t),

Neutral system ∑α

qα = 0.

Dipole moment

µ =∑α

qαrα,

Dipole approximation (|rα| λ):

A(r , t) ≈ A(0, t) and U(r , t) ≈ U(0, t) +r · ∇U(0, t).

Thus∑α

qαU(r , t) ≈ U(0, t)∑α

qα +

(∑α

qαrα

)· ∇U(0, t) = µ · ∇U(0, t).



Gauge transformation

Transformation:

A′(r , t) = A(r , t) + ∇χ(r , t) and U ′(r , t) = U(r , t) − ∂

∂tχ(r , t)

Electric and magnetic field:

E (r , t) = −∇U(r , t) − ∂

∂tA(r , t) and B(r , t) = ∇× A(r , t)

Choice for χ(r , t): χ(r , t) = −r · A(0, t). Then

A′(r , t) = A(r , t) − A(0, t) ≈ 0 and U ′(r , t) = U(r , t) +r · ∂

∂tA(0, t).

or∇U ′(0, t) = ∇U(0, t) +

∂

∂tA(0, t) = −E (0, t).

The Hamiltonian of the system

H =∑α

pα2

2mα+ V (r) − µ · E (0, t).


Driving atoms Rabi oscillations

The Rabi Two-Level Problem

Two states (1 → g and 2 → e):

idcg (t)

dt= ce(t)H′

ge(t)e−iωat and i

dce(t)

dt= cg (t)H′

eg (t)e+iωat

Electric-dipole approximation: H′(t) = −µ · E (r , t)Plane wave:

E (r , t) = E0 cos(kz − ωt) = E0

(e ikz−iωt + e−ikz+iωt

)Substitution (z = 0):

H ′eg (t)e iωat = −e iωat〈e|µ · E |g〉 = −〈e|µ|g〉 ·

(E0e

i(ωa−ω)t + Ee i(ωa+ω)t)

Rotating wave approximation:

H′eg (t) = −〈e|µ|g〉·E0e

+i(ωa−ω)t and H′ge(t) = −〈g |µ|e〉·E0e

−i(ωa−ω)t

Rabi frequency:

Ω =2〈e|µ · E0|g〉



The Rabi Two-Level Problem (cont.)

Final result (δ = ωa − ω):

dcg (t)

dt=

i

2ce(t)Ω

∗e−iδt anddcg (t)

dt=

i

2cg (t)Ωe+iδt

Uncoupling:

d2cg (t)

dt2−iδ

dcg (t)

dt+

Ω2

4cg (t) = 0 and

d2ce(t)

dt2+iδ

dce(t)

dt+

Ω2

4ce(t) = 0

Initial conditions (cg (0) = 1 and ce(0) = 0):

cg (t) =

(cos

Ω′t2

− iδ

Ω′ sinΩ′t2

)e+iδt/2 and ce(t) = −i

Ω

Ω′ sinΩ′t2

e−iδt/2

where Ω′ ≡ √Ω2 + δ2



Rabi oscillations

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Prob

abili

ty

Time

Probability |ce(t)|2 for the atom to be in the excited state for Ω = γ andδ = 0 (solid line), δ = γ (dotted line), and δ = 2.5γ (dashed line). Time isin units of 1/γ.


Driving atoms Two level atom

Two level atom

Hamiltonian for atom-light interaction: H′(t) = −µ · E (r , t). Two coupledstates g and e:

idcg (t)

dt= ce(t)H′

ge(t)e−iωat and i

dce(t)

dt= cg (t)H′

eg (t)e iωat .

Travelling plane wave: E (r , t) = E0ε cos(kz − ωt) with Rabi frequency:

Ω ≡ −eE0

〈e|r |g〉.

Density matrix:

ρ =

(ρee ρeg

ρge ρgg

)=

(cec

∗e cec

∗g

cgc∗e cgc∗g

).



Two level atom

Optical Bloch Equations:

dρgg

dt= +γρee +

i

2(Ω∗ρeg − Ωρge)

dρee

dt= −γρee +

i

2(Ωρge − Ω∗ρeg )

dρge

dt= −

(γ

2+ iδ

)ρge +

i

2Ω∗ (ρee − ρgg )

dρeg

dt= −

(γ

2− iδ

)ρeg +

i

2Ω (ρgg − ρee)



Numerical integration of OBE

0 2 4 6 8 100.00

0.05

0.10

0.15

0.20

0.25

Time

Exc

itatio

npr

obab

ility

Probability |ce(t)|2 for the atom to be in the excited state for Ω = γ andδ = −γ by numerical integration of the OBEs. The solutions are identicalto the Monte Carlo wavefunction method with an infinite number of atomtrajectories. Time is in units of 1/γ.Peter van der Straten (Atom Optics) Photon physics June 5, 2007 25 / 41


Two level atom (continued)

Steady state:

ρee =s

2(1 + s)=

s0/2

1 + s0 + (2δ/γ)2

with the off-resonance saturation parameter s:

s ≡ |Ω|2/2

δ2 + γ2/4≡ s0

1 + (2δ/γ)2

and the on-resonance saturation parameter:

s0 ≡ 2|Ω|2γ2

=I

Is

Saturation intensity:

Is ≡ πhc

3λ3τScattering rate γp:

γp = γρee =s0γ/2

1 + s0 + (2δ/γ)2

Compare to Lorentz modelPeter van der Straten (Atom Optics) Photon physics June 5, 2007 26 / 41


Scattering rate

-10 -5 0 5 100.00

0.10

0.20

0.30

0.40

0.50s0=100.0

s0= 10.0

s0= 1.0

s0= 0.1

Detuning [ ]

Scat

teri

ngra

tep

[]

Excitation rate γp as a function of the detuning δ for several values of thesaturation parameter s0. Note that for s0 > 1 the line profiles start tobroaden substantially from power broadening.


Driving atoms MCWF-method

Monte Carlo Wavefunction Method

Start with the atom in the ground state (cg = 1, ce = 0). After a shortperiod ∆t, determine the new amplitudes by a coherent evolution of thestate vector c = (cg , ce):

c ′ = Hc and H =

(0 Ω∗/2

Ω/2 −(δ + iγ/2)

),

with γ the decay rate due to spontaneous emission. The probability p forthe atom to decay is given by p = |ce |2γ∆t. From a random string ofnumbers take one number x . If x > p, nothing happens. If x < p, theatom decays to the ground state. In the first case, we have to renormalizethe wavefunction to c2

g + c2e = 1. In the second case, we have c = (1, 0).

If we repeat this procedure for one atom, we have a single trajectory of thetime-evolution of the atom. This method can be repeated for severalatoms.


Driving atoms MCWF-method

Excited State Decay and its Effects

0 2 4 6 8 10Time

0.00.1

0.2

0.0

0.1

0.2

0.0

0.1

0.2

0.3

Exc

itatio

n pr

obab

ility

(a)

(b)

(c)

Trajectories for atoms is a radiation field with Ω = γ and δ = −γ, where γis the natural width. The number of atoms averaged over is 1 (a), 10 (b),and 100 (c).


Spontaneous emission

Outline





Spontaneous emission A conversation between father and son


You might wonder what [my father] got out of it at all. I went to MIT.I went to Princeton. I came home, and he said, “Now you’ve got ascience education. I always wanted to know something that I neverunderstood; and so I want you to explain it to me.” I said, “Yes.”He said, “I understand that they say that light is emitted from an atomwhen it goes from one state to another, from an excited state to astate of lower energy.” I said, “That’s right.”“And light is a kind of particle, a photon, I think they call it.” “Yes.”“So, if the photon comes out of the atom when it goes from theexcited state to the lower state, the photon must have been in theatom in the excited state.” I said, “Well, no.”He said, “Well, how do you look at it so you can think of a particlephoton coming out without having been there in the excited state?”I thought a few minutes, and I said, “I’m sorry; I don’t know. I can’texplain it to you.”He was very disappointed after all these years and years of trying toteach me something, that it came out with such poor results. RichardFeynman


Spontaneous emission Einstein relations

Einstein relations

g

Aeg

e

BegBge

Time-derivatives of populations:

dNg

dt= −dNe

dt= NeAeg − NgBgeρ(ω) + NeBegρ(ω)

Steady state:

ρ(ω) =Aeg

(Ng/Ne)Bge − Beg

Boltzmann’s and Planck’s law:

Ng

Ne=

gg

geexp

(ω

kBT

)ρ(ω) =

ω3

π2c2

1

exp(ω/kBT ) − 1

Thus:

Beg =gg

geBge and Aeg =

(ω3

π2c3

)Beg


Spontaneous emission Classical decay rate

Classical decay rate

Radiation of electro-magnetic energy (Larmor formula):

P =2e2a2

3c3.

For an oscillating charge (x = x0 cos ω0t) we get averaged over a period:

P =e2x0

2ω04

3c3.

The energy of an oscillating charge is E = mω02x0

2, we get

dE

dt= −e2x0

2ω04

3c3= −

(e2ω0

2

3mc3

)E .

Thus the decay rate is given by

γrad =e2ω0

2

3mc3.


Spontaneous emission Classical decay rate

Classical decay rate

If we introduce the correction for the oscillator strength:

γrad =e2ω0

2

3mc3f =

e2ω02

3mc3

2mµ2ω0

e2=

2µ2ω03

3c3.

This is half the Einstein A coefficient.


Spontaneous emission Vacuum fluctuations

Field quantization

Free classical field:

−∇2A +1

c2

∂2A

∂t2= 0

Fourier series:

A =∑k

Ak(t)e+ik·r + A∗

k(t)e−ik·r

with kx ,y ,z =2πνx ,y ,z

L

Coulomb gauge: ∇ · A = 0 ⇒ k · Ak(t) = k · A∗k(t) = 0

Thus:∂2Ak(t)

∂t2+ ω2

kAk(t) = 0, ωk = ck

Solution:A(r , t) =

∑k

Ake−iωk t+ik·r + A∗

ke+iωk t−ik·r

or



Field quantization (cont.)

Ek = iωk

Ake−iωk t+ik·r − A∗

ke+iωk t−ik·r

Bk = ik×

Ake−iωk t+ik·r − . . .

Energy density:

εk =1

2

∫cavity

(ε0

E 2k +

B2k

µ0

)dV = 2ε0Vω2

kAk · A∗

k ≡ 1

2

(Pk

2 + ωk2Q2

k

),

with

Ak =1√

4ε0Vωk2

(ωkQk + iPk)εk and A∗k =

1√4ε0Vωk

2(ωkQk − iPk)εk



Harmonic oscillator

Hamiltonian: H = 12

(p2 + ω2q2

)and commutator relation: [q, p] = i

Define destruction and creation operator:

a =ωq + i p√

2ωa† =

ωq − i p√2ω

Thus:

a†a =p2 + ω2q2 + iωqp − iωpq

2ω=

H− 12ω

ωaa† = . . . =

H + 12ω

ω

or[a, a†

]= 1 and H = ω

(a†a + 1

2

).

Quantum algebra:

H|n〉 = ω(a†a + 1

2

) |n〉 = En|n〉× a† ω

(a†a†a + 1

2a†) |n〉 = Ena

†|n〉commutator ω

(a†aa† − a† + 1

2a†) |n〉 = Ena

†|n〉or ω

(a†a + 1

2

)a†|n〉 = (En + ω)a†|n〉 = Ha†|n〉



Harmonic oscillator (cont.)

Creation or destruction:

a†|n〉 = |n + 1〉 (En+1 = En+ω) or a|n〉 = |n − 1〉 (En−1 = En−ω).

Lowest state:

Ha|0〉 = (E0 − ω)a|0〉 ⇒ a|0〉 = 0 ⇒ H|0〉 = ω(a†a + 12)|0〉 = 1

2ω|0〉

orE0 = 1

2ω and En = (n + 1

2)ω.

Proper normalization:

a|n〉 =√

n|n − 1〉 and a†|n〉 =√

n + 1|n + 1〉Introduce the number operator n as

n = a†a, or n|n〉 = a†a|n〉 = n|n〉.



Field quantization (final)

Quantizing the vector field:

Ak =

√

2ε0Vωkakεk and A∗

k =

√

2ε0Vωka†kεk

The electric field:

Ek = i

√ωk

2ε0Vεk

ake−iωk t+ik·r − a†ke+iωk t−ik·r

The magnetic field:

Bk = i

√

2ε0Vk × εk

ake−iωk t+ik·r − a†ke+iωk t−ik·r

The Hamiltonian of the field in the cavity:

HR =∑k

ωk(a†k ak + 12)

The total energy in the cavity: ε =∑

k(nk + 12)ωk .



Vacuum field

Like a harmonic oscillator, the electromagnetic field has a zero-pointenergy ω/2. Use a cube to quantize the electric field:

ki =2πni

L, i = x , y , z , ni = 0, 1, 2, 3, . . .

Thus

dn =

(L

2π

)3

d3k = 2 × Vω2

8π3c3sin θdωdθdφ → ω2V

π2c3dω,

or the “zero-point” spectral density distribution becomes

ρ(ω)Vdω =ω2V

π2c3dω × 1

2ω.

The Einstein B coefficient for a broadband field is given by B = 4π2µ2

32 , and

thus the stimulated rate by the zero-point field: γvac = ρ(ω)B = 2µ2ω3

3c3 .This is half the Einstein A coefficient.Peter van der Straten (Atom Optics) Photon physics June 5, 2007 40 / 41



Total spontaneous emission rate (excited state):

γ ≡ A = γrad + γvac.

Total spontaneous emission rate (ground state):

γ = γrad − γvac ≡ 0.

Symmetric ordening of atomic and field operators:

γ = γrad + γvac.

Normal ordening of atomic and field operators:

γ = 2γrad.

Of course we always get the same expression A for the spontaneousemission rate, . . . but our interpretation of the origin of spontaneousemission may change.

P.W. Milonni, Why spontaneous emission, Am. J. Phys. 52, 340 (1984).Peter van der Straten (Atom Optics) Photon physics June 5, 2007 41 / 41

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