5 Quantum Theory of Radiation - Freie Universitätuserpage.fu-berlin.de/ejb/teaching/aqm/hw5.pdf · 5 Quantum Theory of Radiation Exercise 5.1: Transverse and longitudinal vector

5 Quantum Theory of Radiation

Exercise 5.1: Transverse and longitudinal vector fields

Consider a vector field E(r) defined on a volume of size L3 with periodic boundary conditions.

(a) Show that E(r) can be written the sum E(r) = E⊥(r) + E‖(r), where the transverseand longitudinal components satisfy

∂r · E⊥(r) = 0, ∂r × E‖(r) = 0.

(b) Show that E⊥(r) and E‖(r) are uniquely defined, up to a uniform r-independent term.

Exercise 5.2: Transverse and longitudinal vector fields (2)

(a) Show that the curl of any vector field is a transverse vector field.

(b) Show that the gradient of any scalar field is a longitudinal vector field.

(c) Show that a linear combination of two longitudinal vector fields is a longitudinal vectorfield, and show that a linear combination of two transverse vector fields is a transversevector field.

Exercise 5.3: Classical electromagnetism

(a) Show that the total momentum of particles and electromagnetic fields can be writtenas

P =∑

α

pα +P⊥, (1)

withpα = mαvα +

qαcA⊥(rα) (2)

and

P⊥ =1

4πc2

∫

dr3

∑

i=1

[

A⊥,i(r)∂rA⊥,i(r)]

. (3)

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(b) Show that the total energy of particles and electromagnetic fields can be written as

H =∑

α

1

2mα

|pα − qαA(rα)/c|2 +1

2

∑

α 6=β

qαqβ|rα − rβ|

+H⊥ (4)

with

H⊥ =1

8π

∫

dr

[

|∂rA(r)|2 + 1

c2

∣

∣

∣A(r)

∣

∣

∣

2]

. (5)

(c) Show that the total angular momentum of particles and electromagnetic fields can bewritten as

J =∑

α

r× pα + J⊥,i + J⊥,o, (6)

with

J⊥,i =1

4πc

∫

drE⊥ ×A, (7)

J⊥,o =1

4πc

∫

dr3

∑

j=1

E⊥,j(r× ∂r)A⊥,j . (8)

Exercise 5.4: Linear polarization vectors

The linear polarization vectors ǫk,1 and ǫk,2 are unit vectors, such that the three vectors ǫk,1,ǫk,2, and ek = k/k form a right-handed orthonormal set.

(a) Show that(k× ǫk,λ) · (k× ǫk,λ′) = k2δλλ′ .

(b) Show thatǫk,λ × (k× ǫk,λ′) = kδλ,λ′ .

(c) Show that∑

λ

(ǫk,λ)i(ǫk,λ)j =∑

λ

(ǫk,λ)i(ǫk,λ)j = δij −kikjk2

.

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Exercise 5.5: Quantized electric and magnetic fields in waveguide

Consider the electromagnetic field in a rectangular waveguide with cross section of sizeWy × Wz with Wy > Wz. The walls of the waveguide are perfectly conducting. In thedirection along the waveguide, we apply periodic boundary conditions with period L.

The Maxwell equations allow solutions for the fields inside the waveguide only at discretevalues of the transverse wavenumbers ky and kz. The lowest-frequency modes appear atky = π/Wy and kz = 0 and have vector potential of the form Φk(r) cos(ωkt + θ), whereθ is an arbitrary phase shift, the longitudinal wavenumber k takes the values 2πn/L withn = 0,±1,±2, . . ., the “mode function”

Φk(r) =

√

4

WyLsin

πy

Wy

ez ×

√

12

if k = 0,

cos(kx) if k > 0,sin(kx) if k < 0,

and

ωk = c√

k2 + (π/Wy)2.

In this exercise, no other modes than these lowest-frequency modes need to be considered.In general, the vector potential A(r, t) will be a superposition of the mode function given

above,

A(r, t) =∑

k

Ak(t)Φk(r), (9)

where the coefficients Ak are real numbers. If one defines the “canonical position” Qk and“canonical momentum” Pk as

Qk =1√4πc2

Ak, Pk =1√4πc2

Ak,

the Hamilton function for the lowest-modes fields in the waveguide reads

H =1

2

∑

k

(P 2k + ω2

kQ2k).

(a) Verify that this Hamilton function reproduces the correct equation of motion for thecanonical positions Qk and the canonical momenta Pk.Hint: use the explicit time dependence of the modes of the vector potential given in thetext above.

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(b) In the method of canonical quantization, the canonical positions Qk and the canonicalmomenta Pk are replaced by operators Qk and Pk. What are the fundamental algebraicrelations between these operators?

(c) Instead of using operators Qk and Pk, it is more convenient to use “creation operators”and “annihilation operators” defined as

ak =1√2~ωk

(ωkQk + iPk), a†k =1√2~ωk

(ωkQk − iPk).

What are the fundamental algebraic relations of these operators?

(d) Derive an expression for the vector potential A(r) in terms of the operators a†k and ak.

(e) The vector potential A(r) can be written in the form

A(r) = sinπy

Wy

ez∑

k

√

4πc2

~ωkWyL(ˆake

ikx + ˆa†

ke−ikx), (10)

where the operators ˆak and ˆa†

k satisfy the fundamental algebraic relations of annihila-tion and creation operators,

[ˆak′ , ˆak] = 0, [ˆa†

k′ , ˆa†

k] = 0, [ˆak′ , ˆa†

k] = δk′k.

Show that such an identity follows from the quantum theory you constructed in parts(b)–(d) and derive an expression for the field energy in this formulation.

(f) Bonus question.

Instead of first expanding the vector potential A(r) in terms of real-valued basis func-tions Φk, as was done in Eq. (9), and then transforming to a basis of complex expo-nentials on the level of the quantum theory, as was done in part (e), one may directlyformulate the theory in terms of a Fourier transform of the vector potential A(r),

A(r, t) =

√

2

WyLsin

πy

Wy

∑

k

Ak(t)eikx. (11)

This expansion is the classical analogue of the expansion (10) of the quantum vectorpotential A(r) you constructed in part (e). Express the classical canonical positions

and momenta Qk and Pk that correspond to the operators ˆak and ˆa†

k of Eq. (10) interms of the amplitudes Ak appearing in the Fourier transform (11) of the classicalvector potential A(r).

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Exercise 5.6: Quantum Electrodynamics in a cavity

In this exercise you develop a quantum theory of electromagnetic fields in a cavity withperfectly conducting walls. (In optical language, one would refer to the cavity walls as“mirrors”.)

Starting point is the expression for the combined energy of particles and fields,

H = Hmat +H‖ +H⊥, (12)

with

Hmat =∑

α

1

2mα

∣

∣

∣pα − qα

cA(rα)

∣

∣

∣

2

, (13)

H‖ =1

2

∑

α 6=β

qαqβ|rα − rβ|

, (14)

H⊥ =1

8π

∫

dr

[

|∂r ×A(r)|2 + 1

c2

∣

∣

∣A(r)

∣

∣

∣

2]

. (15)

In order to arrive at a quantum theory for the electromagnetic fields in the cavity, wefirst have to introduce “canonical positions” and “canonical momenta” for the fields in thecavity. Hereto, the transverse vector potential is decomposed as

A⊥(r) =∑

n

A⊥,nΦn(r), (16)

where Φn(r) is a real vector-valued function normalized transverse solution of

∂2rΦn(r) +ω2n

c2Φn(r) = 0, (17)

with the boundary condition that Φn(r) is perpendicular to the boundary of the cavityat the cavity boundary. The set of such vector-valued functions Φn form a complete andorthonormal basis for all transverse vector fields in the cavity,

∫

drΦn′(r) ·Φn(r) = δn′n. (18)

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(a) Argue that the boundary condition that Φn(r) is perpendicular to the cavity boundaryis consistent with the boundary conditions for the electric and magnetic fields at aperfectly conducting cavity wall.

(b) What is the equation of motion for the field component A⊥,n?

(c) If one defines the canonical position and momentum as

Qn =A⊥,n√4πc2

, Pn =A⊥,n√4πc2

, (19)

show that the Hamilton function for the transverse fields reads

H⊥ =1

2

∑

n

(

P 2n + ω2

nQ2n

)

. (20)

(d) Verify that this Hamilton function, together with Hmat and H‖, reproduces the correctequations of motion for particles and fields in the cavity.

(e) Now construct a quantum theory of particles and fields in the cavity using the methodof canonical quantization. Derive expressions for the electric and magnetic fields in thecavity in terms of appropriately defined raising and lowering operators a†n and an.

Exercise 5.7: Vacuum fluctuations

In the vacuum state, the expectation values of the transverse electric field E⊥ and themagnetic field B are zero. However, the fields have nonzero fluctuations.

(a) Express the expectation values of E2⊥ and B2 in the vacuum state as an integration over

the frequency ω of the field modes. Does your integral converge? At what frequencydoes the theory break down?

(b) Calculate the autocorrelation function of the electric field,

Gij(τ) = 〈0|E⊥,i(r, t+ τ)E⊥,j(t)(r, t)|0〉. (21)

Again, express your result as an integral over the frequency ω of the field modes.

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Exercise 5.8: Squeezed states

The uncertainty of the electric and magnetic field components of an electromagnetic waveis often characterized with the help of the quadratures. The quadratures Xk,λ and Yk,λ areobservables which represent the real and imaginary parts of the complex field amplitudesak,λ. In the quantum theory, the corresponding operators are defined as

Xk,λ =1√2(ak,λ + a†k,λ), Yk,λ =

1

i√2(ak,λ − a†k,λ). (22)

Unlike the creation and annihilation operators, the quadratures are hermitian operators, sothat we can speak about their expectation values and their fluctuations. In this exercise wewill consider a single mode only and drop the indices k and λ.

In the vacuum state, the expectation values X and Y both vanish, whereas the uncer-tainties ∆X and ∆Y are given by

∆X = ∆Y =1√2. (23)

(a) Verify Eq. (23).

The so-called “squeezed states” are states of the radiation field where the uncertainties∆X or ∆Y (but not both!) of the quadrature components is smaller than in the vacuum state.Squeezed states can be generated with light fields using methods of nonlinear optics. Becauseof their reduced uncertainty in comparison with the vacuum state, they find application(among others) in high-precision measurement schemes.

Theoretically, squeezed states can be generated from the vacuum state |0〉 by applicationof the “squeeze operator” S(ξ),

S(ξ) = e1

2(ξ∗a2−ξa†2), (24)

where ξ is a complex number.

(b) Show that S(ξ) is a unitary operator. Also show that S(ξ)† = S(−ξ).The complex number ξ is written as ξ = ρeiθ. We introduce “rotated” raising and loweringoperators as

ˆa = ae−iθ/2, ˆa†= a†eiθ/2. (25)

Similarly, rotated quadrature operators are defined as

ˆa =1√2( ˆX + i ˆY ), ˆa

†=

1√2( ˆX − i ˆY ). (26)

7

(c) Show that the rotated raising and lowering operators ˆa†and ˆa satisfy the same com-

mutation relations as the original operators a† and a.

(d) Show that the rotated quadrature operators are obtained from the original operatorsby a rotation in the X-Y plane,

ˆX = X cos(θ/2) + Y sin(θ/2), ˆY = Y cos(θ/2)− X sin(θ/2). (27)

(e) Show that

S(ξ)† ˆXS(ξ) = ˆXe−ρ, S(ξ)† ˆY S(ξ) = ˆY eρ. (28)

(f) Calculate the expectation values of X and Y and calculate their fluctuations ∆X and∆Y in the “squeezed vacuum state” S(ξ)|0〉. How do the uncertainties in X andY compare with the vacuum state? Does this state violate Heisenberg’s uncertaintyprinciple? Can you explain why these states are called “squeezed”?

(g) Describe the time dependence of the squeezed vacuum state S(ξ)|0〉.

Exercise 5.9: Squeezed number states

This exercise is a continuation of Ex. 5.8. In that exercise you have constructed a “squeezedvacuum state” in which the fluctuations of the quadrature components Xk,λ or Yk,λ are lessthan in the vacuum state. The squeezed vacuum state was obtained by acting a “squeezeoperator” S(ξ) on the vacuum state |0〉. In this exercise (as in Ex. 5.8) we will consider asingle mode only and drop the indices k and λ.

In this exercise you construct a set of “squeezed number states” that bear the samerelation to the squeezed vacuum as the standard number states do to the standard vacuum.

(a) Show that the operators b = S(ξ)aS(ξ)† and b† = S(ξ)a†S(ξ)† satisfy the commutationrelations characteristic of lowering and raising operators.

(b) Show that b annihilates the “squeezed vacuum” you constructed in Ex. 5.8.

(c) Show that the “squeezed number states”, which are obtained by repeated applicationof the “squeezed raising operator” b† on the squeezed vacuum state are the same statesas the ones that one obtains by acting the squeeze operator S(ξ) on a regular numberstate |n〉.

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(d) What is the expectation value of the number of (regular) photons in a squeezed numberstate?

Exercise 5.10: Squeezed coherent states

This exercise is a continuation of Exs. 5.8 and 5.9. In Ex. 5.8 you have constructed a“squeezed vacuum state” in which the fluctuations of the quadrature componentsXk,λ or Yk,λare less than in the vacuum state. In Ex. 5.9 you constructed “squeezed number states”. Inboth cases, the squeezed states were obtained from the regular vacuum by acting a “squeezeoperator” S(ξ). Here (as in Exs. 5.8 and 5.9) we will consider a single mode only and dropthe indices k and λ.

In this exercise you construct a set of “squeezed coherent states” that bear the samerelation to the squeezed vacuum as the standard coherent states do to the standard vacuum.

(a) A squeezed coherent state can be obtained by acting the squeeze operator S(ξ) on aregular coherent state |z〉. Find an expression for the state S(ξ)|z〉 in terms of thesqueezed number states.

(b) What is the expectation value of the photon number in the squeezed coherent stateS(ξ)|z〉?

(c) Describe the time dependence of the squeezed coherent state S(ξ)|z〉.

(d) Consider a coherent state |z〉 for which z = x is real at t = 0. In this state, thequadratures X and Y fluctuate equally around X = x and Y = 0. What are thefluctuations of X and Y for the squeezed version of this state? What is its timedependence? Depending on the choice of the phase of the squeeze parameter ξ onerefers to the squeeze as “amplitude squeezing” or “phase squeezing”. Can you explainthis??

Exercise 5.11: Electric field for a squeezed state

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(a) Evaluate the expectation value 〈E⊥(r, t)〉 of the electric field in the squeezed coherentstate S(ξ)|z〉, where the state |z〉 is a coherent state for a single field mode withwavenumber k and linear polarization.

(b) Evaluate the expectation value 〈E⊥(r, t)〉 of the electric field in the squeezed coherentstate S(ξ)|z〉, where the state |z〉 is a coherent state for a single field mode withwavenumber k and circular polarization.

(c) Repeat questions (a) and (b) for the contribution of the mode to the electric fieldvariance 〈∆E⊥(r, t)

2〉.

Exercise 5.12: Translation operator

The operator T (s) displaces a quantum state by a displacement vector s.

(a) For an observable O(r) that depends on position, show that

T (−s)O(r)T (s) = O(r− s).

(b) Show that T (s) = e−iPfield·s/~ translates the photon field by a displacement s, wherePfield =

∑

k,λ ~ka†k,λak,λ is the operator for the total momentum of the fields.

Hint: Show that the equality T (−s)O(r)T (s) = O(r − s) is satisfied for T (s) =

e−iPfield·s/~ and O(r) = A⊥(r) or O(r) = E⊥(r).

Exercise 5.13: Repulsive Casimir force

Zero point fluctuations of the electromagnetic field give an attractive force between twoparallel perfectly conducting plates. In this exercise, you consider the force from zero pointfluctuations of the electromagnetic field between two parallel plates, one of which is a perfectconductor (dielectric constant ǫ → ∞) and one of which has perfect magnetic permeability(magnetic permeability µ→ ∞).

We choose coordinates, such that the z axis is perpendicular to the two plates. Thedistance between the two plates is a. For the x and y directions you may use periodicboundary conditions, with period L≫ a.

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(a) The boundary condition at the perfectly conducting is Ex = Ey = 0. The boundarycondition at the perfectly permeable plate is Bx = By = 0. Show that the allowedmodes of the electromagnetic field between the two plates have wavenumber

kz =

(

n+1

2

)

π

a,

where n = 0, 1, 2, . . ..

(b) Derive an expression for the zero point energy E in0 (a) from the fields between the two

plates.

(c) Calculate the zero point energy Eout0 (a) from the field outside the plates.

(d) Express the a-dependent part of the total zero point energy E0(a) = E in0 (a) + Eout

0 (a)as an integral and/or a sum over the frequency ω of the radiation modes.

(e) In part (d) you should have found an expression that is formally divergent. Thedivergence can be removed by inserting a factor e−ηω in the integrand/summand, whereη is a positive infinitesimal. Find an expression for the total zero point energy as apower series in η.

(f) Now calculate the force between the plates and take the limit η → 0. What is the signof the force?

Exercise 5.14: Dipole approximation

In the dipole approximation one replaces the matter-radiation interaction by its leadingcontribution H1 and neglects the position dependence of the vector potential A(r). In thisapproximation, the radiation-matter interaction takes the form

Hint ≈ −∑

α

qαmα

pα · A⊥(R),

where α labels the particles, qα and mα are the charge and the mass of each particle, and R

is the center-of-mass coordinate. In the lecture this expression for Hint was used to calculatethe Golden-Rule spontaneous transition rate from the initial state |i〉 to the final state |f〉under emission of a photon in the solid angle element dΩ. The result is

dΓsp. em.i→f =

αc

2πe2k3|〈f|D · ǫ∗k,λ|i〉|2dΩ, (29)

11

where D =∑

α qαrα is the electric dipole moment.As an alternative, one may first perform a gauge transformation, after which the inter-

action Hamiltonian takes the form (again in the dipole approximation)

Hint ≈ −D · E⊥(R). (30)

Use Eq. (30) to calculate the Golden Rule transition rate and compare your answer to Eq.(29). Which calculation is easier?

Exercise 5.15: Magnetic dipole radiation

Interactions between matter (particles) and the radiation fields are described by the Hamil-tonian

Hint = H1 + H2 + H3,

with

H1 = −∑

α

qαcmα

pα · A⊥(rα),

H2 = −∑

α

µα · Bα(rα),

H3 =∑

α

q2α2mαc2

|A⊥(rα)|2.

The transverse vector potential A⊥(r) and the magnetic field B(r) are expressed in terms ofphoton creation and annihilation operators as

A⊥(r) =∑

k,λ

√

2π~c2

ωk,λL3

(

ak,λǫk,λeik·r + a†k,λǫ

∗k,λe

−ik·r)

,

B(r) = i∑

k,λ

√

2π~c2

ωk,λL3

(

ak,λ(k× ǫk,λ)eik·r − a†k,λ(k× ǫ

∗k,λ)e

−ik·r)

.

In the dipole approximation, one replaces the exponent e±ik·r by unity. The transitionamplitudes following from the perturbation H1 are then found to be proportional to a matrixelement of the electric dipole moment between the initial matter state |i〉 and final matterstate |f〉. If the electric dipole matrix element vanishes, one has to consider the perturbationH2 or higher orders in a small-k expansion of H1 and H2. In this exercise you analyze the

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next-to-leading order term in the small-k expansion of H1 and the leading-order radiationfrom H2.

(a) Find an expression for the matrix element 〈f, 1kλ|H2|i, 0〉 between a final state consist-ing of the matter state |f〉 and one photon in mode (k, λ) and the initial matter state|i〉 without photons.

The interaction H2 couples to the intrinsic magnetic moment of the particles only. Theintrinsic magnetic moment is the moment associated with spin angular momentum. Hence,the matrix element you calculated in (a) does not involve the total magnetic moment of theparticles. The orbital component has to come from H1, which describes the interaction ofthe charge degrees of freedom with the radiation fields.

(b) Expand the exponents e±ik·r in the expressions for A⊥ and B to first order in k andcalculate the corresponding contribution to the matrix element 〈f, 1kλ|H1|i, 0〉. Arguethat it involves both a contribution from the particles’ orbital magnetic moment anda contribution from the electric quadrupole moment.

(c) Combine the contribution from the orbital magnetic moment with your answer in (a)to find an expression for the magnetic dipole transition rate. How does this rate scalewith the frequency ω of the emitted photon?

(d) What is the frequency dependence of the electric quadrupole transition rate? Whatare the corresponding selection rules? Can one have magnetic dipole radiation andelectric quadrupole radiation at the same time?

Exercise 5.16: The |A|2 perturbation

The interaction Hamiltonian between matter and radiation consists of three contributions,H1, H2, and H3, see Ex. 5.15. When calculating the mass renormalization and the Lambshift, we neglected the contribution from H3. Why is this justified?

Exercise 5.17: Metastable Hydrogen 2s

The Hydrogen 2s state can not decay to the Hydrogen 1s ground state via single photondecay, because such decay would violate the selection rule mf + λ = mi, where λ = ±1 is

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the helicity of the photon and mf and mi are angular momentum quantum numbers of theinitial and final states. In this exercise you investigate alternative decay mechanisms.

(a) The 2s state can decay to the 2p state. The latter has a slightly lower energy becauseof the Lamb shift, and the transition is allowed by electric dipole selection rules. Findan estimate for the lifetime of the 2s state because of this decay mode. Is your lifetimerealistic?

Remark: You can answer this exercise without knowledge of the microscopic origin ofthe Lamb shift.

(b) An alternative decay channel is via two-photon decay: The 2s state decays to the 1sstate by emission of two photons of opposite helicity. Argue that such a process doesnot violate selection rules.

(c) In general, the two-photon decay rate between an initial matter state |i〉 and a finalstate |f〉 can be characterized by a Golden Rule decay rate

dΓfi = 2π|Tfi|2dρ1dρ2dω1,

where ω1 and ω2 are the frequencies of the emitted photons, ~ω1 + ~ω2 = Ei − Ef ,

dρ1 =k21

(2π)3~cdΩ1.

Find an expression for the matrix element Tfi.

(d) Now estimate the resulting lifetime of the Hydrogen 2s state. In your estimate, youmay use the dipole approximation for all matrix elements and you may restrict anysummation over intermediate states to the Hydrogen 2p state. (The latter approxima-tion is not very accurate, though!)

The analysis of two photon decay goes back to Maria Goppert-Mayer, who published herresult in 1931. The first precise calculation of the Hydrogen 2s lifetime took until 1959,see J. Shapiro and G. Breit, Phys. Rev. 113, 179 (1959). The main reason why it took solong was because of the difficult summation over intermediate states. A direct measurementof the 2s lifetime has been possible in ultracold magnetically trapped Hydrogen, see C. L.Cesar et al., Phys. Rev. Lett. 77, 225 (1996).

Exercise 5.18: Radiative transitions in Hydrogen

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In the dipole approximation, the interaction between an H atom and the radiation field isdescribed by the interaction Hamiltonian

Hint = −E · D, (31)

where D = −er is the dipole operator for the electron (charge −e) and E the electric field.

(a) Show that the spontaneous decay rate of an excited state |e〉 = |nl〉 with energy εe = εnlto the Hydrogen ground state |g〉 = |1s〉, with energy εg = ε1s, to leading order in Hint,is given by the expression

Γspe→g =

4π2

L3

∑

k

∑

λ

ωk|〈g|D|e〉 · ǫ∗kλ|2δ(εe − εg − ~ωk).

(b) Calculate the spontaneous decay rate Γsp2p→1s of the Hydrogen 2p state.

(c) At a finite temperature T , the number of photons present in a given mode is givenby the Planck law (or, equivalently, by the Bose-Einstein distribution function at zerochemical potential µ)

nkλ =1

e~ωk/kBT − 1, (32)

where kB is the Boltzmann constant.

The spontaneous decay rate Γsp of parts (a) and (b) describes limit T → 0,

Γsp2p→1s = Γ2p→1s(T = 0).

Give an expression for the finite-temperature decay rate Γ2p→1s(T ) in terms of Γsp2p→1s.

It is sufficient to write down the final answer to this item; no derivation is needed.

(d) Express the rate Γ1s→2p(T ) at which the inverse transition takes place because of ab-sorption of photons in terms of Γsp

2p→1s. It is sufficient to write down the final answerto this item; no derivation is needed.

(e) Use your answers to (c) and (d) to find a relation between the relative probabilitiesthat the H atom is found in the 1s and 2p states at temperature T .

Exercise 5.19: Radiative decay of trapped atoms

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Consider a point particle of charge e and mass m in an anisotropic three-dimensional har-monic oscillator potential. The Hamiltonian is

H =p2

2m+

1

2m(ω2x2 + ω′2y2 + ω′2z2).

The harmonic-oscillator states for the particle have energy

E(nx, ny, nz) = ~ω(nx + 1/2) + ~ω′(ny + nz + 1),

where nx,y,z = 0, 1, 2, . . .. The trap frequency in the y and z directions, ω′, is much larger thanω, the trap frequency in the x direction, so we may restrict ourselves to the case ny = nz = 0for a description of the lowest-lying excited states. In this exercise, we will use the symbol|n〉 to denote the eigenstate with nx = n, ny = nz = 0.The particle is coupled to a radiation field. In the dipole approximation, the interactionHamiltonian reads

Hint = −er · E(0),where electric field E(r) is given in terms of photon creation and annihilation operators as

E(r) = i∑

k

∑

λ=1,2

√

2π~ωk

L3ǫk,λ

(

ak,λeik·r − a†k,λe

−ik·r)

.

(a) The vector ǫk,λ is a real vector. What is its interpretation, and what are the conditionson its direction?

(b) Calculate the rate dΓsp1→0/dΩ that the excited state |1〉 spontaneously decays to the

ground state |0〉, while a photon is emitted within the solid angle element dΩ.

(c) For the spontaneous decay you considered in part (b), what is the polarization of aphoton that is emitted in the y direction?

(d) How does the decay rate you calculated in part (b) change if there is a nonzero averageoccupation n(ω) of photon modes with energy ω? You may express your answer interms of Γsp

1→0.

(e) In general, the emission or absorption of photons can cause transitions between differentharmonic oscillator states |n〉 and |n′〉. What are the selection rules for such transitions?

16

Exercise 5.20: Thomson scattering (1)

In the dipole approximation, the scattering cross section for scattering of a point particle is

dσ

dΩ

∣

∣

∣

∣

Thomson

= r20 |ǫ∗kf ,λf· ǫki,λi

|2.

This result was derived in the dipole approximation. Calculate the differential scatteringcross section without making the dipole approximation. Is the difference with the aboveexpression significant? You may specialize to the case that the particle is at rest initially.

Exercise 5.21: Thomson scattering (2)

A photon with wavevector ki scatters off a free electron, initially at rest. After scattering, thephoton has a different wavevector kf . The Hamiltonian describing the interaction betweenthe electromagnetic field and the electron reads

Hint = − e

mcp · A(r) +

e2

2mc2|A(r)|2,

where A(r) is the vector potential of the quantized electromagnetic field.

(a) Give expressions for the initial and final states of the photon and the electron. For theelectron, specify the wavefunction of the initial and final states. Use a normalizationthat is consistent with the normalization of the photon states.

(b) Explain which of the two terms in the expression for Hint contributes to the scatteringamplitude to first order in Hint (i.e., in the first-order Born approximation).

(c) Calculate the T matrix element for this scattering process in the first-order Born ap-proximation.

(d) Give an expression for the scattering cross section dσ/dΩ in terms of the T matrix.(dΩ refers to the solid angle for the direction of the scattered photon.)

(e) Calculate the scattering cross section dσ/dΩ for this scattering process, according tothe first-order Born approximation.

17

(f) Is the first-order Born approximation sufficient for the calculation of the T matrix? Ifyes, explain. If not, calculate the relevant second-order term.

Exercise 5.22: Rayleigh scattering

If the photon energy ~ω is much smaller than atomic transition energies, the photon scat-tering cross section is given by the Rayleigh formula,

dσ

dΩ

∣

∣

∣

∣

Rayleigh

= (mr0)2 ω4

×∣

∣

∣

∣

∣

∑

n 6=i

〈i|ǫ∗kf ,λf· r|n〉〈n|ǫki,λi

· r|i〉+ 〈i|ǫki,λi· r|n〉〈n|ǫ∗kf ,λf

· r|i〉εi − εn

∣

∣

∣

∣

∣

2

.

(a) Derive the Rayleigh formula from the Kramers-Heisenberg formula for the photonscattering cross section.

(b) Show that the tensor

Dij = e2∑

n 6=i

〈i|ri|n〉〈n|rj|i〉εn − εi

describes the polarizability of the atom, i.e., its electric dipole moment in response toa static electric field.

Exercise 5.23: Photo-electric effect

In the photo-electric effect, a photon is absorbed by an atom and the energy of the photonis used to release an electron initially bound to the atom. In this exercise, you considerthe photo-electric effect for a Hydrogen atom in its ground state. The photo-electric effectmay be considered as a scattering problem, where the initial state consists of a photon withwavevector k and polarization λ and an electron in the Hydrogen ground state and the finalstate consists of a free electron in a state with momentum p.

(a) Give explicit expressions for the initial and final states for this scattering problem.

18

The interaction between the photon and the electron bound to the Hydrogen atom is givenby the Hamiltonian

Hint = − e

mcp · A(r).

(b) Give an expression for the T matrix Tp,kλ for this situation. You may use the Bornapproximation (i.e., you may use first-order perturbation theory in the interactionbetween the photon and the electron). If you prefer, you may also give an expressionfor the scattering matrix Sp,kλ instead of the T matrix Tp,kλ.

(c) What is the relation between the T matrix and the differential cross section dσ/dΩ inthis case?

(d) Calculate the differential scattering cross section dσ/dΩ for the photo-electric effect.

Exercise 5.24: Two-level system in a classical field

Consider a quantum system with two discrete states: an excited state |e〉 and a ground state|g〉. The energy difference between the ground state and the excited state is ~ω0. Withoutloss of generality, you may set the energy of the ground state at −~ω0/2, and the energy ofthe excited state at ~ω0/2, so that the Hamiltonian of the two-level system reads

H0 =~ω0

2(|e〉〈e| − |g〉〈g|). (33)

The system is placed in a classical electric radiation field of frequency ω. In the dipoleapproximation, the Hamiltonian H1 describing the interaction between the radiation fieldand the system is

H1 = −D · E(t),where D is the dipole moment of the two-level system and E(t) = E0 cos(ωt) the electricfield. The matrix elements of H1 between the states |g〉 and |e〉 are then proportional tomatrix elements of the dipole operator,

〈e|H1|g〉 = 〈g|H1|e〉∗ = −〈e|D|g〉 · E0 cos(ωt).

(a) Show that H1 can be written as

H1 = ~ cos(ωt)( Ω|e〉〈g| − Ω∗|g〉〈e| ),and find an expression for the coupling constant Ω. The coupling constant Ω is called“Rabi frequency”. Verify that Ω has the dimension of frequency.

19

(b) The state |Ψ(t)〉 of the atom is a time-dependent linear combination of the states |e〉and |g〉, which can be written in the form

|Ψ(t)〉 = ae(t)e−iωt/2|e〉+ ag(t)e

iωt/2|g〉. (34)

Give explicit equations for the time derivatives ae and ag in terms of the amplitudesae and ag.

(c) Argue that the time evolution of ae and ag is slow (on the scale ω0) if the Rabi frequencyΩ and the detuning δ = ω − ω0 are small in comparision to ω0.

(d) If the conditions mentioned under (c) are fulfilled, terms in the evolution equations forae and ag that oscillate fast on the scale ω0 average to zero and may be left out. Thisapproximation is known as the “rotating wave approximation”. Solve the remainingset of equations for the time-evolution of ae and ag and interpret your answer. If thesystem starts in the ground state |g〉, what is the (time-averaged) probability to findit in the excited state |e〉?

(f) Simplify your expression for |Ψ(t)〉 in the limit that the detuning δ is much larger thanthe Rabi frequency Ω. What is the effective energy of the ground state in this case?

The shift of the ground state energy is known as the “light shift” or “AC Stark effect”.It is an often used tool in quantum optics and atomic physics to manipulate, confine,or guide atoms.

Exercise 5.25: Diffraction of an atomic beam off a standing wave

A beam of two-state atoms in their ground state passes at right angles to a standing waveof light at frequency ω, see the figure below. The standing wave is described by a modefunction Φ(r) as in Ex. 5.6. We choose our coordinates such that the atoms move in thepositive z direction and the standing beam is directed in the x direction. The intensity ofthe standing light wave is maximal near z = 0. Hence, the mode function has the form

Φ(r) ∝ ǫ cos(kx)f(z), k = ω/c,

where the polarization vector ǫ is in the xy plane and f(z) is a slowly varying function of z,peaked near z = 0.

20

x

z

standing light wave

beam of atoms

(a) The detuning δ = ω−ω0, where ~ω0 is the energy difference between the atom’s excitedstate and the ground state, is supposed to be large. Find an effective Hamiltonian forthe atom’s center-of-mass motion.

(b) Upon passing through the standing light wave, the atoms can scatter. Show that theatom’s momentum in the x direction (i.e., perpendicular to their initial propagationdirection) after passing through the standing light wave can only change by integermultiples of 2~k. Interpret this result in terms of absorption and reemission of photons.

(c) Calculate the probability P (l) that the atom’s momentum in the x direction is changedby the amount 2l~k.

Exercise 5.26: Two-slit experiment for atoms

Consider an atom beam incident on two slits in a screen. The slits are a distance d apart.The atoms are detected at a distance L≫ d behind the screen, see the left panel of the figurebelow. Interference between pathes through the two slits gives an interference pattern forthe detected intensity. The incident atom beam consists of atoms of momentum ~k movingperpendicular to the screen.

(a) The probability p(x) to detect an atom at a distance x from a position midway betweenthe two slits is proportional to |ψ(x)|2, where ψ(x) is the wavefunction of the atom ata distance L from the slits. Argue that

p(x) ∝ cos2kxd

2L.

21

x

L L

d

(b) An optical cavity is placed in front of one of the slits, see the right panel of the figure.The cavity contains a field mode at a frequency near an atomic transition, but detunedsufficiently far that no transitions are induced. Argue that the presence of the cavityinduces an additional phase change for atoms that travel through the cavity that isproportional to the number of photons n in the field mode,

δφ = nχ.

(You do not need to find an explicit expression for the proportionality constant χ.)

(c) In the presence of the cavity, the probability p(x) to detect an atom at position x ischanged. Show how the presence of the cavity modifies the x dependence of p(x) withrespect to the probability density given in part (a). Your expression may include anx-independent proportionality constant that you do not need to calculate.

(d) When the state of the field in the cavity before the passage of an atom is given by thelinear combination

|Φ(0)〉 =∑

n

c(0)n |n〉,

where |n〉 is a state with n photons in the field mode, the atom and the field are“entangled” after passage of the atom through the cavity: Its “state” at a distance Lfrom the screen can be written as

|Ψ(0)〉 =∑

n

c(0)n

∫

dxψn(x)|n, x〉,

where ψn(x) is the atomic wavefunction for the case that there are n photons in thecavity. Measuring the atom at position x1 collapses the atom part of the state to |x1〉.What is the state |Φ(1)〉 of the radiation field in the cavity after an atom that passed

22

through the cavity has been detected at a position x1? You may express your answerin terms of the wavefunctions ψn(x).Hint: Recall that a measurement leads to a collaps of the wavefunction.

(e) If initially the radiation field in the cavity was in a state with a well-defined number ofphotons, will the cavity still be in a state with a well-defined number of photons afterone such a measurement? Or will the photon number become uncertain?

(f) If initially the radiation field in the cavity was in a state with an uncertain number ofphotons, will a measurement of the position of the atom change that state? And willit change the uncertainty?

A detailed analysis shows that, after repeated passage of atoms through the cavity and ameasurement of the position of the atom, only a single value of the photon number n will besingled out: The field in the cavity is prepared in a pure number state |n〉. What numberthat is is known from the sequence of atom positions x1, x2, x3, . . . . Such a measurement iscalled a “quantum non-demolition measurement”, because such a measurement preserves thestate of the radiation field after measurement. (It does not measure the number of photonsby absorbing them, as is the standard method to detect photons.) Such a manipulationof the radiation field has been performed experimentally in S. Haroche, M. Brune, and J.M. Raimond, Appl. Phys. B 54, 355 (1992) and M. Brune, S. Haroche, J. M. Raimond, L.Davidovich, and N Zagury, Phys. Rev. A 45, 5193 (1992).

Exercise 5.27: Dicke Superradiance

Consider one atom in a (large) cavity of linear dimension L. The atom is modeled as a twolevel system with Hamiltonian

Ha =~ω0

2(|e〉〈e| − |g〉〈g|) (35)

where |e〉 and |g〉 refer to the excited and ground states of the atom, respectively. The modesof the radiation field in the cavity are labeled with the number n, and the Hamiltonians Hr

and Har describing the radiation field and the coupling between the radiation field and theatom can be taken to be

Hr =∑

n

~ωna†nan, Har =

λ

L3/2

∑

n

(|e〉an〈g|+ |g〉a†n〈e|), (36)

23

respectively, where for simplicity the coupling parameter λ (which is proportional to thedipole matrix element between the states |e〉 and |g〉) is taken to be independent of n.Initially, no photons are present in the cavity and the atom is prepared in the excited state|e〉.

(a) Calculate the (initial) decay rate Γ of the excited state. Express your answer in termsof the coupling parameter λ and the density of states ν(ω) for the radiation modes,

ν(ω) =1

L3

∑

n

δ(ω − ωn).

(b) Use your answer to (a) to find the probability Pe(t) that the atom is in the excitedstate as a function of time.

Now consider two atoms in the same cavity. The atoms are modeled as two level systemswith Hamiltonian

Ha =2

∑

i=1

Ha,i, Ha,i =~ω0

2(|e, i〉〈e, i| − |g, i〉〈g, i|), i = 1, 2, (37)

where |e, i〉 and |g, i〉, i = 1, 2, refer to the excited and ground states of the two atoms,respectively. The separation between the two atoms is assumed to be much less than λ0 =2πc/ω0. The atoms are coupled to the electromagnetic field via the Hamiltonian

Har =2

∑

i=1

Har,i, Har,i =λ

L3/2

∑

n

(|e, i〉an〈g, i|+ |g, i〉a†n〈e, i|) (38)

(c) Explain why the same coupling constant λ can be used for both atoms i = 1, 2. Wouldthat be allowed if their separation is larger than λ0?

(d) If initially only atom no. 1 is in the excited state and there are no photons presentin the cavity, what are the probabilities Pe,1(t) and Pe,2(t) to find atom 1 and atom2 in the excited state (irrespective of the state of the other atom), respectively, as afunction of time?

(e) Answer the same question for the case that all atoms are in the excited state initially.

(f) Generalize your answers to (d) and (e) for the case that there are N atoms, all withina distance λ0 from each other.

24

The effect you find was first discovered by R. H. Dicke, see Phys. Rev. 93, 99 (1954).

Exercise 5.28: Excitations on the surface of liquid Helium

The dispersion relation for waves propagating on the surface of a liquid is

ωq = q3/2

√

A

ρ0, with q = |q|,

where ωq and q = (qx, qy) are, respectively, the frequency and the wave vector of the surfacewave and A and ρ0 are constants that represent the surface tension and the mass density ofthe liquid. A quantum-mechanical theory of surface waves is formulated in terms of bosoniccreation and annihilation operators a†q and aq. These operators describe the creation andannihilation of a “ripplon”, a “quantum of surface oscillation”. We apply periodic boundaryconditions with periods Lx and Ly in the directions parallel to the surface. The amplitudeh(r) of the liquid surface oscillation at point r = (x, y) can then be expressed in terms ofthe operators aq and a†q, and reads

h(r) =

qmax∑

q

√

~q

2ρ0ωqLxLy

(

a†qe−iq·r + aqe

iq·r)

.

The high-momentum cut-off qmax accounts for the fact that for larger values of q, hencesmaller wave lengths, the fluid-dynamical description of the surface ceases to hold. Theenergy of the ripplon excitations is

HS =

qmax∑

q

~ωqa†qaq.

(a) At zero temperature, no ripplon excitations are present. Evaluate the root mean squareamplitude ∆h = (h2)1/2 of the surface at zero temperature.

Electrons can be trapped electrostatically just above the surface of liquid helium. A trappedelectron can move freely along the surface, while in the direction normal to the interface itsmotion is restricted to bound states. A simple model for the Hamiltonian of an electron atcoordinate x = (r, z) is

H =p2

2m+ V (r, z), with V (r, z) =

mω2

2

[

(

z − h(r))2

− h2(r)

]

.

where p is the electron momentum, m its mass, and z the coordinate normal to the surface.

25

(b) Show that the Hamiltonian H may be rewritten as

H =p2

2m+

1

2mω2z2 + Ver(r, z)

where the “electron-ripplon interaction” Ver(r, z) reads

Ver(r, z) = −mω2z∑

q

√

~q

2ρ0ωqLxLy

(

a†qe−iq·r + aqe

iq·r)

. (39)

(c) In the absence of the electron-ripplon interaction Ver(r, z), the electron eigenstates are

ψn,k(r, z) = 〈r, z|n,k〉 = 1√

LxLy

ψn(z)eik·r,

where ψn(z) is an eigenfunction of the one-dimensional Harmonic oscillator and k =(kx, ky) is a two-dimensional wave vector parallel to the surface. What are the corre-sponding energies En,k?

Electrons trapped near the surface can scatter off surface excitations. According to theelectron-ripplon interaction (39), such scattering involves the creation or annihilation of aripplon excitation. At zero temperature, no ripplons are present in the initial state, so thatripplon creation is the only possibility.

(d) Consider the elementary scattering process

|n,k〉 → |n′,k′〉|q〉 (40)

in which a ripplon with wavevector q is created. Give the selection rules and/orconservation laws for the quantum numbers n and n′ as well as for the momenta ~k,~k′, and ~q for this scattering process.

(e) Give the zero-temperature Golden-Rule decay rate Γ1→0(0) for an electron state |n,k〉with n = 1 and k = 0, due to spontaneous emission of a ripplon. You may assumethat the energy of the emitted ripplon ~ωq ≫ ~

2q2/2m.

(f) Express the finite-temperature decay rate Γ1→0(T ) for the same decay process in termsof the zero-temperature rate you calculated in (e).

26