Download 5 Quantum Theory of Radiation

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) + Ek (r), where the transverse
and longitudinal components satisfy
∂r · E⊥ (r) = 0, ∂r × Ek (r) = 0.
(b) Show that E⊥ (r) and Ek (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 vector
field, and show that a linear combination of two transverse vector fields is a transverse
vector field.
Exercise 5.3: Classical electromagnetism
(a) Show that the total momentum of particles and electromagnetic fields can be written
as
X
P=
pα + P⊥ ,
(1)
α
with
pα = mα vα +
and
1
P⊥ =
4πc2
Z
dr
3 h
X
i=1
1
qα
A⊥ (rα )
c
i
A⊥,i (r)∂r Ȧ⊥,i (r) .
(2)
(3)
(b) Show that the total energy of particles and electromagnetic fields can be written as
H=
X 1
1 X qα qβ
+ H⊥
|pα − qα A(rα )/c|2 +
2mα
2 α6=β |rα − rβ |
α
(4)
with
H⊥
1
=
8π
Z
2 1 dr |∂r A(r)| + 2 Ȧ(r) .
c
2
(5)
(c) Show that the total angular momentum of particles and electromagnetic fields can be
written as
X
J=
r × pα + J⊥,i + J⊥,o ,
(6)
α
with
J⊥,i
1
=
4πc
J⊥,o
1
=
4πc
Z
Z
dr E⊥ × A,
dr
3
X
j=1
E⊥,j (r × ∂r )A⊥,j .
(7)
(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′ ) = k 2 δλλ′ .
(b) Show that
ǫk,λ × (k × ǫk,λ′ ) = kδλ,λ′ .
(c) Show that
X
X
ki kj
(ǫk,λ )i (ǫk,λ )j =
(ǫk,λ )i (ǫk,λ )j = δij − 2 .
k
λ
λ
2
Exercise 5.5: Quantized electric and magnetic fields in waveguide
Consider the electromagnetic field in a rectangular waveguide with cross section of size
Wy × Wz with Wy > Wz . The walls of the waveguide are perfectly conducting. In the
direction along the waveguide, we apply periodic boundary conditions with period L.
The Maxwell equations allow solutions for the fields inside the waveguide only at discrete
values of the transverse wavenumbers ky and kz . The lowest-frequency modes appear at
ky = π/Wy and kz = 0 and have vector potential of the form Φk (r) cos(ωk t + θ), where
θ is an arbitrary phase shift, the longitudinal wavenumber k takes the values 2πn/L with
n = 0, ±1, ±2, . . ., the “mode function”
 q
s
1

if k = 0,

2
πy
4
sin
ez ×
Φk (r) =
cos(kx) if k > 0,

Wy L
Wy

sin(kx) if k < 0,
and
q
ωk = c k 2 + (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,
X
A(r, t) =
Ak (t)Φk (r),
(9)
k
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
Ȧk ,
the Hamilton function for the lowest-modes fields in the waveguide reads
1X 2
(Pk + ωk2 Q2k ).
H=
2 k
(a) Verify that this Hamilton function reproduces the correct equation of motion for the
canonical positions Qk and the canonical momenta Pk .
Hint: use the explicit time dependence of the modes of the vector potential given in the
text above.
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(b) In the method of canonical quantization, the canonical positions Qk and the canonical
momenta Pk are replaced by operators Q̂k and P̂k . What are the fundamental algebraic
relations between these operators?
(c) Instead of using operators Qk and Pk , it is more convenient to use “creation operators”
and “annihilation operators” defined as
âk = √
1
1
(ωk Q̂k + iP̂k ), â†k = √
(ωk Q̂k − iP̂k ).
2~ωk
2~ωk
What are the fundamental algebraic relations of these operators?
(d) Derive an expression for the vector potential Â(r) in terms of the operators â†k and âk .
(e) The vector potential A(r) can be written in the form
s
4πc2 ˆ ikx ˆ† −ikx
πy X
),
A(r) = sin
(ãk e + ãk e
ez
Wy
~ω
k Wy L
k
(10)
ˆk and ã
ˆ†k satisfy the fundamental algebraic relations of annihilawhere the operators ã
tion and creation operators,
ˆk′ , ã
ˆ†k ] = δk′ k .
ˆ†k ] = 0, [ã
ˆk′ , ã
ˆk ] = 0, [ã
ˆ†k′ , ã
[ã
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 functions Φk , as was done in Eq. (9), and then transforming to a basis of complex exponentials on the level of the quantum theory, as was done in part (e), one may directly
formulate the theory in terms of a Fourier transform of the vector potential A(r),
s
πy X
2
A(r, t) =
sin
Ãk (t)eikx .
(11)
Wy L
Wy k
This expansion is the classical analogue of the expansion (10) of the quantum vector
potential Â(r) you constructed in part (e). Express the classical canonical positions
ˆk and ã
ˆ†k of Eq. (10) in
and momenta Q̃k and P̃k that correspond to the operators ã
terms of the amplitudes Ãk appearing in the Fourier transform (11) of the classical
vector 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 with
perfectly 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 + Hk + H⊥ ,
(12)
with
2
X 1 qα
pα − A(rα ) ,
2mα
c
α
1 X qα qβ
,
=
2 α6=β |rα − rβ |
Z
2 1
1 2
=
dr |∂r × A(r)| + 2 Ȧ(r) .
8π
c
Hmat =
Hk
H⊥
(13)
(14)
(15)
In order to arrive at a quantum theory for the electromagnetic fields in the cavity, we
first have to introduce “canonical positions” and “canonical momenta” for the fields in the
cavity. Hereto, the transverse vector potential is decomposed as
X
A⊥ (r) =
A⊥,n Φn (r),
(16)
n
where Φn (r) is a real vector-valued function normalized transverse solution of
∂r2 Φn (r) +
ωn2
Φn (r) = 0,
c2
(17)
with the boundary condition that Φn (r) is perpendicular to the boundary of the cavity
at the cavity boundary. The set of such vector-valued functions Φn form a complete and
orthonormal basis for all transverse vector fields in the cavity,
Z
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 boundary
is consistent with the boundary conditions for the electric and magnetic fields at a
perfectly 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
A⊥,n
Ȧ⊥,n
Qn = √
, Pn = √
,
4πc2
4πc2
show that the Hamilton function for the transverse fields reads
1X 2
H⊥ =
Pn + ωn2 Q2n .
2 n
(19)
(20)
(d) Verify that this Hamilton function, together with Hmat and Hk , reproduces the correct
equations of motion for particles and fields in the cavity.
(e) Now construct a quantum theory of particles and fields in the cavity using the method
of canonical quantization. Derive expressions for the electric and magnetic fields in the
cavity in terms of appropriately defined raising and lowering operators â†n and ân .
Exercise 5.7: Vacuum fluctuations
In the vacuum state, the expectation values of the transverse electric field E⊥ and the
magnetic field B are zero. However, the fields have nonzero fluctuations.
(a) Express the expectation values of E⊥2 and B 2 in the vacuum state as an integration over
the frequency ω of the field modes. Does your integral converge? At what frequency
does the theory break down?
(b) Calculate the autocorrelation function of the electric field,
Gij (τ ) = h{0}|Ê⊥,i (r, t + τ )Ê⊥,j (t)(r, t)|{0}i.
Again, express your result as an integral over the frequency ω of the field modes.
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(21)
Exercise 5.8: Squeezed states
The uncertainty of the electric and magnetic field components of an electromagnetic wave
is often characterized with the help of the quadratures. The quadratures Xk,λ and Yk,λ are
observables which represent the real and imaginary parts of the complex field amplitudes
ak,λ . In the quantum theory, the corresponding operators are defined as
1
1
X̂k,λ = √ (âk,λ + â†k,λ ), Ŷk,λ = √ (âk,λ − â†k,λ ).
2
i 2
(22)
Unlike the creation and annihilation operators, the quadratures are hermitian operators, so
that we can speak about their expectation values and their fluctuations. In this exercise we
will 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 uncertainties ∆X and ∆Y are given by
1
∆X = ∆Y = √ .
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. Because
of 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 |0i by application
of the “squeeze operator” Ŝ(ξ),
1 ∗ 2
†2
(24)
Ŝ(ξ) = e 2 (ξ â −ξâ ) ,
where ξ is a complex number.
(b) Show that Ŝ(ξ) is a unitary operator. Also show that Ŝ(ξ)† = Ŝ(−ξ).
The complex number ξ is written as ξ = ρeiθ . We introduce “rotated” raising and lowering
operators as
ˆ = âe−iθ/2 , ã
ˆ† = ↠eiθ/2 .
ã
(25)
Similarly, rotated quadrature operators are defined as
ˆ + iỸˆ ), ã
ˆ − iỸˆ ).
ˆ† = √1 (X̃
ˆ = √1 (X̃
ã
2
2
7
(26)
ˆ† and ã
ˆ satisfy the same com(c) Show that the rotated raising and lowering operators ã
†
mutation relations as the original operators â and â.
(d) Show that the rotated quadrature operators are obtained from the original operators
by a rotation in the X-Y plane,
ˆ = X̂ cos(θ/2) + Ŷ sin(θ/2), Ỹˆ = Ŷ cos(θ/2) − X̂ sin(θ/2).
X̃
(27)
ˆ Ŝ(ξ) = X̃e
ˆ −ρ , Ŝ(ξ)† Ỹˆ Ŝ(ξ) = Ỹˆ eρ .
Ŝ(ξ)† X̃
(28)
(e) Show that
(f) Calculate the expectation values of X̃ and Ỹ and calculate their fluctuations ∆X̃ and
∆Ỹ in the “squeezed vacuum state” Ŝ(ξ)|0i. How do the uncertainties in X̃ and
Ỹ compare with the vacuum state? Does this state violate Heisenberg’s uncertainty
principle? Can you explain why these states are called “squeezed”?
(g) Describe the time dependence of the squeezed vacuum state Ŝ(ξ)|0i.
Exercise 5.9: Squeezed number states
This exercise is a continuation of Ex. 5.8. In that exercise you have constructed a “squeezed
vacuum state” in which the fluctuations of the quadrature components Xk,λ or Yk,λ are less
than in the vacuum state. The squeezed vacuum state was obtained by acting a “squeeze
operator” Ŝ(ξ) on the vacuum state |0i. In this exercise (as in Ex. 5.8) we will consider a
single mode only and drop the indices k and λ.
In this exercise you construct a set of “squeezed number states” that bear the same
relation to the squeezed vacuum as the standard number states do to the standard vacuum.
(a) Show that the operators b̂ = Ŝ(ξ)âŜ(ξ)† and b̂† = Ŝ(ξ)↠Ŝ(ξ)† satisfy the commutation
relations characteristic of lowering and raising operators.
(b) Show that b̂ annihilates the “squeezed vacuum” you constructed in Ex. 5.8.
(c) Show that the “squeezed number states”, which are obtained by repeated application
of the “squeezed raising operator” b̂† on the squeezed vacuum state are the same states
as the ones that one obtains by acting the squeeze operator S(ξ) on a regular number
state |ni.
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(d) What is the expectation value of the number of (regular) photons in a squeezed number
state?
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 components Xk,λ or Yk,λ
are less than in the vacuum state. In Ex. 5.9 you constructed “squeezed number states”. In
both cases, the squeezed states were obtained from the regular vacuum by acting a “squeeze
operator” Ŝ(ξ). Here (as in Exs. 5.8 and 5.9) we will consider a single mode only and drop
the indices k and λ.
In this exercise you construct a set of “squeezed coherent states” that bear the same
relation 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 Ŝ(ξ) on a
regular coherent state |zi. Find an expression for the state Ŝ(ξ)|zi in terms of the
squeezed number states.
(b) What is the expectation value of the photon number in the squeezed coherent state
Ŝ(ξ)|zi?
(c) Describe the time dependence of the squeezed coherent state Ŝ(ξ)|zi.
(d) Consider a coherent state |zi for which z = x is real at t = 0. In this state, the
quadratures X and Y fluctuate equally around X = x and Y = 0. What are the
fluctuations of X and Y for the squeezed version of this state? What is its time
dependence? Depending on the choice of the phase of the squeeze parameter ξ one
refers to the squeeze as “amplitude squeezing” or “phase squeezing”. Can you explain
this??
Exercise 5.11: Electric field for a squeezed state
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(a) Evaluate the expectation value hE⊥ (r, t)i of the electric field in the squeezed coherent
state Ŝ(ξ)|zi, where the state |zi is a coherent state for a single field mode with
wavenumber k and linear polarization.
(b) Evaluate the expectation value hE⊥ (r, t)i of the electric field in the squeezed coherent
state Ŝ(ξ)|zi, where the state |zi is a coherent state for a single field mode with
wavenumber k and circular polarization.
(c) Repeat questions (a) and (b) for the contribution of the mode to the electric field
variance h∆E⊥ (r, t)2 i.
Exercise 5.12: Translation operator
The operator T̂ (s) displaces a quantum state by a displacement vector s.
(a) For an observable O(r) that depends on position, show that
T̂ (−s)Ô(r)T̂ (s) = Ô(r − s).
(b) Show that
e−iP̂field ·s/~ translates the photon field by a displacement s, where
P T̂ (s) =
†
P̂field = k,λ ~kâk,λ âk,λ is the operator for the total momentum of the fields.
Hint: Show that the equality T̂ (−s)Ô(r)T̂ (s) = Ô(r − s) is satisfied for T̂ (s) =
e−iP̂field ·s/~ and Ô(r) = Â⊥ (r) or Ô(r) = Ê⊥ (r).
Exercise 5.13: Repulsive Casimir force
Zero point fluctuations of the electromagnetic field give an attractive force between two
parallel perfectly conducting plates. In this exercise, you consider the force from zero point
fluctuations of the electromagnetic field between two parallel plates, one of which is a perfect
conductor (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. The
distance between the two plates is a. For the x and y directions you may use periodic
boundary conditions, with period L ≫ a.
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(a) The boundary condition at the perfectly conducting is Ex = Ey = 0. The boundary
condition at the perfectly permeable plate is Bx = By = 0. Show that the allowed
modes of the electromagnetic field between the two plates have wavenumber
1 π
kz = n +
,
2 a
where n = 0, 1, 2, . . ..
(b) Derive an expression for the zero point energy E0in (a) from the fields between the two
plates.
(c) Calculate the zero point energy E0out (a) from the field outside the plates.
(d) Express the a-dependent part of the total zero point energy E0 (a) = E0in (a) + E0out (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. The
divergence can be removed by inserting a factor e−ηω in the integrand/summand, where
η is a positive infinitesimal. Find an expression for the total zero point energy as a
power series in η.
(f) Now calculate the force between the plates and take the limit η → 0. What is the sign
of the force?
Exercise 5.14: Dipole approximation
In the dipole approximation one replaces the matter-radiation interaction by its leading
contribution Ĥ1 and neglects the position dependence of the vector potential A(r). In this
approximation, the radiation-matter interaction takes the form
X qα
Ĥint ≈ −
p̂α · Â⊥ (R̂),
m
α
α
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 Ĥint was used to calculate
the Golden-Rule spontaneous transition rate from the initial state |ii to the final state |fi
under emission of a photon in the solid angle element dΩ. The result is
αc 3
sp. em.
dΓi→f
=
k |hf|D · ǫ∗k,λ |ii|2 dΩ,
(29)
2πe2
11
P
where D̂ = α qα r̂α is the electric dipole moment.
As an alternative, one may first perform a gauge transformation, after which the interaction Hamiltonian takes the form (again in the dipole approximation)
Ĥint ≈ −D̂ · 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 Hamiltonian
Ĥint = Ĥ1 + Ĥ2 + Ĥ3 ,
with
X qα
p̂α · Â⊥ (r̂α ),
cmα
α
X
= −
µ̂α · B̂α (r̂α ),
Ĥ1 = −
Ĥ2
α
Ĥ3 =
X
α
qα2
|Â⊥ (r̂α )|2 .
2mα c2
The transverse vector potential Â⊥ (r) and the magnetic field B̂(r) are expressed in terms of
photon creation and annihilation operators as
s
X 2π~c2 †
ik·r
∗
−ik·r
Â⊥ (r) =
â
ǫ
e
+
â
ǫ
e
,
k,λ
k,λ
k,λ k,λ
ωk,λ L3
k,λ
s
X 2π~c2 †
ik·r
∗
−ik·r
.
âk,λ (k × ǫk,λ )e − âk,λ (k × ǫk,λ )e
B̂(r) = i
ωk,λ L3
k,λ
In the dipole approximation, one replaces the exponent e±ik·r by unity. The transition
amplitudes following from the perturbation Ĥ1 are then found to be proportional to a matrix
element of the electric dipole moment between the initial matter state |ii and final matter
state |fi. If the electric dipole matrix element vanishes, one has to consider the perturbation
Ĥ2 or higher orders in a small-k expansion of Ĥ1 and Ĥ2 . In this exercise you analyze the
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next-to-leading order term in the small-k expansion of Ĥ1 and the leading-order radiation
from Ĥ2 .
(a) Find an expression for the matrix element hf, 1kλ |Ĥ2 |i, 0i between a final state consisting of the matter state |fi and one photon in mode (k, λ) and the initial matter state
|ii without photons.
The interaction Ĥ2 couples to the intrinsic magnetic moment of the particles only. The
intrinsic 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 the
particles. The orbital component has to come from H1 , which describes the interaction of
the charge degrees of freedom with the radiation fields.
(b) Expand the exponents e±ik·r in the expressions for Â⊥ and B̂ to first order in k and
calculate the corresponding contribution to the matrix element hf, 1kλ |Ĥ1 |i, 0i. Argue
that it involves both a contribution from the particles’ orbital magnetic moment and
a 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 scale
with the frequency ω of the emitted photon?
(d) What is the frequency dependence of the electric quadrupole transition rate? What
are the corresponding selection rules? Can one have magnetic dipole radiation and
electric quadrupole radiation at the same time?
Exercise 5.16: The |A|2 perturbation
The interaction Hamiltonian between matter and radiation consists of three contributions,
Ĥ1 , Ĥ2 , and Ĥ3 , see Ex. 5.15. When calculating the mass renormalization and the Lamb
shift, we neglected the contribution from Ĥ3 . 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 photon
decay, 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 the
initial 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 because
of the Lamb shift, and the transition is allowed by electric dipole selection rules. Find
an estimate for the lifetime of the 2s state because of this decay mode. Is your lifetime
realistic?
Remark: You can answer this exercise without knowledge of the microscopic origin of
the Lamb shift.
(b) An alternative decay channel is via two-photon decay: The 2s state decays to the 1s
state by emission of two photons of opposite helicity. Argue that such a process does
not violate selection rules.
(c) In general, the two-photon decay rate between an initial matter state |ii and a final
state |fi can be characterized by a Golden Rule decay rate
dΓfi = 2π|Tfi |2 dρ1 dρ2 dω1 ,
where ω1 and ω2 are the frequencies of the emitted photons, ~ω1 + ~ω2 = Ei − Ef ,
dρ1 =
k12
dΩ1 .
(2π)3 ~c
Find an expression for the matrix element Tfi .
(d) Now estimate the resulting lifetime of the Hydrogen 2s state. In your estimate, you
may use the dipole approximation for all matrix elements and you may restrict any
summation over intermediate states to the Hydrogen 2p state. (The latter approximation is not very accurate, though!)
The analysis of two photon decay goes back to Maria Göppert-Mayer, who published her
result 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 so
long was because of the difficult summation over intermediate states. A direct measurement
of 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
14
In the dipole approximation, the interaction between an H atom and the radiation field is
described by the interaction Hamiltonian
Ĥint = −Ê · D̂,
(31)
where D̂ = −er̂ is the dipole operator for the electron (charge −e) and Ê the electric field.
(a) Show that the spontaneous decay rate of an excited state |ei = |nli with energy εe = εnl
to the Hydrogen ground state |gi = |1si, with energy εg = ε1s , to leading order in Ĥint ,
is given by the expression
Γsp
e→g =
4π 2 X X
ωk |hg|D̂|ei · ǫ∗kλ |2 δ(εe − εg − ~ωk ).
L3 k λ
(b) Calculate the spontaneous decay rate Γsp
2p→1s of the Hydrogen 2p state.
(c) At a finite temperature T , the number of photons present in a given mode is given
by the Planck law (or, equivalently, by the Bose-Einstein distribution function at zero
chemical potential µ)
1
,
(32)
nkλ = ~ω /k T
e k B −1
where kB is the Boltzmann constant.
The spontaneous decay rate Γsp of parts (a) and (b) describes limit T → 0,
Γsp
2p→1s = Γ2p→1s (T = 0).
Give an expression for the finite-temperature decay rate Γ2p→1s (T ) in terms of Γsp
2p→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 absorption of photons in terms of Γsp
2p→1s . It is sufficient to write down the final answer
to this item; no derivation is needed.
(e) Use your answers to (c) and (d) to find a relation between the relative probabilities
that the H atom is found in the 1s and 2p states at temperature T .
Exercise 5.19: Radiative decay of trapped atoms
15
Consider a point particle of charge e and mass m in an anisotropic three-dimensional harmonic oscillator potential. The Hamiltonian is
Ĥ =
1
p̂2
+ m(ω 2 x2 + ω ′2 y 2 + ω ′2 z 2 ).
2m 2
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 than
ω, the trap frequency in the x direction, so we may restrict ourselves to the case ny = nz = 0
for a description of the lowest-lying excited states. In this exercise, we will use the symbol
|ni to denote the eigenstate with nx = n, ny = nz = 0.
The particle is coupled to a radiation field. In the dipole approximation, the interaction
Hamiltonian reads
Ĥint = −er̂ · Ê(0),
where electric field Ê(r) is given in terms of photon creation and annihilation operators as
r
X X 2π~ωk
†
ik·r
−ik·r
Ê(r) = i
ǫ
â
e
−
â
e
.
k,λ
k,λ
k,λ
3
L
k λ=1,2
(a) The vector ǫk,λ is a real vector. What is its interpretation, and what are the conditions
on its direction?
(b) Calculate the rate dΓsp
1→0 /dΩ that the excited state |1i spontaneously decays to the
ground state |0i, 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 a
photon that is emitted in the y direction?
(d) How does the decay rate you calculated in part (b) change if there is a nonzero average
occupation n(ω) of photon modes with energy ω? You may express your answer in
terms of Γsp
1→0 .
(e) In general, the emission or absorption of photons can cause transitions between different
harmonic oscillator states |ni and |n′ i. 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 = r02 |ǫ∗kf ,λf · ǫki ,λi |2 .
dΩ Thomson
This result was derived in the dipole approximation. Calculate the differential scattering
cross section without making the dipole approximation. Is the difference with the above
expression 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, the
photon has a different wavevector kf . The Hamiltonian describing the interaction between
the electromagnetic field and the electron reads
Ĥint = −
e2
e
p̂ · Â(r̂) +
|Â(r̂)|2 ,
mc
2mc2
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 the
electron, specify the wavefunction of the initial and final states. Use a normalization
that is consistent with the normalization of the photon states.
(b) Explain which of the two terms in the expression for Ĥint contributes to the scattering
amplitude to first order in Ĥint (i.e., in the first-order Born approximation).
(c) Calculate the T matrix element for this scattering process in the first-order Born approximation.
(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 to
the first-order Born approximation.
17
(f) Is the first-order Born approximation sufficient for the calculation of the T matrix? If
yes, 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 scattering cross section is given by the Rayleigh formula,
dσ = (mr0 )2 ω 4
dΩ Rayleigh
2
X hi|ǫ∗ · r̂|nihn|ǫ
∗
·
r̂|ii
+
hi|ǫ
·
r̂|nihn|ǫ
·
r̂|ii
ki ,λi
ki ,λi
kf ,λf
kf ,λf
×
.
εi − εn
n6=i
(a) Derive the Rayleigh formula from the Kramers-Heisenberg formula for the photon
scattering cross section.
(b) Show that the tensor
Dij = e2
X hi|r̂i |nihn|r̂j |ii
n6=i
εn − εi
describes the polarizability of the atom, i.e., its electric dipole moment in response to
a 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 photon
is used to release an electron initially bound to the atom. In this exercise, you consider
the photo-electric effect for a Hydrogen atom in its ground state. The photo-electric effect
may be considered as a scattering problem, where the initial state consists of a photon with
wavevector k and polarization λ and an electron in the Hydrogen ground state and the final
state 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 given
by the Hamiltonian
e
Ĥint = − p̂ · Â(r̂).
mc
(b) Give an expression for the T matrix Tp,kλ for this situation. You may use the Born
approximation (i.e., you may use first-order perturbation theory in the interaction
between the photon and the electron). If you prefer, you may also give an expression
for 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 in
this 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 |ei and a ground state
|gi. The energy difference between the ground state and the excited state is ~ω0 . Without
loss of generality, you may set the energy of the ground state at −~ω0 /2, and the energy of
the excited state at ~ω0 /2, so that the Hamiltonian of the two-level system reads
~ω0
(|eihe| − |gihg|).
(33)
2
The system is placed in a classical electric radiation field of frequency ω. In the dipole
approximation, the Hamiltonian H1 describing the interaction between the radiation field
and the system is
Ĥ1 = −D̂ · E(t),
Ĥ0 =
where D is the dipole moment of the two-level system and E(t) = E0 cos(ωt) the electric
field. The matrix elements of H1 between the states |gi and |ei are then proportional to
matrix elements of the dipole operator,
he|Ĥ1 |gi = hg|Ĥ1 |ei∗ = −he|D̂|gi · E0 cos(ωt).
(a) Show that Ĥ1 can be written as
Ĥ1 = ~ cos(ωt)( Ω|eihg| − Ω∗ |gihe| ),
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)i of the atom is a time-dependent linear combination of the states |ei
and |gi, which can be written in the form
|Ψ(t)i = ae (t)e−iωt/2 |ei + ag (t)eiωt/2 |gi.
(34)
Give explicit equations for the time derivatives ȧe and ȧg in terms of the amplitudes
ae and ag .
(c) Argue that the time evolution of ae and ag is slow (on the scale ω0 ) if the Rabi frequency
Ω 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 for
ae and ag that oscillate fast on the scale ω0 average to zero and may be left out. This
approximation is known as the “rotating wave approximation”. Solve the remaining
set of equations for the time-evolution of ae and ag and interpret your answer. If the
system starts in the ground state |gi, what is the (time-averaged) probability to find
it in the excited state |ei?
(f) Simplify your expression for |Ψ(t)i in the limit that the detuning δ is much larger than
the 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 wave
of light at frequency ω, see the figure below. The standing wave is described by a mode
function Φ(r) as in Ex. 5.6. We choose our coordinates such that the atoms move in the
positive z direction and the standing beam is directed in the x direction. The intensity of
the 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
standing light wave
beam of atoms
x
z
(a) The detuning δ = ω −ω0 , where ~ω0 is the energy difference between the atom’s excited
state and the ground state, is supposed to be large. Find an effective Hamiltonian for
the atom’s center-of-mass motion.
(b) Upon passing through the standing light wave, the atoms can scatter. Show that the
atom’s momentum in the x direction (i.e., perpendicular to their initial propagation
direction) after passing through the standing light wave can only change by integer
multiples 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 changed
by 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 figure
below. Interference between pathes through the two slits gives an interference pattern for
the detected intensity. The incident atom beam consists of atoms of momentum ~k moving
perpendicular to the screen.
(a) The probability p(x) to detect an atom at a distance x from a position midway between
the two slits is proportional to |ψ(x)|2 , where ψ(x) is the wavefunction of the atom at
a distance L from the slits. Argue that
p(x) ∝ cos2
21
kxd
.
2L
d
111
000
000
111
000
111
000
111
000
111
x
L
L
(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 detuned
sufficiently far that no transitions are induced. Argue that the presence of the cavity
induces an additional phase change for atoms that travel through the cavity that is
proportional 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 is
changed. Show how the presence of the cavity modifies the x dependence of p(x) with
respect to the probability density given in part (a). Your expression may include an
x-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 the
linear combination
X
|Φ(0) i =
c(0)
n |ni,
n
where |ni 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 L
from the screen can be written as
Z
X
(0)
(0)
|Ψ i =
cn
dxψn (x)|n, xi,
n
where ψn (x) is the atomic wavefunction for the case that there are n photons in the
cavity. Measuring the atom at position x1 collapses the atom part of the state to |x1 i.
What is the state |Φ(1) i 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 answer
in 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 of
photons, will the cavity still be in a state with a well-defined number of photons after
one 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 of
photons, will a measurement of the position of the atom change that state? And will
it change the uncertainty?
A detailed analysis shows that, after repeated passage of atoms through the cavity and a
measurement of the position of the atom, only a single value of the photon number n will be
singled out: The field in the cavity is prepared in a pure number state |ni. What number
that is is known from the sequence of atom positions x1 , x2 , x3 , . . . . Such a measurement is
called a “quantum non-demolition measurement”, because such a measurement preserves the
state of the radiation field after measurement. (It does not measure the number of photons
by absorbing them, as is the standard method to detect photons.) Such a manipulation
of 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 two
level system with Hamiltonian
Ĥa =
~ω0
(|eihe| − |gihg|)
2
(35)
where |ei and |gi refer to the excited and ground states of the atom, respectively. The modes
of the radiation field in the cavity are labeled with the number n, and the Hamiltonians Ĥr
and Ĥar describing the radiation field and the coupling between the radiation field and the
atom can be taken to be
Ĥr =
X
n
~ωn â†n ân , Ĥar =
λ X
(|eiân hg| + |giâ†n he|),
L3/2 n
23
(36)
respectively, where for simplicity the coupling parameter λ (which is proportional to the
dipole matrix element between the states |ei and |gi) is taken to be independent of n.
Initially, no photons are present in the cavity and the atom is prepared in the excited state
|ei.
(a) Calculate the (initial) decay rate Γ of the excited state. Express your answer in terms
of the coupling parameter λ and the density of states ν(ω) for the radiation modes,
ν(ω) =
1 X
δ(ω − ωn ).
L3 n
(b) Use your answer to (a) to find the probability Pe (t) that the atom is in the excited
state as a function of time.
Now consider two atoms in the same cavity. The atoms are modeled as two level systems
with Hamiltonian
Ĥa =
2
X
Ĥa,i , Ĥa,i =
i=1
~ω0
(|e, iihe, i| − |g, iihg, i|), i = 1, 2,
2
(37)
where |e, ii and |g, ii, 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
Ĥar =
2
X
i=1
Ĥar,i , Ĥar,i =
λ X
(|e, iiân hg, i| + |g, iiâ†n he, i|)
3/2
L
n
(38)
(c) Explain why the same coupling constant λ can be used for both atoms i = 1, 2. Would
that 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 present
in the cavity, what are the probabilities Pe,1 (t) and Pe,2 (t) to find atom 1 and atom
2 in the excited state (irrespective of the state of the other atom), respectively, as a
function 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 within
a 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
s
A
, with q = |q|,
ωq = q 3/2
ρ0
where ωq and q = (qx , qy ) are, respectively, the frequency and the wave vector of the surface
wave and A and ρ0 are constants that represent the surface tension and the mass density of
the liquid. A quantum-mechanical theory of surface waves is formulated in terms of bosonic
creation and annihilation operators â†q and âq . These operators describe the creation and
annihilation of a “ripplon”, a “quantum of surface oscillation”. We apply periodic boundary
conditions with periods Lx and Ly in the directions parallel to the surface. The amplitude
h(r) of the liquid surface oscillation at point r = (x, y) can then be expressed in terms of
the operators aq and a†q , and reads
s
qmax
X
~q
ĥ(r) =
â†q e−iq·r + âq eiq·r .
2ρ0 ωq Lx Ly
q
The high-momentum cut-off qmax accounts for the fact that for larger values of q, hence
smaller wave lengths, the fluid-dynamical description of the surface ceases to hold. The
energy of the ripplon excitations is
qmax
ĤS =
X
~ωq â†q âq .
q
(a) At zero temperature, no ripplon excitations are present. Evaluate the root mean square
amplitude ∆h = (h2 )1/2 of the surface at zero temperature.
Electrons can be trapped electrostatically just above the surface of liquid helium. A trapped
electron can move freely along the surface, while in the direction normal to the interface its
motion is restricted to bound states. A simple model for the Hamiltonian of an electron at
coordinate x = (r, z) is
2
mω 2 p̂2
2
z − ĥ(r) − ĥ (r) .
+ V̂ (r̂, ẑ), with V̂ (r, z) =
Ĥ =
2m
2
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
Ĥ =
1
p̂2
+ mω 2 ẑ 2 + V̂er (r̂, ẑ)
2m 2
where the “electron-ripplon interaction” V̂er (r, z) reads
s
X
~q
â†q e−iq·r + âq eiq·r .
V̂er (r, z) = −mω 2 z
2ρ0 ωq Lx Ly
q
(39)
(c) In the absence of the electron-ripplon interaction Ver (r, z), the electron eigenstates are
ψn,k (r, z) = hr, z|n, ki = p
1
ψn (z)eik·r ,
Lx Ly
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 corresponding energies En,k ?
Electrons trapped near the surface can scatter off surface excitations. According to the
electron-ripplon interaction (39), such scattering involves the creation or annihilation of a
ripplon excitation. At zero temperature, no ripplons are present in the initial state, so that
ripplon creation is the only possibility.
(d) Consider the elementary scattering process
|n, ki → |n′ , k′ i|qi
(40)
in which a ripplon with wavevector q is created. Give the selection rules and/or
conservation 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, ki
with n = 1 and k = 0, due to spontaneous emission of a ripplon. You may assume
that the energy of the emitted ripplon ~ωq ≫ ~2 q 2 /2m.
(f) Express the finite-temperature decay rate Γ1→0 (T ) for the same decay process in terms
of the zero-temperature rate you calculated in (e).
26