Download Quantum Theory of Light, PY4T02 Problem Set 2 Paul Eastham

Quantum Theory of Light, PY4T02
Problem Set 2
Paul Eastham
1. (a) Consider the electromagnetic cavity discussed in lectures. Show, from Maxwell’s equations, that
for a time-dependent electric field
s
X 2ω 2 m
n
sin(kn z)qn (t)
Ex =
V
0
n
the corresponding magnetic field is
By =
X µ
n
s
k
2ωn2 m
cos(kn z)q̇n (t).
V 0
(b) Show that inserting this decomposition into the electromagnetic field energy
Z
1
1 2
(d3 r)0 Ex2 +
H=
B
2
2µ0 y
gives
H=
X mω 2
n
2
n 2
qn
+
p2n
.
2m
(c) (Optional) Confirm directly that Maxwell’s equations reduce to the equations for a set of harmonic
oscillators.
2. (a) Consider the single-mode electric field operator
Ex = E0 sin(kz)(a + a† ).
Calculate the expectation value and variance of the electric field when this mode is in a number
state |ni.
(b) What is the characteristic scale of the electric field fluctuations for a mode of volume 1µm3 at a
wavelength of 500 nm?
(c) Compare this characteristic field strength to the Coulomb field on the electron in atomic hydrogen.
Hence argue that one expects quantum electrodynamics to introduce corrections of order 1GHz to
the frequencies of atomic transitions. (The best known such correction is the splitting between the
otherwise degenerate degenerate 2 S1/2 and 2 P1/2 levels – the Lamb shift.)
3. Consider the coherent state satisfying
â|λi = λ|λi.
By using commutation relations to rewrite in normal-order, or otherwise, derive expressions for the first
three “moments” of the photon number:
hn̂p i = hλ|(↠â)p |λi,
for p = 1, 2, 3. Show that this gives the result for the variance, hn̂2 i − hn̂i2 , quoted in lectures.
4. (a) Consider a single mode of the electromagentic field, and the associated quadrature operators
1
(â + ↠)
2
1
X̂2 = (â − ↠),
2i
X̂1 =
(5)
where for the ladder operators [â, ↠] = 1. Show that the uncertainties in X1 and X2 satisfy
∆X1 ∆X2 ≥
1
.
4
(You may quote the result that if [Â, B̂] = c, with complex number c, ∆A∆B ≥ |c|/2. ∆A2 =
hÂ2 i − hÂi2 )
(b) Calculate the expectation values and quadrature uncertainties for the state
(5)
|ψi = cos(θ)|0i + sin(θ)|1i,
where θ is a real parameter.
(c) Are there any states of this form for which a quadrature uncertainty is less than in the vacuum
state |0i? If so, show that the state does not violate the uncertainty relation.
5. (a) Consider the coherent state defined by
(5)
(5)
â|λi = λ|λi,
with complex parameter λ, and the quadrature operators
1
(â + ↠)
2
1
X̂2 = (â − ↠).
2i
X̂1 =
Show that the quadrature uncertainties are
2
hX̂12
i − hX̂12 i2 =
1
,
4
and give expressions for the expected values of the quadrature measurements.
(b) Sketch these results on a phase-space portrait. Indicate the size of the uncertainty circle, and the
position of the circle in terms of the mean photon number n̄ in the state.
(5)
(c) When a coherent state propagates in time, the evolution of the quadrature operators corresponds
to a rotation by angle φ = ωt, corresponding to the phase of the associated classical wave. Thus
this phase may be identified with the polar angle in the phase space portrait. By considering your
sketch above when n̄ 1, obtain an expression for the uncertainty ∆φ.
(5)
(d) For the Michelson interferometer shown below, the output intensity classically reflects the phase
difference between the two arms of the interferometer,
(5)
I = I0 cos2 (φ),
where φ = k∆l with k = 2π/λ the wavevector. Use your result above to obtain an order-ofmagnitude expression for the maximum possible sensitivity of this instrument to differences in
distance, when the input source is a laser, in terms of the average number of photons involved in
the experiment, n̄.
(e) Assuming that λ = 514nm, the laser power is 1mW, and the size of the interferometer arms is
approximately 1 m, estimate the maximum sensitivity achievable.
Page 2
(5)
Mirrors
Laser
Detector
Page 3