PHYS5002 Quantum and Atom Optics 2022
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[ PHYS5002 ]
Quantum and Atom Optics
2022
1 (a) In the context of atom cooling and trapping, the scattering force is
Γ I/Is
2 1 + I/Is + (u )2
of this expression, and describe their relevance with respect to the scattering
process. In your answer, highlight under which conditions the scattering force
becomes maximal. What could be experimental limitations? [4]
(b) For small velocities, the scattering force can be expressed as a Taylor
expansion
dFscatt
Discuss
i. the action of a laser beam on an atom with a velocity perpendicular to the laser direction,
ii. the action of two counterpropagating laser beams onto an atom.
Note: One can evaluate Fscatt|k.u=0 = 2方 k .
(c) The Wigner function maps a density operator to phase space and retains all the information of the density operator. Other constructs of this form exist, for example the Q-faηctioη is defined as:
Q(α) = 〈α| |α〉,
with d2 αP (α) = 1.
Show that the Q-function obeys the following inequality:
0 < Q(α) <
and that
tr = d2 α Q(α),
3/7
where tr denotes the trace of an operator. [4]
Hint: For a continuous, complex variable α the trace of an operator in two dimensions is given by
tr = d2 α〈α| |α〉.
(d) Many effects in quantum optics involve the absorption of a single photon
from a quantum field, which is represented by the annihilation operator . If
Calculate the mean number of photons〈n\〉if |ψ〉is a number state |n〉
and coherent state |α〉and explain why the removal of a photon as an isolated process is not a quantum observable. [4]
2 (a) An atom, originally in state |1〉and moving with velocity v = ve塞 to the
left, is exposed to two laser beams as shown in the figure below. The potential energy of state |2〉is 方ω12 larger than that of |1〉.
of having a finite detuning ∆? [2]
ii. Qualitatively, what can you assume about the state of the atom after the
interaction has taken place? [2]
(b) Set up the equations for energy and momentum conservation for this pro- cess, explaining the different terms. From the conservation laws, derive the expression for the change of the atom’s velocity ∆v = v\ _ v . Furthermore, show that the transition is sensitive to the velocity of the atom, i.e. the re- quired detuning 6 depends on the initial velocity of the atom according to
方6 ≈ v方(k1 + k2 ).
[5]
(c) What does the velocity sensitivity mean for thermal atoms in a hot vapour? How would the process change, if both laser beams are co-propagating? For both cases, discuss qualitatively which fraction of the Doppler broadened pop- ulation will particpate in the interaction. [3]
(d) Degenerate parametric down- conversion is a nonlinear optical process in which a laser field with frequency ω3 , called the pump, incident on a crystal results in the generation of two light fields of frequency ω such that ω3 = ω + ω = 2ω .
In a simplified description, taking only a single incident mode into account, the interaction Hamiltonian for this process can be written as
I = _ i方C ╱3十十 _ 3(十)、,
Q 2 continued
that is the annihilation (creation) of a pump photon is accompanied by the creation (annihilation) of two photons of frequency ω . Here, the constant C ∈ R is proportional to the real nonlinear susceptibility.
Argue, under what conditions it is reasonable to ignore the annihilation or creation of photons in the pump field and treat the pump field as a classical light field with constant, real amplitude E ∈ R and frequency ω3 and explain why the interaction Hamiltonian with this assumption can be written as
I = _ i方Ω ╱十十e_i2wt _ ei2wt、,
= 方ω ╱十 + ← + I .
(e) Show that the commutator [ , 十十] = 2十 .
Using this result or otherwise, show that the Heisenberg equation of mo-
tion for (t) is given by
d(d)t = i方(1) ╱ , ] = _iω _ Ω十e_2iwt .
Q 2 continued
(f) Show that
(t) = ╱(0) cosh Ωt _ 十(0) sinh Ωt] e_iwt
Hint: You can use the following identities:
2022-07-21