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Topic: Electromagnetically
Induced Transparency(EIT)
Problem statement: An R-L-C network is
connected in parallel to another by means
of a coupling capacitor.
(a) Determine the mechanical analog of the
circuit
(b) Compare the mechanical model to the
electrical model, in terms of analogous
parameters
(c) Calculate the power spectrum and
absorption spectrum of the mechanical
system

Solutions
(a) The following is the circuit used in
the investigation of
Electromagnetically Induced
Transparency (EIT). It shows the
diagram of an R-L-C network
connected in parallel to another R-L-
C network by means of a coupling
capacitor.

Solutions(contd.)

Solutions(contd.)
The following is the mechanical
analog of the given system. The atom
is modeled as a simple harmonic
oscillator, consisting of a particle 1
with mass 𝑚1 attached to two
springs with spring constants𝑘1,𝑘2
and K respectively.

Solutions(contd.)
The spring with constant k1 is attached to a wall, while the other spring of spring
constant K is attached to a second particle of mass 𝑚2 and initially kept immobile at
a fixed position. Particle 1 is also subject to a harmonic force, given by
ℱ𝑠 = 𝐹𝑒−𝑖(𝜔 𝑠 𝑡+ 𝜙 𝑠)

Solutions(contd.)
(b) An analysis of the power transferred
from the harmonic source to particle 1
as a function of frequency 𝜈 is
performed. It is observed that the
standard resonance absorption profile
discussed above peaked at frequency
𝜔1
2
=
𝑘1 + 𝐾
𝑚

Solutions(contd.)
• If we now allow particle 2 to move, subject only to
the forces from the spring of constant K and a
third spring of constant 𝑘2 attached to a wall, the
absorption profile is modified.
• It is observed that the modified features of the
new absorption profile are similar to
electromagnetically induced transparency evolving
to an Autler–Townes-like doublet as a function of
K.
• Further analysis reveals that this doublet is the
normal-mode splitting.
• For simplicity, we have considered the case 𝑘1 =
𝑘2 = 𝐾 and 𝑚1 = 𝑚2 = 𝑚

Solutions(contd.)
• The physical analogy between the given
electrical circuit and the proposed
mechanical equivalent can be illustrated
as follows.
• To provide a quantitative description of
the system, we write the equations of
motion of particles 1 and 2 in terms of
the displacements x1 and x2 from their
respective equilibrium positions:

Solutions(contd.)
The following assumptions have been made without loss of generality:
• We have set 𝜙𝑠 = 0 for the probe force.
• We also let the frequency of coupling, Ω 𝑟
2
= 𝐾/𝑚.

Solutions(contd.)
Here, the frequency is associated with the coherent
coupling between the pumping oscillator and the
oscillator modeling the atom; 𝛾1 is the friction
constant associated with the energy dissipation acting
on particle 1(which simulates the spontaneous
emission from the atomic excited state); and 𝛾2 is the
energy dissipation rate of the pumping transition. We
now proceed to seek a solution for 𝑥1 𝑡 . Let the
solution of 𝑥1 𝑡 be of the form

Solutions(contd.)
We also assume a similar form of the
solution for 𝑥2 𝑡 , which upon back-
substitution, reveals the solution for
𝑥1 𝑡 to be

Solutions(contd.)
From the probe force defined earlier,
we now compute the mechanical
power P(t) absorbed by the particle
1.

Solutions(contd.)
The power absorbed during one
whole cycle( oscillation) of the probe
force comes out to be

Solutions(contd.)
(c) We have modeled the power spectrum as well as the
absorption spectrum of the mass-spring system as an analog
to the capacitor-coupled R-L-C circuit. The analogies have
been described in vital detail as given under:

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