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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 attached to two springs with spring constants, 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 and initially kept immobile at a fixed position. Particle 1 is also subject to a harmonic force, given by

Solutions(contd.)

(b) An analysis of the power transferred from the harmonic source to particle 1 as a function of frequency is performed. It is observed that the standard resonance absorption profile discussed above peaked at frequency


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 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 and


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 for the probe force. • We also let the frequency of coupling, .

Solutions(contd.)

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


Solutions(contd.)

We also assume a similar form of the solution for , which upon back-substitution, reveals the solution for 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:
