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Coupled oscillators analog to EIT - do I miss something?

  1. Jul 26, 2013 #1
    Hi,

    I have posted this question to "classical physics" forum, but now I think this forum might be more appropriate. I have no idea how to move the thread here, so here is a copy..

    The question seems trivial, but I want to check if I miss something.

    1. The problem statement, all variables and given/known data
    I'm trying to analyze this article : Classical Analog of Electromagnetically Induced Transparency (http://arxiv.org/pdf/quant-ph/0107061.pdf) which gives a classical analog to EIT by two coupled oscillators, both damped and one is driven too.
    The system looks like : ///------[m1]----[m2]----///
    where m1=m2=m are the masses and the -- represent the springs. The strings which attach the masses to the walls have both spring constant k, and the spring which connects both masses has constant K. Mass m1 is driven by an harmonic force : [itex]Fe^{-i\omega_{s}t}[/itex] .

    2. Relevant equations

    The equations given in the article are :
    (1) [itex]\ddot{x_{1}}+\gamma_{1}\dot{x_{1}}+\omega^{2}x_{1}-{\Omega_{r}}^{2}x_{2}=\frac{F}{m}e^{-i\omega_{s}t}[/itex]
    (2) [itex]\ddot{x_{2}}+\gamma_{2}\dot{x_{2}}+\omega^{2}x_{2}-{\Omega_{r}}^{2}x_{1}=0[/itex]
    Where : [itex]x_{1}[/itex] and [itex]x_{2}[/itex] are the displacements of the masses from their equilibrium. [itex]\gamma_{1}[/itex] and [itex]\gamma_{2}[/itex] are the damping constants. And [itex]\omega^{2}=\frac{k}{m}[/itex] and [itex]{\Omega_{r}}^{2}=\frac{K}{m}[/itex].

    3. The attempt at a solution
    The solution for [itex]x_{1,2}[/itex] is trivial by substituting [itex]x_{1,2}=A_{1,2}e^{-i\omega_{s}t}[/itex]. But I'm not sure that the equations are correct.
    My question is : shouldn't there be [itex]{+\Omega_{r}}^{2}x_{1} [/itex] term in eqn. (1) and [itex]{+\Omega_{r}}^{2}x_{2} [/itex] term in eqn. 2? For example if the two masses moved the same distance apparently the spring connecting them should be in it equilibrium length and not influence them.
    I though that's an error in the article, but I see the same in other articles (for example : http://cms.bsu.edu/sitecore/shell/-/media/WWW/DepartmentalContent/Physics/PDFs/Joe/06_1.pdf [Broken] and http://arxiv.org/pdf/1006.5167v3.pdf), so what am I missing?

    Thanks in advance,
    Naftali
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Jul 26, 2013 #2

    Jano L.

    User Avatar
    Gold Member

    Yes, the force is proportional to difference of the two coordinates. I think the paper has it right, but I think that instead of ##\omega## there should be ##\omega_1## (in the next section on the RLC circuit the equation is the same but they write ##\omega_1##). This is defined by

    $$
    \omega_1^2 = \frac{k_1 + K}{m},
    $$

    so there is term ##K/m ~x_1## which is the same as the term you suggest.
     
  4. Jul 26, 2013 #3
    This seems reasonable..

    Thanks!
    Naftali
     
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