Sunday, 12 July 2020

quantum mechanics - Degenerate parametric amplifier: quadratures


The degenerate parametric amplifier is described by the Hamiltonian:


$H=\hbar \omega a^\dagger a-i\hbar \chi /2 (e^{(2i\omega t)}a^2-e^{(-2i\omega t)}(a^\dagger)^2)$


Where $a$ and $a^\dagger$ as just the operators of creation and anhiquilation and $\chi$ is just a real constant.


If we define the quadratures as:


$X_1=a+a^\dagger \ \ \ \ \ \ \ \ \ ; \ \ \ \ \ \ \ \ \ X_2=a-a^\dagger$


How can we calculate the quadratic fluctuations (uncertainties) of these quadratures? In particular, I read that they satisfy the equations:


$(\Delta X_i)^2(t)=e^{(2\chi t)}(\Delta X_i)^2(0)$



I tried applying the Dirac picture with, as we can easily separate:


$H=\underbrace{\hbar \omega a^\dagger a}_{H_0}+\underbrace{-i\hbar \chi /2 (e^{(2i\omega t)}a^2-e^{(-2i\omega t)}(a^\dagger)^2)}_{H_1}$


Where $H_0$ is just the hamiltonian for the harmonic oscillator (with known solution) and $H_1$ is just a perturbation. This allows to find the equations of motion for $a$ and $a^\dagger$, but I'm not sure how to get the form of $(\Delta X_i)^2 (t)$ shown above.


PD: I haven't studied time-dependent perturbation theory so I'm not sure if it's necessary to solve this problem.




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