In cavity optomechanics, if we consider some specific frequency (say, 19 kHz) of moving end mirror, how is the temperature of the moving end mirror related to the mean phonon number?
Hello Muhammad,
There is two way to understand your question:
The first would be to understand how to link the mean phonon number with the temperature:
In this case the mean number of phonon (at thermal equilibrium) is
n=(kB*T)/(hbar*Omega)
With: Kb Boltzmann constant, T in Kelvin, hbar reduced Planck constant and Omega is your 19 KHz. This is deduced from equirepartition theorem and known as thermal occupation.
The second would be to ask how the Mechanical frequency would shift as a function of Temperature ( i.e. Mean Number of phonon).
This is a little bit more tricky and is sample/geometry dependant: One rough estimation is to take the evolution of the Young modulus as a function of temperature, and put it into the Euler-Bernoulli equations of your resonator ( this can be done using simulations in the case of complicated geometries).
Then, using the first answer, you can link it with the mean number of phonons. Anyway in the case of high quality resonators, the boundary conditions are going to change the frequency a lot... it can be sample dependant for high quality mechanical resonators ( the evolution of the Q factor also can be sample dependant.)
Hope that answered your question
Jean
