Oh

I

I think what we did yesterday was after having found a classical Lagrangian

that was drawn to our synthetic electromechanical circuit,

which was some slightly simpler version of what I've drawn here on the board.

And by our drawing on the board, I mean projected on the screen.

We found the Hamiltonian associated with Lagrangian,

went through the description of canonical quantization,

wrote that Hamiltonian, rewrote it using some ladder operators,

and ended up with something that was sort of familiar to us as the

often mechanical Hamiltonian. We added in drive and dissipation coupling

to some external environments, certainly one of which is sort of indicated

by some coaxial cable where we imagine waves running through that cable,

coupling to our electrical circuit, and likewise some environment coupled

to our mechanical oscillator. And then wrote down Heisenberg-Valdevin

equations of motion. We imagined exciting our circuit via some port

with an intense, linear drive with one frequency, and then linearized

the equations of motion around that strong drive. And working in the

rotating frame of that strong drive, we wrote down the linearized

Heisenberg-Valdevin equations of motion. And at that point, I said,

well, we could go to the frequency domain, but let's not.

Let's live in a time domain where we want to. And in particular,

let's live in a time domain because the interaction between the mechanical

oscillator and the resonant circuit is something that we can turn on and

off suddenly. So in the end, we started with this simple interaction

Hamiltonian, and then we played the following game. We imagined driving

our circuits detuned from resonance by the mechanical oscillator's

resonance frequency, and then linearized the Hamiltonian in the

frame of the strong drive, and ended up with some equations of motion

that had been gone backwards and said, what Hamiltonian would have

given me these equations of motion? We would have a form that looked

like this, this kind of beam splitter-like interaction that

annihilates the quantum energy in the mechanics and creates them in the

microwave circuit at resonance. And something that we'll see today is

that when we do that, there's the emergence of a new time scale or a

new rate, something that governs the, if you like, the spectral width of

the peak that we associate with the motion of the mechanical oscillator

modulating the resonant circuit and emitting some photons at the

circuit's resonance frequency, that spectral width is going to be this

new rate capital gamma associated with how strongly we drive, how

strongly our circuit is damped, and the bare electromechanical

interactions. All right. So what I'm going to do for you today is, I

think what we finished with last time, yesterday, was a kind of time

dependent protocol where we imagined very suddenly and very strongly

turning on and off the interaction between the electrical circuit and

the mechanical oscillator that is very strongly turning on and then

turning off. I was going to turn on, probably. Turning on and then

suddenly, and then turning off suddenly, a strong drive which created

an interaction between our mechanical oscillator and our resonant

circuit in such a way that if we left that strong interaction on for