Amplitude damping with harmonic oscillators

[haxel]

Hi

I am new to this community, so don't beat me up too hard :).

I have a question about the Hamiltonian when it will simulate the principal system as a harmonic oscillator interacting with the environment which is also an harmonic oscillator (page 291 in "Quantum computation and Quantum information" Nielsen, Chuang).
Is the communtator [a,b] = 0 ? It seems to me that the operators can't affect each other but I am not sure, should there be a tensor between "a" and "b" in the Hamiltonian?
Do you have any source to back up your statements?

Sincerely
Axel

 

[Jackson Tan]

Okay, I don't have Nielson and Chuang, so I'm not certain what you're talking about, but by conventional notation, it seems that [tex]a[/tex] and [tex]b[/tex] are annihilation operators acting on the system and the environment respectively.

If that is so, then the [tex][a,b] = 0[/tex] because they act on different Hilbert space. [tex]a[/tex] is an operator in the system Hilbert space; [tex]b[/tex] is an operator in the environment Hilbert space.

 

[Entropia]

Hello Axel,

Welcome to the community! Your question about amplitude damping with harmonic oscillators is an interesting one. In this type of system, the Hamiltonian would indeed involve a tensor product between the operators for the principal system and the environment. This is because the two systems are interacting and can affect each other.

Regarding your question about the commutator [a,b], it is not necessarily equal to zero. It depends on the specific form of the Hamiltonian and the operators a and b. In general, the commutator [a,b] represents the extent to which the operators a and b do not commute, and it can have different values depending on the specific system.

As for sources to back up these statements, I would suggest looking at textbooks on quantum mechanics or quantum computation, as well as research papers on amplitude damping with harmonic oscillators. I hope this helps answer your question. Happy researching!