Amplitude damping with harmonic oscillators

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SUMMARY

The discussion centers on the interaction of harmonic oscillators in quantum mechanics, specifically regarding the Hamiltonian that simulates a principal system and its environment as harmonic oscillators. The annihilation operators, denoted as "a" for the system and "b" for the environment, commute, leading to the conclusion that [a, b] = 0 since they operate in separate Hilbert spaces. This confirms that the operators do not influence each other directly. The reference to "Quantum Computation and Quantum Information" by Nielsen and Chuang provides foundational context for these concepts.

PREREQUISITES
  • Understanding of harmonic oscillators in quantum mechanics
  • Familiarity with annihilation and creation operators
  • Knowledge of Hilbert spaces in quantum theory
  • Basic principles of Hamiltonian mechanics
NEXT STEPS
  • Study the properties of commutation relations in quantum mechanics
  • Explore the role of Hamiltonians in quantum systems
  • Investigate the implications of tensor products in quantum mechanics
  • Read "Quantum Computation and Quantum Information" by Nielsen and Chuang for deeper insights
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Quantum physicists, students of quantum mechanics, and researchers interested in the dynamics of harmonic oscillators and their interactions with environments.

haxel
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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
 
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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.
 

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