Few questions about quantum optics\quantum mechanics

[vani11a]

I have some questions concerning quantum optics and quantum mechanics and I'd appreciate if someone could help me with them.(I'm self-studying both so the questions can sound stupid)
1) When treating beam-splitter quantum-mechanically, why we replace classical $$E$$ amplitudes with annihilation operators $$a$$ and don't take both $$a$$ and $$a^{+}$$(since $$E$$ is a sum of them)? I can't understand why we treat the field on the 'out' ports with annihilation operators.
2) Is there any physical meaning for 'negative-frequency' part when quantizing electric field: $$E^{(-)}=\sum_{k}\epsilon_kE_ka_{k}^{+}e^{i\omega_kt-ikr}$$ and why we sometimes use only $$E^{(+)}$$ or $$E^{(-)}$$?
3)Why in down-conversion the Hamiltonian is of the form $$\hbar g(a_{1}^{+}a_{2}^{+}a_{0}+h.c.)$$? Should it be derived replacing $$P$$ with the expression with $$\chi^{(2)}$$?
4)A question about QM: What is actually postulated? Statements about correspondence between isolated physical systems and vectors in Hilbert space or between measurables and operators are usually the same in all the literature I've read, but I've also seen postulating the expression for momentum operator or the relation between commutator and Poisson bracket, though in Landau&Lifgarbagez both of those are derived.
Thanks in advance.

 

[eljose79]

Thank you for your questions regarding quantum optics and quantum mechanics. As a scientist in this field, I would be happy to provide some insight and clarification on these topics.

1) The use of annihilation operators in quantum optics is based on the formalism of quantum mechanics, where the field is described as a collection of harmonic oscillators. These oscillators are represented by the annihilation and creation operators, a and a^{+}, respectively. The reason we do not use both a and a^{+} is because they represent the same physical quantity, just in different directions (annihilation and creation). Therefore, we can choose to use either one of them, and it is more convenient to use the annihilation operator in this case.

2) The negative frequency part of the electric field, E^{(-)}, is used to describe the time-reversed version of the positive frequency part, E^{(+)}. In other words, it represents the same physical quantity but with the time reversed. This is useful in certain situations, such as in quantum optics experiments, where time reversal symmetry is important. We use both E^{(+)} and E^{(-)} because they represent different aspects of the same physical quantity.

3) The Hamiltonian for down-conversion is derived from the interaction between the pump field and the nonlinear medium. This interaction is described by the nonlinearity coefficient, \chi^{(2)}, which is related to the polarization of the medium. Therefore, the Hamiltonian includes the term \hbar g(a_{1}^{+}a_{2}^{+}a_{0}+h.c.), where g is the coupling strength and a_{1}^{+}, a_{2}^{+}, and a_{0} are the annihilation operators for the pump, signal, and idler fields, respectively. This term represents the creation of a pair of photons (signal and idler) from the pump field, and the Hermitian conjugate term represents the annihilation of these photons.

4) In quantum mechanics, there are a few key postulates that form the basis of the theory. These include the postulate of superposition, the postulate of measurement, and the postulate of uncertainty. These postulates are used to describe the behavior of quantum systems and are used to derive the expressions for operators, such as the momentum operator and the relation between commutators and Poisson brackets. These derivations are typically presented