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Single ℂ$^N$ is a state space for single photon, let $H_p$ stands for a Hamiltonian of a single photon.

You are going to struggle here because there is no "Hamiltonian of a single photon". Photons are not conserved particles like electrons or protons, they can only be described as excited states of a quantum field. So the idea of an atom interacting with one photon at a time does not really make sense (if we are considering the free radiation field, not a cavity), since extra photons may be created as a consequence of the interaction.

Nevertheless, a single two-level atom interacting with a heat bath of photons is a canonical theoretical problem, treated extensively in almost any text on quantum optics or open quantum systems. See Breuer & Petruccione, for example. Without going into details, the bound electron in the atom interacts with the quantum and thermal fluctuations of the electric field at the position of the atom. The temperature of the electromagnetic field state determines the mean number of photons flying around for the atom to interact with. After a long time, the internal electronic state of the atom reaches thermal equilibrium with the radiation field via processes of absorption and emission.

The atom-field Hamiltonian describing this situations is a specific instance of the so-called spin-boson model. Study of the spin-boson model is basically a research sub-field in itself. The standard review article for the field is Dynamics of the dissipative two-state system, A. Leggett et al., Rev. Mod. Phys. 59, 1-85 (1987), unfortunately behind a paywall. However, this article is quite old now and many new developments and extensions have been made, so its worth just Googling "spin-boson model" to see what you can get.

Single ℂ$^N$ is a state space for single photon, let $H_p$ stands for a Hamiltonian of a single photon.

You are going to struggle here because there is no "Hamiltonian of a single photon". Photons are not conserved particles like electrons or protons, they can only be described as excited states of a quantum field. So the idea of an atom interacting with one photon at a time does not really make sense (if we are considering the free radiation field, not a cavity), since extra photons may be created as a consequence of the interaction.

Nevertheless, a single two-level atom interacting with a heat bath of photons is a canonical theoretical problem, treated extensively in almost any text on quantum optics or open quantum systems. See Breuer & Petruccione, for example. Without going into details, the bound electron in the atom interacts with the quantum and thermal fluctuations of the electric field at the position of the atom. The temperature of the electromagnetic field state determines the mean number of photons flying around for the atom to interact with. After a long time, the internal electronic state of the atom reaches thermal equilibrium with the radiation field via processes of absorption and emission.

The atom-field Hamiltonian describing this situation is a specific instance of the so-called spin-boson model. Study of the spin-boson model is basically a research sub-field in itself. The standard review article for the field is Dynamics of the dissipative two-state system, A. Leggett et al., Rev. Mod. Phys. 59, 1-85 (1987), unfortunately behind a paywall. However, this article is quite old now and many new developments and extensions have been made, so it's worth just Googling "spin-boson model" to see what you can get.

Single ℂ$^N$ is a state space for single photon, let $H_p$ stands for a Hamiltonian of a single photon.

You are going to struggle here because there is no "Hamiltonian of a single photon". Photons are not conserved particles like electrons or protons, they can only be described as excited states of a quantum field. So the idea of an atom interacting with one photon at a time does not really make sense (if we are considering the free radiation field, not a cavity), since extra photons may be created as a consequence of the interaction.

Nevertheless, a single two-level atom interacting with a heat bath of photons is a canonical theoretical problem, treated extensively in almost any text on quantum optics or open quantum systems. See Breuer & Petruccione, for example. Without going into details, the bound electron in the atom interacts with the quantum and thermal fluctuations of the electric field at the position of the atom. The temperature of the electromagnetic field state determines the mean number of photons flying around for the atom to interact with. After a long time, the internal electronic state of the atom reaches thermal equilibrium with the radiation field via processes of absorption and emission.

The atom-field Hamiltonian describing this situations is a specific instance of the so-called spin-boson model. Study of the spin-boson model is basically a research sub-field in itself. The standard review article for the field is Dynamics of the dissipative two-state system, Rev. Mod. Phys. 59, 1-85 (1987), unfortunately behind a paywall. However, this article is quite old now and many new developments and extensions have been made, so its worth just Googling "spin-boson model" to see what you can get.

Single ℂ$^N$ is a state space for single photon, let $H_p$ stands for a Hamiltonian of a single photon.

You are going to struggle here because there is no "Hamiltonian of a single photon". Photons are not conserved particles like electrons or protons, they can only be described as excited states of a quantum field. So the idea of an atom interacting with one photon at a time does not really make sense (if we are considering the free radiation field, not a cavity), since extra photons may be created as a consequence of the interaction.

Nevertheless, a single two-level atom interacting with a heat bath of photons is a canonical theoretical problem, treated extensively in almost any text on quantum optics or open quantum systems. See Breuer & Petruccione, for example. Without going into details, the bound electron in the atom interacts with the quantum and thermal fluctuations of the electric field at the position of the atom. The temperature of the electromagnetic field state determines the mean number of photons flying around for the atom to interact with. After a long time, the internal electronic state of the atom reaches thermal equilibrium with the radiation field via processes of absorption and emission.

The atom-field Hamiltonian describing this situations is a specific instance of the so-called spin-boson model. Study of the spin-boson model is basically a research sub-field in itself. The standard review article for the field is Dynamics of the dissipative two-state system, A. Leggett et al., Rev. Mod. Phys. 59, 1-85 (1987), unfortunately behind a paywall. However, this article is quite old now and many new developments and extensions have been made, so its worth just Googling "spin-boson model" to see what you can get.

Single ℂ$^N$ is a state space for single photon, let $H_p$ stands for a Hamiltonian of a single photon.

You are going to struggle here because there is no "Hamiltonian of a single photon". Photons are not conserved particles like electrons or protons, they can only be described as excited states of a quantum field. So the idea of an atom interacting with one photon at a time does not really make sense (if we are considering the free radiation field, not a cavity), since extra photons may be created as a consequence of the interaction.

Nevertheless, a single two-level atom interacting with a heat bath of photons is a canonical theoretical problem, and is treated extensively in almost any text on quantum optics and open quantum systems. See, for example, Breuer & Petruccione. Without going into details, the bound electron in the atom interacts with the quantum and thermal fluctuations of the electric field at the position of the atom. The temperature of the electromagnetic field state determines the mean number of photons flying around for the atom to interact with. After a long time, the internal electronic state of the atom reaches thermal equilibrium with the radiation field via processes of absorption and emission.

The atom-field Hamiltonian describing this situations is a specific instance of the so-called spin-boson model. Study of the spin-boson model is basically a research sub-field in itself. The standard review article for the field is Dynamics of the dissipative two-state system, Rev. Mod. Phys. 59, 1-85 (1987), unfortunately behind a paywall. However, this article is quite old now and many new developments and extensions have been made, so its worth just Googling "spin-boson model" to see what you can get.

Single ℂ$^N$ is a state space for single photon, let $H_p$ stands for a Hamiltonian of a single photon.

You are going to struggle here because there is no "Hamiltonian of a single photon". Photons are not conserved particles like electrons or protons, they can only be described as excited states of a quantum field. So the idea of an atom interacting with one photon at a time does not really make sense (if we are considering the free radiation field, not a cavity), since extra photons may be created as a consequence of the interaction.

Nevertheless, a single two-level atom interacting with a heat bath of photons is a canonical theoretical problem, treated extensively in almost any text on quantum optics or open quantum systems. See Breuer & Petruccione, for example. Without going into details, the bound electron in the atom interacts with the quantum and thermal fluctuations of the electric field at the position of the atom. The temperature of the electromagnetic field state determines the mean number of photons flying around for the atom to interact with. After a long time, the internal electronic state of the atom reaches thermal equilibrium with the radiation field via processes of absorption and emission.

The atom-field Hamiltonian describing this situations is a specific instance of the so-called spin-boson model. Study of the spin-boson model is basically a research sub-field in itself. The standard review article for the field is Dynamics of the dissipative two-state system, Rev. Mod. Phys. 59, 1-85 (1987), unfortunately behind a paywall. However, this article is quite old now and many new developments and extensions have been made, so its worth just Googling "spin-boson model" to see what you can get.