I would like to understand better a phenomenon of a quantum heat bath.

Below I present one example, which seems quite clear to me. It would be great to see some less-discrete models, and more specific open quantum systems (with a given concrete Hilbert space).

(Discrete Picture)

Let $S$ be an open quantum system (e.g. an atom), $\mathcal{H}_{S}$ be a Hilbert space of $S$ and $\mathrm{H}_S$ be a Hamiltonian of $S$. Let $\bigotimes_{k \geq 0} \mathbb{C}^{N}$ of copies of $\mathbb{C}^{N}$, where $N\geq 1$ is a fixed integer be the state space for the reservoir, that is a heat bath (e.g., radiation). Let $\{e_i\}_{i = 0}^{N}$ be an orthonormal basis of $\mathbb{C}^{N}$ such that the countable tensor product is taken with respect to the ground state $e_0$. Single $\mathbb{C}^{N}$ is a state space for single photon, let $\mathrm{H}_{p}$ stands for a Hamiltonian of a single photon. When the system and a photon are interacting, we consider the state space $\mathcal{H}_{S} \otimes\mathbb{C}^{N} $ with a interaction Hamiltonian $\mathrm{H}_{I}$.

First question how can we determine the interaction Hamiltonian? How can we describe this interaction?

Now, consider the state of each photon to be given by a density matrix $\rho_{\beta}$ which is a function of $\mathrm{H}_{p}$, e.g., a Gibbs state at inverse temperature $\beta= \frac{1}{T}$: $$ \rho_{\beta}= \frac{1}{\mathrm{Tr}(e^{-\beta\mathrm{H}_{p}})}e^{-\beta\mathrm{H}_{p}}.$$ Denote the Total Hamiltonain by $\mathrm{H}_{T}$, that is, $\mathrm{H}_{T}(n)=\mathrm{H}_{S} \otimes I_{\mathbb{C}^{N}} + I_{\mathcal{H}_S} \otimes \mathrm{H}_{p} + \mathrm{H}_{I}(n) $.

The system $S$ is first in contact with the first photon only and they interact together according to the above Hamiltonian $\mathrm{H}_{T}$. This lasts for a time length $n$ . The system $S$ then stops interacting with the first photon and starts interacting with the second photon only. This second interaction is directed by the same Hamiltonian $\mathrm{H}_{T}$ on the corresponding spaces and it lasts for the same duration $n$ , and so on...

So we can see how the quantum system $S$ interacts with a heat bath. References - Attal and Joye (2007, J. Funct. Anal. 247, 253--288).http://arxiv.org/pdf/math-ph/0612055v1.pdf

Informally speaking, $S$ is a `small' system and as we will go with time to $+\infty$ we should obtain a 'large' system after the interaction with a heat bath.

Some quantum noises should appear as well, if they were be `squeezed' how would we interpret it ? I meant by this that the driving noises form Araki-Woods representations of CCR in a squeezed (quasifree) state. I would like to know more about the physical interpretations of this phenomenon. I would be also grateful if someone could show me some proper pictures presenting that or similar situation.

Thank you very much for any remarks and answers.

I would like to understand better a phenomenon of a quantum heat bath.

Below I present one example, which seems quite clear to me. It would be great to see some less-discrete models, and more specific open quantum systems (with a given concrete Hilbert space).

(Discrete Picture)

Let $S$ be an open quantum system (e.g. an atom), $\mathcal{H}_{S}$ be a Hilbert space of $S$ and $\mathrm{H}_S$ be a Hamiltonian of $S$. Let $\bigotimes_{k \geq 0} \mathbb{C}^{N}$ of copies of $\mathbb{C}^{N}$, where $N\geq 2$ is a fixed integer be the state space for the reservoir, that is a heat bath (e.g., radiation). Let $\{e_i\}_{i = 0}^{N}$ be an orthonormal basis of $\mathbb{C}^{N}$ such that the countable tensor product is taken with respect to the ground state $e_0$. Single $\mathbb{C}^{N}$ is a state space for single photon, let $\mathrm{H}_{p}$ stands for a Hamiltonian of a single photon. When the system and a photon are interacting, we consider the state space $\mathcal{H}_{S} \otimes\mathbb{C}^{N} $ with a interaction Hamiltonian $\mathrm{H}_{I}$.

First question how can we determine the interaction Hamiltonian? How can we describe this interaction?

Now, consider the state of each photon to be given by a density matrix $\rho_{\beta}$ which is a function of $\mathrm{H}_{p}$, e.g., a Gibbs state at inverse temperature $\beta= \frac{1}{T}$: $$ \rho_{\beta}= \frac{1}{\mathrm{Tr}(e^{-\beta\mathrm{H}_{p}})}e^{-\beta\mathrm{H}_{p}}.$$ Denote the Total Hamiltonain by $\mathrm{H}_{T}$, that is, $\mathrm{H}_{T}(n)=\mathrm{H}_{S} \otimes I_{\mathbb{C}^{N}} + I_{\mathcal{H}_S} \otimes \mathrm{H}_{p} + \mathrm{H}_{I}(n) $.

The system $S$ is first in contact with the first photon only and they interact together according to the above Hamiltonian $\mathrm{H}_{T}$. This lasts for a time length $n$ . The system $S$ then stops interacting with the first photon and starts interacting with the second photon only. This second interaction is directed by the same Hamiltonian $\mathrm{H}_{T}$ on the corresponding spaces and it lasts for the same duration $n$ , and so on...

So we can see how the quantum system $S$ interacts with a heat bath. References - Attal and Joye (2007, J. Funct. Anal. 247, 253--288).http://arxiv.org/pdf/math-ph/0612055v1.pdf

Informally speaking, $S$ is a `small' system and as we will go with time to $+\infty$ we should obtain a 'large' system after the interaction with a heat bath.

Some quantum noises should appear as well, if they were be `squeezed' how would we interpret it ? I meant by this that the driving noises form Araki-Woods representations of CCR in a squeezed (quasifree) state. I would like to know more about the physical interpretations of this phenomenon. I would be also grateful if someone could show me some proper pictures presenting that or similar situation.

Thank you very much for any remarks and answers.

I would like to understand better a phenomenon of a quantum heat bath.

Below I present one example, which seems quite clear to me. It would be great to see some less-discrete models, and more specific open quantum systems (with a given concrete Hilbert space).

(Discrete Picture)

Let $S$ be an open quantum system (e.g. an atom), $\mathcal{H}_{S}$ be a Hilbert space of $S$ and $\mathrm{H}_S$ be a Hamiltonian of $S$. Let $\bigotimes_{k \geq 0} \mathbb{C}^{N}$ of copies of $\mathbb{C}^{N}$, where $N\geq 1$ is a fixed integer be the state space for the reservoir, that is a heat bath (e.g., radiation). Let $\{e_i\}_{i = 0}^{N}$ be an orthonormal basis of $\mathbb{C}^{N}$ such that the countable tensor product is taken with respect to the ground state $e_0$. Single $\mathbb{C}^{N}$ is a state space for single photon, let $\mathrm{H}_{p}$ stands for a Hamiltonian of a single photon. When the system and a photon are interacting, we consider the state space $\mathcal{H}_{S} \otimes\mathbb{C}^{N} $ with a interaction Hamiltonian $\mathrm{H}_{I}$.

First question how can we determine the interaction Hamiltonian? How can we describe this interaction?

Now, consider the state of each photon to be given by a density matrix $\rho_{\beta}$ which is a function of $\mathrm{H}_{p}$, e.g., a Gibbs state at inverse temperature $\beta= \frac{1}{T}$: $$ \rho_{\beta}= \frac{1}{\mathrm{Tr}(e^{-\beta\mathrm{H}_{p}})}e^{-\beta\mathrm{H}_{p}}.$$ Denote the Total Hamiltonain by $\mathrm{H}_{T}$, that is, $\mathrm{H}_{T}(n)=\mathrm{H}_{S} \otimes I_{\mathbb{C}^{N}} + I_{\mathcal{H}_S} \otimes \mathrm{H}_{p} + \mathrm{H}_{I}(n) $.

The system $S$ is first in contact with the first photon only and they interact together according to the above Hamiltonian $\mathrm{H}_{T}$. This lasts for a time length $n$ . The system $S$ then stops interacting with the first photon and starts interacting with the second photon only. This second interaction is directed by the same Hamiltonian $\mathrm{H}_{T}$ on the corresponding spaces and it lasts for the same duration $n$ , and so on...

So we can see how the quantum system $S$ interacts with a heat bath.

Informally speaking, $S$ is a `small' system and as we will go with time to $+\infty$ we should obtain a 'large' system after the interaction with a heat bath.

Some quantum noises should appear as well, if they were be `squeezed' how would we interpret it ? I meant by this that the driving noises form Araki-Woods representations of CCR in a squeezed (quasifree) state. I would like to know more about the physical interpretations of this phenomenon. I would be also grateful if someone could show me some proper pictures presenting that or similar situation.

Thank you very much for any remarks and answers.

I would like to understand better a phenomenon of a quantum heat bath.

Below I present one example, which seems quite clear to me. It would be great to see some less-discrete models, and more specific open quantum systems (with a given concrete Hilbert space).

(Discrete Picture)

Let $S$ be an open quantum system (e.g. an atom), $\mathcal{H}_{S}$ be a Hilbert space of $S$ and $\mathrm{H}_S$ be a Hamiltonian of $S$. Let $\bigotimes_{k \geq 0} \mathbb{C}^{N}$ of copies of $\mathbb{C}^{N}$, where $N\geq 1$ is a fixed integer be the state space for the reservoir, that is a heat bath (e.g., radiation). Let $\{e_i\}_{i = 0}^{N}$ be an orthonormal basis of $\mathbb{C}^{N}$ such that the countable tensor product is taken with respect to the ground state $e_0$. Single $\mathbb{C}^{N}$ is a state space for single photon, let $\mathrm{H}_{p}$ stands for a Hamiltonian of a single photon. When the system and a photon are interacting, we consider the state space $\mathcal{H}_{S} \otimes\mathbb{C}^{N} $ with a interaction Hamiltonian $\mathrm{H}_{I}$.

First question how can we determine the interaction Hamiltonian? How can we describe this interaction?

Now, consider the state of each photon to be given by a density matrix $\rho_{\beta}$ which is a function of $\mathrm{H}_{p}$, e.g., a Gibbs state at inverse temperature $\beta= \frac{1}{T}$: $$ \rho_{\beta}= \frac{1}{\mathrm{Tr}(e^{-\beta\mathrm{H}_{p}})}e^{-\beta\mathrm{H}_{p}}.$$ Denote the Total Hamiltonain by $\mathrm{H}_{T}$, that is, $\mathrm{H}_{T}(n)=\mathrm{H}_{S} \otimes I_{\mathbb{C}^{N}} + I_{\mathcal{H}_S} \otimes \mathrm{H}_{p} + \mathrm{H}_{I}(n) $.

The system $S$ is first in contact with the first photon only and they interact together according to the above Hamiltonian $\mathrm{H}_{T}$. This lasts for a time length $n$ . The system $S$ then stops interacting with the first photon and starts interacting with the second photon only. This second interaction is directed by the same Hamiltonian $\mathrm{H}_{T}$ on the corresponding spaces and it lasts for the same duration $n$ , and so on...

So we can see how the quantum system $S$ interacts with a heat bath. References - Attal and Joye (2007, J. Funct. Anal. 247, 253--288).http://arxiv.org/pdf/math-ph/0612055v1.pdf

Informally speaking, $S$ is a `small' system and as we will go with time to $+\infty$ we should obtain a 'large' system after the interaction with a heat bath.

Some quantum noises should appear as well, if they were be `squeezed' how would we interpret it ? I meant by this that the driving noises form Araki-Woods representations of CCR in a squeezed (quasifree) state. I would like to know more about the physical interpretations of this phenomenon. I would be also grateful if someone could show me some proper pictures presenting that or similar situation.

Thank you very much for any remarks and answers.