# Thermalisation - Open quantum systems

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.

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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 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.

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Thank you for the answer. I didn't make this model up, probably I should have written it in my first post. I took it from The Langevin Equation for a Quantum Heat Bath' S. Attal, A. Joye J. Funct. Anal. 247 (2007) arxiv.org/pdf/math-ph/0612055v1.pdf. Probably the radiation which I suggested wasn't a good example, it is not mentioned in that paper, but it seems to be the most natural example of a quantum heat bath. Could you say under what circumstances the above model makes sense? Thank you for suggesting the spin-boson model I will have a look at it. –  m.gn Sep 14 '13 at 9:32
In your answer you wrote since extra photons may be created as a consequence of the interaction', but the model is still fine if the following situation happens, since the heat bath is an infinite tensor product of the particles, am I wrong here? –  m.gn Sep 14 '13 at 9:58
@m.gn Your model makes sense when the incoming "bath" particles possess mass and some other conserved quantum number so that a non-relativistic single-particle description in terms of ordinary quantum mechanics is possible. In particular, you have required these particles to be described individually by a Hamiltonian. As I pointed out, there is no "single-photon Hamiltonian". This is simply an expression of the fact that photons have no mass, so processes involving exchange of arbitrarily small amounts of energy can create additional photons. –  Mark Mitchison Sep 17 '13 at 17:20

I think that I have found that the model which I'm looking for is called `one atom master', so in the above case $S$ is the quantized electromagnetic field in a cavity through which a beam of atoms given bt ($\mathbb{C}^N$, $\mathrm{H}_p$), is shot. Do you agree with that?

It is described in " Repeated Interactions in Open Quantum Systems " by L. Bruneau, A. Joye M. Merkli http://www-fourier.ujf-grenoble.fr/~joye/bruneau_joye_merkli_JMP_special_issue.pdf

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Yep, this model seems perfectly reasonable, and indeed describes a very similar setup to the Nobel-prize winning experiments of Serge Haroche's group. –  Mark Mitchison Sep 17 '13 at 17:14