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In decoherence theory, we explain the decoherence by hamiltonian evolutions between a system and its environment. Calling $H$ the total hamiltonian, I have:

$$H=H_S + H_E + H_{SE} $$

A pointer state $|s\rangle$ is a state of the system $S$ for which the associated observable $|s\rangle \langle s |$ will commute with the total Hamiltonian. Then, coherences between pointer states will be killed under the Hamiltonian evolution.

One particular limit is the quantum-measurement limit for which $H \approx H_{SE}$. As typical interactions between a system and its environment are of the form:

$$H_{SE} = X \otimes E$$

Where $X$ is the position operator of the system, we figure out that the pointer states are the states $|x\rangle$, and thus the coherence between position eigenstates will be killed.

My question:

Now, a particular example of this situation if I understood is the quantum scattering. Basically our system $S$ (a molecule for example) will interact with environmental molecules under $H_{SE}$. And this interaction will kill superposition in the position basis as I explained. And from this we say that it matches the classical behavior in which the molecule doesn't have position superposition.

However I am puzzled by this. Indeed for me the classical limit should be a packet that is narrow both in position and momentum, like a coherent state.

Thus I am not sure to really understand why quantum scattering, as it is a particular case of quantum measurement limit, would fit as a good quantum to classical correspondance. If the particle is localized in space it is spread in momentum and thus highly non classical.

Does that mean that experimentally, we really see that molecules in a gaz for example are well defined in position, but very poorly in momentum (and then this explanation would match the decoherence theory) ?

A paper in which is explicitly said that collisional decoherence is a particular case of the quantum measurement limit (i.e hamiltonian dominated by interacting part) is The quantum-to-classical transition and decoherence on page 5, beginning of second column.

StarBucK
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1 Answers1

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the classical limit should be a packet that is narrow both in position and momentum, like a coherent state.

That's correct. The approximation $H\approx H_{SE}$ can be used to show the tendency for pointer states to be localized in position, but that's just an approximation. The kinetic term $H_S$ tends to make a particle's wavefunction spread out, which competes with the tendency of $H_{SE}$ to make it localized. The balance of these two trends results in the particle being somewhat localized in position, but still spread out enough that it can have a somewhat well-defined momentum, too, so its location can move like a classical particle's location does.

The "measurement limit" is the limit in which the decoherence in one observable's eigenbasis is so rapid and so complete that competing terms don't have a chance. This is the defining feature of a perfect measurement. Natural measurements, like the mutual "measurements" incurred by the molecules in a macroscopic sample of gas, are far enough away from being perfect that both position and momentum can be approximately well-defined (without violating the uncertainty principle $\Delta x\Delta p\gtrsim\hbar$, of course).

Chiral Anomaly
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  • Thank you. Allright then $H \approx H_{SE}$ is a "strong measurement" limit that in a way corresponds to what we expect from projection postulate (the state of the system is brutally projected somewhere). It corresponds to an apparatus super strongly interacting with a system. And it can only occur if the system is small enough (for a too big system $H_S$ would be too big and this limit couldn't be reached). Would you agree ? – StarBucK Jun 08 '20 at 09:22
  • Then, $H=H_S+H_{SE} (+H_E)$ where all term are comparable is something more natural in which spread will both be in momentum and space. It is a more natural situation in which the system is big (so $H_S$ is big), and it interacts with other systems around big as well. But $H_{SE}$ and $H_S$ are then somewhat comparable in size. Then does that mean the author makes a little mistake of interpretation by saying that $H \approx H_{SE}$ would match the scattering of molecules ? – StarBucK Jun 08 '20 at 09:24
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    @StarBucK The conditions under which the approximation $H\approx H_{SE}$ holds depend on the state as well as on things like the size of the system. I've mainly seen it used in a heuristic way: solving the equations with $H_S+H_{SE}$ is too hard, so we look at what would happen if only $H_S$ or only $H_{SE}$ were present and then use intuition to interpolate. Even in a scattering situation, $H\approx H_{SE}$ is only an approximation, and the approximation leaves room for the scattered molecule to still have an approximately-well-defined momentum (and an approximately well-defined position). – Chiral Anomaly Jun 08 '20 at 13:06
  • Hello again, sorry for my late comment but do you have in mind a scattering model I could look at in which we see that indeed the bare Hamiltonian plays a role into localizing in momentum as well. I searched for this for a long time unsuccessfully. In the book "Decoherence and the Quantum-To-Classical Transition", they add the bare Hamiltonian in the dynamic but the pointer states are still exactly position states. I could only see the competition between $H_S$ and $H_{SE}$ that leads to localization both in momentum and position for quantum Brownian motion. I am interested for the same for – StarBucK Jun 27 '20 at 14:44
  • ...scattering models. – StarBucK Jun 27 '20 at 14:46
  • @StarBucK The model I have in mind is for an object moving through a medium like air or sunlight, where scattering is almost continual. The idea is that each scattering event localizes the object to some degree, but loosely enough that it still has an approximately well-defined momentum. The scattering events keep the object's location relatively well defined as it moves. I think I've seen a paper about this somewhere, but I don't remember where. Related papers include Joos and Zeh and Tegmark. – Chiral Anomaly Jun 29 '20 at 02:28
  • Thanks. Super usefull info again. I am reading Joos & Zeh, and what I super barely understand from it (I tried to relate to scattering model in Maximilian book) is that indeed scattering effect tend to localize in position (we kill fastly coherences in positions). But at some point, the coherence length $\Delta x$ becomes lower than the gas particle wavelength. Then the rate of killing coherences reduces progressively, and this is why we don't totally make them vanish in the position basis. Would you agree with that ? Like there is an exponential decrease which becomes less and less – StarBucK Jun 29 '20 at 19:33
  • exponential because at some point $\lambda > \Delta x$ where $\lambda$ are the free particles wavelength and $\Delta x$ the remaining coherence length of the object under study. – StarBucK Jun 29 '20 at 19:34
  • No I think I am over interpreting things. In this paper they seem to say that such behavior is becaused of the recoil-free assumption. To see that we don't fully localize in position we must remove this assumption and use a more refined model. But I don't find any reference in which I can clearly see this. – StarBucK Jun 29 '20 at 19:55
  • @StarBucK It's been a long time since I looked at the Joos and Zeh paper (and I don't have a copy now), but Tegmark's paper deduces an equation that says the scattering environment very quickly localizes the object down to some length scale that depends on the details of the environment, and after that the scattering environment causes further localization relatively slowly. Not sure if that's what you're looking for, but maybe it's related. For convenience, I summarized Tegmark's equation (but not the derivation) in this answer. – Chiral Anomaly Jun 30 '20 at 00:14
  • Allright thanks. I looked in more details to Tegmark paper. What I don't really like about it is the way he derives the equation of motion for the density matrix (he makes some scattering assumptions, I would expect starting from a given general interacting Hamiltonian like $V(x-x_i)$. It is probably equivalent). About the physics, I am not even sure to understand. For any fixed value of $r$, the superposition are killed if one wait long enough. This is the same problem as I have in the other models. I would expect to see a saturation. I want to understand what keeps a few amount – StarBucK Jun 30 '20 at 18:09
  • of superposition in the very end (because I need to be narrow enough in momentum as well). – StarBucK Jun 30 '20 at 18:09
  • He seems to discuss this in his paper on part 4.2 but he uses number from GRW theory as far as I understand to find the $\Delta x$ minimum. I don't understand if/how he gets them for decoherence theory. – StarBucK Jun 30 '20 at 18:16