# Is this decoherence?

I have a very basic understanting of decoherence (i.e. I,ve read the Wikipedia page), but I was recently reading Heisenberg's The Physical Principles of the Quantum Theory and I came across a thought experiment which, I think, is decoherence, as it shows loss of interference and seems analogous to optical decoherence. According to Wikipedia decoherence was first developed in 1952 and this book is from 1930, so I don't know.

This is, not exactly but paraphrased, the experiment.

To begin with, imagine a beam of atoms of width $$d$$ is sent through a field $$F$$ which is inhomogeneous in the $$x$$-direction, and which can separate atoms into states $$n$$ and $$m$$ with energies $$E_n$$ and $$E_m$$ respectively (Like the Stern-Gerlach experiment). The energy of state $$m$$ depends on the field so the force in the $$x$$-direction experienced by an atom in that state will be $$-\frac{\partial E_m}{\partial x} \, .$$ Thus the beam will be deflected by an angle $$\frac{\partial E_m}{\partial x}\frac{T}{p}$$ where $$T$$ is the amount of time the atom spends in the field and $$p$$ is the atom's momentum. The angular separation of the beams in state $$m$$ and state $$n$$ is therefore $$\alpha=\left(\frac{\partial E_m}{\partial x}-\frac{\partial E_n}{\partial x}\right)\frac{T}{p} \, .$$ For the beam to be split into separate beams the angular separation must be larger than the natural scattering of the beam by diffraction, so $$\alpha\ge\frac{\lambda}{d}=\frac{h}{pd}$$ where $$d$$ is the beam width and $$\lambda$$ is the wavelength of the particles. Since $$E_m$$ depends on field strength, there will be a change in the phase of the the atom's state $$|m\rangle$$ associated with the atoms passing through the field, with $$\varphi_m$$ being the phase change and $$\varphi=\varphi_m-\varphi_n$$. However, since the beam has a width and the Field is not uniform, the phase change will vary depending on the atom 'passing' through different parts of the field. This will introduce an uncertainty $$\Delta\varphi$$ in the phase change difference. Since the phase change is $$\varphi_m=\frac{2\pi}{h}E_m T$$ And the uncertainty in position is the width of the beam $$d$$ the uncertainty in the energy will be $$\frac{\partial E_m}{\partial x}d$$ So we have $$\Delta\varphi=2\pi\left(\frac{\partial E_m}{\partial x}-\frac{\partial E_n}{\partial x}\right)\frac{Td}{h}=2\pi \frac{pd}{h}\alpha$$ So $$\Delta\varphi\ge 2\pi$$, so the phase is entirely undetermined and random.

Now imagine a beam of atoms in state $$n$$ is sent to a detector which detects if an atom is in a state $$l$$. The probability of measuring an atom in the state $$l$$ when it is in the state $$n$$ is $$|\langle l | n \rangle|^2=|\sum_m\langle l | m \rangle\langle m | n \rangle|^2=\sum_{m'm''}\langle l | m' \rangle\langle m' | n \rangle\langle m'' | l \rangle\langle n | m'' \rangle$$ Now imagine before the atoms get to the detector, we introduce the field seen previously which separates the atoms in their states $$m$$ now to the probability we must necessarily introduce the phase factor, averaged out over all the possible phase changes introduced by the field $$\sum_{m'm''}\langle l | m' \rangle\langle m' | n \rangle\langle m'' | l \rangle\langle n | m'' \rangle\langle e^{\varphi_{m'}-\varphi_{m''}}\rangle$$ And (assuming the phase difference has the same probability for all possible values, reasonable given its large uncertainty) $$\langle e^{\varphi_{m'}-\varphi_{m''}}\rangle=\delta_{m'm''}$$ And so the probability becomes $$\sum_m|\langle l | m \rangle|^2|\langle m | n \rangle|^2$$ The interference is lost, and the classical probability is obtained.

• Hi. I think the basic answer to your question is "yes", but some of the details are puzzling. I marked one of the equations $(\star)$: can you add some explanation of where that equation comes from? Feb 27 '19 at 2:58
• I've made the edits. Is that clear enough? Feb 27 '19 at 15:02

Formally this is decoherence - loss of coherence that results in the density matrix of the atom becoming mixed.

But assume that the field is static and all its fluctuations are only spatial in its nature. Let's replace the detector with some sort of mirror (that also changes the state of the atom in such a way that the field has the opposite effect) and put detector near the beam emitter. Then on the way back the state of the atom will evolve as if time was reversed. As result the original state of the atom will be restored to the initial one without loss of information.

The decoherence in the general sense can happen due to very different processes and sometimes the distinction is made. What is often called "true decoherence" happens due to the entanglement of the system under consideration with the uncontrollable environmental degrees of freedom. What you describe however is what is called "fake decoherence" which is much more classical in its nature - it is the loss of coherence that happens because of our lack of knowledge of the fundamentally unitary evolution of the system.

Both happen because of certain lack of knowledge however the latter is constrained to the system itself whereas the former involves the whole macroscopic environment. In the "fake decoherence" the information about the initial state is recoverable if only in principle from the knowledge just about the system itself. In contrast in the "true decoherence" the information about the initial state is lost in a much more severe way as to recover it even in principle you need to know the state of the whole macroscopic environment.

• So then, in interpretations without Wave function Collapse where it is only apparent due to decoherence, the 'wave function collapse' is in theory reversible, it just isn't in practice due to the randomness and complexity inherent in the enviroment, similar to entropy, is that right? Feb 28 '19 at 0:24
• @user140323 Well, I would like to know what do you mean when you say "interpretations without Wave function Collapse". People often misunderstand what "collapse" mean and often present rather stupid ways to "get rid of collapse" using decoherence. Decoherence theory indeed explains a lot of stuff about the macroscopic world that was rather poorly understood earlier and in many cases the logic follows what you've said.
– OON
Feb 28 '19 at 13:05
• @user140323 Just a small example of stupid ways. Sometimes people say the following: we consider the system interacting with environment in a certain way and trace out all the environment and get diagonal density matrix of the system in ceratain basis. Then they say - that describes the measurement. It doesn't. What you've done is said "I don't care about all the environment, only about system". But in the measurent you care about certain part of environment - the state of the apparatus. That way you describe as if you turned the apparatus on but didn't care to look at results
– OON
Feb 28 '19 at 13:12
• @user140323 What decoherence theory helps you to understand is how the purely quantum world starts behaving classically when you consider the macroscopic stuff. That's provide the basis for the Copenhagen interpretation idea that we not only can but have to formulate our experiments in the classical terms (because we ourselves are macroscopic)
– OON
Feb 28 '19 at 13:17

Wikipedia is mistaken if it says that decoherence was only developed in 1952. Decoherence is simply part of quantum theory, and it is there in many early discussions of the interactions of small systems with measuring apparatuses, such as the one you describe from Heisenberg. Heisenberg judged that these sorts of physical descriptions were valuable in understanding what is going on, but they do not resolve all the issues concerning measurement and observation. He was right about that.

• Interesting. What are other early examples of decoherence that we can find? Feb 28 '19 at 0:26
• My remark just refers to the fact that whenever one either takes an average over a degree of freedom entangled with the state, or over a phase that cannot be controlled in the lab, then decoherence is simply part of the prediction of the dynamics as described by Schrodinger's equation. This can be found, and was found, by anyone who analyses a system of anything more than modest size of complexity (e.g. try Bohm). Later on people tried to imbue the word 'decoherence' with added significance, as if it settled all the issues concerning measurement. Feb 28 '19 at 8:53