Pure to mixed and back to pure state

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kye

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Let's say you prepare a pure state in a single electron at a time double slit experiment meaning the electron interferes with itself, then you hit the electron with a one photon at the sides (serving as environment) causing decoherence, so the double slit setup decohere and becomes mixed state without the electron interfering with itself. But if you treat the system and environment together by encompassing both the electron and know the photon, you are said to retrieve the phases and get back the interference (become pure state again). What experiments have been done like this? But there is clearly no interference anymore even if you hit it with one photon and know its state to retrieve the phases.. so how can it said to be in pure state again?
 
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I've done similar measurements as the ones you describe. I'm sure there are other examples, but I know the ones I did myself in ion traps the best (see PRL version of arXiv version)

In this situation, A single ion is in a trap and the coherence of its spin can be measured in a spin echo setup. We see first that the coherence is good, then we entangle the spin with the motion/phonon bath of the ion, which represents the environment. If we measure the spin coherence after this, we find it is zero. We then make another measurement where the coupling to the phonon bath is removed again, by coherently stopping all motion. We then find that the original spin coherence is back again.

I guess it was something like this you were wondering about?
 

naima

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I suppose that this occurs because you can manage the whole environment.
 

kye

168
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I've done similar measurements as the ones you describe. I'm sure there are other examples, but I know the ones I did myself in ion traps the best (see PRL version of arXiv version)

In this situation, A single ion is in a trap and the coherence of its spin can be measured in a spin echo setup. We see first that the coherence is good, then we entangle the spin with the motion/phonon bath of the ion, which represents the environment. If we measure the spin coherence after this, we find it is zero. We then make another measurement where the coupling to the phonon bath is removed again, by coherently stopping all motion. We then find that the original spin coherence is back again.

I guess it was something like this you were wondering about?
I don't know how to compare this to the double slit. My simple question is, can you uncollapse the double slit by knowing the states of the decohering environment. If so. How do you recover the interference right in the detector by simply having knowledge of the environment and getting a pure state of the setup?
 
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The answer is probably yes, but you have to define the "environement" a bit more carefully if you want to know How to do it.

In the experiment I described, the phonon bath is the environment, and you can uncollapse it by stopping the motion coherently. You can do the same thing in a double slit, but only after you have reached the same degree of controll over whatever the environment is in that case. I know some experiments have been done with double slits where they move the slit itself with some modulation, but I don't remember the details.

In short, the general and qualitative answer to your question is yes, you can uncollapse the "environment" if you have sufficient control over it, but to answer the question on how to do it in practice you have to dig through the technical details of a specific experiment.
 

naima

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I posted this answer in a wrong thread:

Haroche gives the general answer to the coherence reapparance after a which path meaurement. read p 77 of "Exploring the Quantum: Atoms, Cavities, and Photons".
The information about the path is stored in a huge number of atoms, photons in the environment.
You would have to erase all the subsystems of the environment where the information is imprinted.

Haroche writes that CNOT gates could be used for all the atoms containing the which path information.
This will remain a thought experimrnt.
 
Last edited:

kye

168
2
The answer is probably yes, but you have to define the "environement" a bit more carefully if you want to know How to do it.

In the experiment I described, the phonon bath is the environment, and you can uncollapse it by stopping the motion coherently. You can do the same thing in a double slit, but only after you have reached the same degree of controll over whatever the environment is in that case. I know some experiments have been done with double slits where they move the slit itself with some modulation, but I don't remember the details.

In short, the general and qualitative answer to your question is yes, you can uncollapse the "environment" if you have sufficient control over it, but to answer the question on how to do it in practice you have to dig through the technical details of a specific experiment.

Thinking of this again. I think I'm looking for superposition of positions. What you mentioned is superposition of spin. It's easier to imagine superposition of positions... the interference is positions of a collection of particles... not spin..

Going back to the double slit with environment. So it's like the pure state is the combination of double slit with environment, and there is superposition but not in the sense of interference between the double slit interference but between the double slit and the environment phase. Now I understand we don't perceive the superposition is because we are one of the branches of the decohering terms. Is my understanding correct?
 

naima

Gold Member
936
54
I've done similar measurements as the ones you describe. I'm sure there are other examples, but I know the ones I did myself in ion traps the best (see PRL version of arXiv version)

In this situation, A single ion is in a trap and the coherence of its spin can be measured in a spin echo setup. We see first that the coherence is good, then we entangle the spin with the motion/phonon bath of the ion, which represents the environment. If we measure the spin coherence after this, we find it is zero. We then make another measurement where the coupling to the phonon bath is removed again, by coherently stopping all motion. We then find that the original spin coherence is back again.
There are 3 steps. You says that we begin with a good coherence then we have a zero coherence and at the end coherence is back.
Could you explain what is measured?
Can it be seen in one of the figures of the paper you cited?
How do we recognize a zero coherence result?

Thank you for the answer. The subject is very technical bit i think it would be interesting to understand
the principles
 
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There are 3 steps. You says that we begin with a good coherence then we have a zero coherence and at the end coherence is back.
Could you explain what is measured?
Can it be seen in one of the figures of the paper you cited?
How do we recognize a zero coherence result?

Thank you for the answer. The subject is very technical bit i think it would be interesting to understand
the principles
What was actually measured was the spin coherence through what is called a spin echo sequence, where you vary the phase of the final pi/2 pulse. This gives a sinusoidal signal, where the amplitude of the sinus is proportional to how much "coherence" is left in your system.

Keep in mind that every time we do a measurement to test to spin coherence we collapse any superpostion state, this means that when I say we do three steps, we of course have to make three different experiments, where we start from scratch every time. Those three experiments are:

1) create a superposition state, measure the spin coherence immediately after
2) create a superposition state, entangle the spin with the motion, then measure the spin coherence
3) create a superposition state, entangle with the motion, unentangle with the motion (motion with opposite timings), then measure the spin coherence

In case 1 and 3, the spin echo measurement reveals clear sinusoidals indicating coherence, but in case 2 the same measurement simply gives a flat repsponse of 0. Figures for case 1 and 3 is Fig 4 in the paper I cited. Case two was not included in the paper since it's just a 0 everywhere and trivial. (note the phase shift for the figure corresponding to case 3 is because the ion moved due to the motion entanglement and is proof that it was motion-entangled.)
 

naima

Gold Member
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Great,
merci beaucoup.
.
 

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