Gravitation and Decoherence : New Scientist Article

[Swamp Thing]

It says here that time dilation due to gravity can cause decoherence:

http://www.newscientist.com/article...ce-us-to-do-quantum-experiments-in-space.html

One of Einstein's predictions is that gravity slows down time...
... Lab experiments with atomic clocks have revealed that your head ages slightly faster than your feet, because of the tiny differences in gravitational field strength...

Pikovski's calculations show that molecules placed in a superposition should also experience this time difference, and it can disrupt their quantum state. This happens because the bonds between atoms in a molecule act like springs and constantly vibrate. If a molecule is in a superposition of two states that are at different heights from the ground, each state will vibrate at a different rate, destroying the superposition.

But.. wouldn't the effect of time dilation be more like a modification of the expected unitary evolution, so that there will be a slow change in the probabilities of different measurement results (compared with the zero-g case) ?

 

[jerromyjon]

I doubt there is any way to make any predictions about an entangled pair once they decohere. It doesn't really make logical sense to me, that you could have a pair of "twins" who age at different rates in different environments, but retain some time skewed correlation?

I do wonder if there is a way to do a double slit across a gravitational gradient, would it smear the interference pattern or eliminate it?

 

[andresB]

It sounds like the ideas of Roger Penrose about objective collapse of the wave function due to gravity.

 

[The_Duck]

Actual paper http://www.nature.com/nphys/journal/vaop/ncurrent/full/nphys3366.html.

 

[StevieTNZ]

I asked Caslav this question yesterday morning:

Hi Caslav

I trust you have been well, and are enjoying the summer months in Vienna. Are you teaching any summer semester courses, or do you plan on travelling?

This morning I saw on physorg.com an article about a Nature article which you are a co-author of: “Universal decoherence due to gravitational time dilation”.

I have a question:
When I see the word decoherence, I immediately think that quantum effects are suppressed, but don’t disappear (interference is harder to see, but if you take the system + environment it is in principle in a pure (superposition) state).
Is this the same with the coupling of the quantum system and the external environment (in this case classical general relativity time dilation); that is even though it the quantum system (either micro or macro) is coupled with this environment, the quantum effects are suppressed but in principle still exist?

Many thanks for any clarity you are able to offer.

Best regards
Stevie

Last night before going to bed I checked my emails, he responded:

Hi Stevie,

I am both teaching and traveling – very tiring both.

Yes, you are right, the quantum effects are suppressed but in principle still existent.

All the best,
Caslav

So on the physorg.com article "They calculated that once the small building blocks form larger, composite objects - such as molecules and eventually larger structures like microbes or dust particles -, the time dilation on Earth can cause a suppression of their quantum behavior. The tiny building blocks jitter ever so slightly, even as they form larger objects. And this jitter is affected by time dilation: it is slowed down on the ground and speeds up at higher altitudes. The researchers have shown that this effect destroys the quantum superposition and, thus, forces larger objects to behave as we expect in everyday life."
is wrong - quantum superposition is still present.

 

[The_Duck]

I read part of the paper.

But.. wouldn't the effect of time dilation be more like a modification of the expected unitary evolution, so that there will be a slow change in the probabilities of different measurement results (compared with the zero-g case) ?

Yes, the effect is just a modification of the unitary evolution. The idea seems to be this: Suppose at time $t = 0$ you have an atom in a superposition of two energy eigenstates.

$$|\psi(0)\rangle = \frac{1}{\sqrt 2} |E_1\rangle + \frac{1}{\sqrt 2} |E_2\rangle$$

After a time $t$ the unitary evolution of the state takes the atom to the following state

$$|\psi(t)\rangle = \frac{1}{\sqrt 2} e^{-iE_1 t/\hbar}|E_1\rangle + \frac{1}{\sqrt 2} e^{-iE_1 t/\hbar}|E_2\rangle$$

Now suppose the atom is not only in a superposition of energies but also in a superposition of heights above the ground. Say the initial state is

$$|\psi(0)\rangle = \frac{1}{2} |h_1\rangle (|E_1\rangle + |E_2\rangle) + \frac{1}{2} |h_2\rangle (|E_1\rangle + |E_2\rangle)$$

In a given time interval, the component at height $h_1$ will experience a different elapsed proper time from the component at height $h_2$ because of gravitational time dilation. Let's say the elapsed proper time at $h_1$ is $t_1$, while the elapsed proper time at $h_2$ is $t_2$. So the state will evolve to

$$\frac{1}{2} |h_1\rangle (e^{-iE_1 t_1/\hbar} |E_1\rangle + e^{-iE_2 t_1/\hbar} |E_2\rangle) + \frac{1}{2} |h_2\rangle (e^{-iE_1 t_2/\hbar} |E_1\rangle + e^{-iE_2 t_2/\hbar} |E_2\rangle)$$

(Actually I haven't included the effect of the gravitational potential energy term in the Hamiltonian, but we can actually ignore that for our current purpose). Now we look at the relative phase between the $E_1$ and $E_2$ components of the state. For the $h_1$ part of the wave function this relative phase is $(E_1 - E_2)t_1/\hbar$. For the $h_2$ part of the wave function this relative phase is $(E_1 - E_2)t_2/\hbar$. The relative phases are different because of the different elapsed proper times. This is the effect pointed out by the article.

For a system with more internal degrees of freedom, these slight phase differences will eventually make the $h_1$ component internal state completely orthogonal to the $h_2$ component of the internal state. Once this happens the $h_1$ component can no longer interfere with the $h_2$ component. So we can say that the time dilation is producing decoherence.

I do wonder if there is a way to do a double slit across a gravitational gradient, would it smear the interference pattern or eliminate it?

If the object has no internal degrees of freedom, the interference pattern will be unaffected.

If the object has internal degrees of freedom but it is in an eigenstate of the internal energy, the interference pattern will be unaffected.

The effect discussed here only operates when the object is in a superposition of different values of the internal energy. In this case, yes, the interference pattern will be degraded if the two interfering paths experience different amounts of time dilation. The amount of degradation depends on the amount of time dilation, on the number of internal degrees of freedom, and on the uncertainty of the internal energy.

 

[StevieTNZ]

It sounds like the ideas of Roger Penrose about objective collapse of the wave function due to gravity.

It is mentioned, in the pre-print article -- http://arxiv.org/abs/1311.1095 --
"We also stress that the time-dilation-induced decoherence is entirely within the framework of quantum mechanics and classical general relativity. This is in stark contrast to hypothetical models where gravity leads to spontaneous collapse of the wave function and that require a breakdown of unitarity [2, 3] ..."
[2] Penrose, R. On Gravity's role in Quantum State Reduction. Gen. Relat. Gravit. 28, 581-600 (1996).
[3] Diosi, L. Models for universal reduction of macroscopic quantum uctuations. Phys. Rev. A 40, 1165-1174 (1989)

 

[Swamp Thing]

The effect discussed here only operates when the object is in a superposition of different values of the internal energy. In this case, yes, the interference pattern will be degraded if the two interfering paths experience different amounts of time dilation. The amount of degradation depends on the amount of time dilation, on the number of internal degrees of freedom, and on the uncertainty of the internal energy.

Looking at it as an interferometer whose arms are subject to a differential time dilation,

(1) Is it possible that this process will periodically bring the object back through the initial state? At least in certain situations?

(2) If we let the arms of the interferometer cross over in the middle, so that paths A and B have the same overall "gravitational experience", will the decoherence go away? If so, this is perhaps a sort of "weak" decoherence removal setup, that is sometimes given the dubious distinction of quantum-eraserhood... what say?

 

[The_Duck]

(1) Is it possible that this process will periodically bring the object back through the initial state? At least in certain situations?

Sure. The time for this to happen is probably exponential in the number of internal degrees of freedom.

(2) If we let the arms of the interferometer cross over in the middle, so that paths A and B have the same overall "gravitational experience", will the decoherence go away? If so, this is perhaps a sort of "weak" decoherence removal setup, that is sometimes given the dubious distinction of quantum-eraserhood... what say?

Yes, this "decoherence" is easily reversible if you even out the time dilation between the two legs.

 

[Swamp Thing]

So from the viewpoint of MWI, "No new worlds were created in the making of this experiment" ...

 

[StevieTNZ]

So from the viewpoint of MWI, "No new worlds were created in the making of this experiment" ...

Did you read the email exchange between myself and Caslav? https://www.physicsforums.com/threa...ce-new-scientist-article.819441/#post-5144084 -- I would most certainly say new worlds would be created in the experiment proposed by the article.

Unless of course you are referring to a different experiment, which you might want to clarify.

 

[bhobba]

So from the viewpoint of MWI, "No new worlds were created in the making of this experiment" ...

Why would you think that?

In MW each part of a mixed state after decoherence is a separate world.

If gravity decoheres its no different to anything else doing it.

Thanks
Bill

 

[Swamp Thing]

OK, that was just an imprecise attempt to sum it up in a tag line

New worlds were created when the superposed entity reached the first fork in the interferometer -- agreed.
But are more worlds created with the gravity gradient, compared with the zero-g case ?

 

[bhobba]

But are more worlds created with the gravity gradient, compared with the zero-g case ?

They are created whenever decoherence occurs.

Thanks
Bill

 

[Swamp Thing]

From these replies (which seemed to confirm my own, naive and unreliable, take on it), I gathered that we have to qualify this decoherence with a pair of big quotes around it.

Yes, the effect is just a modification of the unitary evolution.

The effect discussed here only operates when the object is in a superposition of different values of the internal energy. In this case, yes, the interference pattern will be degraded if the two interfering paths experience different amounts of time dilation. The amount of degradation depends on the amount of time dilation, on the number of internal degrees of freedom, and on the uncertainty of the internal energy.

Yes, this "decoherence" is easily reversible if you even out the time dilation between the two legs.

Maybe it hinges on whether one agrees to call this "decoherence" in the same sense as entanglement with a complicated "ancilla", as opposed to a complicated but smooth/gradual smearing process. A weak electrical potential acting near one path through the apparatus would perhaps produce an analogous effect.

An alternative view could be that a thing with complex internal degrees of freedom can become its own ancilla, and the gravity gradient helps this process along.

 

[zonde]

The relative phases are different because of the different elapsed proper times. This is the effect pointed out by the article.

For a system with more internal degrees of freedom, these slight phase differences will eventually make the h1h_1 component internal state completely orthogonal to the h2h_2 component of the internal state. Once this happens the h1h_1 component can no longer interfere with the h2h_2 component. So we can say that the time dilation is producing decoherence.

It's strange to call orthogonal phase a decoherence. There is no interaction with environment and you can easily reverse the process (and I mean really easily, just let the effect last for twice the time and phase goes to opposite instead of orthogonal).

 

[bhobba]

Maybe it hinges on whether one agrees to call this "decoherence" in the same sense as entanglement with a complicated "ancilla", as opposed to a complicated but smooth/gradual smearing process. A weak electrical potential acting near one path through the apparatus would perhaps produce an analogous effect.

It transforms it to a mixed state - its decoherence.

Thanks
Bill

 

[jerromyjon]

The effect discussed here only operates when the object is in a superposition of different values of the internal energy

So this wouldn't affect photons because they only have a fixed internal energy?

 

[Khashishi]

I feel as if we (humans) are close to the point where we can say we finally understand QM.

 

[The_Duck]

It's strange to call orthogonal phase a decoherence. There is no interaction with environment and you can easily reverse the process (and I mean really easily, just let the effect last for twice the time and phase goes to opposite instead of orthogonal).

Yes, for the simple example I gave above, you can just wait longer and the "decoherence" is reversed. But if the internal state is more complicated the decoherence will not spontaneously reverse unless you wait an exponentially long time. (Though as I said above you can reverse the decoherence by evening out the relative time dilation between the two components of the wave function.)

In the case where there are many internal degrees of freedom, I think it really is quite similar to the usual model of decoherence due to interaction between system and environment. Here we can think of the position degree of freedom as the "system" and the internal state as the "environment."

So this wouldn't affect photons because they only have a fixed internal energy?

Right, a double-slit experiment with photons will still work fine even if time dilation is significant.

 

[zonde]

But if the internal state is more complicated the decoherence will not spontaneously reverse unless you wait an exponentially long time.

Complex phase can't be more or less complicated. It is as complicated as complex number can be - either real or imaginary or something in between.

 

[The_Duck]

Complex phase can't be more or less complicated. It is as complicated as complex number can be - either real or imaginary or something in between.

Here's what I mean. In my toy example I considered a very simple internal state
$$|\psi(0)\rangle \frac{1}{\sqrt 2} (|E_1\rangle + |E_2\rangle)$$
Suppose the internal state is more complicated:
$$|\psi(0)\rangle = \sum_i^N c_i |E_i\rangle$$
where the $|E_i\rangle$'s are orthonormal energy eigenstates satisfying $\langle E_i | E_j \rangle = \delta_{ij}$ and proper normalization requires $\sum_i^N |c_i|^2 = 1$.
After a time $t$ this state evolves to
$$|\psi(t)\rangle = \sum_i^N c_i e^{-iE_i t/\hbar} |E_i\rangle$$
Consider the inner product between $|\psi(t_1)\rangle$ and $|\psi(t_2)\rangle$. That is, we ask how much overlap there will be between two branches of the wave function that have experienced different amounts of proper time:
$$\langle\psi(t_1)|\psi(t_2)\rangle = \left(\sum_i^N c_i^* e^{+iE_i t_1/\hbar}\langle E_i|\right) \left(\sum_j^N c_j e^{-iE_j t_2/\hbar} |E_j\rangle \right)$$
$$= \sum_i^N \sum_j^N c_i^* c_j e^{i(E_i t_1-E_j t_2)/\hbar} \langle E_i | E_j \rangle$$
$$= \sum_i^N \sum_j^N c_i^* c_j e^{i(E_i t_1-E_j t_2)/\hbar} \delta_{ij}$$
$$= \sum_i^N |c_i|^2 e^{iE_i(t_1-t_2)/\hbar}$$
For $t_1 = t_2$ (no relative time dilation) this of course gives 1. But if the time difference $t_1 - t_2$ is large enough, the complex phases $e^{i E_i(t_1 - t_2)/\hbar}$ will be essentially random. They will mostly cancel out, and $\langle\psi(t_1)|\psi(t_2)\rangle$ will typically be of order $1/\sqrt{N}$. For large enough $N$ this means that $|\psi(t_1)\rangle$ and $|\psi(t_2)\rangle$ will be essentially orthogonal. If you wait long enough or get very lucky, the phases may miraculously realign and $\langle\psi(t_1)|\psi(t_2)\rangle$ may again become of order one. But the time for this to happen is exponential in $N$ because it requires $N$ phases to align at the same time.

 

[zonde]

That is, we ask how much overlap there will be between two branches of the wave function that have experienced different amounts of proper time:

I can't properly quote your equation. Anyways your math seems incorrect, you have lost crossterms.

 

[The_Duck]

I can't properly quote your equation. Anyways your math seems incorrect, you have lost crossterms.

In my post the $|E_i\rangle$'s are supposed to be orthonormal energy eigenstates, so that $\langle E_i | E_j \rangle = \delta_{ij}$. The cross terms vanish. I edited my post to state this explicitly.

 

[Swamp Thing]

(1) The decoherence effect depends on the idea that different internal degrees of freedom are associated with different trajectories (some "above" and some "below" the center of mass). Now if there are rotational degrees of freedom, would they tend to even out the relative time dilation, so that the proper times of the vibrational DOFs would no longer differ significantly?

(2) If the experiment uses a particle beam of some sort, they would be falling freely along a geodesic -- so no differential time dilation and no decoherence --- is this true? But if we were using optical tweezers / traps etc. then maybe it would be different ... ?

 

[zonde]

Consider the inner product between $|\psi(t_1)\rangle$ and $|\psi(t_2)\rangle$.

I do not recognize inner product between two components of the same wavefunction as meaningful operation. Isn't it sum (and then square result) when it is two components of the same wavefunction? We take projection of the state when performing measurement but then we use operator. But as I see you project one part of the state onto other part.

And from physical perspective I don't understand it either. When we perform interference experiment (double slit experiment or Mach-Zehnder interferometer) different path length change the time of evolution (as I see) for component of wavefunction that is traveling along particular path. So we get different relative phase between different paths.
How is it different in this case? Why effect of longer/shorter evolution time (because of gravity) for two components of coherent state should be any different than longer/shorter path in interference experiment?

 

[jerromyjon]

they would be falling freely along a geodesic -- so no differential time dilation

Falling freely along a geodesic implies you are approaching the source of the gravity, in which its force will increase as you approach it, and the dilation is relative to distance from the source, not the acceleration it does or does not cause. Falling along a geodesic would not be a constant velocity.

 

[fzero]

I do not recognize inner product between two components of the same wavefunction as meaningful operation. Isn't it sum (and then square result) when it is two components of the same wavefunction? We take projection of the state when performing measurement but then we use operator. But as I see you project one part of the state onto other part.

And from physical perspective I don't understand it either. When we perform interference experiment (double slit experiment or Mach-Zehnder interferometer) different path length change the time of evolution (as I see) for component of wavefunction that is traveling along particular path. So we get different relative phase between different paths.
How is it different in this case? Why effect of longer/shorter evolution time (because of gravity) for two components of coherent state should be any different than longer/shorter path in interference experiment?

Here it is clear that

$$ | \psi(t_2) \rangle = e^{-i H(t_2-t_1)} | \psi(t_1)\rangle, $$

so

$$\langle \psi(t_1) | \psi(t_2) \rangle = \langle \psi(t_1) | e^{-i H(t_2-t_1)}| \psi(t_1) \rangle $$

is perfectly well defined. You can even see explicitly this operator if you translate back from the energy eigenvalues in the energy basis that The_Duck used. The expectation value we are computing is a diagonal element of what might be called the transition or scattering matrix, where given a state $|i\rangle$ in the past, we want to compute the amplitude that we find the system in the state $|f\rangle$ in the future.

 

[Swamp Thing]

BTW, Scientific American has an article too: http://www.scientificamerican.com/article/gravity-kills-schroedinger-s-cat/

Meanwhile, back on earth...

If the experiment uses a particle beam of some sort, they would be falling freely along a geodesic -- so no differential time dilation and no decoherence --- is this true? But if we were using optical tweezers / traps etc. then maybe it would be different ... ?

Falling freely along a geodesic implies you are approaching the source of the gravity, in which its force will increase as you approach it, and the dilation is relative to distance from the source, not the acceleration it does or does not cause. Falling along a geodesic would not be a constant velocity.

I may have got this wrong, but I would appreciate help finding the flaw in this logic:

1. A reference frame moving along a geodesic is an inertial frame.
2. A molecule beam follows a parabola in space, and a geodesic in space-time
3. So this molecule's proper reference frame is inertial.
4. If the molecule is dumbbell shaped (say), both parts move in the same inertial frame
5. Hence both parts share the same proper time and their degrees of freedom evolve together

 

[jerromyjon]

I may have got this wrong, but I would appreciate help finding the flaw in this logic:

In general relativity, in any region small enough for the curvature of spacetime to be negligible, one can find a set of inertial frames that approximately describe that region.

I'm pretty sure this is not accurate enough to describe this system, but I am not advanced enough understand this completely. Perhaps someone could provide us both with more details?

 

[zonde]

The expectation value we are computing is a diagonal element of what might be called the transition or scattering matrix, where given a state $|i\rangle$ in the past, we want to compute the amplitude that we find the system in the state $|f\rangle$ in the future.

I looked up scattering matrix and sure operation looks similar. But as you say the two states are past state and future state. But The_Duck applied that operation to two components of past state (future state should be decohered state). So if operation would be carried out using scattering matrix then we would have to have coherent two component past wavefunction (unitary?) evolving into incoherent future wavefunction.

 

[zonde]

I may have got this wrong, but I would appreciate help finding the flaw in this logic:

1. A reference frame moving along a geodesic is an inertial frame.
2. A molecule beam follows a parabola in space, and a geodesic in space-time
3. So this molecule's proper reference frame is inertial.
4. If the molecule is dumbbell shaped (say), both parts move in the same inertial frame
5. Hence both parts share the same proper time and their degrees of freedom evolve together

Can't see flaw in this.

I'm pretty sure this is not accurate enough to describe this system, but I am not advanced enough understand this completely. Perhaps someone could provide us both with more details?

I can give you an argument from the other side. Using Rindler coordinates you can verify that you can get differential time dilation even in flat spacetime region by means of acceleration.
Maybe this can convince you that the opposite is right too (getting rid of differential time dilation in free fall).

 

[fzero]

I looked up scattering matrix and sure operation looks similar. But as you say the two states are past state and future state. But The_Duck applied that operation to two components of past state (future state should be decohered state). So if operation would be carried out using scattering matrix then we would have to have coherent two component past wavefunction (unitary?) evolving into incoherent future wavefunction.

Once the Hamiltonian is specified, and it's eigenstates have been found, then they can be used to write down the states of the system at any time, pure or mixed.

Suppose we prepare the system to be in the state $|\psi_1\rangle$ at time $t_1$, then $|\psi(t_1)\rangle = |\psi_1\rangle$. Physically $|\langle \psi(t_1)|\psi(t_2)\rangle |^2$ is the probability that the system is found in the state $|\psi_1\rangle$ at time $t_2$. This is a perfectly meaningful computation. As The_Duck argued, this probability decreases with time.

It's also just toy model, so it's not going to be exactly the calculation that you might want to see done. Also decoherence is something that is supposed to be derived for the original model, not assumed.

 

[zonde]

Once the Hamiltonian is specified, and it's eigenstates have been found, then they can be used to write down the states of the system at any time, pure or mixed.

Suppose we prepare the system to be in the state $|\psi_1\rangle$ at time $t_1$, then $|\psi(t_1)\rangle = |\psi_1\rangle$. Physically $|\langle \psi(t_1)|\psi(t_2)\rangle |^2$ is the probability that the system is found in the state $|\psi_1\rangle$ at time $t_2$. This is a perfectly meaningful computation. As The_Duck argued, this probability decreases with time.

It's also just toy model, so it's not going to be exactly the calculation that you might want to see done. Also decoherence is something that is supposed to be derived for the original model, not assumed.

Thanks for bearing with me.

So as I understand it, complex phase is specified for each Hamiltonian (energy?) eigenstate. So if we have many energy eigenstates they overlap with different complex phases at different times, right?
And this is similar to white light as opposed to monochromatic light. White light has many frequencies and we can't observe interference. But then again we say that white light is not coherent right at the start.

 

[jerromyjon]

time dilation even in flat spacetime region by means of acceleration.

I thought acceleration doesn't cause time dilation?

White light has many frequencies and we can't observe interference.

Skip to 3:32 and you'll see it...