# Macro superpostions of the metric

Now I would like to apply the reasoning of the last post to the case of verifying macroscopic superpositions of the metric.  It’s been 4 years since I’ve touched GR, so I’m going to rely heavily on E&M concepts and pray I don’t miss any key changes in the translation to gravity.

In the two-slit experiment with light, we don’t take the visibility of interference fringes as evidence of quantum mechanics when there are many photons.  This is because the observations are compatible with a classical field description. We could interfere gravitational waves in a two-slit set up, and this would also have a purely classical explanation.

But in this post I’m not concentrating on evidence for pure quantum mechanics (i.e. a Bell-like argument grounded in locality), or evidence of the discrete nature of gravitons. Rather, I am interested in superpositions of two macroscopically distinct states of the metric as might be produced by a superposition of a large mass in two widely-separated positions.  Now, we can only call a quantum state a (proper) superposition by first identifying a preferred basis that it can be a superposition with respect to.  For now, I will wave my hands and say that the preferred states of the metric are just those metric states produced by the preferred states of matter, where the preferred states of matter are wavepackets of macroscopic amounts of mass localized in phase space (e.g. L/R).  Likewise, the conjugate basis states (e.g. L+R/L-R) are proper superpositions in the preferred basis, and these two bases do not commute.

There are two very distinct ways to produce a superposition with different states of the metric: (1) a coherent superposition of just gravitational radiation Note that we expect to produce this superposition by moving a macroscopic amount of matter into a superposition of two distinct position or momentum states.  Either way, we must bring the matter back into a null configuration coherently.  “Coherently” means the matter must disentangle from the metric (which remains in a superposition) and return to an eigenstate in the preferred basis without producing any other effects (e.g. radiating photons which are correlated with the state of the matter).  If we do not bring the matter back to a null, unentangled state, then we can never hope to directly confirm that the metric itself is in a superposition.  If the matter “knows” the state of the metric, then we can’t see interference.a   or (2) a joint mass-metric entangled state (with no radiation).

Case (1) would look like

(1)

Here, the first evolution produces the matter M superposition, the second evolution is the response of the gravitational field (metric) to the new configuration, and the third evolution is the return of the matter to a null state not entangled with the metric.  Case (2) would stop at the second to last line.

As far as I can tell, it is impossible to create a coherent, unentangled superpositions of static metric configurations for the same reason that, in E&M, one cannot create a superposition of two static configurations of the electric field: different static field configurations will require different configurations of the sources charges (whether mass or electric charge), which will necessarily be entangled with the field There could be a mistake lurking in the semi-classical limit here. Also, could geons would be a counterexample?b  .  The density matrix of the field alone is then diagonal in the preferred basis, i.e. not a local superposition.

Now remember that in order to verify a superposition, it is necessary to make measurements in two non-commuting bases. In both cases (1) and (2), there will always be one basis (the preferred basis) in which it is simple to measure.  Detectors of macroscopic gravitational waves (LIGO, etc.) are able to distinguish between two waves which have different macroscopic properties (e.g. their arrival times, their phase, or their frequency).  Likewise, torsion balances or test masses can be used to distinguish between two macroscopically distinct gravitational fields. But in both these cases, it is not clear how you could ever directly measuring in the conjugate basis. One guess is to probe them with entangled states (i.e. with a test-mass in a superposition), but this can be seen to not work so long as the test mass lives up to its name, i.e. so long as it does not strongly change the macroscopic field configuration.

Here is my intuition for why this is generally so difficult, specializing to case (1).  Two macroscopically distinct gravitational wave states contain an immense number of gravitons, so we cannot guarantee that the hypothetical probe (which is attempting to measure in the conjugate basis) does not minutely change the macroscopic field state (say, by scattering a single graviton). But we do expect the two (possibly mixed) out-states of the field (corresponding to the two preferred basis in-states) to still be orthogonal with each other.  That is, we expect the probe interacting with one macroscopic field configuration to leave the field in a state orthogonal to the situation when it interacts with a different macroscopic field configuration.  Indeed, the only way for this to not be so is if the single bit recorded very redundantly in the macroscopic state (i.e. L or R) were to be completely transferred to the state of the probe. (The necessity of this condition follows just from the linearity of quantum mechanics.)  And if the field out-states are still orthogonal, then one can check that no evidence of an initial coherent superposition of the field in-states can possibly be inferred from the probe.

I know of only one way to get around this: use the unitary self-evolution of the macroscopic system to take a state in the conjugate basis to a state in the preferred basis, as described in the previous post.  This is the solution used in interferometers superposing mesoscopic masses like the OTIMA interferometer, which is predicted to push amu.    I think the feasibility of this technique is tied to the way the particles in an molecule are bound together so that the center-of-mass motion decouples.  (A single photon in a two-slit experiment is just one particle, and is trivially bound together with itself.)  I do not expect the same to be possible for macroscopic waves of gravitons or photons, but I can’t rule it out. Consider the analogous case in E&M, e.g. a superpositions of a laser beam in (say) two anti-parallel directions Note that linearity of the E&M field does not render superpositions of two field configurations (whether static or radiative) trivial.  The zero field configuration is a completely different state than a superposition of having the electric field point in the +z and -z direction.  In the the zero-field state, an electron whose wavefunction is initially a Gaussian wavepacket will be unaffected and remain in a product state.  On the other hand, for the z/-z superposition, the electron will become entangled with the field (with a left-moving/right-moving wavepacket correlated with the respective state of the field, effectively measuring it.)  In other words, the linearity of E&M is with respect to the operation of addition on E&M field vectors, not on QM state vectors.c  . I am not aware of any experiment which has attempted superposed lasers, and presumably it would be much easier than with gravity  One is tempted to consider just superpositions of single gravitons, like the commonly achieved superpositions of photons, but this is in no way macroscopic and is not the subject of this post.d  .

On the other hand, it is sometimes possible to use a probe to evolve two distinct macroscopic in-states to the same out-state, transferring this bit of information into the (microscopic) probe.  This is exemplified by the quantum resonator achieved by Cleland’s group in 2010 and by the optomechanical oscillator proposal being pursued by Bouwmeester’s group.  The idea is to create your macroscopic superposition using a coupling to a single quanta (e.g. a photon in an interferometer) in such a way that the recurrence time of the whole set-up is less than the decoherence time of the macroscopic system.  An initial superpostion in the microscopic system is transferred into the macroscopic system and then–after some revival time–the superposition is transferred back.  While the macroscopic system is in a superposition, it is an example of case (2), an entangled matter-metric state Here, slowness is key.  If the matter radiates any gravitons through gravitational bremsstrahlung (or photons through normal bremsstrahlung), then the matter-metric superposition will decohere.e  .

Take home message: Decoherence is not the only thing that makes Schrödinger’s cat so rarely seen.  In addition to producing the superposition, one must also measure the cat in the conjugate basis (L+R/L-R) to confirm, which is very difficult.  One (“revival”) strategy is to first produce the macroscopic superposition by amplifying a superposed microscopic system and then coherently evolving it back.  Afterwards, the superposition is again represented in a microscopic quantum system, which can be easily measured in two non-commuting bases. I think this is the precise meaning of hand-waving statements like “macroscopic superpositions are hard because of thermodynamics/irreversibility”. The other strategy is to (somehow) effect unitary self-evolution of the cat which takes preferred states to conjugate states, as in the two-slit experiment.  I do not yet understand under what general conditions this is feasible.

Open questions: Are there any other examples of unitaries which evolve preferred basis states to conjugate basis states?  Do all experiments which seek to see macroscopic superpositons either (a) use a center-of-mass variable a la two-slit or (b) a revival technique a la Bouwmeester?

[This post was prompted by conversation with Max Tegmark.]

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### Footnotes

(↵ returns to text)

1. Note that we expect to produce this superposition by moving a macroscopic amount of matter into a superposition of two distinct position or momentum states.  Either way, we must bring the matter back into a null configuration coherently.  “Coherently” means the matter must disentangle from the metric (which remains in a superposition) and return to an eigenstate in the preferred basis without producing any other effects (e.g. radiating photons which are correlated with the state of the matter).  If we do not bring the matter back to a null, unentangled state, then we can never hope to directly confirm that the metric itself is in a superposition.  If the matter “knows” the state of the metric, then we can’t see interference.
2. There could be a mistake lurking in the semi-classical limit here. Also, could geons would be a counterexample?
3. Note that linearity of the E&M field does not render superpositions of two field configurations (whether static or radiative) trivial.  The zero field configuration is a completely different state than a superposition of having the electric field point in the +z and -z direction.  In the the zero-field state, an electron whose wavefunction is initially a Gaussian wavepacket will be unaffected and remain in a product state.  On the other hand, for the z/-z superposition, the electron will become entangled with the field (with a left-moving/right-moving wavepacket correlated with the respective state of the field, effectively measuring it.)  In other words, the linearity of E&M is with respect to the operation of addition on E&M field vectors, not on QM state vectors.
4.  One is tempted to consider just superpositions of single gravitons, like the commonly achieved superpositions of photons, but this is in no way macroscopic and is not the subject of this post.
5. Here, slowness is key.  If the matter radiates any gravitons through gravitational bremsstrahlung (or photons through normal bremsstrahlung), then the matter-metric superposition will decohere.