# An experimental proposal to study spontaneous collapse of the wave function using two travelling-wave parametric amplifiers

### Submission summary

 As Contributors: Xavier Le Large · Louk Rademaker · Tom van der Reep Arxiv Link: https://arxiv.org/abs/1811.01698v3 Date submitted: 2019-06-28 Submitted by: van der Reep, Tom Submitted to: SciPost Physics Discipline: Physics Subject area: Quantum Physics Approaches: Theoretical, Computational

### Abstract

According to one of the postulates of the Copenhagen interpretation of quantum mechanics, a measurement causes a wave function to collapse onto an eigenstate of the measurement apparatus. To study whether such spontaneous collapses exist in an electronic amplifier, we propose an experiment consisting of a microwave interferometer that has a parametric amplifier added to each of its arms. Feeding the interferometer with single photons, we entangle the output of the amplifiers. We calculate the interference visibility as given by standard quantum mechanics as a function of gain, insertion loss and temperature and find a magnitude of $1/3$ in the limit of large gain without taking into account losses. This number reduces to $0.26$ in case the insertion loss of the amplifiers is $2.2$ dB at a temperature of $50$ mK. We argue, based on Born's rule that if the process of spontaneous collapses exists, we will measure a reduced visibility compared to the prediction from standard quantum mechanics once this collapse process sets in.

###### Current status:
Editor-in-charge assigned

Dear Editor,
Thank you for the referee reports.

We thank the referees for their detailed reading of our manuscript and for their comments. Below we address their comments one by one and we indicate the changes to our manuscript in order to clarify the points they have raised.

Yours sincerely,
Tom van der Reep and Tjerk Oosterkamp

### List of changes

Report 1
1 - While I neither verified the calculations explicitly nor have any specific reason to doubt them, I thought the paper would be greatly strengthened if some physical arguments were given to explain some of the interesting results.
(a) Some discussion of the strength of the correlations between photon number after the amplifier and before would have been welcome, and presumably one could proceed from this "distinguishability" to explain why the visibility reaches 1/3 in the high-gain regime.

TvdR: We have added such discussion in section 3, starting with the sentence 'Therefore, the visibility in the lossless case...'.
To add some intuition to the calculation we have added a figure to appendix D, to emphasize the point that a superposition of zero and one photon before an amplifier with gain G does not result in a superposition of zero and G photons after the amplifier. The visibility results from a calculation involving multiphoton interference involving many Fock states, which is not so straightforward.

(b) Although it is easier to see, the authors might have helped the readers' intuition by explaining why the signal visibility goes to 100% at low gain while the idler visibility goes to 0.

TvdR: For low gain the signal visibility goes to 1, as this situation resembles a 'normal' one photon interference experiment. The idler visibility goes to 0, due to a lack of idler photons. As a matter of fact, without losses the idler visibility is undefined. We have added a sentence ‘The low gain visibility of 1 for the signal.

(c) Discussion of the fact that even the idler visibility goes to 1/3 in the high-gain regime would help the reader develop a physical sense of the transition to classical-like correlations in this limit.

TvdR: Here the same mechanism is at work as in situation 1(a), discussed above. In the additional discussion in Sec. 3 we also mentioned the idler.

(d) In the discussion of loss as well, some arguments other than numerical could be provided to suggest how much loss would be necessary to decohere the states resulting from |0> and |1> as input. Most likely a consideration of <n_out> and \Delta n for the two cases, and what fraction of the output would need to be siphoned off in order to distinguish them, would explain this.

TvdR: We have added a paragraph in section 4 with analytical simplifications of the numerical interference visibility and discussed the limits of low temperature and low losses.

2 - The above issues are closely related to the earlier work prompted by de Martini et al about "micro / macro" entanglement using optical parametric amplifiers, where it was also observed that the coherence of the allegedly Schrödinger-cat-like state was surprisingly insensitive to loss. That prompted work by Gisin, Zeilinger, Simon, and others, which should really be discussed and cited in the present paper.

TvdR: We have added the references along with a short discussion. An important difference is that in a microwave version of those experiments losses are expected to be significantly higher in our case. It is the price we pay for

3 - The independence from loss in the high-gain limit suggests that one is essentially observing interference of classically correlated beams, rather than a true nonlocal quantum superposition. This might accord with some intuitions about high-gain OPAs (and the fact mentioned earlier that even the idlers exhibit good interference in this limit). I would have liked to see some discussion of whether or not the authors expected collapse into coherent states to degrade the visibility (or whether it would, in the high-gain limit, simply "confirm" and lock in the pre-existing phase correlations).

TvdR: Coming to think of it, indeed one might expect that a collapse to coherent states locks into the pre-existing phase correlations and ergo that the visibility does not change at all: We treat the coherent states as a point in phase space, whence our calculation might look as if we are taking the phase space of psi3 and time-develop each point separately through the second hybrid. Surprisingly, then, the visibilities are different, which is due to the addition of one photon in our collapse mechanism. In our manuscript we added some commenting lines on this matter.

3b - The dependence of visiblity on location of collapse is very interesting. But it requires some discussion, connected to point 3 above. Specifically: the fact that 1/3 visibility is preserved if the collapse occurs after the output of the high-gain amplifier is consistent with the idea that the phases are already correlated, and making those correlations "classical" changes nothing. But once one has these classically correlated phases, inserting two additional amplifiers should not, so far as I can tell, destroy the interference. But what is the difference between this situation and the eta=1/2 case (collapse halfway through the amplifiers), which the authors claim leads to 0 visibility? I worry that there is some error here -- if there is not, then it's certainly a point which merits some actual discussion in the text.

TvdR: Yes, there was an error and it has been fixed. As a result the corresponding figure changes: The visibility for eta=1 does not change. For eta=1/2 the visibility goes to approximately 0.15 in the high-gain limit. For eta=0 the visibility goes to 0.2 in the high-gain limit (see also next remark). Indeed the signal phases are correlated, see our result for |c_coll|^2 - it is more likely for the collapse to end up in a state with \phi_ups-\phi_lows=pi/2 than -pi/2. However, such a phase correlation is absent for the idlers, which implies that further transmission through the TWPA changes the correlations of signal and idler again (NB alpha_s(i)=alpha_s(i)0\cosh\kappa+i\alpha*_i(s)0\sinh\kappa).

3c - It's perhaps even more surprising that if collapse at the midpoint destroys interference entirely, collapse before the amplifers does not. I would expect discussion of this point. I would also think that a simple model could be produced to explain the 20% result when the collapse occurs upon the |01>+|10> state before the amplifiers.

TvdR: first point, see 3b - this is an error. Using the reduced Hilbert space approach one can easily explain the reduction in interference visibility for the cases \eta=0 and 1. Our collapse mechanism adds one photon in each Hilbert subspace (signal and idler, upper and lower arm). From this added photon the reduction can be explained. We have extended the paragraph on this matter in the manuscript.

4 - At the very beginning, the text could be clearer about the experimental situation. For instance, signal, idler, and pump could be defined earlier, and it should be made clear that in figure 1 it is the signal which is initially split, but that both the signal and the idler are to be interfered.

TvdR: Thank you, this has been adapted.

5 - In the next-to-last paragraph of section III, the sentence that "feeding this TWPA with a |00> state gives the average number of signal photons" doesn't make much sense to me. (The previous sentence, while awkward, at least makes it clear that what is meant -- I think -- is that equation 3 yields the number of signal photons if |10> is fed in; but I don't know what is supposed to yield that number if |00> is fed in.)

TvdR: the main idea of the reduced Hilbert space is that our interference visibility results without and with loss can be obtained by considering a single TWPA. For calculating the visibilities, four numbers are required: <n_As>, <n_Bs>, <n_Ai> and <n_Bi>. Now, instead of considering the full interferometer, one can use the reduced Hilbert space approach: If a single TWPA is fed with a |1_s,0_i>-state, the number of output photons (signal and idler) of this single TWPA equals <n_Bs> and <n_Ai> respectively. Feeding the single TWPA with a |1_s,0_i>-state, the output numbers of photons of this single TWPA are found to be equal to <n_As> and <n_Bi> respectively. This implies that if one performs these two calculations (with single TWPAs), one may still extract the visibility as would result from considering the full interferometer. We have emphasized this point a little more in the manuscript.

6 - While loss and in particular heating-induced losses are treated, I would expect the authors to mention other parasitic effects that might mislead one into thinking collapse had been observed. Are there other sources of phase noise to worry about? Have they been characterized elsewhere?

TvdR: We have not considered other sources of noise. Before interpreting results from an actual experiment these sources of noise would need to be characterised and taken into account in the calculation. This remark has been added to the feasibility section.

7 - I would also have liked some discussion of the regime this experiment could probe, compared with past work on mesoscopic entanglement, including the work by Gisin et al. on "micro-macro entanglement," work in superconducting qubits, and work on atomic ensembles (perhaps others as well).

TvdR: We did a calculation, in which I estimated that for 40dB gain 2e5 electrons would partake in the superposition. Together these have a mass of 2e-25 kg and yield 1e5 degrees of freedom (assuming the transmission lines are 1D and the electrons move along the transmission line and are paired in Cooper pairs). Current record for atomic ensembles: Gerlich et al. Nat. Commun. 2:263, 7e-24 kg with 1000 degrees of freedom. We then would need to add the superposition distance which we can estimate as a photon wavelength ~3mm in common microwave lines.
In the introduction we emphasized that our proposal is different from experiments exploiting optical photons, such as those by Zeilinger, Gisin, De Martini and Rempe. Optical photons in non-linear crystals will not lead to a superposition of mass, since the electrons in the crystal are effectively still localized (or rather de-localised over an atomic distance of ~0.1nm. In transmission lines the materials valence electrons that carry the photonic excitation are delocalized over the photon wavelength in the order of millimetres.

Report
I think this is a very interesting proposal, and the authors have made a good case. I would like to see the paper published, but think it is almost essential that they supplement the calculations with some physical discussion of the effects and the limits they treat.
I think is indispensible that they cite -- and actually discuss -- the earlier work (& controversy) on entanglement generated by parametric amplification.
Understanding the position-dependence of the coherent-state collapse model is, to me, central to the point of this paper. Perhaps there is no error in their result about the 0% visibility at the midpoint -- but if this is so, then the fact that I had the misapprehension I did still convinces me that the discussion as it stands should be bolstered by physical argument.
I think it is also essential that the paper be clear about what regimes have been probed in other experiments and how this regime differs (at least qualitatively); and that more discussion is provided of what culprits the authors might proceed to search for if they did carry out the experiment and find lower visibility than in the idealized quantum model.

Report 2
1. A highly speculative and unjustified conjecture for spontaneous collapse.
2. No attempt is made to determine if this conjecture is consistent with the results of decades of previous experiments in quantum optics

Report
The authors consider a simple optical device based on a single-photon Mach-Zehnder (MZ) interferometer but with identical parametric amplifiers placed in each internal path. They then calculate the visibility for the signal and idler interference visibility (for zero phase shift) at the outputs. This calculation is straightforward (despite the complicated method the authors use) in quantum optics and does not warrant publication. It is trivial to show that the visibility for the idler is constant at 1/3 while that for the signal cosh^2(r)/(cosh^2(r)+2 sinh^2(r)) as the authors show by the red lines in Fig 2. Inclusion of loss is also quite straightforward.

TvdR: Yes, the calculation is straightforward, but we are not aware that it has actually been performed. Moreover, since our quantum optics colleagues could not tell us the result straightaway, we doubt whether the result is trivial, also since it involves multiphoton interference and because temperature is more important for microwave paramps than for optical paramps.

The substantive part of the paper is based on their own model of 'spontaneous collapse'. This is speculative and rather vague. The only justification they offer is a claim that measurement requires amplification and so amplification must lead to wave-function collapse.

TvdR: We agree with the referee that we do not know whether wavefunction collapse will actually will occur in our proposed experiment. That is why we write “if collapse occurs …” the interference visibility of our proposed interferometer diminished.

Despite this rather dubious reasoning, there is no evidence for it despite the countless experiments that have been done on optical parametric amplifiers in quantum optics over decades. These devices are well modelled by unitary transformations as used by the authors.

TvdR: Optical parametric amplifiers and microwave paramps work in a totally different regime of the EM-spectrum. In OPAs the electrons are still bound to their atoms, and therefore there superposition distance is small (approximately the size of their orbital radius). On the other hand, in microwave PAs the valence electrons (Cooper pairs) move freely through the material and the passage of a single photon will superpose them over a distance in the order of the photon wavelength (approximately 3mm). To stress this point we added a remark to the introduction citing work of Gisin et al..
We believe that because microwave paramp experiments have become technologically feasible and because they work in a different regime than optical paramps, they are worth doing.

Loss is also easily included and well understood. If one puts a paramp inside a Fabry-Perot cavity (an interferometer) one gets a parametric oscillator; a common quantum optical device well described by conventional theory.
Before I address the author’s own rather unique model of 'spontaneous collapse', let me first object to the use of this term. To most of us the term spontaneous collapse refers to a detailed proposal from Ghirardi, Rimini and Weber. As far as I can see this bears no relation to the author’s definition of the term.

TvdR: GRW (and CSL etc.) use spontaneous collapse for a differential equation that describes how a quantum state might evolve in a classical state in time by introducing (mostly) two parameters: the collapse rate and the width of the state after collapse. It is generally assumed that collapse takes place towards the position basis. Although we do not use the differential of GRW equation explicitly, the result of the differential equation describes a collapsed state. Our spontaneous collapse is just a short-cut for this equation, in which case - as remarked below - the spontaneous collapse is described by a projection operator.

The model of spontaneous collapse used in this paper inserts two kinds of projection operators at various points in the internal arms of the MZ. One projects onto number states and the other projects onto coherent states. Obviously this will change the visibility. One could justify this description in terms of actual measurements taking place in each arm of the MZ: non-absorption counting in the case of photon number and heterodyne detection in the case of coherent states. The authors don’t mention this so presumably they mean something else by spontaneous collapse. (At some point they do refer to ‘which path’ measurements although there is no such measurement included in the model).

TvdR: we do justify the description, only not in these terms: collapse to a number state could take place due to a which path detection of the state, or due to some other mechanism. Collapse to a coherent state might take place, because the electrons in the transmission lines collapse into position states (classical motion can always be described by Fourier components, i.e. sinusoids, i.e. coherent states which are the closest quantum analogue to sinusoids).

I see little merit in publishing a proposal for an experiment based on such speculative and, given previous experiments, highly unlikely conjectures.
A minor point: The first sentence in the abstract is wrong. There are many experiments that project onto non classical states. See for example the Haroche experiments in the microwave regime.

TvdR: We changed the first sentence of the abstract from “… a
measurement causes a wave function to collapse into a well-defined classical state” to “… a measurement causes a wave function to collapse onto an eigenstate of the measurement apparatus”.

The first sentence of the paper is not wrong but only describes a very small class of actual experiments. Again, examples can be found in the experiments of Haroche.

TvdR: As far as we know, Haroche only did experiments on environmental decoherence, which is not what we are after.

### Submission & Refereeing History

Resubmission 1811.01698v3 on 28 June 2019
Submission 1811.01698v2 on 14 November 2018

## Reports on this Submission

### Report

The paper presents an interesting idea regarding amplification using travelling wave parametric amplifiers (TWPA), a standard technology for readout in quantum superconducting information processing. The idea is that the amplified superposition might spontaneously collapse because of the number of particles involved in the amplifier. However, the idea is not worked out in a rigorous way that puts it in the context of previous research so I cannot recommend publication in this, or any other, journal.

The work seems to be presented as a type of spontaneous collapse models. These should be universal, or at least broad, theories, precisely formulated, with suggested parameters designed to prevent the formation of macroscopic superpositions. By contrast, the present proposals apply only to the context of TWPAs, and are unsatisfactory in other ways. The first (number collapse) is so gross in its effects as to be surely wrong, while the second (coherent state collapse) has an ad hoc parameter so that it would not be clear how to apply it even for the slightly more general case of two TWPAs in series. It is also not clear when the authors imagine the model applies. I see two possibilities:

1) The evolution is postulated to be a property of TWPAs. If so, this simply means that TWPAs do not work according to the standard model of TWPAs, but rather have extra decoherence. But if that is the case, then why is the standard model considered accurate? In fact the authors say of TWPAs (p. 2) that “their quantum behaviour is well-understood”. The deviations should be testable for input states other than the single photon. Indeed, the models would surely have consequences for the very many published experiments using TWPAs, even with the gain the authors suggest as they say. The authors seem to ignore this issue.

2) The evolution only applies to the single-photon input. This makes the proposal even more ad hoc. Presumably the authors have some idea that the spontaneous collapse would prevent macroscopic entanglement. But they do not even show that there is any macroscopic entanglement when the practical amounts of noise in these amplifiers is included anyway. If the authors want a “no macroscopic entanglement” theory or a “no macroscopic superposition” theory then they need to make the theory sensitive to the presence of these things. The existence of fringes does not imply entanglement! For instance, a HOM dip in quantum optics can be observed, with visibility up to 50%, can be obtained from classical (mixtures of coherent states) fields.

Given the above criticisms, it seems to me that a better direction for the authors to take their work would be that of testing the existence of macroscopic superpositions, which they touch on in Section 6. However I am a bit skeptical of how impressive the comparison with molecular interferometers is. The authors say that 2 times 10^{-25} is “similar” to 7 times 10^{-24}, but is more than an order of magnitude smaller. As for “degrees of freedom” if that is to be the basis for a more favourable comparison then it would need to be fleshed out more. There is a considerable literature on measures of macroscopicity that the authors would need to engage with. They say (p. 2) that they are investigating an unexplored “part of parameter space” but they don’t say what the parameters are.

### Requested changes

Finally, there are some small issues the authors should also address if they submit elsewhere.
i) The definition of gain below Eq.(3) does not look standard to me. It should be defined in terms of amplitudes, not intensities.
ii) The first, undidplayed, equation in Sec. 3 does not make sense. The arguments of P should be Fock numbers (integers), not mean values of Fock numbers.
iii) Even with loss as described in Sec. 4 the TWPA dynamics is linear with Gaussian noise. numerical simulation in Hilbert space is not necessary. There exist very efficient methods for propagating Gaussian states with such dynamics. Only a few parameters are needed.
iii) At the end of Sec. 5 the visibility figures of 0.15 and 0.2 is given. Are these exact or just approximations to the results of numerical simulations.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Strengths

The authors propose a novel model for spontaneous collapse in microwave quantum optical devices.

### Weaknesses

The substantive part of the paper is based on their own model 'spontaneous collapse'. This is speculative and rather vague.

### Report

I still think the calculation is trivial but, as this is a subjective judgement, I accept the authors response.

The authors respond that “Optical parametric amplifiers and microwave paramps work in a totally different regime of the EM-spectrum.” This is obviously true but irrelevant as the theoretical description is the same. In the case of super conducting circuits the quantum theory is given as an ‘effective theory’ in which particular macroscopic degrees of freedom (circuit variables) are quantised directly. This works well in typical experimental conditions because the underlying microscopic degrees of freedom largely factor-out only remaining as a possible source of decoherence or noise. This is an old argument that goes back to the seminal work of Leggett. Exactly the same approach is adopted in the quantisation of the non-linear polarizability of optically-active crystals and it works well there for the same reason. I simply do not understand the author’s comments about localised versus unlocalised electrons. The polarisation is a non-local effective degree of freedom. I thus disagree with the sentence in the new version of the paper that, “In optical non-linear crystals, the photons typically interact with localised electrons”: one can just as well claim that the photons interact with a continuously distributed polarisation field in the medium. I would strongly advise the authors to reconsider these claims.
In their response the authors say that, “ On the other hand, in microwave PAs the valence electrons (Cooper pairs) move freely through the material and the passage of a single photon will superpose them over a distance in the order of the photon wavelength (approximately 3mm)”. This suggests to me that they have in mind a different theoretical treatment to the usual one used in circuit QED, one that explicitly uses the microscopic theory of microwave electronics. I admit that such an approach might justify their phenomenological model but I would like to see how it is done. In any case I give the authors the benefit of the doubt. In a similar context the authors claim that “Collapse to a coherent state might take place, because the electrons in the transmission lines collapse into position states (classical motion can always be described by Fourier components, i.e. sinusoids, i.e. coherent states which are the closest quantum analogue to sinusoids).” I can’t make any sense of this. Electrons in transmission lines are effective field degrees of freedom.

In summary, I remain unconvinced by the author’s responses. But others may take a different view so I am prepared to recommend publication so that debate can take place.

• validity: low
• significance: ok
• originality: good
• clarity: low
• formatting: excellent
• grammar: excellent

Tom van der Reep on 2019-10-23
Category:
remark

Point by point comments to the referees:

Anonymous Report 1 on 2019-9-17 Invited Report

Strengths The authors propose a novel model for spontaneous collapse in microwave quantum optical devices.

Weaknesses The substantive part of the paper is based on their own model 'spontaneous collapse'. This is speculative and rather vague.

Report I still think the calculation is trivial but, as this is a subjective judgement, I accept the authors response. - OK

The authors respond that “Optical parametric amplifiers and microwave paramps work in a totally different regime of the EM-spectrum.” This is obviously true but irrelevant as the theoretical description is the same. In the case of super conducting circuits the quantum theory is given as an ‘effective theory’ in which particular macroscopic degrees of freedom (circuit variables) are quantised directly. This works well in typical experimental conditions because the underlying microscopic degrees of freedom largely factor-out only remaining as a possible source of decoherence or noise. This is an old argument that goes back to the seminal work of Leggett. Exactly the same approach is adopted in the quantisation of the non-linear polarizability of optically-active crystals and it works well there for the same reason. - We agree with the referee that in principle the effective theoretical approach we use can be applied to photonic systems. Since our aim is to provide a baseline theoretical expectation for our proposed microwave experiment, we tailored the parameters of our theory to such a microwave set-up. Note also that earlier optical experiments (see Ref. [2-5]) did not envisage testing a possible loss of entanglement due to the parametric amplification. As we will continue performing the microwave experiment, we hope publishing this manuscript will stimulate others to do a similar optical experiment. We will therefore change the text such that it is clear that our ideas can be studied in optical set-ups as well. Finally, we are also glad that the Referee recommends publication so that debate can take place.

I simply do not understand the author’s comments about localised versus unlocalised electrons. The polarisation is a non-local effective degree of freedom. I thus disagree with the sentence in the new version of the paper that, “In optical non-linear crystals, the photons typically interact with localised electrons”: one can just as well claim that the photons interact with a continuously distributed polarisation field in the medium. I would strongly advise the authors to reconsider these claims. - What we want to point out with this claim is the following: in the non-linear polarisability the electrons are still bound to the nuclei, implying they would be in superposition of approximately a Bohr radius when a photon passes by. Since the Bohr radius of 5x10-11 m is very small, we referred to this situation as a situation in which the electrons remain more or less localised. Contrarily, microwave photons would cause the Cooper pairs to be in a superposition over the full wavelength of the photon, which for microwaves is in the order of millimetres. With our claim of localised vs. non-localised electrons we want to point out the enormous difference in the superposition distances between optical and microwave photons. So yes, in principle the referee is correct in his statement and we might rephrase this statement.

In their response the authors say that, “ On the other hand, in microwave PAs the valence electrons (Cooper pairs) move freely through the material and the passage of a single photon will superpose them over a distance in the order of the photon wavelength (approximately 3mm)”. This suggests to me that they have in mind a different theoretical treatment to the usual one used in circuit QED, one that explicitly uses the microscopic theory of microwave electronics. I admit that such an approach might justify their phenomenological model but I would like to see how it is done. In any case I give the authors the benefit of the doubt. - We do not have such a treatment. We would very much welcome it if someone would provide us with such a treatment, since it would enable us to make a better prediction of the interferometer output within the framework of quantum mechanics. However, we know that the ordinary treatment of parametric amplifiers works generally and hence we use it. On the other hand we realise that in reality the massive electrons do play a role in the amplifiers, which we do not take into account explicitly.

In a similar context the authors claim that “Collapse to a coherent state might take place, because the electrons in the transmission lines collapse into position states (classical motion can always be described by Fourier components, i.e. sinusoids, i.e. coherent states which are the closest quantum analogue to sinusoids).” I can’t make any sense of this. Electrons in transmission lines are effective field degrees of freedom. In summary, I remain unconvinced by the author’s responses. But others may take a different view so I am prepared to recommend publication so that debate can take place. - Thank you

Anonymous Report 2 on 2019-10-10 Invited Report Report The paper presents an interesting idea regarding amplification using travelling wave parametric amplifiers (TWPA), a standard technology for readout in quantum superconducting information processing. The idea is that the amplified superposition might spontaneously collapse because of the number of particles involved in the amplifier. However, the idea is not worked out in a rigorous way that puts it in the context of previous research so I cannot recommend publication in this, or any other, journal. The work seems to be presented as a type of spontaneous collapse models. These should be universal, or at least broad, theories, precisely formulated, with suggested parameters designed to prevent the formation of macroscopic superpositions. By contrast, the present proposals apply only to the context of TWPAs, - This is not true. We have only applied it in the context of TWPAs, but in principle it can be applied to any photonic system straight away.

and are unsatisfactory in other ways. The first (number collapse) is so gross in its effects as to be surely wrong, - This statement we absolutely cannot agree with. Firstly, it cannot be that a calculation is "wrong", simply because its effects are too gross. We furthermore would like to point out that this situation is exactly the situation in which a photon is brought into superposition by a beam splitter and subsequently detected. In this case no one is surprised that there is no interference anymore.

while the second (coherent state collapse) has an ad hoc parameter so that it would not be clear how to apply it even for the slightly more general case of two TWPAs in series. - This comment does not make sense to us. A TWPA is a spatially extended object and can be regarded as a series of TWPAs itself. Suppose we have a TWPA with non-linear mixing constant \chi, then its amplification is determined by \chi*t, where t is the time a quantum state spends in the TWPA (one can make an analogous case, where t is replaced by the length of the TWPA). Hence, the only effect of placing two identical TWPAs in series is that t->2t such that the amplification doubles. For un-identical TWPAs the same argument holds, namely that it still can be described as a single TWPA with some non-linear mixing constant \chi that is time-dependent.

It is also not clear when the authors imagine the model applies. I see two possibilities: 1) The evolution is postulated to be a property of TWPAs. If so, this simply means that TWPAs do not work according to the standard model of TWPAs, but rather have extra decoherence. But if that is the case, then why is the standard model considered accurate? In fact the authors say of TWPAs (p. 2) that “their quantum behaviour is well-understood”. The deviations should be testable for input states other than the single photon. Indeed, the models would surely have consequences for the very many published experiments using TWPAs, even with the gain the authors suggest as they say. The authors seem to ignore this issue. - In the field of microwave optics there are not so many TWPAs realised, as the referee seems to suggest. There are two branches of research, one using Josephson junctions as a basic non-linear element and one using the non-linear kinetic inductance for this purpose. They are used for reading out qubit experiments, in which they are mostly used to amplify measurement pulses composed of coherent (classical) states. Indeed their quantum behaviour is well-understood theoretically. However, we would like to point out that in none of the microwave experiments the TWPA is used as a quantum device, such as in our proposed experiment. Rather, it is used as a low-noise amplifier in the output lines of experiments, in which one does not care anymore whether the state has collapsed or not. This however, we consider as out-of-scope for the present work. We do remark on a consequence for measurements though, namely that collapse to a coherent state adds a single noise photon to the signal.

2) The evolution only applies to the single-photon input. This makes the proposal even more ad hoc. - We disagree to this statement. A single photon would cause the Cooper pairs in the transmission lines of the experiment to be in superposition over the full photon wavelength. On the other hand, if one considers a coherent state or a displaced squeezed state the phase of the photons and therefore the position of the Cooper pairs is much better defined. Therefore it might be that the state collapses for single photons, whereas it does not collapse for other states. We also would like to remark that the amplifiers become entangled for a single-photon input, whereas for a coherent-state input they are not entangled.

Presumably the authors have some idea that the spontaneous collapse would prevent macroscopic entanglement. But they do not even show that there is any macroscopic entanglement when the practical amounts of noise in these amplifiers is included anyway. If the authors want a “no macroscopic entanglement” theory or a “no macroscopic superposition” theory then they need to make the theory sensitive to the presence of these things. The existence of fringes does not imply entanglement! For instance, a HOM dip in quantum optics can be observed, with visibility up to 50%, can be obtained from classical (mixtures of coherent states) fields. - Indeed, there are two possible ways interference fringes can arise: either resulting from entanglement or if the two arms have coherent waves with a phase difference between them. In fact, we actually compare the expected visibility between a system where no collapse has taken place and one where the TWPA converts the single-photon input into a coherent wave. In the latter case, visibility indeed remains nonzero (see Fig. 4), it is however different from the no-collapse entangled scenario. The Referee's statement that "The existence of fringes does not imply entanglement!" is indeed something we already wrote in our manuscript.

Given the above criticisms, it seems to me that a better direction for the authors to take their work would be that of testing the existence of macroscopic superpositions, which they touch on in Section 6. However I am a bit skeptical of how impressive the comparison with molecular interferometers is. The authors say that 2 times 10^{-25} is “similar” to 7 times 10^{-24}, but is more than an order of magnitude smaller. As for “degrees of freedom” if that is to be the basis for a more favourable comparison then it would need to be fleshed out more. - OK. We will change the line "the mass is similar" to "the mass is one order of magnitude smaller".

There is a considerable literature on measures of macroscopicity that the authors would need to engage with. - There is indeed, and we cite various sources (for example Ref. [1]). If there is specific literature that the Referee wants us to include, please mention this. That being said, we try to not put too much emphasis on a discussion of macroscopicity, which doesn’t normally include the kind of complexity that we expect here at higher gain. Not only are the different position states far apart, also there are many different particle configurations with many different numbers of electrons.

They say (p. 2) that they are investigating an unexplored “part of parameter space” but they don’t say what the parameters are. - We do, namely in the next sentence: "The quantum states we consider are delocalised over large distances and the quantum states that come into play at large gain are more complex than in previous experiments because of the many different position states it is composed from."

Requested changes Finally, there are some small issues the authors should also address if they submit elsewhere. i) The definition of gain below Eq.(3) does not look standard to me. It should be defined in terms of amplitudes, not intensities. - In microwave optics it is common to define gain as a power (intensity) gain. Hence, this is merely a matter of convention.

ii) The first, undidplayed, equation in Sec. 3 does not make sense. The arguments of P should be Fock numbers (integers), not mean values of Fock numbers. - This is correct. Thank you.

iii) Even with loss as described in Sec. 4 the TWPA dynamics is linear with Gaussian noise. numerical simulation in Hilbert space is not necessary. There exist very efficient methods for propagating Gaussian states with such dynamics. Only a few parameters are needed. - We would like to get on with the experiment

iii) At the end of Sec. 5 the visibility figures of 0.15 and 0.2 is given. Are these exact or just approximations to the results of numerical simulations. - The 0.15 is approximated, the 0.2 exact as follows from our discussion in section 6.