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Quantum Gross-Pitaevskii Equation

by Jutho Haegeman, Damian Draxler, Vid Stojevic, J. Ignacio Cirac, Tobias J. Osborne, Frank Verstraete

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Submission summary

Authors (as registered SciPost users): Jutho Haegeman · Frank Verstraete
Submission information
Preprint Link:  (pdf)
Date submitted: 2017-03-09 01:00
Submitted by: Haegeman, Jutho
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
  • Quantum Physics
Approach: Theoretical


We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of many body system ---including entanglement and correlations--- and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for (quasi) one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov -- de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential.

List of changes

* We now already discuss the quantum Bogoliubov de Gennes equations (and its application) in the abstract and introduction.
* We have expanded our discussion regarding the comparison of cMPS results with results from phase space methods for dynamical simulations.
* We have adapted Figure 1 and the surrounding discussion.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2017-4-1 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1501.06575v4, delivered 2017-04-01, doi: 10.21468/SciPost.Report.103


1 - Clear and interesting message
2 - Concisely written


1 - Very technical


The article demonstrates how a continuous matrix product state ansatz in combination with the time-dependent variational principle gives rise to the Gross-Pitaevskii equation (GPE) for a matrix dimension of $D=1$. For $D>1$ this ansatz provides quantum corrections and results in a generalized quantum GPE. Furthermore, this quantum GPE can be linearized into a quantum version of Bogoliubov - de Gennes equations. This is a nice and interesting result and is certainly worth publishing.

I disagree with the second referee on the issues of “length” and “scope” of the paper. More details from the supplementary material in the main text would only obscure the message and make it even harder to read. Furthermore, the paper clearly avoids standard MPS language and is sufficiently general for readers without a tensor network background.

However, a clear disadvantage of the paper is that it is indeed very technical and some parts are hard to follow. For example, I don’t think that phrases like “… since any complex submanifold of Hilbert space is automatically Kähler …” speak to many people. The paper falls short in addressing readers without the appropriate mathematical background. Unfortunately, the last part of the paper that presents a specific example calculation (and that would be accessible to a broader audience) seems unmotivated.

Ideally, the authors would simply show a comparison to an exact calculation as suggested by the referee 2. Then for a general reader the ideas would not seem “insufficiently verified” as referee 2 remarked. At least, however, the authors should explain the physics of Fig. 1 in more detail. Since the authors write in the abstract that their method includes “entanglement and correlations”, they should also explain how this can be seen from Fig. 1. At the moment they only refer to [45] and write that “This response cannot be explained using standard GPE, as interactions are a key ingredient for exciting Type II excitations.” Why is this peak at $2k_F$? What are these excitations? Why are they not described in the GPE and why are they present in the exact solution?

I recommend publication, if the authors add an explanation of the physics.

Requested changes

1 - Add explanation of the physics of Fig. 1

  • validity: good
  • significance: high
  • originality: high
  • clarity: good
  • formatting: good
  • grammar: good

Anonymous Report 1 on 2017-3-30 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1501.06575v4, delivered 2017-03-30, doi: 10.21468/SciPost.Report.102






The authors have made appropriate changes with respect to my previous comments. Specifically, they have added more focus on the application to the QBdG equations, commented in more detail on comparisons to pre-existing techniques and made miscellaneous corrections.

With respect to the correspondence of the second referee, I am supportive of the authors response. I believe the paper is suitably written as to outline the method as a generalization of the conventional GPE. Whilst it is true that some of the more technical details will only be fully appreciated by an MPS-focused audience, the authors do do write the manuscript at a sufficient level to allow the reader to appreciate their conclusions. Lastly, I disagree with the referees assertion that the paper is too short. Much of the calculation details are indeed attached in the supplemental material, and I believe this is the appropriate place for them. I believe the paper in it's current form is suitably concise for the subject matter. Inclusion of further technical details would only serve to obfuscate the message of the paper.

Requested changes


  • validity: top
  • significance: high
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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