# Quantum Gross-Pitaevskii Equation

### Submission summary

 Authors (as Contributors): Jutho Haegeman · Frank Verstraete
Submission information
Date submitted: 2017-05-16 02:00
Submitted by: Haegeman, Jutho
Submitted to: SciPost Physics
Ontological classification
Specialties:
• Atomic, Molecular and Optical Physics - Theory
• Quantum Physics
Approach: Theoretical

### Abstract

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.

Published as SciPost Phys. 3, 006 (2017)

### List of changes

We have extended the discussion of the periodic potential example. We clearly indicate how our framework improves upon the mean field prediction (Bogoliubov theory), which is also shown in the plot. We also discuss the underlying physics that shows up in the response amplitude, namely the signature of the low-lying excitations in the system, which are known to be Lieb's Type I and Type II excitations. Whereas Bogoliubov theory can only access the first type, our method also captures the effect of the second type, which is important at stronger interactions.