Quantum Gross-Pitaevskii Equation

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

SciPost Phys. 3, 006 (2017) · published 28 July 2017


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.

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Bogoliubov-de Gennes equations Continuous matrix product states Entanglement Gross-Pitaevskii equation Matrix product states (MPS) One-dimensional Bose gas One-dimensional systems Time-dependent variational principle (TDVP)

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