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Nonlinear Inverse Iterations for Spin-Orbit Coupled Quantum Gases

Patrick Henning, Laura Huynh

TL;DR

This work tackles the computation of ground states for two-component spin--orbit coupled Bose--Einstein condensates by formulating a constrained nonlinear eigenproblem. It introduces the J-method, a nonlinear inverse-iteration scheme based on the Jacobian of the operator, and leverages spectral shifting to achieve fast local convergence, including linear rates for fixed shifts and potential superlinear growth with adaptive shifts. The authors develop a rigorous local convergence theory, establish the spectral properties of the J-operator, and provide an auxiliary phase-fixed analysis to circumvent gauge degeneracy. Numerical experiments on 2D domains with varying k0 demonstrate substantial speedups over traditional A-methods, validating the method’s effectiveness for challenging supersolid regimes and its practical relevance for high-precision ground-state computations.

Abstract

This work concerns the computation of ground states of two-component spin-orbit coupled Bose-Einstein condensates (SO-coupled BECs), modelled by a coupled nonlinear eigenvalue problem of Gross-Pitaevskii type. Spin-orbit coupling gives rise to fascinating phenomena, including supersolid-like phases with spatially modulated densities. However, in such complex settings, conventional numerical approaches, such as generalized inverse iterations or gradient descent, often converge very slowly. To overcome this issue, we apply the concept of the J-method [E.~Jarlebring, S.~Kvaal, W.~Michiels. SIAM~J.~Sci.~Comput.~36-4,~2014] to construct a nonlinear inverse iteration scheme whose convergence can be accelerated through spectral shifting, analogous to techniques used for linear eigenproblems. For a fixed shift parameter, we establish local linear convergence rates determined by spectral gaps in the neighbourhood of each quasi-unique ground state. With adaptively chosen shifts, superlinear convergence is observed, which we verify through numerical experiments.

Nonlinear Inverse Iterations for Spin-Orbit Coupled Quantum Gases

TL;DR

This work tackles the computation of ground states for two-component spin--orbit coupled Bose--Einstein condensates by formulating a constrained nonlinear eigenproblem. It introduces the J-method, a nonlinear inverse-iteration scheme based on the Jacobian of the operator, and leverages spectral shifting to achieve fast local convergence, including linear rates for fixed shifts and potential superlinear growth with adaptive shifts. The authors develop a rigorous local convergence theory, establish the spectral properties of the J-operator, and provide an auxiliary phase-fixed analysis to circumvent gauge degeneracy. Numerical experiments on 2D domains with varying k0 demonstrate substantial speedups over traditional A-methods, validating the method’s effectiveness for challenging supersolid regimes and its practical relevance for high-precision ground-state computations.

Abstract

This work concerns the computation of ground states of two-component spin-orbit coupled Bose-Einstein condensates (SO-coupled BECs), modelled by a coupled nonlinear eigenvalue problem of Gross-Pitaevskii type. Spin-orbit coupling gives rise to fascinating phenomena, including supersolid-like phases with spatially modulated densities. However, in such complex settings, conventional numerical approaches, such as generalized inverse iterations or gradient descent, often converge very slowly. To overcome this issue, we apply the concept of the J-method [E.~Jarlebring, S.~Kvaal, W.~Michiels. SIAM~J.~Sci.~Comput.~36-4,~2014] to construct a nonlinear inverse iteration scheme whose convergence can be accelerated through spectral shifting, analogous to techniques used for linear eigenproblems. For a fixed shift parameter, we establish local linear convergence rates determined by spectral gaps in the neighbourhood of each quasi-unique ground state. With adaptively chosen shifts, superlinear convergence is observed, which we verify through numerical experiments.
Paper Structure (17 sections, 21 theorems, 217 equations, 4 figures, 4 tables)

This paper contains 17 sections, 21 theorems, 217 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Mathematical setting and problem formulation
  3. Minimization problem
  4. First and second order conditions for minimizers
  5. The $A$-method
  6. The J-method
  7. The $J$-formulation of the SO-GPE
  8. $J$-inverse iteration with spectral shifting
  9. Local convergence rate of the shifted $J$-method
  10. Properties of the $J$-operator
  11. Proof of local convergence
  12. Canonical fixed-point formulation
  13. Auxiliary fixed-point formulation
  14. Numerical experiments
  15. The case $k_0=10$
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $\boldsymbol{u} = (u_1,u_2) \in {\mathbf H}^1_0(\mathcal{D})$, and suppose that assumptions A1--A3 hold. Then the energy functional $E$ defined in eq:SO-energy is nonnegative, i.e.,

Figures (4)

  • Figure 1: Difference to reference energy in each iteration for $k_0=10$.
  • Figure 2: Density of first and second component of ground state for $k_0=10$.
  • Figure 3: Difference to reference energy in each iteration for $k_0=50$. The solid lines represent the iterations for the A1-A2-J2 version, while the dashed line represents the A2-J2 case.
  • Figure 4: Density of first and second component of ground state for $k_0=50$ from a bird's view perspective (top row) and side view (bottom row).

Theorems & Definitions (43)

  • Lemma 2.1: Positivity of the energy functional
  • proof
  • Lemma 3.1: Coercivity of $\mathcal{A}(\boldsymbol{u})$
  • Definition 3.2: $A$-method
  • Definition 3.3: Damped $A$-method
  • Theorem 3.4
  • Definition 4.1: $J$-method with shifting
  • Theorem 4.2: Local convergence rate
  • Lemma 5.1: Eigenspace to $\lambda$
  • proof
  • ...and 33 more