Symplectic Split-Operator Propagators from Tridiagonalized Multi-Mode Bosonic Hilbert Spaces for Bose-Hubbard Hamiltonians

Abstract

In this methods paper, we show how to tridia\-go\-nalize two families of bosonic multimode systems: optomechanical and Bose-Hubbard hamiltonians. Using tools from number theory, we devise a rendering of these systems in the form of exact $D \times D$ tridiagonal symmetric matrices with real-valued entries. Such matrices can subsequently be exactly diagonalized using specialized sparse-matrix algorithms that need on the order of $D \ln(D)$ steps. This makes it possible to describe systems with much larger numbers of basis states than available to date. It also allows for efficient diagonal representation of large, accurate, symplectic split-operator propagators for which we moreover show that the required basis changes can be implemented by simple re-indexing, at marginal computational cost.

Figures (2)

• Figure 1: The ratio $R = {\cal N}_{N}^{K} / N^K$, see Eq. (\ref{['eq:NumberInIsland']}), quantifies the size of an excitation island as compared to the naïvely chosen size $N^K$ of a hilbert space for $N$ excitations across $K$ sites. $R$ be-comes relatively very small for moderate values of $K$ and $N$.
• Figure 2: For a Bose-Hubbard system QuSpinBHM, with $U = 1$, $J = 1,$$\mu = 0,$ for $N = 100$ bosons on $K = 4$ sites ($\psi(t=0)=|100,0,0,0\rangle$), and time-steps $dt = 0.01$, the plots demonstrate that using QuSpin-software the size of the state, in the Euclidian $l^2$-norm, grows linearly with time (away from unity). Instead, our Skolem-approach uses a symplectic integrator preserving norm at machine precision (note: scale on ordinate axis multiplied by $10^{15}$). Also, for this example, our code runs roughly ten times faster Codes.