Simulating Bosonic Baths with Error Bars

We derive rigorous truncation-error bounds for the spin-boson model and its generalizations to arbitrary quantum systems interacting with bosonic baths. For the numerical simulation of such baths, the truncation of both the number of modes and the local Hilbert-space dimensions is necessary. We derive superexponential Lieb-Robinson-type bounds on the error when restricting the bath to finitely many modes and show how the error introduced by truncating the local Hilbert spaces may be efficiently monitored numerically. In this way we give error bounds for approximating the infinite system by a finite-dimensional one. As a consequence, numerical simulations such as the time-evolving density with orthogonal polynomials algorithm (TEDOPA) now allow for the fully certified treatment of the system-environment interaction.

Figure 1

A system coupled to a bosonic bath. Red lines indicate the truncations: the spatial truncation to a chain of finite length $L$ and the truncation of the local Hilbert space dimensions to $m_i$.

Figure 2

Fock space truncation error [Eq. (17)] for the particle mapping and power-law spectral densities as in Eq. (19), with $\mathrm{\Delta }/{\omega }_c=1$, $\alpha =0.8$, $s=3$ for initial state ${\widehat{\varrho }}_0={\widehat{\varrho }}_S^0\otimes {\widehat{\varrho }}_B^0$, ${\widehat{\varrho }}_S^0=|\uparrow ⟩⟨\uparrow |$ and ${\widehat{\varrho }}_B^0$ the vacuum. We truncate each local Hilbert space at the same value $m_i=m$ and $L$ has the values 3 to 6, but they are indistinguishable (e.g., the difference between the $L=6$ and $L=3$ curve at the point denoted by a red square is $4.95\times 10^{-6}$). Lines are guides for the eye. The log-log plot on the left suggests algebraic increase in time and the plot on the right (plotted for a constant time $t=2.45/{\omega }_c$) suggests a better than exponential decrease with $m$.