Accurate Truncations of Chain Mapping Models for Open Quantum Systems

Abstract

The dynamics of open quantum systems are of great interest in many research fields, such as for the interaction of a quantum emitter with the electromagnetic modes of a nanophotonic structure. A powerful approach for treating such setups in the non-Markovian limit is given by the chain mapping where an arbitrary environment can be transformed to a chain of modes with only nearest-neighbor coupling. However, when long propagation times are desired, the required long chain lengths limit the utility of this approach. We study various approaches for truncating the chains at manageable lengths while still preserving an accurate description of the dynamics. We achieve this by introducing losses to the chain modes in such a way that the effective environment acting on the system remains unchanged, using a number of different strategies. Furthermore, we demonstrate that extending the chain mapping to allow next-nearest neighbor coupling permits the reproduction of an arbitrary environment, and adding longer-range interactions does not further increase the effective number of degrees of freedom in the environment.

Few-mode Field Quantization for Multiple Emitters

“…In Ref. [35], we recently showed that for the one-emitter case, it is generally sufficient to use a chain form with only next-nearest neighbor coupling between the modes (i.e., H𝑖𝑗 = 0 if |𝑖 − 𝑗| > 2), although this choice makes it more challenging to obtain converged fits. We here instead choose a block-diagonal form,…”

“…It would thus seem that this form is more convenient for fitting than the non-diagonal form. However, this turns out not to be the case [35]: First, it is not straightforward to enforce that the fit parameters correspond to a physical system where 𝒥 mod (𝜔) is real positive semidefinite for all frequencies. Second, even when that constraint is achieved, implementation of the dynamics through a Lindblad master equation (which as discussed above is the final goal of this approach) requires a form in which the imaginary part of the complex symmetric matrix H is negative semidefinite, while g is real.…”

“…Second, even when that constraint is achieved, implementation of the dynamics through a Lindblad master equation (which as discussed above is the final goal of this approach) requires a form in which the imaginary part of the complex symmetric matrix H is negative semidefinite, while g is real. This would require an algorithm that finds a complex orthogonal matrix V which "undiagonalizes" the system and can be used to obtain physical H = V ΩV 𝑇 and g = gV 𝑇 for any given Ω and g. To the best of our knowledge, no algorithm that achieves this has been found [35,36]. It is thus more straightforward to directly use the non-diagonal form Eq.…”

Few-mode field quantization for multiple emitters

“…It would thus seem that this form is more convenient for fitting than the nondiagonal form. However, this turns out not to be the case [ 59 ]: first, in this form, it is not straightforward to enforce that the fit parameters correspond to a physical system where is real positive semidefinite for all frequencies. Second, even when that constraint is achieved, implementation of the dynamics through a Lindblad master equation (which as discussed above is the final goal of this approach) requires a form in which the imaginary part of the complex symmetric matrix is negative semidefinite, while g is real.…”

Efficient method to simulate non-perturbative dynamics of an open quantum system using a quantum computer

“…Additional applications are Hamiltonian dynamics simulations of general quantum biological systems [40,52] and condensed matter systems [53][54][55] where perturbative approaches fail to provide answers. In the future, it would be interesting to implement optimizations to reduce the harmonic oscillator chain length in the same spirit as the ones reported in reference [56] and to find applications of the Q-TEDOPA to more general quantum computing problems besides quantum simulation. In this section, we show how to map bosonic creation and annihilation operators to Pauli operators using unary and binary qubit encondings for d-dimensional harmonic oscillators.…”