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# Embedding strategies for delay problems in different fields

Friday, 17. May 2019

Location: Technische Universität Berlin

Main building, Room H 3005

Straße des 17. Juni 135, 10623 Berlin

Guests are welcome!

# Programme

**Friday, 17. May 2019**

15:00 | Transfer-tensor-method and Markovian embeddings based on orthogonal polynomialsDr. Javier Cerrillo Technische Universität Berlin |

15:25 | The pseudospectral approximation method for delay differential equationsBabette de Wolff Freie Universität Berlin |

15:50 | Coffee Break |

16:10 | Fokker-Planck description of a delayed stochastic process via Markovian embeddingSarah A.M. Loos Technische Universität Berlin |

16:35 | Non-Markovian quantum feedback in the presence of finite temperaturesDr. Alexander Carmele Technische Universität Berlin |

17:00 | Informal get-together ("Stammtisch") |

# Abstracts

**Transfer-tensor-method and Markovian embeddings based on orthogonal polynomials**

Dr. Javier Cerrillo

Comprehensive simulation methods of general open quantum systems tend to be numerically

demanding, in particular in the presence of non-Markovian effects and strong coupling to the

environment. It is generally the case that the size of the propagator or of the stochastic sample scales unfavorably with the time length of the simulation or the corresponding perturbative expansion order, and can be interpreted in terms of the exponential growth of the relevant Hilbert space. The question arises whether there are regimes where this scaling can be mitigated in some form, i.e. if an effective

propagator of a reduced size can be extracted that facilitates long-time simulations. This question was

addressed with the creation of a tool known as the transfer-tensor-method (TTM), which has been

shown to provide extraordinary acceleration of non-Markovian open quantum system simulations. This

is achieved by blackbox learning from sample exact trajectories for some short initial period

and subsequent generation of a compact multiplicative propagator for the system degrees of freedom

alone. For a learning period longer than the environment correlation time, the propagator accurately

reproduces the long time system dynamics with linear effort. TTM is a general and flexible approach

that does not depend on the form of the environment or the interaction, and has generated widespread

interest. In particular, it has been shown to be a useful tool for the reproduction of absorption and

emission spectra of atomic or molecular systems dressed with environmental vibrations and in the

context of laser cooling experiments.

**The pseudospectral approximation method for delay differential equations**

*Babette de Wolff*

The pseudospectral approximation method for delay equations was introduced by Breda et al. in 2005

as a method to approximate eigenvalues of delay differential equations (DDEs) by eigenvalues of a

family of ordinary differential equations (ODEs). Because of the specific structure of the family of

ODEs, it has been proposed that also the bifurcation behaviour of the DDEs is approximated by the

bifurcation behaviour of the ODEs. This would allow us to use ODE bifurcation tools to analyze the

bifurcation behaviour of delay equations.

In this talk, we will introduce the pseudospectral method and discuss its bifurcation behaviour. In

particular, we will discuss the convergence of the Lyapunov coefficient in the Hopf bifurcation.

**Fokker-Planck description of a delayed stochastic process via Markovian embedding**

*Sarah A.M. Loos*

A discrete time delay in the Langevin equation naturally leads to an infinite hierarchy of Fokker-Planck

(FP) equations for the n-time joint probability distribution functions [1]. Finding a probabilistic

description is hence challenging, especially for systems subject to nonlinear forces. One major issue is

that the higher members of the hierarchy contain unknown functional derivatives between noise and the

stochastic state variable.

In this talk, I will introduce a new way to derive the Fokker-Planck equation via a Markovian

embedding technique. In particular, I will discuss an extended Markovian system with auxiliary

variables which generates the same dynamics as the original (delayed) system in the limit of an

infinitely large system. This extended system can further be studied under a stochastic

thermodynamical [2] perspective, allowing to find a closed expression for the entropy production,

which is a nontrivial problem in the presence of delay.

[1] Loos & Klapp, ArXiv:1903.02322 (2019).

[2] Loos & Klapp, Sci. Rep. 9, 2491 (2019).

**Non-Markovian quantum feedback in the presence of finite temperatures**

*Dr. Alexander Carmele*

Feedback introduces additional non-Markovian memory and noise into open quantum system

dynamics. In this talk, a detailed discussion of non-Markovian feedback is presented in the case of

quantum coherence control for exotic pure dephasing dynamics in acoustic cavities. It is shown that

feedback allows to stabilize initial coherences in the system up to room temperature due to quantum

interference effects. Furthermore, an outlook is given how to implement non-Markovian contributions

in an augmented density matrix approach relying on real-time Feynman path integrals.