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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.