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Saratov Fall Meeting: Laser Physics, Photonic Technologies, and Molecular Modeling

Symmetry, coherent states, and control of quantum dynamics

Alexander V. Gorokhov

Published Jun 3, 2019• 0 citations
Publisher

Abstract

A problem of the coherent control of a quantum system in an external electromagnetic field is considered. We use the technique of dynamical symmetry Lie groups and relevant coherent states. The general approach is illustrated by examples of qubit and qudit dynamics in an external classical field, and transitions between the Rydberg states of a hydrogen-like atom with the fixed principal quantum number. The problem of generalization of the approach for taking into account additional interaction with a large dissipative environment is discussed.

DOI
10.1117/12.2524430

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A problem of the coherent control of a quantum system in an external electromagnetic field is considered. We use the technique of dynamical symmetry Lie groups and relevant coherent states. The general approach is illustrated by examples of qubit and qudit dynamics in an external classical field, and transitions between the Rydberg states of a hydrogen-like atom with the fixed principal quantum number. The problem of generalization of the approach for taking into account additional interaction with a large dissipative environment is discussed.
Transitions probabilities for highly excited Rydberg states of atoms in electromagnetic traps under the influence of an external electromagnetic fields are calculated. We used the method of groups of dynamical symmetry for hydrogenlike atoms and the technique of relevant coherent states. This allows us to calculate the transition probabilities between atomic levels outside the framework of perturbation theory. The problem of coherent controlling the quantum dynamics of atoms in cavities and some possible applications for quantum information theory are briefly discussed.
We critically evaluate the most widespread assumption in the theoretical description of coherent control strategies for open quantum systems. We show that, for non-Markovian open systems dynamics, this fixed-dissipator assumption leads to a serious pitfall generally causing difficulties in the effective modeling of the controlled system. We show that at present, to avoid these problems, a full microscopic description of the controlled system in the presence of noise may often be necessary. We illustrate our findings with a paradigmatic example.
All physical quantum systems are in contact with the external world which is inevitably a source of noise. It is therefore necessary to take into account this dissipation when designing controls that accomplish useful tasks in quantum information processing. This thesis is on that overlap of open quantum systems and control theory; it looks at what dynamics can happen with the use of external driving, and how this driving can be chosen to accomplish a desired goal. The first result looks at how a given quantum dissipator can be manipulated, using coherent controls, into replicating the action of a different type of noise. This results in no-go theorems for how noise can be transformed based on its isotropy, with applications in simulating open systems. Another way of doing simulations is to use only unitary dynamics over the system and a finite dimensional ancilla, and it is proved that there always exists a dilation Hamiltonian that replicates the noisy dynamics continuously in time. This also highlights the fact that adding controls on noise can result in a different evolution than if the controls were done on the under- lying system-environment level. A conjecture is introduced and studied which states that both approaches are equivalent in the special case of the controls commuting with the Lindbladian. A way around this difficulty is to use a quantum system to compute in situ which controls are best for achieving a desired task on itself. This problem is studied in the context of reaching entangling gates on a quantum simulator in order to upgrade it into a quantum computer. The experimental cost of doing so is found to be polynomial in the number of qubits in simulations. The same underlying principle is also used to find error correcting codes tailored to the dissipation in a system.
The review is devoted to an analysis of definite overcomplete non-orthogonal state systems that are connected with irreducible representations of Lie groups–the so called systems of generalized coherent states. These systems, which the author is the first to propose, are generalizations of Glauber's coherentstate system and arise in natural fashion in physical problems that have dynamic symmetry. They permit a considerable simplification of the solution of the quantum problem by reducing it to a simpler classical problem. The review deals with the properties of generalized-coherent-state systems connected with the simplest Lie groups.
For many purposes it is desirable to have an easily understandable and accurate picture of the atomic states. This is especially true for the highly excited states which exhibit features not present in the well known states hydrogen-like orbitals with usual values of the quantum numbers. It could be expected that such visualizations are readily available. Unfortunately, that is not the case. We illustrate the problems by showing several less fortunate earlier presentations in some scientifically most valuable works, and show more suitable visualizations for those cases. The selected cases are not chosen to criticize the authors' approach. Rather, we have taken these very important papers to underline the need for serious work with graphical representations which this work attempts to be a part of. In this text we discuss the problems encountered when visualizing Rydberg states, review some existing presentations and propose guidelines for applications. The focus of this work are so called Stark states and coherent elliptic states of Rydberg atoms. In the sections on elliptic states we show some quite novel geometric interpretations which should be very helpful in analysis of Rydberg state manipulations in external fields. Our demonstration examples involve only hydrogen-like states. However, the techniques and analysis are also relevant for Rydberg molecules and more complicated highly excited atomic and molecular systems.