Applications of tensor networks toward dynamics of quantum systems тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Изотова Екатерина Александровна

• Изотова Екатерина Александровна
• кандидат наук
• 2023

Оглавление диссертации кандидат наук Изотова Екатерина Александровна

Contents

1 Abstract

2 Publications

3 Acknowledgements

4 Introduction

5 Review of the literature

5.1 Closed quantum systems

5.2 Open quantum systems

6 Closed quantum systems

6.1 The physical setting

6.2 Periods of thermalization process

6.2.1 Xon-integrable systems

6.2.2 Integrable systems

6.2.3 Xon-integrable systems close to an integrable point , ,

6.3 Two definitions of the slowest operator

6.3.1 "Local" definition

6.3.2 "Translationally-invariant" definition

6.4 Finding the slowest operator using tensor networks

6.4.1 Local slowest operator

6.4.2 Translationally-invariant slowest operator

6.5 Entanglement entropy of the slowest operator

6.6 Dependence of the slowest operator on the parameters in Hamil-tonian

6.6.1 The physical quantities

6.6.2 The results

6.7 Dependence of the slowest operator on support size N , , , ,

6.7.1 The physical quantities

6.7.2 The results

6.8 Time evolution

6.8.1 The physical quantities

6.8.2 The results

7 Open quantum systems

7.1 The physical setting

7.2 The first steps

7.3 The perturbation scheme of calculation of FGKLS pointers , ,

7.3.1 Xon-degenerate Hamiltonian

7.3.2 Degenerate Hamiltonian

7.4 Examples

7.4.1 First oscillator example

7.4.2 Second oscillator example

7.4.3 Third example: two-dimensional space of states , , , ,

7.5 Pointers for the FGKLS Equation in Two Dimensions

7.5.1 Diagonal Lindblad Operator

7.5.2 Lindblad Operator of a Jordan Block Form

7.6 Expression for a Pointer for Weak Interaction with Environment

7.7 The General Solution of the FGKLS Equation in Two-Dimensional Hilbert Space

7.7.1 The Case of Diagonal Lindblad Operator

7.7.2 The Case of the Lindblad Operator of the Jordan Block Form

7.8 Other Forms of the Solution of the FGKLS Equation

7.9 Positivity of the Solution of FGKLS Equation

7.10 The Behavior of Solutions for Weak Interaction with Environment

7.11 Unitons

8 Conclusion

Введение диссертации (часть автореферата) на тему «Applications of tensor networks toward dynamics of quantum systems»

4 Introduction

In this thesis we are interested in the process of how closed and open systems reach equilibrium.

The famous setting is a quantum system connected to a thermal bath. The bath drives the system to a thermal equilibrium. This a primary example of an open quantum system.

The similar situation is observed in closed quantum systems. One needs to consider a many-body system, usually a spin chain, and look at some finite subsystem that has a size much smaller than that of a whole system. Then one will observe how this subsystem is driven to equilibrium by its complement.

The latter process can be described using averages of local operators, having support where the subsystem of interest is. If these averages reach thermal averages (with respect to some thermal ensemble) during time evolution, then one can state that the system thermalizes. I.e. the subsystem appears to forget all the information about the initial state and reaches thermal equilibrium. But there are exceptions to this picture: a system can fail to thermalize and reach instead an equilibrium with respect to another, not thermal, ensemble (Generalized Gibbs Ensemble (GGE), see below). It is observed, for instance, in integrable and many-body localized systems. Therefore, it is of much interest to us to study the vicinity of an integrable point - to observe how a system changes its dynamics from thermalization to equilibration with respect to GGE,

On the other hand, we aim to look at evolution of open quantum systems and equilibration in this setting, as well. We are interested in final states of evolution - pointers. The example, that we keep in mind, is a measuring apparatus. When one measures a quantum system, a measuring apparatus plays a role of a bath and drives a system to a particular quantum state shown on a panel of this measuring device. Here we also observe the loss of information during time evolution: the system forgets about initial state and reaches some pointer states. Therefore, we aim to understand what are these pointer states and how they are connected to the form and strength of interaction with an environment. We also ask if there are situations, when a system does not reach a unique final state, and what are the possible scenarios of its time evolution.

Заключение диссертации по теме «Другие cпециальности», Изотова Екатерина Александровна

8 Conclusion

In the first part of the thesis we studied late-time dynamics of closed quantum systems. We considered the quantum Ising model in external magnetic field close to an integrable point. We focused on the slowest operator, as it plays an important role in the final period of dynamics. We introduced local and translationallv-invariant definitions of the slowest operator. We showed that both operators have low entanglement, and, therefore, we were able to construct them using tensor networks. We conclude that the local slowest operator is not an integral of motion of the integrable system (h = 0). As one increases h, there is a transition from integrable to thermalizing behavior. In the integrable system, there are revivals (of full and half amplitude). As one h

rate of derealization changes from extremely slow to slower than diifusion. The operator has a significant overlap with diifusion mode/energy flux. On the other hand, the translationallv-invariant slowest operator corresponds to

h=0

the exponent PgcE = at prethermalization period. It changes its

nature at a specific value h*: before the transition (h < h*) it does not have an overlap with any magnetization and expands over the chain faster than diifusion; after the transition (h > h*) it has non-zero overlap with magnetization!. and magnetization3 and expands slower than diifusion. The time evolution shows no fluctuations. The two definitions have common features in the dynamics (consider non-integrable system): the initial period of dependence on one parameter — Tr[H, O]2, then derealization and approaching the boundaries and final thermalization.

In the second part of the thesis we studied late-time dynamics of open quantum systems. Our first goal was to construct FGKLS pointers, given an interaction with an environment is weak, and perturbation theory can be applied. We succeeded in presenting the formulas for finding FGKLS pointers in each order of perturbation theory for non-degenerate and degenerate Hamiltonians, We found that turning an interaction with an environment on completely changes the final states. When the system is closed, they are much more arbitrary. If the system becomes open, its final states obey some specific sets of equations. It means that an interaction directs the system toward a set of fixed states, pointers, that we were looking for throughout this work. We also studied particular examples of quantum harmonic oscillator with spin interacting with external magnetic field. The pointers are predicted by our perturbation theory scheme and are shown to coincide with the exact solution. Then, we exhaustively studied the evolution of an open quantum system for a Hilbert space of dimension 2, We obtained final fixed states of an evolution of an open system (called pointers), and then we found a solution to the FGKLS equation and proved that it always converges to a pointer. It signifies a decoherenee process, when information about an initial

state becomes lost during an evolution as a result of an interaction with an environment.

In both, closed and open, quantum systems, we observed a similar feature: loss of information about initial states (thermalization in the closed case, decoherence in the open case). In the case of the closed systems, this process can be inhibited by approaching an integrable point, while in the case of open systems, one needs to reduce interaction with environment.

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