Open quantum entanglement: a study of two atomic system in static patch of de Sitter space

Open Quantum Cosmology: A study

of two body quantum entanglement

in static patch of De Sitter space

Samim Akhtar,

Sayantan Choudhury,

Satyaki

Chowdhury,

Debopam Goswami,

Sudhakar Panda,

Abinash Swain

a_{Department of Physics, Indian Institute of Technology, Madras, Chennai 600036, India.}

b_{Quantum Gravity and Unified Theory and Theoretical Cosmology Group, Max Planck Institute} for Gravitational Physics (Albert Einstein Institute), Am M¨uhlenberg 1, 14476 Potsdam-Golm, Germany.

c_{National Institute of Science Education and Research, Jatni, Bhubaneswar, Odisha -} 752050,In-dia.

d_{Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai-400085,} India.

e_{Department of Physics, Indian Institute of Technology, Kanpur- 208 016, India.}

f_{Department of Physics, Indian Institute of Technology, Gandhinagar, Palaj, Gandhinagar - 382355,} India.

E-mail: [email protected], [email protected],

[email protected], [email protected], [email protected], [email protected]

Abstract: In this work, our prime objective is to study non-locality and long-range effect of two-body correlation using quantum entanglement from the various information-theoretic measure in the static patch of De Sitter space using a two-body Open Quantum System (OQS). The OQS is described by two entangled atoms which are surrounded by a thermal bath, which is modelled by a massless probe scalar field. Firstly, we partially trace over the bath field and construct the Gorini Kossakowski Sudarshan Lindblad (GSKL) master equation, which describes the time evolution of the reduced subsystem density matrix. This GSKL master equation is characterized by two components, these are-Spin chain interaction Hamiltonian and the Lindbladian. To fix the form of both of them, using Schwinger-Keldysh formalism we compute the Wightman functions for probe massless scalar field. Using this result along with the large time equilibrium behaviour we obtain the analytical solution for reduced density matrix. Further using this solution we compute Von-Neumann entropy,

Re0_{nyi entropy, logarithmic negativity, entanglement of formation, concurrence and quantum}

discord in a static patch of De Sitter space. Finally, we have studied the violation of Bell-CHSH inequality, which is the key ingredient to study non-locality in primordial cosmology. Keywords: Open Quantum Systems, Quantum dissipation, Quantum entanglement, QFT of De Sitter space, Quantum Information Theory, Theoretical Cosmology.

1

Contents

1

Introduction

2

Modelling two atomic Open Quantum System (OQS)

3

Non unitary time evolution of the reduced subsystem

4

Reduced sub system construction

4.1 Density Matrix construction 10

4.2 Partial Trace operation 11

5

Effective Hamiltonian Construction

6

Quantum Dissipator or Lindbladian Construction

7

Bloch Sphere representation of density matrix

8

Time evolution of the reduced subsystem density matrix

8.1 Large Scale time dependent solution 20

8.2 Arbitrary time dependent solution 22

8.3 Solution of the evolution equations 23

8.4 General solution 24

9

Von Neumann Entanglement Entropy

9.1 Bloch sphere representation of Von Neumann Entropy 32

9.2 Von Neumann entropy from OQS of two entangled atoms 33

10

Re

nyi Entropy

10.1 Bloch sphere representation of Re0_{nyi Entropy 37}

10.2 Re0_{nyi entropy from OQS of two entangled atoms 37}

11

Logarithmic Negativity

11.1 Bloch sphere representation of Logarithmic negativity 43

11.2 Logarithmic negativity from OQS of two entangled atoms 44

12

Entanglement of formation and Concurrence

12.1 Bloch sphere representation of Concurrence 47

12.2 Concurrence from OQS of two entangled atoms 48

12.3 Entanglement of formation from OQS of two entangled atoms 50

13

Quantum Discord

13.1 Bloch sphere representation of Quantum Discord 53

14

Non Locality from Bell CHSH inequality in De Sitter space

14.1 Non-locality in Quantum Mechanics 56

14.2 Bell’s CHSH inequality violation in De-Sitter Space 59

15

Conclusion

A

Geometry of Static Patch of De Sitter space time

B

Solution for the bath field equation for probe massless scalar

field in Static Patch of De Sitter Space

B.1 Finding un-normalized total solution for probe scalar field 72

B.2 Finding un-normalized regular solution for probe scalar field 74

B.3 Finding the normalization constant for the regular solution for probe scalar

field 75

B.4 Finding normalized regular solution for probe scalar field 77

C

Quantization of bath modes for probe massless scalar field in

Static Patch of De Sitter Space

D

Bath Hamiltonian for probe massless scalar field in Static Patch

of De Sitter Space

D.1 Constructing classical bath Hamiltonian 81

D.2 Constructing quantized bath Hamiltonian 83

E

Quantum states for many body (two atomic) entangled states

F

Many body (two atomic) Wightman function for probe

mass-less scalar field in Static Patch of De Sitter Space

G

Gorini Kossakowski Sudarshan Lindblad (GSKL) (C

) matrix

92

G.1 Calculation of ˜Aαβ 92

G.2 Calculation of ˜Bαβ 94

G.3 Calculation of Cαβ

ij matrix elements 94

H

Effective Hamiltonian (H

) matrix

H.1 Calculation of Aαβ ₉₅

H.2 Calculation of Bαβ ₉₆

H.3 Calculation of Hαβ

ij matrix elements 97

I

Calculation of useful integrals

I.1 Integral I 98

J

Functions appearing in the solution of GSKL master equations102

1

Introduction

The theory of closed quantum systems is a very popular topic and has already been firmly established. But in practical situations no quantum system can be ideally treated as closed and its interactions with the surroundings cannot be neglected, hence it becomes essential to develop a theoretical framework to treat these non adiabatic interactions and develop a proper understanding of the quantum mechanical system. The knowledge of complete time evolutionary dynamics of a quantum mechanical system requires incorporation of the details of the thermal environment, which paves the way towards the study of Open Quantum

System [1] (OQS), where the physical system weakly interacts with the environment.In

general these interactions with the environment significantly controls the time evolution of the quantum mechanical system and induces the phenomenon of quantum dissipation. The dynamics of the reduced subsystem of OQS cannot be described using the unitary time evolution operators after integrating out the bath degrees of freedom from the theory. Correct time evolution of the system requires solving the effective master equation, which describes the non-unitary time evolution of the reduced density matrix of the system.

To deal with the subsystem in the context of OQS the entire combination of the system and its surroundings (thermal bath) is together treated as a closed quantum system and hence the evolution equations can be assumed to follow the unitary transformation rules. The prime assumption used in the present context is that the entire system environment combination forms a large closed quantum system. Therefore, its time evolution is governed by a unitary transformation generated by a global Hamiltonian which is made up of subsys-tem Hamiltonian, bath Hamiltonian and interaction Hamiltonian. Moreover the interaction between the system and the bath is assumed to depend on only the present moment, it carries no past memories at all. In short, the interaction is Markovian in nature. The inter-action between the system and the surroundings is assumed to be weak which justifies the argument that the only effective change that can be seen over time occurs in the context of OQS. This assumption is generally useful in treating the evolution of the system when the OQS has sufficient time to relax to the equilibrium before being perturbed in presence of interaction between the system and thermal environment. However, in a special situation where the system has very fast or frequent perturbation in presence of system- thermal bath coupling, one needs to consider Non Markovian approximation. In this treatment addition-ally it has been assumed that the system is completely uncorrelated with the surroundings at initial time scale, provided the coupling between the subsystem and the environment is sufficiently weak in nature. The techniques developed in the context of OQS have proven very powerful in the context of quantum optics, statistical mechanics, information theory, thermodynamics, cosmology and biology.

On the other hand, quantum entanglement [2] is probably the most fascinating

manifes-tation of quantum theory in which the beauty of quantum mechanics is truly realized. It is a physical phenomenon that occurs when pairs or groups of particles are generated, interact, or share spatial proximity in ways such that the quantum state of each particle cannot be

Atom 1 Atom 2 Inter-atomic interaction

Two Body Quantum Entanglement Two Body Quantum Entanglement

Two entangled system

Environment (Thermal bath)

Non-adiabatic interaction (heat+matter exchange)=RCPI Two atomic Open Quantum System

Fix Quantum Dissipator or Lindbladian Matrix

Two atomic Two point correlation for probe massless scalar field which is placed at the thermal bath

(Wightman function)

Static De Sitter Space in D=4

Fourier+Hilbert transformation Fix Effective Lamb Shift Hamiltonian Matrix

Quantum Liouville Equation for

Total Density Matrix

Reduced Density Matrix of the Subsystem (two atoms) Gorini Kassakowski Sudarshan Lindblad (GSKL) Master Equation

for Reduced Density Matrix

From the solution compute different entanglement measure used in QIT (x)

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(x)

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From the solution study Non-locality in QM from the violation of Bell’s CHSH inequality Background Metric Fourier Transformation H↵ ij

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C↵ ij

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Partial Trace over bath modes

+

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Static patch of De Sitter Space

Open two atomic (many-body) quantum system

South Pole

North Pole

I

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I

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Figure 1. Penrose diagram representing the static patch of the De Sitter space in which we have placed a two atomic open quantum system which is interacting with a thermal bath non-adiabatically. This diagram actually represents the causal patch of an observer sitting at the north pole, which is represented byr = 0 is static coordinate in De Sitter space. Equivalently this can be described in global coordinate withθ = 0. Here the bifurcation Killing horizon for ∂tis represented byr = α where the parameter α =q3

Λ > 0 as the cosmological constant Λ > 0 in De Sitter space. In this context the bifurcation sphere appears just at the middle of the Penrose diagram and in global coordinates it is described byt = 0 time slice. However, the other three regions can also be filled by the static coordinate system, which is just like Schwarzschild black holes.

described independently of the state of the others, even when the particles are separated by a large distance. Equivalently quantum entanglement of a state which is shared by two parties is necessary but not sufficient for that state to be non-local. Quantum entanglement

plays a significant role in the context of quantum computation [3], information and

tele-portation theory [4, 5], quantum error correction [6, 7],etc.There are huge applications of

quantum entanglement in various contexts. In interferometry the process of entanglement

is used to achieve the Heisenberg limit [8]. In multi electron atomic system the electronic

shells always consists of electrons in entangled state. [9]. In the process of

photosynthe-sis,entanglement is seen in the transfer of energy between light harvesting complexes and