OPEN QUANTUM SYSTEMS AND ITS APPLICATIONS

Open Quantum Systems and their Applications TAN DA YANG B.Sc (Hons), NUS A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN SCIENCE DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2016 Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety I have duly acknowledged all the sources of information which have been used in the thesis This thesis has also not been submitted for any degree in any university previously Tan Da Yang 19 August 2016 Contents Summary i Acknowledgements ii List of Publications iii List of Figures ix Introduction 1.1 What are Quantum Open Systems? 1.2 Overview of Main Fields of Research 1.2.1 Protection of Quantum Systems 1.2.2 Decoherence in Adiabatic Transport 1.2.3 Non-Markovianity in Open Quantum Systems 1.2.4 Aspects of Open Quantum Systems in Biological Systems Outline of the Thesis 10 1.3 Master Equations 2.1 Overview 13 13 CONTENTS 2.2 Derivation of Master Equation by Perturbation Theory 15 2.2.1 Master Equation in Integral Form 15 2.2.2 Master Equation in Integro-Di↵erential Form 18 2.2.3 Pure Dephasing Master Equation 20 2.2.4 Further Remarks 20 2.3 Driven Systems and its Challenges 21 2.4 Time-Dependent Master Equation in Lindblad Form 22 2.4.1 Dissipative Lindblad Equation 23 2.4.2 Dephasing Lindblad Equation 26 Concluding Remarks 27 2.A Alternative Derivation of the Master Equation in Eq (2.13) 29 2.B Derivation of the Interaction Unitary Operator UI (t) 32 2.5 Environment Induced Entanglement 34 3.1 The Spin-Boson Model 35 3.2 Bath Correlator and Spectral Density 37 3.3 Extension of The Spin-Boson Model 40 3.4 Concurrence as an Entanglement Measure 44 3.5 Entanglement Dynamics 46 3.5.1 Pure Dephasing Dynamics 46 3.5.2 More General Dynamics 49 3.5.3 Dependence with Temperature 54 Conclusion 55 3.A Entanglement dynamics with hard cuto↵ function 56 3.B Entanglement with respect to !c 57 3.C Numerical Check of the Master Equation 58 3.6 CONTENTS Environmental Induced Spin Squeezing 59 4.1 Basic Concept of Spin Squeezing 61 4.2 One-Axis Twisting Hamiltonian 63 4.3 Squeezing Dynamics in Bosonic Environment 64 4.3.1 The Model 64 4.3.2 Optimization of spin squeezing 66 Conclusion 69 4.A Derivation of squeezing parameter ⇠S2 70 4.4 Population Transfer in Dephasing and Dissipation 5.1 5.2 5.3 5.4 71 Problem of Avoided Crossings 72 5.1.1 An Simple Illustration of the Problem 72 5.1.2 Significance of the Problem 75 5.1.3 Problem Statement 76 Population Transfer in Presence of Dissipation 77 5.2.1 The Derivation 77 5.2.2 Landau Zener Problem - An Example 81 Population Transfer under Dephasing 83 5.3.1 Example 85 Concluding Remarks 87 5.A Derivation of Density Matrix Elements for N Levels Systems Under Dissipation 89 5.B Proof of Generality of Eq (5.20) 91 5.C Alternative Derivation of the Dephasing Lindblad Equation 92 CONTENTS Adiabatic Pumping in Dissipative Environment 94 6.1 Introduction 94 6.2 Derivation of Pumping Formula 97 6.3 Chern Insulator - An Example 102 6.4 6.3.1 Transport Across Phase Transition Point 109 6.3.2 E↵ects of Initial State Preparation 111 Conclusion 118 6.A Comparison of the charge transport formula with numerics 119 6.B Relaxation of Even Function of k Assumption in Initial States 120 6.C Adiabatic Pumping Under Dephasing 121 Conclusion and Future Perspective 125 7.1 What Have We Achieved? 125 7.2 Outlook 127 Bibliography 129 Dedicated to my family Summary In this thesis, we will investigate various aspects of open quantum systems, i.e systems that are interacting with an external environment We will first study how the phenomenon of entanglement between two qubits and spin squeezing of a large spin system can be optimised by the environment, and find that contrary to conventional wisdom, the environment may sometimes assist with the formation of these quantum e↵ects We will then turn our attention to driven systems, whereby we first investigate the e↵ects of dissipation and dephasing on population transfer between energy levels as a result of adiabatic driving We will then extend these results to investigate the e↵ects of dissipation on adiabatic quantum pumping i Acknowledgements I would like to first thank my supervisor Prof Gong Jiangbin for his unwavering support during my entire candidature Thank you for being such an inspiring teacher and understanding supervisor who goes all the way out to help all your students I would also like to thank my research group mates, past and present, Adam, Derek, Yon Shin, Hailong, Qi Fang, Longwen, Gaoyang, Neresh and Jia Wen, for all the meaningful discussions in both office and over meal table In particular, I would like to especially thank Adam and Longwen for both of your guidance and pointers over the past few years, and helping out with the technical difficulties that I encountered along the way Special thanks also goes out to Junkai and Kendra for all the interesting and random discussions over meal table Thank you for being a big part in my life Last, but not least, this Ph.D journey could not have been possible without the support of my family I will eternally be grateful for that ii List of Publications Da Yang Tan, Adam Zaman Chaudhry, and Jiangbin Gong Optimization of the environment for generating entanglement and spin squeezing, Journal of Physics B: Atomic, Molecular and Optical Physics 48, 11 (2015): 115505 Longwen Zhou, Da Yang Tan and Jiangbin Gong E↵ects of dephasing on quantum adiabatic pumping with nonequilibrium initial states, Physical Review B 92, 24 (2015): 245409 iii 7.1 WHAT HAVE WE ACHIEVED? and rotating wave approximations, in order to arrive at the master equation in the Lindblad form With these formalism in place, we began our exploration of the e↵ects of the environment in entanglement and spin squeezing in Chapter and respectively We started from an exactly solvable pure dephasing spin-boson model and found that there existed optimal conditions with the environment in which the degree of entanglement could be maximised Using the master equation in Chapter 2, we found a similar optimisation with respect to the environment Thus, we concluded that by correctly tuning the environment, one could in fact maximise the entanglement that could be generated We then applied similar ideas to the large spin systems and investigated if spin squeezing could be generated While we found that the environment did not exactly improve the amount of squeezing as compared to the one-axis twisting Hamiltonian in closed system, we can nonetheless find a similar optimisation where maximum squeezing can be achieved by tuning the Ohmicity parameter of the environment In Chapter 5, our exploration took a di↵erent turn and we started to study the dynamics of driven system in the context of finite-time Landau Zener problem Here, for reasons explained above and in Chapter 2, we began our studies of the population dynamics in dissipation and dephasing by using the Lindblad master equation, and by recasting it in the instantaneous eigenbasis, we were able to derive a formula for adiabatic population transfer in both dissipation and dephasing In particular, we found that under the influence of dissipation, the system will always decay to a final state that is dictated by the Lindblad operators at the end of the driving regardless of the choice of the initial state The population dynamics also displayed interesting non-monotonic dynamics with respect to the systemenvironment coupling under dephasing After using two-level systems as our test bed to investigate the problem of population transfer in adiabatic setting, we extended our results to two-band systems and studied the pumping phenomenon in presence of dissipation in Chapter We found that for initial states that were prepared with its distribution independent of quasimomentum k or contain 126 CHAPTER CONCLUSION AND FUTURE PERSPECTIVE even functions of k, the charge transport was independent of the choice of initial state However, for particular values of energy bias , the number of pumped particles in fact displayed a non-monotonic behaviour with the coupling strength Such a behaviour could be attributed to the distribution of the Berry curvature, which was dictated by the energy bias Furthermore, unlike population transfer of two-level systems, the choice of initial state preparation may still a↵ect the charge transport property and we had discussed the conditions for the state preparation to play a significant role in the chapter In the following concluding section, we will stock take and discuss some of the possible directions that the research can take as a result of our findings 7.2 Outlook In Chapter and 4, we considered the optimisation of the environment-induced entanglement and spin squeezing A natural extension of this problem will be to investigate means to generate such form of controlled environment As mentioned in Section 1.2.1, there have been proposals to control the Ohmicity of the environment through Bose-Einstein condensate However, a general way to control such parameter is still not completely known Another possible direction will be to consider the e↵ects of initial system-environment correlation on the entanglement and spin squeezing It has been found [89, 90] that by preparing the initial state of the system as a correlated state, rather than a factorised state, the dynamics of the large spin system displays a strong oscillatory behaviour, which is amplified upon the increase in the number of spins In a factorised state, the system is ideally assumed to have no interaction initially However in practice, systems ought to have interacted with the environment and equilibrate, thus the initial states discussed in Refs [89] and [90] should be a more realistic starting point Furthermore, it has been hinted that the contribution1 due to the initial system-environment correlation may display non-trivial contribution, especially in the super-Ohmic regime Such interesting findings may lead to results that further our understanding of such optimisation problem 127 7.2 OUTLOOK In Chapter 5, we investigated how the population transfer is a↵ected by dissipation and dephasing In our example, we only limited ourselves to the standard Landau-Zener driving protocol Hence, it may be possible to ask how a di↵erent protocol will a↵ect the population transfer While we expect the choice of protocol to have negligible e↵ect under dissipation, the dynamics under dephasing might be more interesting Our earlier work [19] provided some hints to the problem, but a more comprehensive treatment is needed Also, in the dissipation problem, we investigated only two level systems, a multi-level systems treatment may be needed to give us further insights In Chapter 6, we investigated the e↵ects of dissipation on charge transport, and the effects of dephasing was also investigated in one of our earlier works [19] A natural extension of this problem will be to extend this results to a more general form of coupling, for example, the one described by Eq (2.13) In this case, we can include both the non-Markovian e↵ects and the e↵ects of other systems parameters such as temperature into our investigations A current limitation is that any computational studies of such slow process requires a sufficiently small time step and therefore the computational time may make the problem intractable Hence, another simple extension of our problem is to investigate the limit of fast driving on the population transfer and charge transport dynamics We have tried to investigate the behaviour of fast driving during the formation of this thesis and find that the population transfer exhibits interesting oscillatory dynamics with respect to the coupling strength1 Hence, the e↵ects of fast driving under dissipation may be worth pursuing, and one can possibly extend these results to a more general master equation set up since it is expected that the computational 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destruction of... 1.2.4 Aspects of Open Quantum Systems in Biological Systems The union between quantum mechanics and biological systems is indeed one that is intriguing For a long time, the warm and wet environment... Transport 1.2.3 Non-Markovianity in Open Quantum Systems 1.2.4 Aspects of Open Quantum Systems in Biological Systems Outline of the Thesis