Quantum simulations with photons in one dimensional nonlinear waveguides 1

59 5.2.2 Two-Component Lieb-Liniger Model and Spin-Charge Separation.. 60 5.3 Spin-Charge Separation with Differently Colored Photons.. 71 5.4 Spin-Charge Separation with Differently Pol

QUANTUM SIMULATIONS WITH

I hereby declare that the 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.

MING-XIA HUO

24 July 2013

First and foremost, I am deeply grateful to Kwek Leong Chuan Towork with him has been a real pleasure to me He has been oriented andsupported me with promptness and care He has always been patient andencouraging in times of new ideas and difficulties He has also providedinsightful discussions and suggestions I appreciate all his contributions oftime, ideas, and funding to make my PhD experience excellent Above all,

he made me feel a friend, which I appreciate from my heart

Furthermore, I am immensely grateful to Dimitris Angelakis His highlevel of comprehension and sharpness on physical subjects have taught me

to be rigorous and to tackle aspects of physics with a level of confidencethat I never had before I am also very grateful for his scientific advice andknowledge and many insightful discussions and suggestions

In addition, I have been very privileged to get to know and to collaboratewith David Hutchinson I learned a lot from him about research, language,how to tackle new problems and how to develop techniques to solve them

He has been a pleasure to work with Thanks for helping me a lot

I would also like to give thanks to Wenhui Li for her unwavering port professionally and personally at every important moment of my PhDexperience I will truly miss those discussions and conversations Thanksfor all the good times

sup-I also had the great pleasure of meeting Christian Miniatura From thevery beginning of my PhD career, he supported me Thanks for the funand encouraging discussions over the last several years

I would like to thank my collaborator Darrick Chang, whose physicalunderstanding in the research area is tremendous and whose scientific workinspired me a lot Our collaborated work has also benefited from sugges-

i

tions and kind encouragement from Vladimir Korepin His technical depthand attention to details are amazing I also want to thank David Wilkowskifor providing me the opportunity to have fruitful collaborations with hisexperimental group in a near future.

I would like to thank the collaborators I had the pleasure to work with.Special thanks to the postdocs Changsuk Noh, Blas M Rodriguez-Lara,Elica Kyoseva, and the student Nie Wei They have helped me a lot

I am grateful to my committee members: Berge Englert, Wenhui Li,and Chorng Haur Sow I have also immeasurably benefited from the course

"Quantum Information and Computation", for which I thank the instructorDagomir Kaszlikowski

I wish to thank Rosario Fazio and Davide Rossini for making DMRGavailable to me I have been interested in DMRG for a long time, andthey provided me with a big help I also want to thank Kerson Huangfor interesting and illuminating discussions when I first started my PhDcareer

I am grateful to our physics group members for providing me with thebest working environment In particular, I like to thank Dai Li, Setiawan,Thi Ha Kyaw for their willingness to share their research experience andmany helpful information I also want to thank Chunfeng Wu for heradvice, support, and encouragement A special acknowledgement goes toHui Min Evon Tan and Ethan Lim, who were very nice and always ready

to help

I will forever be thankful to my former Bachelor and Master Degreeadvisor Zhi Song He has been helpful in providing advice many timesduring my stay there He remains my best role model for a scientist andteacher I am also very grateful to Changpu Sun, for his tremendous and

ii

invaluable suggestions, advices, inspiration, and guidence Thanks for allthese helps.

This dissertation is dedicated to my father Jujiang Hu, my motherShulan Chen, my husband Ying Li, and my daughter Daiyao Li, for theirinfinite support throughout everything Words cannot express my gratitude

of love A final thanks goes to my friends, not previously mentioned, whosupported me and influenced me a lot

iii

3 Pinning Quantum Phase Transition of Photons 21

3.1 Bose-Hubbard and Sine-Gordon Models 21

3.2 Quantum Optical Simulator with One-Species Four-LevelAtoms 22

3.3 Polaritons Trapped in an Effective Periodic Lattice 27

3.4 Reaching Correlated Bose-Hubbard and Sine-Gordon Regimes 30

3.5 Polaritonic/Photonic Pinning Transitions 31

3.6 Characteristic First- and Second-Order Correlations of sitions 35

Tran-v

Fermi-4.5 Witnesses of BCS-BEC-BB Crossover 53

5 Spin-Charge Separation in a Photonic Luttinger Liquid 57

5.1 Spin-Charge Separation 57

5.2 From Luttinger Liquid to Spin-Charge Separation 59

5.2.1 Bosonization Approach and Single-Component

Lieb-Liniger Model 59

5.2.2 Two-Component Lieb-Liniger Model and

Spin-Charge Separation 60

5.3 Spin-Charge Separation with Differently Colored Photons 62

5.3.1 Polaritonic Spin-Charge Separation with

Two-Species Four-Level Atoms 62

5.3.2 Spinon and Holon Velocities 71

5.4 Spin-Charge Separation with Differently Polarized Lights 76

5.4.1 Polaritonic Spin-Charge Separation with

Single-Species Multi-Level Atoms 78

5.4.2 Spinon and Holon Velocities 82

6 Simulating Interacting Relativistic Quantum Field

6.1 Thirring Model 85

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6.2 Photons for Interacting Fermions 87

6.3 Nonlinear Dynamics of Relativistic Stationary Polaritons 88

6.4 Thirring Model with Stationary Pulses of Light 91

A.1 Atomic Operators 115

A.2 Quantum Light Evolution 120

B Derivation of Nonlinear Evolution Equation for Species Photons with Different Frequencies 123

Two-B.1 Atomic Operators 123

B.2 Quantum Light Evolution 131

C Derivation of Nonlinear Evolution Equation for Two-speciesPhotons with Different Polarizations 133

C.1 Aomtic operators 133

C.2 Quantum Light Evolution 138

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The study of one-dimensional (1D) strongly correlated quantum liquids is

a fascinating area and has attracted a lot of attention recently In spite

of their apparent conceptual simplicity, both their ground states and tated states exhibit a large number of exotic strong-correlated effects Onthe other hand, the achievement of strongly correlated quantum opticalsystems which accurately describe condensed matter physics and quan-tum field theory models has opened many new perspectives for research

exci-on strexci-ongly correlated systems These quantum optical systems with themost famous examples being optical lattices and ion traps share manyfeatures with conventional systems, while several of their properties distin-guish them from the traditional setups like, their long coherence times andthe ability to control Hamiltonian parameters over a wide range Recently,the 1D optical nonlinear waveguide as a promising optical system with atight field confinement and coherent photon trapping techniques has beenproposed, where the dark-state polaritons are formed as a combination oflight and matter excitations Their highly nonlinear behavior has provided

a platform to enable amazing experiments in the field on quantum nonlinearoptics and slow-light applications It is the purpose of our work to employsuch setups to emulate and efficiently observe some of the well-known mod-els and phenomena in condensed matter physics and quantum field theory

by using the dark-state polaritons, where the polaritons are shown to low the dynamics of quantum systems such as the Lieb-Liniger model, theBose-Hubbard model, the quantum sine-Gordon model, the Fermi-Hubbardmodel, and the relativistic Thirring model These quantum simulators pro-vide a new platform in the field of understanding complex condensed matter

fol-ix

phenomena and quantum field models with currently accessible quantumoptical techniques The main tools used in our proposals involving sta-tionary light-matter polaritons, condensed matter physics, and quantumfield models, are textbook examples in the fields of quantum optics andstrongly correlated systems However, this inter-disciplinary combinationcan be used to resolve some of the long standing problems such as a directobservation of spin-charge separation, the correlations as witnesses of pin-ning transition and BCS-BEC crossover, and the realization of interactingDirac particles in a continuous system, which have been proven undoubt-edly difficult, and which have been demonstrated to be possible in ourproposed schemes

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List of Publications

Publications:

[1] “Luttinger Liquid of Photons and Spin-Charge Separation in Core Fibers”, D G Angelakis, M.-X Huo, E Kyoseva, and L C Kwek,Phys Rev Lett 106, 153601 (2011)

Hollow-[2] “Sine-Gordon and Bose-Hubbard dynamics with photons in a core fiber”, M.-X Huo, and D G Angelakis, Phys Rev A 85, 023821(2012)

hollow-[3] “Spinons and Holons with Polarized Photons in a Nonlinear uide”, M.-X Huo, D G Angelakis, and L C Kwek, New J Phys 14,

Waveg-075027 (2012)

[4] “Probing the BCS-BEC crossover with photons in a nonlinear opticalfiber”, M.-X Huo, C Noh, B M Rodríguez-Lara, and D G Angelakis,Phys Rev A 86, 043840 (2012)

[5] “Mimicking interacting relativistic theories with stationary pulses

of light”, D G Angelakis, M.-X Huo, D Chang, L C Kwek, and V.Korepin, Phys Rev Lett 110, 100502 (2013)

Preprints:

[1] “Interference Signatures of Abelian and Non-Abelian Bohm Effect on Neutral Atoms in Optical Lattices”, M.-X Huo, W Nie,

Aharonov-D Hutchinson, and L C Kwek, arXiv:1210.8008

Previous Publications in Other Directions:

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[3] “The Peierls Distorted Chain as a Quantum Data Bus for QuantumState Transfer”, M.-X Huo, Y Li, Z Song, and C.-P Sun, Europhys Lett.

84, 30004 (2008)

[4] “Atomic Entanglement versus Visibility of Photon Interference forQuantum Criticality of a Hybrid System”, M.-X Huo, Y Li, Z Song, andC.-P Sun, Phys Rev A 77, 022103 (2008)

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List of Figures

2.1 The ladder-, Vee-, and lambda-structure atoms 10

3.1 The model setup for simulating sG and BH models 23

3.2 The scheme for modulating the atomic density 28

3.3 The Lieb-Liniger interaction parameter and the lattice depth 30 3.4 The phase diagrams for sG model and BH model 33

3.5 The interaction and tunneling strengths 35

3.6 The correlation functions for the polaritonic gas 36

4.1 The model setup for simulating BCS-BEC-BB crossover 41

4.2 The inter-species interaction and hoppling strengths 53

4.3 The correlation functions of polaritons 55

4.4 The cross-species second-order correlations 56

5.1 A schematic diagram of the spin-charge separation 63

5.2 A schematic diagram of the system under consideration 64

5.3 The model setup for simulating spin-charge separation with differently colored lights 66

5.4 The Fourier transform of density-density correlation func-tion for the system with differently colored light 72

5.5 The model setup for simulating spin-charge separation with differently polarized lights 77

5.6 The Fourier transform of density-density correlation func-tion for the system with differently polarized lights 83

6.1 The model setup for simulating Thirring model 89

6.2 Regimes of bosonic and fermionic Thirring models 95

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List of Figures

6.3 The momentum cutoff and the log-scaled two-point tions 97

corerla-xiv

atomic densityphotonic densityone-photon detuningtwo-photon detuningcoupling strength between photon and atomwavevector of quatum light

wavevector of classical lightcentral frequency of quantum lightcentral frequency of classical lightvelocity of light in an empty mediumgroup velocity of light in a nonlinear mediummass

interaction strengthtotal spontaneous emission ratespontaneous emission rate into the waveguideoptical depth

single-atom cooperativityLieb-Lineger parameterLuttinger parameter

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1

Chapter 1 Introduction

tum system which is computationally difficult or experimently inaccessible.The desired quantities are either detected or measured in a real experiment,where the uncharted regimes which are originally deemed impossible areexplored and investigated

The idea that a quantum system is best simulated with a quantummechanical device stems from a seminal presentation by Richard Feyn-man thirty years ago [4] However, at that time, the state of technology

in manipulating cold atoms and molecules and constructing optical tups have not reached the current level of sophistication Recent advances

se-in technology, especially trappse-ing technology, have demonstrated convse-inc-ingly that such the possibility for a scalable quantum simulator could beachieved in the foreseeable future In principle, quantum simulators arecontrollable quantum systems that could be used to simulate other quan-tum systems They are essentially analog (quantum) computers capable ofstudying other quantum systems directly through measurements and ob-servations In some sense, they behave some primitive computer of nature[4,5] This direction also creates a motivation for the need to advance thecurrent state of technology for cooling, addressing and manipulating atomsand molecules

convinc-Many platforms for experimental realization of quantum simulatorshave been proposed and tested to some extent Among the potential can-didates for quantum simulators, the trapping of ultracold atomic gases inoptical lattices has been widely used for a number of schemes, includingthe study of Mott transition from superfluid (SF) to Mott insulator (MI)phase [6,7], the Tonks-Giradeau gas for strongly interacting bosons [8], thecrossover from Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein conden-sate (BEC) in both fermionic and bosonic systems [9,10,11,12,13,14,15],

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the realizations of field theory sine-Gordon (sG) [16] and Thirring models[17], and so on While some marvelous achievements have been obtainedexperimentally for quantum simulations, there are still some inherent chal-lenges in these schemes, for instance, fermionic atoms have been found to

be difficult to cool to a sufficiently low temperature due to the existence ofPauli exclusion principle and measurements of certain correlations are stillintractable despite advances in single-site addressing of atoms [18, 19]

A new research direction with strongly interacting dark-state tons through light-matter coupling has been proposed in recent years,where the polaritons are formed between the lower two stable levels ofthree-level atoms and resonant probe light in a nonlinear optical waveg-uide [20, 21, 22, 23, 24], based on a typical electromagnetically inducedtransparency (EIT) effect [21] In the nonlinear waveguide case, thelight beams are injected into a hollow-core waveguide doped with atoms[25, 26,27, 28, 29,30] or a nanofiber with atoms brought close to the sur-face of the fiber [31, 32] A natural quasi 1D system is formed due to thetight-confinement of the waveguides With a pair of counter-propagatingclassical control lasers in the waveguide, a weak quantum pulse is trapped

polari-as a standing wave formed by control lpolari-asers due to the Bragg scattering[22, 23, 24] The principal differences between the quantum and classicallights are their frequencies and intensities The classical lights are in or-ders of magnitude stronger and contain macroscopically large number ofphotons They are also detuned from each other, by several GHz Thistrapping allows enough time for polaritons to evolve according to like, e.g.,the nonlinear Schrödinger equations of a designed system During the evo-lution, the polaritons obey approximate bosonic statistics and can interactstrongly with each other After reaching the desired state, one of the con-

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