STUDIES OF QUANTUM SPIN CHAIN DYNAMICS AND THEIR POTENTIAL APPLICATIONS IN QUANTUM INFORMATION VINITHA BALACHANDRAN NATIONAL UNIVERSITY OF SINGAPORE 2011 STUDIES OF QUANTUM SPIN CHAIN DYNAMICS AND THEIR POTENTIAL APPLICATIONS IN QUANTUM INFORMATION VINITHA BALACHANDRAN (M. Sc., Cochin University of Science And Technology, India) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN SCIENCE DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2011 Acknowledgements I would like to express my sincere gratitude to my supervisor Assoc. Prof. Gong Jiangbin for his constant support and guidance through out the course of my PhD. I am very much indebted to him for imparting me various skills and techniques to carry out effective and quality research. I am indeed fortunate to work under such a wonderful teacher and researcher. I also express by deepest gratitude to Prof. Giulio Casati and Asst. Prof. Giuliano Benenti for providing me an opportunity to work on a collaborative project. Their valuable guidance and timely suggestions helped a lot for the successful completion of my PhD. My thanks go to all staff in Physics department, especially in Centre for Computational Science and Engineering, for their valuable assistance. I acknowledge National University of Singapore (NUS) and Faculty of Science for providing graduate student fellowship. I am also grateful to my father Balachandran, mother Vimala, and siblings Vipin and Smitha for their prayers, inspiration and support. I express special thanks to my mother, to whom I dedicate this thesis. Finally, I thank my friend Alwyn for his encouragement and support, especially at times of adversities in research. i Contents Acknowledgements i Contents ii Summary vii List of Publications ix List of Figures x Introduction 1.1 Quantum information science (QIS) . . . . . . . . . . . . . . . . . . . 1.2 Basics of quantum information processing . . . . . . . . . . . . . . . 1.2.1 Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Quantum computation . . . . . . . . . . . . . . . . . . . . . . Prospects of quantum information processing . . . . . . . . . . . . . . 1.3.1 Quantum algorithms . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Quantum communication . . . . . . . . . . . . . . . . . . . . . 1.3 ii 1.3.3 Quantum control . . . . . . . . . . . . . . . . . . . . . . . . . Realizing quantum information processing . . . . . . . . . . . . . . . 1.4.1 Trapped ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Trapped atoms . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Nuclear magnetic resonance (NMR) . . . . . . . . . . . . . . . 10 1.4.4 Quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4.5 Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . 11 Spin chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.1 Heisenberg spin chain . . . . . . . . . . . . . . . . . . . . . . . 13 1.5.2 XY spin chain . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5.3 Ising spin chain . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Applications of spin chains in quantum information processing . . . . 15 1.6.1 Universal quantum computation . . . . . . . . . . . . . . . . . 15 1.6.2 Quantum state transfer . . . . . . . . . . . . . . . . . . . . . . 16 1.6.3 Quantum state amplification . . . . . . . . . . . . . . . . . . . 17 Manifestations of quantum many-body nature in spin chains . . . . . 18 1.7.1 Quantum phase transitions . . . . . . . . . . . . . . . . . . . . 18 1.7.2 Quantum chaos . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.7.3 Quantum complexity . . . . . . . . . . . . . . . . . . . . . . . 22 1.7.4 Quantum many-body localization . . . . . . . . . . . . . . . . 23 1.8 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.9 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4 1.5 1.6 1.7 Adiabatic quantum transport in spin chains using a moving poten- iii tial 29 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2 Adiabatic quantum transport in spin chains: A pendulum perspective 32 2.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.2 Mapping of spin chain to pendulum . . . . . . . . . . . . . . . 33 2.2.3 Mechanism of adiabatic quantum transport scheme . . . . . . 34 Adiabatic transport by moving potential: Computational results . . . 36 2.3.1 Single spin excitation . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.2 Gaussian excitation profile . . . . . . . . . . . . . . . . . . . . 42 2.3 2.4 Speed of adiabatic quantum transport . . . . . . . . . . . . . . . . . 44 2.5 Robustness of adiabatic transport . . . . . . . . . . . . . . . . . . . . 46 2.5.1 Static disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5.2 Dynamic disorder . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.6 Adiabatic transport in a dual spin chain . . . . . . . . . . . . . . . . 52 2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Controlled quantum state amplification in spin chains 57 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Spin chain model of controlled quantum state amplification . . . . . . 59 3.2.1 Quantum state amplification . . . . . . . . . . . . . . . . . . . 60 3.2.2 Mapping quantum state transfer and amplification . . . . . . 62 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3.1 Idealized model . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3.2 Realistic model without disorder . . . . . . . . . . . . . . . . . 68 3.3 iv 3.3.3 Realistic model with disorder . . . . . . . . . . . . . . . . . . 73 3.4 Controlled growth of Schr¨odinger cat states . . . . . . . . . . . . . . 74 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Complexity in quantum many-body dynamics 77 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Harmonics of the Wigner function . . . . . . . . . . . . . . . . . . . . 80 4.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.4 Phase-space characterization of complexity . . . . . . . . . . . . . . . 85 4.4.1 Initial growth of S(t) . . . . . . . . . . . . . . . . . . . . . . . 85 4.4.2 Wigner harmonics and entanglement . . . . . . . . . . . . . . 88 4.4.3 Wigner harmonics, chaos, and thermalization . . . . . . . . . 95 4.4.4 Advantages of Wigner harmonics . . . . . . . . . . . . . . . . 99 4.5 4.6 Dynamics of disordered Ising chains . . . . . . . . . . . . . . . . . . . 100 4.5.1 Short term dynamics . . . . . . . . . . . . . . . . . . . . . . . 101 4.5.2 Long term dynamics . . . . . . . . . . . . . . . . . . . . . . . 102 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Engineering of multipartite entangled states in spin chains 107 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2.1 Heisenberg spin chain . . . . . . . . . . . . . . . . . . . . . . . 109 5.2.2 The quantum kicked rotor model and the Heisenberg spin chain model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3 Techniques for angular focusing of quantum rotors . . . . . . . . . . . 112 v 5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.4.1 Dynamics of the kicked spin chain vs dynamics of the quantum kicked rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.5 5.4.2 W state generation in a finite spin chain . . . . . . . . . . . . 120 5.4.3 Quasimomentum state generation . . . . . . . . . . . . . . . . 124 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Conclusions 129 Bibliography 132 Summary Quantum information processing has been the subject of intense research due to its potential applications in computation, communication, and fundamental science. In this regard, numerous physical systems like superconducting circuits, quantum dots etc., have been proposed for realizing quantum processors. In spite of the wide variety of the proposals, there exist a few classes of models that may describe the relevant properties of most such devices. One dimensional quantum spin systems called spin chains is one such class. In this thesis, we investigate the dynamics of spin chains from the perspective of quantum information processing. In particular, we consider three specific applications: quantum state transfer, quantum state amplification and quantum state engineering. First, we study the feasibility of using spin chain as a quantum wire. We propose an adiabatic scheme for robust high fidelity quantum transport. The scheme is studied both numerically and theoretically with a detailed discussion of its advantages. Next, by extending the ideas of this transport scheme, we propose a scheme for controlled measurement of a single spin state. We investigate the scheme in detail both in idealistic and realistic models. In addition, using the correspondence between spin chain and a well studied quantum dynamical system, we have vii come up with a scheme to engineer arbitrary quasimomentum states of spin chains. The scheme can also be used to efficiently generate entangled W states in spin chains. In addition to these applications, we have investigated the dynamics of spin chains for gaining insights into intriguing properties of quantum many-body systems. Along this line, we introduce a phase space based complexity measure to characterize the complex dynamics of a quantum many-body system. The use of this measure is investigated in a spin chain model. Furthermore, we have investigated the interplay between non-integrability and disorder in the quantum many-body dynamics of spin chains. viii period T0 = 1. As in the earlier case, we evolve the chain for time t = and then apply a field of strength C that minimizes the expectation value [ cos(6π/20) − cos(k) ]2 for each kick. For kicks where the expectation value cannot be minimized by field in the specified range, a weak field of strength c = 0.00001 is used. As the particular minimizing function is zero for k = ±6π/20 states, both quasimomentum states with k = ±6π/20 are equally populated. This is clear from Fig. 5.7(b) where the probability of both k = 6π/20 states are equal. Also, more than 49% probability in each of the two target states implies the efficiency of our scheme in generating quasimomentum states. Another illustration of our scheme is shown in Fig. 5.7(c) where the results are achieved by minimizing [ cos(14π/20) − cos(k) ]2 for 300 kicks. Here also we see about 49% population probabilities in both ±14π/20 states. Note that these are non-stationary states and hence evolves in time after turning off the parabolic field. Nevertheless, the population distribution of these states remains the same. Inspired by the success of our scheme in engineering different quasimomentum k states, we further test our scheme for generating superposition of states. For instance, let the state be engineered is an equal superposition state [|k1 + | − k1 + eiφ (|k2 + | − k2 )]/2. We found from our numerical investigations that any combination of previously used minimizing function of two k states does not work. Surprisingly, we could arrive at the target state by using a minimizing function discussed below. We minimized for each kick the function (2|cnk1 |2 −0.5)2 +(2|cnk2 |2 −0.5)2 +2(θ−φ)2 , where cnk1 and cnk2 are the probability amplitudes of the quasimomentum states |k1 and |k2 states respectively at the nth kick, and θ is the phase difference between them. Note that when the target state is reached, the value of the above minimizing function 126 Figure 5.8: Probability distribution of the quasimomentum states of a Heisenberg chain of 10 spins. Panel (a) corresponds to the initial state. Panel (b) and (c) corresponds to the resultant state obtained after applying 350 pulses and by using (2|cnk1 |2 − 0.5)2 + (2|cnk2 |2 − 0.5)2 + 2(θ − φ)2 as the minimizing function. In particular, for panel (b) k1 = 2π/10, k2 = 8π/10 and φ = and for (c) k1 = 4π/10, k2 = 8π/10 and φ = π/2. Pulse strength C is varied between [1 : 40] whereas the pulse duration T0 is kept at constant value of 1. is zero. Results of one such calculation are illustrated in Fig 5.8 with the initial state |5 . In the panel (b) of Fig. 5.8 an equal superposition state (|k = 2π/10 + |k = −2π/10 + |k = 8π/10 + |k = −8π/10 )/2 is engineered using our scheme. The field strength were limited to the interval [1:40] and kicking period T0 to 1. Initially, the minimizing function has a value of 0.09. After applying 350 kicks, the function has a value of 0.004. The probability distribution corresponding to this is shown in Fig. 350 5.8(b). We obtain |c350 ±k1 | = 0.2346 and |c±k2 | = 0.2086, quite close to the probability distribution required for the desired state. Also, our numerical calculation shows that 127 the phase difference θ is 0.0078. Another example for our scheme is illustrated in bottom panel where a state [|k = 4π/10 + |k = −4π/10 + eiπ/2 (|k = 8π/10 + |k = −8π/10 )]/2, an equal superposition of k states with a π/2 phase difference between the different k states, is targeted. Before applying the kick to the initial state |5 , the minimizing function has a value 2.557. Using our scheme, the value reduces to 350 0.0008 for the 350th kick with |c350 ±k1 | = 0.251 and |c±k2 | = 0.229. This is shown in Fig. 5.8(c). Calculated phase difference between the engineered superposition states was found to be 1.557, a value very close to π/2. These illustrate that our scheme can be generalized to engineer any arbitrary superposition of quasimomentum states. 5.5 Conclusions In short, we proposed a scheme for the generation of entangled W states in a spin chain by taking advantage of the mapping between the kicked Heisenberg spin chain and quantum kicked rotor model. In particular, the scheme is based on the accumulative angular squeezing technique used in the context of quantum rotor model [130]. We have numerically illustrated our scheme and showed that W state can be generated with high fidelity. The global entanglement of the state generated by our scheme is in close agreement with the targeted value. Furthermore, we have showed that by generalizing the minimizing function used in our scheme, it is possible to engineer arbitrary quasimomentum states of spin chains. 128 Chapter Conclusions In the previous chapters, we have presented our results on the studies of quantum spin chain dynamics and their potential applications in quantum information. The main findings of our study are summarized below. In efforts to achieve robust quantum state transfer, an adiabatic quantum population transfer scheme is proposed in chapter 2. The proposed scheme makes use of a slowly moving external parabolic potential and is qualitatively explained in terms of the adiabatic following of a quantum state with a moving separatrix structure in the classical phase space of a pendulum analogy. Detailed aspects of our adiabatic population transfer scheme, including its robustness, is studied computationally. Applications of our adiabatic scheme in quantum information transfer are also discussed, with emphasis placed on the usage of a dual spin chain to encode quantum phases. In chapter 3, we propose a simple scheme to realize controlled measurements of the state of a single spin, via controlled quantum signal amplification in a onedimensional spin chain model. It is shown that by adiabatically moving an external field applied to a spin chain that is also irradiated with a transverse driving field, a robust spin amplifier may be realized with little dispersion. In addition to the spin 129 based quantum information processing, our results also find applications in controlled growth of “Schr¨odinger cat” states. We propose a phase-space Wigner harmonics entropy measure for many-body quantum dynamical complexity in chapter 4. This measure, which reduces to the well known measure of complexity in classical systems and which is valid for both pure and mixed states in single-particle and many-body systems, takes into account the combined role of chaos and entanglement in the realm of quantum mechanics. The effectiveness of the measure is illustrated in the example of the Ising chain in a homogeneous tilted magnetic field. We provide numerical evidence that the multipartite entanglement generation leads to a linear increase of entropy until saturation in both integrable and chaotic regimes, so that in both cases the number of harmonics of the Wigner function grows exponentially with time. The entropy growth rate can be used to detect quantum phase transitions. The proposed entropy measure can also distinguish between integrable and chaotic many-body dynamics by means of the size of long term fluctuations which become smaller when quantum chaos sets in. In the latter part of chapter 4, we studied the effects of non-integrability on many-body localization-delocalization transitions using disordered Ising chain in a homogeneous tilted field and investigated the generation of global entanglement in the chain. Our findings suggest the existence of the localization-delocalization transition for weak disorders even in non-integrable regimes. In addition, the transition point was found to depend strongly on the integrability breaking term. Also, we observed the existence of many-body localization in the presence of strong disorder in our model. In chapter 5, we introduce a scheme to engineer W state in a spin chain. The scheme is mainly built on the accumulative angular squeezing technique used in the 130 context of quantum kicked rotor for focussing rotors to a delta-like angular distribution. Our studies shows that efficient generation of W state can be achieved using our scheme. Also, we have shown that the scheme can be further generalized to engineer arbitrary quasimomentum superposition states of a spin chain. Our studies on spin chain dynamics may be extended to gain more insights into many-body aspects of quantum system. For instance, our complexity measure works equally well with both pure and mixed states. Hence, it could be studied in relation to mixed state entanglement. This would be particularly interesting as mixed state entanglement is at present not well understood and is the focus of ongoing research. Also, the scheme we introduced can be exploited for other interesting applications. For instance, as the parabolic external field used in our adiabatic transport scheme can enclose more than one excitation, it is interesting to investigate the usefulness of our scheme to transfer entangled states. 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E 81, 046201 (2010). 141 [...]... the two spins i and j Depending upon the directions of spin- spin interaction, spin chains can be classified into three types: Heisenberg spin chain, XY spin chain and Ising spin chain In the following, we will describe each of the spin chains in detail along with their possible physical realization 12 1.5.1 Heisenberg spin chain In the Heisenberg spin chain, the spin- spin interaction exists in all the... have the same spin state, the second spin can be flipped by the driving field only if the first spin (acting as the object spin) is up The flipped second spin can then cause the flipping of the third spin, and so on, thus triggering a wave of spin excitation in the spin chain and realizing quantum signal amplification The fruitful interplay between the studies of spin chains and quantum information theory... a spin chain 1.6 Applications of spin chains in quantum information processing Researchers all over the world have come up with numerous proposals considering spin chains as a promising candidate for quantum information processing Using spin chains for universal quantum computation [36], for quantum communication [37], for measuring quantum states [38], for generating quantum entanglement [39], and. .. disorder in spin chain 48 x 2.10 Adiabatic transport of an initial single excitation in the presence of dynamic disorder in spin chain 50 2.11 Same as in Fig 2.10 but with strong dynamics disorder 51 3.1 Spin polarization profile for an idealized spin chain model 65 3.2 Time dependence of the total polarization for an idealized model of spin chain with a moving... implementation of quantum computers There lies also 17 a deep theoretical aspect of gaining insight into intriguing properties of quantum many-body systems This is discussed in detail in the later section 1.7 Manifestations of quantum many-body nature in spin chains Spin chains have been used as a paradigmatic model for studying the many-body effects like entanglement and its scaling at the critical points of quantum. .. as explained in Heisenberg spin chain subsection [29] Also, Ising interactions can be realized using trapped ions [34] and superconducting qubits [35] Motivated by the large number of possible physical realizations, there has been an explosion of interest in spin chains over the last few years, mainly in the quantum information community In the next section, we briefly discuss some interesting applications. .. computing requires the transfer of quantum information between different quantum processors Hence, the linking of quantum processors efficiently is essential Spin chains can be used as a coherent data bus and the natural evolution of the spin chain can be used to transfer quantum states [37] The basic transport protocol consists of the following steps 1 Initialize the spin chain in its ground state 2 Put the...List of Publications 1 Vinitha Balachandran, and Jiangbin Gong, Adiabatic quantum transport in a spin chain with a moving potential, Phys Rev A 77, 012303 (2008) 2 Vinitha Balachandran, and Jiangbin Gong, Controlled measurement processes: Simple spin- chain model of controlled quantum- state amplification, Phys Rev A 79, 012317 (2009) 3 Vinitha Balachandran, Giuliano Benenti, Giulio Casati, and Jiangbin... Also, their Hamiltonian is very simple and thereby allows the easy identification of the parameters responsible for specific effects In the next section, we elaborate the spin chain and its potential advantages in the context of quantum information processing 11 1.5 Spin chain Spins are systems endowed with tiny quantized magnetic moments Bulk materials usually consist of large collection of spins permanently... for cloning quantum states [40] are a few examples 1.6.1 Universal quantum computation A spin chain can be used as a processor core for a quantum computer This is because universal quantum gates [36], accomplishing all types of quantum computation, can be constructed using a spin chain For instance, consider the most common Heisenberg spin chain Encode the qubit in a subspace made up of three spins as . STUDIES OF QUANTUM SPIN CHAIN DYNAMICS AND THEIR POTENTIAL APPLICATIONS IN QUANTUM INFORMATION VINITHA BALACHANDRAN (M. Sc., Cochin University of Science And Technology, India). STUDIES OF QUANTUM SPIN CHAIN DYNAMICS AND THEIR POTENTIAL APPLICATIONS IN QUANTUM INFORMATION VINITHA BALACHANDRAN NATIONAL UNIVERSITY OF SINGAPORE. relevant properties of most such devices. One dimensional quantum spin systems called spin chains is one such class. In this thesis, we investigate the dynamics of spin chains from the perspective of quantum information