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Dynamics and Simulation of Open Quantum Systems

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Dynamics and Simulation of Open Quantum Systems Dynamics and Simulation of Open Quantum Systems Thesis by Michael Philip Zwolak In Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2008 (Defended June 12, 2007) ii °c 2008 Michael Philip Zwolak All Rights Reserved iii To my family and friends iv Acknowledgements I would like to thank John Preskill, Guifre Vidal, Massimiliano Di Ventra, Gil Refael, and Y. A. Liu for support, guidance, and/or collaboration. Also, I am appreciative of all of my collaborators over the years for fruitful scienti¯c discourse. And of course, I am ever grateful to my family and friends. v Abstract All systems are open to an essentially uncontrollable environment that acts as a source of decoherence and dissipation. In some cases the environment's only e®ect is to add a weak relaxation mechanism and thus can be ignored for short timescales. In others, however, the presence of the environment can fundamentally alter the behavior of the system. Such is the case in mesoscopic superconductors where the environment can stabilize superconductivity and in spin-boson systems where the environment induces a localization transition. Likewise, in technological applications we are often interested in systems operating far from equilibrium. Here the environment might act as a particle reservoir or strong driving force. In all these examples, we need accurate methods to describe the influence of the environment on the system and to solve for the resulting dynamics or equilibrium states. In this thesis, we develop computational and conceptual approaches to e±ciently simulate quantum systems in contact with an environment. Our starting point is the use of numerical renormalization techniques. Thus, we restrict our attention to one-dimensional lattices or small quantum systems coupled to an environment. We have developed several complementary algorithms: a superoperator renormalization algorithm for simulating real-time Markovian dynamics and for calculating states in thermal equilibrium; a blocking algorithm for simulating integro-di®erential equations with long-time memory; and a tensor network algorithm for branched lattices, which can be used to simulate strongly dissipative systems. Further, we provide support for an idea that to generically and accurately simulate the real-time dynamics of strongly dissipative systems, one has to include all or part of the environment within the simulation. In addition, we discuss applications and open questions. vi Contents Acknowledgements iv Abstract v 1 Introduction 1 1.1 Some examples . 2 1.1.1 Strongly dissipative quantum systems . 2 1.1.2 Driven quantum systems . 4 1.2 Dissipative quantum systems . 4 1.2.1 Representing the environment . 5 1.2.2 Generic path integral treatment . 7 1.2.3 Dissipative quantum phase transitions . 9 1.3 Matrix product state simulations . 13 1.3.1 Representation . 14 1.3.2 Simulation . 17 1.4 Summary . 19 2 Mixed state quantum dynamics in one dimension 20 2.1 Introduction . 20 2.2 Superoperator renormalization . 21 2.2.1 Reduced superoperators . 22 2.2.2 Renormalization of reduced superoperators . 23 2.2.3 Matrix product decomposition and TEBD . 23 2.3 Examples . 24 2.3.1 Thermal states . 24 2.3.2 Time-dependent Markovian master equation . 25 2.3.3 Unequal-time correlators . 26 2.3.4 Pure state density matrices . 27 2.4 Conclusions . 29 vii 3 Numerical ansatz for integro-di®erential equations 30 3.1 Introduction . 30 3.2 Algorithm . 32 3.2.1 Increasing Smoothness . 33 3.2.2 Blocking algorithm for a polynomially decaying kernel . 34 3.2.3 Growing blocks in real time . 38 3.2.4 Errors . 38 3.3 Examples . 42 3.3.1 Oscillating integro-di®erential equation . 42 3.3.2 NIBA equations . 43 3.4 Conclusions . 46 3.5 Subsidiary Calculations . 46 3.5.1 Higher-order blocking algorithms . 46 3.5.2 Two-stage Runge-Kutta method . 47 3.5.3 Simple derivation of the NIBA equations . 49 4 Untrotterized: adding matrix product states 53 4.1 Introduction . 53 4.2 Addition of matrix product states . 54 4.2.1 Variational method for ¯nding an optimal MPS . 54 4.2.2 Adding together many MPS . 61 4.3 Integrating with MPS addition . 62 4.3.1 Integration . 65 4.3.2 Locally updatable integrals . 66 4.3.3 Computational cost . 68 4.3.3.1 Independent reservoirs . 68 4.3.3.2 Single reservoir . 68 4.3.3.3 Arbitrary reservoir . 69 4.4 Conclusions . 70 5 Generic approach for simulating real-time, non-Markovian dynamics 71 5.1 Introduction . 71 5.2 Generic Approach . 72 5.3 Another approach to generic non-Markovian dynamics . 81 5.3.1 Non-Markovian amplitude damping . 82 5.3.1.1 Exact . 82 5.3.1.2 Born Approximation . 85 viii 5.3.1.3 Shabani-Lidar Equation . 87 5.3.2 Solution . 87 5.3.3 Summary . 88 5.3.4 Some subsidiary calculations . 89 5.3.4.1 Reservoir correlation function . 89 5.3.4.2 Damping Basis . 90 5.4 Keldysh approach for non-Markovian equations . 91 5.4.1 Equations of motion . 91 5.4.2 Full equation for the density matrix . 96 5.4.3 Shabani-Lidar revisited . 97 5.4.4 Breakdown of positivity . 98 5.5 Conclusions . 98 6 In¯nite system tensor network simulations 101 6.1 Introduction . 101 6.2 Algorithm . 102 6.2.1 Examples . 104 6.2.1.1 One-dimensional Ising . 104 6.2.1.2 Ising with branches . 106 6.2.2 Logarithmic covering of reservoir modes . 106 6.3 Conclusions . 107 7 Dynamical approach to transport 108 7.1 Introduction . 108 7.2 Transport with MPS . 109 7.3 Dynamical corrections to the DFT-LDA electron conductance . 112 7.4 Steady-state solutions in quantum transport . 119 7.4.1 Variational method . 120 7.4.2 Particles on a lattice . 122 7.4.2.1 Single fermion on the lattice . 123 7.4.2.2 Many noninteracting particles . 124 7.4.2.3 Interacting particle states . 125 7.4.3 Variational Procedure . 125 7.4.3.1 Free parameters . 125 7.4.3.2 Minimization for noninteracting fermions . 127 7.4.4 Example . 129 7.4.5 Conclusions . 131 ix 7.4.6 Subsidiary calculations . 132 7.5 Conclusions . 134 8 Entropy of \multi-channel" models 136 8.1 Introduction . ..
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