# Using Matrix-Product States for Open Quantum Many-Body Systems: Efficient Algorithms for Markovian and Non-Markovian Time-Evolution

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Time Evolution with Matrix-Product States

## 3. Modeling Markovian System–Reservoir Interaction

#### 3.1. Model

#### 3.2. Quantum Stochastic Schrödinger Equation (QSSE)

#### 3.3. Algorithm

## 4. Modeling Non-Markovian System—Reservoir Interaction

#### 4.1. Model

#### 4.2. Algorithm

## 5. Application Examples

#### 5.1. A Dissipative Spin Chain with Markovian Interaction

#### 5.2. An Open Spin Chain in a Semi-Infinite Waveguide

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Breuer, H.P.; Petruccione, F.F. The Theory of Open Quantum Systems; Oxford University Press: Oxford, UK, 2002. [Google Scholar]
- Crispin Gardiner, P.Z. Quantum Noise—A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics; Springer Verlag: Berlin, Germany, 2002. [Google Scholar]
- Carmichael, H. An Open Systems Approach to Quantum Optics; Springer: Berlin, Germany, 1993. [Google Scholar]
- Weiss, U. Quantum Dissipative Systems, 4th ed.; World Scientific Publishing Co.: Singapore, 2012. [Google Scholar]
- Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- de Vega, I.; Porras, D.; Ignacio Cirac, J. Matter-Wave Emission in Optical Lattices: Single Particle and Collective Effects. Phys. Rev. Lett.
**2008**, 101, 260404. [Google Scholar] [CrossRef] [PubMed] - Navarrete-Benlloch, C.; de Vega, I.; Porras, D.; Cirac, J.I. Simulating quantum-optical phenomena with cold atoms in optical lattices. New J. Phys.
**2011**, 13, 023024. [Google Scholar] [CrossRef][Green Version] - Häffner, H.; Roos, C.; Blatt, R. Quantum computing with trapped ions. Phys. Rep.
**2008**, 469, 155–203. [Google Scholar] [CrossRef][Green Version] - Blatt, R.; Wineland, D. Entangled states of trapped atomic ions. Nature
**2008**, 453, 1008–1015. [Google Scholar] [CrossRef] - Rotter, I.; Bird, J.P. A review of progress in the physics of open quantum systems: Theory and experiment. Rep. Prog. Phys.
**2015**, 78, 114001. [Google Scholar] [CrossRef] - Nogues, G.; Rauschenbeutel, A.; Osnaghi, S.; Brune, M.; Raimond, J.M.; Haroche, S. Seeing a single photon without destroying it. Nature
**1999**, 400, 239–242. [Google Scholar] [CrossRef] - Prokof, N.V.; Stamp, P.C.E. Giant spins and topological decoherence: A Hamiltonian approach. J. Condens. Matter Phys.
**1993**, 5, L663–L670. [Google Scholar] [CrossRef] - Lambert, N.; Chen, Y.N.; Cheng, Y.C.; Li, C.M.; Chen, G.Y.; Nori, F. Quantum biology. Nat. Phys.
**2013**, 9, 10–18. [Google Scholar] [CrossRef] - Daley, A. Quantum trajectories and open many-body quantum systems. Adv. Phys.
**2014**, 63, 77–149. [Google Scholar] [CrossRef][Green Version] - de Vega, I.; Alonso, D. Dynamics of non-Markovian open quantum systems. Rev. Mod. Phys.
**2017**, 89, 015001. [Google Scholar] [CrossRef][Green Version] - Koch, C.P. Controlling open quantum systems: Tools, achievements, and limitations. J. Condens. Matter Phys.
**2016**, 28, 213001. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cirac, J.I.; Zoller, P. Goals and opportunities in quantum simulation. Nat. Phys.
**2012**, 8, 264–266. [Google Scholar] [CrossRef] - Droenner, L.; Carmele, A. Boundary-driven Heisenberg chain in the long-range interacting regime: Robustness against far-from-equilibrium effects. Phys. Rev. B
**2017**, 96, 184421. [Google Scholar] [CrossRef][Green Version] - Žnidarič, M.; Scardicchio, A.; Varma, V.K. Diffusive and Subdiffusive Spin Transport in the Ergodic Phase of a Many-Body Localizable System. Phys. Rev. Lett.
**2016**, 117, 040601. [Google Scholar] [CrossRef] [PubMed][Green Version] - Heyl, M.; Polkovnikov, A.; Kehrein, S. Dynamical Quantum Phase Transitions in the Transverse-Field Ising Model. Phys. Rev. Lett.
**2013**, 110, 135704. [Google Scholar] [CrossRef] [PubMed] - Huber, J.; Kirton, P.; Rabl, P. Non-equilibrium magnetic phases in spin lattices with gain and loss. arXiv
**2019**, arXiv:1908.02290. [Google Scholar] - Huber, J.; Rabl, P. Active energy transport and the role of symmetry breaking in microscopic power grids. Phys. Rev. A
**2019**, 100, 012129. [Google Scholar] [CrossRef][Green Version] - Pizzi, A.; Nunnenkamp, A.; Knolle, J. Bistability and time crystals in long-ranged directed percolation. arXiv
**2020**, arXiv:2004.13034. [Google Scholar] - Bertini, B.; Heidrich-Meisner, F.; Karrasch, C.; Prosen, T.; Steinigeweg, R.; Znidaric, M. Finite-temperature transport in one-dimensional quantum lattice models. arXiv
**2020**, arXiv:2003.03334. [Google Scholar] - Hauke, P.; Tagliacozzo, L. Spread of Correlations in Long-Range Interacting Quantum Systems. Phys. Rev. Lett.
**2013**, 111, 207202. [Google Scholar] [CrossRef] - Trautmann, N.; Hauke, P. Trapped-ion quantum simulation of excitation transport: Disordered, noisy, and long-range connected quantum networks. Phys. Rev. A
**2018**, 97, 023606. [Google Scholar] [CrossRef][Green Version] - Prosen, T.C.V. Exact Nonequilibrium Steady State of a Strongly Driven Open XXZ Chain. Phys. Rev. Lett.
**2011**, 107, 137201. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ljubotina, M.; Žnidari, M.; Prosen, T. Spin diffusion from an inhomogeneous quench in an integrable system. Nat. Commun.
**2017**, 8, 16117. [Google Scholar] [CrossRef][Green Version] - Lange, F.; Ejima, S.; Shirakawa, T.; Yunoki, S.; Fehske, H. Spin transport through a spin-1/2 XXZ chain contacted to fermionic leads. Phys. Rev. B
**2018**, 97, 245124. [Google Scholar] [CrossRef][Green Version] - Žnidarič, M. Spin Transport in a one-dimensional anisotropic Heisenberg model. Phys. Rev. Lett.
**2011**, 106, 220601. [Google Scholar] [CrossRef] [PubMed] - Žnidarič, M. Transport in a one-dimensional isotropic Heisenberg model at high temperature. J. Stat. Mech. Theory Exp.
**2011**, 2011, P12008. [Google Scholar] [CrossRef][Green Version] - Katzer, M.; Knorr, W.; Finsterhölzl, R.; Carmele, A. Long-range interaction in an open boundary-driven Heisenberg spin lattice—A far-from-equilibrium transition to ballistic transport. arXiv
**2020**, arXiv:2004.12738. [Google Scholar] [CrossRef] - Wang, T.; Wang, X.; Sun, Z. Entanglement oscillations in open Heisenberg chains. Phys. A
**2007**, 383, 316–324. [Google Scholar] [CrossRef][Green Version] - Wu, Y.Z.; Ren, J.; Jiang, X.F. Dynamics of entanglement in Heisenberg chains with asymmetric DzyaloShinskii-moriya interactions. Int. J. Quantum Inf.
**2011**, 09, 751–761. [Google Scholar] [CrossRef] - Lindblad, G. On the generators of quantum dynamical semigroups. Comm. Math. Phys.
**1976**, 48, 119–130. [Google Scholar] [CrossRef] - Pollet, L. Recent developments in quantum Monte Carlo simulations with applications for cold gases. Rep. Prog. Phys.
**2012**, 75, 094501. [Google Scholar] [CrossRef] [PubMed] - Kimble, H.J.; Dagenais, M.; Mandel, L. Photon Antibunching in Resonance Fluorescence. Phys. Rev. Lett.
**1977**, 39, 691–695. [Google Scholar] [CrossRef][Green Version] - Dalibard, J.; Castin, Y.; Mølmer, K. Wave-function approach to dissipative processes in quantum optics. Phys. Rev. Lett.
**1992**, 68, 580–583. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dum, R.; Zoller, P.; Ritsch, H. Monte Carlo simulation of the atomic master equation for spontaneous emission. Phys. Rev. A
**1992**, 45, 4879–4887. [Google Scholar] [CrossRef] - Zoller, P.; Gardiner, C.W. Quantum Noise in Quantum Optics: The Stochastic Schrödinger Equation. arXiv
**1997**, arXiv:quant-ph/9702030. [Google Scholar] - Alonso, D.; de Vega, I. Multiple-Time Correlation Functions for Non-Markovian Interaction: Beyond the Quantum Regression Theorem. Phys. Rev. Lett.
**2005**, 94, 200403. [Google Scholar] [CrossRef][Green Version] - Piilo, J.; Maniscalco, S.; Härkönen, K.; Suominen, K.A. Non-Markovian Quantum Jumps. Phys. Rev. Lett.
**2008**, 100, 180402. [Google Scholar] [CrossRef][Green Version] - Pichler, H.; Zoller, P. Photonic Circuits with Time Delays and Quantum Feedback. Phys. Rev. Lett.
**2016**, 116, 093601. [Google Scholar] [CrossRef][Green Version] - Heisenberg, W. Zur Theorie des Ferromagnetismus. Z. Phys.
**1928**, 49, 619–636. [Google Scholar] [CrossRef] - Bethe, H. Zur Theorie der Metalle—I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Zeitschrift für Physik
**1931**, 71, 205–226. [Google Scholar] [CrossRef] - Dupont, M.; Moore, J.E. Universal spin dynamics in infinite-temperature one-dimensional quantum magnets. Phys. Rev. B
**2020**, 101, 121106. [Google Scholar] [CrossRef][Green Version] - Hild, S.; Fukuhara, T.; Schauß, P.; Zeiher, J.; Knap, M.; Demler, E.; Bloch, I.; Gross, C. Far-from-Equilibrium Spin Transport in Heisenberg Quantum Magnets. Phys. Rev. Lett.
**2014**, 113, 147205. [Google Scholar] [CrossRef] [PubMed] - Tang, Y.; Kao, W.; Li, K.Y.; Seo, S.; Mallayya, K.; Rigol, M.; Gopalakrishnan, S.; Lev, B.L. Thermalization near Integrability in a Dipolar Quantum Newton’s Cradle. Phys. Rev. X
**2018**, 8, 021030. [Google Scholar] [CrossRef][Green Version] - Langen, T.; Erne, S.; Geiger, R.; Rauer, B.; Schweigler, T.; Kuhnert, M.; Rohringer, W.; Mazets, I.E.; Gasenzer, T.; Schmiedmayer, J. Experimental observation of a generalized Gibbs ensemble. Science
**2015**, 348, 207–211. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kinoshita, T.; Wenger, T.; Weiss, D.S. A quantum Newton’s cradle. Nature
**2006**, 440, 900–903. [Google Scholar] [CrossRef] - Maier, C.; Brydges, T.; Jurcevic, P.; Trautmann, N.; Hempel, C.; Lanyon, B.P.; Hauke, P.; Blatt, R.; Roos, C.F. Environment-Assisted Quantum Transport in a 10-qubit Network. Phys. Rev. Lett.
**2019**, 122, 050501. [Google Scholar] [CrossRef] [PubMed][Green Version] - Derrida, B.; Evans, M.R.; Hakim, V.; Pasquier, V. Exact solution of a 1D asymmetric exclusion model using a matrix formulation. J. Phys. A Math. Theor.
**1993**, 26, 1493–1517. [Google Scholar] [CrossRef] - Fannes, M.; Nachtergaele, B.; Werner, R.F. Finitely correlated states on quantum spin chains. Comm. Math. Phys.
**1992**, 144, 443–490. [Google Scholar] [CrossRef] - Kolezhuk, A.K.; Mikeska, H.J. Finitely Correlated Generalized Spin Ladders. Int. J. Mod. Phys. B
**1998**, 12, 2325–2348. [Google Scholar] [CrossRef][Green Version] - Schollwöck, U. The density-matrix renormalization group. Rev. Mod. Phys.
**2005**, 77, 259–315. [Google Scholar] [CrossRef][Green Version] - Schollwöck, U. The density-matrix renormalization group in the age of matrix product states. Ann. Phys. N. Y.
**2011**, 326, 96–192. [Google Scholar] [CrossRef][Green Version] - White, S.R. Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B
**1993**, 48, 10345–10356. [Google Scholar] [CrossRef] [PubMed] - Vidal, G. Efficient Classical Simulation of Slightly Entangled Quantum Computations. Phys. Rev. Lett.
**2003**, 91, 147902. [Google Scholar] [CrossRef] [PubMed][Green Version] - Vidal, G. Efficient Simulation of One-Dimensional Quantum Many-Body Systems. Phys. Rev. Lett.
**2004**, 93, 040502. [Google Scholar] [CrossRef] [PubMed][Green Version] - White, S.R.; Feiguin, A.E. Real-Time Evolution Using the Density Matrix Renormalization Group. Phys. Rev. Lett.
**2004**, 93, 076401. [Google Scholar] [CrossRef] [PubMed][Green Version] - Orús, R.; Vidal, G. Infinite time-evolving block decimation algorithm beyond unitary evolution. Phys. Rev. B
**2008**, 78, 155117. [Google Scholar] [CrossRef][Green Version] - Suzuki, M. Relationship between d-Dimensional Quantal Spin Systems and (d+1)-Dimensional Ising Systems: Equivalence, Critical Exponents and Systematic Approximants of the Partition Function and Spin Correlations. Prog. Theor. Phys.
**1976**, 56, 1454–1469. [Google Scholar] [CrossRef] - Suzuki, M. General theory of fractal path integrals with applications to many-body theories and statistical physics. J. Math. Phys.
**1991**, 32, 400–407. [Google Scholar] [CrossRef] - Paeckel, S.; Köhler, T.; Swoboda, A.; Manmana, S.R.; Schollwöck, U.; Hubig, C. Time-evolution methods for matrix-product states. Ann. Phys.
**2019**, 411, 167998. [Google Scholar] [CrossRef] - Benenti, G.; Casati, G.; Prosen, T.; Rossini, D. Negative differential conductivity in far-from-equilibrium quantum spin chains. EPL
**2009**, 85, 37001. [Google Scholar] [CrossRef][Green Version] - Prosen, T. Matrix product solutions of boundary driven quantum chains. J. Phys. A Math. Theor.
**2015**, 48, 373001. [Google Scholar] [CrossRef][Green Version] - Prosen, T.; Žnidarič, M. Matrix product simulations of non-equilibrium steady states of quantum spin chains. J. Stat. Mech. Theory Exp.
**2009**, 2009, P02035. [Google Scholar] [CrossRef][Green Version] - Karevski, D.; Popkov, V.; Schütz, G.M. Exact Matrix Product Solution for the Boundary-Driven Lindblad XXZ Chain. Phys. Rev. Lett.
**2013**, 110, 047201. [Google Scholar] [CrossRef][Green Version] - Cai, Z.; Barthel, T. Algebraic versus Exponential Decoherence in Dissipative Many-Particle Systems. Phys. Rev. Lett.
**2013**, 111, 150403. [Google Scholar] [CrossRef] [PubMed][Green Version] - Xu, X.; Guo, C.; Poletti, D. Interplay of interaction and disorder in the steady state of an open quantum system. Phys. Rev. B
**2018**, 97, 140201. [Google Scholar] [CrossRef][Green Version] - Mendoza-Arenas, J.J.; Žnidarič, M.; Varma, V.K.; Goold, J.; Clark, S.R.; Scardicchio, A. Asymmetry in energy versus spin transport in certain interacting disordered systems. Phys. Rev. B
**2019**, 99, 094435. [Google Scholar] [CrossRef][Green Version] - Popkov, V.; Prosen, T.C.V.; Zadnik, L. Inhomogeneous matrix product ansatz and exact steady states of boundary-driven spin chains at large dissipation. Phys. Rev. E
**2020**, 101, 042122. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mascarenhas, E.; Flayac, H.; Savona, V. Matrix-product-operator approach to the nonequilibrium steady state of driven-dissipative quantum arrays. Phys. Rev. A
**2015**, 92, 022116. [Google Scholar] [CrossRef][Green Version] - Strathearn, A.; Kirton, P.; Kilda, D.; Keeling, J.; Lovett, B.W. Efficient non-Markovian quantum dynamics using time-evolving matrix product operators. Nat. Commun.
**2018**, 9, 3322. [Google Scholar] [CrossRef] - Droenner, L.; Naumann, N.L.; Schöll, E.; Knorr, A.; Carmele, A. Quantum Pyragas control: Selective control of individual photon probabilities. Phys. Rev. A
**2019**, 99, 023840. [Google Scholar] [CrossRef][Green Version] - Carmele, A.; Nemet, N.; Canela, V.; Parkins, S. Pronounced non-Markovian features in multiply excited, multiple emitter waveguide QED: Retardation induced anomalous population trapping. Phys. Rev. Res.
**2020**, 2, 013238. [Google Scholar] [CrossRef][Green Version] - Német, N.; Carmele, A.; Parkins, S.; Knorr, A. Comparison between continuous- and discrete-mode coherent feedback for the Jaynes-Cummings model. Phys. Rev. A
**2019**, 100, 023805. [Google Scholar] [CrossRef][Green Version] - Eisert, J.; Cramer, M.; Plenio, M.B. Colloquium: Area laws for the entanglement entropy. Rev. Mod. Phys.
**2010**, 82, 277–306. [Google Scholar] [CrossRef][Green Version] - Orus, R. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Ann. Phys. N. Y.
**2014**, 349, 117–158. [Google Scholar] [CrossRef][Green Version] - McCulloch, I.P. From density-matrix renormalization group to matrix product states. J. Stat. Mech. Theory Exp.
**2007**, 2007, P10014. [Google Scholar] [CrossRef][Green Version] - Collins, B.; González-Guillén, C.E.; Pérez-García, D. Matrix Product States, Random Matrix Theory and the Principle of Maximum Entropy. Comm. Math. Phys.
**2013**, 663, 677. [Google Scholar] [CrossRef][Green Version] - García-Ripoll, J.J. Time evolution of Matrix Product States. New J. Phys.
**2006**, 8, 305. [Google Scholar] [CrossRef] - Wolf, M.M.; Ortiz, G.; Verstraete, F.; Cirac, J.I. Quantum Phase Transitions in Matrix Product Systems. Phys. Rev. Lett.
**2006**, 97, 110403. [Google Scholar] [CrossRef][Green Version] - Chan, G.K.-L.; Keselman, A.; Nakatani, N.; Li, Z.; White, S.R. Matrix product operators, matrix product states, and ab initio density matrix renormalization group algorithms. J. Chem. Phys.
**2016**, 145, 014102. [Google Scholar] [CrossRef] - Wang, Z.; Jaako, T.; Kirton, P.; Rabl, P. Supercorrelated Radiance in Nonlinear Photonic Waveguides. Phys. Rev. Lett.
**2020**, 124, 213601. [Google Scholar] [CrossRef] - Ramos, T.; Pichler, H.; Daley, A.J.; Zoller, P. Quantum Spin Dimers from Chiral Dissipation in Cold-Atom Chains. Phys. Rev. Lett.
**2014**, 113, 237203. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ramos, T.; Vermersch, B.; Hauke, P.; Pichler, H.; Zoller, P. Non-Markovian dynamics in chiral quantum networks with spins and photons. Phys. Rev. A
**2016**, 93, 062104. [Google Scholar] [CrossRef][Green Version] - Tufarelli, T.; Ciccarello, F.; Kim, M.S. Dynamics of spontaneous emission in a single-end photonic waveguide. Phys. Rev. A
**2013**, 87, 013820. [Google Scholar] [CrossRef] - Tufarelli, T.; Kim, M.S.; Ciccarello, F. Non-Markovianity of a quantum emitter in front of a mirror. Phys. Rev. A
**2014**, 90, 012113. [Google Scholar] [CrossRef][Green Version] - Karrasch, C.; Moore, J.E.; Heidrich-Meisner, F. Real-time and real-space spin and energy dynamics in one-dimensional spin-1/2 systems induced by local quantum quenches at finite temperatures. Phys. Rev. B
**2014**, 89, 075139. [Google Scholar] [CrossRef][Green Version] - Ilievski, E.; De Nardis, J.; Medenjak, M.; Prosen, T.c.v. Superdiffusion in One-Dimensional Quantum Lattice Models. Phys. Rev. Lett.
**2018**, 121, 230602. [Google Scholar] [CrossRef] [PubMed][Green Version] - Medenjak, M.; Karrasch, C.; Prosen, T.c.v. Lower Bounding Diffusion Constant by the Curvature of Drude Weight. Phys. Rev. Lett.
**2017**, 119, 080602. [Google Scholar] [CrossRef] [PubMed][Green Version] - Benenti, G.; Casati, G.; Prosen, T.C.V.; Rossini, D.; Žnidarič, M. Charge and spin transport in strongly correlated one-dimensional quantum systems driven far from equilibrium. Phys. Rev. B
**2009**, 80, 035110. [Google Scholar] [CrossRef][Green Version] - Hughes, S. Coupled-Cavity QED Using Planar Photonic Crystals. Phys. Rev. Lett.
**2007**, 98, 083603. [Google Scholar] [CrossRef] - Fang, Y.L.L.; Baranger, H.U. Waveguide QED: Power spectra and correlations of two photons scattered off multiple distant qubits and a mirror. Phys. Rev. A
**2015**, 91, 053845. [Google Scholar] [CrossRef][Green Version] - Calajó, G.; Fang, Y.L.L.; Baranger, H.U.; Ciccarello, F. Exciting a Bound State in the Continuum through Multiphoton Scattering Plus Delayed Quantum Feedback. Phys. Rev. Lett.
**2019**, 122, 073601. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dorner, U.; Zoller, P. Laser-driven atoms in half-cavities. Phys. Rev. A
**2002**, 66, 023816. [Google Scholar] [CrossRef][Green Version] - Trautmann, N.; Alber, G. Dissipation-enabled efficient excitation transfer from a single photon to a single quantum emitter. Phys. Rev. A
**2016**, 93, 053807. [Google Scholar] [CrossRef][Green Version] - Faulstich, F.M.; Kraft, M.; Carmele, A. Unraveling mirror properties in time-delayed quantum feedback scenarios. J. Mod. Opt.
**2018**, 65, 1323–1331. [Google Scholar] [CrossRef][Green Version] - Cook, R.; Schuster, D.I.; Cleland, A.N.; Jacobs, K. Input-output theory for superconducting and photonic circuits that contain weak retroreflections and other weak pseudocavities. Phys. Rev. A
**2018**, 98, 013801. [Google Scholar] [CrossRef][Green Version] - Cook, R.J.; Milonni, P.W. Quantum theory of an atom near partially reflecting walls. Phys. Rev. A
**1987**, 35, 5081–5087. [Google Scholar] [CrossRef][Green Version] - Milonni, P.W.; Knight, P.L. Retardation in the resonant interaction of two identical atoms. Phys. Rev. A
**1974**, 10, 1096–1108. [Google Scholar] [CrossRef] - Német, N.; Parkins, S. Enhanced optical squeezing from a degenerate parametric amplifier via time-delayed coherent feedback. Phys. Rev. A
**2016**, 94, 023809. [Google Scholar] [CrossRef][Green Version] - Crowder, G.; Carmichael, H.; Hughes, S. Quantum trajectory theory of few-photon cavity-QED systems with a time-delayed coherent feedback. Phys. Rev. A
**2020**, 101, 023807. [Google Scholar] [CrossRef][Green Version] - Barkemeyer, K.; Finsterhölzl, R.; Knorr, A.; Carmele, A. Revisiting Quantum Feedback Control: Disentangling the Feedback-Induced Phase from the Corresponding Amplitude. Adv. Quantum Technol.
**2020**, 3, 1900078. [Google Scholar] [CrossRef] - Lu, Y.; Naumann, N.L.; Cerrillo, J.; Zhao, Q.; Knorr, A.; Carmele, A. Intensified antibunching via feedback-induced quantum interference. Phys. Rev. A
**2017**, 95, 063840. [Google Scholar] [CrossRef][Green Version] - Guimond, P.O.; Pletyukhov, M.; Pichler, H.; Zoller, P. Delayed coherent quantum feedback from a scattering theory and a matrix product state perspective. Quantum Sci. Technol.
**2017**, 2, 044012. [Google Scholar] [CrossRef][Green Version] - Guimond, P.O.; Pichler, H.; Rauschenbeutel, A.; Zoller, P. Chiral quantum optics with V-level atoms and coherent quantum feedback. Phys. Rev. A
**2016**, 94, 033829. [Google Scholar] [CrossRef][Green Version] - Kabuss, J.; Krimer, D.O.; Rotter, S.; Stannigel, K.; Knorr, A.; Carmele, A. Analytical study of quantum-feedback-enhanced Rabi oscillations. Phys. Rev. A
**2015**, 92, 053801. [Google Scholar] [CrossRef][Green Version] - Kabuss, J.; Katsch, F.; Knorr, A.; Carmele, A. Unraveling coherent quantum feedback for Pyragas control. J. Opt. Soc. Am. B
**2016**, 33, C10–C16. [Google Scholar] [CrossRef][Green Version] - Carmele, A.; Kabuss, J.; Schulze, F.; Reitzenstein, S.; Knorr, A. Single Photon Delayed Feedback: A Way to Stabilize Intrinsic Quantum Cavity Electrodynamics. Phys. Rev. Lett.
**2013**, 110, 013601. [Google Scholar] [CrossRef] - Finsterhölzl, R.; Katzer, M.; Carmele, A. Non-equilibrium non-Markovian steady-states in open quantum many-body systems: Persistent oscillations in Heisenberg quantum spin chains. arXiv
**2020**, arXiv:2006.03324. [Google Scholar]

**Figure 1.**Sketch of an open quantum system with Markovian type of interaction. The total system consists of a microscopic region ${|\psi \left(t\right)\rangle}_{\mathrm{sys}}$ which couples to its surrounding environment or reservoir ${|\psi \left(t\right)\rangle}_{\mathrm{res}}$ with a coupling strength $\mathsf{\Gamma}$. During time evolution for one time step $\Delta t$, the Markov approximation requires that the reservoir recovers instantly from the interaction and relaxes again into its previous state, thus ${|\psi (t+\Delta t)\rangle}_{\mathrm{res}}={|\psi \left(t\right)\rangle}_{\mathrm{res}}$.

**Figure 2.**Block diagram for the time evolution of an open quantum system. Each tensor is depicted as a box, while the lines correspond to the indices of the respective tensor. Connecting lines between boxes indicate shared indices of the tensors. (

**a**) MPS of the open quantum system: ${n}_{i}$ labels the site indices of the many-body system, while ${m}_{k}$ labels the time bins, and ${l}_{1}\cdots {l}_{{N}_{T-1}}$ labels the link indices. Initially, all system tensors are placed on the left in the MPS, followed by the time bins. During the time evolution, the system bins must be moved through the MPS to the right, a numerically very demanding procedure. (

**b**) MPO for the time evolution of the dissipative many-body system where exemplary the last site is subject to dissipation. Please note that the MPO affects the entire many-body system, but only the reservoir time bin of the present time step.

**Figure 3.**Block diagram of an algorithm with MPS for open quantum systems where one site of the many-body system is subject to dissipation. The diagram demonstrates the calculation of the kth time step. Blue boxes indicate left-orthogonality of the tensors, while green boxes indicate right-orthogonality and the red box marks the position of the orthogonality center of the MPS. The MPS contains the wave vector of the many-body system and of the reservoir. One time step is computed by applying the MPO on the many-body system, where the present state of the reservoir is contracted into the dissipative site. This step is illustrated in the lower figure and consists of contracting all tensor over their link indices while keeping the relevant site indices. Afterwards, the tensors are decomposed, and the present time bin must be swapped through the MPS to the left of the many-body system.

**Figure 4.**Block diagram of an efficient algorithm for open quantum systems where one site of the system is subject to dissipation. The diagram demonstrates the calculation of the kth time step. Blue boxes indicate left-orthogonality of the tensors, while green boxes indicate right-orthogonality and the red box marks the position of the orthogonality center of the MPS. One time step is computed as follows: the present state of the reservoir is modeled with a single time bin initialized in the vacuum state, while the MPS only contains the system bins. The time bin is multiplied into the dissipative site of the many-body system and the MPO is applied according to Equation (1). Afterwards, the tensors are being decomposed and the time bin is dropped, including its link to the past.

**Figure 5.**Sketch of an open quantum system with non-Markovian type of interaction. The total system consists of a microscopic region ${|\psi \left(t\right)\rangle}_{\mathrm{sys}}$ which couples to its surrounding environment or reservoir ${|\psi \left(t\right)\rangle}_{\mathrm{res}}$ with a coupling strength $\mathsf{\Gamma}$. Contrary to the Markovian case, the state of the reservoir at the time $t+\Delta t$ remains influenced by the interaction with the system which has occurred during the time step $\Delta t$, thus ${|\psi (t+\Delta t)\rangle}_{\mathrm{res}}\ne {|\psi \left(t\right)\rangle}_{\mathrm{res}}$.

**Figure 6.**Block diagram for the computation of one time step. Blue boxes indicate left-orthogonality of the tensors, while green boxes indicate right-orthogonality and the red box marks the position of the orthogonality center of the MPS. The many-body system MPS and the reservoir MPS are connected at the kth time bin ${m}_{k}$. Also, the past time bin ${m}_{k-l}$ has been brought next to them with swapping operations. For the application of the MPO, the Nth chain bin, the present and feedback time bin ${m}_{k}$ and ${m}_{k-l}$ are contracted and the chain MPO is multiplied into the MPS. Afterwards, the tensors are decomposed and moved back to their original position in the chain.

**Figure 7.**Block diagram of MPS and MPO of a many-body system in non-Markovian interaction with a reservoir. (

**a**) the MPS is 2-dimensional and consists of one MPS with the system bins labeled with the site indices ${n}_{i}$ and the link indices ${l}_{i}$, and one MPS containing the reservoir bins labeled with the site indices ${m}_{k}$ and the link indices ${p}_{k}$. The two MPS are stuck together at the Nth system bin where the interaction occurs. (

**b**) MPO for the computation of the time evolution according to Equation (1). Please note that it affects all system bins, yet only the present time bin ${m}_{k}$ and relevant time bin describing the past state of the reservoir ${m}_{k-l}$, where $l\in \mathbb{N}$ denotes the number of time steps between present and relevant past state of the reservoir.

**Figure 8.**Algorithm benchmark with the full master equation: (

**a**) Time dynamics of the spin current between the first and second site in a chain with $N=4$ sites in a chain initialized in the Neel state, thus $|\uparrow \downarrow \uparrow \downarrow \rangle $. Clearly, initially the current oscillates irregularly and finally equilibrates out to a non-equilibrium steady state (NESS). Please note that the curve is plotted twice, as this figure furthermore serves as a benchmark using the full solution for $|\psi \left(t\right)\rangle $ with the Lindblad master equation (black dotted line). (

**b**) Relative current ${\langle j\rangle}_{\mathrm{rel}}$ through the chain as a function of the number of sites N of the many-body system (black triangles). The data is fitted with a power law function, where we obtain the parameter $\gamma =0.01\pm 0.0006$, indicating a superdiffusive behavior corresponding to [30]. We demonstrate the benchmark for small system sizes using the full solution of the master equation (red crosses).

**Figure 9.**Dynamics of the time-dependent occupation densities ${\langle {\sigma}_{11}\left(t\right)\rangle}^{i}$ in a Heisenberg chain of different lengths N and for different trapping conditions ${\varphi}_{c}$, ${\tau}_{c}$. (

**a**) Dynamics for population trapping: Regular and periodic oscillations in a chain of $N=30$ sites. Parameters for this plot are $\mathsf{\Gamma}=1.5$, $J=0.1$. (

**b**) Dynamics without population trapping: Clearly, the initial state quickly dissipates into the environment and no excitation remains within the chain. Please note that we only plot a few selected sites in the chain. Parameters for this plot are $\mathsf{\Gamma}=0.24$ and $J=0.1$.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Finsterhölzl, R.; Katzer, M.; Knorr, A.; Carmele, A.
Using Matrix-Product States for Open Quantum Many-Body Systems: Efficient Algorithms for Markovian and Non-Markovian Time-Evolution. *Entropy* **2020**, *22*, 984.
https://doi.org/10.3390/e22090984

**AMA Style**

Finsterhölzl R, Katzer M, Knorr A, Carmele A.
Using Matrix-Product States for Open Quantum Many-Body Systems: Efficient Algorithms for Markovian and Non-Markovian Time-Evolution. *Entropy*. 2020; 22(9):984.
https://doi.org/10.3390/e22090984

**Chicago/Turabian Style**

Finsterhölzl, Regina, Manuel Katzer, Andreas Knorr, and Alexander Carmele.
2020. "Using Matrix-Product States for Open Quantum Many-Body Systems: Efficient Algorithms for Markovian and Non-Markovian Time-Evolution" *Entropy* 22, no. 9: 984.
https://doi.org/10.3390/e22090984