Analyzing the positivity preservation of numerical methods for the Liouville-von Neumann equation

Abstract The density matrix is a widely used tool in quantum mechanics. In order to determine its evolution with respect to time, the Liouville-von Neumann equation must be solved. However, analytic solutions of this differential equation exist only for simple cases. Additionally, if the equation is coupled to Maxwell's equations to model light-matter interaction, the resulting equation set – the Maxwell-Bloch or Maxwell-Liouville-von Neumann (MLN) equations – becomes nonlinear. In these advanced cases, numerical methods are required. Since the density matrix has certain mathematical properties, the numerical methods applied should be designed to preserve those properties. We establish the criterion that only methods that have a completely positive trace preserving (CPTP) update map can be used in long-term simulations. Subsequently, we assess the three most widely used methods – the matrix exponential (ME) method, the Runge-Kutta (RK) method, and the predictor-corrector (PC) approach – whether they provide this feature, and demonstrate that only the update step of the matrix exponential method is a CPTP map.

[1]  E. Sudarshan,et al.  Completely Positive Dynamical Semigroups of N Level Systems , 1976 .

[2]  C. Jirauschek,et al.  Modeling techniques for quantum cascade lasers , 2014, 1412.3563.

[3]  Abraham Nitzan,et al.  Numerical studies of the interaction of an atomic sample with the electromagnetic field in two dimensions , 2011, 1104.3325.

[4]  Ronnie Kosloff,et al.  Solution of the time-dependent Liouville-von Neumann equation: dissipative evolution , 1992 .

[5]  K. Busch,et al.  Time-Domain Simulations of the Nonlinear Maxwell Equations Using Operator-Exponential Methods , 2009, IEEE Transactions on Antennas and Propagation.

[6]  Timothy F. Havel.  Robust procedures for converting among Lindblad, Kraus and matrix representations of quantum dynamical semigroups , 2002, quant-ph/0201127.

[7]  Olivier Saut,et al.  Numerical methods for the bidimensional Maxwell-Bloch equations in nonlinear crystals , 2006, J. Comput. Phys..

[8]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[9]  Ziółkowski,et al.  Ultrafast pulse interactions with two-level atoms. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[10]  G. Lindblad.  On the generators of quantum dynamical semigroups , 1976 .

[11]  Ulf Osterberg,et al.  Multilevel Maxwell-Bloch simulations in inhomogeneously broadened media. , 2011, Optics express.

[12]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[13]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .

[14]  K. Kraus.  General state changes in quantum theory , 1971 .

[15]  C. Jirauschek,et al.  Dynamic Simulations of Quantum Cascade Lasers Beyond the Rotating Wave Approximation , 2018, 2018 2nd URSI Atlantic Radio Science Meeting (AT-RASC).

[16]  Brigitte Bidégaray-Fesquet,et al.  Nonstandard finite-difference schemes for the two-level Bloch model , 2018, Int. J. Model. Simul. Sci. Comput..

[17]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[18]  I. Kuprov,et al.  Efficient simulation of ultrafast magnetic resonance experiments. , 2017, Physical chemistry chemical physics : PCCP.

[19]  A. Bourgeade,et al.  Introducing physical relaxation terms in Bloch equations , 2001 .

[20]  Awad H. Al-Mohy,et al.  Computing the Action of the Matrix Exponential, with an Application to Exponential Integrators , 2011, SIAM J. Sci. Comput..

[21]  H. Metiu,et al.  An efficient method for solving the quantum liouville equation: applications to electronic absorption spectroscopy , 1986 .

[22]  Richard W. Ziolkowski,et al.  Coupled Maxwell-Pseudospin equations for investigation of self-induced transparency effects in a degenerate three-level quantum system in two dimensions: Finite-difference time-domain study , 2002 .

[23]  Knight,et al.  Comparison of quantum-state diffusion and quantum-jump simulations of two-photon processes in a dissipative environment. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[24]  Man-Duen Choi.  Completely positive linear maps on complex matrices , 1975 .

[25]  Brigitte Bidégaray,et al.  Time discretizations for Maxwell‐Bloch equations , 2003 .

[26]  R. Kosloff.  Propagation Methods for Quantum Molecular Dynamics , 1994 .

[27]  C. Lubich,et al.  On Krylov Subspace Approximations to the Matrix Exponential Operator , 1997 .

[28]  L. Marder,et al.  Dissipative quantum dynamics and optimal control using iterative time ordering: an application to superconducting qubits , 2018, The European Physical Journal B.

[29]  Hans-Joachim Bungartz,et al.  Performance evaluation of numerical methods for the Maxwell–Liouville–von Neumann equations , 2018 .

[30]  K. Schafer,et al.  Ultrafast time-dependent absorption in a macroscopic three-level helium gas , 2013 .

[31]  Alexei Deinega,et al.  Self-interaction-free approaches for self-consistent solution of the Maxwell-Liouville equations , 2014 .

[32]  Marlis Hochbruck,et al.  Exponential Integrators for Large Systems of Differential Equations , 1998, SIAM J. Sci. Comput..

[33]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[34]  C. Tang.  Fundamentals of Quantum Mechanics: For Solid State Electronics and Optics , 2005 .

[35]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[36]  Jesper Mørk,et al.  Self-consistent Maxwell-Bloch model of quantum-dot photonic-crystal-cavity lasers , 2017 .