Quantum trajectories for time-dependent adiabatic master equations

The chemical master equation is a continuous time discrete space Markov model of chemical reactions. The chemical master equation is derived mathematically and it is shown that the corresponding initial value problem has a unique solution. Conditions are given under which this solution is a probability distribution. We present finite state and aggregation-disaggregation approximations and provide error bounds for the case of piecewise constant disaggregation. The aggregation-disaggregation approximation allows the solution of the chemical master equation for larger state spaces and is also an important tool for the solution of multidimensional problems. References Adam Arkin, John Ross, and Harley H. McAdams. {Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Ph age lambda-Infected Escherichia coli Cells}. Genetics, 149(4):1633--1648, 1998. http://www.genetics.org/cgi/content/abstract/149/4/1633 P. Deuflhard, W. Huisinga, T. Jahnke, and M. Wulkow. Adaptive discrete Galerkin methods applied to the chemical master equation. Technical Report ZIB-Report 07-04, Konrad--Zuse Zentrum fuer Informationstechnik Berlin, 2007. http://www.zib.de/Publications/Reports/ZR-07-04.pdf Stefan Engblom. Computing the moments of high dimensional solutions of the master equation. Appl. Math. Comput., 180(2):498--515, 2006. doi:10.1016/j.amc.2005.12.032 Lars Ferm and Per Loetstedt. Numerical method for coupling the macro and meso scales in stochastic chemical kinetics. BIT Numerical Mathematics, 47:735--762, 2007. doi:10.1007/s10543-007-0150-z Daniel T. Gillespie. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry, 81(25):2340--2361, December 1977. Markus Hegland, Conrad Burden, Lucia Santoso, Shev MacNamara, and Hilary Booth. A solver for the stochastic master equation applied to gene regulatory networks. J. Comput. Appl. Math., 205(2):708--724, 2007. doi:10.1016/j.cam.2006.02.053 Leo Huber and Markus Hegland. Dimension adaptive sparse grids for the chemical master equations. submitted, 2008. B. Munsky and M. Khammash. The finite state projection algorithm for the solution of the chemical master equation. The Journal of Chemical Physics, 124:044104--1--044104--1, 2006. doi:10.1063/1.2145882 Roger B. Sidje. EXPOKIT: Software package for computing matrix exponentials. ACM Transactions on Mathematical Software, 24(1):130--156, March 1998. doi:10.1145/285861.285868 T. Tian and K. Burrage. Bistability and switching in the lysis/lysogeny genetic regulatory network of bacteriophage-$\lambda $. Journal of Theoretical Biology, 227:229--237, 2004. doi:10.1016/j.jtbi.2003.11.003 N. G. van Kampen. Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam, the Netherlands, 1981.

Recent experiments with increasingly larger numbers of qubits have sparked renewed interest in adiabatic quantum computation, and in particular quantum annealing. A central question that is repeatedly asked is whether quantum features of the evolution can survive over the long time-scales used for quantum annealing relative to standard measures of the decoherence time. We reconsider the role of decoherence in adiabatic quantum computation and quantum annealing using the adiabatic quantum master equation formalism. We restrict ourselves to the weak-coupling and singular-coupling limits, which correspond to decoherence in the energy eigenbasis and in the computational basis, respectively. We demonstrate that decoherence in the instantaneous energy eigenbasis does not necessarily detrimentally affect adiabatic quantum computation, and in particular that a short single-qubit $T_2$ time need not imply adverse consequences for the success of the quantum adiabatic algorithm. We further demonstrate that boundary cancellation methods, designed to improve the fidelity of adiabatic quantum computing in the closed system setting, remain beneficial in the open system setting. To address the high computational cost of master equation simulations, we also demonstrate that a quantum Monte Carlo algorithm that explicitly accounts for a thermal bosonic bath can be used to interpolate between classical and quantum annealing. Our study highlights and clarifies the significantly different role played by decoherence in the adiabatic and circuit models of quantum computing.

The quantum trajectory method (QTM) is extended to the dynamics of electronic nonadiabiatic collisions. Equations of motion are first derived for the probability density, velocity, and action function for wave packets moving on each of the coupled electronic potential surfaces. These discretized equations are solved in the Lagrangian (moving with the fluid) picture to give the trajectory dynamics of fluid elements evolving on each potential surface. This trajectory method is fully quantum mechanical and does not involve “trajectory surface hopping.” The method is applied to nonadiabiatic collision models involving two coupled electronic states. The quantum trajectory results are in excellent agreement with solutions computed (using space-fixed grid methods) directly from the time-dependent Schrödinger equation.

A system of N spin-1/2 particles interacting with a thermal reservoir is used as a pedagogical example for advanced undergraduate and graduate students. We introduce and illustrate some methods, approximations, and phenomena related to dissipation and nonlinearity in many-particle physics. We start our analysis from the dynamical Sudarshan-Lindblad quantum master equation for the density operator of a system S interacting with a thermal reservoir R. We derive the quantum version of the so-called Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) equations such that the master equation can be decomposed in a hierarchical set of N - 1 equations (N > 1). The hierarchy is broken by introducing the mean-field approximation and reducing the problem to a nonlinear single particle system. In this scenario, the Hamiltonian is nonlinear (i.e., it depends on the state of <S), although the superoperator responsible for the dissipation and decoherence of S remains unaffected. To provide a useful tool to students: (1) we discuss the physical approximations involved, (2) we derive the analytical solution to the mean values equations of motion resulting from the Hamiltonian, (3) we solve analytically the master equation in the stationary regime, (4) we obtain and discuss the solution of the nonlinear master equation, numerically, and finally, (5) we discuss the master equation beyond the mean-field approximation and show how to introduce higher order quantum correlations that have been previously neglected.

The use of quantum mechanical approximation methods in the solution of master equations

Approximate solutions for multistep master equations describing the time evolution of product formation in multiphoton or thermal unimolecular reactions are investigated. In particular, a method based on fitting the first few moments of the passage time distribution associated with the given stochastic process to proposed simple expressions for the product yield function is studied. It is shown that reasonable agreement with the exact numerical solution of the corresponding master equation is obtained with a two parameter fit (using two passage time moments) and an excellent agreement is obtained with a three parameter fit (using three passage time moments). In no case studied does a need arise for more than a three-moment description and the quality of available experimental results makes the simpler two-moment description sufficient in most cases. Analytical solutions for the first and second passage time moments are obtained for simple discrete and continuous master equation models. Expressions for the incubation time and the reaction rate are obtained in terms of these solutions. The validity of discretizing a continuous master equation (which is an important simplifying step in evaluating the time evolution associated with multiphoton dissociations in the presence of collisions, or with thermal unimolecular reactions involving large molecules) is studied using both the approximate two-moment solutions and exact numerical solutions. It is concluded that a proper discretization of a continuous master equation may be carried out provided ε≪kBT, where ε is the discretization energy step, kB the Boltzmann constant, and T the effective (density of states weighted) temperature. A larger discretization step can be used if only the incubation time is required. Using the approximately discretized master equation, we next calculate the effect of collisions on the incubation time and the rate of multiphoton dissociation using a model constructed to correspond to the unimolecular dissociation of tetramethyldioxethane. Incubation times are found to be less sensitive to collisions then the reaction rates. Finally, we investigate the applicability of the passage time moments method to describe the time evolution of product formation in a system whose dynamics is determined by a quantum mechanical Liouville equation. Again the two-moment description provides a reasonable and the three-moment approximation a good approximation to the exact solution. The three-moment approximations, however, cannot be used when the pressure (i.e., the dephasing rate) is too low.

We obtain the finite-temperature unconditional master equation of the density matrix for two coupled quantum dots (CQD) when one dot is subjected to a measurement of its electron occupation number using a point contact (PC). To determine how the CQD system state depends on the actual current through the PC device, we use the so-called quantum trajectory method to derive the zero-temperature conditional master equation. We first treat the electron tunneling through the PC barrier as a classical stochastic point process (a quantum-jump model). Then we show explicitly that our results can be extended to the quantum-diffusive limit when the average electron tunneling rate is very large compared to the extra change of the tunneling rate due to the presence of the electron in the dot closer to the PC. We find that in both quantum-jump and quantum-diffusive cases, the conditional dynamics of the CQD system can be described by the stochastic Schr\"{o}dinger equations for its conditioned state vector if and only if the information carried away from the CQD system by the PC reservoirs can be recovered by the perfect detection of the measurements.

We investigate the Su-Schrieffer-Heeger model in presence of an injection and removal of particles, introduced via a master equation in Lindblad form. It is shown that the dynamics of the density matrix follows the predictions of calculations in which the gain and loss are modeled by complex $\mathcal{PT}$-symmetric potentials. In particular it is found that there is a clear distinction in the dynamics between the topologically different cases known from the stationary eigenstates.

We define the hitting time for a model of continuous-time open quantum walks in terms of quantum jumps. Our starting point is a master equation in Lindblad form, which can be taken as the quantum analogue of the rate equation for a classical continuous-time Markov chain. The quantum jump method is well known in the quantum optics community and has also been applied to simulate open quantum walks in discrete time. This method however, is well-suited to continuous-time problems. It is shown here that a continuous-time hitting problem is amenable to analysis via quantum jumps: The hitting time can be defined as the time of the first jump. Using this fact, we derive the distribution of hitting times and explicit exressions for its statistical moments. Simple examples are considered to illustrate the final results. We then show that the hitting statistics obtained via quantum jumps is consistent with a previous definition for a measured walk in discrete time [Phys.~Rev.~A~{\bf 73}, 032341 (2006)] (when generalised to allow for non-unitary evolution and in the limit of small time steps). A caveat of the quantum-jump approach is that it relies on the final state (the state which we want to hit) to share only incoherent edges with other vertices in the graph. We propose a simple remedy to restore the applicability of quantum jumps when this is not the case and show that the hitting-time statistics will again converge to that obtained from the measured discrete walk in appropriate limits.

The continuous coupling function in quantum impurity problems is exactly partitioned into a part represented by a finite size Wilson chain and a part represented by a set of additional reservoirs, each coupled to one Wilson chain site. These additional reservoirs represent high-energy modes of the environment neglected by the numerical renormalization group and are required to restore the continuum limit of the original problem. We present a hybrid time-dependent numerical renormalization group approach which combines an accurate numerical renormalization group treatment of the non-equilibrium dynamics on the finite size Wilson chain with a Bloch-Redfield formalism to include the effect of these additional reservoirs. Our approach overcomes the intrinsic shortcoming of the time-dependent numerical renormalization group approach induced by the bath discretization with a Wilson parameter $\Lambda > 1$. We analytically prove that for a system with a single chemical potential, the thermal equilibrium reduced density operator is the steady-state solution of the Bloch-Redfield master equation. For the numerical solution of this master equation a Lanczos method is employed which couples all energy shells of the numerical renormalization group. The presented hybrid approach is applied to the real-time dynamics in correlated fermionic quantum-impurity systems. An analytical solution of the resonant-level model serves as a benchmark for the accuracy of the method which is then applied to non-trivial models, such as the interacting resonant-level model and the single impurity Anderson model.

Solution of the master equation for unimolecular reactions

QmeQ 1.0: An open-source Python package for calculations of transport through quantum dot devices

Cellular reactions have a multi-scale nature in the sense that the abundance of molecular species and the magnitude of reaction rates can vary across orders of magnitude. This diversity naturally leads to hybrid models that combine continuous and discrete modeling regimes. In order to capture this multi-scale nature, we proposed jump-diffusion approximations in a previous study. The key idea was to partition reactions into fast and slow groups, and then to combine a Markov jump updating scheme for the slow group with a diffusion (Langevin) updating scheme for the fast group. In this study we show that the joint probability density function of the jump-diffusion approximation over the reaction counting process satisfies a hybrid master equation that combines terms from the chemical master equation and from the Fokker–Planck equation. Inspired by the method of conditional moments, we propose a efficient method to solve this master equation using the moments of reaction counters of the fast reactions given the reaction counters of the slow reactions. For each time point of interest, we then solve a set of maximum entropy problems in order to recover the conditional probability density from its moments. This finally allows us to reconstruct the complete joint probability density over all reaction counters and hence obtain an approximate solution of the hybrid master equation. Finally, we show the accuracy of the method applied to a simple multi-scale conversion process.

Theoretical study on optical response in nanostructures in the Born–Markov regime: The role of spontaneous emission and dephasing

The shapes of dust self‐organized structures in presence of volume ionization is investigated. Master Equations for small diffusions and presence of an external ionization describing spherical dust structures are solved numerically using asymptotic solutions at the center of the structures. The structures are determined by two parameters, the external plasma flux at the surface of the structures and the power of volume ionizations. It is shown that the range of possible equilibrium states is reduced with an increase of the ionization power. This reduction is due mainly to an increase of the minimum possible value of ion density at the center of the structure for which the equilibrium is possible (an increase of minimum possible external flux). It is found that for certain large ionization power the equilibrium structure without void at the center cannot exist. The critical value of ionization power is found numerically. In the range of ionization power where the equilibria is possible the distribution of the parameters inside the structures are investigated by solving the Master Equations in the limits of small and large ionization power (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)