Stochastic Liouville-von Neumann Equation Research Articles

Stochastic entropy production for continuous measurements of an open quantum system

We investigate the total stochastic entropy production of a two-level bosonic open quantum system under protocols of time dependent coupling to a harmonic environment. These processes are intended to represent the measurement of a system observable, and consequent selection of an eigenstate, whilst the system is also subjected to thermalising environmental noise. The entropy production depends on the evolution of the system variables and their probability density function, and is expressed through system and environmental contributions. The continuous stochastic dynamics of the open system is based on the Markovian approximation to the exact, noise-averaged stochastic Liouville-von Neumann equation, unravelled through the addition of stochastic environmental disturbance mimicking a measuring device. Under the thermalising influence of time independent coupling to the environment, the mean rate of entropy production vanishes asymptotically, indicating equilibrium. In contrast, a positive mean production of entropy as the system responds to time dependent coupling characterises the irreversibility of quantum measurement, and a comparison of its production for two coupling protocols, representing connection to and disconnection from the external measuring device, satisfies a detailed fluctuation theorem.

Assessment of weak-coupling approximations on a driven two-level system under dissipation

The standard weak-coupling approximations associated to open quantum systems have been extensively used in the description of a two-level quantum system, qubit, subjected to relatively weak dissipation compared with the qubit frequency. However, recent progress in the experimental implementations of controlled quantum systems with increased levels of on-demand engineered dissipation has motivated precision studies in parameter regimes that question the validity of the approximations, especially in the presence of time-dependent drive fields. In this paper, we address the precision of weak-coupling approximations by studying a driven qubit through the numerically exact and non-perturbative method known as the stochastic Liouville–von Neumann equation with dissipation. By considering weak drive fields and a cold Ohmic environment with a high cutoff frequency, we use the Markovian Lindblad master equation as a point of comparison for the SLED method and study the influence of the bath-induced energy shift on the qubit dynamics. We also propose a metric that may be used in experiments to map the regime of validity of the Lindblad equation in predicting the steady state of the driven qubit. In addition, we study signatures of the well-known Mollow triplet and observe its meltdown owing to dissipation in an experimentally feasible parameter regime of circuit electrodynamics. Besides shedding light on the practical limitations of the Lindblad equation, we expect our results to inspire future experimental research on engineered open quantum systems, the accurate modeling of which may benefit from non-perturbative methods.

A variance reduction technique for the stochastic Liouville–von Neumann equation

The Stochastic Liouville-von Neumann equation provides an exact numerical simulation strategy for quantum systems interacting with Gaussian reservoirs [J.T. Stockburger & H. Grabert, PRL 88, 170407 (2002)]. Its scaling with the extension of the time interval covered has recently improved dramatically through time-domain projection techniques [J.T. Stockburger, EPL 115, 40010 (2016)]. Here we present a sampling strategy which results in a significantly improved scaling with the strength of the dissipative interaction, based on reducing the non-unitary terms in sample propagation through convex optimization techniques.

Currents and fluctuations of quantum heat transport in harmonic chains

Heat transport in open quantum systems is particularly susceptible to the modeling of system–reservoir interactions. It thus requires us to consistently treat the coupling between a quantum system and its environment. While perturbative approaches are successfully used in fields like quantum optics and quantum information, they reveal deficiencies—typically in the context of thermodynamics, when it is essential to respect additional criteria such as fluctuation-dissipation theorems. We use a non-perturbative approach for quantum dissipative dynamics based on a stochastic Liouville–von Neumann equation to provide a very general and extremely efficient formalism for heat currents and their correlations in open harmonic chains. Specific results are derived not only for first- but also for second-order moments, which requires us to account for both real and imaginary parts of bath–bath correlation functions. Spatiotemporal patterns are compared with weak coupling calculations. The regime of stronger system–reservoir couplings gives rise to an intimate interplay between reservoir fluctuations and heat transfer far from equilibrium.

Partition-free approach to open quantum systems in harmonic environments: An exact stochastic Liouville equation

We present a partition-free approach to the evolution of density matrices for open quantum systems coupled to a harmonic environment. The influence functional formalism combined with a two-time Hubbard-Stratonovich transformation allows us to derive a set of exact differential equations for the reduced density matrix of an open system, termed the Extended Stochastic Liouville-von Neumann equation. Our approach generalises previous work based on Caldeira-Leggett models and a partitioned initial density matrix. This provides a simple, yet exact, closed-form description for the evolution of open systems from equilibriated initial conditions. The applicability of this model and the potential for numerical implementations are also discussed.

Time-correlated blip dynamics of open quantum systems

The non-Markovian dynamics of open quantum systems is still a challenging task, particularly in the non-perturbative regime at low temperatures. While the Stochastic Liouville-von Neumann equation (SLN) provides a formally exact tool to tackle this problem for both discrete and continuous degrees of freedom, its performance deteriorates for long times due to an inherently non-unitary propagator. Here we present a scheme which combines the SLN with projector operator techniques based on finite dephasing times, gaining substantial improvements in terms of memory storage and statistics. The approach allows for systematic convergence and is applicable in regions of parameter space where perturbative methods fail, up to the long time domain. Findings are applied to the coherent and incoherent quantum dynamics of two- and three-level systems. In the long time domain sequential and super-exchange transfer rates are extracted and compared to perturbative predictions.

On the structure of the master equation for a two-level system coupled to a thermal bath

We derive a master equation from the exact stochastic Liouville–von-Neumann (SLN) equation (Stockburger and Grabert 2002 Phys. Rev. Lett. 88 170407). The latter depends on two correlated noises and describes exactly the dynamics of an oscillator (which can be either harmonic or present an anharmonicity) coupled to an environment at thermal equilibrium. The newly derived master equation is obtained by performing analytically the average over different noise trajectories. It is found to have a complex hierarchical structure that might be helpful to explain the convergence problems occurring when performing numerically the stochastic average of trajectories given by the SLN equation (Koch et al 2008 Phys. Rev. Lett. 100 230402, Koch 2010 PhD thesis Fakultät Mathematik und Naturwissenschaften der Technischen Universitat Dresden).

Application of stochastic Liouville–von Neumann equation to electronic energy transfer in FMO complex

A stochastic Liouville–von Neumann approach to solving a spin-boson model is applied to electronic energy transfer in Fenna–Matthews–Olson (FMO) complexes as a case study of the dynamics in biological systems. We modify a noise generation method to treat an experimentally obtained highly structured spectral density. By considering the population dynamics in a two-site system with a model structured spectral density, we numerically observe two kinds of coherent motions associated with inter-site coupling and system–bath coupling, the latter of which is mainly attributed to the peak structure of the spectral density.

Intermediate motions as studied by solid-state separated local field NMR experiments

In this report, the application of a class of separated local field NMR experiments named dipolar chemical shift correlation (DIPSHIFT) for probing motions in the intermediate regime is discussed. Simple analytical procedures based on the Anderson-Weiss (AW) approximation are presented. In order to establish limits of validity of the AW based formulas, a comparison with spin dynamics simulations based on the solution of the stochastic Liouville-von-Neumann equation is presented. It is shown that at short evolution times (less than 30% of the rotor period), the AW based formulas are suitable for fitting the DIPSHIFT curves and extracting kinetic parameters even in the case of jumplike motions. However, full spin dynamics simulations provide a more reliable treatment and extend the frequency range of the molecular motions accessible by DIPSHIFT experiments. As an experimental test, molecular jumps of imidazol methyl sulfonate and trimethylsulfoxonium iodide, as well as the side-chain motions in the photoluminescent polymer poly[2-methoxy-5-(2'-ethylhexyloxy)-1,4-phenylenevinylene], were characterized. Possible extensions are also discussed.

Exact and approximate many-body dynamics with stochastic one-body density matrix evolution

We show that the dynamics of interacting fermions can be exactly replaced by a quantum jump theory in the many-body density matrix space. In this theory, jumps occur between densities formed of pairs of Slater determinants, $D_{ab}=| \Phi_a > < \Phi_b |$, where each state evolves according to the Stochastic Schr\"odinger Equation (SSE) given in ref. \cite{Jul02}. A stochastic Liouville-von Neumann equation is derived as well as the associated Bogolyubov-Born-Green-Kirwood-Yvon (BBGKY) hierarchy. Due to the specific form of the many-body density along the path, the presented theory is equivalent to a stochastic theory in one-body density matrix space, in which each density matrix evolves according to its own mean field augmented by a one-body noise. Guided by the exact reformulation, a stochastic mean field dynamics valid in the weak coupling approximation is proposed. This theory leads to an approximate treatment of two-body effects similar to the extended Time-Dependent Hartree-Fock (Extended TDHF) scheme. In this stochastic mean field dynamics, statistical mixing can be directly considered and jumps occur on a coarse-grained time scale. Accordingly, numerical effort is expected to be significantly reduced for applications.

Anisotropic relaxation and motion of half-integer quadrupole nuclei studied by central transition nuclear magnetic resonance spectroscopy

An efficient theoretical formalism and advanced experimental methods are presented for studying the effects of anisotropic molecular motion and relaxation on solid-state central transition NMR spectra of half-integer quadrupole nuclei. The theoretical formalism is based on density operator algebra and involves the stochastic Liouville–von Neumann equation. In this approach the nuclear spin interactions are represented by the Hamiltonian while the motion is described by a discrete stochastic operator. The nuclear spin interactions fluctuate randomly in the presence of molecular motion. These fluctuations may stimulate the relaxation of the system and are represented by a discrete relaxation operator. This is derived from second-order perturbation theory and involves the spectral densities of the system. Although the relaxation operator is valid only for small time intervals it may be used recursively to obtain the density operator at any time. The spectral densities are allowed to be explicitly time dependent making the approach valid for all motional regimes. The formalism has been applied to simulate partially relaxed central transition 17 O NMR spectra of representative model systems. The results have revealed that partially relaxed central transition lineshapes are defined not only by the nuclear spin interactions but also by anisotropic motion and relaxation. This has formed the basis for the development of central transition spin-echo and inversion-recovery NMR experiments for investigating molecular motion in solids. As an example we have acquired central transition spin-echo and inversion-recovery 17 O NMR spectra of polycrystalline cristobalite (SiO 2) at temperatures both below and above the α–β phase transition. It is found that the oxygen atoms exhibit slow motion in α-cristobalite. This motion has no significant effects on the fully relaxed lineshapes but may be monitored by studying the partially relaxed spectra. The α–β phase transition is characterized by structural and motional changes involving a slight increase in the Si–O–Si bond angle and a substantial increase in the mobility of the oxygen atoms. The increase in the Si–O–Si angle is supported by the results of 17 O and 29 Si NMR spectroscopy. The oxygen motion is shown to be orders of magnitude faster in β-cristobalite resulting in much faster relaxation and characteristic lineshapes. The measured oscillation frequencies are consistent with the rigid unit mode model. This shows that solid-state NMR and lattice dynamics simulations agree and may be used in combination to provide more detailed models of solid materials.

The Application of MAS Recoupling Methods in the Intermediate Motional Regime

We present investigations concerning the effect of molecular motions on the experimental timescale upon the recoupling of anisotropic interactions under magic-angle spinning conditions. An approach for the efficient simulation of spin dynamics occurring during complex pulse sequences, based on a linearization of the general solution of the stochastic Liouville–von Neumann equation, was developed. Using 13C CSA recoupling of the methyl carbon in dimethylsulfon as a sample interaction, we observed a characteristic signal decay under recoupling upon entering the intermediate motional regime, which can be well described by an apparent transverse relaxation time, T 2 rcpl . This quantity does not depend on the spinning frequency to a first approximation. Specific recoupling experiments, namely the measurement of tensor parameters by spinning sideband analysis, and the determination of rate constants with the CODEX experiment, are discussed with respect to possibilities and limits of their application in the intermediate motional regime. Important conclusions are drawn with regards to the limited applicability of popular recoupling methods like REDOR to samples exhibiting intermediate mobility.

Computational aspects of motional symmetry in nuclear magnetic resonance spectroscopy

The most fundamental approach to calculate the effects of molecular motion on nuclear magnetic resonance spectra involves the density operator and the stochastic Liouville–von Neumann equation. In order to obtain a solution to this equation it is useful to expand the density operator into a set of infinitesimal group generators representing the different alignments and coherences of the nuclear spin system. In this representation the stochastic Liouville–von Neumann equation may be rewritten in the form of a linear homogeneous system of coupled first-order differential equations among the alignments and coherences. The dimension of this system is usually very large and the presence of molecular motion makes the equations highly stiff. This implies that the numerical solution is very difficult and can only be obtained by semi-implicit or implicit integration methods that are sufficiently accurate and stable. The computational efficiency may be improved significantly for any integration method by reducing the dimension of the system. In this paper we introduce a group theoretical methodology to reduce the stochastic Liouville–von Neumann equation for any system exhibiting sufficient motional symmetry. The approach is based on the concept of motional graphs and may be applied to any system independently of the choice of group generators. The results include a set of rules to determine the reducibility of the stochastic Liouville–von Neumann equation. The formalism is exemplified by application to different molecular systems with particular emphasis on demonstrating the results of reducibility.

Design and Implementation of Runge–Kutta Methods for MAS NMR Lineshape Calculations

The magic angle spinning (MAS) nuclear magnetic resonance (NMR) technique has proven successful for obtaining information about molecular structure and motion in solids. The most advanced description of the MAS NMR experiment involves the density operator and the stochastic Liouville–von Neumann equation. In this equation, the effects of the anisotropic nuclear spin interactions are represented by the Hamiltonian, while the motion involves a stochastic operator. For many systems, molecular motion may be described by a discrete Markov process. In order to obtain a solution for a discrete Markov process we have used a Lie algebra formalism to rewrite the stochastic Liouville–von Neumann equation in the form of a linear homogeneous system of coupled first-order differential equations. This system has periodically time-dependent coefficients and can only be solved by numerical methods. In this paper, we discuss the use of different Runge–Kutta methods to approximate the solution. These methods have the advantages of simplicity and relatively high efficiency and may be implemented with automatic stepsize control. The study involves the most important classes of Runge–Kutta methods including explicit, semi-implicit, and implicit schemes. The results have shown that the choice of the Runge–Kutta method depends on the motion. It is found that explicit Runge–Kutta methods are the most efficient schemes in the slow and intermediate motion regimes. However, in the fast motion regime, the stochastic Liouville–von Neumann equation becomes stiff. In this regime, we have used semi-implicit and implicit Runge–Kutta methods with better stability characteristics. For many semi-implicit or implicit Runge–Kutta methods, the stiff components are represented inaccurately, and the methods are shown to be relatively inefficient. In order to improve the efficiency we have designed and implemented stiffly A-stable Runge–Kutta methods. These have better stability and accuracy characteristics and are useful in the fast motion regime. In another approach, we have rewritten the stochastic Liouville–von Neumann equation in a form with a vanishing stiffness ratio. Based on this form we have designed modified explicit Runge–Kutta methods, which are stable and accurate in the fast motion regime. The results have shown that the Runge–Kutta methods improve the computational efficiency for small systems by a factor between two and five compared with the eigenvalue method. The efficiency increases with the dimension of the system, and for large systems the improvement approaches an order of magnitude. This is an important result because of the long computation times involved in MAS NMR lineshape calculations.

Measurement of molecular motion in solids by nuclear magnetic resonance spectroscopy of half-integer quadrupole nuclei

An accurate and efficient formalism is presented for simulating the effects of molecular motion on satellite and central transition nuclear magnetic resonance (NMR) spectra of half-integer quadrupole nuclei. The approach is based on the principles of the density operator and the stochastic Liouville–von Neumann equation and may be applied for both rotating and nonrotating samples. The symmetry properties of nuclear spin ensembles have been used to rewrite the stochastic Liouville–von Neumann equation in the form of a linear homogeneous system of coupled first-order differential equations among the alignments and coherences. This system is highly stiff and can only be solved by methods that are sufficiently accurate and stable. The properties of Cartan–Weyl operators have been used to obtain the most efficient solution for secular interactions. The methodology has been incorporated into computer programs to simulate the effects of motion for any half-integer quadrupole nucleus. These programs include the first- and second-order quadrupole and first-order shielding interactions. The formalism has been used to calculate central transition O17 NMR spectra of representative model systems. The calculations have revealed several interesting and important properties of central transition NMR spectra that have been discussed in terms of the functional form of the line shape. The validity of the methodology has been demonstrated experimentally by simulating variable temperature central transition O17 NMR spectra of the silicate (SiO2) mineral cristobalite for both rotating and nonrotating samples. The simulations have allowed the structural and dynamical details of the α–β phase transition in cristobalite to be separated. The line shapes can only be simulated if the effects of motion are included and are consistent with a model where the oxygen atoms reorient between six different orientations. It is found that the oxygen motion is present both before and after the α–β phase transition and does not change significantly at the transition temperature. The observed changes in the quadrupole and shielding parameters are shown not to be the results of motional averaging but derive from an abrupt structural change associated with the first-order character of the α–β phase transition. The structural changes may be interpreted in terms of a model where the Si–O–Si bond angle increases slightly.

Investigation of multiaxial molecular dynamics by [formula omitted] MAS NMR spectroscopy

The technique of 2 H MAS NMR spectroscopy is presented for the investigation of multiaxial molecular dynamics. To evaluate the effects of discrete random reorientation a Lie algebraic formalism based on the stochastic Liouville–von Neumann equation is developed. The solution to the stochastic Liouville–von Neumann equation is obtained both in the presence and absence of rf irradiation. This allows effects of molecular dynamics to be evaluated during rf pulses and extends the applicability of the formalism to arbitrary multiple pulse experiments. Theoretical methods are presented for the description of multiaxial dynamics with particular emphasis on the application of vector parameters to represent molecular rotations. Numerical time and powder integration algorithms are presented that are both efficient and easy to implement computationally. The applicability of 2 H MAS NMR spectroscopy for investigating molecular dynamics is evaluated from theoretical spectra. To demonstrate the potential of the technique the dynamics of thiourea– 2 H 4 is investigated experimentally. From a series of variable temperature MAS and quadrupole echo spectra it has been found that the dynamics can be described by composite rotation about the CS and CN bonds. Both experiments are sensitive to the fast CS rotation which is shown to be described by the Arrhenius parameters E CS=46.4±2.3 kJ mol −1 and ln( A CS)=32.6±0.9. The MAS experiment represents a significant improvement by simultaneously allowing the dynamics of the slow CN rotation to be fully characterized in terms of E CN=56.3±3.4 kJ mol −1 and ln( A CN)=25.3±1.1.