Master Equation approaches to open quantum systems

A core strength of the Theoretical Chemical and Quantum Physics Group


Every experimental realisation of a controllable quantum system suffers from the effects of its environment. The result of these environmental perturbations is to modify the (coherent) quantum mechanical evolution. This results in a loss of phase coherence or decoherence and is the main obstacle on the way to large scale quantum electronics, such as quantum computing. Therefore a lot of research world-wide is focused on decreasing the influence of environmental decoherence on quantum systems. At the same time decoherence is an interesting physical phenomenon which is not yet fully understood and which promises deeper understanding of basic quantum mechanical axioms such as the collapse of the wave function during a measurement.

Decay rates in a multi-level quantum system

Master equations are differential rate equations which describe the process of decoherence as a perturbative extension to the system's coherent evolution, as described by the Schrödinger equation or von Neumann equation. We work with two common forms of these differential equations.

Bloch Redfield equations

To derive the Bloch-Redfield equations, the environment is treated as a weakly coupled (often Bosonic) bath. The quantum system of interest evolves (and decoheres) due to its coupling to this bath. The decoherence rates are derived from this coupling operator and the properties of the bath. The resulting equations are a helpful tool to describe complex quantum systems as they help to link system-environment-couplings (the physical origin) with the decoherence rates of the system.

The interaction Hamiltonian is assumed to be a product of an operator acting on the system with an operator acting on the bath
. For the elements of the density matrix ρ the Bloch Redfield equations read in the eigenbasis of the system Hamiltonian:
\[\dot{\rho} = -i\omega_{nm}\rho_{nm}+\sum_{n^\prime m^\prime} R_{nmn^\prime m^\prime}\rho_{n^\prime m^\prime}\]
with the Redfield tensor:
\[\begin{align} R_{nmn^\prime m^\prime} &= \Lambda_{m^\prime mnn^\prime} + \widetilde{\Lambda}_{nn^\prime m^\prime m} - \sum_k\left(\Lambda_{nkkn^\prime} \delta_{mm^\prime}+\widetilde{\Lambda}_{kmm^\prime k} \delta_{nn^\prime}\right)\\ \Lambda_{m^\prime mnn^\prime} &= s_{nm}s_{n^\prime m^\prime}Q_B\left(\omega=\omega_{m^\prime n^\prime}\right)\\ \widetilde{\Lambda}_{m^\prime mnn^\prime} &= s_{nm}s_{n^\prime m^\prime}\widetilde{Q}_B\left(\omega=\omega_{m^\prime n^\prime}\right)\\ Q_B(\omega) &= \int_0^\infty \mathrm{d}\tau \; e^{i\omega\tau}\langle B(\tau)B(0)\rangle \\ \widetilde{Q}_B(\omega) &= \int_0^\infty \mathrm{d}\tau \; e^{i\omega\tau}\langle B(0)B(\tau)\rangle \end{align}\]
\[\omega_{nm} \equiv \omega_n - \omega_m\]
i.e. the energy difference of the states of the system and
is taken in the interaction picture.
is related to the Fourier transform of the correlation function of the bath operator B:
\[C(\omega) \equiv \int_{-\infty}^\infty \mathrm{d}\tau \; e^{i\omega\tau}\langle B(\tau)B(0)\rangle\]
\[Q_B(\omega) = \frac{1}{2}C(\omega), \qquad \widetilde{Q}_B(\omega) = \frac{1}{2}C(-\omega)\]

Lindblad equations

The Lindblad equations allow a more phenomenological approach to the decoherence process by the ad hoc assumption of decoherence rates. Their strength lies in the mathematical form which is proven to be the most general form describing physically sensible (i.e. completely positive) evolution.

The Lindblad equations read:

\[\dot{\rho} =-i[H,\rho]+\sum_{j=1}^N \Gamma_{L_j}\left(L_j\rho L_j^\dagger-\frac{1}{2}\left\{L_j^\dagger L_j, \rho\right\}\right)\]
with a number of decoherence rates
and corresponding Lindbladian operators

Recent Publications

Jan Jeske, David J. Ing, Martin B. Plenio, Susana F. Huelga and Jared H. Cole, Bloch-Redfield equations for modeling light-harvesting complexes, Journal of Chemical Physics 142, 064104 (2015)

J. Jeske and J. H. Cole, Derivation of Markovian master equations for spatially correlated decoherence, Physical Review A 87, 052138 (2013)

For more information about this technique, please contact Jared Cole.

Molecular Mechanics

A core strength of the Theoretical Chemical and Quantum Physics Group


In Molecular Mechanics a group of molecules can be considered as classical collections of balls and springs rather than quantum collections of electrons and nuclei. Models vary in complexity; the most simplistic case approximates atoms as hard spheres with a radius equal to that of their covalent radii, and then uses Valence Shell Electron Pair Repulsion Theory to predict the shape of molecules based upon the extent of the associated electron-pair electrostatic repulsion. This case is obviously trivial to calculate, and as this method is a classical approximation, terms tend to simply append to the original expression - rather than couple in a Gordian-esque knot like many quantum theories. Aided with modern computing power, even the most variegated system can be constructed in a fraction of the time it would take to generate the same system using Density Funcitonal Theory (DFT).

It has been shown that under homogeneous compression, solids can be described by an expression of pair-potentials and contributions from non-pair interactions can be expanded in terms of the pair interactions. This effectively includes any non-pair interactions into the pair terms and therefore the E(V) and P(V) states of the solid can be described on a unified basis. These interactions are mathematically constructed from well known classical mechanics formulae, and in Molecular Mechanics are known as force fields.

The CQP Group primarily uses The General Utility Lattice Program (GULP) when implementing Molecular Mechanics. GULP has the capability of simulating molecules and clusters (0D), polymers (1D), surfaces and slabs (2D), and bulk solids (3D). It uses empirical forcefield methods to simulate a variety of materials, such as the shell model for ionic systems, molecular mechanics for organic systems, embedded atom method (EAM) for metals, and reactive REBO potential for hydrocarbons.

GULP also allows the calculation of properties such as energies, bulk moduli, elastic constants, vibrational density of states, and heat capacity. In addition, it also allows one to perform NVE, NVT, and NPT molecular dynamics (MD) simulations. The CQP group typically uses GULP to perform tasks such as geometry optimisation of large systems, annealing of materials, and preparation of systems for further refinement using ab-initio methods.

TiSiO2 Structure

Atomic structure of a titanium defect in a SiO2 crystal modelled using GULP.

Oxide growth on aluminium

GULP simulation of the controlled growth of aluminium-oxide.

For more information about this technique, please contact Jared Cole or Salvy Russo.

Density Functional Theory

A core strength of the Theoretical Chemical and Quantum Physics Group


Density Functional Theory (DFT) is presently the most successful (and also the most promising) approach to modelling the electronic structure of matter. It can predict a variety of molecular properties: vibrational frequencies, atomisation energies, ionisation energies, electric and magnetic properties, molecular structures etc. The original theory has also been generalised and extended to deal with phenomenon like spin polarisation and superconductivity, as well as the introduction of time and temperature dependencies.

DFT is a ground state (GS) theory, which describes an interacting system of fermions via its density rather than its many-body wavefunction. The GS properties of a system, including the energy E, can be expressed as functionals of the GS electron density. Practical applications of DFT are based on approximations of the so-called exchange-correlation potential. The exchange-correlation potential describes the effects of the Pauli principle and the Coulomb potential beyond a pure electrostatic interaction of the electrons. Possessing the exact exchange-correlation potential means that the many-body problem could be solved exactly, which is clearly not feasible in solids.

The most common approximation is the local density approximation (LDA), which locally substitutes the exchange-correlation energy density of an inhomogeneous system by that of an electron gas evaluated at the local density. While many ground state properties (lattice constants, bulk moduli, etc.) are well described in the LDA, the dielectric constant is overestimated by 10-40% compared to experiment. This overestimation stems from the neglect of a polarisation-dependent exchange correlation field. This, and other limitations have forced the invention of other exchange-correlation functionals such as the Generalised Gradient Approximation (GGA). The TCQP Group employ a number of different potentials in their work and are selected on a per system basis.

Bonding in diamond nanowires

Analysis of bonding in Diamond nanowires.  Left: Electron charge densities; Right: sp3 iso-surface; Top: De-Hydrogenated; Bottom: Hydrogenated.

Recent Publications

N.C. Wilson and S.P. Russo, Hybrid density functional theory study of the high-pressure polymorphs of α-Fe2O3 hematite, Phys. Rev. B 79, 094113 (2009)

S.P. Russo, I.E. Grey, and N.C. Wilson, Nitrogen/Hydrogen Codoping of Anatase: A DFT Study, J. Phys. Chem. C 112(20) (2008)

For more information about this project, please contact Salvy Russo