## Master Equation approaches to open quantum systems

A core strength of the Theoretical Chemical and Quantum Physics Group

## Outline

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.

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
$H_{int}=sB$
. 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}
where
$\omega_{nm} \equiv \omega_n - \omega_m$
i.e. the energy difference of the states of the system and
$B(\tau)$
is taken in the interaction picture.
$Q_B(\omega)$
is related to the Fourier transform of the correlation function of the bath operator B:
$C(\omega) \equiv \int_{-\infty}^\infty \mathrm{d}\tau \; e^{i\omega\tau}\langle B(\tau)B(0)\rangle$
$Q_B(\omega) = \frac{1}{2}C(\omega), \qquad \widetilde{Q}_B(\omega) = \frac{1}{2}C(-\omega)$

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.

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