Nakajima–Zwanzig equation

The Nakajima–Zwanzig equation (named after the physicists who developed it, Sadao Nakajima^[1] and Robert Zwanzig^[2]) is an integral equation describing the time evolution of the "relevant" part of a quantum-mechanical system. It is formulated in the density matrix formalism and can be regarded as a generalization of the master equation.

Four approaches to obtaining the memory kernel of the GQME considering the system-bath form of the Hamiltonian.

The equation belongs to the Mori-Zwanzig formalism within the statistical mechanics of irreversible processes (named after Hazime Mori). By means of a projection operator, the dynamics is split into a slow, collective part (relevant part) and a rapidly fluctuating irrelevant part. The goal is to develop dynamical equations for the collective part.

The Nakajima-Zwanzig (NZ) generalized master equation is a formally exact approach for simulating quantum dynamics in condensed phases. This framework is particularly designed to address the dynamics of a reduced system interact with a larger environment, often represented as a system coupled to a bath.  Within the NZ framework, one can choose between time convolution (TC) and time convolution less (TCL) forms of the quantum master equations.

The TC approach involves memory effects, where the future state of the system depends on its entire history (Non-Markovian dynamics). The TCL approach formulates the dynamics where the system's rate of change at any moment depends only on its current state, simplifying calculations by neglecting memory effects (Markovian dynamics).

Derivation

The total Hamiltonian of a system interacting with its environment (or bath) is typically expressed in system-bath form,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \hat{H} = \hat{H}_S + \hat{H}_B + \hat{H}_{SB}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \hat{H}_S} is the system Hamiltonian, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \hat{H}_B} is the bath Hamiltonian, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \hat{H}_{SB}} describes the coupling between them.

The starting point^{[note 1]} is the quantum mechanical version of the von Neumann equation, also known as the Liouville equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \partial_t \rho = \frac{i}{\hbar}[\rho,H] = L \rho,}

where the Liouville operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle L} is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle L A = \frac{i}{\hbar}[A,H]} .

In the Nakajima-Zwanzig formulation, a key step involves defining a projection operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{P}} that projects the total density operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \rho} onto the subspace of the system of interest. The complementary operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{Q}\equiv 1-\mathcal{P}} projects onto the orthogonal subspace, effectively separating the system from the bath.

The Liouville – von Neumann equation can thus be represented as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle {\partial_t}\left( \begin{matrix} \mathcal{P} \\ \mathcal{Q} \\ \end{matrix} \right)\rho =\left( \begin{matrix} \mathcal{P} \\ \mathcal{Q} \\ \end{matrix} \right)L\left( \begin{matrix} \mathcal{P} \\ \mathcal{Q} \\ \end{matrix} \right)\rho +\left( \begin{matrix} \mathcal{P} \\ \mathcal{Q} \\ \end{matrix} \right)L\left( \begin{matrix} \mathcal{Q} \\ \mathcal{P} \\ \end{matrix} \right)\rho.}

The dynamics of the projected state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{P}\rho} , under any idempotent projection operator (whereFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{P}^2=\mathcal{P}} ), is described by the NZ generalized master equation (GQME). This equation can be used to obtain a closed equation of motion for the reduced system dynamics, focusing solely on the dynamics within the subsystem of interest. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \frac{d}{dt}\mathcal{P}\hat{\rho}(t) = -\frac{i}{\hbar}\mathcal{P} L\mathcal{P}\hat{\rho}(t)-\frac{i}{\hbar^2}\int_{0}^{t} d\tau\mathcal{P}Le^{-i\mathcal{Q}L\tau/\hbar}\mathcal{Q}L\mathcal{P}\hat{\rho}(t-\tau)-\frac{i}{\hbar}\mathcal{P}Le^{-i\mathcal{Q}L\tau/\hbar}\mathcal{Q}\hat{\rho}(0) }

In practice, the specific form of the projection operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{P}} can be chosen based on the problem at hand. One common choice involves defining Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{P}} using a reference nuclear density operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \hat{\rho}_{n}^{\text{ref}}} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \text{Tr}_{B} \{ \hat{\rho}_{n}^{\text{ref}} \} = 1} .

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{P}(\rho) = \hat{\rho}_n^{\text{ref}} \otimes \text{Tr}_n \{\rho\} }

This ensures that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{P}} remains idempotent. Using this projection, tracing over the nuclear Hilbert space leads to a generalized quantum master equation that describes the reduced electronic density operator which accounts for both Markovian dynamics generated by the Hamiltonian and non-Markovian dynamics due to coupling between electronic and nuclear degrees of freedom.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \frac{d}{dt} \hat{\sigma}(t) = -\frac{i}{\hbar} (L)_n^{\text{ref}} \hat{\sigma}(t) - \int_0^t d\tau \,\mathcal{K}(\tau) \hat{\sigma}(t-\tau) + \hat{I}(t)}

This Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \langle L \rangle_{n}^{\text{ref}} = \text{Tr}_{n} \{ L \hat{\rho}_{n}^{\text{ref}} \}}

describes the dynamics driven by the Hamiltonian, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \langle \hat{H} \rangle_n^{\text{ref}} = \text{Tr}_n \{ \hat{H} \rho_n^{\text{ref}} \}} which are Hamiltonian and Markovian in nature, while the other two terms on the right-hand side represent the non-Hamiltonian and non-Markovian dynamics that arise from the interactions between the electronic and nuclear degrees of freedom.

The memory kernel Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{K}(\tau)} captures the effects of the bath on the system over the time interval from (0, t), reflecting non-Markovian dynamics where the system's history influences its future evolution.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{K}(\tau) = \frac{i}{\hbar^2} \operatorname{Tr}_n \{ L e^{-i\mathcal{Q}L\tau/\hbar}\mathcal{Q}L \hat{\rho}_n^{\text{ref}} \}}

The inhomogeneous term Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \hat{I}(t)} represents the influence of the initial state of the bath on the system at time t, which is crucial for accurately describing the system dynamics from an initial condition.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \hat{I}(t) = \frac{i}{\hbar} \text{Tr}_n \left\{ L e^{-i\mathcal{Q}\tau/\hbar}\mathcal{Q} \hat{\rho}_n(0) \right\}}

The memory kernel is crucial for simulating the dynamics of the electronic degrees of freedom. However, calculating Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{K}(\tau)} presents difficulties due to its time-dependent nature. Additionally, the time dependency of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{K}(\tau)} is complex because it is governed by the projection-dependent propagator, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle e^{-i\mathcal{Q}L\tau/\hbar}} . Therefore, the exact memory kernel is difficult to calculate except for several analytically solvable models proposed by Shi-Geva to remove the projection operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathcal{Q}} .

See also

• Redfield equation

Notes

1. A derivation analogous to that presented here is found, for instance, in Breuer, Petruccione The theory of open quantum systems, Oxford University Press 2002, S.443ff