A quantum master equation is a generalization of the idea of a

master equation In physics, chemistry, and related fields, master equations are used to describe the time evolution of a system that can be modeled as being in a probabilistic combination of states at any given time, and the switching between states is determi ...

. Rather than just a system of differential equations for a set of probabilities (which only constitutes the diagonal elements of a

density matrix In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while th ...

), quantum master equations are differential equations for the entire density matrix, including

off-diagonal element In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagona ...

s. A density matrix with only diagonal elements can be modeled as a classical random process, therefore such an "ordinary" master equation is considered classical. Off-diagonal elements represent

quantum coherence Coherence expresses the potential for two waves to interfere. Two monochromatic beams from a single source always interfere. Wave sources are not strictly monochromatic: they may be ''partly coherent''. When interfering, two waves add togethe ...

which is a physical characteristic that is intrinsically quantum mechanical. A formally exact quantum master equation is the

Nakajima–Zwanzig equation The Nakajima–Zwanzig equation (named after the physicists who developed it, Sadao Nakajima and Robert Zwanzig) is an integral equation describing the time evolution of the "relevant" part of a quantum-mechanical system. It is formulated in th ...

, which is in general as difficult to solve as the full quantum problem. The

Redfield equation Redfield may refer to: People * Redfield (surname) Places ;United Kingdom *Redfield, Bristol, an area within the City of Bristol ;United States * Mount Redfield, a mountain in Essex County, New York *Redfield, Arkansas, a city in northwestern ...

and

Lindblad equation In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation (named after Vittorio Gorini, Andrzej Kossakowski, E. C. George Sudarshan, George Sudarshan and Göran Lindblad (physicist), Göran Lindblad), master equation in ...

are examples of approximate Markovian quantum master equations. These equations are very easy to solve, but are not generally accurate. Some modern approximations based on quantum master equations, which show better agreement with exact numerical calculations in some cases, include the polaron transformed quantum master equation and the VPQME (variational polaron transformed quantum master equation). Numerically exact approaches to the kinds of problems to which master equations are usually applied include numerical

Feynman integral The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or ...

quantum Monte Carlo Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution (or an accurate approximation) of the ...

, DMRG and NRG,

MCTDH Multi-configuration time-dependent Hartree (MCTDH) is a general algorithm to solve the time-dependent Schrödinger equation for multidimensional dynamical systems consisting of distinguishable particles. MCTDH can thus determine the quantal motion o ...

, and HEOM.

References

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See also

References