In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation, named after
Vittorio Gorini,
Andrzej Kossakowski
Andrzej Marek Kossakowski (20 February 1938 – 31 January 2021) was a Polish theoretical physicist and a professor at the Nicolaus Copernicus University. He was best known for his work on open quantum systems.
Education
Andrzej Kossakowski was ...
,
George Sudarshan
Ennackal Chandy George Sudarshan (also known as E. C. G. Sudarshan; 16 September 1931 – 13 May 2018) was an Indian American theoretical physicist and a professor at the University of Texas. Sudarshan has been credited with numerous contrib ...
and
Göran Lindblad Göran Lindblad may refer to:
* Göran Lindblad (politician) (born 1950), Swedish politician
* Göran Lindblad (physicist)
Göran Lindblad (9 July 1940 - 30 November 2022) was a Swedish theoretical physicist and a professor at the KTH Royal Inst ...
), master equation in Lindblad form, quantum Liouvillian, or Lindbladian is one of the general forms of
Markovian and
time-homogeneous master equations describing the (in general non-unitary) evolution of the
density matrix
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
that preserves the laws of quantum mechanics (i.e., is
trace-preserving and completely positive for any initial condition).
[
]
The
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
is a special case of the more general Lindblad equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation. The Schrödinger equation deals with
state vectors, which can only describe
pure quantum states and are thus less general than
density matrices
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any Measurement in quantum mechanics, measurement ...
, which can describe
mixed states as well.
Motivation
In the canonical formulation of quantum mechanics, a system's time evolution is governed by unitary dynamics. This implies that there is no decay and phase coherence is maintained throughout the process, and is a consequence of the fact that all participating degrees of freedom are considered. However, any real physical system is not absolutely isolated, and will interact with its environment. This interaction with degrees of freedom external to the system results in dissipation of energy into the surroundings, causing decay and randomization of phase. More so, understanding the interaction of a quantum system with its environment is necessary for understanding many commonly observed phenomena like the spontaneous emission of light from excited atoms, or the performance of many quantum technological devices, like the laser.
Certain mathematical techniques have been introduced to treat the interaction of a quantum system with its environment. One of these is the use of the
density matrix
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
, and its associated master equation. While in principle this approach to solving quantum dynamics is equivalent to the
Schrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may ...
or
Heisenberg picture
In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, bu ...
, it allows more easily for the inclusion of incoherent processes, which represent environmental interactions. The density operator has the property that it can represent a classical mixture of quantum states, and is thus vital to accurately describe the dynamics of so-called open quantum systems.
Definition
More generally, the Lindblad master equation for an -dimensional system's density matrix can be written as
(for a pedagogical introduction you can refer to)
:
where is a (
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature meth ...
)
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
part, and
is an arbitrary orthonormal
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
of the
Hilbert–Schmidt operators on the system's
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
with the restriction that is proportional to the identity operator (i.e.,
).
Our convention implies that the other are traceless, and note that the summation only runs to thus excluding the only basis-matrix with a non-zero trace.
The coefficient matrix , together with the Hamiltonian, determines the system dynamics. The matrix must be
positive semidefinite to ensure that the equation is trace-preserving and completely positive. The
anticommutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...
is defined as
If the are all zero, then this reduces to the
quantum Liouville equation for a closed system,