Lindblad equation
Encyclopedia
In quantum mechanics
, the Kossakowski–Lindblad equation (after Andrzej Kossakowski and Göran Lindblad) or master equation in the Lindblad form is the most general type of markovian and time-homogeneous master equation
describing non-unitary evolution of the density matrix
that is trace preserving and completely positive for any initial condition.
The Lindblad master equation for an
-dimensional system's reduced density matrix 
can be written:
where
is a (Hermitian
) Hamiltonian
part, the
are an arbitrary orthonormal basis
of the operators on the system's Hilbert space, and the
are constants which determine the dynamics. The coefficient matrix 
must be positive
to ensure that the equation is trace preserving and completely positive. The summation only runs to
because we have taken 
to be proportional to the identity operator, in which case the summand vanishes. Our convention implies that the 
are traceless for 
. The terms in the summation where 
can be described in terms of the Lindblad superoperator
,
.
If the
terms are all zero, then this is the quantum Liouville equation (for a closed system), which is the quantum analog of the classical Liouville equation. A related equation describes the time evolution of the expectation values of observables, it is given by the Ehrenfest theorem
.
Note that
is not necessarily equal to the self-Hamiltonian of the system. It may also incorporate effective unitary dynamics arising from the system-environment interaction.
is positive, it can be diagonalized
with a unitary
transformation u:
where the eigenvalues
are non-negative. If we define another orthonormal operator basis
we can rewrite the Lindblad equation in diagonal form
This equation is invariant under a unitary transformation of the Lindblad operators and constants,
and also under the inhomogenous transformation
However, the first transformation destroys the orthonormality of the operators
(unless all the 
are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the 
, the 
of the diagonal form of the Lindblad equation are uniquely determined by the dynamics so long as we require them to be orthonormal and traceless.
, it has
, 
, 
, 
with all others 
. Here 
is the mean number of excitations in the reservoir damping the oscillator and 
is the decay rate. Additional Lindblad operators can be included to model various forms of dephasing and vibrational relaxation. These methods have been incorporated into grid-based density matrix
propagation methods.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the Kossakowski–Lindblad equation (after Andrzej Kossakowski and Göran Lindblad) or master equation in the Lindblad form is the most general type of markovian and time-homogeneous master equation
Master equation
In physics and chemistry and related fields, master equations are used to describe the time-evolution of a system that can be modelled as being in exactly one of countable number of states at any given time, and where switching between states is treated probabilistically...
describing non-unitary evolution of the density matrix
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
that is trace preserving and completely positive for any initial condition.The Lindblad master equation for an
-dimensional system's reduced density matrix can be written:where
is a (HermitianHermitian
A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite:*Hermitian adjoint*Hermitian connection, the unique connection on a Hermitian manifold that satisfies specific conditions...
) Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
part, the
are an arbitrary orthonormal basisOrthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
of the operators on the system's Hilbert space, and the
are constants which determine the dynamics. The coefficient matrix must be positivePositive-definite matrix
In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....
to ensure that the equation is trace preserving and completely positive. The summation only runs to
because we have taken to be proportional to the identity operator, in which case the summand vanishes. Our convention implies that the are traceless for . The terms in the summation where can be described in terms of the Lindblad superoperatorLindblad superoperator
The Lindblad superoperator is often used to express the quantum master equation for a dissipative system.In the canonical formulation of quantum mechanics, a system's time evolution is governed by unitary dynamics. This implies that phase coherence is maintained throughout the process, and is a...
,
.If the
terms are all zero, then this is the quantum Liouville equation (for a closed system), which is the quantum analog of the classical Liouville equation. A related equation describes the time evolution of the expectation values of observables, it is given by the Ehrenfest theoremEhrenfest theorem
The Ehrenfest theorem, named after Paul Ehrenfest, the Austrian physicist and mathematician, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system...
.
Note that
is not necessarily equal to the self-Hamiltonian of the system. It may also incorporate effective unitary dynamics arising from the system-environment interaction.Diagonalization
Since the matrix is positive, it can be diagonalizedDiagonalizable matrix
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...
with a unitary
Unitary
Unitary may refer to:* Unitary construction, in automotive design, another common term for a unibody or monocoque construction**Unitary as chemical weapons opposite of Binary...
transformation u:
where the eigenvalues
are non-negative. If we define another orthonormal operator basiswe can rewrite the Lindblad equation in diagonal form
This equation is invariant under a unitary transformation of the Lindblad operators and constants,
and also under the inhomogenous transformation
However, the first transformation destroys the orthonormality of the operators
(unless all the are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the , the of the diagonal form of the Lindblad equation are uniquely determined by the dynamics so long as we require them to be orthonormal and traceless.Harmonic oscillator example
The most common Lindblad equation is that describing the damping of a quantum harmonic oscillatorQuantum harmonic oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...
, it has
, , , with all others . Here is the mean number of excitations in the reservoir damping the oscillator and is the decay rate. Additional Lindblad operators can be included to model various forms of dephasing and vibrational relaxation. These methods have been incorporated into grid-based density matrixDensity matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
propagation methods.