In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation, named after Vittorio Gorini, Andrzej Kossakowski, George Sudarshan and Göran Lindblad), master equation in Lindblad form, quantum Liouvillian, or Lindbladian is one of the general forms of Markovian master equations describing open quantum systems. It generalizes the Schrödinger equation to open quantum systems; that is, systems in contacts with their surroundings. The resulting dynamics is no longer unitary, but still satisfies the property of being trace-preserving and completely positive for any initial condition.[1]
The Schrödinger equation or, actually, the von Neumann equation, is a special case of the GKSL 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.[2] The Schrödinger equation deals with state vectors, which can only describe pure quantum states and are thus less general than density matrices, which can describe mixed states as well.
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, and its associated master equation. While in principle this approach to solving quantum dynamics is equivalent to the Schrödinger picture or Heisenberg picture, 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.
The Lindblad master equation for system's density matrix can be written as[1] (for a pedagogical introduction you may refer to[3])
\rho=-{i\over\hbar}[H,\rho]+\sum |
i |
\gammai\left(Li\rho
\dagger | ||
L | - | |
i |
1 | |
2 |
\dagger | |
\left\{L | |
i |
Li,\rho\right\}\right)
where
\{a,b\}=ab+ba
H
Li
\gammai\geq0
\gammai=0
\rho=-(i/\hbar)[H,\rho] |
More generally, the GKSL equation has the form
\rho=-{i\over\hbar}[H,\rho]+\sum |
n,mhnm\left(An\rho
| ||||
A | ||||
m |
\dagger | |
\left\{A | |
m |
An,\rho\right\}\right)
where
\{Am\}
Am
N
N2-1
Since the matrix is positive semidefinite, it can be diagonalized with a unitary transformation :
u\daggerhu=\begin{bmatrix} \gamma1&0& … &0\\ 0&\gamma2& … &0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0& … &
\gamma | |
N2-1 |
\end{bmatrix}
where the eigenvalues are non-negative. If we define another orthonormal operator basis
Li=\sumjujiAj
This reduces the master equation to the same form as before:
\rho=-{i\over\hbar}[H,\rho]+\sum |
i |
\gammai\left(Li\rho
\dagger | ||
L | - | |
i |
1 | |
2 |
\dagger | |
\left\{L | |
i |
Li,\rho\right\}\right)
See main article: Quantum Markov semigroup.
The maps generated by a Lindbladian for various times are collectively referred to as a quantum dynamical semigroup - a family of quantum dynamical maps
\phit
t\ge0
\phis(\phit(\rho))=\phit+s(\rho), t,s\ge0.
l{L}(\rho)=lim\Delta
\phi\Delta(\rho)-\phi0(\rho) | |
\Deltat |
\phit
\phit+s(\rho)=el{Ls}\phit(\rho).
The Lindblad equation is invariant under any unitary transformation of Lindblad operators and constants,
\sqrt{\gammai}Li\to\sqrt{\gammai'}Li'=\sumjvij\sqrt{\gammaj}Lj,
and also under the inhomogeneous transformation
Li\toLi'=Li+aiI,
H\toH'=H+
1 | |
2i |
\sumj\gammaj\left
* | |
(a | |
j |
Lj-aj
\dagger | |
L | |
j |
\right)+bI,
where are complex numbers and is a real number.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.
The Lindblad-type evolution of the density matrix in the Schrödinger picture can be equivalently described in the Heisenberg pictureusing the following (diagonalized) equation of motion[4] for each quantum observable :
X |
=
i | |
\hbar |
[H,X]+\sumi\gammai
\dagger | |
\left(L | |
i |
XLi-
1 | |
2 |
\dagger | |
\left\{L | |
i |
Li,X\right\}\right).
The Lindblad master equation describes the evolution of various types of open quantum systems, e.g. a system weakly coupled to a Markovian reservoir.[1] Note that the appearing in the equation is not necessarily equal to the bare system Hamiltonian, but may also incorporate effective unitary dynamics arising from the system-environment interaction.
A heuristic derivation, e.g., in the notes by Preskill,[5] begins with a more general form of an open quantum system and converts it into Lindblad form by making the Markovian assumption and expanding in small time. A more physically motivated standard treatment[6] [7] covers three common types of derivations of the Lindbladian starting from a Hamiltonian acting on both the system and environment: the weak coupling limit (described in detail below), the low density approximation, and the singular coupling limit. Each of these relies on specific physical assumptions regarding, e.g., correlation functions of the environment. For example, in the weak coupling limit derivation, one typically assumes that (a) correlations of the system with the environment develop slowly, (b) excitations of the environment caused by system decay quickly, and (c) terms which are fast-oscillating when comparedto the system timescale of interest can be neglected. These three approximations are called Born,Markov, and rotating wave, respectively.[8]
The weak-coupling limit derivation assumes a quantum system with a finite number of degrees of freedom coupled to a bath containing an infinite number of degrees of freedom. The system and bath each possess a Hamiltonian written in terms of operators acting only on the respective subspace of the total Hilbert space. These Hamiltonians govern the internal dynamics of the uncoupled system and bath. There is a third Hamiltonian that contains products of system and bath operators, thus coupling the system and bath. The most general form of this Hamiltonian is
H=HS+HB+HBS
The dynamics of the entire system can be described by the Liouville equation of motion,
\chi |
=-i[H,\chi]
\rho=\operatorname{tr}B\chi
\tilde{M}=U0MU
\dagger | |
0 |
M
i(HS+HB)t | |
U | |
0=e |
U(t,t0)
\tilde{\chi |
where the Hamiltonian
\tilde{H}BS
i(HS+HB)t | |
=e |
HBS
-i(HS+HB)t | |
e |
\tilde{\chi}=UBS(t,t0)\chi
\dagger | |
U | |
BS |
(t,t0)
UBS
\dagger | |
=U | |
0 |
U(t,t0)
\tilde{\chi}(t)=\tilde{\chi}(0)
t | |
-i\int | |
0 |
dt'[\tilde{H}BS(t'),\tilde{\chi}(t')]
This implicit equation for
\tilde{\chi}
\tilde{\chi |
We proceed with the derivation by assuming the interaction is initiated at
t=0
\chi(0)=\rho(0)R0
R0
Tracing over the bath degrees of freedom,
\operatorname{tr}R\tilde{\chi}=\tilde{\rho}
\tilde{\rho |
This equation is exact for the time dynamics of the system density matrix but requires full knowledge of the dynamics of the bath degrees of freedom. A simplifying assumption called the Born approximation rests on the largeness of the bath and the relative weakness of the coupling, which is to say the coupling of the system to the bath should not significantly alter the bath eigenstates. In this case the full density matrix is factorable for all times as
\tilde{\chi}(t)=\tilde{\rho}(t)R0
\tilde{\rho |
The equation is now explicit in the system degrees of freedom, but is very difficult to solve. A final assumption is the Born-Markov approximation that the time derivative of the density matrix depends only on its current state, and not on its past. This assumption is valid under fast bath dynamics, wherein correlations within the bath are lost extremely quickly, and amounts to replacing
\rho(t') → \rho(t)
\tilde{\rho |
If the interaction Hamiltonian is assumed to have the form
HBS=\sumi\alphai\Gammai
for system operators
\alphai
\Gammai
\tilde{H}BS=\sumi\tilde{\alpha}i\tilde{\Gamma}i
\tilde{\rho |
which can be expanded as
\tilde{\rho |
The expectation values
\langle\Gammai\Gammaj\rangle=\operatorname{tr}\{\Gammai\GammajR0\}
\langle\Gammai(t)\Gammaj(t')\rangle\propto\delta(t-t')
F
\rho
l{D}[F](\rho)={F\rhoF\dagger}-
1 | |
2 |
\left(F\daggerF\rho+\rhoF\daggerF\right)
Such a term is found regularly in the Lindblad equation as used in quantum optics, where it can express absorption or emission of photons from a reservoir. If one wants to have both absorption and emission, one would need a jump operator for each. This leads to the most common Lindblad equation describing the damping of a quantum harmonic oscillator (representing e.g. a Fabry–Perot cavity) coupled to a thermal bath, with jump operators
\begin{align}F1&=a,&\gamma1&=\tfrac{\gamma}{2}\left(\overline{n}+1\right),\\ F2&=a\dagger,&\gamma2&=\tfrac{\gamma}{2}\overline{n}. \end{align}
Here
\overline{n}
\omegac
\rho |
=-i[\omegaca\daggera,\rho]+\gamma1l{D}[F1](\rho)+\gamma2l{D}[F2](\rho).
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.