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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 the most general type of Markovian and time-homogeneous master equation describing (in general non-unitary) evolution of the density matrix ? that preserves the laws of quantum mechanics (i.e., is trace-preserving and completely positive for any initial condition).[1]

The Schrödinger equation 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.[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.

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. These effects are the reasons quantum mechanics is difficult to observe on a macroscopic scale. 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.

Definition

More generally, the Lindblad master equation for an N-dimensional system's density matrix ? can be written as[1] (for a pedagogical introduction you can refer to[3])

${\displaystyle {\dot {\rho }}=-{i \over \hbar }[H,\rho ]+\sum _{n,m=1}^{N^{2}-1}h_{nm}\left(A_{n}\rho A_{m}^{\dagger }-{\frac {1}{2}}\left\{A_{m}^{\dagger }A_{n},\rho \right\}\right)}$

where H is a (Hermitian) Hamiltonian part, and ${\displaystyle \{A_{m}\}}$ is an arbitrary orthonormal basis of the Hilbert-Schmidt operators on the system's Hilbert space with the restriction that AN2 is proportional to the identity operator. Our convention implies that the other Am are traceless, and note that the summation only runs to N2 - 1 thus excluding the only basis-matrix with a non-zero trace. The coefficient matrix h, together with the Hamiltonian, determines the system dynamics. The matrix h must be positive semidefinite to ensure that the equation is trace-preserving and completely positive. The anticommutator is defined as ${\displaystyle \{a,b\}=ab+ba.}$

If the hmn are all zero, then this reduces to the quantum Liouville equation for a closed system, ${\displaystyle {\dot {\rho }}=-(i/\hbar )[H,\rho ]}$. This is also known as the von Neumann equation, and is the quantum analog of the classical Liouville equation.

Since the matrix h is positive semidefinite, it can be diagonalized with a unitary transformation u:

${\displaystyle u^{\dagger }hu={\begin{bmatrix}\gamma _{1}&0&\cdots &0\\0&\gamma _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\gamma _{N^{2}-1}\end{bmatrix}}}$

where the eigenvalues ?i are non-negative. If we define another orthonormal operator basis

${\displaystyle L_{i}=\sum _{j=1}^{N^{2}-1}u_{ji}A_{j}}$

we can rewrite the Lindblad equation in diagonal form

${\displaystyle {\dot {\rho }}=-{i \over \hbar }[H,\rho ]+\sum _{i=1}^{N^{2}-1}\gamma _{i}\left(L_{i}\rho L_{i}^{\dagger }-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},\rho \right\}\right).}$

The new operators Li are commonly called the Lindblad or jump operators of the system.

Quantum dynamical 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 ${\displaystyle \phi _{t}}$ on the space of density matrices indexed by a single time parameter ${\displaystyle t\geq 0}$ that obey the semigroup property

${\displaystyle \phi _{s}(\phi _{t}(\rho ))=\phi _{t+s}(\rho ),\qquad t,s\geq 0.}$

The Lindblad equation can be obtained by

${\displaystyle {\mathcal {L}}(\rho )=\mathrm {lim} _{\Delta t\to 0}{\frac {\phi _{\Delta t}(\rho )-\phi _{0}(\rho )}{\Delta t}}}$

which, by the linearity of ${\displaystyle \phi _{t}}$, is a linear superoperator. The semigroup can be recovered as

${\displaystyle \phi _{t+s}(\rho )=e^{{\mathcal {L}}s}\phi _{t}(\rho ).}$

Invariance properties

The Lindblad equation is invariant under any unitary transformation v of Lindblad operators and constants,

${\displaystyle {\sqrt {\gamma _{i}}}L_{i}\to {\sqrt {\gamma _{i}'}}L_{i}'=\sum _{j=1}^{N^{2}-1}v_{ij}{\sqrt {\gamma _{j}}}L_{j},}$

and also under the inhomogeneous transformation

${\displaystyle L_{i}\to L_{i}'=L_{i}+a_{i}I,}$
${\displaystyle H\to H'=H+{\frac {1}{2i}}\sum _{j=1}^{N^{2}-1}\gamma _{j}\left(a_{j}^{*}L_{j}-a_{j}L_{j}^{\dagger }\right)+bI,}$

where ai are complex numbers and b is a real number. However, the first transformation destroys the orthonormality of the operators Li (unless all the ?i are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the ?i, the Li 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.

Heisenberg picture

The Lindblad-type evolution of the density matrix in the Schrödinger picture can be equivalently described in the Heisenberg picture using the following (diagonalized) equation of motion[] for each quantum observable X:

${\displaystyle {\dot {X}}={\frac {i}{\hbar }}[H,X]+\sum _{i=1}^{N^{2}-1}\gamma _{i}\left(L_{i}^{\dagger }XL_{i}-{\frac {1}{2}}\left\{L_{i}^{\dagger }L_{i},X\right\}\right).}$

A similar equation describes the time evolution of the expectation values of observables, given by the Ehrenfest theorem. Corresponding to the trace-preserving property of the Schrödinger picture Lindblad equation, the Heisenberg picture equation is unital, i.e. it preserves the identity operator.

Physical derivation

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 H 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,[4] 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[5][6] 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 compared to the system timescale of interest can be neglected. These three approximations are called Born, Markov, and rotating wave, respectively.[7]

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

${\displaystyle H=H_{S}+H_{B}+H_{BS}\,}$

The dynamics of the entire system can be described by the Liouville equation of motion, ${\displaystyle {\dot {\chi }}=-i[H,\chi ]}$. This equation, containing an infinite number of degrees of freedom, is impossible to solve analytically except in very particular cases. What's more, under certain approximations, the bath degrees of freedom need not be considered, and an effective master equation can be derived in terms of the system density matrix, ${\displaystyle \rho =\operatorname {tr} _{B}\chi }$. The problem can be analyzed more easily by moving into the interaction picture, defined by the unitary transformation ${\displaystyle {\tilde {M}}=U_{0}MU_{0}^{\dagger }}$, where ${\displaystyle M}$ is an arbitrary operator, and ${\displaystyle U_{0}=e^{i(H_{S}+H_{B})t}}$. Also note that ${\displaystyle U(t,t_{0})}$is the total unitary operator of the entire system. It is straightforward to confirm that the Liouville equation becomes

${\displaystyle {\dot {\tilde {\chi }}}=-i[{\tilde {H}}_{BS},{\tilde {\chi }}]\,}$

where the Hamiltonian ${\displaystyle {\tilde {H}}_{BS}=e^{i(H_{S}+H_{B})t}H_{BS}e^{-i(H_{S}+H_{B})t}}$ is explicitly time dependent. Also, according to the interaction picture, ${\displaystyle {\tilde {\chi }}=U_{BS}(t,t_{0})\chi U_{BS}^{\dagger }(t,t_{0})}$, where ${\displaystyle U_{BS}=U_{0}^{\dagger }U(t,t_{0})}$. This equation can be integrated directly to give

${\displaystyle {\tilde {\chi }}(t)={\tilde {\chi }}(0)-i\int _{0}^{t}dt'[{\tilde {H}}_{BS}(t'),{\tilde {\chi }}(t')]}$

This implicit equation for ${\displaystyle {\tilde {\chi }}}$ can be substituted back into the Liouville equation to obtain an exact differo-integral equation

${\displaystyle {\dot {\tilde {\chi }}}=-i[{\tilde {H}}_{BS},{\tilde {\chi }}(0)]-\int _{0}^{t}dt'[{\tilde {H}}_{BS}(t),[{\tilde {H}}_{BS}(t'),{\tilde {\chi }}(t')]]}$

We proceed with the derivation by assuming the interaction is initiated at ${\displaystyle t=0}$, and at that time there are no correlations between the system and the bath. This implies that the initial condition is factorable as ${\displaystyle \chi (0)=\rho (0)R_{0}}$, where ${\displaystyle R_{0}}$ is the density operator of the bath initially.

Tracing over the bath degrees of freedom, ${\displaystyle \operatorname {tr} _{R}{\tilde {\chi }}={\tilde {\rho }}}$, of the aforementioned differo-integral equation yields

${\displaystyle {\dot {\tilde {\rho }}}=-\int _{0}^{t}dt'\operatorname {tr} _{R}\{[{\tilde {H}}_{BS}(t),[{\tilde {H}}_{BS}(t'),{\tilde {\chi }}(t')]]\}}$

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 ${\displaystyle {\tilde {\chi }}(t)={\tilde {\rho }}(t)R_{0}}$. The master equation becomes

${\displaystyle {\dot {\tilde {\rho }}}=-\int _{0}^{t}dt'\operatorname {tr} _{R}\{[{\tilde {H}}_{BS}(t),[{\tilde {H}}_{BS}(t'),{\tilde {\rho }}(t')R_{0}]]\}}$

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 ${\displaystyle \rho (t')\rightarrow \rho (t)}$ on the right hand side of the equation.

${\displaystyle {\dot {\tilde {\rho }}}=-\int _{0}^{t}dt'\operatorname {tr} _{R}\{[{\tilde {H}}_{BS}(t),[{\tilde {H}}_{BS}(t'),{\tilde {\rho }}(t)R_{0}]]\}}$

If the interaction Hamiltonian is assumed to have the form

${\displaystyle H_{BS}=\sum \alpha _{i}\Gamma _{i}}$

for system operators ${\displaystyle \alpha _{i}}$ and bath operators ${\displaystyle \Gamma _{i}}$, the master equation becomes

${\displaystyle {\dot {\tilde {\rho }}}=-\sum \int _{0}^{t}dt'\operatorname {tr} _{R}\{[\alpha _{i}(t)\Gamma _{i}(t),[\alpha _{j}(t')\Gamma _{j}(t'),{\tilde {\rho }}(t)R_{0}]]\}}$

which can be expanded as

${\displaystyle {\dot {\tilde {\rho }}}=-\sum \int _{0}^{t}dt'\left(\alpha _{i}(t)\alpha _{j}(t')\rho -\alpha _{j}(t')\rho (t)\alpha _{i}(t)\right)\langle \Gamma _{i}(t)\Gamma _{j}(t')\rangle +\left(\rho (t)\alpha _{j}(t')\alpha _{i}(t)-\alpha _{i}(t)\rho (t)\alpha _{j}(t')\right)\langle \Gamma _{j}(t')\Gamma _{i}(t)\rangle }$

The expectation values ${\displaystyle \langle \Gamma _{i}\Gamma _{j}\rangle =\operatorname {tr} \{\Gamma _{i}\Gamma _{j}R_{0}\}}$ are with respect to the bath degrees of freedom. By assuming rapid decay of these correlations (ideally ${\displaystyle \langle \Gamma _{i}(t)\Gamma _{j}(t')\rangle \propto \delta (t-t')}$), above form of the Lindblad superoperator L is achieved.

Examples

For one jump operator ${\displaystyle F}$ and no unitary evolution, the Lindblad superoperator, acting on the density matrix ${\displaystyle \rho }$, is

${\displaystyle {\mathcal {D}}[F](\rho )=F\rho F^{\dagger }-{\frac {1}{2}}\left(F^{\dagger }F\rho +\rho F^{\dagger }F\right)}$

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

{\displaystyle {\begin{aligned}F_{1}&=a,&\gamma _{1}&={\tfrac {\gamma }{2}}\left({\overline {n}}+1\right),\\F_{2}&=a^{\dagger },&\gamma _{2}&={\tfrac {\gamma }{2}}{\overline {n}}.\end{aligned}}}

Here ${\displaystyle {\overline {n}}}$ is the mean number of excitations in the reservoir damping the oscillator and ? is the decay rate. If we also add additional unitary evolution generated by the quantum harmonic oscillator Hamiltonian with frequency ${\displaystyle \omega _{c}}$, we obtain

${\displaystyle {\dot {\rho }}=-i[\omega _{c}a^{\dagger }a,\rho ]+{\mathcal {D}}[F_{1}](\rho )+{\mathcal {D}}[F_{2}](\rho ).}$

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.

References

1. ^ a b c Breuer, Heinz-Peter; Petruccione, F. (2002). The Theory of Open Quantum Systems. Oxford University Press. ISBN 978-0-1985-2063-4.
2. ^ Weinberg, Steven (2014). "Quantum Mechanics Without State Vectors". Phys. Rev. A. 90: 042102. arXiv:1405.3483. doi:10.1103/PhysRevA.90.042102.
3. ^ Manzano, Daniel (2020). "A short introduction to the Lindblad master equation". AIP Advances. 10: 025106. arXiv:1906.04478. doi:10.1063/1.5115323.
4. ^ Preskill, John. Lecture notes on Quantum Computation, Ph219/CS219 (PDF).
5. ^ Alicki, Robert; Lendi, Karl (2007). Quantum Dynamical Semigroups and Applications. Springer. doi:10.1007/b11976790.
6. ^ Carmichael, Howard. An Open Systems Approach to Quantum Optics. Springer Verlag, 1991
7. ^ This paragraph was adapted from Albert, Victor V. "Lindbladians with multiple steady states: theory and applications". arXiv:1802.00010.

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