The resulting dynamics are no longer unitary, but still satisfy the property of being trace-preserving and completely positive for any initial condition.
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.
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 will interact with its environment, and is not absolutely isolated.
The interaction with degrees of freedom that are external to the system results in dissipation of energy into the surroundings, causing decay and randomization of phase.
Certain mathematical techniques have been introduced to treat the interaction of a quantum system with its environment.
The Lindblad master equation for system's density matrix ρ can be written as[1] (for a pedagogical introduction you may refer to[3]) where
are a set of non-negative real coefficients called damping rates.
describing unitary dynamics, which 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: where the eigenvalues γi are non-negative.
If we define another orthonormal operator basis This reduces the master equation to the same form as before:
on the space of density matrices indexed by a single time parameter
that obey the semigroup property The Lindblad equation can be obtained by which, by the linearity of
The semigroup can be recovered as The Lindblad equation is invariant under any unitary transformation v of Lindblad operators and constants, and also under the inhomogeneous transformation where ai are complex numbers and b is a real number.
Therefore, up to degeneracies among the γi, 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.
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[4] for each quantum observable X: A similar equation describes the time evolution of the expectation values of observables, given by the Ehrenfest theorem.
[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,[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.
The system and bath each possess a Hamiltonian written in terms of operators acting only on the respective subspace of the total Hilbert space.
The most general form of this Hamiltonian is The dynamics of the entire system can be described by the Liouville equation of motion,
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,
The problem can be analyzed more easily by moving into the interaction picture, defined by the unitary transformation
, of the aforementioned differo-integral equation yields 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 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 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: Here
is the mean number of excitations in the reservoir damping the oscillator and γ is the decay rate.
of the photons, we can add a further unitary evolution: Additional Lindblad operators can be included to model various forms of dephasing and vibrational relaxation.