Motivation
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. It is a consequence of the fact that all participating degrees of freedom are considered. However, any real physical system will interact with its environment as it cannot be perfectly isolated. The interaction with external degrees of freedom results in the dissipation of energy into its surroundings, which causes decay and randomization of phase. 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 this approach to solving
quantum dynamics is in principle equivalent to the
Schrödinger picture or
Heisenberg picture, it facilitates us to account for the incoherent processes that 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==
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. 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 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. 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 {{NumBlk|::| H= H_S + H_B + H_{BS} \, |}} The dynamics of the entire system can be described by the Liouville equation of motion, \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, \rho=\operatorname{tr}_B \chi . The problem can be analyzed more easily by moving into the interaction picture, defined by the unitary transformation \tilde{M}= U_0MU_0^\dagger, where M is an arbitrary operator, and U_0=e^{i(H_S+H_B)t} . Also note that U(t,t_0) is the total unitary operator of the entire system. It is straightforward to confirm that the Liouville equation becomes {{NumBlk|::| \dot{\tilde{\chi}}=-i[\tilde{H}_{BS},\tilde{\chi}] \, |}} where the Hamiltonian \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, \tilde{\chi}= U_{BS}(t,t_0)\chi U_{BS}^\dagger (t,t_0), where U_{BS}=U_0 ^\dagger U(t,t_0). This equation can be integrated directly to give {{NumBlk|::| \tilde{\chi}(t)=\tilde{\chi}(0) -i\int^t_0 dt' [\tilde{H}_{BS}(t'),\tilde{\chi}(t')] |}} This implicit equation for \tilde{\chi} can be substituted back into the Liouville equation to obtain an exact differo-integral equation {{NumBlk|::| \dot{\tilde{\chi}}=-i[\tilde{H}_{BS}(t),\tilde{\chi}(0)] - \int^t_0 dt' [\tilde{H}_{BS}(t),[\tilde{H}_{BS}(t'),\tilde{\chi}(t')|}} We proceed with the derivation by assuming the interaction is initiated at 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 \chi(0) = \rho(0) R_0 , where R_0 is the density operator of the bath initially. Tracing over the bath degrees of freedom, \operatorname{tr}_R \tilde{\chi} = \tilde{\rho} , of the aforementioned differo-integral equation yields {{NumBlk|::| \dot{\tilde{\rho}}= - \int^t_0 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 \tilde{\chi}(t)=\tilde{\rho}(t)R_0 . The master equation becomes {{NumBlk|::| \dot{\tilde{\rho}}= - \int^t_0 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 \rho(t')\rightarrow \rho(t) on the right hand side of the equation. {{NumBlk|::| \dot{\tilde{\rho}}= - \int^t_0 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 {{NumBlk|::|H_{BS}=\sum_i \alpha_i \Gamma_i|}} for system operators \alpha_i and bath operators \Gamma_i then \tilde{H}_{BS}=\sum_i \tilde{\alpha}_i \tilde{\Gamma}_i. The master equation becomes {{NumBlk|::| \dot{\tilde{\rho}}= - \sum_{i,j} \int^t_0 dt' \operatorname{tr}_R\{[\tilde{\alpha}_i(t) \tilde{\Gamma}_i(t),[\tilde{\alpha}_j(t') \tilde{\Gamma}_j(t'),\tilde{\rho}(t)R_0\} |}} which can be expanded as {{NumBlk|::|\dot{\tilde{\rho}} = - \sum_{i,j} \int^t_0 dt' \left[ \left( \tilde{\alpha}_i(t) \tilde{\alpha}_j(t') \tilde{\rho}(t) - \tilde{\alpha}_i(t) \tilde{\rho}(t) \tilde{\alpha}_j(t') \right) \langle\tilde{\Gamma}_i(t)\tilde{\Gamma}_j(t')\rangle + \left( \tilde{\rho}(t) \tilde{\alpha}_j(t') \tilde{\alpha}_i(t) - \tilde{\alpha}_j(t') \tilde{\rho}(t) \tilde{\alpha}_i(t) \right) \langle\tilde{\Gamma}_j(t')\tilde{\Gamma}_i(t)\rangle \right] |}} The expectation values \langle \Gamma_i\Gamma_j \rangle=\operatorname{tr}\{\Gamma_i\Gamma_jR_0\} are with respect to the bath degrees of freedom. By assuming rapid decay of these correlations (ideally \langle \Gamma_i(t)\Gamma_j(t') \rangle \propto \delta(t-t') ), above form of the Lindblad superoperator L is achieved. ==Examples==
Examples
In the simplest case, there is just one jump operator F and no unitary evolution. In this case, the Lindblad equation is {{NumBlk|::| \mathcal{L}(\rho) ={F\rho F^\dagger} -\frac{1}{2}\left( F^\dagger F \rho + \rho F^\dagger F\right)|}} This case is often used in
quantum optics to model either absorption or emission of photons from a reservoir. To model 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} 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{align} Here \overline{n} is the mean number of excitations in the reservoir damping the oscillator and is the decay rate. To model the
quantum harmonic oscillator Hamiltonian with frequency \omega_c of the photons, we can add a further unitary evolution: {{NumBlk|::| \dot{\rho}=-i[\omega_c a^\dagger a,\rho]+\gamma_1\mathcal{D}[F_1](\rho) + \gamma_2\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. ==See also==