Lindblad State-to-State Method in Open Quantum Systems

• The Lindblad state-to-state method is a rigorous framework for modeling quantum transitions, ensuring trace conservation and embedding non-Hermitian dynamics.
• It computes state-to-state decay rates and partial widths, bridging quantum kinetics with classical rate equations in systems like auto-ionization.
• The method employs quantum-jump unraveling and Markov chain techniques to simulate open quantum systems and derive nonequilibrium steady states.

The Lindblad state-to-state method is a general, rigorously trace-conserving framework for describing quantum systems coupled to reservoirs or exhibiting decay, rooted in the Lindblad master equation. This formalism calculates state-to-state transition rates, partial decay widths, and nonequilibrium steady states (NESS) by embedding the underlying non-Hermitian dynamics into a completely positive, Markovian evolution. The method preserves probability and recovers classical rate equations in the appropriate limits. It is central for modeling auto-ionization, impurity transport, and open quantum kinetics.

1. Lindblad Master Equation: Structure and Interpretations

The Lindblad master equation governs the time evolution of the density matrix $\rho$ for a finite-dimensional quantum system,

$$\frac{d\rho}{dt} = -i[H,\rho] + \sum_k \gamma_k \left(L_k\rho L_k^\dagger - \frac{1}{2}\{L_k^\dagger L_k, \rho\}\right),$$

where $H$ is the Hermitian system Hamiltonian, $\{L_k\}$ are Lindblad (jump) operators, and $\gamma_k$ are positive rates. This canonical form is completely positive and Markovian (Fernengel et al., 2022), and guarantees trace conservation over time.

For systems subject to decay, as in auto-ionization, the Hamiltonian includes an anti-Hermitian part $H_{ah} \geq 0$ to model irreversible loss, e.g.,

$H = H_h - i H_{ah}, \quad H_h = H_h^\dagger.$

Embedding this non-Hermitian structure within the Lindblad equation,

$$i\hbar\,\frac{d\rho}{dt} = [H_h,\rho] - i\{H_{ah}, \rho\} + 2i\sum_k A_k \rho A_k^\dagger,$$

with the consistency condition $\sum_k A_k^\dagger A_k = H_{ah}$ (Selstø, 2012), ensures the physically required trace conservation.

2. State-to-State Transition Rates and Partial Widths

A central feature of this methodology is the derivation and calculation of state-to-state rates and partial widths in open quantum systems.

For auto-ionization, starting from a resonant $N$-particle state $|\Psi_{\rm res}\rangle$, the time-dependent population of each $(N-1)$-particle bound channel $|\varphi_p\rangle$ is governed by the coupled dynamics: \begin{align} i\hbar \frac{d\rho_N}{dt} &= [H_h, \rho_N] - i{H_{ah}, \rho_N}, \ i\hbar \frac{d\rho_{N-1}}{dt} &= [H_h, \rho_{N-1}] - i{H_{ah}, \rho_{N-1}} + S[\rho_N], \end{align} with the source term $S[\rho_N] = 2\sum_k r(x_k) c_k \rho_N c_k^\dagger$ (Selstø, 2012).

Projecting onto the channel states, the time derivative of the population $$P_p(t) = \langle \varphi_p | \rho_{N-1}(t) | \varphi_p \rangle$$ gives the partial width: $$\frac{dP_p}{dt} = \frac{\Gamma_p}{\hbar} e^{-(\Gamma/\hbar)t},$$ where

$$\Gamma_p = 2\sum_k r(x_k) |\langle \varphi_p | c_k | \Psi_{\rm res}\rangle|^2,$$

and the total width $\Gamma = 2\Im(E_{\rm res})$ is exhausted by the sum over channels, $\sum_p \Gamma_p = \Gamma$ (Selstø, 2012).

In general open-system scenarios, each Lindblad operator $L_k$ mediates transitions between energy eigenstates $|E_m\rangle \to |E_n\rangle$ at rates

$$W_{m\to n} = \sum_k |\langle E_n | L_k | E_m\rangle|^2,$$

directly embedding quantum kinetics within the master equation formalism (Yuge et al., 2014).

3. Quantum-Jump Unraveling and Markov Chain Interpretation

Quantum-jump unraveling interprets the Lindblad evolution in terms of stochastic pure-state trajectories, each evolving deterministically between jumps under the non-Hermitian "conditional" Hamiltonian,

$$H_c = H - \frac{i}{2} \sum_k \gamma_k L_k^\dagger L_k,$$

with random, state-dependent waiting times for the next jump. The waiting time density for jump $k$ from state $|\psi_n\rangle$ reads

$$f^{(k)}(t|\psi_n) = \gamma_k \|L_k e^{-iH_c t} \psi_n\|^2,$$

and the post-jump state is

$$|\psi_{n+1}\rangle = \frac{L_k e^{-iH_c t} |\psi_n\rangle}{\|L_k e^{-iH_c t} |\psi_n\rangle\|}.$$

The set of visited post-jump states corresponds to a finite discrete-time Markov chain. Its stationary distribution yields explicit expressions for the steady-state density matrix, averaging over ergodic trajectories (Fernengel et al., 2022).

4. Correspondence with Classical Rate Equations and Nonequilibrium Steady State

The Lindblad state-to-state method reduces exactly to classical first-order kinetics under exponential decay,

$$\dot{P}_{\rm res} = -k_{\rm tot} P_{\rm res}, \quad \dot{P}_p = k_p P_{\rm res},$$

with $k_p = \Gamma_p/\hbar$ and $k_{\rm tot} = \Gamma/\hbar$. This correspondence ensures that, in regimes dominated by incoherent transitions, quantum and classical approaches coincide (Selstø, 2012).

For nonequilibrium steady-state analysis of open systems weakly coupled to reservoirs, the method leads to a perturbative expansion: $\mathcal{L} = \mathcal{L}_L + v \mathcal{R},$ where $\mathcal{L}_L$ is the secular Lindblad part and $v \mathcal{R}$ is the residual. The steady-state solution is

$\rho_{\rm NESS} = \rho_0 + \rho_1 + O(v^2),$

with $\rho_0$ the Lindblad steady state, and $\rho_1 = -\mathcal{L}_L^{-1} \mathcal{R}[\rho_0]$ gives first-order corrections (Yuge et al., 2014).

5. Practical Computation: Complex Absorbing Potentials and Numerical Strategies

The methodology accommodates systems equipped with complex absorbing potentials (CAP) or exterior complex scaling (ECS), generalizing the anti-Hermitian part:

• CAP: $H_{ah} = \sum_k r(x_k) c_k^\dagger c_k$, $A_k = \sqrt{r(x_k)} c_k$,
• ECS: $H_{ah}$ includes additional one-body and two-body anti-Hermitian terms, split into jump operators $A_\alpha$ (Selstø, 2012).

For numerical computations in quantum impurity and transport models, the Lindblad state-to-state method is implemented via Lindblad-driven discretized leads (LDDL) or renormalized Lindblad-NESS (RL-NESS) frameworks. These employ local Lindblad drive rates $\gamma \sim \delta$ to blur discrete levels into a continuum, hybrid log-linear discretization, Wilson chain mapping, and matrix-product density operator (MPDO) real-time evolution to reach numerically exact NESS with bounded operator entanglement (Lotem et al., 2020, Schwarz et al., 2016). Analytical solutions exist for quadratic models, e.g., the resonant level model, with explicit Green's function formulas and Landauer-type current expressions in the continuum limit.

6. Trace Conservation, Population Dynamics, and Rigorous Implementation

Trace conservation is fundamental: the Lindblad equation ensures $\mathrm{Tr}[\rho(t)] = 1$ at all times, and outflow from decaying sectors is matched by inflow into lower-particle-number channels. Explicitly,

$$\frac{d}{dt}\mathrm{Tr}[\rho_N] + \sum_p \frac{d}{dt}P_p = 0,$$

with $\sum_p \Gamma_p = \Gamma$ (Selstø, 2012). Population dynamics are therefore fully governed by the partial widths and total decay rates, and time-dependent solutions (e.g., $P_{\rm res}(t) = e^{-(\Gamma/\hbar)t}$, $$P_p(t) = \frac{\Gamma_p}{\Gamma}(1 - e^{-(\Gamma/\hbar)t})$$) follow directly from the formalism.

Rigorous implementation strategies involve:

• Calculation of resonance states and eigenvalues.
• Diagonalization of the bound-state spectrum for the residual sector.
• Evaluation of jump operator matrix elements to extract partial rates.
• Verification of width sum rules.
• Structural mapping to numerical chain representations for large system simulation.

7. Extensions and Analytical Steady-State Formulae

Explicit analytical steady-state solutions are accessible for finite state spaces using quantum-jump unraveling and ergodic Markov chain theory. Given a finite set of pure states $S = \{|\psi_1\rangle, \ldots, |\psi_{|S|}\rangle\}$ and transition matrix $P_{rs}$, the stationary distribution yields the steady-state density matrix as a convex sum over time-averaged pure-state trajectories, with uniqueness depending on the irreducibility of the Markov chain (Fernengel et al., 2022).

The classical limit is achieved for $H = 0$ and completely depolarizing Lindblad terms, where the Markov process reduces to a jump process with rates $\gamma_k |\langle \psi_r| L_k | \psi_s\rangle|^2$.

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The Lindblad state-to-state method unifies rigorous trace-conserving quantum master equations, explicit transition-rate computations, and practical mapping to kinetic and continuum-limit models. It provides a comprehensive framework for partial width extraction, steady-state analysis, and numerical simulation in open many-body quantum systems (Selstø, 2012, Lotem et al., 2020, Yuge et al., 2014, Schwarz et al., 2016, Fernengel et al., 2022).