What is "Lindblad Superoperator" in Stochastic Master Equation

Question

I was reading a paper titled "Using a Recurrent Neural Network to Reconstruct Quantum Dynamics of a Superconducting Qubit from Physical Observations" and was confused about a stochastic master equation governing the evolution of the single-qubit system. According to the paper, they are denoting the non-unitary evolution of a superconducting qubit. The equation in question is the following: $$ d \rho_{t}=\underbrace{\left(i\left[H_{R}, \rho_{t}\right]+\mathcal{L}\left[\sqrt{\frac{\gamma}{2}} \sigma_{Z}\right] \rho_{t}\right)}_{\text {dissipative evolution }} d t+\underbrace{\sqrt{\eta} \mathcal{H}\left[\sqrt{\frac{\gamma}{2}} \sigma_{Z}\right] \rho_{t}}_{\text {backaction }} d w_{t} $$ In this case, $H_{\mathrm{R}}=\frac{\hbar \Omega_{R}}{2} \sigma_{X}$, "describes a microwave drive at the qubit transition frequency which induces unitary evolution of the qubit state characterized by the Rabi frequency $\Omega_R$." I understand and was able to implement (in MATLAB) the first half of the dissipative evolution, $i[H_R,p_T]$, as this is fairly simple; however, what do the $\mathcal{L}$ and $\mathcal{H}$ mean? The paper describes $\mathcal{L}$ as "the Lindblad superoperator describing the qubit dephasing induced by the measurement of strength $\gamma$"; however I have little idea as to what this means. What exactly does this mean and, if possible, what are methods of encoding this computationally (in something like MATLAB or python)

Answer 1

$\newcommand\dag\dagger$ The Lindblad superoperator is shorthand for a longer operation that generically reads $$\mathcal{L}[\hat{L}]\hat{\rho} = \hat{L} \hat{\rho} \hat{L}^\dag - \tfrac{1}{2}\left(\hat{L}^\dag\hat{L}\hat{\rho} + \hat{\rho}\hat{L}^\dag\hat{L} \right),$$ where the $\dag$ denotes the conjugate transpose (i.e. Hermitian adjoint) of the Lindblad operators $\hat{L}$.
Likewise, the last term (measurement / backaction superoperator) connected to the Wiener noise stands for $$\mathcal{H}[\hat{L}]\hat{\rho} = \hat{L}\hat{\rho} + \hat{\rho}\hat{L}^\dag - \hat{\rho}\,tr(\hat{L}\hat{\rho} + \hat{\rho}\hat{L}^\dag),$$ The stochastic master equation listed in the initial question contains three terms. The commutator (resonant drive) captures unitary evolution of the qubit, wherein it behaves like a closed quantum system. (You would just have Heisenberg's equation of motion, equivalent to the Schrodinger equation, if you truncated there). The two superoperator terms are related to the fact that this qubit is not actually a closed quantum system, but is open to an environmental channel through which it is being continuously and weakly monitored. In particular, $\mathcal{L}$ captures the average / dissipative evolution of the open channel, and the term $\mathcal{H}$ captures conditional evolution due to monitoring with efficiency $\eta$. The particular choice of $\hat{L} = \sqrt{\tfrac{\gamma}{2}}\hat{\sigma}_z$ characterizes the environmental channel through which measurement is being performed, and indicates that the Pauli-z observable is being tracked at a rate $\gamma$ (i.e. a larger $\gamma$ denotes a stronger measurement power, which is more invasive and ``collapses'' the qubit state faster). You can imagine that this is modeling the time-continuum limit of a sequence of very short but weak quantum measurements of $\hat{\sigma}_z$, each of which necessarily returns a random outcome. The SME gives the evolution of the qubit density matrix conditioned on the noisy measurement readout $r \propto \sqrt{\eta}\,tr(\hat{L}\hat{\rho} + \hat{\rho}\hat{L}^\dag) + dw_t/dt$, where the randomness / noise is characterized by $dw_t$, and the resulting individual trajectories undergo something that looks like Brownian motion.

As a further note: The Lindblad form $\mathcal{L}$ generically appears for any open quantum system which interacts with environmental channel(s) such that the resulting dissipation is Markovian. It dates to https://link.springer.com/article/10.1007/BF01608499, and a more recent introduction appears in, for example, https://aip.scitation.org/doi/pdf/10.1063/1.5115323. If you want further context or details about the case where the environment is monitored (i.e. continuous quantum measurement / stochastic quantum trajectories, which is the where $\mathcal{H}$ becomes crucial), you might also consider perusing some of the following...

1. https://arxiv.org/abs/quant-ph/0108132
2. https://arxiv.org/abs/quant-ph/0611067
3. https://arxiv.org/abs/1410.5345
4. https://arxiv.org/abs/1908.04720