A modified Lindblad equation for a Rabi driven electron-spin qubit with tunneling to a Markovian lead

We derive a modified Lindblad equation for the state of quantum dot tunnel coupled to a Markovian lead when the spin state of the dot is driven by an oscillating magnetic field. We show that the equation is a completely positive, trace-preserving map and find the jump operators. This is a driven-dissipative regime in which coherent driving is relevant to the tunneling and cannot be treated as simply a rotation modifying the system with a bath derived under a static magnetic field. This work was motivated by an experimental desire to determine the Zeeman splitting of an electron spin on a quantum dot (a spin qubit), and in a related work we show that this splitting energy can be found by measuring the charge occupancy of the dot while sweeping the frequency of the driving field \ arXiv:2503.17481. Here we cover the full derivation of the equation and give the jump operators. These jump operators are potentially useful for describing the stochastic behavior of more complex systems with coherent driving of a spin capable of tunneling on or off of a device, such as in electron spin resonance scanning tunneling microscopy. The jump operators have the interesting feature of combining jumps of electrons onto and off of the device.

💡 Research Summary

The paper presents a rigorous derivation of a Lindblad‑type master equation for a single‑electron spin qubit confined in a quantum dot that is simultaneously driven by an oscillating magnetic field (Rabi driving) and tunnel‑coupled to a Markovian electronic lead. The authors start by defining the system: the dot can host N‑1, N, or N+1 electrons, with the N‑electron subspace containing a single electron whose spin is split by a static magnetic field B₀ into Zeeman levels separated by ℏω₀. An AC magnetic field B₁(t) in the rotating‑wave approximation induces coherent Rabi oscillations at frequency ω with coupling ℏω₁. The dot Hamiltonian H_S(t) therefore contains the charging energy, the Zeeman term, the Coulomb interaction U for the doubly‑occupied singlet, and the time‑dependent Rabi term.

The lead is modeled as a non‑interacting electron reservoir at chemical potential μ_L and temperature T, described by H_E. Tunneling between dot and lead is spin‑conserving and characterized by a (initially spin‑ and orbital‑independent) amplitude λ. The total Hamiltonian is H(t)=H_S(t)+H_E+H_I, with H_I containing the creation/annihilation operators for dot and lead electrons.

To obtain the reduced dynamics of the dot, the authors move to the interaction picture with respect to H_S(t) and H_E, write the von‑Neumann equation for the total density matrix, and formally integrate it to second order in λ. The first‑order term vanishes because the lead is in a thermal diagonal state. They then invoke the Born approximation (the lead remains in its equilibrium state) and the Markov approximation (the system density matrix varies slowly compared to the bath correlation time), which reduces the integro‑differential equation to a time‑local form involving the bath correlation functions N_{ls}(t₁,t₂) and G_{ls}(t₁,t₂).

A major technical obstacle is that H_S(t) is explicitly time‑dependent, so the interaction‑picture tunneling operators acquire multiple frequency components (e^{±iωt}, e^{±i(ω±ω₀)t}, etc.). The authors evaluate these operators by a sequence of unitary transformations (including the rotating‑frame transformation that removes the fast oscillation at ω) and express them in matrix form. This allows them to perform a single secular (rotating‑wave) approximation on the full second‑order expression: only terms that are resonant (i.e., have zero net frequency after combining the exponentials) are retained, while rapidly oscillating terms average to zero.

After the secular approximation the master equation assumes the Gorini‑Kossakowski‑Sudarshan‑Lindblad (GKSL) form:

\dot{ρ}=−i