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

Emily Townsend, Joshua Pomeroy, Garnett W. Bryant

TL;DR

The paper develops a driven-dissipative open-quantum-system framework for a four-level quantum dot tunnel-coupled to a Markovian lead under AC driving. By explicit time-evolution of electron creation/annihilation operators and a tailored secular approximation, it derives a CPTP master equation and casts the dissipator in Lindblad form with six jump operators, whose rates depend on the drive via parameters a=ω−ω_0 and Ω=√(ω_1^2+a^2). The authors analyze a low-temperature, single-lead regime to show how the drive reshapes transition energies and steady-state occupations, enabling Zeeman splitting determination from charge readout while driving the spin. The work connects to dressed-state concepts but extends them to a tunneling bath, offering jump-operator-based descriptions applicable to ESR-STM and more complex driven spin-transport systems. Potential extensions include multiple reservoirs, current calculations, and applying the framework to larger spin networks with local tunneling events.

Abstract

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.

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

TL;DR

The paper develops a driven-dissipative open-quantum-system framework for a four-level quantum dot tunnel-coupled to a Markovian lead under AC driving. By explicit time-evolution of electron creation/annihilation operators and a tailored secular approximation, it derives a CPTP master equation and casts the dissipator in Lindblad form with six jump operators, whose rates depend on the drive via parameters a=ω−ω_0 and Ω=√(ω_1^2+a^2). The authors analyze a low-temperature, single-lead regime to show how the drive reshapes transition energies and steady-state occupations, enabling Zeeman splitting determination from charge readout while driving the spin. The work connects to dressed-state concepts but extends them to a tunneling bath, offering jump-operator-based descriptions applicable to ESR-STM and more complex driven spin-transport systems. Potential extensions include multiple reservoirs, current calculations, and applying the framework to larger spin networks with local tunneling events.

Abstract

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.
Paper Structure (20 sections, 94 equations, 1 figure)

This paper contains 20 sections, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. System, Environment and Interaction Hamiltonian
  4. Derivation of driven-dissipative dynamics
  5. Born and Markov approximations
  6. Secular approximation
  7. Structure due to non-commuting time evolved creation and destruction operators
  8. Performing the sum and integration
  9. Jump operators and demonstratng that the map is CPTP
  10. Conclusion
  11. Comparison to the dressed state picture of Cohen-Tannoudji, et al. and Elouard et al.
  12. Future work
  13. Acknowledgements
  14. Frame Transformations to solve the system Hamiltonian
  15. Interaction picture
  16. ...and 5 more sections

Figures (1)

  • Figure 1: (from reference Townsend2025) Relative one-electron-addition transition energies of different states of the dot and of the lead (i.e. chemical potential of the lead).