Validity of the Lindblad master equations

Abstract

Theory of open quantum systems, which studies quantum systems interacting with their environments, has a wide variety of applications in physics, ranging from quantum optics and condensed matter physics to quantum informatics and quantum computation. One important feature of open quantum systems is quantum mechanical treatment for damping, which is usually described by Lindblad master equations derived using the Born–Markov approximation. However, because of their perturbative nature, such master equations can only model systems with weak system-environment coupling. For example, in superconducting quantum circuits the environmental coupling can be increased to values for which the master equations are no longer accurate. A more accurate equation of motion, the formally exact stochastic Liouville–von Neumann equation, can be obtained by using the path integral formalism of quantum mechanics.

We study a simple but important model where a two state system, qubit, is coupled to a quantum harmonic oscillator, which is further coupled to a harmonic oscillator bath. The qubit-oscillator system can be described by the Rabi Hamiltonian. We solve the dynamics of this system numerically with the stochastic Liouville–von Neumann equation and two different Lindblad master equations, the quantum optical master equation and the eigenstate master equation. The former treats the qubit and the oscillator separately and the transitions occur between the eigenstates of the oscillator, whereas the latter treats them as a single system with transitions between the eigenstates of the whole system Hamiltonian. Numerical solutions of the stochastic Liouville–von Neumann equation are unstable with long simulation times. Because of this we are only able to solve it in the case where the qubit and the oscillator are in resonance. In this case we find parameters with which all three equations produce nearly the same results. We also see that there exists cases where only one of the master equations agrees with the stochastic Liouville–von Neumann equation. Further on, we find parameters for which both master equations produce results that deviate notably from those given by the stochastic Liouville–von Neumann equation. In off-resonance, we find that the solutions of the two master equations do not agree with each other for any parameters we study.

These results suggest that the current state of the numerical simulations of open quantum systems can be improved by using the formally exact methods instead of the approximate ones for situations where the environmental coupling is of the order of the qubit-cavity coupling or stronger.