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arXiv:2304.07956 · Physical Review A

Driven Markovian master equation based on the Lewis-Riesenfeld-invariant theory

S. L. Wu, X. L. Huang, X. X. Yi

Published Nov 29, 2022• 9 citations
arXivHTMLPDF

Abstract

We derive a Markovian master equation for driven open quantum systems based on the Lewis-Riesenfeld-invariants theory, which is available for arbitrary driving protocols. The role of the Lewis-Riesenfeld invariants is to help us bypass the time-ordering obstacle in expanding the propagator of the free dynamics, such that the Lindblad operators in our driven Markovian master equation can be determined easily. We also illustrate that, for the driven open quantum systems, the spontaneous emission and the thermal excitation induce the transitions between eigenstates of the Lewis-Riesenfeld invariant, but not the system Hamiltonian's. As an example, we present the driven Markovian master equation for a driven two-level system coupled to a heat reservoir. By comparing to the exactly solvable models, the availability of the driven Markovian master equation is verified. Meanwhile, the adiabatic limit and inertial limit of the driven Markovian master equation are also discussed, which result in the same Markovian master equations as those presented before in the corresponding limits.

DOI
10.1103/physreva.106.052217

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We derive a Markovian master equation for driven open quantum systems based on the Lewis-Riesenfeld-invariants theory, which is available for arbitrary driving protocols. The role of the Lewis-Riesenfeld invariants is to help us bypass the time-ordering obstacle in expanding the propagator of the free dynamics, such that the Lindblad operators in our driven Markovian master equation can be determined easily. We also illustrate that, for the driven open quantum systems, the spontaneous emission and the thermal excitation induce the transitions between eigenstates of the Lewis-Riesenfeld invariant, but not the system Hamiltonian's. As an example, we present the driven Markovian master equation for a driven two-level system coupled to a heat reservoir. By comparing to the exactly solvable models, the availability of the driven Markovian master equation is verified. Meanwhile, the adiabatic limit and inertial limit of the driven Markovian master equation are also discussed, which result in the same Markovian master equations as those presented before in the corresponding limits.
We investigate the dynamics of the driven open double two-level system by deriving a driven Markovian master equation based on the Lewis-Riesenfeld invariant theory. The transitions induced by coupling to the heat reservoir occur between the instantaneous eigenstates of the Lewis-Riesenfeld invariant. Therefore, different driving protocols associated with corresponding Lewis-Riesenfeld invariants result in different open system dynamics and symmetries. In particular, we show that since the instantaneous steady state of the driven double two-level system is one of eigenstates of the Lewis-Riesenfeld invariant at ultralow reservoir temperature, the inverse engineering method based on the Lewis-Riesenfeld invariants has a good performance in rapidly preparing the quantum state of open quantum systems. As an example, a perfect entangled state is generated by means of the inverse engineering method.
We construct a quantum Markovian master equation for a driven system coupled to a thermal bath. The derivation utilizes an explicit solution of the propagator of the driven system. This enables the validity of the master equation to be extended beyond the adiabatic limit. The nonadiabatic master equation (NAME) is derived employing the weak system-bath coupling limit. The NAME is valid when a separation of timescales exists between the bath dynamics and the external driving. In contrast to the adiabatic master equation, the NAME leads to coupled equations of motion for the population and coherence. We employ the NAME to solve the example of an open driven time-dependent harmonic oscillator. For the harmonic oscillator the NAME predicts the emergence of coherence associated with the dissipation term. As a result of the nonadiabatic driving the thermalization rate is suppressed. The solution is compared with both numerical calculations and the adiabatic master equation.
We employ a generalized approach to the master equation for driven open NNN-level (N>2N>2N>2) quantum systems using Lewis-Riesenfeld invariants, which avoids the driving-strength restrictions inherent to conventional approaches. We show that the invariant-based master equation provides a unifying generalized framework, which reduces to the frequently employed laboratory-frame master equations and the less frequently employed rotating-frame master equation framework under appropriate simplifications. Extending the prototypical two-level system, we show that the inclusion of another state coupled to the ground state via reservoir-induced dephasing gives rise to qualitatively new dissipative behaviors that are, in general, not captured by standard approximations. We also apply the invariant-based master equation framework to a driven quantum dot coupled to a leaky cavity, demonstrating the framework's ability to capture relevant dissipative dynamics without additional assumptions. Our work paves the way for quantum-control applications in the presence of dissipation.
Abstract We propose using the dynamical invariants, also known as the Lewis–Riesenfeld invariants, to speed-up the equilibration of a driven open quantum system. This allows us to reverse engineer the time-dependent master equation that describes the dynamics of the open quantum system and systematically derive a protocol that realizes a shortcut to equilibration. The method does not require additional constraints on the timescale of the dynamics beside the Born–Markov approximation and can be generically applied to boost single particle quantum engines significantly. We demonstrate this with the damped harmonic oscillator, and show that our protocol can achieve high-fidelity control on shorter timescales than simple non-optimized protocols. We find that the system is heated during the dynamics to speed-up the equilibration, which can be considered as an analogue of the Mpemba effect in quantum control.
We study the dynamics of open quantum systems driven by an external time-periodic force. Combining Floquet theory and the time-convolutionless projection operator technique we derive a time-local quantum master equation which exactly takes into account the periodic driving, while treating the system-environment interaction within second order in the coupling strength without performing the Markov approximation. The resulting equation of motion for the reduced density matrix is called Floquet time-convolutionless master equation. Employing the example of the driven spin-boson system, we demonstrate that this master equation is capable of describing strong non-Markovian effects, while yielding the Floquet-Lindblad master equation in the Markovian limit. A characteristic feature of memory effects in such driven dissipative systems is the emergence of sharp peaks of the trace-distance based non-Markovianity measure as a function of the driving amplitude, which can be traced back to quasienergy crossings leading to almost decoherence-protected subspaces through a quasienergy-induced dissipative decoupling mechanism.
We derive a Markovian master equation that models the evolution of systems subject to driving and control fields. Our approach combines time rescaling and weak-coupling limits for the system-environment interaction with a secular approximation. The derivation makes use of the adiabatic time-evolution operator in a manner that allows for the efficient description of strong driving, while recovering the well-known adiabatic master equation in the appropriate limit. To illustrate the effectiveness of our approach, firstly we apply it to the paradigmatic case of a two-level (qubit) system subject to a form of periodic driving that remains unsolvable using a Floquet representation and lastly we extend this scenario to the situation of two interacting qubits, the first driven while the second one directly in contact with the environment. We demonstrate the reliability and broad scope of our approach by benchmarking the solutions of the derived reduced time evolution against numerically exact simulations using tensor networks. Our results provide rigorous conditions that must be satisfied by phenomenological master equations for driven systems that do not rely on first-principles derivations.