The urgent need for reliable simulation tools to match the extreme accuracy needed to control tailored quantum devices highlights the importance of understanding open quantum systems and their modeling. To this end, we compare here the commonly used Redfield and Lindblad master equations against numerically exact results in the case of one and two resonant qubits transversely coupled at a single point to a Drude-cut ohmic bath. All the relevant parameters are varied over a broad range, which allows us to give detailed predictions about the validity and physically meaningful applicability of the weak-coupling approaches. We characterize the accuracy of the approximate approaches by comparing the maximum difference of their system evolution superoperators with numerically exact results. After optimizing the parameters of the approximate models to minimize the difference, we also explore if and to what extent the weak-coupling equations can be applied at least as phenomenological models. Optimization may lead to an accurate reproduction of experimental data, but yet our results are important to estimate the reliability of the extracted parameter values such as the bath temperature. Our findings set general guidelines for the range of validity of the usual Born-Markov master equations and indicate that they fail to accurately describe the physics in a surprisingly broad range of parameters, in particular, at low temperatures. Since quantum-technological devices operate there, their accurate modeling calls for a careful choice of methods.
The urgent need for reliable simulation tools to match the extreme accuracy needed to control tailored quantum devices highlights the importance of understanding open quantum systems and their modeling. To this end, we compare here the commonly used Redfield and Lindblad master equations against numerically exact results in the case of one and two resonant qubits transversely coupled at a single point to a Drude-cut ohmic bath. All the relevant parameters are varied over a broad range, which allows us to give detailed predictions about the validity and physically meaningful applicability of the weak-coupling approaches. We characterize the accuracy of the approximate approaches by comparing the maximum difference of their system evolution superoperators with numerically exact results. After optimizing the parameters of the approximate models to minimize the difference, we also explore if and to what extent the weak-coupling equations can be applied at least as phenomenological models. Optimization may lead to an accurate reproduction of experimental data, but yet our results are important to estimate the reliability of the extracted parameter values such as the bath temperature. Our findings set general guidelines for the range of validity of the usual Born-Markov master equations and indicate that they fail to accurately describe the physics in a surprisingly broad range of parameters, in particular, at low temperatures. Since quantum-technological devices operate there, their accurate modeling calls for a careful choice of methods.
The dynamics of open quantum systems is commonly described by quantum master equations derived under the assumption of weak system-bath coupling and a separation of timescales between system and bath. When the system is additionally subjected to a periodic driving, the validity of the resulting Floquet master equations is further restricted to regimes of weak or high-frequency driving. Here, we benchmark a set of commonly used Floquet master equations for a model of two locally driven spins coupled to a shared Ohmic reservoir at finite temperature. We systematically probe the accuracy of the equations as a function of the driving parameters, thus identifying limits of their applicability. Dynamical maps predicted by each master equation are compared against numerically exact non-Markovian simulations, tracking the full relaxation dynamics. We find that the accuracy of each master equation closely reflects the assumptions underlying its derivation. For the Floquet-Lindblad equation, errors can be strongly amplified near resonances where the secular approximation breaks down, while approaches that avoid the secular approximation perform better and exhibit a more systematic dependence of the error on driving frequency and amplitude.
We present the Born-Markov approximated Redfield quantum master equation (RQME) description for an open system of noninteracting particles (bosons or fermions) on an arbitrary lattice of N sites in any dimension and weakly connected to multiple reservoirs at different temperatures and chemical potentials. The RQME can be reduced to the Lindblad equation, of various forms, by making further approximations. By studying the N=2 case, we show that RQME gives results which agree with exact analytical results for steady-state properties and with exact numerics for time-dependent properties over a wide range of parameters. In comparison, the Lindblad equations have a limited domain of validity in nonequilibrium. We conclude that it is indeed justified to use microscopically derived full RQME to go beyond the limitations of Lindblad equations in out-of-equilibrium systems. We also derive closed-form analytical results for out-of-equilibrium time dynamics of two-point correlation functions. These results explicitly show the approach to steady state and thermalization. These results are experimentally relevant for cold atoms, cavity QED, and far-from-equilibrium quantum dot experiments.
The Lindblad theory of open quantum systems has been successfully applied in various fields ranging from quantum optics, condensed matter physics, and quantum thermodynamics to quantum chemistry. Nevertheless, there are situations where we can compare its predictions to those of exact methods or numerically exact results and find discrepancies between them. One example is the case where a single harmonic oscillator is coupled to a thermal bath of independent harmonic oscillators. In this case, for certain parameters, the exact master equation describes equilibration of the system to a thermal state with a temperature different from the one predicted by the Lindblad master equation. The Lindblad theory, based on a weak coupling between the system and environment as well as the Born-Markov approximation, predicts that the system will evolve to a thermal state whose temperature is equal to the initial temperature of the bath. This result is a consequence of the Born approximation, which supposes that throughout the evolution, the bath only fluctuates around its initial equilibrium state. In this master thesis we study the validity of the Born approximation as well as its consistency with other approximations. By starting from a Hamiltonian for the total system we show that, if the the Born approximation is not invoked, the dynamics of the system is determined by a hierarchy of equations for matrices which can be used to reconstruct the reduced density matrix of the system. This hierarchy approach is fundamentally different from the standard approach, which describes the system dynamics with a weak-coupling master equation, because the hierarchy does not restrict the amount of entanglement possible between the system and the environment. Furthermore, we show how invoking the Born approximation reduces to the usual result and how a generalised ansatz, which we refer to as a generalised Born approximation, reduces the hierarchy to a master equation where the bath temperature is an additional degree of freedom. Finally, we analyse the conservation of the energy of the total system and its relationship to the Markov and secular approximations, and discuss whether imposing it as a condition can fix the time-dependent temperature.
Away from equilibrium, the properties of open quantum systems depend on the details of their environment. A microscopic derivation of a master equation (ME) is therefore crucial. Of particular interest are Lindblad-type equations, not only because they provide the most general class of Markovian MEs, but also since they are the starting point for efficient quantum trajectory simulations. Lindblad-type MEs are commonly derived from the Born-Markov-Redfield equation via a rotating-wave approximation (RWA). However the RWA is valid only for ultraweak system-bath coupling and often fails to accurately describe nonequilibrium processes. Here we derive an alternative Lindbladian approximation to the Redfield equation, which does not rely on ultraweak system-bath coupling. Applying it to an extended Hubbard model coupled to Ohmic baths, we show that, especially away from equilibrium, it provides a good approximation in large parameter regimes where the RWA fails.
The known quantum error-correcting codes are typically built on approximative open-quantum-system models such as Born-Markov master equations. However, it is an open question how such codes perform in actual physical systems that, to some extent, necessarily exhibit phenomena beyond the limits of these models. To this end, we employ numerically exact open-quantum-system dynamics to analyze the performance of a five-qubit error-correction code where each qubit is coupled to its own bath. We first focus on the performance of a single error-correction cycle covering timescales and coupling strengths beyond those of Born-Markov models. We observe distinct power-law behavior of the error-corrected channel infidelity $\ensuremath{\propto}{t}^{2a}$: $a\ensuremath{\lesssim}2$ in the ultrashort times $t<3/{\ensuremath{\omega}}_{\mathrm{c}}$ and $a\ensuremath{\approx}1/2$ in the short-time range $3/{\ensuremath{\omega}}_{\mathrm{c}}<t<30/{\ensuremath{\omega}}_{\mathrm{c}}$, where ${\ensuremath{\omega}}_{\mathrm{c}}$ is the cutoff angular frequency of the bath. Furthermore, the observed scaling of the performance of error correction is found to be robust against various imperfections in physical qubit systems, such as perturbations in qubit-qubit coupling strengths and parametric disorder. For repeated error correction, we demonstrate the breaking of the five-qubit error-correction code and of the Born-Markov models if the repetition rate of the error-correction cycles exceeds $2\ensuremath{\pi}/\ensuremath{\omega}$ or the coupling strength $\ensuremath{\kappa}\ensuremath{\gtrsim}\ensuremath{\omega}/10$, where $\ensuremath{\omega}$ is the angular frequency of the qubit. Our results provide bounds of validity for the standard quantum error-correction codes and pave the way for applying numerically exact open-quantum-system methods for further studies of error correction beyond simple error models and for other strongly coupled many-body models.
Fast and reliable manipulation with qubits is fundamental for any quantum technology. The implementation of these manipulations in physical systems is the focus of studies involving optimal control theory. Realistic physical devices are open quantum systems. So far, studies in optimal control theory have primarily utilized the Redfield/Lindblad quantum master equation to simulate the dynamics of such systems. However, this Markov description is not always sufficient. Here, we present a study of qubit control utilizing the nonequilibrium Green's function method. We compare the traditional master equation with more general Green's function results and demonstrate that even in the parameter regime suitable for the application of the Redfield/Lindblad approach, the two methods yield drastically different results when addressing evolution involving mixed states. In particular, we find that, in addition to predicting different optimal driving profiles, a more accurate description of system evolution enables the system to reach the desired final state much more quickly. We argue that the primary reason for this is the significance of the non-Markov description of driven system dynamics due to the effect of time-dependent driving on dissipation.