Markovian Master Equation Research Articles

Exploiting automatic differentiation for quantum optimal control of driven dissipative two-level systems

In principle one can use Pontryagin's minimum principle to treat an optimal control problem and derive the gradient for the cost functional. However, for the driven spin-boson model studied here, the response of the system to the variation of the control is determined by the master equation and the equation of motion for the propagator of the coherent system dynamics. As a result, the application of Pontryagin's minimum principle is less straightforward. An alternative to this approach is the technique of automatic differentiation which in principle amounts to doing calculus on the fully discretized form of the optimal control problem. Automatic differentiation tools can be viewed as black boxes taking as input a program computing the cost functional and giving as output another program computing its gradient. First we derive for the polaron transformed Hamiltonian a Born-Markov master equation using the Bloch-Redfield formalism. By combining the latter with automatic differentiation we are able to implement Z-gate and coherent destruction of tunnelling with high fidelity. Optimization of a dissipative quantum gate poses a more complex numerical problem, since it should occur independent of the input state. To overcome this difficulty, we apply optimal control to the concept of resolvent of the master equation which is the generalisation of quantum unitary evolution operator in the case of dissipative dynamics.

Quantum–Classical Hybrid Dynamics: Coupling Mechanisms and Diffusive Approximation

In this paper we demonstrate that any Markovian master equation defining a completely positive evolution for a quantum–classical hybrid state can always be written in terms of four basic coupling mechanisms. Each of them is characterized by a different “backaction” on each subsystem. On this basis, for each case, we find the conditions under which a diffusive limit is approached, that is, the time evolution can be approximated in terms of the first and second derivatives of the hybrid state with respect to a classical coordinate. In this limit, the restricted class of evolutions that guaranty the positivity of the hybrid state at all times (quantum Fokker-Planck master equations) emerges when the coupling mechanisms lead to infinitesimal (non-finite) changes in both the quantum and classical subsystems. A broader class of diffusive evolutions is obtained when positivity is only granted after a transient time or alternatively is granted after imposing an initial finite width on the state of the classical subsystem. A set of representative examples support these results.

General framework for quantifying dissipation pathways in open quantum systems. II. Numerical validation and the role of non-Markovianity.

In the previous paper [C. W. Kim and I. Franco, J. Chem. Phys. 160, 214111-1-214111-13 (2024)], we developed a theory called MQME-D, which allows us to decompose the overall energy dissipation process in open quantum system dynamics into contributions by individual components of the bath when the subsystem dynamics is governed by a Markovian quantum master equation (MQME). Here, we contrast the predictions of MQME-D against the numerically exact results obtained by combining hierarchical equations of motion (HEOM) with a recently reported protocol for monitoring the statistics of the bath. Overall, MQME-D accurately captures the contributions of specific bath components to the overall dissipation while greatly reducing the computational cost compared to exact computations using HEOM. The computations show that MQME-D exhibits errors originating from its inherent Markov approximation. We demonstrate that its accuracy can be significantly increased by incorporating non-Markovianity by exploiting time scale separations (TSS) in different components of the bath. Our work demonstrates that MQME-D combined with TSS can be reliably used to understand how energy is dissipated in realistic open quantum system dynamics.

Markovian quantum master equation with Poincaré symmetry

Markovian quantum master equation with Poincaré symmetry

Modelling non-Markovian noise in driven superconducting qubits

Non-Markovian noise can be a significant source of errors in superconducting qubits. We develop gate sequences utilising mirrored pseudoidentities that allow us to characterise and model the effects of non-Markovian noise on both idle and driven qubits. We compare three approaches to modelling the observed noise: (i) a Markovian noise model, (ii) a model including interactions with a two-level system (TLS), (iii) a model utilising the post Markovian master equation, which we show to be equivalent to the qubit-TLS model in certain regimes. When running our noise characterisation circuits on a superconducting qubit device we find that purely Markovian noise models cannot reproduce the experimental data. Our model based on a qubit-TLS interaction, on the other hand, is able to closely capture the observed experimental behaviour for both idle and driven qubits. We investigate the stability of the noise properties of the hardware over time, and find that the parameter governing the qubit-TLS interaction strength fluctuates significantly even over short time-scales of a few minutes. Finally, we evaluate the changes in the noise parameters when increasing the qubit drive pulse amplitude. We find that although the hardware noise parameters fluctuate significantly over different days, their drive pulse induced relative variation is rather well defined within computed uncertainties: both the phase error and the qubit-TLS interaction strength change significantly with the pulse strength, with the phase error changing quadratically with the amplitude of the applied pulse. Since our noise model can closely describe the behaviour of idle and driven qubits, it is ideally suited to be used in the development of quantum error mitigation and correction methods.

Tunability in 3-Site Electronic Excitation Transfer Dynamics: Insights into the Role of Perturbative Coupling.

Electronic excitation transfer dynamics in photosynthetic systems, including the Fenna-Matthews-Olson complex, are often modeled using the interaction picture of three two-level systems, also known as the 3-site system. Among the two possible configurations, uphill and downhill, a recent publication reported an intriguing correlation between population dynamics and the intersite coupling. Specifically, the uphill configuration has been shown to have a pronounced dependence on perturbations in the intersite coupling, whereas the downhill configuration displays negligible dependence. In this study, we consider a generic 3-site configuration and model site interactions through the Markovian master equation. Through this approach, we derive succinct analytical expressions for the population dynamics between the sites, shedding light on the differences in behavior between the two configurations. Using these analytical expressions, we demonstrate the range of tunability in population dynamics achievable with minimal changes in intersite coupling, and we validate these findings through simulation results. These insights into the population dynamics of a 3-site system are expected to play a crucial role in facilitating the design of efficient energy-transfer pathways in molecular systems.

A pedagogical introduction to continuously monitored quantum systems and measurement-based feedback

In this manuscript we present a pedagogical introduction to continuously monitored quantum systems. We start by giving a simplified derivation of the Markovian master equation in Lindblad form, in the spirit of collision models and input-output theory, which describes the unconditional dynamics of a continuously monitored system. The same formalism is then exploited to derive stochastic master equations that describe the conditional dynamics. We focus on the two most paradigmatic examples of continuous monitoring: continuous photodetection, leading to a discontinuous dynamics with “quantum jumps”, and continuous homodyne measurements, leading to a diffusive dynamics. We then present a derivation of feedback master equations that describe the dynamics (either conditional or unconditional) when the continuous measurement photocurrents are fed back to the system as a linear driving Hamiltonian, a paradigm known as linear Markovian feedback. In the second part of the manuscript we focus on continuous-variable Gaussian systems: we first present the equations for first and second moments describing the dynamics under continuous general-dyne measurements, and we then discuss in more detail the conditional and unconditional dynamics under Markovian and state-based feedback.

Fitting quantum noise models to tomography data

The presence of noise is currently one of the main obstacles to achieving large-scale quantum computation. Strategies to characterise and understand noise processes in quantum hardware are a critical part of mitigating it, especially as the overhead of full error correction and fault-tolerance is beyond the reach of current hardware. Non-Markovian effects are a particularly unfavourable type of noise, being both harder to analyse using standard techniques and more difficult to control using error correction. In this work we develop a set of efficient algorithms, based on the rigorous mathematical theory of Markovian master equations, to analyse and evaluate unknown noise processes. In the case of dynamics consistent with Markovian evolution, our algorithm outputs the best-fit Lindbladian, i.e., the generator of a memoryless quantum channel which best approximates the tomographic data to within the given precision. In the case of non-Markovian dynamics, our algorithm returns a quantitative and operationally meaningful measure of non-Markovianity in terms of isotropic noise addition. We provide a Python implementation of all our algorithms, and benchmark these on a range of 1- and 2-qubit examples of synthesised noisy tomography data, generated using the Cirq platform. The numerical results show that our algorithms succeed both in extracting a full description of the best-fit Lindbladian to the measured dynamics, and in computing accurate values of non-Markovianity that match analytical calculations.

Markovian master equations for quantum-classical hybrid systems

The problem of constructing a consistent quantum-classical hybrid dynamics is afforded in the case of a quantum component in a separable Hilbert space and a continuous, finite-dimensional classical component. In the Markovian case, the problem is formalized by the notion of hybrid dynamical semigroup. A classical component can be observed without perturbing the system and information on the quantum component can be extracted, thanks to the quantum-classical interaction. This point is formalized by showing how to introduce positive operator valued measures and operations compatible with the hybrid dynamical semigroup; in this way the notion of hybrid dynamics is connected to quantum measurements in continuous time. Then, the case of the most general quasi-free generator is presented and the various quantum-classical interaction terms are discussed. To bee quasi-free means to send, in the Heisenberg description, hybrid Weyl operators into multiples of Weyl operators; the results on the structure of quasi-free semigroups were proved in Reference [12]. Even in the pure quantum case, a quasi-free semigroup is not restricted to have only a Gaussian structure, but also jump-type terms are allowed. An important result is that, to have interactions producing a flow of information from the quantum component to the classical one, suitable dissipative terms must be present in the generator. Finally, some possibilities are discussed to go beyond the quasi-free case.

The Josephson junction as a quantum engine

We treat the Cooper pairs in the superconducting electrodes of a Josephson junction (JJ) as an open system, coupled via Andreev scattering to external baths of electrons. The disequilibrium between the baths generates the direct-current bias applied to the JJ. In the weak-coupling limit we obtain a Markovian master equation that provides a simple dynamical description consistent with the main features of the JJ, including the form of the current–voltage characteristic, its hysteresis, and the appearance under periodic voltage driving of discrete Shapiro steps. For small dissipation, our model also exhibits a self-oscillation of the JJ’s electrical dipole with frequency around mean voltage V. This self-oscillation, associated with ‘hidden attractors’ of the nonlinear equations of motion, explains the observed production of monochromatic radiation with frequency Ω and its harmonics. We argue that this picture of the JJ as a quantum engine resolves open questions about the Josephson effect as an irreversible process and could open new perspectives in quantum thermodynamics and in the theory of dynamical systems.

Relaxation dynamics of nonresonant excitation transfer processes assisted by coherent phonon environment

In this study, the steepest-entropy-ascent quantum thermodynamics (SEAQT) framework is used to investigate the excitation transfer (ET) dynamics of two-level nanosystems (TLSs), focusing on non-resonant processes that involve a dynamic local phonon system. In contrast to other methods based on Markovian or non-Markovian quantum master equations, SEAQT analysis always guarantees the positivity of the density operators, thus enabling the discussion of both transient and long-term dynamics of nanosystems. The findings of this study demonstrate that the relaxation time of coherent phonons, relative to the Rabi oscillation period in two TLSs, significantly affects the relaxation process of non-resonant ETs. Moreover, the degree of mutual synchronization between ET dynamics of TLSs and local (coherent) phonons can either prolong or shorten the decoherence time, presenting a way to control the coherence in nanosystems and stimulate quantum device applications.

Thermodynamics of Markovian Open Quantum Systems with Application to Lasers

Göran Lindblad was one of the pioneers of what is now called quantum thermodynamics. From this vast and rapidly developing field, we have selected a sample of results concerning quantum open systems described by Markovian master equations of the Lindblad (Gorini–Kossakowski–Sudarshan) type, which are applied to models of lasers. One can study their thermodynamics using the properties of quantum relative entropy, also introduced by Lindblad.

Exceptional point in self-consistent Markovian master equations

Exceptional point in self-consistent Markovian master equations

Violation of Detailed Balance in Quantum Open Systems.

We consider the dynamics of a quantum system immersed in a dilute gas at thermodynamic equilibrium using a quantum Markovian master equation derived by applying the low-density limit technique. It is shown that the Gibbs state at the bath temperature is always stationary while the detailed balance condition at this state can be violated beyond the Born approximation. This violation is generically related to the absence of time-reversal symmetry for the scattering T matrix, which produces a thermalization mechanism that allows the presence of persistent probability and heat currents at thermal equilibrium. This phenomenon is illustrated by a model of an electron hopping between three quantum dots in an external magnetic field.

Hybrid completely positive Markovian quantum-classical dynamics

A concise and self-contained derivation of hybrid quantum-classical dynamics is given in terms of Markovian master equations. Many previously known results are rederived and revised and some of them are completed or corrected. Using a method as simple as possible, our goal is a brief introduction to the state of the art of hybrid dynamics, with a limited discussion of the implications for foundations and without discussion of further relevance in the measurement problem, quantum gravity, chemistry, numeric methods, etc. Hybrid dynamics is defined as a special case of composite quantum dynamics where the observables of one of the two subsystems are restricted to the commuting set of diagonal operators in a fixed basis. With this restriction, the derivation of hybrid dynamical equations is clear conceptually and simple technically. Jump and diffusive dynamics follow in the form of hybrid master equations. Their stochastic interpretation (called unravelings) is derived. We discuss gauge-type ambiguities, problems of uniqueness, and covariance of the diffusive master equation. Also conditions of minimum noise and of monitoring the quantum trajectory are derived. We conclude that the hybrid formalism is equivalent to the standard Markovian theory of time-continuous quantum measurement (monitoring) on the one hand and is a motivating alternative formalism on the other.

Spontaneous symmetry breaking in nonsteady modes of open quantum many-body systems

In a quantum many-body system coupled to the environment, its steady state can exhibit spontaneous symmetry breaking when a control parameter exceeds a critical value. In this study, we consider spontaneous symmetry breaking in non-steady modes of an open quantum many-body system. Assuming that the time evolution of the density matrix of the system is described by a Markovian master equation, the dynamics of the system is fully characterized by the eigenmodes and spectrum of the corresponding time evolution superoperator. Among the non-steady eigenmodes with finite lifetimes, we focus on the eigenmodes with the highest frequency, which we call the most coherent mode. For a dissipative spin model, it is shown that the most coherent mode exhibits a transition from a disordered phase to a symmetry-broken ordered phase, even if the steady state does not show singular behavior. We further argue that the phase transition of the most coherent mode induces a qualitative change in the decoherence dynamics of highly entangled states, i.e., the Schr\"odinger's cat states.

Optimal figureof merit of low-dissipation quantum refrigerators.

The Drazin inverse of the Liouvillian superoperator provides a solution to determine the dynamics of a time-dependent system governed by the Markovian master equation. Under the condition of slow driving, the perturbation expansion of the density operator of the system in powers of time can be derived. As an application, a finite-time cycle model of the quantum refrigerator driven by a time-dependent external field is established. The method of the Lagrange multiplier is adopted as a strategy to find the optimal cooling performance. The figureof merit given by the product of the coefficient of performance and the cooling rate is taken as a new objective function, and, consequently, the optimally operating state of the refrigerator is revealed. The effects of the frequency exponent determining dissipation characteristics on the optimal performance of the refrigerator are discussed systemically. The results obtained show that the adjacent areas of the state of the maximum figureof merit are the best operation region of low-dissipative quantum refrigerators.

Dynamics of a driven open double two-level system and its entanglement generation

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.

Identical damped harmonic oscillators described by coherent states

Some aspects of quantum damped harmonic oscillator (DHO) obeying a Markovian master equation are considered in the absence of thermal noise. The continuity equation is derived and Bohmian trajectories are constructed. As a solution of the master equation, we take a single coherent state and compute analytically the relative entropy of coherence, [Formula: see text], in the energy, position and momentum bases. Although [Formula: see text] is constant in both the position and the momentum bases, it is a decreasing function of time in the energy basis becoming zero at long times, revealing its role as the preferred basis. Then, quantum coherence is computed for a superposition of two coherent states, a cat state, and also a superposition of two cat states in the energy basis as a function of separation, in the complex plane, between the two superposed states. It is seen that the quantum coherence increases with this separation. Furthermore, quantum coherence of superposition is compared to that of decomposed states in the superposition. Finally, by considering a system of two noninteracting DHOs, the effect of quantum statistics is studied on the coherence of reduced single-particle states, the joint detection probability and the mean square separation of particles. Our computations show that the single-particle coherence for antisymmetric states is always less than that of symmetric ones. Furthermore, boson anti-bunching and fermion bunching is seen in this open system. This behavior of bosons is the matter-wave analogue of photon anti-bunching seen in a modified Hanbury Brown–Twiss (HBT) interferometer.

Optimal nonequilibrium thermometry in Markovian environments

What is the minimum time required to take a temperature? In this paper, we solve this question for a large class of processes where temperature is inferred by measuring a probe (the thermometer) weakly coupled to the sample of interest, so that the probe's evolution is well described by a quantum Markovian master equation. Considering the most general control strategy on the probe (adaptive measurements, arbitrary control on the probe's state and Hamiltonian), we provide bounds on the achievable measurement precision in a finite amount of time, and show that in many scenarios these fundamental limits can be saturated with a relatively simple experiment. We find that for a general class of sample-probe interactions the scaling of the measurement uncertainty is inversely proportional to the time of the process, a shot-noise like behaviour that arises due to the dissipative nature of thermometry. As a side result, we show that the Lamb shift induced by the probe-sample interaction can play a relevant role in thermometry, allowing for finite measurement resolution in the low-temperature regime. More precisely, the measurement uncertainty decays polynomially with the temperature as T→0, in contrast to the usual exponential decay with T−1. We illustrate these general results for (i) a qubit probe interacting with a bosonic sample, where the role of the Lamb shift is highlighted, and (ii) a collective superradiant coupling between a N-qubit probe and a sample, which enables a quadratic decay with N of the measurement uncertainty.