Born-Markov 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.

Decoherence and maintenance of an entangled state in three-spin system coupled to ferromagnetic electrodes

We study the model of three-spin system by using Born-Markov master equation for density matrix. Three-spin system consists of a spin-1/2 quantum dot and two spin-1 magnetic particles arranged in triangular geometry and a quantum dot is also tunnel-coupled to ferromagnetic electrodes, respectively. In this case, we are interested in decoherence of entangled state of the system. Decoherence occurs due to tunneling of electrons through coupling to ferromagnetic electrodes. Therefore, an entangled state undergoes decoherence dependent on the spin polarization as well as magnetization orientations of electrodes. We have established master-equation formalism for the density matrix of three-spin system and calculated spin-spin correlation function between two magnetic particles corresponding to entangled state. The results show that with the increase of spin polarization, spin-spin correlation function for parallel configuration of electrodes’ magnetizations decays more rapidly, while spin-spin correlation function for antiparallel configuration decays more slowly.

Deriving Generalized Born-Markov Optical Master Equation for Analysis of Inelastic Tunneling Under Non Equilibrium Condition

Deriving Generalized Born-Markov Optical Master Equation for Analysis of Inelastic Tunneling Under Non Equilibrium Condition

Single-junction quantum-circuit refrigerator

We propose a quantum-circuit refrigerator (QCR) based on photon-assisted quasiparticle tunneling through a single normal-metal–insulator–superconductor (NIS) junction. In contrast to previous studies with multiple junctions and an additional charge island for the QCR, we directly connect the NIS junction to an inductively shunted electrode of a superconducting microwave resonator making the device immune to low-frequency charge noise. At low characteristic impedance of the resonator and parameters relevant to a recent experiment, we observe that a semiclassical impedance model of the NIS junction reproduces the bias voltage dependence of the QCR-induced damping rate and frequency shift. For high characteristic impedances, we derive a Born–Markov master equation and use it to observe significant non-linearities in the QCR-induced dissipation and frequency shift. We further demonstrate that, in this regime, the QCR can be used to initialize the linear resonator into a non-thermal state even in the absence of any microwave drive.

Coherent time-dependent oscillations and temporal correlations in triangular triple quantum dots

The fluctuation behavior of triple quantum dots (TQDs) has, so far, largely focused on current cumulants in the long-time limit via full counting statistics. Given that (TQDs) are non-trivial open quantum systems with many interesting features, such as Aharonov-Bohm interference and coherent population blocking, new fluctuating-time statistics, such as the waiting time distribution (WTD), may provide more information than just the current cumulants alone. In this paper, we use a Born-Markov master equation to calculate the standard and higher-order WTDs for coherentlycoupled TQDs arrayed in triangular ring geometries for several transport regimes. In all cases we find that the WTD displays coherent oscillations that correspond directly to individual time-dependent dot occupation probabilities, a result also reported recently in Ref.[1]. Our analysis, however, goes beyond the single-occupancy and single waiting time regimes, investigating waiting time behavior for TQDs occupied by multiple electrons and with finite electron-electron interactions. We demonstrate that, in these regimes of higher occupancy, quantum coherent effects introduce correlations between successive waiting times, which we can tune via an applied magnetic field. We also show that correlations can be used to distinguish between TQD configurations that have identical FCS and that dark states can be tuned with Aharonov-Bohm interference for more complicated regimes than single-occupancy.

The Lindblad and Redfield forms derived from the Born–Markov master equation without secular approximation and their applications

In this paper, we derive the Lindblad and Redfield forms of the master equation based on the Born–Markov master equation with and without the secular approximation for open multi-level quantum systems. The coefficients of the equations are re-evaluated according to the scheme in [(2019), Phys. Rev. A 99, 022118]. They are complex numbers rather than the real numbers obtained from traditional simplified methods. The dynamics of two models (one is an open three-level quantum system model, and the other is the model of phycoerythrin 545 (PE545) in a photosynthesis system) are studied. It is shown that the secular approximation and the simplified real coefficients may cause a small distortion of the dynamics in some environments, but a large distortion of the dynamics in others. These effects are discussed and characterized by studying the dynamics of nontrivial instances of multi-level systems in the presence of dissipation.

Theoretical study of optical rectification of a nanostructure inside an ideal photonic crystal cavity

A theoretical method is applied to calculate the optical rectification (OR) in a cavity-quantum dot system based on open quantum systems theory. This method allows studying the linear and non-linear optical response of a nanostructure found within an ideal photonic crystal cavity by solving a Born–Markov master equation in the Lindblad form, including spontaneous emission processes and dephasing due to interaction with the environment. For a quantum dot (QD) interacting with non-classical light in a very high-quality factor cavity, an analytical expression for the OR was found when the average number of photons n¯ is small, which explicitly depends on the quantum dot-cavity system’s initial state. The expression found was applied to a nanostructure constituted by a cylindrical quantum dot (CQD) of GaAs/Ga0.6Al0.4As with a donor impurity inside. It was found that the OR in nanostructures that interact with a quantum field of light exhibits an absorption peak at lower frequencies than in the case of interaction with classical light. The physical reason for this behavior is attributed to the number of quantum states involved in the optical transitions and, therefore, to the number of photons involved in the transitions.

Optical rectification in self-assembled quantum dots: The role of incoherent pumping

The effect of the incoherent pumping of excitons on the optical rectification of a nanostructure that interacts with a monochromatic classical optical field is studied theoretically, by solving a Born-Markov master equation in the Lindblad form that includes the processes of spontaneous emission, dephasing, and incoherent pumping of excitons between two levels of the nanostructured quantum system. In addition, the quantum vacuum of the electromagnetic field that affects the nanostructure was considered as an environment for the system. An analytical expression for the optical rectification was found in terms of the incoherent pumping rate of excitons P in which a weak pumping regime and a strong pumping regime are evidenced.

Effective quantum dynamics induced by a driven two-level-system bath

We derive a Born-Markov master equation describing the dissipation induced by a bath of lossy but coherently driven two-level systems (TLS) coupled to a bosonic system via Jaynes-Cummings interaction. We analytically derive all the master equation rates. We characterize these rates for the particular case of a single-mode system coupled to identical TLS. We study the steady state of the system and its exotic properties stemming from the non-thermal stationary state of the driven TLS bath. These properties include dissipative amplification, bath-induced linear instability, and both coherent and dissipative squeezing. The master equation is valid for arbitrarily strong TLS driving, and it can be generalized to include multi-level systems or other system-bath interaction terms, among others. Our work provides a tool to study and characterize TLS-induced decoherence, a key limiting factor in quantum technological devices based on, for instance, superconducting circuits, magnonic systems, or quantum acoustics.

Electronic statistics on demand: Bunching, antibunching, positive, and negative correlations in a molecular spin valve

One of the long-standing goals of quantum transport is to use the noise, rather than the average current, for information processing. However, achieving this requires on-demand control of quantum fluctuations in the electric current. In this paper, we demonstrate theoretically that transport through a molecular spin-valve provides access to many different statistics of electron tunneling events. Simply by changing highly tunable parameters, such as electrode spin-polarization, magnetization angle, and voltage, one is able to switch between Poisson behavior, bunching and anti-bunching of electron tunnelings, and positive and negative temporal correlations. The molecular spin-valve is modeled by a single spin-degenerate molecular orbital with local electronic repulsion coupled to two ferromagnetic leads with magnetization orientations allowed to rotate relative to each other. The electron transport is described via Born-Markov master equation and fluctuations are studied with higher-order waiting time distributions. For highly magnetized parallel-aligned electrodes, we find that strong positive temporal correlations emerge in the voltage range where the second transport channel is partially open. These are caused by a spin-induced electron-bunching, which does not manifest in the stationary current alone.

Conductor's elastic response to the vacuum-field radiation pressure

The momentum transfer from an electromagnetic field to reflecting and absorbing surfaces was asserted by the Maxwell equations. This phenomenon, with important implications for the development of the cosmos, has also been investigated in the discretized energy scale using optomechanical devices. With the quantization of the electromagnetic field, it was discovered that the vacuum field may influence the dynamics of some physical systems, such as in the Casimir effect and the radiative decay of an atom. Here, the effect of the radiation pressure by the electromagnetic vacuum on the surface of a compressible conductor is analyzed, and the model based on the Born-Markov master equation predicts a harmonic strain and momentum analog to the classical counterpart. A fundamental difference observed is the oscillating purity of the deformation state. In addition, it was demonstrated that the time-averaged force originated from the elastic reflection of the vacuum-field modes is comparable to the Casimir force for two ideal metallic plates separated by a distance proportional to the reflectivity cutoff wavelength.

Engineered thermalization and cooling of quantum many-body systems

We develop a scheme for engineering genuine thermal states in analog quantum simulation platforms by coupling local degrees of freedom to driven, dissipative ancilla pseudospins. We demonstrate the scheme in a many-body quantum spin lattice simulation setting. A Born-Markov master equation describing the dynamics of the many-body system is developed, and we show that if the ancilla energies are periodically modulated, with a carefully chosen hierarchy of timescales, one can effectively thermalize the many-body system. Through analysis of the time-dependent dynamical generator, we determine the conditions under which the true thermal state is an approximate dynamical fixed point for general system Hamiltonians. Finally, we evaluate the thermalization protocol through numerical simulation and discuss prospects for implementation on current quantum simulation hardware.

High-fidelity quantum state transfer through a dissipative quantum data bus**Supported by the Shandong Provincial Natural Science Foundation under Grant No. ZR2016AM18 and the National Natural Science Foundation of China under Grant No. 61675118.

We propose and analyze a robust quantum state transfer protocol through a scalable quantum data bus that consists of a network of controlled dissipative modules. In particular, we first demonstrate the ability to achieve perfect state transfer between two distinct quantum sites which are adiabatically coupled to the data bus in non-dissipative situation. We then consider the role of dissipation in adiabatic quantum state transfer via using Born–Markov master equation in the standard Lindblad form. Numerical simulation shows that the dissipation effect on the quality of transmission can be suppressed by engineering the network couplings of data bus properly.

Erratum: Computable form of the Born-Markov master equation for open multilevel quantum systems [Phys. Rev. A 99 , 022118 (2019)

Erratum: Computable form of the Born-Markov master equation for open multilevel quantum systems [Phys. Rev. A <b>99</b> , 022118 (2019)

Computable form of the Born-Markov master equation for open multilevel quantum systems

The Redfield and Lindblad forms of the master equations are obtained under not only the Born and Markov approximations but also the secular approximation. In this paper, we propose a computable form of the master equation for open multilevel quantum systems beyond the secular approximation. The proposed form can retain some dynamical effects that are lost due to use of the secular approximation. Two kinds of open multilevel quantum systems are investigated by using this method, and the results are compared to the ones obtained from the Redfield equation and an exactly numerical path integral approach. It is shown that the results obtained from this computable form match those from the exact numerical path integral approach better than results from the Redfield equation. The normal-diffusion coefficient, momentum-damping coefficient, anomalous-diffusion coefficient, etc. in the master equation are also reevaluated.

From quantum optimal control theory to coherent destruction of tunneling

In this paper, we study coherent destruction of tunneling in driven spin-boson model. We formulate this physical process as a quantum optimal control problem. A running cost functional has been used to address the optimal control problem at hand. The dynamic is described by a Born–Markov master equation derived using the Bloch–Redfield formalism. We use the techniques of automatic differentiation to compute the gradient for the cost functional. We write the optimal control field as monochromatic plane wave. The robustness of the control profile against temperature and the system–bath coupling variations are also analyzed.

Theoretical study on optical response in nanostructures in the Born–Markov regime: The role of spontaneous emission and dephasing

We theoretically study the linear and nonlinear optical susceptibility in a nanostructure that interacts with a classical monochromatic optical field, by means of the solution of a Born–Markov master equation in the Lindblad form, considering as environment for the system the quantum vacuum of the continuum of modes of the external electromagnetic field. The processes of spontaneous emission and dephasing between two levels of the nanostructured quantum system are studied formally, within the framework of the quantum theory of open systems. It was found that the phenomenological master equation corresponds to a Born–Markov master equation in the Lindblad form that considers only the dissipative terms of spontaneous emission and dephasing, provided that the transverse coherence time T2 is smaller than spontaneous emission time T1. Finally, comparative results are presented between the proposed methodology and the conventional method for: optical rectification, optical absorption, generation of second harmonic and generation of third harmonic in a cylindrical quantum dot (CQD) of AsGa∕AsGaAl with a hydrogen-like impurity inside it and under the action of a uniform magnetic field.

Lindblad model of quantum Brownian motion

The theory of quantum Brownian motion describes the properties of a large class of open quantum systems. Nonetheless, its description in terms of a Born-Markov master equation, widely used in the literature, is known to violate the positivity of the density operator at very low temperatures. We study an extension of existing models, leading to an equation in the Lindblad form, which is free of this problem. We study the dynamics of the model, including the detailed properties of its stationary solution, for both constant and position-dependent coupling of the Brownian particle to the bath, focusing in particular on the correlations and the squeezing of the probability distribution induced by the environment

Resonance fluorescence of strongly driven two-level system coupled to multiple dissipative reservoirs

We present a theoretical formalism for resonance fluorescence radiating from a two-level system (TLS) driven by any periodic driving and coupled to multiple reservoirs. The formalism is derived analytically based on the combination of Floquet theory and Born–Markov master equation. The formalism allows us to calculate the spectrum when the Floquet states and quasienergies are analytically or numerically solved for simple or complicated driving fields. We can systematically explore the spectral features by implementing the present formalism. To exemplify this theory, we apply the unified formalism to comprehensively study a generic model that a harmonically driven TLS is simultaneously coupled to a radiative reservoir and a dephasing reservoir. We demonstrate that the significant features of the fluorescence spectra, the driving-induced asymmetry and the dephasing-induced asymmetry, can be attributed to the violation of detailed balance condition, and explained in terms of the driving-related transition quantities between Floquet-states and their steady populations. In addition, we find the distinguished features of the fluorescence spectra under the biharmonic and multiharmonic driving fields in contrast with that of the harmonic driving case. In the case of the biharmonic driving, we find that the spectra are significantly different from the result of the RWA under the multiple resonance conditions. By the three concrete applications, we illustrate that the present formalism provides a routine tool for comprehensively exploring the fluorescence spectrum of periodically strongly driven TLSs.

Spin uncertainty relation under decoherence

The evolution of a spin 1/2 particle subjected to a general Hamiltonian in interaction with its environment is considered. Using a new method, the exact solution of the spin-boson model for a general Hamiltonian is applied to investigate the dynamics of the open system. Born–Markov master equation in Caldeira–Leggett regime is used to derive an exact form of the expectation values of spin Pauli operators for a double-well potential. Subsequently, a definite form of spin uncertainty relation is calculated under the influence of the environment effects. Then, we scrutinize that how non-Unitary character of the evolution of open quantum systems can affect the traditional form of the spin uncertainty relation.