Hierarchy Equations Of Motion Research Articles

Noise suppression of transport through double quantum dots by feedback control

A controllable low noise current lies at the heart of high-precision measurements in quantum transport and metrology. While the previous research dealt with the suppression of noise in transport through a tunneling junction or a single quantum dot (QD) device, the present work investigates noise inhibition of a double quantum dot (DQD) transport system based on closed-loop feedback control. The unique advantage of a DQD device is that bidirectional transport at low bias can be measured by a nearby quantum point contact. However, the continuous monitoring of the DQD states inevitably leads to a measurement-induced dephasing. To appropriately characterize its transport properties in the presence of feedback action, we here develop a numerical method dubbed auxiliary density matrix approach, motivated by the hierarchical expansion of the moment-generating function in the hierarchy equations of motion [J. Cerrillo et al., Phys. Rev. B 94, 214308 (2016)]. This generic method has no restriction on the system structure and parameters and is able to evaluate the feedback current cumulants to an arbitrary order. It is revealed that the feedback control of the tunnel coupling between the two dots is the most effective to suppress the noise under various tunnel coupling configurations. The influence of interdot Coulomb interaction, measurement-induced dephasing, and finite time delay on feedback is also analyzed in detail.

Basic mechanisms in the laser control of non-Markovian dynamics

Referring to a Fano-type model qualitative analogy we develop a comprehensive basic mechanism for the laser control of the non-Markovian bath response in strongly coupled Open Quantum Systems (OQS). A converged Hierarchy Equations Of Motion (HEOM) is worked out to numerically solve the master equation of a spin-boson Hamiltonian to reach the reduced electronic density matrix of a heterojunction in the presence of strong THz laser pulses. Robust and efficient control is achieved increasing by a factor ?2 non-Markovianity measured by the time evolution of the volume of accessible states. The consequences of such fields on the central system populations and coherence are examined, putting the emphasis on the relation between the increase of non- Markovianity and the slowing down of decoherence processes.

Quantum kinetic expansion in the spin-boson model: Matrix formulation and system-bath factorized initial state.

Within the framework of the hierarchy equation of motion (HEOM), the quantum kinetic expansion (QKE) method of the spin-boson model is reformulated in the matrix representation. The equivalence between the two formulations (HEOM matrices and quantum operators) is numerically verified from the calculation of the time-integrated QKE rates. The matrix formulation of the QKE is extended to the system-bath factorized initial state. Following a one-to-one mapping between HEOM matrices and quantum operators, a quantum kinetic equation is rederived. The rate kernel is modified by an extra term following a systematic expansion over the site-site coupling. This modified QKE is numerically tested for its reliability by calculating the time-integrated rate and non-Markovian population kinetics. For an intermediate-to-strong dissipation strength and a large site-site coupling, the population transfer is found to be significantly different when the initial condition is changed from the local equilibrium to system-bath factorized state.

Zero-temperature localization in a sub-Ohmic spin-boson model investigated by an extended hierarchy equation of motion

With a decomposition scheme for the bath correlation function, the hierarchy equation of motion (HEOM) is extended to the zero-temperature sub-Ohmic spin-boson model, providing a numerically accurate prediction of quantum dynamics. As a dynamic approach, the extended HEOM determines the delocalized-localized (DL) phase transition from the extracted rate kernel and the coherent-incoherent dynamic transition from the short-time oscillation. As the bosonic bath approaches from the strong to weak sub-Ohmic regimes, a crossover behavior is identified for the critical Kondo parameter of the DL transition, accompanied by the transition from the coherent to incoherent dynamics in the localization.

Comparisons of different witnesses of non-Markovianity

Abstract In this paper, the evolutions of two kinds of witnesses of the non-Markovianity and their rates of changes with time are investigated and compared. Four definitions, the trace distance, fidelity, quantum relative entropy, and quantum Fisher information are used for the first kind of witnesses which are based on the completely positive maps (CPM). Three definitions, the quantum entanglement, quantum mutual information, and quantum discord are used for the second kind of witnesses, and they are based on the local completely positive maps (LCPM). An open two-level quantum system model and a numerically quantum dissipative dynamics method, hierarchy equation of motion (HEM) are used in the investigations. It is shown that the evolutions of the witnesses and their rates of the changes calculated with different definitions clearly show the characteristics of the non-Markovianity and they are in agreement with each other.

Complete positivity of non-Markovian quantum dynamics

We derive a purely algebraic framework for the identification of hierarchy equations of motion that induce completely positive dynamics. As we show with several explicit examples, this permits to construct well-behaved phenomenological models with strongly non-Markovian revivals of quantum coherence.

Extended hierarchy equation of motion for the spin-boson model.

An extended hierarchy equation of motion (HEOM) is proposed and applied to study the dynamics of the spin-boson model. In this approach, a complete set of orthonormal functions are used to expand an arbitrary bath correlation function. As a result, a complete dynamic basis set is constructed by including the system reduced density matrix and auxiliary fields composed of these expansion functions, where the extended HEOM is derived for the time derivative of each element. The reliability of the extended HEOM is demonstrated by comparison with the stochastic Hamiltonian approach under room-temperature classical ohmic and sub-ohmic noises and the multilayer multiconfiguration time-dependent Hartree theory under zero-temperature quantum ohmic noise. Upon increasing the order in the hierarchical expansion, the result obtained from the extended HOEM systematically converges to the numerically exact answer.

Using non-Markovian measures to evaluate quantum master equations for photosynthesis.

When dealing with system-reservoir interactions in an open quantum system, such as a photosynthetic light-harvesting complex, approximations are usually made to obtain the dynamics of the system. One question immediately arises: how good are these approximations, and in what ways can we evaluate them? Here, we propose to use entanglement and a measure of non-Markovianity as benchmarks for the deviation of approximate methods from exact results. We apply two frequently-used perturbative but non-Markovian approximations to a photosynthetic dimer model and compare their results with that of the numerically-exact hierarchy equation of motion (HEOM). This enables us to explore both entanglement and non-Markovianity measures as means to reveal how the approximations either overestimate or underestimate memory effects and quantum coherence. In addition, we show that both the approximate and exact results suggest that non-Markonivity can, counter-intuitively, increase with temperature, and with the coupling to the environment.

Spins Dynamics in a Dissipative Environment: Hierarchal Equations of Motion Approach Using a Graphics Processing Unit (GPU).

A system with many energy states coupled to a harmonic oscillator bath is considered. To study quantum non-Markovian system-bath dynamics numerically rigorously and nonperturbatively, we developed a computer code for the reduced hierarchy equations of motion (HEOM) for a graphics processor unit (GPU) that can treat the system as large as 4096 energy states. The code employs a Padé spectrum decomposition (PSD) for a construction of HEOM and the exponential integrators. Dynamics of a quantum spin glass system are studied by calculating the free induction decay signal for the cases of 3 × 2 to 3 × 4 triangular lattices with antiferromagnetic interactions. We found that spins relax faster at lower temperature due to transitions through a quantum coherent state, as represented by the off-diagonal elements of the reduced density matrix, while it has been known that the spins relax slower due to suppression of thermal activation in a classical case. The decay of the spins are qualitatively similar regardless of the lattice sizes. The pathway of spin relaxation is analyzed under a sudden temperature drop condition. The Compute Unified Device Architecture (CUDA) based source code used in the present calculations is provided as Supporting Information .

Exciton dissociation in the presence of phonons: A reduced hierarchy equations of motion approach

Combining the reduced hierarchy equations of motion (HEOM) approach with the Wigner-function formalism, we investigate nonperturbatively exciton dissociation under the influence of a phonon bath in an organic heterojunction. The exciton is modeled by an electron-hole pair with the electron moving in the presence of both an external electric field and the Coulomb attraction potential from the hole. In the absence of a phonon bath, calculated HEOM results reproduce those from the Onsager-Braun theory in weak electric fields. In the presence of a phonon bath, substantial deviations from the Onsager-Braun theory are found, signaling phonon-induced quantum effects. Furthermore, time evolution of the spatial current distribution is examined, and an initial spike followed by a polarity change of the transient photocurrent have been recovered.

Vibrationally mediated transport in molecular transistors

We investigate the steady-state electronic transport through a suspended dimer molecule coupled to leads. When strongly coupled to a vibrational mode, the electron transport is enhanced at the phonon resonant frequency and higher-order resonances. The temperature and bias determines the nature of the phonon-assisted resonances, with clear absorption and emission peaks. The strong coupling also induces a Frank-Condon-like blockade, suppressing the current between the resonances. We compare an analytical polaron transformation method to two exact numerical methods: the Hierarchy equations of motion and an exact diagonalization in the Fock basis. In the steady--state, our two numerical results are an exact match and qualitatively reflect the main features of the polaron treatment. Our results indicate the possibility of a new type of molecular transistor or sensor where the current can be extremely sensitive to small changes in the energies of the electronic states in the dimer.

Multipartite quantum entanglement evolution in photosynthetic complexes

We investigate the evolution of entanglement in the Fenna-Matthew-Olson (FMO) complex based on simulations using the scaled hierarchical equations of motion approach. We examine the role of entanglement in the FMO complex by direct computation of the convex roof. We use monogamy to give a lower bound for entanglement and obtain an upper bound from the evaluation of the convex roof. Examination of bipartite measures for all possible bipartitions provides a complete picture of the multipartite entanglement. Our results support the hypothesis that entanglement is maximum primary along the two distinct electronic energy transfer pathways. In addition, we note that the structure of multipartite entanglement is quite simple, suggesting that there are constraints on the mixed state entanglement beyond those due to monogamy.

Open Quantum Dynamics Calculations with the Hierarchy Equations of Motion on Parallel Computers.

Calculating the evolution of an open quantum system, i.e., a system in contact with a thermal environment, has presented a theoretical and computational challenge for many years. With the advent of supercomputers containing large amounts of memory and many processors, the computational challenge posed by the previously intractable theoretical models can now be addressed. The hierarchy equations of motion present one such model and offer a powerful method that remained under-utilized so far due to its considerable computational expense. By exploiting concurrent processing on parallel computers the hierarchy equations of motion can be applied to biological-scale systems. Herein we introduce the quantum dynamics software PHI, that solves the hierarchical equations of motion. We describe the integrator employed by PHI and demonstrate PHI's scaling and efficiency running on large parallel computers by applying the software to the calculation of inter-complex excitation transfer between the light harvesting complexes 1 and 2 of purple photosynthetic bacteria, a 50 pigment system.