Bath Correlation Research Articles

Reduced quantum dynamics with arbitrary bath spectral densities: Hierarchical equations of motion based on several different bath decomposition schemes

We investigated applications of the hierarchical equation of motion (HEOM) method to perform high order perturbation calculations of reduced quantum dynamics for a harmonic bath with arbitrary spectral densities. Three different schemes are used to decompose the bath spectral density into analytical forms that are suitable to the HEOM treatment: (1) The multiple Lorentzian mode model that can be obtained by numerically fitting the model spectral density. (2) The combined Debye and oscillatory Debye modes model that can be constructed by fitting the corresponding classical bath correlation function. (3) A new method that uses undamped harmonic oscillator modes explicitly in the HEOM formalism. Methods to extract system-bath correlations were investigated for the above bath decomposition schemes. We also show that HEOM in the undamped harmonic oscillator modes can give detailed information on the partial Wigner transform of the total density operator. Theoretical analysis and numerical simulations of the spin-Boson dynamics and the absorption line shape of molecular dimers show that the HEOM formalism for high order perturbations can serve as an important tool in studying the quantum dissipative dynamics in the intermediate coupling regime.

Non-Markovian response of ultrafast coherent electronic ring currents in chiral aromatic molecules in a condensed phase

Results of a theoretical study on non-Markov response for femtosecond laser-driven coherent ring currents in chiral aromatic molecules embedded in a condensed phase are presented. Coherent ring currents are generated by coherent excitation of a pair of quasi-degenerated π-electronic excited states. The coherent electronic dynamical behaviors are strongly influenced by interactions between the electronic system and phonon bath in a condensed phase. Here, the bath correlation time is not instantaneous but should be taken to be a finite time in ultrashort time-resolved experiments. In such a case, Markov approximation breaks down. A hierarchical master equation approach for an improved semiclassical Drude dissipation model was adopted to examine the non-Markov effects on ultrafast coherent electronic ring currents of (P)-2,2'-biphenol in a condensed phase. Time evolution of the coherent ring current derived in the hierarchical master equation approach was calculated and compared with those in the Drude model in the Markov approximation and in the static limit. The results show how non-Markovian behaviors in quantum beat signals of ring currents depend on the Drude bath damping constant. Effects of temperatures on ultrafast coherent electronic ring currents are also clarified.

A hybrid stochastic hierarchy equations of motion approach to treat the low temperature dynamics of non-Markovian open quantum systems

The hierarchical equations of motion technique has found widespread success as a tool to generate the numerically exact dynamics of non-Markovian open quantum systems. However, its application to low temperature environments remains a serious challenge due to the need for a deep hierarchy that arises from the Matsubara expansion of the bath correlation function. Here we present a hybrid stochastic hierarchical equation of motion (sHEOM) approach that alleviates this bottleneck and leads to a numerical cost that is nearly independent of temperature. Additionally, the sHEOM method generally converges with fewer hierarchy tiers allowing for the treatment of larger systems. Benchmark calculations are presented on the dynamics of two level systems at both high and low temperatures to demonstrate the efficacy of the approach. Then the hybrid method is used to generate the exact dynamics of systems that are nearly impossible to treat by the standard hierarchy. First, exact energy transfer rates are calculated across a broad range of temperatures revealing the deviations from the Förster rates. This is followed by computations of the entanglement dynamics in a system of two qubits at low temperature spanning the weak to strong system-bath coupling regimes.

A new method to account for the difference between classical and quantum baths in quantum dissipative dynamics

We investigate the difference between a quantum and classical bath within the Caldeira–Leggett model for dissipative quantum dynamics. It is well known that a Markovian equation of motion for the reduced dynamics can be derived by taking the classical approximation of the harmonic bath correlation function. However, such approximation is only valid at high temperatures, and it is necessary to include the non-Markovian effect of the quantum bath in more general cases. We show that the equation of motion derived for the classical bath can be extended to the exact quantum one, by simply adding a real stochastic process to take account of the difference between the quantum and classical bath correlation functions. Numerical examples in calculating electron and excitation energy transfer dynamics, as well as absorption spectra of molecular aggregates indicate that the proposed method is a valid approach to extend the existing theories to include the quantum effect of the harmonic bath. The possibility of applying a similar idea to account for the difference between zero and high temperature quantum dissipative dynamics is discussed.

Quantum discord dynamics in the presence of initial system–bath correlations

We investigate the influence of the initial correlations between a two-qubit system and a family of baths on the dynamics of quantum discord of these two qubits. We show that the initial states with entanglement between the two-qubit system and the bath can prompt two qubits to gain more quantum discord. Moreover, for the initial states only with discord but null entanglement between the two-qubit system and the bath, the effects depend on whether two qubits or one of the two qubits interact with the bath. We also find that there exist some specific initial states for which the entanglement between two qubits is invariant as time evolves, but the quantum discord is not, and for some other initial states although the states evolve, both the entanglement and the quantum discord remain invariant.

Coherent versus incoherent excitation energy transfer in molecular systems

We investigate the Markovian limit of a polaronic quantum master equation for coherent resonance energy transfer proposed recently by Jang et al. [J. Chem. Phys. 129, 101104 (2008)]. An expression for the rate of excitation energy transfer (EET) is derived and shown to exhibit both coherent and incoherent contributions. We then apply this theory to calculated EET rates for model dimer systems, and demonstrate that the small-polaron approach predicts a variety of dynamical behaviors. Notably, the results indicate that the EET dynamical behaviors can be understood by the interplay between noise-assisted EET and dynamical localization, while both are well captured by the polaron theory. Finally, we investigate bath correlation effects on the rate of EET and show that bath correlations (or anti-correlations) can either enhance or suppress EET rate depending on the strength of individual system-bath couplings. In summary, we introduce the small-polaron approach as an intuitive physical framework to consolidate our understanding of EET dynamics in the condensed phase.

Non-Markovian reduced dynamics based upon a hierarchical effective-mode representation

A reduced dynamics representation is introduced which is tailored to a hierarchical, Mori-chain type representation of a bath of harmonic oscillators which are linearly coupled to a subsystem. We consider a spin-boson system where a single effective mode is constructed so as to absorb all system-environment interactions, while the residual bath modes are coupled bilinearly to the primary mode and among each other. Using a cumulant expansion of the memory kernel, correlation functions for the primary mode are obtained, which can be suitably approximated by truncated chains representing the primary-residual mode interactions. A series of reduced-dimensional bath correlation functions is thus obtained, which can be expressed as Fourier-Laplace transforms of spectral densities that are given in truncated continued-fraction form. For a master equation which is second order in the system-bath coupling, the memory kernel is re-expressed in terms of local-in-time equations involving auxiliary densities and auxiliary operators.

Efficient computational approach to the non-Markovian second order quantum master equation: electronic energy transfer in model photosynthetic systems

An efficient numerical algorithm for solving the time-non-local non-Markovian master equation in the second Born approximation with exponentially decaying bath correlation function is introduced. Specifically, the coupled integro-differential equations for the reduced density matrix are solved by an efficient auxiliary function method in both the energy and site representations. As an application we consider the traditional dimer system that is used to model excitation energy transfer in photosynthetic systems, examining exact second order results and using them to assess the range of validity of the traditional Markov approximation. The method is also used to examine the dependence of the dynamics on the initial coherences in the case of two coupled oscillators. The computational results are augmented by a set of analytic inequalities obtained for the regime of validity of the Markov approximation in the cases of weak and strong resonance coupling, allowing for a rapid determination of the utility of the Markovian dynamics in various parameter regions.

Protein Dynamics and the Diversity of an Antibody Response

The immune system is remarkable in its ability to produce antibodies (Abs) with virtually any specificity from a limited repertoire of germ line precursors. Although the contribution of sequence diversity to this molecular recognition has been studied for decades, recent models suggest that protein dynamics may also broaden the range of targets recognized. To characterize the contribution of protein dynamics to immunological molecular recognition, we report the sequence, thermodynamic, and time-resolved spectroscopic characterization of a panel of eight Abs elicited to the chromophoric antigen 8-methoxypyrene-1,3,6-trisulfonate (MPTS). Based on the sequence data, three of the Abs arose from unique germ line Abs, whereas the remaining five comprise two sets of siblings that arose by somatic mutation of a common precursor. The thermodynamic data indicate that the Abs recognize MPTS via a variety of mechanisms. Although the spectroscopic data reveal small differences in protein dynamics, the anti-MPTS Abs generally show similar levels of flexibility and conformational heterogeneity, possibly representing the convergent evolution of the dynamics necessary for function. However, one Ab is significantly more rigid and conformationally homogeneous than the others, including a sibling Ab from which it differs by only five somatic mutations. This example of divergent evolution demonstrates that point mutations are capable of fixing significant differences in protein dynamics. The results provide unique insight into how high affinity Abs may be produced that bind virtually any target and possibly, from a more general perspective, how new protein functions are evolved.

Efficient estimation of energy transfer efficiency in light-harvesting complexes

The fundamental physical mechanisms of energy transfer in photosynthetic complexes is not yet fully understood. In particular, the degree of efficiency or sensitivity of these systems for energy transfer is not known given their realistic with surrounding photonic and phononic environments. One major problem in studying light-harvesting complexes has been the lack of an efficient method for simulation of their dynamics in biological environments. To this end, here we revisit the second order time-convolution (TC2) master equation and examine its reliability beyond extreme Markovian and perturbative limits. In particular, we present a derivation of TC2 without making the usual weak system-bath coupling assumption. Using this equation, we explore the long-time behavior of exciton dynamics of Fenna-Matthews-Olson (FMO) portein complex. Moreover, we introduce a constructive error analysis to estimate the accuracy of TC2 equation in calculating energy transfer efficiency, exhibiting reliable performance for system-bath interactions with weak and intermediate memory and strength. Furthermore, we numerically show that energy transfer efficiency is optimal and robust for the FMO protein complex of green sulfur bacteria with respect to variations in reorganization energy and bath correlation time scales.

System Bath Correlations and the Nonlinear Response of Qubits

In this letter we study qubits coupled to the bath formed by their environment. Although entanglement of the qubits is a well-known topic, much less effort has gone into the description of the correlations between the qubits and the bath. Here, we investigate these correlations, and study their effect on the qubits in equilibrium and their dynamics following the interaction with one or several external pulses. We find that a correct description of the correlations at the moment of these interactions is essential for a correct understanding of the dynamics.

Parametric projection operator technique for second order non-linear response

We demonstrate the application of the recently introduced parametric projector operator technique to a calculation of the second order non-linear optical response of a multilevel molecular system. We derive a parametric quantum master equation (QME) for the time evolution of the excited state of an excitonic system after excitation by the first two pulses in the usual spectroscopic four-wave-mixing scheme. This master equation differs from the usual QME by a correction term which depends on the delay τ between the pulses. In the presence of environmental degrees of freedom with finite bath correlation time and in the presence of intramolecular vibrations we find distinct dynamics of both the excited state populations and the electronic coherence for different delays τ.

Optimal control for non-Markovian open quantum systems

An efficient optimal-control theory based on the Krotov method is introduced for a non-Markovian open quantum system with a time-nonlocal master equation in which the control parameter and the bath correlation function are correlated. This optimal-control method is developed via a quantum dissipation formulation that transforms the time-nonlocal master equation to a set of coupled linear time-local equations of motion in an extended auxiliary Liouville space. As an illustration, the optimal-control method is applied to find the control sequences for high-fidelity Z gates and identity gates of a qubit embedded in a non-Markovian bath. Z gates and identity gates with errors less than 10^{-5} for a wide range of bath decoherence parameters can be achieved for the non-Markovian open qubit system with control over only the {\sigma}z term. The control-dissipation correlation and the memory effect of the bath are crucial in achieving the high-fidelity gates.

New features of entanglement dynamics with initial system–bath correlations

We investigate the influence of initial correlations between two qubits and a family of baths on the entanglement dynamics of these two qubits. We show that initial system–bath correlations can effectively avoid the occurrence of entanglement sudden death, and for the initial states with quantum correlations the entanglement between two qubits can be larger than its initial value. Significantly, we find that there exist initial states which we called entanglement preserving states, such that, although the state of the qubit subsystem evolves the entanglement of two qubits does not evolves at all.

Combining dynamical decoupling with fault-tolerant quantum computation

We study how dynamical decoupling (DD) pulse sequences can improve the reliability of quantum computers. We prove upper bounds on the accuracy of DD-protected quantum gates and derive sufficient conditions for DD-protected gates to outperform unprotected gates. Under suitable conditions, fault-tolerant quantum circuits constructed from DD-protected gates can tolerate stronger noise, and have a lower overhead cost, than fault-tolerant circuits constructed from unprotected gates. Our accuracy estimates depend on the dynamics of the bath that couples to the quantum computer, and can be expressed either in terms of the operator norm of the bath's Hamiltonian or in terms of the power spectrum of bath correlations; we explain in particular how the performance of recursively generated concatenated pulse sequences can be analyzed from either viewpoint. Our results apply to Hamiltonian noise models with limited spatial correlations.

Performance comparison of dynamical decoupling sequences for a qubit in a rapidly fluctuating spin bath

Avoiding the loss of coherence of quantum mechanical states is an important prerequisite for quantum information processing. Dynamical decoupling (DD) is one of the most effective experimental methods for maintaining coherence, especially when one can access only the qubit-system and not its environment (bath). It involves the application of pulses to the system whose net effect is a reversal of the system-environment interaction. In any real system, however, the environment is not static, and therefore the reversal of the system-environment interaction becomes imperfect if the spacing between refocusing pulses becomes comparable to or longer than the correlation time of the environment. The efficiency of the refocusing improves therefore if the spacing between the pulses is reduced. Here, we quantify the efficiency of different DD sequences in preserving different quantum states. We use 13C nuclear spins as qubits and an environment of 1H nuclear spins as the environment, which couples to the qubit via magnetic dipole-dipole couplings. Strong dipole-dipole couplings between the proton spins result in a rapidly fluctuating environment with a correlation time of the order of 100 us. Our experimental results show that short delays between the pulses yield better performance if they are compared with the bath correlation time. However, as the pulse spacing becomes shorter than the bath correlation time, an optimum is reached. For even shorter delays, the pulse imperfections dominate over the decoherence losses and cause the quantum state to decay.

Biexponential theory of Drude dissipation via hierarchical quantum master equation

A nonperturbative quantum dissipation theory is developed based on an optimal construction of biexponential Drude bath correlation function for its influence on the system dynamics. It is an advanced hierarchical quantum master equation approach, aiming at a numerically efficient non-Markovian quantum dissipation propagator, with the support of a convenient criterion to estimate in advance its accuracy for general systems. Compared to its low level, single-exponential counterpart [R. X. Xu et al., J. Chem. Phys. 131, 214111 (2009)], the present theory remarkably improves the applicability range over all-parameter space, as tested critically with electron transfer and frequency-dispersed transient absorption of exciton dimer model systems.

Nonclassical correlations in non-Markovian continuous-variable systems

We consider two identical and noninteracting harmonic oscillators coupled to either two independent bosonic baths or to a common bosonic bath. Under the only assumption, weak coupling, we analyze in detail the non-Markovian short-time-scale evolution of intensity correlations, entanglement, and quantum discord for initial two-mode squeezed-thermal vacuum states. In the independent reservoirs case, we observe the detrimental effect of the environment for all these quantities and we establish a hierarchy for their robustness against the environmental noise. In the common reservoir case, for initial uncorrelated states, we find that only quantum discord can be created via interaction with the bath, while entanglement and subshot noise intensity correlations remain absent.

The importance of initial correlations in rate dynamics: A consistent non-Markovian master equation approach

The non-Markovian master equation is applied to the calculation of reaction rates. Starting from the flux-side correlation function form, we treat both the thermal and real time evolution consistently within second order perturbation theory in the system–bath coupling. It is shown that the non-Markovian dynamics enter formally not only in the time propagation but also in the expressions for the initial system–bath correlations. We show that these initial correlations can have a significant effect on the reaction rate. The method presented, although approximate, is an effective way to calculate reaction rates for weakly coupled systems over a wide range of temperatures. As such it provides a complementary approach to the exact treatment based on the ML-MCTDH method of Craig et al. [1], which serves as reference in this work.

Systematic perturbation theory for dynamical coarse-graining

We demonstrate how the dynamical coarse-graining approach can be systematically extended to higher orders in the coupling between system and reservoir. Up to second order in the coupling constant, we explicitly show that dynamical coarse-graining unconditionally preserves positivity of the density matrix---even for bath density matrices that are not in equilibrium and also for time-dependent system Hamiltonians. By construction, the approach correctly captures the short-time dynamics; i.e., it is suitable for analyzing non-Markovian effects. We compare the dynamics with the exact solution for highly non-Markovian systems and find a remarkable quality of the coarse-graining approach. The extension to higher orders is straightforward but rather tedious. The approach is especially useful for bath correlation functions of simple structure and for small system dimensions.