System-bath Coupling Strength Research Articles

Quantum heat current under non-perturbative and non-Markovian conditions: Applications to heat machines.

We consider a quantum system strongly coupled to multiple heat baths at different temperatures. Quantum heat transport phenomena in this system are investigated using two definitions of the heat current: one in terms of the system energy and the other in terms of the bath energy. When we consider correlations among system-bath interactions (CASBIs)-which have a purely quantum mechanical origin-the definition in terms of the bath energy becomes different. We found that CASBIs are necessary to maintain the consistency of the heat current with thermodynamic laws in the case of strong system-bath coupling. However, within the context of the quantum master equation approach, both of these definitions are identical. Through a numerical investigation, we demonstrate this point for a non-equilibrium spin-boson model and a three-level heat engine model using the reduced hierarchal equations of motion approach under the strongly coupled and non-Markovian conditions. We observe the cyclic behavior of the heat currents and the work performed by the heat engine, and we find that their phases depend on the system-bath coupling strength. Through consideration of the bath heat current, we show that the efficiency of the heat engine decreases as the strength of the system-bath coupling increases, due to the CASBI contribution. In the case of a large system-bath coupling, the efficiency decreases further if the bath temperature is increased, even if the ratio of the bath temperatures is fixed, due to the discretized nature of energy eigenstates. This is also considered to be a unique feature of quantum heat engines.

Green’s functions for spin boson systems: Beyond conventional perturbation theories

Unraveling general properties of Green’s functions of quantum dissipative systems is of both experimental relevance and theoretical interest. Here, we study the spin-boson model as a prototype. By utilizing the Majorana-fermion representation together with the polaron transformation, we establish a theoretical approach to analyze Green’s functions of the spin-boson model. In contrast to conventional perturbation theories either in the tunneling energy or in the system-bath coupling strength, the proposed scheme gives reliable results over wide regimes of the coupling strength, bias, as well as temperature. To demonstrate the utility of the approach, we consider the susceptibility as well as the symmetrized spin correlation function (SSCF) which can be expressed in terms of Green’s functions. Thorough investigations are made on systems embedded in Ohmic or sub-Ohmic bosonic baths. We found the so-obtained SSCF is the same as that of the non-interacting blip approximation (NIBA) in unbiased systems while it is applicable for a wider range of temperature in the biased systems compared with the NIBA. We also show that a previous perturbation result is recovered as a weak coupling limit of the so-obtained SSCF. Furthermore, by studying the quantum criticality of the susceptibility, we confirm the validity of the quantum-to-classical mapping in the whole sub-Ohmic regime.

Quantum Thermodynamics in Strong Coupling: Heat Transport and Refrigeration

The performance characteristics of a heat rectifier and a heat pump are studied in a non-Markovian framework. The device is constructed from a molecule connected to a hot and cold reservoir. The heat baths are modelled using the stochastic surrogate Hamiltonian method. The molecule is modelled by an asymmetric double-well potential. Each well is semi-locally connected to a heat bath composed of spins. The dynamics are driven by a combined system–bath Hamiltonian. The temperature of the baths is regulated by a secondary spin bath composed of identical spins in thermal equilibrium. A random swap operation exchange spins between the primary and secondary baths. The combined system is studied in various system–bath coupling strengths. In all cases, the average heat current always flows from the hot towards the cold bath in accordance with the second law of thermodynamics. The asymmetry of the double well generates a rectifying effect, meaning that when the left and right baths are exchanged the heat current follows the hot-to-cold direction. The heat current is larger when the high frequency is coupled to the hot bath. Adding an external driving field can reverse the transport direction. Such a refrigeration effect is modelled by a periodic driving field in resonance with the frequency difference of the two potential wells. A minimal driving amplitude is required to overcome the heat leak effect. In the strong driving regime the cooling power is non-monotonic with the system–bath coupling.

Non-canonical distribution and non-equilibrium transport beyond weak system-bath coupling regime: A polaron transformation approach

The concept of polaron, emerged from condense matter physics, describes the dynamical interaction of moving particle with its surrounding bosonic modes. This concept has been developed into a useful method to treat open quantum systems with a complete range of system-bath coupling strength. Especially, the polaron transformation approach shows its validity in the intermediate coupling regime, in which the Redfield equation or Fermi’s golden rule will fail. In the polaron frame, the equilibrium distribution carried out by perturbative expansion presents a deviation from the canonical distribution, which is beyond the usual weak coupling assumption in thermodynamics. A polaron transformed Redfield equation (PTRE) not only reproduces the dissipative quantum dynamics but also provides an accurate and efficient way to calculate the non-equilibrium steady states. Applications of the PTRE approach to problems such as exciton diffusion, heat transport and light-harvesting energy transfer are presented.

Electron Pumping under Non-Markovian Dissipation: The Role of the Self-Consistent Field

Focusing on electron transport through a periodically driven resonant tunneling diode, we study the generation of a non-vanishing dc-current by applying symmetry breaking external ac fields with phase difference φ in a statically unbiased system. The effect of an environment is investigated using the system-bath Hamiltonian represented by the electron system coupled to harmonic oscillator modes with a Drude–Lorentz spectral density. To carry out simulations, we use the hierarchal equations of motion approach in the Wigner representation including a self-consistently constructed electric field that is determined from the electron distribution using the Poisson equation. We show that the maximal pumping current at a phase difference near φ = π/2 is strongly influenced by the system-bath coupling strength. The effect of dissipation is diminished if the self-consistent part of the potential is ignored.

Dynamics of a Landau–Zener transitions in a two-level system driven by a dissipative environment

The paper investigates the effects of a two-level quantum system coupled to transversal and longitudinal dissipative environment. The time-dependent phase accumulation, LZ transition probability and entropy in the presence of fast-ohmic, sub-ohmic and super-ohmic quantum noise are derived. Analytical results are obtained in terms of temperature, dissipation strength, LZ parameter and bath cutoff frequency. The bath is observed to modify the standard occupation difference by a decaying random phase factor and also produces dephasing during the transfer of population. The dephasing characteristics or the initial non-zero decoherence rate are observed to increase in time with the bath temperature and depend on the system-bath coupling strength and cutoff frequency. These parameters are found to strongly affect the memory and thus tailor the coherence process of the system.

Quantum entanglement at high temperatures? Bosonic systems in nonequilibrium steady state

This is the second of a series of three papers examining how viable it is for entanglement to be sustained at high temperatures for quantum systems in thermal equilibrium (Case A), in nonequilibrium (Case B) and in nonequilibrium steady state conditions (Case C). The system we analyze here consists of two coupled quantum harmonic oscillators each interacting with its own bath described by a scalar field, set at temperatures $T_1 > T_2$. For \textit{constant bilinear inter-oscillator coupling} studied here (Case C1) owing to the Gaussian nature, the problem can be solved exactly at arbitrary temperatures even for strong coupling. We find that the valid entanglement criterion in general is not a function of the bath temperature difference, in contrast to thermal transport in the same NESS setting [1]. Thus lowering the temperature of one of the thermal baths does not necessarily help to safeguard the entanglement between the oscillators. Indeed, quantum entanglement will disappear if any one of the thermal baths has a temperature higher than the critical temperature $T_c$. With the Langevin equations derived we give a full display of how entanglement dynamics in this system depends on $T_{1}$, $T_{2}$ , the inter-oscillator coupling and the system-bath coupling strengths. For weak oscillator-bath coupling the critical temperature $T_c$ is about the order of the inverse oscillator frequency, but for strong oscillator-bath coupling it will depend on the bath cutoff frequency. We conclude that in most realistic circumstances, for bosonic systems in NESS with constant bilinear coupling, `hot entanglement' is largely a fiction. In Paper III we will examine the case (C2) of \textit{time-dependent driven coupling } which contains the parametric pumping type described in [2] wherein entanglement was first shown to sustain at high temperatures.

Quantum heat transport of a two-qubit system: Interplay between system-bath coherence and qubit-qubit coherence.

We consider a system consisting of two interacting qubits that are individually coupled to separate heat baths at different temperatures. The quantum effects in heat transport are investigated in a numerically rigorous manner with a hierarchial equations of motion (HEOM) approach for non-perturbative and non-Markovian system-bath coupling cases under non-equilibrium steady-state conditions. For a weak interqubit interaction, the total system is regarded as two individually thermostatted systems, whereas for a strong interqubit interaction, the two-qubit system is regarded as a single system coupled to two baths. The roles of quantum coherence (or entanglement) between the two qubits (q-q coherence) and between the qubit and bath (q-b coherence) are studied through the heat current calculated for various strengths of the system-bath coupling and interqubit coupling for high and low temperatures. The same current is also studied using the time convolutionless (TCL) Redfield equation and using an expression derived from the Fermi golden rule (FGR). We find that the HEOM results exhibit turnover behavior of the heat current as a function of the system-bath coupling strength for all values of the interqubit coupling strength, while the results obtained with the TCL and FGR approaches do not exhibit such behavior, because they do not possess the capability of treating the q-b and q-q coherences. The maximum current is obtained in the case that the q-q coherence and q-b coherence are balanced in such a manner that coherence of the entire heat transport process is realized. We also find that the heat current does not follow Fourier's law when the temperature difference is very large, due to the non-perturbative system-bath interactions.

Förster resonance energy transfer, absorption and emission spectra in multichromophoric systems. III. Exact stochastic path integral evaluation.

A numerically exact path integral treatment of the absorption and emission spectra of open quantum systems is presented that requires only the straightforward solution of a stochastic differential equation. The approach converges rapidly enabling the calculation of spectra of large excitonic systems across the complete range of system parameters and for arbitrary bath spectral densities. With the numerically exact absorption and emission operators, one can also immediately compute energy transfer rates using the multi-chromophoric Förster resonant energy transfer formalism. Benchmark calculations on the emission spectra of two level systems are presented demonstrating the efficacy of the stochastic approach. This is followed by calculations of the energy transfer rates between two weakly coupled dimer systems as a function of temperature and system-bath coupling strength. It is shown that the recently developed hybrid cumulant expansion (see Paper II) is the only perturbative method capable of generating uniformly reliable energy transfer rates and emission spectra across a broad range of system parameters.

Linear-algebraic bath transformation for simulating complex open quantum systems

In studying open quantum systems, the environment is often approximated as a collection of non-interacting harmonic oscillators, a configuration also known as the star-bath model. It is also well known that the star-bath can be transformed into a nearest-neighbor interacting chain of oscillators. The chain-bath model has been widely used in renormalization group approaches. The transformation can be obtained by recursion relations or orthogonal polynomials. Based on a simple linear algebraic approach, we propose a bath partition strategy to reduce the system-bath coupling strength. As a result, the non-interacting star-bath is transformed into a set of weakly coupled multiple parallel chains. The transformed bath model allows complex problems to be practically implemented on quantum simulators, and it can also be employed in various numerical simulations of open quantum dynamics.

Iterative calculation of energy eigenstates employing the multilayer multiconfiguration time-dependent Hartree theory.

The improved relaxation approach of Meyer and co-workers is extended by employing the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory. Within this approach the trial wave function is expressed by a recursive, multilayered expansion that is the same as in a normal ML-MCTDH simulation. Calculation of an energy eigenstate then proceeds by iteratively diagonalizing the Boltzmann operator in the top layer representation using a Lanczos/Arnoldi method while relaxing the single particle functions of all layers using the ML-MCTDH imaginary time propagation. Numerical iteration is stopped until the relative energy and/or density of the target state meet the convergence criteria. The application of this multilayer improved relaxation method is illustrated by computing the energy splitting for the lowest pair of eigenstates as well as the spin observables and entanglement entropy of the spin-boson model. A large number of degrees of freedom, up to 10,000 bath modes, can be treated, and thus the appropriate phase transition at a critical system-bath coupling strength may be predicted via appropriate extrapolations.

Coherent quantum dynamics in donor–bridge–acceptor systems: beyond the hopping and super-exchange mechanisms

The population transfer dynamics of model donor–bridge–acceptor systems is studied by comparing a recently developed polaron-transformed quantum master equation (PQME) with the well-known Redfield and Förster theories of quantum transport. We show that the PQME approach reduces to these two theories in their respective limits of validity and naturally interpolates between them as a function of the system–bath coupling strength. By exploring the parameter space of our model problem, we identify novel regimes of transport dynamics in bridged systems like those encountered in biological and organic energy transfer problems. Furthermore, we demonstrate that three-level systems like the ones studied herein represent ideal minimal models for the identification of quantum coherent transport as embodied in super-exchange phenomena that cannot be captured by Förster-like hopping approaches.

Dynamics of a two-level system coupled to a bath of spins

The dynamics of a two-level system coupled to a spin bath is investigated via the numerically exact multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory. Consistent with the previous work on linear response approximation [N. Makri, J. Phys. Chem. B 103, 2823 (1999)], it is demonstrated numerically that this spin-spin-bath model can be mapped onto the well-known spin-boson model if the system-bath coupling strength obeys an appropriate scaling behavior. This linear response mapping, however, may require many bath spin degrees of freedom to represent the practical continuum limit. To clarify the discrepancies resulted from different approximate treatments of this model, the population dynamics of the central two-level system has been investigated near the transition boundary between the coherent and incoherent motions via the ML-MCTDH method. It is found that increasing temperature favors quantum coherence in the nonadiabatic limit of this model, which corroborates the prediction in the previous work [J. Shao and P. Hanggi, Phys. Rev. Lett. 81, 5710 (1998)] based on the non-interacting blip approximation (NIBA). However, the coherent-incoherent boundary obtained by the exact ML-MCTDH simulation is slightly different from the approximate NIBA results. Quantum dynamics in other physical regimes are also discussed.

Generalized Gibbs state with modified Redfield solution: Exact agreement up to second order

A novel scheme for the steady state solution of the standard Redfield quantum master equation is developed which yields agreement with the exact result for the corresponding reduced density matrix up to second order in the system-bath coupling strength. We achieve this objective by use of an analytic continuation of the off-diagonal matrix elements of the Redfield solution towards its diagonal limit. Notably, our scheme does not require the provision of yet higher order relaxation tensors. Testing this modified method for a heat bath consisting of a collection of harmonic oscillators we assess that the system relaxes towards its correct coupling-dependent, generalized quantum Gibbs state in second order. We numerically compare our formulation for a damped quantum harmonic system with the nonequilibrium Green's function formalism: we find good agreement at low temperatures for coupling strengths that are even larger than expected from the very regime of validity of the second-order Redfield quantum master equation. Yet another advantage of our method is that it markedly reduces the numerical complexity of the problem; thus, allowing to study efficiently large-sized system Hilbert spaces.

Consistent treatment of coherent and incoherent energy transfer dynamics using a variational master equation

We investigate the energy transfer dynamics in a donor-acceptor model by developing a time-local master equation technique based on a variational transformation of the underlying Hamiltonian. The variational transformation allows a minimisation of the Hamiltonian perturbation term dependent on the system parameters, and consequently results in a versatile master equation valid over a range of system-bath coupling strengths, temperatures, and environmental spectral densities. While our formalism reduces to the well-known Redfield, Förster and polaron forms in the appropriate limits, in general it is not equivalent to perturbing in either the system-environment or donor-acceptor coupling strengths, and hence can provide reliable results between these limits as well. Moreover, we show how to include the effects of both environmental correlations and non-equilibrium preparations within the formalism.

Separation-dependent localization in a two-impurity spin-boson model

Using a variational approach we investigate the delocalized to localized crossover in the ground state of an Ohmic two-impurity spin-boson model, describing two otherwise noninteracting spins coupled to a common bosonic environment. We show that a competition between an environment-induced Ising spin interaction and externally applied fields leads to variations in the system-bath coupling strength, ${\ensuremath{\alpha}}_{c}$, at which the delocalized-localized crossover occurs. Specifically, the crossover regime lies between ${\ensuremath{\alpha}}_{c}=0.5$ and ${\ensuremath{\alpha}}_{c}=1$ depending upon the spin separation and the strength of the transverse tunneling field. This is in contrast to the analogous single-spin case, for which the crossover occurs (in the scaling limit) at fixed ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}1$. We also discuss links between the two-impurity spin-boson model and a dissipative two-spin transverse Ising model, showing that the latter possesses the same qualitative features as the Ising strength is varied. Finally, we show that signatures of the crossover may be observed in single impurity observables, as well as in the behavior of the system-environment entanglement.

From coherent motion to localization: II. Dynamics of the spin-boson model with sub-Ohmic spectral density at zero temperature

The dynamics of the spin-boson model at zero temperature is studied for a bath characterized by a sub-Ohmic spectral density. Using the numerically exact multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method, the population dynamics of the two-level subsystem has been investigated in a broad range of parameter space. The results show the transition of the dynamics from weakly damped coherent motion to localization upon increase of the system–bath coupling strength. Comparison of the exact ML-MCTDH simulations with the non-interacting blip approximation (NIBA) shows that the latter performs rather poorly in the weak coupling regime with small Kondo parameters. However, NIBA improves significantly upon increase in the coupling strength and is quantitatively correct in the strong coupling, nonadiabatic limit. The transition from coherent motion to localization as a function of the different parameters of the model is analyzed in some detail.

Exact quantum dissipative dynamics under external time-dependent driving fields

An exact and nonperturbative quantum master equation can be constructed via the calculus on the path integral. It results in hierarchical equations of motion for the reduced density operator. Involved also are a set of well-defined auxiliary density operators that resolve not just the system–bath coupling strength but also memory. In this work, we scale these auxiliary operators individually to achieve a uniform error tolerance, as set by the reduced density operator. An efficient propagator is then proposed to the hierarchical Liouville-space dynamics of quantum dissipation. Numerically exact studies are carried out on the dephasing effect on population transfer in the simple stimulated Raman adiabatic passage scheme. We also make assessments on several perturbative theories for their applicabilities in the present system of study.

From coherent motion to localization: dynamics of the spin-boson model at zero temperature

The dynamics of the spin-boson model at zero temperature is investigated using the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method. This method allows a numerically exact description of the dynamics of the spin-boson model in a broad range of coupling strengths. The results show the transition of the dynamics from weakly damped coherent motion to incoherent decay and finally to localization upon increase of the system–bath coupling strength. A detailed analysis of the incoherent decay for stronger coupling reveals that multiple timescales are involved in the dynamics. Furthermore, the applicability of the scaling limit for large characteristic frequencies of the bath as well as the validity of the non-interacting blip approximation are studied in some detail.

Role of vibrational energy relaxation in the photoinduced nonadiabatic dynamics of pyrrole at the [formula omitted] conical intersection

The influence of intramolecular vibrational relaxation on the nonadiabatic dynamics at the 1 A 2 ( π σ ∗ ) – S 0 conical intersection of pyrrole has been investigated within the reduced-density-matrix formalism. The system Hamiltonian represents the 1 A 2 ( π σ ∗ ) – S 0 conical intersection of pyrrole, taking the dominant tuning and coupling modes into account. Within the system-plus-bath model, the system is assumed to couple with a harmonic bath, which represents the many other photoinactive vibrational modes. The system–bath coupling results in vibrational energy transfer from the system to the bath. An extension of Redfield theory has been developed, which is suitable for the description of quantum dissipative dynamics of a dissociative system. The electronic population probabilities, the dissociation probability, the internal-conversion probability and their dependence on the system–bath coupling strength have been explored by the calculations. It is shown that the system–bath coupling reduces the dissociation probability and enhances the internal-conversion probability.