System-bath Coupling Strength Research Articles

Thermodynamic quantum Fokker–Planck equations and their application to thermostatic Stirling engine

We developed a computer code for the thermodynamic quantum Fokker–Planck equations (T-QFPE), derived from a thermodynamic system–bath model. This model consists of an anharmonic subsystem coupled to multiple Ohmic baths at different temperatures, which are connected to or disconnected from the subsystem as a function of time. The code numerically integrates the T-QFPE and their classical expression to simulate isothermal, isentropic, thermostatic, and entropic processes in both quantum and classical cases. The accuracy of the results was verified by comparing the analytical solutions of the Brownian oscillator. In addition, we illustrated a breakdown of the Markovian Lindblad-master equation in the pure quantum regime. As a demonstration, we simulated a thermostatic Stirling engine employed to develop non-equilibrium thermodynamics [S. Koyanagi and Y. Tanimura, J. Chem. Phys. 161, 114113 (2024)] under quasi-static conditions. The quasi-static thermodynamic potentials, described as intensive and extensive variables, were depicted as work diagrams. In the classical case, the work done by the external field is independent of the system–bath coupling strength. In contrast, in the quantum case, the work decreases as the coupling strength increases due to quantum entanglement between the subsystem and bath. The codes were developed for multicore processors using Open Multi-Processing (OpenMP) and for graphics processing units using the Compute Unified Device Architecture. These codes are provided in the supplementary material.

Energetics of the dissipative quantum oscillator

In this paper, we discuss some aspects of the energetics of a quantum Brownian particle placed in a harmonic trap, also known as the dissipative quantum oscillator. Based on the fluctuation–dissipation theorem, we analyze two distinct notions of thermally-averaged energy that can be ascribed to the oscillator. These energy functions, respectively dubbed hereafter as the mean energy and the internal energy, are found to be inequivalent for arbitrary system–bath coupling strength when the dissipation kernel has a non-zero memory timescale, as in the case of a Drude bath. Remarkably, both the energy functions satisfy the quantum counterpart of energy equipartition theorem, but with different probability distributions on the frequency domain of the heat bath. Moreover, the Gibbs approach to thermodynamics provides us with yet another thermally-averaged energy function. In the weak-coupling limit, all the above-mentioned energy expressions reduce to ϵ=ħω02coth(ħω02kBT), which is the familiar result. We generalize our analysis to the case of the three-dimensional dissipative charged oscillator placed in a spatially-uniform magnetic field.

Microscopic contributions to the entropy production at all times: from nonequilibrium steady states to global thermalization

Based on exact integration of the Schrödinger equation, we numerically study microscopic contributions to the entropy production for the single electron transistor, a paradigmatic model describing a single Fermi level tunnel coupled to two baths of free fermions. To this end, we decompose the entropy production into a sum of information theoretic terms and study them across all relevant time scales, including the nonequilibrium steady state regime and the final stage of global thermalization. We find that the entropy production is dominated for most times by microscopic deviations from thermality in the baths and the correlation between (but not inside) the baths. Despite these microscopic deviations from thermality, the temperatures and chemical potentials of the baths thermalize as expected, even though our model is integrable. Importantly, this observation is confirmed for both initially mixed and pure states. We further observe that the bath-bath correlations are quite insensitive to the system-bath coupling strength contrary to intuition. Finally, the system-bath correlation, small in an absolute sense, dominates in a relative sense and displays pure quantum correlations for all studied parameter regimes.

Thermo-Field Dynamics Approach to Photo-induced Electronic Transitions Driven by Incoherent Thermal Radiation.

The effects of thermal light-matter interaction on the dynamics of photo-induced electronic transitions in molecules are investigated using a novel first principles approach based on the thermo-field dynamics description of both the molecular vibrational modes and of the radiation field. The developed approach permits numerically accurate simulations of quantum dynamics of electronic/excitonic systems coupled to nuclear and photonic baths kept at different temperatures. The baths can be described by arbitrary spectral densities and can have any system-bath coupling strengths. In agreement with the results obtained previously by less rigorous methods, we show that the excitation process obtained by the continuous interaction with the suddenly turned-on thermal radiation field creates a mixed ensemble having a nonnegligible component consisting of a superposition of vibronic eigenstates which can sustain coherent oscillations for relatively long times. The results become especially relevant for the dynamics of electronic transitions upon sunlight excitation. Analytical results based on time-dependent perturbation theory support the numerical simulations and provide a simple interpretation of the time evolution of quantum observables.

Stability and dynamics of many-body localized systems coupled to a small bath

It is known that strong disorder in closed quantum systems leads to many-body localization (MBL), and that this quantum phase can be destroyed by coupling to an infinitely large Markovian environment. However, the stability of the MBL phase is less clear when the system and environment are of finite and comparable size. Here, we study the stability and eventual localization properties of a disordered Heisenberg spin chain coupled to a finite environment, and extensively explore the effects of environment disorder, geometry, initial state, and system-bath coupling strength, using the steady-state value of magnetization as a probe. Focusing on nonequilibrium dynamics and steady-state properties, our results indicate that within system sizes amenable to exact diagonalization, a strongly localized system interacting in a junction configuration retains remnant information on its initial state at long times despite coupling to a finite ergodic environment. In contrast, in a ladder configuration, strong dependencies on the initial state and coupling strength are observed, which can lead to either the loss or retention of information. Finally, we highlight and discuss discrepancies that can arise when similar methodologies are employed to infer localization or thermalization, revealing the need for careful interpretation.

Collapse and revival structure of information backflow for a central spin coupled to a finite spin bath

The Markovianity of quantum dynamics is an important property of open quantum systems determined by various ingredients of the system and bath. Apart from the system-bath interaction, the initial state of the bath, etc., the dimension of the bath plays a critical role in determining the Markovianity of quantum dynamics, as a strict decay of the bath correlations requires an infinite dimension for the bath. In this work, we investigate the role of finite bath dimension in the Markovianity of quantum dynamics by considering a simple but nontrivial model in which a central spin is isotropically coupled to a finite number of bath spins, and show how the dynamics of the central spin transits from non-Markovian to Markovian as the number of the bath spins increases. The non-Markovianity is characterized by the information backflow from the bath to the system in terms of the trace distance of the system states. We derive the time evolution of the trace distance analytically, and find periodic collapse-revival patterns in the information flow. The mechanism underlying this phenomenon is investigated in detail, and it shows that the period of the collapse-revival pattern is determined by the competition between the number of the bath spins, the system-bath coupling strength, and the frequency detuning. When the number of bath spins is sufficiently large, the period of the collapse-revival structure as well as the respective collapse and revival times increase in proportion to the number of the bath spins, which characterizes how the information backflow decays with a large dimension of the bath. We also analyze the effect of the system-bath interaction strength and frequency detuning on the collapse-revival patterns of the information flow, and obtain the condition for the existence of the collapse-revival structure. The results are illustrated by numerical computation.

Stochastic learning control of adiabatic speedup in a non-Markovian open qutrit system

Precise and efficient control of quantum systems is essential to perform quantum information-processing tasks. In terms of adiabatic speedup via leakage elimination operator approach, for a closed system, the ideal pulse control conditions have been theoretically derived by P-Q partitioning technique. However, it is a challenge to design the corresponding control pulses for an open system, which requires noisy environments to be addressed. In this paper, we apply the stochastic search procedures to an open qutrit system and successfully obtain the optimal control pulses for significant adiabatic speedup. The calculation results show that these optimal pulses allow us to acquire higher fidelities than the ideal pulses. The improvement of fidelity is large for relatively strong system-bath coupling strength and high bath temperature. For certain coupling strength and bath temperature, the maximal improvement can be achieved for a critical characteristic frequency which represents the memory time of the environment. Our investigation indicates that the stochastic search procedures are powerful tools to design control pulses for combating the detrimental effects of the environment.

Suppression of Crosstalk in Superconducting Qubits Using Dynamical Decoupling

Currently available superconducting quantum processors with interconnected transmon qubits are noisy and prone to various errors. The errors can be attributed to sources such as open quantum system effects and spurious inter-qubit couplings (crosstalk). The ZZ-coupling between qubits in fixed frequency transmon architectures is always present and contributes to both coherent and incoherent crosstalk errors. Its suppression is therefore a key step towards enhancing the fidelity of quantum computation using transmons. Here we propose the use of dynamical decoupling to suppress the crosstalk, and demonstrate the success of this scheme through experiments performed on several IBM quantum cloud processors. In particular, we demonstrate improvements in quantum memory as well as the performance of single-qubit and two-qubit gate operations. We perform open quantum system simulations of the multi-qubit processors and find good agreement with the experimental results. We analyze the performance of the protocol based on a simple analytical model and elucidate the importance of the qubit drive frequency in interpreting the results. In particular, we demonstrate that the XY4 dynamical decoupling sequence loses its universality if the drive frequency is not much larger than the system-bath coupling strength. Our work demonstrates that dynamical decoupling is an effective and practical way to suppress crosstalk and open system effects, thus paving the way towards higher-fidelity logic gates in transmon-based quantum computers.

A simple improved low temperature correction for the hierarchical equations of motion.

The study of open system quantum dynamics has been transformed by the hierarchical equations of motion (HEOM) method, which gives the exact dynamics for a system coupled to a harmonic bath at arbitrary temperature and system-bath coupling strength. However, in its standard form, this method is only consistent with the weak-coupling quantum master equation at all temperatures when many auxiliary density operators are included in the hierarchy, even when low temperature corrections are included. Here, we propose a new low temperature correction scheme for the termination of the hierarchy based on Zwanzig projection, which alleviates this problem and restores consistency with the weak-coupling master equation with a minimal hierarchy. The utility of the new correction scheme is demonstrated on a range of model systems, including the Fenna-Matthews-Olson complex. The new closure is found to improve convergence of the HEOM even beyond the weak-coupling limit and is very straightforward to implement in existing HEOM codes.

Ground-state and dynamical properties of a spin-S Heisenberg star

We generalize the Heisenberg star consisting of a spin-1/2 central spin and a homogeneously coupled spin bath modeled by the XXX ring [Richter J and Voigt A 1994 J. Phys. A: Math. Gen. 27 1139-1149] to the case of arbitrary central-spin size S < N/2, where N is the number of bath spins. We describe how to block-diagonalize the model based on the Bethe ansatz solution of the XXX ring, with the dimension of each block Hamiltonian ≤ 2S + 1. We obtain all the eigenenergies and explicit expressions of the sub-ground states in each l-subspace with l being the total angular momentum of the bath. Both the eigenenergies and the sub-ground states have distinct structures depending whether S ≤ l or l < S. The absolute ground-state energy and the corresponding l as functions of the intrabath coupling are numerically calculated for N = 16 and S = 1, 2, ⋯ ,7 and their behaviors are quantitatively explained in the weak and strong intrabath coupling limits. We then study the dynamics of the antiferromagnetic order within an XXX bath prepared in the Néel state. Effects of the initial state of the central spin, the value of S, and the system-bath coupling strength on the staggered magnetization dynamics are investigated. By including a Zeeman term for the central spin and the anisotropy in the intrabath coupling, we also study the polarization dynamics of the central spin for a bath prepared in the spin coherent state. Under the resonant condition and at the isotropic point of the bath, the polarization dynamics for S > 1/2 exhibit collapse-revival behaviors with fine structures. However, the collapse-revival phenomena are found to be fragile with respect to the anisotropy of the intrabath coupling.

Nonequilibrium quantum thermodynamics in non-Markovian adiabatic speedup

Understanding heat transfer between a quantum system and its environment is of undisputed importance if reliable quantum devices are to be constructed. Here, we investigate the heat transfer between system and bath in non-Markovian open systems in the process of adiabatic speedup. Using the quantum state diffusion equation method, the heat current, energy current, and power are calculated during free evolution and under external control of the system. While the heat current increases with increasing system–bath coupling strength and bath temperature, it can be restricted by the non-Markovian nature of the bath. Without pulse control, the heat current is nearly equal to the energy current. On the other hand, with pulse control, the energy current turns out to be nearly equal to the power. In this scenario, we show that non-Markovianity is a useful tool to drive the system through an approximate adiabatic dynamics, with pulse control acting in the conversion between heat current and power throughout the evolution.

Generalized master equation for charge transport in a molecular junction: Exact memory kernels and their high order expansion.

We derive a set of generalized master equations (GMEs) to study charge transport dynamics in molecular junctions using the Nakajima-Zwanzig-Mori projection operator approach. In the new GME, time derivatives of population on each quantum state of the molecule, as well as the tunneling current, are calculated as the convolution of time non-local memory kernels with populations on all system states. The non-Markovian memory kernels are obtained by combining the hierarchical equations of motion (HEOM) method and a previous derived Dyson relation for the exact kernel. A perturbative expansion of these memory kernels is then calculated using the extended HEOM developed in our previous work [M. Xu et al., J. Chem. Phys. 146, 064102 (2017)]. By using the resonant level model and the Anderson impurity model, we study properties of the exact memory kernels and analyze convergence properties of their perturbative expansions with respect to the system-bath coupling strength and the electron-electron repulsive energy. It is found that exact memory kernels calculated from HEOM exhibit short memory times and decay faster than the population and current dynamics. The high order perturbation expansion of the memory kernels can give converged results in certain parameter regimes. The Padé and Landau-Zener resummation schemes are also found to give improved results over low order perturbation theory.

A low-temperature quantum Fokker-Planck equation that improves the numerical stability of the hierarchical equations of motion for the Brownian oscillator spectral density.

We investigate the numerical stability of the hierarchical equations of motion (HEOM) method applied to systems with the Brownian oscillator (BO) and multimode BO (MBO) spectral densities. It is shown that, with the increase in the system-bath coupling strength, the standard HEOM may become unstable, and a simple increase in the truncation depth of the HEOM cannot remove the instability at long times. To solve this problem, we first show that the high-temperature approximation of the HEOM with the BO spectral density is equivalent to the celebrated quantum Fokker-Planck equation (QFPE). By starting from the HEOM, we then derive a new multidimensional phase space differential equation that generalizes the QFPE to arbitrary temperature. It is further shown that the numerical instability can be removed if the new low-temperature QFPE is expanded in a basis set different than the one that leads to the conventional HEOM. The matrix product state method is also employed to propagate the new equation based on the low-temperature QFPE and to resolve the numerical instability problem for an electron transfer model with the MBO spectral density presented in the recent literature.

Quantum counterpart of energy equipartition theorem for a dissipative charged magneto-oscillator: Effect of dissipation, memory, and magnetic field.

In this paper, we formulate and study the quantum counterpart of the energy equipartition theorem for a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a passive quantum heat bath through coordinate variables. The bath is modeled as a collection of independent quantum harmonic oscillators. We derive closed form expressions for the mean kinetic and potential energies of the charged dissipative magneto-oscillator in the forms E_{k}=〈E_{k}〉 and E_{p}=〈E_{p}〉, respectively, where E_{k} and E_{p} denote the average kinetic and potential energies of individual thermostat oscillators. The net averaging is twofold; the first one is over the Gibbs canonical state for the thermostat, giving E_{k} and E_{p}, and the second one, denoted by 〈·〉, is over the frequencies ω of the bath oscillators which contribute to E_{k} and E_{p} according to probability distributions P_{k}(ω) and P_{p}(ω), respectively. The relationship of the present quantum version of the equipartition theorem with that of the fluctuation-dissipation theorem (within the linear-response theory framework) is also explored. Further, we investigate the influence of the external magnetic field and the effect of different dissipation processes through Gaussian decay and Drude and radiation bath spectral density functions on the typical properties of P_{k}(ω) and P_{p}(ω). Finally, the role of system-bath coupling strength and the memory effect is analyzed in the context of average kinetic and potential energies of the dissipative charged magneto-oscillator.

Fluctuation–dissipation relation for a quantum Brownian oscillator in a parametrically squeezed thermal field

In this paper we study the nonequilibrium evolution of a quantum Brownian oscillator, modeling the internal degree of freedom of a harmonic atom or an Unruh–DeWitt detector, coupled to a nonequilibrium and nonstationary quantum field bath and inquire whether a fluctuation–dissipation relation (FDR) can exist after/if it approaches equilibration. This is a nontrivial issue because a squeezed field bath cannot reach equilibration and yet, as this work shows, the system oscillator indeed can, which is a necessary condition for FDRs. We discuss three different settings: (A) The bath field essentially remains in a squeezed thermal state throughout, whose squeeze parameter is a mode- and time-independent constant. This situation is often encountered in quantum optics and quantum thermodynamics. (B) The bath field is initially in a thermal state, but is subjected to a parametric process leading to mode- and time-dependent squeezing. This scenario is encountered in cosmology and dynamical Casimir effects. The squeezing in the bath in both types of processes will affect the oscillator’s nonequilibrium evolution. We show that at late times it approaches equilibration and this stationarity condition warrants the existence of a FDR. The trait of squeezing is marked by the oscillator’s effective equilibrium temperature, and the proportionality factor in the FDR is only related to the stationary component of the noise kernel of the bath field. Setting (C) is more subtle: A finite system–bath coupling strength can set the oscillator in a squeezed state even though the bath field is stationary and does not engage in any parametric process. The squeezing of the system in this case is in general time-dependent but becomes constant when the internal dynamics is fully relaxed. We begin with comments on the broad range of physical processes involving squeezed thermal baths and end with some remarks on the significance of FDRs in capturing the essence of quantum backreaction in nonequilibrium and stochastic systems.

Non-Markovian effects in the spin-boson model at zero temperature

We investigate memory effects in the spin-boson model using a recently proposed measure for non-Markovian behavior based on the information exchange between an open system and its environment. Employing the numerical exact multilayer multiconfguration time-dependent Hartree approach, we simulate the dynamics of the spin-boson model at zero temperature for a broad range of parameters. For a fast bath, i.e. in the scaling limit, we find non-Markovian dynamics for a coherently decaying spin at weak system-bath coupling, whereas memory effects are absent for stronger coupling in the regimes of incoherent decay and localization. If the time scales of system and bath are comparable, a complex, non-monotonic dependence of non-Markovianity on the system-bath coupling strength is observed.

Environmentally Improved Coherent Light Harvesting.

Coherence-enhanced light harvesting has not been directly observed experimentally, despite theoretical evidence that coherence can significantly enhance light-harvesting performance. The main experimental obstacle has been the difficulty in isolating the effect of coherence in the presence of confounding variables. Recent proposals for externally controlling coherence by manipulating the light's degree of polarization showed that coherent efficiency enhancements would be possible, but they were restricted to light-harvesting systems weakly coupled to their environment. Here, we show that increases in system-bath coupling strength can amplify coherent efficiency enhancements, rather than suppress them. This result dramatically broadens the range of systems that could be used to conclusively demonstrate coherence-enhanced light harvesting or to engineer coherent effects into artificial light-harvesting devices.

Strong coupling effects in quantum thermal transport with the reaction coordinate method

We present a semi-analytical approach for studying quantum thermal energy transport at the nanoscale. Our method, which is based on the reaction coordinate method, reveals the role of strong system-bath coupling effects in quantum energy transport. Considering as a case study the nonequilibrium spin-boson model, a collective coordinate is extracted from each thermal environment and added into the system to construct an enlarged system (ES). After performing additional Hamiltonian’s truncation and transformation, we obtain an effective two-level system with renormalized parameters, resulting from the strong system-bath coupling. The ES is weakly coupled to its environments, thus can be simulated using a perturbative Markovian quantum master equation approach. We compare the heat current characteristics of the effective two-state model to other techniques, and demonstrate that we properly capture strong system-bath signatures such as the turnover behavior of the heat current as a function of system-bath coupling strength. We further investigate the thermal diode effect and demonstrate that strong couplings moderately improve the rectification ratio relative to the weak coupling limit. The effective Hamiltonian method that we developed here offers fundamental insight into the strong coupling behavior, and is computationally economic. Applications of the method toward studying multi-level quantum thermal machines are anticipated.

Quantum state transmission through a spin chain in finite-temperature heat baths

Transmission of a quantum state is essential for performing quantum information processing tasks. The communication channel will be inevitably immersed in its surrounding environment under realistic conditions. In this paper, we investigate the influence of environment noise on the transmission fidelity when transferring a quantum state through a spin chain. The non-Markovian open system dynamics is systematically analyzed by using the quantum state diffusion equation method. With each spin immersed in its own finite temperature and non-Markovian heat bath, we consider three types of system–bath interaction: dephasing, dissipation and spin-boson. The transmission fidelity is found to decrease with the increasing bath temperature and system–bath coupling strength. Interestingly, we find that the bath non-Markovianity can help enhancing the transmission fidelity.

Effect of logarithmic perturbations in ohmic like spectral densities in dynamics of electronic excitation using variational polaron transformation approach

Electronic excitation energy transfer is a ubiquitous process that has generated prime research interest since its discovery. Recently developed variational polaron transformation-based second-order master equation is capable of interpolating between Förster and Redfield limits with exceptional accuracy. Forms of spectral density functions studied so far through the variational approach provide theoretical support for various experiments. Recently introduced ohmic like spectral density function that can account for logarithmic perturbations provides generality and exposition to a unique and practical set of environments. In this paper, we exploit the energy transfer dynamics of a two-level system attached to an ohmic like spectral density function with logarithmic perturbations using a variational polaron transformed master equation. Our results demonstrate that even for a relatively large bath coupling strength, quantum coherence effects can be increased by introducing logarithmic perturbations of the order of one and two in super-ohmic environments. Moreover, for particular values of the ohmicity parameter, the effect of logarithmic perturbations is observed to be insignificant for the overall dynamics. In regard to ohmic environments, as logarithmic perturbations increase, damping characteristics of the coherent transient dynamics also increase in general. It is also shown that, having logarithmic perturbations of the order of one in an ohmic environment can result in a less efficient energy transfer for relatively larger system bath coupling strengths.