We study the bath correlation functions (BCFs) of open quantum systems interacting with thermal baths, in case the spectral densities (SDs) exhibit removable logarithmic singularities at low frequencies and are arbitrarily shaped at higher frequencies. The singularities consist in arbitrarily positive or negative powers of logarithmic functions, as additional factors for the power laws of the Ohmic-like SDs. If the SD vanishes sufficiently fast at high frequencies the short time behavior of the BCF is algebraic. The long time behavior of the BCF exhibits a variety of relaxations that involve inverse power laws and arbitrary powers of logarithmic forms. The imaginary part of the BCF shows over long times regular dependence on the low frequency structure of the SD, except for certain conditions where the ohmicity parameter takes odd natural values. Same dependence holds for the real part of the BCF at non-vanishing temperatures. At zero temperature the real part of the BCF exhibits over long times the same regular relationship with the low frequency structure of the SD, except for certain conditions involving even natural values of the ohmicity parameter. The exceptional conditions provide relaxations that are faster than those obtained via the regular dependence. In this way, various long time relaxations of the BCF that are slower than exponential decays and arbitrarily faster or slower than inverse power laws, can be interpreted in terms of removable logarithmic singularities in the low frequency structure SD of an open quantum system.
We study the bath correlation functions (BCFs) of open quantum systems interacting with thermal baths, in case the spectral densities (SDs) exhibit removable logarithmic singularities at low frequencies and are arbitrarily shaped at higher frequencies. The singularities consist in arbitrarily positive or negative powers of logarithmic functions, as additional factors for the power laws of the Ohmic-like SDs. If the SD vanishes sufficiently fast at high frequencies the short time behavior of the BCF is algebraic. The long time behavior of the BCF exhibits a variety of relaxations that involve inverse power laws and arbitrary powers of logarithmic forms. The imaginary part of the BCF shows over long times regular dependence on the low frequency structure of the SD, except for certain conditions where the ohmicity parameter takes odd natural values. Same dependence holds for the real part of the BCF at non-vanishing temperatures. At zero temperature the real part of the BCF exhibits over long times the same regular relationship with the low frequency structure of the SD, except for certain conditions involving even natural values of the ohmicity parameter. The exceptional conditions provide relaxations that are faster than those obtained via the regular dependence. In this way, various long time relaxations of the BCF that are slower than exponential decays and arbitrarily faster or slower than inverse power laws, can be interpreted in terms of removable logarithmic singularities in the low frequency structure SD of an open quantum system.
We analyze the short- and long-time behavior of the bath correlation function for open quantum systems interacting with thermal baths. The spectral densities under study are ohmic-like at low frequencies, exhibit possible perturbations of the low-frequency power-law profiles and are arbitrarily shaped at higher frequencies. The low-frequency perturbations are provided by arbitrarily positive or negative powers of logarithmic forms as additional factors for the power laws of the ohmic-like spectral densities. If the spectral density decays sufficiently fast at high frequencies the short-time behavior of the bath correlation function is algebraic. The model provides long-time relaxations of the bath correlation function which are arbitrarily faster or slower than inverse power laws. In fact, the long-time relaxations are described by inverse power laws and by arbitrary powers of logarithmic forms as additional factors. Such relaxations are regularly related to the low-frequency structure of the spectral density, except for even or odd natural values of the ohmicity parameter. In these exceptional conditions the long-time relaxations of the bath correlation function are faster than those obtained via the regular relation.
Gaussian baths are widely used to model non-Markovian environments, yet the cost of accurate simulation at long times remains poorly understood, especially when spectral densities exhibit nonanalytic behavior as in a range of realistic models. We rigorously bound the complexity of representing bath correlation functions on a time interval [0,T] by sums of complex exponentials, as employed in recent variants of pseudomode and hierarchical equations of motion methods. These bounds make explicit the dependence on the maximal simulation time T, inverse temperature β, and the type and strength of singularities in an effective spectral density. For a broad class of spectral densities, the required number of exponentials is bounded independently of T, achieving time-uniform complexity. The T-dependence emerges only as polylogarithmic factors for spectral densities with strong singularities, such as step discontinuities and inverse power-law divergences. The temperature dependence is mild for bosonic environments and disappears entirely for fermionic environments. Thus, the true bottleneck for long-time simulation is not the simulation duration itself, but rather the presence of sharp nonanalytic features in the bath spectrum. Our results are instructive both for long-time simulation of non-Markovian open quantum systems, as well as for Markovian embeddings of classical generalized Langevin equations with memory kernels.
The loss of coherence in a general open quantum system interacting with a bosonic environment is analyzed. The reservoir is initially in a thermal state. The reduced dynamics is described by a non-Markovian time-local master equation. We consider spectral densities that are sub- or super-Ohmic at low frequencies and arbitrarily shaped at high frequencies. In the super-Ohmic regime, for noninteger frequency powers larger than 2, long time survival of coherence appears. In the latter regime, at vanishing temperature, the asymptotic amount of surviving coherence is stabilized to its initial value, up to a phase factor, by properly increasing the bandwidth and decreasing the low-frequency profile of the spectral density. For noninteger positive frequency powers less than 2, stretched exponential-like decoherence is found over long times. The relaxations to the asymptotic configurations become arbitrarily slow by approaching the frequency power 2 of the super-Ohmic regime. The same dependence on temperature, spectral density, and scale frequency appears for purity and concurrence of two qubits and coherence of a qubit.
This study investigates the decomposition of bath correlation functions (BCFs) in terms of complex exponential functions, with an eye on the realistic modeling of open quantum systems based on the hierarchical equations of motion. We introduce the theoretical background of various decomposition schemes in both time and frequency domains and assess their efficiency and accuracy by demonstrating the decomposition of various BCFs. We further develop a new procedure for the decomposition of BCFs originating from highly structured spectral densities with a high accuracy and compare it with existing fitting techniques. Advantages and disadvantages of each methodology are discussed in detail with special attention to their application to the corresponding quantum dynamical problem. This work provides fundamental tools for choosing and using a variety of decomposition techniques of BCFs for the study of open quantum systems in structured environments.
The energy of the bosonic bath and the flow of quantum information in local dephasing channels are studied over short and long times in case the distribution of frequency modes of the bosonic bath exhibits a low-frequency gap. The initial conditions consist in special correlations between the qubit and the bosonic bath or are factorized, and involve thermal states of the whole system or of the bath. The low-frequency gap generates damped oscillations of the bath energy around the asymptotic value, for the correlated initial conditions, and induces the open system to alternately loose and gain information, for the factorized initial configurations. The long-time oscillations of the bath energy become regular and the frequency of the oscillations coincides with the upper cut-off frequency of the spectral gap. Regular long-time sequences of intervals are found over which the bath energy increases (decreases), for the correlated initial conditions, and information is lost (gained) by the open system, for the factorized initial configurations, even at different temperatures. This relation is reversed, if compared to the one obtained without the low-frequency gap, and can fail if the spectral density is tailored near the spectral gap according to power laws with odd natural powers.
Variations of the bath energy are compared with the information flow in local dephasing channels. Special correlated initial conditions are prepared from the thermal equilibrium of the whole system, by performing a selective measurement on the qubit. The spectral densities under study are ohmic-like at low frequencies and include logarithmic perturbations of the power-law profiles. The bath and the correlation energy alternately increase or decrease, monotonically, over long times, according to the value of the ohmicity parameter, following logarithmic and power laws. Consider initial conditions such that the environment is in a thermal state, factorized from the state of the qubit. In the superohmic regime the long-time features of the information flow are transferred to the bath and correlation energy, by changing the initial condition from the factorized to the specially correlated, even with different temperatures. In fact, the low-frequency structures of the spectral density that provide information backflow with the factorized initial condition, induce increasing (decreasing) bath (correlation) energy with the specially correlated initial configuration. By performing the same change of initial conditions, the spectral properties providing information loss, produce decrease (increase) of the bath (correlation) energy.