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.
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.
The flow of quantum information in local dephasing channels is analyzed over short and long times in case the structured reservoir of frequency modes exhibits a spectral gap in the density of modes over low frequencies. The presence of the low-frequency gap with upper cut-off frequency [Formula: see text] produces an infinite sequence of long-time intervals over which information backflow appears. Though generally irregular, the time intervals exhibit, under certain conditions, upper bounds: the [Formula: see text]th episode of information backflow has already started at the instant [Formula: see text], and already ended at the instant [Formula: see text], for every [Formula: see text]. The intervals become regular over long times, tend to the asymptotic length [Formula: see text], as the supremum value, and are described analytically in terms of the structure of the spectral density near the upper cut-off frequency of the spectral gap. Consequently, engineering structured reservoirs of frequency modes with a spectral gap over low frequencies produces in local dephasing channels regular sequences of information backflow and recoherence over long times, along with non-Markovian evolution.
Consider local dephasing processes of a qubit that interacts with a structured reservoir of frequency modes or a thermal bath, with Ohmic-like spectral density (SD). It is known that non-Markovian evolution appears uniquely above a temperature-dependent critical value of the Ohmicity parameter and non-Markovianity can be induced by properly engineering the external environment. In the same scenario, we find regular patterns in the flow of quantum information: Alternate directions appear in correspondence with periodic intervals of the Ohmicity parameter ${\ensuremath{\alpha}}_{0}$. The information flows back into the system over long times at zero temperature for $2+4n<{\ensuremath{\alpha}}_{0}<4+4n$, where n=0,1,2,..., and at nonvanishing temperatures for $3+4n<{\ensuremath{\alpha}}_{0}<5+4n$. Under special conditions, backflow of information appears also for nonvanishing, even natural values of the Ohmicity parameter, at zero temperature, and for odd natural values at nonvanishing temperatures. Otherwise, the long-time information flows into the environment. In the transition from vanishing to arbitrary nonvanishing temperature, the long-time backflow of information is stable for $3+4n<{\ensuremath{\alpha}}_{0}<4+4n$, while it is reversed for $2+4n<{\ensuremath{\alpha}}_{0}<3+4n$ and $4+4n<{\ensuremath{\alpha}}_{0}<5+4n$. The patterns in the information flow are not altered if the low-frequency Ohmic-like profiles of the SDs are perturbed with additional factors that consist in arbitrary powers of logarithmic forms. Consequently, the flow of information can be controlled, directed, and reversed over long times by engineering a wide variety of reservoirs that includes and continuously departs from the Ohmic-like structure at low frequencies. Non-Markovianity and recoherence appear according to the same rules along with the backflow of information.
The evolution of quantum coherence in a qubit is analyzed, over short and long times, in case the qubit interacts with a bosonic environment. The distribution of the frequency modes exhibits a low-frequency band gap. The initial conditions under study of the qubit and the bosonic environment are factorized or prepared with selective or nonselective measurements. The modulus of the coherence term of the reduced density matrix of the qubit evolves algebraically over short times and, due to the band gap, tends to the nonvanishing asymptotic value with damped oscillations. Depending on the measurement scheme, the nonselective preparation measurement can enhance quantum coherence or accelerate the decoherence process, over short times, according to dominant linear laws. Over long times, the nonselective preparation measurement produces the slowest-decaying envelope of the damped oscillations among the decoherence processes under study. Similar short- and long-time evolution is exhibited by the trace distance measure of two general states of the qubit.
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.
We investigate the dynamical quantum Fisher information of a two-level system coupled to a bosonic environment, focusing on the estimation of the qubit gap. We combine analytical calculations with numerically controlled matrix-product-state simulations. In the exactly solvable pure-dephasing Ohmic regime at zero temperature, the long-time quantum Fisher information displays a coupling-dependent algebraic behavior, leading to a dynamical crossover: it grows without bound at weak coupling, approaches a finite asymptotic value at the crossover coupling, and vanishes at strong coupling. At finite temperature, thermal fluctuations suppress the long-time growth and generate a finite-time maximum, whose dependence on the dephasing coupling retains a clear signa- ture of the zero-temperature crossover. We show that this crossover is absent at zero temperature in non-Ohmic baths: the long-time quantum Fisher information vanishes in the sub-Ohmic and diverges in the super-Ohmic regimes. At non-zero temperature instead, the crossover appears for a super-Ohmic quadratic bath and the crossover coupling becomes temperature-dependent. Moreover, in the zero-temperature Ohmic regime, we introduce an additional amplitude-damping system-bath coupling that induces energy relaxation. This relaxation channel replaces the unbounded long-time growth with a finite asymptotic quantum Fisher information associated with the reduced interacting ground state, while a signature of the pure-dephasing crossover persists in the early-time dynamics. These results establish a direct connection between the low-frequency structure of the bath and the asymptotic metrological behavior of dynamical gap sensing, and show how thermal fluctuations and energy relaxation regularize the ideal pure-dephasing dynamical crossover of the quantum Fisher information.