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Physical Review A

Energy range for quantum coherence

Filippo Giraldi

Published Jun 11, 2015• 3 citations
Publisher

Abstract

Decoherence-free states have been found in Markovian open system dynamics and in the framework of non-Markovian fermionic environments. In the latter scenario we determine an energy range where quantum coherence is maintained over long times for every initial coherent configuration. In particular, for a two-state fermionic system interacting with a non-Markovian fermionic environment, undamped oscillating or stable coherence between the two energy eigenstates appears if the degenerate energy level is below a critical energy. If these energies coincide, the same behavior is observed in the superohmic regime and, under special conditions, the initial coherence is approached asymptotically. Equivalently, coherence persists if a defined dressed energy of the system is less than the band-gap energy of the reservoir. Same properties are found in the framework of bosonic environments.

DOI
10.1103/physreva.91.062112

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Decoherence-free states have been found in Markovian open system dynamics and in the framework of non-Markovian fermionic environments. In the latter scenario we determine an energy range where quantum coherence is maintained over long times for every initial coherent configuration. In particular, for a two-state fermionic system interacting with a non-Markovian fermionic environment, undamped oscillating or stable coherence between the two energy eigenstates appears if the degenerate energy level is below a critical energy. If these energies coincide, the same behavior is observed in the superohmic regime and, under special conditions, the initial coherence is approached asymptotically. Equivalently, coherence persists if a defined dressed energy of the system is less than the band-gap energy of the reservoir. Same properties are found in the framework of bosonic environments.
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.
Decoherence-free subspaces (DFSs) provide a strategy for protecting the dynamics of an open system from decoherence induced by the system-environment interaction. So far, DFSs have been primarily studied in the framework of Markovian master equations. In this work, we study decoherence-free (DF) states in the general setting of a non-Markovian fermionic environment. We identify the DF states by diagonalizing the nonunitary evolution operator for a two-level fermionic system attached to an electron reservoir. By solving the exact master equation, we show that DF states can be stabilized dynamically.
Understanding the non-Markovian dynamics of dissipative processes induced by memory effects of the environment is a fundamental subject of open quantum systems. Because of the complexity of open quantum systems, e.g., the multiple energy scales involving that of the system, the environment, and their mutual coupling, it is generally a challenging task to characterize the relationship among the parameters of the system dynamics and the reservoir spectra. For the two-level spontaneous emission model within structured environments, it was shown in a recent literature (Opt. Lett. 38, 3650) that a functional relation could be established between the asymptotically non-decaying population and the spectral density of the reservoir as the system undergoes a long-time evolution. It hence renders a distinct perspective to look into the character of long-lived quantum coherence in the corresponding non-Markovian process. This article is devoted to further investigate the phenomena of limit cycle oscillations possibly occurring in such non-Markovian dissipative systems in a long-time evolution. For a two-level system subjected to an environment with Ohmic class spectra, due to the presence of a unique bound-state mode of the system, the evolution trajectory of the given initial states will converge to a limit cycle in the Bloch space. The dependence of the radius and the location of the limit cycle on the spectral density function of the reservoir are manifested by virtue of the described functional relation. For the model subjected to a photonic crystal environment with multiple bands, our studies reveal that, owing to the presence of two or more bound states, the evolution trajectory of the system will converge to a toric curve of a paraboloid in the Bloch space and the phenomena of periodic or quasi-periodic oscillations could exhibit. While the equation of the parabolic curve is fully determined by the initial values of the state vector in the Bloch space, our results reveal that the scope of the evolution trajectory inside the toric curve is related to the spectral density of the reservoir and their quantified relation is distinctly characterized. Finally, the asymptotic dynamics of the correlations of a two-qubit system is discussed when it is subjected locally to the non-Markovian dissipative process.
The quantum-to-classical transition is due to environment-induced decoherence, and it depicts how classical dynamics emerges from quantum systems. Previously, the quantum-to-classical transition has mainly been described with memory-less (Markovian) quantum processes. Here we study the complexity of the quantum-to-classical transition through general non-Markovian memory processes. That is, the influence of various reservoirs results in a given initial quantum state evolving into one of the following four scenarios: thermal state, thermal-like state, quantum steady state, or oscillating quantum nonstationary state. In the latter two scenarios, the system maintains partial or full quantum coherence due to the strong non-Markovian memory effect, so that in these cases, the quantum-to-classical transition never occurs. This unexpected new feature provides a new avenue for the development of future quantum technologies because the remaining quantum oscillations in steady states are decoherence-free.
The behavior of quantum coherence is studied under Markovian and non-Markovian dynamics for open quantum systems. For single qubit systems, we show that the coherence depending on the off-diagonal elements of the density matrix is the upper bound of the coherence depending on the relative entropy under both Markovian and non-Markovian processes. For two-qubit systems, in both Markovian and non-Markovian processes, quantum discord and coherence show less sensitivity to the initial state than quantum entanglement. We also find that the quantum discord has similar behaviors with coherence under both Markovian and non-Markovian dynamics.
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.