The decoherence of a qubit is studied for a general class of spectral densities with an arbitrary band gap. The exact dynamics reveals the existence of critical values of the qubit transition frequency, such that relaxations of coherence show discontinuities, jumping from decays with powers arbitrarily larger than unity to powers arbitrarily larger than 0. The critical frequency is the sum of the band-gap frequency and the first negative moment of the spectral density. In the superohmic case, coherence is not entirely lost and the ratio of the trapped population approaches unity as the second negative moment of the spectral density becomes negligible with respect to the scale frequency. Arbitrarily slow decoherence processes are obtained in the critical configuration by approaching the boundary between subohmic and superohmic regimes. The setup of critical configurations constitutes a different way to control qubit decoherence.
The decoherence of a qubit is studied for a general class of spectral densities with an arbitrary band gap. The exact dynamics reveals the existence of critical values of the qubit transition frequency, such that relaxations of coherence show discontinuities, jumping from decays with powers arbitrarily larger than unity to powers arbitrarily larger than 0. The critical frequency is the sum of the band-gap frequency and the first negative moment of the spectral density. In the superohmic case, coherence is not entirely lost and the ratio of the trapped population approaches unity as the second negative moment of the spectral density becomes negligible with respect to the scale frequency. Arbitrarily slow decoherence processes are obtained in the critical configuration by approaching the boundary between subohmic and superohmic regimes. The setup of critical configurations constitutes a different way to control qubit decoherence.
The dissipative effects of a quantum harmonic oscillator, initially set in a coherent state and linearly coupled to a continuous distribution of frequency modes, are analyzed over long time scales in relation to the behavior of the spectral density near an arbitrary band gap, arbitrarily shaped at the higher frequencies. The reservoir is initially set either in the vacuum state or in continuous distributions of coherent states. These distributions are arbitrarily shaped at high frequencies and structured in sub- or super-ohmic configurations near an arbitrary band gap frequency. Similarly to certain decoherence processes of a qubit, critical conditions emerge, such that arbitrarily slow inverse power law relaxations of the expectation values of the observables, are obtained by approaching the boundary between the sub- and the super-ohmic regimes. Also, in such critical conditions, a trapping of the number of excitations appears in the super-ohmic regime. The technique of critical frequency control, emerging in the scenario of the environment-induced decoherence of a qubit via the reservoir engineering approach, is extended to the harmonic quantum Brownian motion.
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.
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.
We study decoherence effects in qubits coupled to environments that exhibit resonant frequencies in their spectral function. We model the coupling of the qubit to its environment via the Caldeira-Leggett formulation of quantum dissipation/coherence, and study the simplest example of decoherence effects in circuits with resonances such as a dc SQUID phase qubit in the presence of an isolation circuit, which is designed to enhance the coherence time. We emphasize that the spectral density of the environment is strongly dependent on the circuit design, and can be engineered to produce longer decoherence times. We begin with a general discussion of superconducting qubits such as the flux qubit, the Cooper pair box and the phase qubit and show that in these kinds of systems appropriate circuit design can greatly modify the spectral density of the environment and lead to enhancement of decoherence times. In the particular case of the phase qubit, for instance, we show that when the frequency of the qubit is at least two times larger than the resonance frequency of the environmental spectral density, the decoherence time of the qubit is a few orders of magnitude larger than that of the typical ohmic regime, where the frequency of the qubit is much smaller than the resonance frequency of the spectral density. In addition, we demonstrate that the environment does not only affect the decoherence time, but also the frequency of the transition itself, which is shifted from its environment-free value. Second, we show that when the qubit frequency is nearly the same as the resonant frequency of the environmental spectral density, an oscillatory non-Markovian decay emerges, as the qubit and its environment self-generate Rabi oscillations of characteristic time scales shorter than the decoherence time.
Anomalous decoherence in the Jaynes-Cummings model emerges for a certain class of bosonic reservoirs, described by spectral densities with a band gap edge frequency coinciding with the qubit transition frequency. The special reservoirs are piecewise similar to those usually adopted in quantum optics, i.e., sub-ohmic at low frequencies and inverse power laws at high frequencies. The exact dynamics of the qubit is described analytically through Fox H-functions. Over estimated long time scales, decoherence results in inverse power laws with powers decreasing continuously to unity, according to the particular choice of the special reservoir. The engineering reservoir approach is a new way of strongly delaying the decoherence process with possible applications to quantum technologies, due to the simple form of the designed reservoirs.
The exact dynamics of a Jaynes-Cummings model for a qubit interacting with a continuous distribution of bosons, characterized by a special form of the spectral density, is evaluated analytically. The special reservoir is designed to induce anomalous decoherence, resulting in an inverse power-law relaxation, of power 3/2, over an evaluated long time scale. If compared to the exponential-like relaxation obtained from the original Jaynes-Cummings model for Lorentzian-type spectral density functions, decoherence is strongly suppressed. The special reservoir exhibits a band-edge frequency coinciding with the qubit transition frequency. Known theoretical models of photonic band-gap media suitable for the realization of the designed reservoir are proposed.