Reservoir for inverse-power-law decoherence of a qubit

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.

Figure 1

The coherent term $|{\rho }_{1,0}(t)|$ of the reduced density matrix of a qubit interacting with a reservoir, described either by $J_L(\omega )$, both in strong (thin solid line) and weak (dashed line) coupling regimes, or by $J(\omega )$ (thick solid line), and spectral density function, given by Eqs. (13) and (19), respectively, for $0⩽t/\tau ⩽5.9$. The values of the parameters are $A=0.8\hspace{0.5em}{a}^{5/2}$, ${\omega }_0=a/2$, $\tau \simeq 0.974/a$, ${\tau }_B=1/a$, and ${\tau }_R=1/(10\hspace{0.5em}a)$ in the strong coupling regime and ${\tau }_B=1/(20\hspace{0.5em}a)$ and ${\tau }_R=10/(13\hspace{0.5em}a)$ in the weak coupling regime. The initial conditions read ${\rho }_{1,1}(0)=1/2$, ${\rho }_{1,0}(0)=1/5$, and $C_0$ labels the cross point of the largest time coordinate between the thin and thick solid lines.

Figure 2

The relaxation of coherent term $|{\rho }_{1,0}(t)|$ over long time scales of the reduced density matrix of a qubit, interacting with a reservoir, described either by $J_L(\omega )$, both in strong (thin solid line) and weak (dashed line) coupling regimes, or by $J(\omega )$ (thick solid line), and spectral density function, given by Eqs. (15) and (19), respectively, for $3.2⩽t/\tau ⩽30$. The values of the parameters are $A=0.8\hspace{0.5em}{a}^{5/2}$, ${\omega }_0=a/2$, $\tau \simeq 0.974/a$, ${\tau }_B=1/a$, and ${\tau }_R=1/(10\hspace{0.5em}a)$ in the strong coupling regime and ${\tau }_B=1/(20\hspace{0.5em}a)$ and ${\tau }_R=10/(13\hspace{0.5em}a)$ in the weak coupling regime. The magnitude of the decoherence factor reads $\left|\mathcal{D}\right|\simeq 0.112\hspace{0.5em}{a}^{-3/2}$; the initial conditions are ${\rho }_{1,1}(0)=1/2$, ${\rho }_{1,0}(0)=1/5$; and $C_0$ labels the cross point of the largest time coordinate between the thin and thick solid lines.