Coherence protection in coupled quantum systems

The interaction of a quantum system with its environment causes decoherence, setting a fundamental limit on its suitability for quantum information processing. However, we show that if the system consists of coupled parts with different internal energy scales then the interaction of one part with a thermal bath need not lead to loss of coherence from the other. Remarkably, we find that the protected part can remain coherent for longer when the coupling to the bath becomes stronger or the temperature is raised. Our theory will enable the design of decoherence-resistant hybrid quantum computers.

Figure 1

(a) A coupled nuclear-spin–electron-spin system with electron Zeeman splitting of ${\omega }_0$ and hyperfine coupling $g$. We neglect the Zeeman splitting of the nuclear spin, and only the electron interacts with a thermal reservoir. (b) The eigenenergies of the two-spin system and allowed transitions ${\omega }_{1,2}$. When the hyperfine coupling is small (c), the transitions overlap and nuclear-spin coherences can survive. Increasing the hyperfine coupling (d) resolves the transition lines and coherences are lost.

Figure 2

The magnitude of the steady-state coherence in the lower electron state, ${\rho }_{\downarrow \downarrow ,\downarrow \uparrow }$, at zero temperature as a function of the spin-spin coupling, $g$. The bath has an Ohmic spectral density peaked on resonance with the bare electron splitting ${\omega }_0$ and peak rate ${\gamma }_0$. Solid lines represent the exact solutions, while dashed lines show the solutions using the Born-Markov approximation.

Figure 3

The quasi-steady-state nuclear coherences for the upper (orange) and lower (red, long dashed) electron state, plotted as a function of inverse temperature for $g=0.1{\omega }_0$, ${\gamma }_0=0.02{\omega }_0$. Also plotted are the two characteristic decay rates, $Re\left\{{\kappa }_{\pm }\right\}$. A quasi steady state is only observed when $Re\left\{{\kappa }_{-}\right\}$ is sufficiently small. The three colored areas of background indicate three of the regimes of behavior discussed in the text: Green (far right shaded area) is low but finite temperature, in which there is a QSS; blue (middle shaded area) is at an intermediate temperature where there is no QSS; red (far left shaded area) indicates very high temperature in which a QSS returns.

Figure 4

The classical decay rate, based on the RTS model, compared with the two quantum decay rates ${\kappa }_{+}$ and ${\kappa }_{-}$ defined in Eq. (13). Good agreement between ${\kappa }_{-}$ and the classical rate ${\kappa }_{\mathrm{SC}}$ can be seen in the high-temperature limit, and the two theories also agree across a temperature range where the Markov approximation is still valid (see text for details).