Experimental realization of time-dependent phase-modulated continuous dynamical decoupling

The coherence times achieved with continuous dynamical decoupling techniques are often limited by fluctuations in the driving amplitude. In this work, we use time-dependent phase-modulated continuous driving to increase the robustness against such fluctuations in a dense ensemble of nitrogen-vacancy centers in diamond. Considering realistic experimental errors in the system, we identify the optimal modulation strength and demonstrate an improvement of an order of magnitude in the spin preservation of arbitrary states over conventional single continuous driving. The phase-modulated driving exhibits results comparable to those found with previously examined amplitude-modulated techniques and is expected to outperform them in experimental systems having higher phase accuracy. The proposed technique could open new avenues for quantum information processing and many-body physics in systems dominated by high-frequency spin-bath noise, for which pulsed dynamical decoupling is less effective.

Figure 1

(a) Conventional double driving scheme, utilizing two microwave sources: first driving with amplitude ${\mathrm{\Omega }}_1$ along the $x$ axis and second driving with amplitude ${\mathrm{\Omega }}_2$ along the $z$ axis. For ${\mathrm{\Omega }}_2<{\mathrm{\Omega }}_1$, the effective Hamiltonian (3) is reproduced, significantly reducing amplitude fluctuations of the first driving. (b) Phase-modulated double driving scheme, utilizing a single microwave source and time-dependent phase modulation $\delta \left(t\right)=2{\mathrm{\Omega }}_2\text{sinc}\left(2{\mathrm{\Omega }}_{1}t\right)$, reproducing the same effective Hamiltonian (3) (Appendix pp1).

Figure 2

Fidelity of the ensemble spin state (Appendix pp4) after performing continuous driving along the $x$ axis. (a) Conventional continuous driving with no modulation, and double driving with phase modulation strength $\alpha =0.1$, for a state initialized along the $y$ axis $(T_2$ decay with “Rabi” oscillations). (b) Double driving with different phase modulation strengths for a state initialized along the $x$ axis $(T_{1\rho }$ decay).

Figure 3

Decoherence time of the ensemble spin state after performing phase-modulated continuous driving, as a function of the phase modulation strength. The state was initialized along the $y$ axis $(T_2$ decay).

Figure 4

(a) Experimental results and (b) simulations of the relaxation time of the spin ensemble state after performing phase-modulated continuous driving as a function of the phase modulation strength. The state was initialized along the driving axis. (c) Initial fluorescence signal contrast measured in the same experiment.

Figure 5

Comparison between the fidelity of NV ensemble spin state initialized (a) along the $x$ axis and (b) along the $y$ axis for amplitude [Eq. (4)] and phase [Eq. (5)] modulations with equal strength $\alpha =0.1$. Other modulation strengths produced comparable results as well.