Controlling the magnetic field sensitivity of atomic clock states by microwave dressing

We demonstrate control of the differential Zeeman shift between clock states of ultracold rubidium atoms by means of non-resonant microwave dressing. Using the dc-field dependence of the microwave detuning, we suppress the first and second order differential Zeeman shift in magnetically trapped $^{87}$Rb atoms. By dressing the state pair 5S$_{1/2} F= 1, m_F = -1$ and $F= 2, m_F = 1$, a residual frequency spread of <0.1 Hz in a range of 100 mG around a chosen magnetic offset field can be achieved. This is one order of magnitude smaller than the shift of the bare states at the magic field of the Breit-Rabi parabola. We further identify double magic points, around which the clock frequency is insensitive to fluctuations both in the magnetic field and the dressing Rabi frequency. The technique is compatible with chip-based cold atom systems and allows the creation of clock and qubit states with reduced sensitivity to magnetic field noise.