Spin echo is a fundamental tool for quantum registers and biomedical imaging. It is believed that a strong magnetic field is needed for the spin echo to provide long memory and high resolution, since a degenerate spin cannot be controlled or addressed under a zero magnetic field. While a degenerate spin is never subject to dynamic control, it is still subject to geometric control. Here we show the spin echo of a degenerate spin subsystem, which is geometrically controlled via a mediating state split by the crystal field, in a nitrogen vacancy centre in diamond. The demonstration reveals that the degenerate spin is protected by inherent symmetry breaking called zero-field splitting. The geometric spin echo under zero field provides an ideal way to maintain the coherence without any dynamics, thus opening the way to pseudo-static quantum random access memory and non-invasive biosensors.

Quantum bits or qubits naturally decohere by becoming entangled with uncontrollable environments. Dynamical decoupling is thereby required to disentangle qubits from an environment by periodically reversing the qubit bases, but this causes rotation error to accumulate. Whereas a conventional qubit is rotated within the SU(2) two-level system, a geometric qubit defined in the degenerate subspace of a V-shaped SU(3) three-level system is geometrically rotated via the third ancillary level to acquire a geometric phase. We here demonstrate that, simply by introducing detuning, the dynamical decoupling of the geometric qubit on a spin triplet electron in a nitrogen-vacancy center in diamond can be made to spontaneously suppress error accumulation. The geometric dynamical decoupling extends the coherence time of the geometric qubit up to 1.9 ms, limited by the relaxation time, with 128 decoupling gates at room temperature. Our technique opens a route to holonomic quantum memory for use in various quantum applications requiring sequential operations. 2 Main textQuantum information technology is becoming a reality in the form of quantum computers, simulators, sensors, as well as the repeaters required for the quantum internet. Long memory time and high-fidelity gates are the key factors that must be scaled up to finally achieve real applications.The widely used dynamical decoupling technique [1][2][3][4][5][6][7][8], which in principle extends the memory time or the coherence time, in practice faces the problem of error accumulation after a large number of decoupling gates, which eventually degrades the state fidelity. The Carr-Purcell-Meiboom-Gill (CPMG) sequence [1] has thus been developed to suppress the accumulation of gate errors, while the initial state is restricted to the eigenstate of the driving field of the decoupling gate. As alternatives, a composite pulse technique for achieving high-fidelity gates [3,4] and a specially designed gate sequence [2,5,6] have been developed to be independent of the initial state.A qubit is typically defined as being in a two-level system with an energy gap, which allows direct transition within the bases to implement dynamic quantum gates. Another type of qubit can also be defined in a two-level system without an energy gap; this type requires an indirect transition via a third ancillary level, and thus constitutes a V-shaped three-level system to implement geometric quantum gates [9][10][11][12]. Geometric quantum gates can be either adiabatic [13][14][15] or non-adiabatic [9][10][11][12][16][17][18][19]. In contrast to the adiabatic geometric gate, the non-adiabatic geometric gate enables faster gate operation to reduce the influence of the environmental noise, thereby resulting in high fidelity.Moreover, the degenerate two-level system is independent of the global phase of the driving field, as seen in the polarization or time-bin encoding of a photon, enabling post selection of successful operations to exclude a population loss from the qubit space spanned by the degenerate...