he geometric phase acquired by one system can be employed to manipulate the quantum state of the others [@Zheng2004Unconventional; @Pechal2012Geometric]. The geometric phase of a harmonic vibrational mode of trapped ions has been utilized for implementing high-fidelity entangling gates for the ionic qubits [@Leibfried2003Experimental]. In a recent experiment [@Song2017Geometric], the geometric phase of a continuous-variable field mode was observed through Ramsey interference and used for realizing controlled phase gates with up to four qubits in a superconducting circuit. On the other hand, it has been shown that the geometric phase of a superconducting qubit can be used for realizing Selective Number-dependent Arbitrary Phase (SNAP) gates on a cavity [@Krastanov2015]. This kind of gates has been experimentally demonstrated and used to produce a single-photon state [@Heeres2015]. Recently, a quantum controlled-NOT (CNOT) gate between two cavity systems has been demonstrated by use of both the dynamical and AA phases produced by controllably coupling these cavities to a qubit  [@Rosenblum2018]. This gate requires the logic states of the control qubit to be respectively encoded on the vacuum state and a nonzero photon-number state, which renders it incompatible with quantum error correction schemes; on the occurrence of single-photon loss the control qubit will collapse to a Fock state, leading to complete loss of the stored information.

![Geometric manipulation of a photonic cat state. (a) Schematic of the nonadiabatic AA phase of a qubit. Two successive $\pi$ rotations of the qubit produce a geometric phase $\gamma = \pi + \varphi$, where $\varphi$ is the angle between the two rotation axes. (b) Experimental sequence to manipulate the cat state. A cavity is dispersively coupled to the qubit and initialized in a cat state $\left({\ensuremath{\left|0\right\rangle}}+{\ensuremath{\left|2\sqrt{2}\right\rangle}}\right)/\sqrt{2}$ with the help of an ancillary qubit $Q_2$. The AA phase produced by the rotations of