  A dynamical quantum phase transition can occur during time evolution of
sudden quenched quantum systems across a phase transition. It corresponds to
the nonanalytic behavior at a critical time of the rate function of the quantum
state return amplitude, analogous to nonanalyticity of the free energy density
at the critical temperature in macroscopic systems. A variety of many-body
systems can be represented in momentum space as a spin-1/2 state evolving on
the Bloch sphere, where each momentum mode is decoupled and thus can be
simulated independently by a single qubit. Here, we report the observation of a
dynamical quantum phase transition in a superconducting qubit simulation of the
quantum quench dynamics of many-body systems. We take the Ising model with a
transverse field as an example for demonstration. In our experiment, the spin
state, which is initially polarized longitudinally, evolves based on a
Hamiltonian with adjustable parameters depending on the momentum and strength
of the transverse magnetic field. The time evolving quantum state is read out
by state tomography. Evidence of dynamical quantum phase transitions, such as
paths of time evolution states on the Bloch sphere, non-analytic behavior of
the dynamical free energy and the emergence of Skyrmion lattice in
momentum-time space, is observed. The experimental data agrees well with
theoretical and numerical calculations. The experiment demonstrates for the
first time explicitly the topological invariant, both topologically trivial and
non-trivial, for dynamical quantum phase transitions. Our results show that the
quantum phase transitions of this class of many-body systems can be simulated
successfully with a single qubit by varying certain control parameters over the
corresponding momentum range.
