  We show that Feynman's Clock construction, in which the time-evolution of a
closed quantum system is encoded as a ground state problem, can be extended to
open quantum systems. In our formalism, the ground states of an ensemble of
non-Hermitian Feynman Clock Hamiltonians yield stochastic trajectories, which
unravel the evolution of a Lindblad master equation. In this way, one can use
Feynman's Clock not only to simulate the evolution of a quantum system, but
also it's interaction with an environment such as a heat bath or measuring
apparatus. A simple numerical example of a two-level atom undergoing
spontaneous emission is presented and analyzed.
