Quantum trajectories – revisiting the past

The time evolution of quantum systems is governed, on the one hand, by the system Hamiltonian and damping terms and, on the other hand, by the back action associated with the random outcome of measurements on the system. The stochastic evolution of a continuously monitored quantum system is known as a quantum trajectory [1].
After a brief tutorial on quantum trajectories, we shall discuss a new element in the theory of monitored quantum systems where random measurement outcomes do not only cause an update of the current state of the quantum system, but also of the past state, i.e., of what we (now) believe was the state of the quantum system in the past.The most complete information about the quantum state of a system at time t is formally given by the conventional density matrix and an additional matrix, that solve conditioned master equations that depend on the measurement data obtained before and after t, respectively [2]. The interpretation of this new element in quantum theory will be discussed, and its validity and potential uses will be illustrated by application to recent experiments.
[1] H. Carmichael, An Open Systems Approach to Quantum Optics, Springer, 1993; J. Dalibard, Y. Castin and K. Mølmer, Phys. Rev. Lett. 68, 580 (1992)
[2] S. Gammelmark, B. Julsgaard, and K. Mølmer, Phys. Rev. Lett. 111, 160401 (2013).