Evolution of quantum superpositions in open environments: Quantum trajectories, jumps, and localization in phase space

The decay of coherence when a quantum system interacts with a much larger environment is usually described by a master equation for the system reduced density matrix and emphasizes the evolution of an entire ensemble. We consider two methods that have been developed recently to simulate the evolution of single realizations. Quantum-state diffusion involves both diffusion, where the individual quantum trajectory fluctuates through a Wiener process deriving from the environment, and localization to a coherent state, an eigenstate of the relevant Lindblad operator describing the coupling of the system to the environment. We demonstrate the localization process for different initial states and utilize the Wigner function to depict this localization in phase space. We concentrate on quantum states that can be expressed as a superposition of appropriate coherent states. For an initial superposition of two coherent states (a Schrödinger ``cat''), one of the two components will dominate the evolution. For initial Fock states, which can be described as a continuous superposition of coherent states on a ring, localization takes place when one coherent state is selected from that ring where each component has nearly the same energy as the original Fock state. We also consider the localization from a nonclassical squeezed ground state, which can be expressed as a superposition of coherent states along a line in phase space. The second simulation method considered is the state vector Monte Carlo, or ``quantum jump,'' approach, which relates to the direct counting of decay quanta. In the case of an initial Schrödinger ``cat,'' we find that when no quantum is detected the ``cat'' shrinks, but when a quantum is detected, the Schrödinger ``cat'' ``jumps'' from one type of ``cat'' to another with different internal phase. For an initial squeezed state we show how quantum jumps lead to individual realizations which are superpositions of two squeezed states.