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arXiv:1706.07967 · Physical Review A

Quantum trajectories for a system interacting with environment in a single-photon state: Counting and diffusive processes

Anita Magdalena Dąbrowska, Gniewomir Sarbicki, Dariusz Chruściński

Published Nov 8, 2017• 30 citations
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Abstract

We derived quantum trajectories for a system interacting with the environment prepared in a continuous mode single-photon state as the limit of a discrete filtering model with an environment defined as a series of independent qubits prepared initially in the entangled state being an analog of a continuous mode state. The environment qubits interact with the quantum system and they are subsequently measured. The initial correlation between the bath qubits is the source of the non-Markovianity. The conditional evolutions of the quantum system for the limit of the continuous in time observations together with the formulas for the photon counting probabilities are given.

DOI
10.1103/physreva.96.053819

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We derived quantum trajectories for a system interacting with the environment prepared in a continuous mode single-photon state as the limit of a discrete filtering model with an environment defined as a series of independent qubits prepared initially in the entangled state being an analog of a continuous mode state. The environment qubits interact with the quantum system and they are subsequently measured. The initial correlation between the bath qubits is the source of the non-Markovianity. The conditional evolutions of the quantum system for the limit of the continuous in time observations together with the formulas for the photon counting probabilities are given.
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We study the dynamics of quantum systems interacting with a stream of entangled-qubit pairs in the weak coupling limit. For a large class of two-qubit bath states, we present a detailed framework describing conditional dynamical maps for the system, known as quantum trajectories, that arise when the qubits are measured. Depending on the measurement basis, these quantum trajectories can be jump-type or diffusive-type, and can successively transfer entanglement from the bath qubits other quantum systems. They also exhibit features not present in standard quantum optical trajectories due to the fact that collisional models are not confined by the standard white-noise limit for optical-mode baths. As an example of this formalism, we consider the case of two remote two-level systems, where jump-type quantum trajectories herald the birth and death of entanglement.
The excitation of atomic and molecular systems by propagating light in a two-photon state within the Wigner-Weisskopf approximation has been described using stochastic tools. The problem of a stochastic evolution of the quantum system, depending on the results of the measurement of the output field, was formulated and solved making use of the model of repeated interactions and measurement. We defined the discrete in-time interaction between the quantum system and its environment being the electromagnetic field approximated by a chain or chains of harmonic oscillators. We determined analytical formulae for quantum trajectories associated with one-dimensional and two-dimensional counting processes, corresponding respectively to unidirectional or bidirectional input field prepared in the two-photon states. We derived the formulae for the exclusive probability densities of photon counts that allow us to completely characterize the photon statistics of the output field. Finally, we showed how to apply the quantum trajectories to obtain the formula for the probability of the two-photon absorption for a three-level atom in a ladder configuration. The paper also includes a discussion on the optimal two-photon state that maximizes the two-photon absorption probability.