Solving quantum trajectories for systems with linear Heisenberg-picture dynamics and Gaussian measurement noise

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Author(s)
Warszawski, Prahlad
Wiseman, Howard M
Doherty, Andrew C
Griffith University Author(s)
Year published
2020
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We study solutions to the quantum trajectory evolution of N-mode open quantum systems possessing a time-independent Hamiltonian, linear Heisenberg-picture dynamics, and Gaussian measurement noise. In terms of the mode annihilation and creation operators, a system will have linear Heisenberg-picture dynamics under two conditions. First, the Hamiltonian must be quadratic. Second, the Lindblad operators describing the coupling to the environment (including those corresponding to the measurement) must be linear. In cases where we can solve the 2N-degree polynomials that arise in our calculations, we provide an analytical solution ...
View more >We study solutions to the quantum trajectory evolution of N-mode open quantum systems possessing a time-independent Hamiltonian, linear Heisenberg-picture dynamics, and Gaussian measurement noise. In terms of the mode annihilation and creation operators, a system will have linear Heisenberg-picture dynamics under two conditions. First, the Hamiltonian must be quadratic. Second, the Lindblad operators describing the coupling to the environment (including those corresponding to the measurement) must be linear. In cases where we can solve the 2N-degree polynomials that arise in our calculations, we provide an analytical solution for initial states that are arbitrary (i.e., they are not required to have Gaussian Wigner functions). The solution takes the form of an evolution operator, with the measurement-result dependence captured in 2N stochastic integrals over these classical random signals. The solutions also allow the positive operator-valued measure (POVM), which generates the probabilities of obtaining measurement outcomes, to be determined. To illustrate our results, we solve some single-mode example systems, with the POVMs being of practical relevance to the inference of an initial state, via quantum state tomography. Our key tool is the representation of mixed states of quantum mechanical oscillators as state vectors rather than state matrices (albeit in a larger Hilbert space). Together with methods from Lie algebra, this allows a more straightforward manipulation of the exponential operators comprising the system evolution than is possible in the original Hilbert space.
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View more >We study solutions to the quantum trajectory evolution of N-mode open quantum systems possessing a time-independent Hamiltonian, linear Heisenberg-picture dynamics, and Gaussian measurement noise. In terms of the mode annihilation and creation operators, a system will have linear Heisenberg-picture dynamics under two conditions. First, the Hamiltonian must be quadratic. Second, the Lindblad operators describing the coupling to the environment (including those corresponding to the measurement) must be linear. In cases where we can solve the 2N-degree polynomials that arise in our calculations, we provide an analytical solution for initial states that are arbitrary (i.e., they are not required to have Gaussian Wigner functions). The solution takes the form of an evolution operator, with the measurement-result dependence captured in 2N stochastic integrals over these classical random signals. The solutions also allow the positive operator-valued measure (POVM), which generates the probabilities of obtaining measurement outcomes, to be determined. To illustrate our results, we solve some single-mode example systems, with the POVMs being of practical relevance to the inference of an initial state, via quantum state tomography. Our key tool is the representation of mixed states of quantum mechanical oscillators as state vectors rather than state matrices (albeit in a larger Hilbert space). Together with methods from Lie algebra, this allows a more straightforward manipulation of the exponential operators comprising the system evolution than is possible in the original Hilbert space.
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Journal Title
Physical Review A
Volume
102
Issue
4
Copyright Statement
© 2020 American Physical Society. This is the author-manuscript version of this paper. Reproduced in accordance with the copyright policy of the publisher. Please refer to the journal's website for access to the definitive, published version.
Subject
Physical Sciences
Science & Technology
Optics
Physics, Atomic, Molecular & Chemical