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arXiv:1203.3831 · Physical Review A

Optimal feedback control of linear quantum systems in the presence of thermal noise

Marco G. Genoni, Stefano Mancini, Alessio Serafini

Published Apr 26, 2013• 37 citations
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Abstract

We study the possibility of taking bosonic systems subject to quadratic Hamiltonians and a noisy thermal environment to nonclassical stationary states by feedback loops based on weak measurements and conditioned linear driving. We derive general analytical upper bounds for the single-mode squeezing and multimode entanglement at steady state, depending only on the Hamiltonian parameters and on the number of thermal excitations of the bath. Our findings show that, rather surprisingly, larger number of thermal excitations in the bath allow for larger steady-state squeezing and entanglement if the efficiency of the optimal continuous measurements conditioning the feedback loop is high enough. We also consider the performance of feedback strategies based on homodyne detection and show that, at variance with the optimal measurements, it degrades with increasing temperature.

DOI
10.1103/physreva.87.042333

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We study the possibility of taking bosonic systems subject to quadratic Hamiltonians and a noisy thermal environment to nonclassical stationary states by feedback loops based on weak measurements and conditioned linear driving. We derive general analytical upper bounds for the single-mode squeezing and multimode entanglement at steady state, depending only on the Hamiltonian parameters and on the number of thermal excitations of the bath. Our findings show that, rather surprisingly, larger number of thermal excitations in the bath allow for larger steady-state squeezing and entanglement if the efficiency of the optimal continuous measurements conditioning the feedback loop is high enough. We also consider the performance of feedback strategies based on homodyne detection and show that, at variance with the optimal measurements, it degrades with increasing temperature.
We develop a general framework to describe interferometric coherent feedback loops and prove that, under any such scheme, the steady-state squeezing of a bosonic mode subject to a rotating wave coupling with a white noise environment and to any quadratic Hamiltonian must abide by a noise-dependent bound that reduces to the 3dB limit at zero temperature. Such a finding is contrasted, at fixed dynamical parameters, with the performance of homodyne continuous monitoring of the output modes. The latter allows one to beat coherent feedback and the 3dB limit under certain dynamical conditions, which will be determined exactly.
It has recently been shown that finding the optimal measurement on the environment for stationary linear quadratic Gaussian control problems is a semidefinite program. We apply this technique to the control of the Einstein-Podolsky-Rosen correlations between two bosonic modes interacting via a parametric Hamiltonian at steady state. The optimal measurement turns out to be nonlocal homodyne measurement---the outputs of the two modes must be combined before measurement. We also find the optimal local measurement and control technique. This gives the same degree of entanglement but a higher degree of purity than the local technique previously considered [S. Mancini, Phys. Rev. A 73, 010304(R) (2006)].
We determine a general upper bound for the steady-state entanglement achievable by continuous feedback for a system of any number of bosonic degrees of freedom. We apply such a bound to the specific case of parametric interactions--the most common practical way to generate entanglement in quantum optics--and single out optimal feedback strategies that achieve the maximal entanglement. We also consider the case of feedback schemes entirely restricted to local operations and compare their performance to the optimal, generally nonlocal, schemes.
We consider the problem of frequency estimation for a single bosonic field evolving under a squeezing Hamiltonian and continuously monitored via homodyne detection.In particular, we exploit reinforcement learning techniques to devise feedback control strategies achieving increased estimation precision.We show that the feedback control determined by the neural network greatly surpasses in the long-time limit the performances of both the "no-control" strategy and the standard "open-loop control" strategy, which we considered as benchmarks.We indeed observe how the devised strategy is able to optimize the nontrivial estimation problem by preparing a large fraction of trajectories corresponding to more sensitive quantum conditional states.
We develop a feedback strategy based on optimal quantum feedback control for Gaussian systems to maximize the likelihood of steady-state entanglement detection between two directly interacting masses. We employ linear quadratic Gaussian control to engineer the phase-space dynamics of the two masses and propose Einstein-Podolsky-Rosen-type variance minimization constraints for the feedback to facilitate unconditional entanglement generation. This scheme allows for stationary entanglement in parameter regimes where strategies based on total energy minimization () would fail.
Abstract The relaxation of a system to a steady state is a central point of interest in many attempts to advance control over the quantum world. In this paper, we consider control through instantaneous Gaussian unitary operations on the ubiquitous lossy channel, and find locally optimal conditions for the cooling and heating of a multimode Gaussian state subject to losses and possibly thermal noise. This is done by isolating the parameters that encode entropy and temperature and by deriving an equation for their evolution. This equation is in such a form that it grants clear insight into how relaxation may be helped by instantaneous quantum control. It is thus shown that squeezing is a crucial element in optimising the rate of change of entropic properties under these channels. Exact relaxation times for heating and cooling are derived, up to an arbitrarily small distance from the fixed point of the lossy channel with locally optimal strategies.