Both in classical and quantum stochastic control theory a major role is played by the filtering equation, which recursively updates the information state of the system under observation. Unfortunately, the theory is plagued by infinite-dimensionality of the information state which severely limits its practical applicability, except in a few select cases (e.g. the linear Gaussian case.) One solution proposed in classical filtering theory is that of the projection filter. In this scheme, the filter is constrained to evolve in a finite-dimensional family of densities through orthogonal projection on the tangent space with respect to the Fisher metric. Here we apply this approach to the simple but highly nonlinear quantum model of optical phase bistability of a stongly coupled two-level atom in an optical cavity. We observe near-optimal performance of the quantum projection filter, demonstrating the utility of such an approach.Both in classical and quantum stochastic control theory a major role is played by the filtering equation, which recursively updates the information state of the system under observation. Unfortunately, the theory is plagued by infinite-dimensionality of the information state which severely limits its practical applicability, except in a few select cases (e.g. the linear Gaussian case.) One solution proposed in classical filtering theory is that of the projection filter. In this scheme, the filter is constrained to evolve in a finite-dimensional family of densities through orthogonal projection on the tangent space with respect to the Fisher metric. Here we apply this approach to the simple but highly nonlinear quantum model of optical phase bistability of a stongly coupled two-level atom in an optical cavity. We observe near-optimal performance of the quantum projection filter, demonstrating the utility of such an approach.

Jan 1, 200534 citations

Quantum projection filtering for open quantum systems

Qing Zhong Gao, Guofeng Zhang

This paper presents an approximation quantum projection filtering strategy, aiming to reduce the computational cost in calculating the standard quantum filter equation in time. By using a differential geometric approach, the trajectory of the resulting quantum projection filter is constrained to evolve in a finite-dimensional differentiable manifold consisting of an exponential family of quantum density operators. A convenient design of the differentiable manifold is then developed through reduction of the local approximation errors, which allows simplification of the quantum projection filter equations. Finally, simulation results from a two-level quantum system example illustrate the approximation performance of the proposed filtering scheme. The proposed approach is expected to be of practical use in developing more efficient quantum control methods.This paper presents an approximation quantum projection filtering strategy, aiming to reduce the computational cost in calculating the standard quantum filter equation in time. By using a differential geometric approach, the trajectory of the resulting quantum projection filter is constrained to evolve in a finite-dimensional differentiable manifold consisting of an exponential family of quantum density operators. A convenient design of the differentiable manifold is then developed through reduction of the local approximation errors, which allows simplification of the quantum projection filter equations. Finally, simulation results from a two-level quantum system example illustrate the approximation performance of the proposed filtering scheme. The proposed approach is expected to be of practical use in developing more efficient quantum control methods.

IEEE Conference on Decision and Control (CDC)Dec 1, 20170 citations

Design of a Quantum Projection Filter

Qing Zhong Gao, Daoyi Dong, Ian R. Petersen +1 more

This article develops a quantum projection filtering approach to approximating the quantum filter equation based on It $\hat{\rm o}$ stochastic Taylor expansions and quantum information geometric techniques. The proposed approximation scheme is designed so that the truncated Taylor expansion of the difference between the true quantum trajectory, and its approximation on a lower dimensional differential submanifold is minimized, through an orthogonal projection operation in the quantum Fisher metric. In addition, a convenient design for a special class of open quantum systems is formulated. Simulation results from a numerical example demonstrate the approximation capability of the proposed quantum projection filter.This article develops a quantum projection filtering approach to approximating the quantum filter equation based on It$\hat{\rm o}$ stochastic Taylor expansions and quantum information geometric techniques. The proposed approximation scheme is designed so that the truncated Taylor expansion of the difference between the true quantum trajectory, and its approximation on a lower dimensional differential submanifold is minimized, through an orthogonal projection operation in the quantum Fisher metric. In addition, a convenient design for a special class of open quantum systems is formulated. Simulation results from a numerical example demonstrate the approximation capability of the proposed quantum projection filter.

IEEE Transactions on Automatic ControlNov 14, 201915 citations

Derivation of Maxwell-Bloch-type equations by projection of quantum models

Hideo Mabuchi

A simple algebraic procedure is described for deriving Maxwell-Bloch-type equations from single-atom cavity quantum electrodynamics (cavity QED) master equations via orthogonal projection onto a manifold of semiclassical states. In particular, the usual Maxwell-Bloch equations are obtained---up to a state-dependent correction factor of order unity---straightforwardly from the unconditional Jaynes-Cummings master equation. The technique of projecting onto a semiclassical manifold can also be applied with conditional master equations (quantum filters), leading to stochastic simulation models that include multiplicative noise terms associated with fluctuations of the atomic dipole. The utility of such models is briefly explored in the context of single-atom absorptive bistability.A simple algebraic procedure is described for deriving Maxwell-Bloch-type equations from single-atom cavity quantum electrodynamics (cavity QED) master equations via orthogonal projection onto a manifold of semiclassical states. In particular, the usual Maxwell-Bloch equations are obtained---up to a state-dependent correction factor of order unity---straightforwardly from the unconditional Jaynes-Cummings master equation. The technique of projecting onto a semiclassical manifold can also be applied with conditional master equations (quantum filters), leading to stochastic simulation models that include multiplicative noise terms associated with fluctuations of the atomic dipole. The utility of such models is briefly explored in the context of single-atom absorptive bistability.

arXiv:0803.2887Physical Review AJul 15, 200826 citations

Quantum filter reduction for measurement-feedback control via unsupervised manifold learning

Anne E. B. Nielsen, Asa Hopkins, Hideo Mabuchi

We derive simple models for the dynamics of a single atom coupled to a cavity field mode in the absorptive bistable parameter regime by projecting the time evolution of the state of the system onto a suitably chosen nonlinear low-dimensional manifold, which is found by use of local tangent space alignment. The output field from the cavity is detected with a homodyne detector allowing observation of quantum jumps of the system between states with different average numbers of photons in the cavity. We find that the models, which are significantly faster to integrate numerically than the full stochastic master equation, largely reproduce the dynamics of the system, and we demonstrate that they are sufficiently accurate to facilitate feedback control of the state of the system based on the predictions of the models alone.We derive simple models for the dynamics of a single atom coupled to a cavity field mode in the absorptive bistable parameter regime by projecting the time evolution of the state of the system onto a suitably chosen nonlinear low-dimensional manifold, which is found by use of local tangent space alignment. The output field from the cavity is detected with a homodyne detector allowing observation of quantum jumps of the system between states with different average numbers of photons in the cavity. We find that the models, which are significantly faster to integrate numerically than the full stochastic master equation, largely reproduce the dynamics of the system, and we demonstrate that they are sufficiently accurate to facilitate feedback control of the state of the system based on the predictions of the models alone.

arXiv:0905.1139New Journal of PhysicsOct 30, 200931 citations

Regularized linearization for quantum nonlinear optical cavities: application to degenerate optical parametric oscillators

Carlos Navarrete-Benlloch, Eugenio Roldán, Yue Chang +1 more

Nonlinear optical cavities are crucial both in classical and quantum optics; in particular, nowadays optical parametric oscillators are one of the most versatile and tunable sources of coherent light, as well as the sources of the highest quality quantum-correlated light in the continuous variable regime. Being nonlinear systems, they can be driven through critical points in which a solution ceases to exist in favour of a new one, and it is close to these points where quantum correlations are the strongest. The simplest description of such systems consists in writing the quantum fields as the classical part plus some quantum fluctuations, linearizing then the dynamical equations with respect to the latter; however, such an approach breaks down close to critical points, where it provides unphysical predictions such as infinite photon numbers. On the other hand, techniques going beyond the simple linear description become too complicated especially regarding the evaluation of two-time correlators, which are of major importance to compute observables outside the cavity. In this article we provide a regularized linear description of nonlinear cavities, that is, a linearization procedure yielding physical results, taking the degenerate optical parametric oscillator as the guiding example. The method, which we call self-consistent linearization, is shown to be equivalent to a general Gaussian ansatz for the state of the system, and we compare its predictions with those obtained with available exact (or quasi-exact) methods. Apart from its operational value, we believe that our work is valuable also from a fundamental point of view, especially in connection to the question of how far linearized or Gaussian theories can be pushed to describe nonlinear dissipative systems which have access to non-Gaussian states.Nonlinear optical cavities are crucial both in classical and quantum optics; in particular, nowadays optical parametric oscillators are one of the most versatile and tunable sources of coherent light, as well as the sources of the highest quality quantum-correlated light in the continuous variable regime. Being nonlinear systems, they can be driven through critical points in which a solution ceases to exist in favour of a new one, and it is close to these points where quantum correlations are the strongest. The simplest description of such systems consists in writing the quantum fields as the classical part plus some quantum fluctuations, linearizing then the dynamical equations with respect to the latter; however, such an approach breaks down close to critical points, where it provides unphysical predictions such as infinite photon numbers. On the other hand, techniques going beyond the simple linear description become too complicated especially regarding the evaluation of two-time correlators, which are of major importance to compute observables outside the cavity. In this article we provide a regularized linear description of nonlinear cavities, that is, a linearization procedure yielding physical results, taking the degenerate optical parametric oscillator as the guiding example. The method, which we call self-consistent linearization, is shown to be equivalent to a general Gaussian ansatz for the state of the system, and we compare its predictions with those obtained with available exact (or quasi-exact) methods. Apart from its operational value, we believe that our work is valuable also from a fundamental point of view, especially in connection to the question of how far linearized or Gaussian theories can be pushed to describe nonlinear dissipative systems which have access to non-Gaussian states.

arXiv:1402.7250Optics ExpressSep 23, 201415 citations

This article develops a quantum projection filtering approach to approximating the quantum filter equation based on It

\hat{\rm o}

stochastic Taylor expansions and quantum information geometric techniques. The proposed approximation scheme is designed so that the truncated Taylor expansion of the difference between the true quantum trajectory, and its approximation on a lower dimensional differential submanifold is minimized, through an orthogonal projection operation in the quantum Fisher metric. In addition, a convenient design for a special class of open quantum systems is formulated. Simulation results from a numerical example demonstrate the approximation capability of the proposed quantum projection filter.