On Exponential Stabilization of N-Level Quantum Angular Momentum Systems

In this paper, we consider the feedback stabilization problem for N-level quantum angular momentum systems undergoing continuous-time measurements. By using stochastic and geometric control tools, we provide sufficient conditions on the feedback control law ensuring almost sure exponential convergence to a predetermined eigenstate of the measurement operator. In order to achieve these results, we establish general features of quantum trajectories which are of interest by themselves. We illustrate the results by designing a class of feedback control laws satisfying the above-mentioned conditions and finally we demonstrate the effectiveness of our methodology through numerical simulations for three-level quantum angular momentum systems.

[1]  Ramon van Handel,et al.  Feedback control of quantum state reduction , 2005, IEEE Transactions on Automatic Control.

[2]  Viacheslav P. Belavkin,et al.  Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes , 1989 .

[3]  Jie Xiong,et al.  An Introduction to Stochastic Filtering Theory , 2008 .

[4]  Mazyar Mirrahimi,et al.  Real-time quantum feedback prepares and stabilizes photon number states , 2011, Nature.

[5]  S. L. Adler,et al.  Martingale models for quantum state reduction , 2001 .

[6]  Weichao Liang,et al.  On Exponential Stabilization of Spin $-\frac{1}{2}$ Systems , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[7]  S. Varadhan,et al.  On the Support of Diffusion Processes with Applications to the Strong Maximum Principle , 1972 .

[8]  V. P. Belavkin,et al.  A new wave equation for a continuous nondemolition measurement , 1989 .

[9]  Mazyar Mirrahimi,et al.  Stabilizing Feedback Controls for Quantum Systems , 2005, SIAM J. Control. Optim..

[10]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[11]  Kai Liu.  Stochastic Stability of Differential Equations in Abstract Spaces , 2022 .

[12]  Ramon van Handel,et al.  On the separation principle of quantum control , 2005 .

[13]  E. Dynkin.  Functionals of Markov processes , 1965 .

[14]  Mazyar Mirrahimi,et al.  Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays , 2012, Autom..

[15]  V. P. Belavkin,et al.  Quantum stochastic calculus and quantum nonlinear filtering , 1992 .

[16]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[17]  P. Rouchon,et al.  Quantum feedback by discrete quantum nondemolition measurements: Towards on-demand generation of photon-number states , 2009, 0905.0114.

[18]  Mazyar Mirrahimi,et al.  Design of strict control-Lyapunov functions for Quantum systems with QND measurements , 2011, IEEE Conference on Decision and Control and European Control Conference.

[19]  John K. Stockton,et al.  Adaptive homodyne measurement of optical phase. , 2002, Physical review letters.

[20]  S. Hara,et al.  Analysis on Behaviors of Controlled Quantum Systems via Quantum Entropy , 2008 .

[21]  Mazyar Mirrahimi,et al.  Feedback generation of quantum Fock states by discrete QND measures , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[22]  Andrew J. Landahl,et al.  Continuous quantum error correction via quantum feedback control , 2002 .

[23]  P. Protter.  Stochastic integration and differential equations , 1990 .

[24]  Pierre Rouchon,et al.  Exponential Stochastic Stabilization of a Two-Level Quantum System via Strict Lyapunov Control , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[25]  Robin L. Hudson,et al.  Quantum Ito's formula and stochastic evolutions , 1984 .

[26]  Matthew R. James,et al.  An Introduction to Quantum Filtering , 2006, SIAM Journal of Control and Optimization.

[27]  M. Sentís.  Quantum theory of open systems , 2002 .

[28]  P. Rouchon,et al.  Deterministic submanifolds and analytic solution of the quantum stochastic differential master equation describing a monitored qubit , 2016, 1603.05402.

[29]  Shinji Hara,et al.  Feedback control of quantum entanglement in a two-spin system , 2005, CDC 2005.

[30]  V. P. Belavkin,et al.  Quantum Filtering of Markov Signals with White Quantum Noise , 2005, quant-ph/0512091.

[31]  K. Tsumura.  Global stabilization at arbitrary eigenstates of N-dimensional quantum spin systems via continuous feedback , 2008, 2008 American Control Conference.

[32]  Hideo Mabuchi,et al.  Principles and applications of control in quantum systems , 2005 .

[33]  E. Davies,et al.  Quantum stochastic processes , 1969 .

[34]  X. Mao,et al.  Stochastic Differential Equations and Applications , 1998 .