Quantum Control Theory; The basics. Naoki Yamamoto Keio Univ.

Transcription

1 Quantum ontrol Theory; The basics Naoki Yamamoto Keio Univ.

2 ontents lassical control theory Quantum control theory --- ontinuous meas.

3 1. control (1) : Various systems and purposes ignal (c-number) A M A M

4 1. control (2) : Linear feedback control --- stabilization Linear system Target value : Feedback control law : Design system so that the closed-loop is stable. (e.g.) is unstable when ; Via the FB law Thus choosing, the system becomes, the system is stabilized ;

5 1. control (3) : Optimal linear feedback control Linear system uppose the input ontrol purpose : is a function of FB control law that minimizes the above cost function is given by : Target trajectory :

6 1. control (4) : Nonlinear FB control --- Lyapunov method A nonlinear system uppose is a function of A M ontrol purpose : et a non-negative function then we have FB control law yields Thus always decreases in time. In particular, we have

7 1. control (5) : tochastic FB control tochastic system A M Before discussing how to control White noise approximation: Noise added in = Wiener increment Formally, and subjected to hence Dynamics in = stochastic differential equation (DE)

8 1. control (5) : tochastic FB control tochastic system It would be a good idea to use the estimate value of, say, and design an estimate-based FB control A M Need the conditional probability : Actually, We want to have the quantum version of this scheme.

9 2. Q control (1) : Prelimi (i) onditional prob. & Measurement lassical conditional probability : Dice as an example (i.e., ) Prob. distribution conditioned on the result of even Prob. distribution conditioned on the result of odd

10 2. Q control (1) : Prelimi (i) onditional prob. & Measurement Represent using quantum mechanics state Projection onto even Projection onto odd observable

11 2. Q control (1) : Prelimi (i) onditional prob. & Measurement tate reduction = conditional probability even odd

12 2. Q control (1) : Prelimi (ii) Generalized measurement tate preparation and interaction Projection measurement on the ancilla --- state reduction Output probability

13 2. Q control (3) : ontinuous meas. --- field as an ancilla Interaction with the vacuum field Quantum white noise field system Quantum Wiener increment Formally, and the output prob. of is Interaction Hamiltonian in

14 2. Q control (3) : ontinuous meas. --- interaction Interaction Unitary in field system interaction

15 2. Q control (3) : ontinuous meas. --- projection meas. Homodyne meas. of the field = field Projection onto the quadrature basis system (instantaneous value) (increment value) tate reduction ; The unnormalized ket vector is given by Recall:

16 2. Q control (3) : ontinuous meas. --- output probability tate reduction ; The unnormalized ket vector is: Output probability Recall: is subjected to ummary : time evolution of the unnormalized ket vector

17 2. Q control (4) : tochastic chrodinger and Master Eqs. Time evolution of the normalized ket vector = E Time evolution of the density operator = ME

18 2. Q control (5) : General framework (classical) tochastic system A M Time evolution of the conditional probability density : Via a state-dependent FB control We aim to attain a desirable state convergence:

19 2. Q control (5) : General framework (quantum) system field Time evolution of the cond. density operator: A Via a state-dependent FB control We aim to attain a desirable state convergence :

20 2. Q control (5) : General framework (, Heisenberg pic.) lassical stochastic system (DE): Time evolution of the estimate of is given by the following filtering equation (use ) A M Estimation-based FB controller can be a solution to e.g. some optimal control problem:

21 2. Q control (5) : General framework (Q, Heisenberg pic.) Quantum stochastic system (QDE): A M Time evolution of the estimate of is given by the following filtering equation (use ) Estimation-based FB controller can be a solution to e.g. some optimal control problem: