Quantum gravity with photon interactions, cavity resonators with inhomogeneities, classical and quantum optimal control of fields
Harish Parthasarathy
We wish j t ( X ) to track the noiseless trajectory X d ( t )∈ L ( h ). Assume non-demoltion measurements Y o ( t )= U ( t )* Y i ( t ) U ( t ) are made and the Belavkin filter for π t ( X ) = j t ( X ) | η t ) , η t = σ ( Y 0 ( s ) : s ≤ t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1006.tif"/> has been constructed as d π t ( X ) = F t ( X ) d t + G t ( X ) d Y o ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1007.tif"/> trajectory estimation error X d ( t )− π t ( X ) is given as feedback to the state equations in the form d j t ( X ) = j t ( θ 0 ( X ) ) d t + j t ( θ 1 ( X ) ) d A ( t ) + j t ( θ 2 ( X ) ) d A ( t ) * + j t ( θ 3 ( X ) ) d Λ ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1008.tif"/> 192 + K ( t ) ( X d ( t ) − π t ( X ) ) d t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1009.tif"/> Alternately, if we assume that X d follows a noiseless trajectory, ie, it evolves according to the equation X d ( t ) , = j t ( 0 ) ( X d ) = U 0 ( t ) * X d U 0 ( t ) , U 0 ( t ) = e x p ( − i t H 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1010.tif"/> then we can give as feedback the estimation error E ( t ) = X d ( t ) − T r 2 ( π t ( X ) ( I ⊗ | ϕ ( u ) > < ϕ ( u ) | ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1011.tif"/> which is a system observable. This error feedback is given to the above state equations, ie, to the Evans-Hudson flow resulting in the dynamics d j t ( X ) = j t ( θ 0 ( X ) ) d t + j t ( θ 1 ( X ) ) d A ( t ) + j t ( θ 2 ( X ) ) d A ( t ) * + j t ( θ 3 ( X ) ) d Λ ( t ) + K ( t ) E ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1012.tif"/> Alternately, we can incorporate this error feedback in the original HP equation as d U ( t ) = ( - ( i H + P + K ( t ) E ( t ) ) d t + L 1 d A ( t ) + L 2 d A ( t ) * + S d Λ ( t ) ) U ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1013.tif"/> resulting in the Evans-Hudson flow the same as above but with θ 0 (X) replaced by θ 0 ( X )+ iK ( t )[ E ( t ), X ]. Luc-Bouten’s method of control is slightly different. Here, we choose a system observable Z and give an infinitesimal control unitary U c ( t , t + d t ) = U ( t + d t ) * e x p ( - i Z d Y i ( t ) ) U ( t + d t ) = e x p ( i Z ( t + d t ) d Y o ( t ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1014.tif"/> Note that Y o ( t ) = U ( t + d t ) * Y i ( t ) U ( t + d t ) , Y o ( t + d t ) = U ( t + d t ) * Y i ( t + d t ) U ( t + d t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1015.tif"/> and hence, forming the difference, we get d Y 0 ( t ) = U ( t + d t ) * d Y i ( t ) U ( t + d t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1016.tif"/> Also, Z commutes with dY i ( t ) and hence Z ( t + dt )= U ( t + dt )* ZU ( t + dt ) commutes with U ( t + dt )* dY o ( t ) U ( t + dt ). Let ρ c ( t ) denote the controlled state at time t, ie after applying the Belavkin filter and control upto time t to the HP evolved state. Then, we apply the Belavkin filter from t to t + dt , via ρ B ( t + d t ) - ρ c ( t ) + δ ρ B ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1017.tif"/> where δ ρ B ( t ) = L t * ( ρ c ( t ) ) d t + ( M t ρ c ( t ) + ρ c ( t ) M t * - T r ( M t + M t * ) ρ c ( t ) ) ρ c ( t ) ) ( d Y o ( t ) - T r ( ρ c ( t ) ( M t + M t * ) ) d t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1018.tif"/> Finally, the filtered and controlled state at time t + dt is given by ρ c ( t + d t ) - ρ c ( t ) + δ ρ c ( t ) = https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1019.tif"/>
CRC Press eBooks•Feb 3, 2021•0 citations