Modeling and Control of the LKB Photon-Box: 1 From discrete to continuous-time systems

Modeling and Control of the LKB Photon-Box:

From discrete to continuous-time systems

Pierre Rouchon

pierre.rouchon@mines-paristech.fr

Würzburg, July 2011

1LKB: Laboratoire Kastler Brossel, ENS, Paris.

Several slides have been used during the IHP course (fall 2010) given with Mazyar Mirrahimi (INRIA) see:

http://cas.ensmp.fr/~rouchon/QuantumSyst/index.html

Stochastic models for open-quantum systems Discrete-time modelsareMarkov chains

ρ_k+1= 1

pν(ρ_k)Mνρ_kM_ν^† with proba. pν(ρ_k) =Tr Mνρ_kM_ν^† associated toKraus maps(ensemble average, open quantum channels)

E^{(ρk+1/ρk) =K(ρk) =X}

ν

Mνρ_kM_ν^† with X

ν

M_ν^†Mν =1

Continuous-time modelsarestochastic differential systems dρ=

−i[H, ρ] +LρL^†−¹

2(L^†Lρ+ρL^†L) dt +

Lρ+ρL^†−Tr (L+L^†)ρ ρ

dw driven byWiener processes²dw =dy− Tr (L+L^†)ρ

dt with measurey and associated toLindbaldmaster equations:

d dtρ=−ⁱ

~[H, ρ] +LρL^†−¹

2(L^†Lρ+ρL^†L)

2Another common possibility not considered here: SDE driven by Poisson processes.

Numerics based on Ito rules and Markov chains

For Monte-Carlo simulations of

dρ=

−i[H, ρ] +LρL^†−¹

2(L^†Lρ+ρL^†L) dt +

Lρ+ρL^†−Tr

(L+L^†)ρ ρ

dw

take a small sampling timedt, generate a random numberdwt

according to a Gaussian law ofstandard deviation√

dt, and set ρ_t+dt =Mdyt(ρt)where the jump operatorMdyt is labelled by the measurement outcomedyt =Tr (L+L^†)ρt

dt+dwt:

Mdyt(ρ_t) = ^I+(−iH−

1

2L^†L)dt+Ldyt

ρt I+(iH−¹₂L^†L)dt+L^†dyt

Tr

I+(−iH−¹₂L^†L)dt+Ldyt

ρt I+(iH−¹₂L^†L)dt+L^†dyt

.

Thenρt+dt remains always a density operator and using the Ito rules (dw of order√

dt anddw²≡dt) we get the good dρ=ρ_t+dt −ρ_t up toO((dt)^3/2)terms.

Quantum filtering

Assumet7→y_t is the detector signal. Then the quantum filter equation providing an estimationρbof the stateρreads³

dρb=

−i[H,ρ] +b LbρL^†−¹

2(L^†Lρb+ρLb ^†L) dt +

Lρb+ρLb ^†−Tr (L+L^†)ρb ρb

dy−Tr (L+L^†)ρb dt sincedw=dy−Tr (L+L^†)ρ

dt. To get this filter take a small sampling timedt, use the measure outcomedy_tand the jump operator labelled bydy_t to updateρaccording toρb_t+dt =Mdyt(ρb_t).

You recover then up toO(dt^3/2)terms the above filter that reads also dρb=

−i[H,ρ] +b LbρL^†−¹

2(L^†Lρb+ρLb ^†L) dt +

Lbρ+bρL^†−Tr (L+L^†)ρb ρb

dw

−

Lρb+ρLb ^†−Tr (L+L^†)ρb bρ

Tr (L+L^†)(ρb−ρ) dt.

3See Belavkin theoretical work

Numerics based on Kraus maps

For the Lindblad equation

d

dtρ=−ⁱ

~[H, ρ] +LρL^†−¹

2(L^†Lρ+ρL^†L) take a small sampling timedtand set

ρ_t+dt = ^I+(−iH−

1 2L^†L)dt

ρt I+(iH−¹₂L^†L)dt

+dtLρtL^† Tr

I+(−iH−¹

2L^†L)dt

ρt I+(iH−¹

2L^†L)dt

+dtLρtL^†.

Thenρ_t+dt remains always a density operator and

d

dtρ= (ρt+dt−ρt)/dt up toO(dt)terms.

The controlled SDE of a two-level system

Attached toH= ^u₂σx,L= qγ

2σz, thecontrolled SDE

dρ= −i^u₂[σx, ρ] +^γ₂σzρσz−γρ dt +

qγ 2

σzρ+ρσz−2Tr(σzρ)ρ

dw

whereu ∈Ris the control input and dy =√

2γ Tr(σ_zρ)dt+dw is the measured output.

For open-loop almost-sure convergence towards|gi hg|or

|ei he|takeV(ρ) =Tr(σ_zρ)²as Lyapunov function.

For closing the loop take . . . or see M. Mirrahimi and R.

Van Handel: Stabilizing feedback controls for quantum systems. SIAM Journal on Control and Optimization, 2007

Few references

L. Arnold, Stochastic differential equations: theory and applications. Wiley-Interscience, New York, 1974.

N.G. Van Kampen, Stochastic processes in physics and chemistry. Elsevier, 1992.

The lectures on quantum circuits given by Michel Devoret at Collège de France:

http://www.physinfo.fr/lectures.html