Modeling and Control of the LKB Photon-Box: 1 Feedback stabilization with Quantum Non-Demolition (QND) Measurements

Modeling and Control of the LKB Photon-Box:

¹

Feedback stabilization with Quantum Non-Demolition (QND) Measurements

Pierre Rouchon

[email protected]

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

The controlled non-linear Markov chain

Attached to M

_g

= cos(ϕ

₀

+ ϑN) and M

_e

= sin(ϕ

₀

+ ϑN) we have the controlled Markov chain:

ρ

_k+1

= D

αk

(ρ

_k+1 2

), ρ

_k+1 2

= M

sk

(ρ

_k

) = M

_sk

ρ

_k

M

^†_s

k

Tr

M

_sk

ρ

_k

M

^†_s

k

where

input: α

_k

∈ R drives a unitary operation on the

cavity-field: D

α

(ρ) := D

_α

ρD

α^†

, D

_α

= exp(α(a

^†

− a)).

state: ρ

k

the density matrix of the cavity-field; it resumes all the past.

output: s

_k

∈ {g, e} is a stochastic variable, associated to probabilities p

_g,k

and p

_e,k

depending on ρ

_k

, p

_g,k

= Tr

M

_g

ρ

_k

M

^†_g

and p

_e,k

= Tr

M

_e

ρ

_k

M

^†_e

,

and given by the detector outcome at time k .

Outline

1

Open-loop convergence properties

2

Feedback stabilization A first feedback scheme

Construction of strict control Lyapunov functions Quantum filter and separation principle

3

Stability, Lyapunov functions and martingales

Deterministic systems: continuous-time ODE

Stochastic systems: discrete-time Markov chain

Truncation to n

^max

photons

Restriction to finite dimensional subspace spanned by the n

^max

+ 1 first modes {|0i , |1i , . . . , |n

^max

i}.

N = diag(0, 1, . . . , n

^max

), a |0i = 0, a |ni = √

n |n − 1i . The truncated creation operator a

^†

is the Hermitian conjugate of a. We still have N = a

^†

a, but truncation does not preserve the usual commutation [a, a

^†

] = 1 (this is only valid when n

^max

= ∞).

The Markov chain of state ρ (ρ

^†

= ρ, ρ ≥ 0 and Tr (ρ) = 1):

ρ

_k+1

=



 



 



M

g

(ρ

k

) =

^MgρkM

† g

Tr

Mgρ_kM^†_g

, prob. p

_g,k

= Tr

M

_g

ρ

k

M

^†_g

; M

e

(ρ

_k

) =

^MeρkM

† e

Tr

MeρkM^†_e

, prob. p

_e,k

= Tr

M

_e

ρ

_k

M

^†_e

. with M

_g

and M

_e

diagonal operators (dispersive atom/cavity interaction)

M

_g

= cos(ϕ

₀

+ Nϑ), M

_e

= sin(ϕ

₀

+ Nϑ)

100 Monte-Carlo simulations with α

_k

≡ 0 (h3|ρ

_k

|3i versus k )

50 100 150 200 250 300 350 400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Step number

Fidelity between ρ and the goal Fock state

Open-loop convergence of the truncated model

³

Theorem

Consider the Markov process defined above with an initial density matrixρ0. Assume that the parametersϕ0,ϑare chosen in order to have

Mg=cos(ϕ0+Nϑ),Me=sin(ϕ0+Nϑ)invertible and such that the spectrum ofM^†gMg=M²gandM^†eMe=M²eare not degenerate. Then

1 for any n∈ {0, . . . ,n^max}, Tr(ρk|ni hn|) =hn|ρk|niis a martingale 2 ρk converges with probability1to one of the n^max+1Fock state|ni hn|

with n∈ {0, . . . ,n^max}.

3 the probability to converge towards the Fock state|ni hn|is given by Tr(ρ0|ni hn|) =hn|ρ0|ni.

Proof²: for stat.2 useV^open-loop(ρ) =Pn^max

n=0(Tr(|ni hn|ρ))²and∀xµ, θµ∈[0,1]

X

µ

θµ=1 =⇒ X

µ

θµ(xµ)²= X

µ

θµxµ

!2

+X

µ,ν

θµθν(xµ−xν)² 2

2See H.Amini, M. Mirrahimi, PR: http://arxiv.org/abs/1103.1365.

3For theinfinite dimensionalMarkov chain see R. Somaraju, M.

Mirrahimi, PR: http://arxiv.org/abs/1103.1724

Lyapunov control for stabilizing ρ ¯ = | ni h ¯ n| ¯

Choosing α

_k

such that E ^{(Tr (ρ}

k

ρ)) ¯ is increasing.

We have

ρ

_k+1

2

=



 



 



MgρkM_g^† Tr

MgρkM_g^†

, with probability Tr

M

_g

ρ

_k

M

_g^†

,

MeρkM_e^† Tr

MeρkM_e^†

, with probability Tr

M

e

ρ

_k

M

_e^†

,

So E

Tr

ρk+1

2

¯ ρ

|ρk

=Tr

|¯ni h¯n|MgρkM_g^† +Tr

|¯ni h¯n|MeρkM_e^†

=Tr(|¯ni h¯n|ρk), as

M_g^†|¯ni h¯n|Mg+M_e^†|¯ni h¯n|Me=

cos²+sin²

|¯ni h¯n|=|¯ni h¯n|.

Lyapunov control: continued

Furthermore

ρ_k+1=D(α_k)ρ

k+¹₂D(−α_k), and BCH formula

DαρD_α^† =e^{αa†−α∗a}ρe^{−(αa†−α∗a)} =ρ+ [αa^†−α^∗a, ρ] +O(|α|²).

So

Tr(ρk+1ρ) =¯ Tr

ρk+1 2

¯ ρ

+αkTr

[|¯ni h¯n|,a^†]ρ

k+1 2

−α^∗kTr

[|¯ni h¯n|,a]ρ

k+1 2

+O(|αk|²).

Therefore, taking α

_k

= Tr

|¯ ni h¯ n| [ρ

_k+1 2

, a]

= Tr

[|¯ ni h¯ n| , a

^†

]ρ

_k+1 2

∗

,

for sufficiently small > 0, we have

Tr (ρ

k+1

ρ) ¯ ≥ Tr (ρ

k

ρ) ¯ = ⇒ E ^{(Tr (ρ}

^k+1

^{ρ) ¯ | ρ}

^k

^{) ≥ Tr (ρ}

^k

^{ρ) ¯}

Tr (ρ

_k

ρ) ¯ is a sub-martingale

Bad attractors

We do not have semi-global stabilization ...

50 100 150 200 250 300 350 400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Step number

Fidelity between and the goal Fock state

Tr(ρ_kρ)¯ converges almost surely towards a random variable with values 0 or 1

Modified feedback law

⁴

α

_k

=



 

 

Tr

¯ ρ[ρ

_k+¹

2

, a]

if Tr

¯ ρρ

_k+¹

2

≥ η argmax

|α|≤α¯

Tr

¯

ρ D

α

(ρ

_k+1 2

)

if Tr

¯ ρρ

_k+1

2

< η

50 100 150 200 250 300 350 400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Step number Fidelity between and the goal Fock state

4See I. Dotsenko et al., Phys. Rev. A, 2009. See also, M. Mirrahimi, R.

Van Handel, SIAM JCO 2007, for a similar feedback in continuous time.

Closed-loop convergence

Closed-loop Markov chain:

, ρ_k+1=Dα_k(ρ

k+¹₂), ρ

k+¹ 2

=Msk(ρ_k) with

α_k =





 Tr

¯

ρ[ρk+¹₂,a]

if Tr

¯ ρρk+¹₂

≥η argmax

|α|≤α¯

Tr

¯ ρDα(ρ

k+¹₂)

if Tr

¯ ρρk+¹₂

< η

Theorem

Consider the above closed-loop quantum system. For small enough parameters, η >0 in the feedback scheme, the trajectories

converge almost surelytoward the target Fock stateρ.¯

Proof’s scheme

Four steps:

1

First, we show that for small enough η, the trajectories starting within the set S

_<η

= {ρ | Tr ( ¯ ρρ) < η} always reach in one step the set S

_≥2η

= {ρ | Tr ( ¯ ρρ) ≥ 2η};

2

next, we show that the trajectories starting within the set S

_≥2η

, will never hit the set S

_<η

with a uniformly non-zero probability p

_η

> 0 (Doob’s inequality);

3

we prove an inequality showing that, for small enough , V (ρ

k

) = f(Tr ( ¯ ρρ

k

)) with f (x ) =

^x2₂^+x

is a sub-martingale within S

_≥η

= {ρ | Tr ( ¯ ρρ) ≥ η};

4

finally, we combine the previous step and the Kushner’s

invariance principle, to prove that almost all trajectories

remaining inside S

≥η

converge towards ρ. ¯

Step 2: Doob’s inequality

Doob’s Inequality

Let{Xn}be a Markov chain on state spaceX. Suppose that there is a non-negative functionV(x)satisfying

E

^{(V(X1)|X0=x)−V(x) =−k(x),}

wherek(x)≥0 on the set{x:V(x)< λ} ≡Qλ. Then

P sup

∞>n≥0

V(Xn)≥λ

X0=x

!

≤V(x) λ .

Here we takeV(ρk) =1−Tr( ¯ρρk)which is a super-martingale. We have:

P sup

k⁰≥k

(1−Tr( ¯ρρk⁰))≥1−η

ρk ∈ S≥2η

!

≤ 1−Tr( ¯ρρk)

1−η ≤ 1−2η 1−η , and thus

P

inf

k⁰≥kTr( ¯ρρk⁰)> η

Tr( ¯ρρk)≥2η

=1−P sup

k⁰≥k

(1−Tr( ¯ρρ_k0))≥1−η

Tr( ¯ρρk)≥2η

!

≥1−1−2η 1−η = η

1−η =pη.

Strict control-Lyapunov function

⁵

(1)

For any functionλ, consider the open-loop martingale

Vλ(ρ) =Tr(λ(N)ρ) =

d

X

n=1

λ_nTr(|ni hn|ρ) =

d

X

n=1

λ_nhn|ρ|ni.

(λ(N)is a fixed point of the adjoint Kraus map).

For each Fock stateρ|ni hn|,α=0 is a critical point of α7→Vλ(Dα(ρ), ^{dVλ Dα(|nihn|)}

dα

α=0

=0, and

d²V_λ Dα(|nihn|)

dα²

α=0

=Tr

[[a^†−a,[a^†−a, λ(N)]]|ni hn|

=Tr(Rλ(N)|ni hn|) whereR=is a tridiagonal Laplacian matrixwith dim(kerR) =1 with entries

Rn−1,n=2n, ,Rn,n=−4n−2, Rn+1,n=2n+2.

5See H.Amini, M. Mirrahimi, PR: CDC/ECC 2011 http://arxiv.org/abs/1103.1365.

Strict control-Lyapunov function (2)

Take a goal Fock state |¯ ni and, for each n 6= ¯ n, σ

n

> 0. By inverting The Laplacian R, we define λ

n

such that, for any n 6= 0,

d²Vλ Dα(|nihn|)

dα²

α=0

= σ

n

> 0.

Then

^d

2Vλ Dα(|¯nih¯n|)

dα²

α=0

= − P

n6=¯n

σ

n

< 0. Moreover n 7→ λ(n) is strictly increasing from 0 to n ¯ and strictly decreasing from n ¯ to n

^max

.

Then, for > 0 small enough

W

(ρ) = V

^open-loop

(ρ) + V

_λ

(ρ)

becomes a strict Lyapunov function with the feedback α

_k

= K (ρ

_k+1

2

) = argmax

α∈[−α,¯¯α]

W

D

α

ρ

_k+1 2

,

for any α > ¯ 0.

Strict control-Lyapunov function (3)

In closed-loopWis a strict sub-martingale since, forρ_k 6=|¯ni h¯n|,

E

^{(W(ρk+1)|ρk)>W(ρk)}

because we have

E

^(W(ρk+1)|ρ_k)−W(ρ_k) = X

µ∈{g,e}

pµ,ρ_k

α∈[−maxα,¯¯u]

W Dα(Mµ(ρ_k))

−W(ρ_k)

=

X

µ∈{g,e}

pµ,ρ_k

W Mµ(ρ_k)

−W(ρ_k) +

X

µ∈{g,e}

pµ,ρk

max

α∈[−α,¯α]¯

W Dα(Mµ(ρ_k))

−W Mµ(ρ_k)

Theblue sumis strictly positive, excepted whenρ_k is a Fock state (see open-loop convergence). Thered sumis always non-negative.

Whenρ_k is a Fock state, thered sumvanishes only forρ_k =|¯ni h¯n|.

Quantum filter for feedback control

ρ_k+1=Msk(ρ

k+¹ 2

), ρ

k+¹₂ =Dα_k(ρ_k).

We wish to find the controlα_k as a function of thek first measured jumps.In this aim we need to estimate the state of the system.

We consider here the ideal case (no measurement uncertainties nor decoherence):Best estimate is given by the system dynamics itself.

Quantum filter

ρ^est_k+1=Msk(ρ^est

k+¹₂), ρ^est

k+¹₂ =Dα_k(ρ^est_k ),

where the values fors_k ∈ {g,e}are given by the measurement results andα_k is a function ofρ^est_k : α_k =α(ρ^est_k ).

A quantum separation principle

⁶

System+Filter dynamics:

ρk+¹ 2

=Msk(ρ_k), ρ_k+1=Dα_k(ρ

k+¹₂), ρ^est

k+¹₂ =Msk(ρ^est_k ), ρ^est_k+1=Dα_k(ρ^est

k+¹₂)

wheresk takes the valuesg orewith probabilitiespg,k andpe,k given by

p_g,k =Tr

M_gρ_kM^†_g

, p_e,k =Tr

M_eρ_kM^†_e and whereα_k =α(ρ^est

k+¹₂).

Theorem: a quantum separation principle

Consider a closed-loop system of the above form. Assume moreover that, wheneverρ^est₀ =ρ₀(so that the quantum filter coincides with the closed-loop dynamics,ρ^est≡ρ), the closed-loop system converges almost surelytowards a fixedpure stateρ. Then, for any choice of the¯ initial stateρ^est₀ , such thatkerρ^est₀ ⊂kerρ₀, the trajectories of the system-filter converge almost surely towards the same pure state:

ρ_k, ρ^est_k →ρ.¯

6See R. Van Handel: Filtering, Stability, and Robustness. PhD thesis, CalTech, 2007.

Proof (1)

E

^{Tr(ρkρ)¯ |ρ0, ρest0} depends linearly onρ0even though we are applying a feedback control.

Indeed, we can write

αk =α(ρ^est₀ ,s0, . . . ,sk−1), and simple computations imply

E

Tr( ¯ρρk)|ρ0, ρ^est0

= X

s₀,...,s_k−1

Tr

¯

ρMe^sk−1◦D^αk−1◦. . .◦Me^s0◦D^α0(ρ0)

where

Me^sρ=MsρM^†s.

So, we easily havethe linearity of

E

^{Tr(ρkρ)¯ |ρ0, ρest0} with respect toρ0. The rest of the proof follows from the assumption kerρ^est₀ ⊂kerρ0which implies the existence of a constantγ >0 and a density matrixρ^c₀, such that

ρ^est0 =γρ0+ (1−γ)ρ^c0.

Proof (2)

We know thatif both the system and filter start atρ^est₀ , we have the almost sure convergence. This, together with dominated convergence theorem implies

lim

k→∞

E

Tr(ρkρ)¯ |ρ^est0 , ρ^est0

=1.

By thelinearity of

E

^{Tr(ρkρ)¯ |ρ0, ρest0} with respect toρ0, we have

E

Tr(ρkρ)¯ |ρ^est0 , ρ^est0

=γ

E

Tr(ρkρ)¯ |ρ0, ρ^est0

+(1−γ)

E

Tr(ρkρ)¯ |ρ^c0, ρ^est0

, and as both

E

^{Tr(ρkρ)¯ |ρ0, ρest0 and}

E

^{Tr(ρkρ)¯ |ρc0, ρest0} are less than or equal to one, we necessarily have that both of them converge to 1:

k→∞lim

E

Tr(ρkρ)¯ |ρ0, ρ^est0

=1.

This implies the almost sure convergence of the physical system towards the pure stateρ.¯

Lyapunov stability for ODE

¯ x ∈ R

ⁿ

is an equilibrium of

_dt^d

x = v(x ), when v (¯ x ) = 0.

Stability

Equilibrium x ¯ ∈ R

ⁿ

is stable iff ∀ > 0, ∃η > 0 such that ∀x

⁰

, kx

⁰

− x ¯ k ≤ η, the solution of the Cauchy problem

_dt^d

x = v (x , t) starting from x

⁰

at t = 0 satisfies

kx (t) − x ¯ k ≤ , ∀t ≥ 0

Asymptotic stability

The equilibrium x ¯ ∈ R

ⁿ

is said locally asymptotically stable iff it is stable and moreover, ∃η > 0 such that

kx

⁰

− x ¯ k ≤ η, implies x (t) −→ x ¯

when t −→ +∞

First Lyapunov method

⁷

Spectrum and local stability

The equilibrium x ¯ of

_dt^d

x = v (x ) is locally asymptotically stable if the eigenvalues of the Jacobian matrix at x ¯ ,

∂v

_i

∂x

_j

¯x

, are all with strictly negative real parts.

The equilibrium x ¯ is unstable if at least one of the eigenvalues of the Jacobian matrix admits a strictly positive real part

7See H.K. Khalil, Nonlinear Systems (Prentice Hall, 2001).

Second Lyapunov method

⁸

Lyapunov functions and Lasalle’s invariance principle

Rⁿ3x 7→v(x)∈RⁿC¹versusx. TakeRⁿ3x 7→V(x)∈R⁺aC¹ function ofx.Assume that

1 limkxk7→+∞V(x) = +∞

2 V decreases along all solutions of _dt^dx =v(x):

d

dtV(x) =∇V(x)·v(x) =

n

X

i=1

∂V

∂xi

(x)vi(x)≤0, for allx.

Then,for all initial conditionx⁰, the solution_dt^dx =v(x)is defined for anyt >0 (no finite-time explosion) and converges towardsthe largest invariant setcontained in

x ∈Rⁿ| _dt^dV(x) =0 .

8See H.K. Khalil, Nonlinear Systems (Prentice Hall, 2001).

Stability and convergence of stochastic processes (1)

Convergence of a random process

Consider(Xk)k∈N, a discrete-time sequence of random variables defined on the probability space(Ω,F,P)and taking values in a Banach spaceX. The random processXkis said to,

1 convergein probabilitytowards the constantx¯∈ X if for all >0, lim

k→∞P(kXk−¯xk> ) = lim

k→∞P(ω∈Ω| kXk(ω)−¯xk> ) =0;

2 convergealmost surelytowards the constantx¯if P

lim

k→∞Xk = ¯x

=P

ω∈Ω| lim

k→∞Xk(ω) = ¯x

=1;

3 convergein meantowards the constant¯xif lim

k→∞

E

^{(kXk−¯xk) =0.}

Mean convergence implies convergence in probability.

Almost sure convergence implies convergence in probability.

Stability and convergence of stochastic processes (2)

Markov process

The sequence(Xk)^∞_k=1is called a Markov process, if fork⁰>kand any measurable real functionf(x)with sup_x|f(x)|<∞,

E

^{(f(Xk0)|X1, . . . ,Xk) =}

E

^(f(Xk0)|Xk).

Martingales

Consider a measurable real functionV(x)and(Xk)k∈Na Markov chain onX. V(Xk)^∞_k=1is asuper-martingale, asub-martingaleor amartingale, if

E

^{(kV(Xk)k)<∞fork>}0, and if, respectively,

E

^{(V(Xk+1)|Xk)≤V(Xk) (P}almost surely), ∀k>0, or

E

^{(V(Xk+1)|Xk)≥V(Xk) (P}almost surely), ∀k>0, or finally,

E

^{(V(Xk+1)|Xk) =V(Xk) (P}almost surely), ∀k>0,

Stability and convergence of stochastic processes (3)

Doob’s Inequality

Let{Xk}be a Markov chain on state spaceX. Suppose that there is a non-negative functionV(x)satisfying

E

^{(V(X1)|X0=x)−V(x) =−k(x),}

wherek(x)≥0 on the set{x:V(x)< λ} ≡Qλ. Then, for allx∈Qλ,

P sup

∞>k≥0

V(Xk)≥λ

X0=x

!

≤V(x) λ .

Corollary: stability in probability

Consider the same assumptions as in the above theorem. Assume moreover that there exists¯x∈ Xsuch thatV(¯x) =0 and thatV(X)6=0 for allx different fromx. Then the Doob’s inequality implies that the Markov process¯ Xk isstable in probabilityaround¯x, i.e.

x→¯limxP

sup

k

kXk−xk ≥¯ |X0=x

=0, ∀ >0.

Stability and convergence of stochastic processes (4)

Kushner’s invariance Theorem

Consider the same assumptions as that of the Doob’s inequality. Letµ0=σ be concentrated on a statex0∈Qλ, i.e.σ(x0) =1. Assume that

0≤f(Xk)→0 inQλimplies thatXk → {x|f(x) =0} ∩Qλ≡Fλ. For the trajectories never leavingQλ,Xk converges toFλalmost surely. Also, the associated conditioned probability measuresµ˜k tend to the largest invariant set of measuresM∞⊂Mwhose support set is inFλ. Finally, for the trajectories never leavingQλ,Xk converges, in probability, to the support set ofM∞.

Corollary: global stability

Consider the same assumptions as in the above theorem and assume moreover that¯x∈ Xis the only point inQλsuch thatV(¯x) =0 and

furthermore that the setFλis reduced to{¯x}(strict Lyapunov function). Then the equilibrium¯xisglobally stable in probabilityin the setQλ, i.e.¯x

is stable in probability and moreover P

k→∞lim Xk = ¯x|Xk never leavesQλ

=1.