Stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays

Stabilization of discrete-time quantum systems subject to non-demolition

measurements with imperfections and delays

Pierre Rouchon, Mines-ParisTech, Centre Automatique et Syst `emes, [email protected]

Delayed Complex Systems June 2012, Palma de Majorque

Joint work with

Hadis Amini (Mines-ParisTech), Mazyar Mirrahimi (INRIA),

Ram Somaraju (INRIA/Vrije Universiteit Brussel)

Outline

Feedback and quantum systems

Measurement-based feedback of photons The LKB experiment

The ideal Markov model Open-loop convergence Closed-loop experimental data

Observer/controller design for the photon box A strict control Lyapunov control

Quantum filter, separation principle and delay

Conclusion: measurement-based versus coherent feedbacks Generalization to any discrete-time QND systems

Controlled QND Markov chains Open-loop convergence

Feedback, delay and closed-loop convergence Imperfect measurements

1H. Amini et al. Preprint arxiv:1201.1387, 2012.

Model of classical systems

control

perturbation

measure system

For the harmonic oscillator of pulsation

with

, controlled by the force

and subject to an additional unknown force

x = (x₁,x₂)∈R², y =x₁

d

dtx₁=x₂, _dt^dx₂=−ω²x₁+u+w

Feedback for classical systems

feedback observer/controller

perturbation

set point

control

measure system

Proportional Integral Derivative (PID) for

with the set point

u=−K_p y −y_sp

−K_ddt^d y−y_sp

−K_int Z

y −y_sp

with the positive

tuned as follows (0

1, 0

1:

Kp= Ω²₀, K_d =

2ξΩ

Feedback for the quantum system

Key issue: back-action due to the measurement process.

Measurement-based feedback: measurement back-action onS is stochastic (collapse of the wave-packet); controller is classical; the control inputuis a classical variable appearing in some controlled Schr ¨odinger equation;u depends on the past measures.

Coherent feedback: the systemSis coupled to another quantum system (the controller); the composite system,S ⊗ controller, is an open-quantum system relaxing to some target (separable) state (related toreservoir

engineering).

This talk is devoted tothe first experimental realizationof a measurement-based state feedback. It has been done at Laboratoire Kastler Brossel of Ecole Normale Sup ´erieure by the Cavity Quantum ElectroDynamics (CQED) group of Serge Haroche.²

2C. Sayrin et al.: Real-time quantum feedback prepares and stabilizes photon number states. Nature, 477:73–77, 2011.

The closed-loop CQED experiment

3

•

Control input

; measure output

•

Sampling time 80

3Courtesy of Igor Dotsenko

The ideal Markov chain for the wave function

k

g

g _k

k 1

y_k g e

g M_{g k} e M_{e k} M_{yk k} M_{yk k} UR1

U_C U_R2

D_uk

Inputu_k, state|ψ_ki=P

n≥0ψⁿ_k|ni, outputy_k: ψ_k+1/2

= Myk|ψ_ki

kM_yk|ψ_ki k, |ψ_k+1i=D_uk ψ_k+1/2

with

I yk =g(resp.e) with probabilitykM_g|ψ_ki k²(resp.kM_e|ψ_ki k²);

I measurement Kraus operatorsM_g =cos_φ

0N+φR

2

and Me=sin_φ

0N+φR

2

: M_g^†Mg+M_e^†Me=1with

N=a^†a=diag(0,1,2, . . .)the photon number operator;

I displacement unitary operator(u∈R):D_u=e^ua†−uawith a=upper diag(√

1,√

2, . . .)the photon annihilation operator.

4S. Haroche and J.M. Raimond.Exploring the Quantum: Atoms, Cavities and Photons.Oxford Graduate Texts, 2006.

The ideal Markov chain for the density operator

Diagonal elements ofρ,ρⁿⁿ=hn|ρ|ni=|ψⁿ|², form the photon number distribution.

ρ_k+1=



















Duk Mgρ_kM_g^†D_u^†_k Tr

M_gρ_kM_g^† yk =gwith probabilitypg,k =Tr

Mgρ_kM_g^†

D_uk M_eρ_kM_e^†D^†_uk Tr

M_eρ_kM_e^† yk =ewith probabilitype,k =Tr

Meρ_kM_e^†

I Displacement unitary operator(u∈R):D_u=e^ua†−uawith a=upper diag(√

1,√

2, . . .)the photon annihilation operator.

I Measurement Kraus operatorsMg =cos_φ

0N+φR

2

and M_e=sin_φ

0N+φR

2

: M_g^†M_g+M_e^†M_e=1with

N=a^†a=diag(0,1,2, . . .)the photon number operator.

Open-loop behavior (u

0)

Anexperimental open-loop trajectorystarting from coherent state ρ₀=|ψ₀i hψ₀|with¯n=3 photons:|ψ₀i=e^−¯n/2P

n≥0

q¯nⁿ n!|ni.

I Afast convergencetowards|ni hn|for somen,

I followed by aslow relaxation towards vacuum|0i h0|:

decoherence due to finite photon life time around 70 ms (not included into the ideal model).

Open-loop stability ofρ_k+1= ^MykρkM

† yk

Tr(M_ykρ_kM_yk^†)explaining thisfast convergencewhenφ₀/πis irrational⁵

I for anyn,ρⁿⁿ_k =hn|ρ_k|niis amartingale:

E

=ρⁿⁿ_k ;

I almost all realizations starting fromρ₀converge towards a photon number state|ni hn|; the probability to converge towards

|ni hn|is given by the initial populationρⁿⁿ₀ .

This convergence characterizes aQuantum Non Demolition (QND) measurementof photons (counting photons without destroying them).

5H. Amini et al., IEEE Trans. Automatic Control, in press, 2012.

Closed-loop experimental data

•Initial state coherent state withn¯=3 photons

•State estimation via a quantum filterof stateρ^est_k .

•Lyapunov state feedback uk =f(ρ_k)stabilizing towards|¯ni hn|¯

•ρ_k is replaced by its estimateρ^est_k in the feedback (quantum separation principle) Sampling period 80µs Experience imperfections:

•detection efficiency 40%

•detection error rate 10%

•delay: 4 steps

•truncation to 9 photons

•finite photon life time

•atom occupancy 30%

Stabilization around 3-photon state

Fidelity as control Lyapunov function

In

we propose the following stabilizing state feedback law based on the fidelity towards the target state

u=f(ρ) =:

Argmin

υ∈[−¯u,¯u]

V(DυρD^†_υ)

where

1

1

and

0 is small.

Two important issues.

I

The state

is not directly measured; output delay is of 4 steps: it was solved by a quantum filter taking into account the delay.

I V

is maximum and equal to 1 for any

with

no distinction between

1 (close to the target) and

1000 (far from the target). This issue has been solved by

.

6I. Dotsenko et al.: Quantum feedback by discrete quantum non-demolition measurements: towards on-demand generation of photon-number states.

Physical Review A80:013805, 2009.

Lyapunov-based feedback (goal photon number

n

−hn|ρ|ni²+σ_nhn|ρ|ni

is astrict control Lyapunov functionwith >0 small enough,

σ_n=











1 4+P^¯n

ν=11

ν −_ν¹2, ifn=0;

P^¯n ν=n+11

ν −_ν¹2, ifn∈[1,¯n−1];

0, ifn= ¯n;

Pn ν=¯n+11

ν +_ν¹2, ifn∈[¯n+1,+∞[, and thefeedbacku=f(ρ) =:Argmin

υ∈[−¯u,¯u]

V

DυρD_υ^†

(u¯>0 small).

In closed-loop,V(ρ)becomes astrict super-martingale:

E

withQ(ρ)continuous, positive and vanishing only whenρ=|¯ni h¯n|.

This feedback law yields

I global stabilizationfor any finite dimensional approximation consisting in truncation ton^max<+∞photons.

I global approximate stabilizationforn^max= +∞.

7H. Amini et al.: CDC-2011.

The control Lyapunov function used for the photon box

9.

0 2 4 6 8

0 0.2 0.4 0.6 0.8 1

photon number n

Coefficients σ_n of the control Lyapunov function

V(ρ) =P9 n=0

−hn|ρ|ni²+σnhn|ρ|ni

Global approximate stabilization (n

I

The feedback

Argmin

υ∈[−¯u,¯u]

V

DυρD^†υ

ensures a strict closed-loop Lyapunov function

V(ρ) =X

n≥0

−hn|ρ|ni²+σnhn|ρ|ni

with

log

large (high photon-number cut-off).

I

For any

0 and

0, exist

0 and

0 (small), such that, for any initial value

with

converges almost surely towards a number inside

1].

With Tr

1, and

0, this means, that almost surely, for

large enough,

is close (weak-* topology) to the goal Fock state

8R. Somaraju, M. Mirrahimi, P.R.: CDC 2011 http://arxiv.org/abs/1103.1724

Design of the strict control Lyapunov function

Exploit open-loop stability: for eachn,hn|ρ|niis a martingale;

V(ρ) =−¹₂P

nhn|ρ|ni²is a super-martingale with

E

whereQ(ρ)≥0 andQ(ρ) =0 iff,ρis a Fock state.

For closing the looptakeσ_nsuch that u 7→X

n

σ_nD n

D_uρD^†_u

nE

1. is strongly convex forρ=|ni h¯ ¯n|

2. is strongly concave forρ=|ni hn|,n6= ¯n.

This is achieved by inverting theLaplacian matrix associated to the control HamiltionanH=ı(a−a^†). Remember thatDu=e^−ıuH.

9H. Amini et al., CDC 2011,http://arxiv.org/abs/1103.1365

Estimation of

from the past measures

via a quantum filter



















ρ_k+1=Duk Mykρ_kM_y^†_kD^†_uk Tr

M_ykρ_kM_y^†_k

ρ^est_k+1=D_uk M_ykρ^est_k M_y^†_kD_u^†_k Tr

Mykρ^est_k My^†k

I Assume we knowρ_k andu_k. Outcome of measure nok,y_k, defines the jump operatorMyk and we can computeρ_k+1.

I Quantum filter and real-time estimation:initialize the estimation ρ^estto some initial valueρ^est₀ and update at stepk with measured jumpsyk and the known controlsuk.

I Quantum separation principle for stabilization towards a pure state¹⁰: assume that the feedbacku=f(ρ)ensures global asymptotic convergence towards a pure state; then, if

ker(ρ^est₀ )⊂ker(ρ₀), the feedbacku_k =f(ρ^est_k )ensures also global asymptotic convergence towards the same pure state.

10Bouten, van Handel, 2008.

A modified quantum filter with a measure delayed by one step

uk =Argmin V Dv

M_ykρ^est_k M_yk^† Tr(^M_yk^ρestk M_yk^†)D_v^†

With delay, we have only access toyk−1and the stabilizing feedback uses the Kraus mapK(ρ) =M_gρM_g^†+M_eρM_e^†:

u_k =Argmin V

DvK(ρ^est_k )D_v^†

This is the same feedback law but with another state estimation at stepk:K(ρ^est_k )instead of ^Mykρ

est k M_yk^† Tr(M_ykρ^est_k M_yk^†).

I System theoretical interpretation: K(ρ^est_k )stands for the the prediction of cavity state at stepk. This prediction is in average (expectation value) sinceyk ∈ {g,e}can take two values.

I Quantum physics interpretation:K(ρ^est_k )corresponds to tracing over the atom that has already interacted with the cavity (entangled with cavity state) but that has not been measured at stepk.

A delay of two steps involves two iterations of such Kraus maps, . . .

Conclusion: measurement-based versus coherent feedback.

I Classical state-feedback stabilization: continuous time systems with QND measurement (possible extension of M. Mirrahimi and R. van Handel, SIAM JOC, 2007), filtering stability (Belavkin seminal contributions, see also van Handel, ...).

I Stabilization bycoherent feedback: similarly to theWatt regulatorwhere a mechanical system is controlled by another one, the controller is a quantum system coupled to the original one (Mabuchi, Nurdin, Gough, James, Petersen, ...); related to

”quantum circuit” theory (see last chapters of Gardiner-Zoller book and the courses of Michel Devoret at Coll `ege de France);

I Coherent feedback is closely related toreservoir engineering:

exploit and design themeasurementprocess (here operators Mµ) and itsintrinsic back-actionto ensure convergence of the ensemble-average dynamics towards a unique pure state (Ticozzi, Viola, . . . )

Watt regulator: a classical analogue of quantum coherent feedback.

The first variations of speed

and governor angle

obey to

d

dtδω=−aδθ

d²

dt²δθ=−Λ_dt^dδθ−Ω²(δθ−bδω)

with

positive param- eters.

Third order system

d³

dt³δω=−Λ_dt^d2₂δω−Ω²_dt^dδω−abΩ²δω=

0

Characteristic polynomial

with roots having negative real parts iff

governor damping must be strong enough to ensure asymptotic stability of the closed-loop system.

11J.C. Maxwell: On governors. Proc. of the Royal Society, No.100, 1868.

Reservoir engineering stabilizing Schr ¨odinger cats for the photon box

Wigner functions of the various states that can be produced by such reservoir based on composite dispersive/resonant

atom/cavity interaction.

0 0.2 0.4 0.6

-0.2 -0.4 -0.6

(a)

(e) (f)

(b)

(g) (c)

(h) (d)

-3 -3

3 3

0 0

Re(γ)

W(γ)

Im(γ)

12Sarlette et al: PRL 107:010402,2011 and PRA to appear in 2012.

Control of a QND Markov chain with delay

ρk+1=M^uµ^kk^−τ(ρk) =: M_µ^uk−τ_k ρkM_µ^u_k^k−τ†

Tr

Mµ^uk−τk ρ_kMµ^uk−τk

†

I To each measurement outcomeµis attached the Kraus operator M_µ^u∈C^d×d depending onµand also on a scalar control input u∈R. For eachu,Pm

µ=1M_µ^u†M_µ^u=I, and we havethe Kraus mapK^u(ρ) =Pm

µ=1M_µ^uρM_µ^u†

I µ_k is a random variable taking valuesµin{1,· · ·,m}with probabilityp^uµ,ρ^k−τ_k =Tr

Mµ^uk−τρ_kMµ^uk−τ

† .

I Foru=0, the measurement operatorsM_µ⁰are diagonal in the same orthonormal basis{ |ni | n∈ {1,· · ·,d}}, therefore M_µ⁰=Pd

n=1c_µ,n|ni hn|withc_µ,n∈C.

I For alln16=n2in{1,· · · ,d},there existsµ∈ {1,· · ·,m}such that|c_µ,n1|²6=|c_µ,n2|².

Open-loop convergence

For any initial conditionρ₀,

I with probability one,ρ_k converges to one of thed states|ni hn|

withn∈ {1,· · · ,d}.

I the probability of convergence towards the state|ni hn|depends only onρ₀and is given byhn|ρ₀|ni.

Proof based on

I the martingaleshn|ρ|ni

I the super-martingaleV(ρ) :=−P

n

hn|ρ|ni2

2 satisfying

E

withQ(ρ) =¹₄P

n,µ,νp⁰_µ,ρp⁰_ν,ρ|c

µ,n|²hn|ρ|ni

p⁰_µ,ρ −^|cν,n|_p²0^hn|ρ|ni ν,ρ

2

.

I Q(ρ) =0 iff existsn∈ {1, . . . ,d}such thatρ=|ni hn|.

Feedback stabilization of

towards

I V0(ρ) =Pd

n=1σ_nhn|ρ|niwithσ_n≥0 chosen such thatσ¯n=0 and for anyn6= ¯n, the second-orderu-derivative of

V₀(K^u(|ni hn|))atu=0 is strictly negative (K^uis the Kraus map): set of linear equations inσ_nsolved by inverting an irreducibleM-matrix(Perron-Frobenius theorem).

I The function ( >0 small enough):

V(ρ) =V₀(ρ)−₂Pd

n=1(hn|ρ|ni)²still admits a unique global minimum at|¯ni hn|; for¯ u close to 0,u7→V(K^u(|ni hn|))is strongly concave for anyn6= ¯nand strongly convexe forn= ¯n.

I The delay ofτ steps: stabilize the stateχ= (ρ, β₁,· · ·, βτ)(β_l control inputudelayedlsteps) towardsχ¯= (|¯ni hn|¯ ,0, . . . ,0) using thecontrol-Lyapunov function

W(χ) =V(K^β1(K^β2(. . . .K^βτ(ρ). . .))).

For¯uandsmall enough, the feedback u_k =f(χ_k) =:argmin

ξ∈[−u,¯¯u]

E

ensuresglobal stabilizationtowardsχ.¯

Quantum separation principle

I

Estimate the hidden state

by

satisfying

where

obeys to

with the stabilizing feedback

computed using

instead of

I

If ker(ρ

ker(ρ

and

converge almost surely towards the target state

Proof based on

:

I h¯n|ρ_k|ni ∈¯ [0,1],

I linearity of

E

versusρ₀,

I decompositionρ^est₀ =γρ₀+ (1−γ)ρ^c₀withγ∈]0,1[.

13Bouten, van Handel, 2008.

Imperfect measurements: the new ”observable” state

I Theleft stochastic matrixη:ηµ⁰,µ∈ |0,1]is the probability of having the imperfect outcomeµ⁰∈ {1, . . . ,m⁰}knowing that the perfect one isµ∈ {1, . . . ,m}.

I ρb_k =

E

obeys to¹⁴ ρb_k+1=L^u_µ^k−τ0

k

(ρb_k), where

I L^uµ⁰(ρ) =b ^L

u µ0(ρ)b Tr

L^u_µ0(bρ) withL^u_µ0(ρ) =b Pm

µ=1ηµ⁰,µM_µ^uρMb _µ^u†;

I µ⁰_k is a random variable taking valuesµ⁰ in{1,· · ·,m⁰}with probabilityp^u_µ^k−τ0,ρbk =Tr

L^u_µ^k−τ0 (ρb_k) .

I

E

I Assumption: for alln16=n2in{1,· · ·,d},there exists µ⁰ ∈ {1,· · · ,m⁰}, s.t. Tr

L⁰_µ0(|n₁i hn₁|) 6=Tr

L⁰_µ0(|n₂i hn₂|) . Open-loop convergence ofρb_k towards|ni hn|with prob.hn|ρb₀|ni.

14R. Somaraju et al., ACC 2012 (http://arxiv.org/abs/1109.5344)

Feedback stabilization of

towards

I With the previous functionV(ρ) =V₀(ρ)−₂Pd

n=1(hn|ρ|ni)² stabilizeχb= (ρ, βb ₁,· · · , βτ)towardsχ¯= (|ni h¯ n|¯ ,0, . . . ,0)using thecontrol-Lyapunov function

W(χ) =b V(K^β1(K^β2(. . . .K^βτ(ρ)b . . .))).

I Foru¯andsmall enough, the feedback u_k =f(χb_k) =:argmin

ξ∈[−¯u,¯u]

E

I Sinceρb_k =

E

convergences towards the pure state|¯ni h¯n|,ρ_k converges also towards the same pure state.

Quantum separation principle

I

Estimate the hidden state

by

satisfying

k (ρb^est_k )

where

I ρ_k obeys to

ρ_k+1=M^uµk−τk (ρ_k) with the stabilizing feedback

uk =f(bρ^est_k ,uk−1, . . . ,uk−τ) computed usingρb^estinstead ofρ.

I

µ

= µ

with probability η

.

I

Filter stability:

Tr

qp

ρb_kρb^est_k p ρb_k

2

is always a sub-martingale

.

I

If ker(

ker(ρ

and

converge almost surely towards the target state

15P.R., IEEE Trans. Automatic Control, 2011.

Closed-loop experimental data

•Initial state coherent state withn¯=3 photons

•State estimation via a quantum filterof stateρ^est_k .

•Lyapunov state feedback uk =f(ρ_k)stabilizing towards|¯ni hn|¯

•ρ_k is replaced by its estimateρ^est_k in the feedback (quantum separation principle) Sampling period 80µs Experience imperfections:

•detection efficiency 40%

•detection error rate 10%

•delay 4 sampling periods

•truncation to 9 photons

•finite photon life time

•atom occupancy 30%

Stabilization around 3-photon state

The left stochastic matrix for the LKB photon box

For each control input

I

we have a total of

3

7

21 Kraus operators. The jumps are labeled by

with

µ^a∈ {no,g,e,gg,ge,eg,ee}

labeling atom related jumps and

cavity decoherence jumps.

I

we have only

6 real detection possibilities

µ⁰ ∈ {no,g,e,gg,ge,ee}

corresponding respectively to no detection, a single detection in

double detection both in

and the other in

µ⁰\µ (no, µ^c) (g, µ^c) (e, µ^c) (gg, µ^c) (ee, µ^c) (ge, µ^c)or(eg, µ^c)

no 1 1-_d 1-_d (1-_d)² (1-_d)² (1-_d)²

g 0 d(1-ηg) dηe 2d(1-d)(1-ηg) 2d(1-d)ηe d(1-d)(1-ηg+ηe) e 0 dηg d(1-ηe) 2d(1-d)ηg 2d(1-d)(1-ηe) d(1-d)(1-ηe+ηg)

gg 0 0 0 ²_d(1-ηg)^{2 2}_dη²_e ²_dηe(1-ηg)

ge 0 0 0 2²_dηg(1-ηg) 2²_dηe(1-ηe) ²_d((1-ηg)(1-ηe) +ηgηe)

ee 0 0 0 ²_dη_g^{2 2}_d(1-ηe)^{2 2}_dηg(1-ηe)

16R. Somaraju et al.: ACC 2012 (http://arxiv.org/abs/1109.5344)