Stabilization of discrete-time quantum systems subject to non ...
Stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays Pierre Rouchon, Mines-ParisTech, ` Centre Automatique et Systemes, [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) ` Igor Dotsenko, Serge Haroche (ENS, College de France) ´ Michel Brune, Jean-Michel Raimond, Clement Sayrin (ENS).
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 1 Controlled QND Markov chains Open-loop convergence Feedback, delay and closed-loop convergence Imperfect measurements 1
H. Amini et al. Preprint arxiv:1201.1387, 2012.
Model of classical systems
perturbation system
control
measure For the harmonic oscillator of pulsation ω with measured position y , controlled by the force u and subject to an additional unknown force w. x = (x1 , x2 ) ∈ R2 , d dt x1
= x2 ,
d dt x2
y = x1
= −ω 2 x1 + u + w
Feedback for classical systems perturbation
set point
observer/controller
measure
system control
feedback
Proportional Integral Derivative (PID) for with the set point v = ysp
d2 y dt 2
d u = −Kp y − ysp − Kd dt y − ysp − Kint
= −ω 2 y + u + w Z y − ysp
with the positive gains (Kp , Kd , Kint ) tuned as follows (0 < Ω0 ∼ ω, 0 < ξ ∼ 1, 0 < 1: Kp = Ω20 ,
Kd = 2ξΩ0 ,
, Kint = Ω30 .
Feedback for the quantum system S Key issue: back-action due to the measurement process. Measurement-based feedback: measurement back-action on S is stochastic (collapse of the wave-packet); controller is classical; the control input u is a classical variable ¨ appearing in some controlled Schrodinger equation; u depends on the past measures. Coherent feedback: the system S is coupled to another quantum system (the controller); the composite system, S ⊗ controller, is an open-quantum system relaxing to some target (separable) state (related to reservoir engineering). This talk is devoted to the first experimental realization of a measurement-based state feedback. It has been done at ´ Laboratoire Kastler Brossel of Ecole Normale Superieure by the Cavity Quantum ElectroDynamics (CQED) group of Serge Haroche.2 2
C. 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 u = AeıΦ ; measure output y ∈ {g, e}. • Sampling time 80 µs long enough for numerical computations. 3
Courtesy of Igor Dotsenko
The ideal Markov chain for the wave function |ψi 4 g
UR 1
yk
UR 2
g e
UC D uk
k
g
k
Input uk , state |ψk i =
g
P
Mg
n≥0
k
e
Me
k
M yk M yk
k 1
k k
ψkn |ni, output yk :
ψk+1/2 = Myk |ψk i , kMyk |ψk i k
|ψk+1 i = Duk ψk+1/2
with
yk = g (resp. e) with probability kMg |ψk i k2 (resp. kMe |ψk i k2 ); I measurement Kraus operators Mg = cos φ0 N+φR and 2 † † φ0 N+φR Me = sin : Mg Mg + Me Me = 1 with 2 I
N = a† a = diag(0, 1, 2, . . .) the photon number operator; I 4
†
ua −ua displacement unitary with √ √operator (u ∈ R): Du = e a = upper diag( 1, 2, . . .) the photon annihilation operator.
S. Haroche and J.M. Raimond. Exploring the Quantum: Atoms, Cavities and Photons. Oxford Graduate Texts, 2006.
The ideal Markov chain for the density operator ρ = |ψi hψ| Diagonal elements of ρ, ρnn = hn| ρ |ni = |ψ n |2 , form the photon number distribution. † † Duk Mg ρk Mg Duk † y = g with probability p = Tr M ρ M g k g,k k g Tr M ρ M † g k g ρk +1 = Duk Me ρk Me† Du†k † y = e with probability p = Tr M ρ M e k e k e,k Tr Me ρk Me† †
ua −ua Displacement unitary with √ √operator (u ∈ R): Du = e a = upper diag( 1, 2, . . .) the photon annihilation operator. I Measurement Kraus operators Mg = cos φ0 N+φR and 2 † † φ0 N+φR Me = sin : Mg Mg + Me Me = 1 with 2
I
N = a† a = diag(0, 1, 2, . . .) the photon number operator.
Open-loop behavior (u = 0) An experimental open-loop trajectory starting from coherent state q P ¯ = 3 photons: |ψ0 i = e−n¯/2 n≥0 n¯n!n |ni . ρ0 = |ψ0 i hψ0 | with n I
A fast convergence towards |ni hn| for some n,
I
followed by a slow relaxation towards vacuum |0i h0|: decoherence due to finite photon life time around 70 ms (not included into the ideal model).
Open-loop stability of ρk +1 =
Myk ρk My†
k
Tr(Myk ρk My†k ) 5
explaining this fast
convergence when φ0 /π is irrational I
for any n, ρnn k = hn| ρk |ni is a martingale:
nn E ρnn k+1 | ρk = ρk ;
I
almost all realizations starting from ρ0 converge towards a photon number state |ni hn|; the probability to converge towards |ni hn| is given by the initial population ρnn 0 .
This convergence characterizes a Quantum Non Demolition (QND) measurement of photons (counting photons without destroying them). 5
H. Amini et al., IEEE Trans. Automatic Control, in press, 2012.
Closed-loop experimental data • Initial state coherent state ¯ = 3 photons with n • State estimation via a quantum filter of state ρest k . • Lyapunov state feedback uk = f (ρk ) stabilizing ¯ i hn ¯| towards |n • ρk is replaced by its estimate ρest k in the feedback (quantum separation principle) Sampling period 80 µs 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 6 we propose the following stabilizing state feedback law ¯ i, based on the fidelity towards the target state |n u = f (ρ) =: Argmin V (Dυ ρDυ† ) ¯ ,u ¯] υ∈[−u
¯ i hn ¯ | , ρ) = 1 − ρn¯n¯ and u ¯ > 0 is small. where V (ρ) = 1 − F (|n Two important issues. I
I
6
The state ρ is not directly measured; output delay is of 4 steps: it was solved by a quantum filter taking into account the delay. ¯: V is maximum and equal to 1 for any ρ = |ni hn| with n 6= n ¯ + 1 (close to the target) and no distinction between n = n ¯ + 1000 (far from the target). This issue has been solved n by changing the Lyapunov function V .
I. Dotsenko et al.: Quantum feedback by discrete quantum non-demolition measurements: towards on-demand generation of photon-number states. Physical Review A80:013805, 2009.
¯) 7 Lyapunov-based feedback (goal photon number n P 2
V (ρ) = n −hn |ρ| ni + σn hn| ρ |ni function with > 0 small enough, 1 Pn¯ 1 1 ν=1 ν − ν 2 , 4 + P ¯ n 1 1 ν=n+1 ν − ν 2 , σn = 0, Pn 1 1 ¯+1 ν + ν 2 , ν=n
is a strict control Lyapunov
if n = 0; ¯ − 1]; if n ∈ [1, n ¯; if n = n ¯ + 1, +∞[, if n ∈ [n ¯ > 0 small). and the feedback u = f (ρ) =: Argmin V Dυ ρDυ† (u ¯ ,u ¯] υ∈[−u
In closed-loop, V (ρ) becomes a strict super-martingale:
E (V (ρk+1 | ρk ) = V (ρk ) − Q(ρk ) ¯ i hn ¯ |. with Q(ρ) continuous, positive and vanishing only when ρ = |n This feedback law yields I
global stabilization for any finite dimensional approximation consisting in truncation to nmax < +∞ photons.
I
global approximate stabilization for nmax = +∞.
7
H. Amini et al.: CDC-2011.
The control Lyapunov function used for the photon box nmax = 9. Coefficients σn of the control Lyapunov function 1
0.8
0.6
0.4
0.2
0
0
V (ρ) =
2
4 6 photon number n
P9
n=0
−hn |ρ| ni2 + σn hn| ρ |ni
8
Global approximate stabilization (nmax = +∞) 8 I
The feedback u = Argmin V Dυ ρDυ† ensures a strict ¯ ,u ¯] υ∈[−u
closed-loop Lyapunov function X V (ρ) = −hn |ρ| ni2 + σn hn| ρ |ni n≥0
I
8
with σn ∼ log n, for n large (high photon-number cut-off). ¯ > 0 (small), For any η > 0 and C > 0, exist > 0 and u such that, for any initial value ρ0 with V (ρ0 ) ≤ C, ρnk¯n¯ converges almost surely towards a number inside [1 − η, 1]. With Tr (ρk ) = 1, and ρk = ρ†k ≥ 0, this means, that almost surely, for k large enough, ρk is close (weak-* topology) to ¯ i hn ¯ |. the goal Fock state ρ¯ = |n
R. Somaraju, M. Mirrahimi, P.R.: CDC 2011 http://arxiv.org/abs/1103.1724
Design of the strict control Lyapunov function9 Exploit open-loop stability: for each n, hn |ρ| ni is a martingale; P 2 V (ρ) = − 12 n hn |ρ| ni is a super-martingale with
E (V (ρk+1 ) / ρk ) = V (ρk ) − Q(ρk ) where Q(ρ) ≥ 0 and Q(ρ) = 0 iff, ρ is a Fock state. For closing the loop take σn such that E X D u 7→ σn n Du ρDu† n n
¯ i hn ¯| 1. is strongly convex for ρ = |n ¯. 2. is strongly concave for ρ = |ni hn|, n 6= n This is achieved by inverting the Laplacian matrix associated to the control Hamiltionan H = ı(a − a† ). Remember that Du = e−ıuH . 9
H. Amini et al., CDC 2011,http://arxiv.org/abs/1103.1365
Estimation of ρk from the past measures yν≤k via a quantum filter Duk Myk ρk My†k Du†k ρ = k+1 Tr Myk ρk My†k † † Duk Myk ρest k Myk D uk ρest = k+1 † Tr Myk ρest k Myk
I
Assume we know ρk and uk . Outcome of measure no k , yk , defines the jump operator Myk and we can compute ρk+1 .
I
Quantum filter and real-time estimation: initialize the estimation ρest to some initial value ρest 0 and update at step k with measured jumps yk and the known controls uk .
I
Quantum separation principle for stabilization towards a pure state10 : assume that the feedback u = f (ρ) ensures global asymptotic convergence towards a pure state; then, if est ker(ρest 0 ) ⊂ ker(ρ0 ), the feedback uk = f (ρk ) ensures also global asymptotic convergence towards the same pure state.
10
Bouten, van Handel, 2008.
A modified quantum filter with a measure delayed by one step Without delay the stabilizing feedback reads My ρest M † uk = Argmin V Dv Tr Mk ρk est Myk† Dv† ( yk k yk ) With delay, we have only access to yk −1 and the stabilizing feedback uses the Kraus map K(ρ) = Mg ρMg† + Me ρMe† : † uk = Argmin V Dv K(ρest k )Dv This is the same feedback law but with another state estimation at step k : K(ρest k ) instead of
† Myk ρest k My
k
Tr(Myk ρest My†k ) k
.
I
System theoretical interpretation: K(ρest k ) stands for the the prediction of cavity state at step k. This prediction is in average (expectation value) since yk ∈ {g, e} can take two values.
I
Quantum physics interpretation: K(ρest 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 step k .
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 by coherent feedback: similarly to the Watt regulator where 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 College de France);
I
Coherent feedback is closely related to reservoir engineering: exploit and design the measurement process (here operators Mµ ) and its intrinsic back-action to ensure convergence of the ensemble-average dynamics towards a unique pure state (Ticozzi, Viola, . . . )
Watt regulator: a classical analogue of quantum coherent feedback.
11
The first variations of speed δω and governor angle δθ obey to d dt δω = −aδθ d2 d δθ = −Λ dt δθ dt 2
− Ω2 (δθ − bδω)
with (a, b, Λ, Ω) positive parameters. Third order system d3 δω dt 3
2
2 2d d = −Λ dt 2 δω − Ω dt δω − abΩ δω = 0
Characteristic polynomial P(s) = s3 + Λs2 + Ω2 s + abΩ2 with roots having negative real parts iff Λ > ab: governor damping must be strong enough to ensure asymptotic stability of the closed-loop system. 11
J.C. Maxwell: On governors. Proc. of the Royal Society, No.100, 1868.
¨ Reservoir engineering stabilizing Schrodinger cats for the photon box 12 Wigner functions of the various states that can be produced by such reservoir based on composite dispersive/resonant atom/cavity interaction. (a)
(b)
(c)
(d)
0.6
W(γ) 0.4 0.2
3 (e)
(g)
(h)
Im(γ)
(f)
0 -0.2
0
-0.4
-0.6
-3 -3
0 12
Re(γ) 3
Sarlette et al: PRL 107:010402,2011 and PRA to appear in 2012.
Control of a QND Markov chain with delay τ u
ρk+1 =
u Mµkk−τ (ρk )
u
†
Mµk−τ ρ Mµk −τ =: ku k ku † Tr Mµkk −τ ρk Mµkk−τ
I
To each measurement outcome µ is attached the Kraus operator Mµu ∈ Cd×d depending on µ and also on a scalar control input Pm u ∈ R. For each u, µ=1 Mµu † Mµu = I, and we have the Kraus P m map Ku (ρ) = µ=1 Mµu ρMµu †
I
µk is a random variable in {1, · · · , m} with taking values µ u
u
u
k −τ probability pµ,ρ Mµk −τ ρk Mµk−τ k = Tr
†
.
I
For u = 0, the measurement operators Mµ0 are diagonal in the same orthonormal basis { |ni | n ∈ {1, · · · , d}}, therefore Pd Mµ0 = n=1 cµ,n |ni hn| with cµ,n ∈ C.
I
For all n1 6= n2 in {1, · · · , d}, there exists µ ∈ {1, · · · , m} such that |cµ,n1 |2 6= |cµ,n2 |2 .
Open-loop convergence ρk+1 = M0µk (ρk ) For any initial condition ρ0 , I
with probability one, ρk converges to one of the d states |ni hn| with n ∈ {1, · · · , d}.
I
the probability of convergence towards the state |ni hn| depends only on ρ0 and is given by hn| ρ0 |ni .
Proof based on I
the martingales hn| ρ |ni
I
the super-martingale V (ρ) := −
P
n
hn|ρ|ni 2
2 satisfying
E (V (ρk+1 )|ρk ) − V (ρk ) = −Q(ρk ) ≤ 0 with Q(ρ) = I
1 4
0 0 n,µ,ν pµ,ρ pν,ρ
P
|cµ,n |2 hn|ρ|ni 0 pµ,ρ
−
|cν,n |2 hn|ρ|ni 0 pν,ρ
Q(ρ) = 0 iff exists n ∈ {1, . . . , d} such that ρ = |ni hn|.
2
.
u ¯ i hn ¯| Feedback stabilization of ρk+1 = Mµkk−τ (ρk ) towards |n I
Pd V0 (ρ) = n=1 σn hn| ρ |ni with σn ≥ 0 chosen such that σn¯ = 0 ¯ , the second-order u-derivative of and for any n 6= n u V0 (K (|ni hn|)) at u = 0 is strictly negative (Ku is the Kraus map): set of linear equations in σn solved by inverting an irreducible M-matrix (Perron-Frobenius theorem).
I
The function ( > 0 small enough): Pd V (ρ) = V0 (ρ) − 2 n=1 (hn| ρ |ni)2 still admits a unique global ¯ i hn ¯ |; for u close to 0, u 7→ V (Ku (|ni hn|)) is minimum at |n ¯ and strongly convexe for n = n ¯. strongly concave for any n 6= n
I
The delay of τ steps: stabilize the state χ = (ρ, β1 , · · · , βτ ) (βl ¯ i hn ¯ | , 0, . . . , 0) control input u delayed l steps) towards χ ¯ = (|n using the control-Lyapunov function W (χ) = V (Kβ1 (Kβ2 (. . . . . . Kβτ (ρ) . . .))).
¯ and small enough, the feedback For u uk = f (χk ) =: argmin ¯ ,u ¯] ξ∈[−u
E (W (χk +1 )|χk , uk = ξ)
ensures global stabilization towards χ. ¯
Quantum separation principle
I
Estimate the hidden state ρ by ρest satisfying u
k−τ est ρest k+1 = Mµk (ρk )
u
where ρ obeys to ρk+1 = Mµkk−τ (ρk ) with the stabilizing est feedback uk = f (ρest k , uk−1 , . . . , uk −τ ) computed using ρ instead of ρ. I
est If ker(ρest 0 ) ⊂ ker(ρ0 ), ρk and ρk converge almost surely ¯ i hn ¯ |. towards the target state |n 13 Proof based on : I I I
13
¯ | ρk |n ¯ i ∈ [0, 1], hn ¯ | ρk |n ¯ i |ρ0 , ρest linearity of E hn versus ρ0 , 0 decomposition ρest = γρ + (1 − γ)ρc0 with γ ∈]0, 1[. 0 0
Bouten, van Handel, 2008.
Imperfect measurements: the new ”observable” state ρb The left stochastic matrix η: ηµ0 ,µ ∈ |0, 1] is the probability of having the imperfect outcome µ0 ∈ {1, . . . , m0 } knowing that the perfect one is µ ∈ {1, . . . , m}. I ρ bk = E ρk |ρ0 , µ00 , . . . , µ0k−1 , u−τ , . . . , uk−τ −1 obeys to14 I
u
ρbk+1 = Lµk0−τ (b ρk ),
where
k
I I
ρ) = Luµ0 (b
Lu 0 (b ρ) µ Tr Luµ0 (b ρ)
with Luµ0 (b ρ) =
Pm
µ=1
ηµ0 ,µ Mµu ρbMµu † ;
µ0k is a random variable µ0 in {1, · · · , m0 } with taking values u
u
k −τ probability pµk0−τ ρk ) . ,b ρk = Tr Lµ0 (b
I
I
E (b ρk+1 |b ρk = ρ, uk−τ = u) = Ku (b ρ)
Assumption: for all n1 6=n2 in {1, · · · , d}, there exists µ0 ∈ {1, · · · , m0 }, s.t. Tr L0µ0 (|n1 i hn1 |) 6= Tr L0µ0 (|n2 i hn2 |) .
Open-loop convergence of ρbk towards |ni hn| with prob. hn| ρb0 |ni. 14
R. Somaraju et al., ACC 2012 (http://arxiv.org/abs/1109.5344)
u ¯ i hn ¯| Feedback stabilization of ρbk+1 = Lµk−τ (b ρk ) towards |n k
I
Pd With the previous function V (ρ) = V0 (ρ) − 2 n=1 (hn| ρ |ni)2 ¯ i hn ¯ | , 0, . . . , 0) using stabilize χ b = (b ρ, β1 , · · · , βτ ) towards χ ¯ = (|n the control-Lyapunov function W (b χ) = V (Kβ1 (Kβ2 (. . . . . . Kβτ (b ρ) . . .))).
I
¯ and small enough, the feedback For u uk = f (b χk ) =: argmin ¯ ,u ¯] ξ∈[−u
E (W (b χk +1 )|b χk , uk = ξ)
ensures global stabilization of χ bk towards χ. ¯ I
Since ρbk = E ρk |ρ0 , µ00 , . . . , µ0k −1 , u−τ , . . . , uk−τ −1 ¯ i hn ¯ |, ρk converges also convergences towards the pure state |n towards the same pure state.
Quantum separation principle I
Estimate the hidden state ρb by ρbest satisfying u
k−τ ρest ρbest k ) k+1 = Lµ0 (b k
where I
ρk obeys to
u
ρk+1 = Mµkk−τ (ρk ) with the stabilizing feedback uk = f (b ρest k , uk−1 , . . . , uk−τ ) computed using ρbest instead of ρ. I
µ0k = µ0 with probability ηµ0 ,µk . est
2 q p p est ρbk ρbk ρbk is , Tr
I
Filter stability: F ρbk , ρbk
I
always a sub-martingale15 . If ker(b ρest best 0 ) ⊂ ker(ρ0 ), ρk and ρ k converge almost surely ¯ ¯ towards the target state |ni hn|.
15
P.R., IEEE Trans. Automatic Control, 2011.
Closed-loop experimental data • Initial state coherent state ¯ = 3 photons with n • State estimation via a quantum filter of state ρest k . • Lyapunov state feedback uk = f (ρk ) stabilizing ¯ i hn ¯| towards |n • ρk is replaced by its estimate ρest k in the feedback (quantum separation principle) Sampling period 80 µs 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 box16 For each control input u,
µ0 \ µ
I
we have a total of m = 3 × 7 = 21 Kraus operators. The jumps are labeled by µ = (µa , µc ) with µa ∈ {no, g, e, gg, ge, eg, ee} labeling atom related jumps and µc ∈ {o, +, −} cavity decoherence jumps.
I
we have only m0 = 6 real detection possibilities µ0 ∈ {no, g, e, gg, ge, ee} corresponding respectively to no detection, a single detection in g, a single detection in e, a double detection both in g, a double detection one in g and the other in e, and a double detection both in e.
(no, µc )
no g e gg ge ee
1 0 0 0 0 0 16
(g, µc ) 1-d d (1-ηg ) d ηg 0 0 0
(e, µc ) 1-d d ηe d (1-ηe ) 0 0 0
(gg, µc )
(ee, µc )
(ge, µc ) or (eg,
)2
)2
(1-d )2 d (1-d )(1-ηg + d (1-d )(1-ηe + 2d ηe (1-ηg ) 2d ((1-ηg )(1-ηe ) + 2d ηg (1-ηe )
(1-d 2d (1-d )(1-ηg ) 2d (1-d )ηg 2d (1-ηg )2 22d ηg (1-ηg ) 2d ηg2
(1-d 2d (1-d )ηe 2d (1-d )(1-ηe ) 2d ηe2 22d ηe (1-ηe ) 2d (1-ηe )2
R. Somaraju et al.: ACC 2012 (http://arxiv.org/abs/1109.5344)
Delayed Complex Systems June 2012, Palma de Majorque Joint work with Hadis Amini (Mines-ParisTech), Mazyar Mirrahimi (INRIA), Ram Somaraju (INRIA/Vrije Universiteit Brussel) ` Igor Dotsenko, Serge Haroche (ENS, College de France) ´ Michel Brune, Jean-Michel Raimond, Clement Sayrin (ENS).
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 1 Controlled QND Markov chains Open-loop convergence Feedback, delay and closed-loop convergence Imperfect measurements 1
H. Amini et al. Preprint arxiv:1201.1387, 2012.
Model of classical systems
perturbation system
control
measure For the harmonic oscillator of pulsation ω with measured position y , controlled by the force u and subject to an additional unknown force w. x = (x1 , x2 ) ∈ R2 , d dt x1
= x2 ,
d dt x2
y = x1
= −ω 2 x1 + u + w
Feedback for classical systems perturbation
set point
observer/controller
measure
system control
feedback
Proportional Integral Derivative (PID) for with the set point v = ysp
d2 y dt 2
d u = −Kp y − ysp − Kd dt y − ysp − Kint
= −ω 2 y + u + w Z y − ysp
with the positive gains (Kp , Kd , Kint ) tuned as follows (0 < Ω0 ∼ ω, 0 < ξ ∼ 1, 0 < 1: Kp = Ω20 ,
Kd = 2ξΩ0 ,
, Kint = Ω30 .
Feedback for the quantum system S Key issue: back-action due to the measurement process. Measurement-based feedback: measurement back-action on S is stochastic (collapse of the wave-packet); controller is classical; the control input u is a classical variable ¨ appearing in some controlled Schrodinger equation; u depends on the past measures. Coherent feedback: the system S is coupled to another quantum system (the controller); the composite system, S ⊗ controller, is an open-quantum system relaxing to some target (separable) state (related to reservoir engineering). This talk is devoted to the first experimental realization of a measurement-based state feedback. It has been done at ´ Laboratoire Kastler Brossel of Ecole Normale Superieure by the Cavity Quantum ElectroDynamics (CQED) group of Serge Haroche.2 2
C. 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 u = AeıΦ ; measure output y ∈ {g, e}. • Sampling time 80 µs long enough for numerical computations. 3
Courtesy of Igor Dotsenko
The ideal Markov chain for the wave function |ψi 4 g
UR 1
yk
UR 2
g e
UC D uk
k
g
k
Input uk , state |ψk i =
g
P
Mg
n≥0
k
e
Me
k
M yk M yk
k 1
k k
ψkn |ni, output yk :
ψk+1/2 = Myk |ψk i , kMyk |ψk i k
|ψk+1 i = Duk ψk+1/2
with
yk = g (resp. e) with probability kMg |ψk i k2 (resp. kMe |ψk i k2 ); I measurement Kraus operators Mg = cos φ0 N+φR and 2 † † φ0 N+φR Me = sin : Mg Mg + Me Me = 1 with 2 I
N = a† a = diag(0, 1, 2, . . .) the photon number operator; I 4
†
ua −ua displacement unitary with √ √operator (u ∈ R): Du = e a = upper diag( 1, 2, . . .) the photon annihilation operator.
S. Haroche and J.M. Raimond. Exploring the Quantum: Atoms, Cavities and Photons. Oxford Graduate Texts, 2006.
The ideal Markov chain for the density operator ρ = |ψi hψ| Diagonal elements of ρ, ρnn = hn| ρ |ni = |ψ n |2 , form the photon number distribution. † † Duk Mg ρk Mg Duk † y = g with probability p = Tr M ρ M g k g,k k g Tr M ρ M † g k g ρk +1 = Duk Me ρk Me† Du†k † y = e with probability p = Tr M ρ M e k e k e,k Tr Me ρk Me† †
ua −ua Displacement unitary with √ √operator (u ∈ R): Du = e a = upper diag( 1, 2, . . .) the photon annihilation operator. I Measurement Kraus operators Mg = cos φ0 N+φR and 2 † † φ0 N+φR Me = sin : Mg Mg + Me Me = 1 with 2
I
N = a† a = diag(0, 1, 2, . . .) the photon number operator.
Open-loop behavior (u = 0) An experimental open-loop trajectory starting from coherent state q P ¯ = 3 photons: |ψ0 i = e−n¯/2 n≥0 n¯n!n |ni . ρ0 = |ψ0 i hψ0 | with n I
A fast convergence towards |ni hn| for some n,
I
followed by a slow relaxation towards vacuum |0i h0|: decoherence due to finite photon life time around 70 ms (not included into the ideal model).
Open-loop stability of ρk +1 =
Myk ρk My†
k
Tr(Myk ρk My†k ) 5
explaining this fast
convergence when φ0 /π is irrational I
for any n, ρnn k = hn| ρk |ni is a martingale:
nn E ρnn k+1 | ρk = ρk ;
I
almost all realizations starting from ρ0 converge towards a photon number state |ni hn|; the probability to converge towards |ni hn| is given by the initial population ρnn 0 .
This convergence characterizes a Quantum Non Demolition (QND) measurement of photons (counting photons without destroying them). 5
H. Amini et al., IEEE Trans. Automatic Control, in press, 2012.
Closed-loop experimental data • Initial state coherent state ¯ = 3 photons with n • State estimation via a quantum filter of state ρest k . • Lyapunov state feedback uk = f (ρk ) stabilizing ¯ i hn ¯| towards |n • ρk is replaced by its estimate ρest k in the feedback (quantum separation principle) Sampling period 80 µs 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 6 we propose the following stabilizing state feedback law ¯ i, based on the fidelity towards the target state |n u = f (ρ) =: Argmin V (Dυ ρDυ† ) ¯ ,u ¯] υ∈[−u
¯ i hn ¯ | , ρ) = 1 − ρn¯n¯ and u ¯ > 0 is small. where V (ρ) = 1 − F (|n Two important issues. I
I
6
The state ρ is not directly measured; output delay is of 4 steps: it was solved by a quantum filter taking into account the delay. ¯: V is maximum and equal to 1 for any ρ = |ni hn| with n 6= n ¯ + 1 (close to the target) and no distinction between n = n ¯ + 1000 (far from the target). This issue has been solved n by changing the Lyapunov function V .
I. Dotsenko et al.: Quantum feedback by discrete quantum non-demolition measurements: towards on-demand generation of photon-number states. Physical Review A80:013805, 2009.
¯) 7 Lyapunov-based feedback (goal photon number n P 2
V (ρ) = n −hn |ρ| ni + σn hn| ρ |ni function with > 0 small enough, 1 Pn¯ 1 1 ν=1 ν − ν 2 , 4 + P ¯ n 1 1 ν=n+1 ν − ν 2 , σn = 0, Pn 1 1 ¯+1 ν + ν 2 , ν=n
is a strict control Lyapunov
if n = 0; ¯ − 1]; if n ∈ [1, n ¯; if n = n ¯ + 1, +∞[, if n ∈ [n ¯ > 0 small). and the feedback u = f (ρ) =: Argmin V Dυ ρDυ† (u ¯ ,u ¯] υ∈[−u
In closed-loop, V (ρ) becomes a strict super-martingale:
E (V (ρk+1 | ρk ) = V (ρk ) − Q(ρk ) ¯ i hn ¯ |. with Q(ρ) continuous, positive and vanishing only when ρ = |n This feedback law yields I
global stabilization for any finite dimensional approximation consisting in truncation to nmax < +∞ photons.
I
global approximate stabilization for nmax = +∞.
7
H. Amini et al.: CDC-2011.
The control Lyapunov function used for the photon box nmax = 9. Coefficients σn of the control Lyapunov function 1
0.8
0.6
0.4
0.2
0
0
V (ρ) =
2
4 6 photon number n
P9
n=0
−hn |ρ| ni2 + σn hn| ρ |ni
8
Global approximate stabilization (nmax = +∞) 8 I
The feedback u = Argmin V Dυ ρDυ† ensures a strict ¯ ,u ¯] υ∈[−u
closed-loop Lyapunov function X V (ρ) = −hn |ρ| ni2 + σn hn| ρ |ni n≥0
I
8
with σn ∼ log n, for n large (high photon-number cut-off). ¯ > 0 (small), For any η > 0 and C > 0, exist > 0 and u such that, for any initial value ρ0 with V (ρ0 ) ≤ C, ρnk¯n¯ converges almost surely towards a number inside [1 − η, 1]. With Tr (ρk ) = 1, and ρk = ρ†k ≥ 0, this means, that almost surely, for k large enough, ρk is close (weak-* topology) to ¯ i hn ¯ |. the goal Fock state ρ¯ = |n
R. Somaraju, M. Mirrahimi, P.R.: CDC 2011 http://arxiv.org/abs/1103.1724
Design of the strict control Lyapunov function9 Exploit open-loop stability: for each n, hn |ρ| ni is a martingale; P 2 V (ρ) = − 12 n hn |ρ| ni is a super-martingale with
E (V (ρk+1 ) / ρk ) = V (ρk ) − Q(ρk ) where Q(ρ) ≥ 0 and Q(ρ) = 0 iff, ρ is a Fock state. For closing the loop take σn such that E X D u 7→ σn n Du ρDu† n n
¯ i hn ¯| 1. is strongly convex for ρ = |n ¯. 2. is strongly concave for ρ = |ni hn|, n 6= n This is achieved by inverting the Laplacian matrix associated to the control Hamiltionan H = ı(a − a† ). Remember that Du = e−ıuH . 9
H. Amini et al., CDC 2011,http://arxiv.org/abs/1103.1365
Estimation of ρk from the past measures yν≤k via a quantum filter Duk Myk ρk My†k Du†k ρ = k+1 Tr Myk ρk My†k † † Duk Myk ρest k Myk D uk ρest = k+1 † Tr Myk ρest k Myk
I
Assume we know ρk and uk . Outcome of measure no k , yk , defines the jump operator Myk and we can compute ρk+1 .
I
Quantum filter and real-time estimation: initialize the estimation ρest to some initial value ρest 0 and update at step k with measured jumps yk and the known controls uk .
I
Quantum separation principle for stabilization towards a pure state10 : assume that the feedback u = f (ρ) ensures global asymptotic convergence towards a pure state; then, if est ker(ρest 0 ) ⊂ ker(ρ0 ), the feedback uk = f (ρk ) ensures also global asymptotic convergence towards the same pure state.
10
Bouten, van Handel, 2008.
A modified quantum filter with a measure delayed by one step Without delay the stabilizing feedback reads My ρest M † uk = Argmin V Dv Tr Mk ρk est Myk† Dv† ( yk k yk ) With delay, we have only access to yk −1 and the stabilizing feedback uses the Kraus map K(ρ) = Mg ρMg† + Me ρMe† : † uk = Argmin V Dv K(ρest k )Dv This is the same feedback law but with another state estimation at step k : K(ρest k ) instead of
† Myk ρest k My
k
Tr(Myk ρest My†k ) k
.
I
System theoretical interpretation: K(ρest k ) stands for the the prediction of cavity state at step k. This prediction is in average (expectation value) since yk ∈ {g, e} can take two values.
I
Quantum physics interpretation: K(ρest 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 step k .
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 by coherent feedback: similarly to the Watt regulator where 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 College de France);
I
Coherent feedback is closely related to reservoir engineering: exploit and design the measurement process (here operators Mµ ) and its intrinsic back-action to ensure convergence of the ensemble-average dynamics towards a unique pure state (Ticozzi, Viola, . . . )
Watt regulator: a classical analogue of quantum coherent feedback.
11
The first variations of speed δω and governor angle δθ obey to d dt δω = −aδθ d2 d δθ = −Λ dt δθ dt 2
− Ω2 (δθ − bδω)
with (a, b, Λ, Ω) positive parameters. Third order system d3 δω dt 3
2
2 2d d = −Λ dt 2 δω − Ω dt δω − abΩ δω = 0
Characteristic polynomial P(s) = s3 + Λs2 + Ω2 s + abΩ2 with roots having negative real parts iff Λ > ab: governor damping must be strong enough to ensure asymptotic stability of the closed-loop system. 11
J.C. Maxwell: On governors. Proc. of the Royal Society, No.100, 1868.
¨ Reservoir engineering stabilizing Schrodinger cats for the photon box 12 Wigner functions of the various states that can be produced by such reservoir based on composite dispersive/resonant atom/cavity interaction. (a)
(b)
(c)
(d)
0.6
W(γ) 0.4 0.2
3 (e)
(g)
(h)
Im(γ)
(f)
0 -0.2
0
-0.4
-0.6
-3 -3
0 12
Re(γ) 3
Sarlette et al: PRL 107:010402,2011 and PRA to appear in 2012.
Control of a QND Markov chain with delay τ u
ρk+1 =
u Mµkk−τ (ρk )
u
†
Mµk−τ ρ Mµk −τ =: ku k ku † Tr Mµkk −τ ρk Mµkk−τ
I
To each measurement outcome µ is attached the Kraus operator Mµu ∈ Cd×d depending on µ and also on a scalar control input Pm u ∈ R. For each u, µ=1 Mµu † Mµu = I, and we have the Kraus P m map Ku (ρ) = µ=1 Mµu ρMµu †
I
µk is a random variable in {1, · · · , m} with taking values µ u
u
u
k −τ probability pµ,ρ Mµk −τ ρk Mµk−τ k = Tr
†
.
I
For u = 0, the measurement operators Mµ0 are diagonal in the same orthonormal basis { |ni | n ∈ {1, · · · , d}}, therefore Pd Mµ0 = n=1 cµ,n |ni hn| with cµ,n ∈ C.
I
For all n1 6= n2 in {1, · · · , d}, there exists µ ∈ {1, · · · , m} such that |cµ,n1 |2 6= |cµ,n2 |2 .
Open-loop convergence ρk+1 = M0µk (ρk ) For any initial condition ρ0 , I
with probability one, ρk converges to one of the d states |ni hn| with n ∈ {1, · · · , d}.
I
the probability of convergence towards the state |ni hn| depends only on ρ0 and is given by hn| ρ0 |ni .
Proof based on I
the martingales hn| ρ |ni
I
the super-martingale V (ρ) := −
P
n
hn|ρ|ni 2
2 satisfying
E (V (ρk+1 )|ρk ) − V (ρk ) = −Q(ρk ) ≤ 0 with Q(ρ) = I
1 4
0 0 n,µ,ν pµ,ρ pν,ρ
P
|cµ,n |2 hn|ρ|ni 0 pµ,ρ
−
|cν,n |2 hn|ρ|ni 0 pν,ρ
Q(ρ) = 0 iff exists n ∈ {1, . . . , d} such that ρ = |ni hn|.
2
.
u ¯ i hn ¯| Feedback stabilization of ρk+1 = Mµkk−τ (ρk ) towards |n I
Pd V0 (ρ) = n=1 σn hn| ρ |ni with σn ≥ 0 chosen such that σn¯ = 0 ¯ , the second-order u-derivative of and for any n 6= n u V0 (K (|ni hn|)) at u = 0 is strictly negative (Ku is the Kraus map): set of linear equations in σn solved by inverting an irreducible M-matrix (Perron-Frobenius theorem).
I
The function ( > 0 small enough): Pd V (ρ) = V0 (ρ) − 2 n=1 (hn| ρ |ni)2 still admits a unique global ¯ i hn ¯ |; for u close to 0, u 7→ V (Ku (|ni hn|)) is minimum at |n ¯ and strongly convexe for n = n ¯. strongly concave for any n 6= n
I
The delay of τ steps: stabilize the state χ = (ρ, β1 , · · · , βτ ) (βl ¯ i hn ¯ | , 0, . . . , 0) control input u delayed l steps) towards χ ¯ = (|n using the control-Lyapunov function W (χ) = V (Kβ1 (Kβ2 (. . . . . . Kβτ (ρ) . . .))).
¯ and small enough, the feedback For u uk = f (χk ) =: argmin ¯ ,u ¯] ξ∈[−u
E (W (χk +1 )|χk , uk = ξ)
ensures global stabilization towards χ. ¯
Quantum separation principle
I
Estimate the hidden state ρ by ρest satisfying u
k−τ est ρest k+1 = Mµk (ρk )
u
where ρ obeys to ρk+1 = Mµkk−τ (ρk ) with the stabilizing est feedback uk = f (ρest k , uk−1 , . . . , uk −τ ) computed using ρ instead of ρ. I
est If ker(ρest 0 ) ⊂ ker(ρ0 ), ρk and ρk converge almost surely ¯ i hn ¯ |. towards the target state |n 13 Proof based on : I I I
13
¯ | ρk |n ¯ i ∈ [0, 1], hn ¯ | ρk |n ¯ i |ρ0 , ρest linearity of E hn versus ρ0 , 0 decomposition ρest = γρ + (1 − γ)ρc0 with γ ∈]0, 1[. 0 0
Bouten, van Handel, 2008.
Imperfect measurements: the new ”observable” state ρb The left stochastic matrix η: ηµ0 ,µ ∈ |0, 1] is the probability of having the imperfect outcome µ0 ∈ {1, . . . , m0 } knowing that the perfect one is µ ∈ {1, . . . , m}. I ρ bk = E ρk |ρ0 , µ00 , . . . , µ0k−1 , u−τ , . . . , uk−τ −1 obeys to14 I
u
ρbk+1 = Lµk0−τ (b ρk ),
where
k
I I
ρ) = Luµ0 (b
Lu 0 (b ρ) µ Tr Luµ0 (b ρ)
with Luµ0 (b ρ) =
Pm
µ=1
ηµ0 ,µ Mµu ρbMµu † ;
µ0k is a random variable µ0 in {1, · · · , m0 } with taking values u
u
k −τ probability pµk0−τ ρk ) . ,b ρk = Tr Lµ0 (b
I
I
E (b ρk+1 |b ρk = ρ, uk−τ = u) = Ku (b ρ)
Assumption: for all n1 6=n2 in {1, · · · , d}, there exists µ0 ∈ {1, · · · , m0 }, s.t. Tr L0µ0 (|n1 i hn1 |) 6= Tr L0µ0 (|n2 i hn2 |) .
Open-loop convergence of ρbk towards |ni hn| with prob. hn| ρb0 |ni. 14
R. Somaraju et al., ACC 2012 (http://arxiv.org/abs/1109.5344)
u ¯ i hn ¯| Feedback stabilization of ρbk+1 = Lµk−τ (b ρk ) towards |n k
I
Pd With the previous function V (ρ) = V0 (ρ) − 2 n=1 (hn| ρ |ni)2 ¯ i hn ¯ | , 0, . . . , 0) using stabilize χ b = (b ρ, β1 , · · · , βτ ) towards χ ¯ = (|n the control-Lyapunov function W (b χ) = V (Kβ1 (Kβ2 (. . . . . . Kβτ (b ρ) . . .))).
I
¯ and small enough, the feedback For u uk = f (b χk ) =: argmin ¯ ,u ¯] ξ∈[−u
E (W (b χk +1 )|b χk , uk = ξ)
ensures global stabilization of χ bk towards χ. ¯ I
Since ρbk = E ρk |ρ0 , µ00 , . . . , µ0k −1 , u−τ , . . . , uk−τ −1 ¯ i hn ¯ |, ρk converges also convergences towards the pure state |n towards the same pure state.
Quantum separation principle I
Estimate the hidden state ρb by ρbest satisfying u
k−τ ρest ρbest k ) k+1 = Lµ0 (b k
where I
ρk obeys to
u
ρk+1 = Mµkk−τ (ρk ) with the stabilizing feedback uk = f (b ρest k , uk−1 , . . . , uk−τ ) computed using ρbest instead of ρ. I
µ0k = µ0 with probability ηµ0 ,µk . est
2 q p p est ρbk ρbk ρbk is , Tr
I
Filter stability: F ρbk , ρbk
I
always a sub-martingale15 . If ker(b ρest best 0 ) ⊂ ker(ρ0 ), ρk and ρ k converge almost surely ¯ ¯ towards the target state |ni hn|.
15
P.R., IEEE Trans. Automatic Control, 2011.
Closed-loop experimental data • Initial state coherent state ¯ = 3 photons with n • State estimation via a quantum filter of state ρest k . • Lyapunov state feedback uk = f (ρk ) stabilizing ¯ i hn ¯| towards |n • ρk is replaced by its estimate ρest k in the feedback (quantum separation principle) Sampling period 80 µs 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 box16 For each control input u,
µ0 \ µ
I
we have a total of m = 3 × 7 = 21 Kraus operators. The jumps are labeled by µ = (µa , µc ) with µa ∈ {no, g, e, gg, ge, eg, ee} labeling atom related jumps and µc ∈ {o, +, −} cavity decoherence jumps.
I
we have only m0 = 6 real detection possibilities µ0 ∈ {no, g, e, gg, ge, ee} corresponding respectively to no detection, a single detection in g, a single detection in e, a double detection both in g, a double detection one in g and the other in e, and a double detection both in e.
(no, µc )
no g e gg ge ee
1 0 0 0 0 0 16
(g, µc ) 1-d d (1-ηg ) d ηg 0 0 0
(e, µc ) 1-d d ηe d (1-ηe ) 0 0 0
(gg, µc )
(ee, µc )
(ge, µc ) or (eg,
)2
)2
(1-d )2 d (1-d )(1-ηg + d (1-d )(1-ηe + 2d ηe (1-ηg ) 2d ((1-ηg )(1-ηe ) + 2d ηg (1-ηe )
(1-d 2d (1-d )(1-ηg ) 2d (1-d )ηg 2d (1-ηg )2 22d ηg (1-ηg ) 2d ηg2
(1-d 2d (1-d )ηe 2d (1-d )(1-ηe ) 2d ηe2 22d ηe (1-ηe ) 2d (1-ηe )2
R. Somaraju et al.: ACC 2012 (http://arxiv.org/abs/1109.5344)
