On the first experimental realization of a quantum state feedback

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On the first experimental realization of a quantum state feedback

55th IEEE Conference on Decision and Control Las Vegas, USA, December 12-14, 2016

Pierre Rouchon

Centre Automatique et Systèmes, Mines ParisTech, PSL Research University Quantic Research Team, Inria

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Technologies for quantum simulation and computation

¹

Superconduc�ng circuits

Photons

Ultra-cold neutral/Rydberg

atoms

Trapped ions

Quantum dots

Requirement:

Scalable modular architecture

Control software from the very beginning

1Courtesy of Walter Riess, IBM Research - Zurich.

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Nobel Prize in Physics 2012

Serge Haroche David J. Wineland

" This year’s Nobel Prize in Physics honours the experimental inventions and discoveries that have allowed themeasurement and control of individual quantum systems. They belong to two separate but related technologies: ions in a harmonic trap and photons in a cavity"

From the Scientific Background on the Nobel Prize in Physics 2012 compiled by the Class for Physics of the Royal Swedish Academy of Sciences, October 9, 2012.

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Outline

The photon-box experiment of the Haroche group Modelling based on three quantum rules Measuring photons without destroying them Stabilizing measurement-based feedback

Model structures (decoherence, measurement back-action) Discrete-time models (hidden Markov chain)

Diffusive models (Wiener process) Jump models (Poisson process) Feedback for quantum systems

Measurement-based feedback

Coherent (autonomous) feedback and dissipation engineering

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The first experimental realization of a quantum-state feedback microwave photons

(10 GHz)

Experiment:C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini,M. Brune, J.M. Raimond,S. Haroche:

Real-time quantum feedback prepares and stabilizes photon number states.

Nature,2011, 477, 73-77.

Theory:I. Dotsenko,M. Mirrahimi,M. Brune,S. Haroche, J.M. Raimond,P. Rouchon:

Quantum feedback by discrete quantum non-demolition measurements: towards on-demand generation of photon-number states.Physical Review A,2009, 80:

013805-013813.

M. Mirrahimi et al. CDC 2009, 1451-1456,2009.

H. Amini et al. IEEE Trans. Automatic Control, 57 (8): 1918–1930,2012.

R. Somaraju et al., Rev. Math. Phys., 25, 1350001,2013.

H. Amini et. al., Automatica, 49 (9): 2683-2692,2013.

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Three quantum features emphasized by the photon box

² 1. Schrödinger: wave funct.|ψiof norm one in Hilbert spaceH

d

dt|ψi=−ⁱ

~H|ψi, H=H0+uH1, uclassical control input Dirac notations:|ψi ≡

•

,|ψi^†=hψ| ≡ • •

,H=H^†≡ • •

• •

. 2. Origin of dissipation: collapse of the wave packetinduced by the

measurement of observableOwith spectral decomp.P

yλ_yP_y:

I measured outputy with probabilityPy =hψ|P_y|ψi;

I back-action:

I Hilbert spaceH=H_S⊗ H_M

I HamiltonianH=H_S⊗IM+Hint+I_S⊗HM I observable on sub-systemM only:O=I_S⊗OM.

2S. Haroche and J.M. Raimond.Exploring the Quantum: Atoms, Cavities and Photons.Oxford Graduate Texts, 2006.

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Composite system

(S,M): harmonic oscillator⊗

qubit.

I SystemS

corresponds to a quantized harmonic oscillator:

H_S = (_∞

X

n=0

xn|ni

(xn)^∞_n=0∈l²(C) )

,

where

|ni

is the photon-number state with

n

photons (hn

₁|n₂i=δn1,n2

).

I MeterM

is a qubit, a 2-level system:

H_M =

xg |gi+xe|ei

xg,xe∈C

,

where

|gi

(resp.

|ei) is the ground (resp. excited) state

(hg|gi

=he|ei=

1 and

hg|ei=

0)

I State of the composite system|Ψi∈ H_S⊗ H_M

:

|Ψi=

+∞

X

n=0

xng |ni ⊗ |gi+xne |ni ⊗ |ei, xne,xng ∈C.

Ortho-normal basis:

|ni ⊗ |gi,|ni ⊗ |ei

n∈N

.

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The hidden Markov chain (1)

C

B

D R

1

R

₂

B R

₂

I When atom comes outB, the quantum state|Ψi_Bof the composite system isseparable:|Ψi_B=|ψi ⊗ |gi.

I Just before the measurement inD, the state is in general entangled(not separable):

|Ψi_R

2=U_SM |ψi ⊗ |gi

= M_g|ψi

⊗ |gi+ M_e|ψi

⊗ |ei whereU_SM is a unitary transformation (Schrödinger propagator) defining measurement operatorsMgandMeonH_S.

U_SM unitary impliesM^†_gM_g+M^†_eM_e≡I.

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The hidden Markov chain (2)

Just beforethe atom detectorDthe quantum state isentangled:

|Ψi_R

2 =M_g|ψi ⊗ |gi+M_e|ψi ⊗ |ei

Just afteroutcome³y ∈ {g,e}, the state becomesseparable:

|Ψi_D= q ^M^y

hψ|M^†_yMy|ψi|ψi

!

⊗ |yi with probabilityhψ|M^†_yM_y|ψi.

Hidden Markov chain:|ψi ⊗ |gi −→ q ^M^y

hψ|M^†_yMy|ψi|ψi

!

⊗ |yi

|ψ_k₊₁i=







Mg

q

hψ_k|M^†_gMg|ψ_ki|ψ_ki, yk =gwith probabilityhψ_k|M^†_gMg|ψ_ki;

Me

√

hψk|M^†_eMe|ψki|ψ_ki, yk =ewith probabilityhψ_k|M^†_eM_e|ψ_ki;

with state|ψ_kiand outputyk ∈ {g,e}at time-stepk:

3Measurement operatorO=IS⊗(|eihe| − |gihg|).

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Why density operators ρ instead of wave functions |ψi

Assume known|ψ₀ianddetector out of order:what about|ψ₁i?

I Expectation value of|ψ₁ihψ₁|knowing|ψ₀i:⁴ E

|ψ₁ihψ₁|

| {z }

ρ₁

|ψ₀i

=Mg|ψ₀ihψ₀|

| {z }

ρ₀

M^†_g+Me|ψ₀ihψ₀|

| {z }

ρ₀

M^†_e.

I SetK(ρ),MgρM^†_g+MeρM^†_efor any operatorρ.

I ρ_k expectation of|ψ_kihψ_k|knowingρ₀=|ψ₀ihψ₀|:

ρ₁=K(ρ₀), ρ₂=K(ρ₁), . . . ,ρ_k =K(ρ_k₋₁).

Linear mapK: positivity and trace preserving (M^†_gM_g+M^†_eM_e=I) calledKraus mapor quantum channel.

Density operatorsρ: Hermitian non-negative operators of trace one.

4|ψihψ|:orthogonal projector on line spanned by unitary vector|ψi.

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The Markov chain with

ρ

as hidden state

Detector efficiencyη∈[0,1]. Measured outputy ∈ {g,e,∅}:

ρ_k₊₁=











Kg(ρ_k)

Tr K_g(ρ_k),y_k =gwith probability Tr K_g(ρ_k)

; Ke(ρ_k)

Tr(K_e(ρ_k)),yk =ewith probability Tr(Ke(ρ_k));

K_∅(ρ_k)

Tr(K_∅(ρ_k)),y_k =∅with probability Tr(K_∅(ρ_k));

with Kraus maps

K_g(ρ) =ηM_gρM^†_g, K_e(ρ) =ηM_eρM^†_e K_∅(ρ) = (1−η)

M_gρM^†_g+M_eρM^†_e . We still have:

E

^ρ^k+1^ρ^k^,^K(ρ^k^{) =}^M^g^ρ^k^M^†^g⁺^M^e^ρ^k^M^†^e⁼^X

y

K_y(ρ_k).

Imperfections, decoherence modeled similarly

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A controlled Markov process (input

u, hidden stateρ, outputy

)

Inputu: classical amplitude of a coherent micro-wave pulse Stateρ: the density operator of the photon(s) trapped in the cavity Outputy: measurement of the probe atom

ρ_k₊₁=











D_u_kK_g(ρ_k)D^†_u_k

Tr Kg(ρ_k) ,yk =gwith probability Tr Kg(ρ_k)

; D_u_kK_e(ρ_k)D^†_u_k

Tr(Ke(ρ_k)) ,y_k =ewith probability Tr(K_e(ρ_k));

D_u_kK_∅(ρ_k)D^†_u_k

Tr(K_∅(ρ_k)) ,yk =∅with probability Tr(K_∅(ρ_k));

Controlled displacement unitary operator(u∈R):D_u=e^u^(a^†^−a)with a=upper diag(√

1,√

2, . . .)the photon annihilation operator.

Measurement Kraus operators with linear dispersive interaction Mg =cos

φ0N+φR

2

andMe=sin

φ0N+φR

2

:M^†_gMg+M^†_eMe=I with N=a^†a=diag(0,1,2, . . .)the photon number operator.

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Open-loop dynamics (u

=

0): experimental data

y

diag(ρ) ρ

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u =

0: Quantum Non Demolition (QND) measurement of photons.

ρ_k+1=











Kg(ρ_k)

Tr(Kg(ρ_k)),y_k =gwith probability Tr K_g(ρ_k)

;

Ke(ρ_k)

Tr(Ke(ρ_k)),yk =ewith probability Tr(Ke(ρ_k));

K∅(ρ_k)

Tr(K∅(ρ_k)),y_k =∅with probability Tr(K_∅(ρ_k));

MartingalesWg(ρ) = Tr(g(N)ρ) for any functiong:

E

^W^g^(ρ^k⁺¹⁾^/^ρ^k⁼^W^g^(ρ^k^).

Convergence:W(ρ) =1−P

nhn|ρ|ni²super-martingale,

E

^W^(ρk+1)/ρ_k

=W(ρ_k)−Q(ρ_k)

whereQ(ρ)≥0 andQ(ρ) =0 iff∃¯nsuch thatρ=|¯nih¯n|.

Probability to converge to|nih¯ n|: Tr¯ (|nih¯ n|ρ¯ ₀) =hn|ρ¯ ₀|¯ni.

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Feedback stabilization around 3-photon state: experimental data

y

u

diag(ρ) ρ

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Structure of the stabilizing quantum-state feedback scheme

With a sampling time of 80µs, the controller is classical

I Goal: stabilization towards|¯nih¯n|.

I Stepk−1 to stepk

1. readyk−1the measurement outcome for probe atomk−1.

2. computeρ_kfromρ_k−1viaρ_k =Duk−1Kyk−1(ρ_k−1)D^†_u_k−1 Tr K_y_k−1(ρ_k₋₁) , 3. computeu_k as a function ofρ_k(state feedback).

4. apply the micro-wave pulse of amplitudeu_k. Observer/controllerstructure:

1. real-time state estimation: the quantumBelavkinfilter 2. u=f(ρ)based on astrict control Lyapunov functionV(ρ)

derived from open-loop martingales Tr(g(N)ρ):

E

^V^(ρ^k+1⁾ρ_k,uk =f(ρ_k)

=V(ρ_k)−Q(ρ_k) whereQ(ρ)≥0 andQ(ρ) =0 iffρ=|¯nih¯n|.

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Experimental closed-loop data

C. Sayrin et. al., Nature 477, 73-77, Sept. 2011.

Decoherence due to finite photon life-time (70 ms) Detection efficiency 40%

Detection error rate 10%

Delay 4 sampling periods The quantum filter includes decoherence, detector imperfections and delays (Bayes law).

Truncation to 9 photons

Stabilization around 3-photon state

V u y

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Outline

The photon-box experiment of the Haroche group Modelling based on three quantum rules Measuring photons without destroying them Stabilizing measurement-based feedback

Model structures (decoherence, measurement back-action) Discrete-time models (hidden Markov chain)

Diffusive models (Wiener process) Jump models (Poisson process)

Feedback for quantum systems Measurement-based feedback

Coherent (autonomous) feedback and dissipation engineering

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Discrete-time Stochastic Master Equations (SME)

Trace preserving Kraus mapKudepending onu:

Ku(ρ) =X

ξ

Mu,ξρM^†_u,ξ with X

ξ

M^†_u,ξMu,ξ=I.

Take aleft stochastic matrix ηy,ξ

and setKu,y(ρ) =P

ξηy,ξMu,ξρM^†_u,ξ. Associated Markov chain reads:

ρ_k+1= Ku_k,y_k(ρ_k)

Tr(Ku_k,y_k(ρ_k)) withyk with probability Tr(Ku_k,y_k(ρ_k)). Classical inputu, hidden stateρ, measured outputy.

Ensemble average given byKusince

E

^ρ^k+1^ρ^k^,^u^k⁼^K^u^k^(ρ^k^).

Markov model useful for:

1. Monte-Carlo simulations of quantum trajectories(decoherence, measurement back-action).

2. quantum filteringto get the quantum stateρ_k fromρ₀and(y0, . . . ,yk−1) (Belavkin quantum filterdeveloped for diffusive models).

3. feedback design and Monte-Carlo closed-loop simulations.

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Stochastic master equation driven by Wiener process

Lindblad equation(HHermitian,Larbitrary, depending onu):

d

dtρ,L(ρ) = −ⁱ

~[H,ρ]

| {z } Schrödinger

+LρL^†−¹

2(L^†Lρ+ρL^†L)

| {z } ,DL(ρ)decoherence For anyt≥0,ρ₀7→e^tL(ρ₀) =ρ_t is a trace preserving Kraus map.

Continuous-time outputt7→ytwithdyt=√

ηTr (L+L^†)ρ_t

dt+dWt: dρ_t =

−ⁱ

~[H,ρ_t] +Lρ_tL^†−¹

2(L^†Lρ_t+ρ_tL^†L)

dt +√

η

Lρ_t+ρ_tL^†− Tr

(L+L^†)ρ_t ρ_t

dWt

driven by Wiener processdWt (ηdetection efficiency).

Another formulation withIt¯o differentiation rule:

ρ_t_+dt = Kdy_t(ρ)

Tr(Kdy_t(ρ))with proba. density Tr(Kdy_t(ρ))fordyt

Kdy(ρ) =e⁻^dy

2 2dt

MdyρM^†_dy+ (1−η)LρtL^†dt Mdy =I+ −ⁱ

~H−¹₂ L^†L dt+√

ηdyL.

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Stochastic master equation driven by Poisson process

Lindblad equation: _dt^dρ,L(ρ) =−ⁱ

~[H,ρ] +LρL^†−¹₂(L^†Lρ+ρL^†L) Countert7→y(t)∈N(detection imperfectionsϑ≥0,η∈[0,1]):

dy(t) =0or1:

E

^dy^(t)^ρ^t⁼^ϑ⁺^η^Tr^Lρ^t^L^† ^dt

Dynamics:

dρ_t =

−ⁱ

~[H,ρ_t] +LρtL^†−¹

2(L^†Lρt+ρ_tL^†L)

dt

+ ϑρ_t+ηLρ_tL^† ϑ+ηTr LρtL^†−ρ_t

!

dy(t)−

ϑ+ηTr

Lρ_tL^† dt

dy(t) =0:ρ_t+dt = K0(ρ_t)

Tr(K0(ρ_t))with probability Tr(K0(ρ_t)) dy(t) =1:ρ_t+dt = K1(ρ_t)

Tr(K1(ρ_t))with probability Tr(K1(ρ_t)) K0(ρ) =

M0ρM^†₀+ (1−η)LρL^†dt

(1−ϑdt), K1(ρ) = ϑρ+ηLρL^† dt M0=I+ −ⁱ

~H−¹₂L^†L dt

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Outline

The photon-box experiment of the Haroche group Modelling based on three quantum rules Measuring photons without destroying them Stabilizing measurement-based feedback

Model structures (decoherence, measurement back-action) Discrete-time models (hidden Markov chain)

Diffusive models (Wiener process) Jump models (Poisson process)

Feedback for quantum systems Measurement-based feedback

Coherent (autonomous) feedback and dissipation engineering

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Measurement-based feedback

system

controller

quantum world

classical world

y

u

decoherence

P-controller (Markovian feedback^a):

foru(t) =f(y(t)), average closed-loop dynamics ofρremains governed by a Lindblad master equation.

PID controller:no Lindblad master equation in closed-loop;

Nonlinear hidden-state stochastic systems:convergence analysis, Lyapunov exponents, dynamic output feedback, delays, robustness, . . .

aH.M. Wiseman:Quantum Trajectories and Feedback. PhD Thesis,

University of Queensland, 1994.

Short sampling times limit feedback complexity

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First SISO measurement-based feedback for a superconducting qubit⁵

Analog Multiplier

d Feedback circuit

X*Y

Rabi reference 3.0 MHz 10 MHz

Phase error = Feedback signal _X ^Y

a Signal generation

Rabi drive c Homodyne setup

I LO Readout

drive Q

LO Digitizer/

computer Q RF

RF I

b Dilution refrigerator

Transmon qubit

IN

(x2) Paramp HEMT

Q Q

OUT ^|0>

I Readout

cavity

|1>

|0>

|1> ^I

5R. Vijay, . . . , I. Siddiqi.Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback.Nature 490, 77-80, October 2012.

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First MIMO measurement-based feedback for a superconducting qubit⁶

FM Rabi

b)

~ ~

FM

~ ~

~

Rabi

JPC out in

Drift

a) ,

c) 3 inputs

2 outputs

6P. Campagne-Ibarcq, . . . , B. Huard:Using Spontaneous Emission of a Qubit as a Resource for Feedback Control.Phys. Rev. Lett. 117(6), 2016.

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Measurement-based feedback stabilization of ^√¹

2(|gi⊗|ei+|ei⊗|gi) ⁷

Alice Bob

7D. Ristè,. . . , L. DiCarlo:Deterministic entanglement of superconducting qubits by parity measurement and feedback. Nature 502, 350-354 (2013).

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Coherent (autonomous) feedback (dissipation engineering)

Quantum analogue of Watt speed governor: adissipative mechanical system controls another mechanical system⁸

system

controller quantum world classical world

decoherence

u

y

u

c

y

c

Optical pumping (Kastler1950), coherent population trapping (Arimondo1996) Dissipation engineering, autonomous feedback: (Zoller, Cirac, Wolf, Verstraete, Devoret, Schoelkopf, Siddiqi, Lloyd, Viola, Ticozzi, Mirrahimi, Sarlette, ...)

(S,L,H) theoryandlinear quantum systems: quantum feedback networks based on stochastic Schrödinger equation, Heisenberg picture (Gardiner, Yurke, Mabuchi, Genoni, Serafini, Milburn, Wiseman, Doherty,Gough, James, Petersen, Nurdin, Yamamoto, Zhang, Dong, . . . )

Stability analysis: Kraus maps and Lindblad propagators are always contractions (non commutative diffusion and consensus).

8J.C. Maxwell:On governors.Proc. of the Royal Society, No.100, 1868.

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Reservoir engineering to stabilize "Schrödinger cats"|αi+|-αi⁹ Cavity mode (system)

atom (controller) R1

R2

Box of atoms

Aim:

engineer atom-mode interaction, to stabilize |-α +|α

DC field:

(controls atom frequency) ENS experiment

Jaynes-Cumming Hamiltionian

H(t)/~=ω_ca^†a⊗IM+ω_q(t)I_S⊗σ_z/2+iΩ(t) a^†⊗σ_-−a⊗σ₊ /2 with the open-loop controlt7→ω_q(t)combiningdispersiveω_q6=ω_c andresonantω_q=ω_cinteractions.

Key issues:convergenceofρ_k+1=K(ρk) =Mgρ_kM^†_g+Meρ_kM^†_e.

9A. Sarlette et al:Stabilization of nonclassical states of the radiation field in a cavity by reservoir engineering.Phys. Rev. Lett. 107(1), 2011.

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Autonomous feedback stabilization of

^√¹

2(|gi⊗|ei−|ei⊗|gi)¹⁰

Lindblad master equation:

d

dtρ=−i[H(t),ρ] +κDa(ρ) + ¹

T₁^AD_σA

−(ρ) + ¹

2T_φ^AD_σA z(ρ) + ¹

T₁^BD

σ^B₋(ρ) + ¹

2T_φ^BD

σ^B_z(ρ) with

H(t)/~=_χ

A

2σ_z^A+^χ₂^Bσ^B_z a^†a +2ccos

χ_A+χ_B

2 t a+a^† + Ω0

σ^A_x+σ^B_x +Ωn

e⁻ⁱⁿ

χ_A+χ_B 2 t

(σ₊^A−σ₊^B) +h.c.

10S. Shankar, . . . , M.H. Devoret.Autonomously stabilized entanglement between two superconducting quantum bits.Nature, 504: 419-422, 2013.

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Combining measurement-based feedback with dissipation engineering¹¹

11Y. Liu, . . . , M.H. Devoret:Comparing and combining measurement-based and driven-dissipative entanglement stabilization.Phys. Rev. X 6, 2016.

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Concluding remarks

I Three key quantum rules (highlighted by Haroche photon-box) 1. Schrödinger equationsdefining unitary transformations, 2. partial collapseof the wave packet: dissipation induced by

measurement of observables withdegeneratespectra, 3. tensor productforcomposite systems,

explain structure ofstochastic master equations, illustrate importance ofspin/spring models.

I Feedback control ofcoherenceandentanglementin composite systems: important issues for quantum computing, simulation, metrology and communication.

I Composite systems:curse of dimensionalityfor quantum networks with delays in coherent feedback loops.

Importance of collaborations with experimental quantum physicists for defining relevant control engineering questions

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