Stabilisation par feedback de systèmes quantiques ouverts (

Stabilisation par feedback de systèmes quantiques ouverts ( Collège de France, 14 mars 2014)

Pierre Rouchon

Centre Automatique et Systèmes Mines ParisTech

1 / 52

Feedback for quantum systems: the back-action of the measure.

A typical stabilizing feedback-loop for a classical system

system

controller

Two kinds of stabilizing feedbacks for quantum systems

1. Measurement-based feedback: measurement back-action onS is stochastic (collapse of the wave-packet); controller is

classical; the control inputu is a classical variable appearing in some controlled Schrödinger equation;udepends on the past measures.

2. Coherent/autonomous feedback and reservoir engineering:the systemSis coupled to another quantum system (the controller);

the composite system,HS⊗ Hcontroller, is an open-quantum system relaxing to some target (separable) state or decoherence free subspace.

2 / 52

Outline

Feedback stabilization of photons: the LKB photon box The closed-loop experiment (2011)

Quantum stochastic model

QND measurement and the quantum-state feedback Dynamical models of open quantum systems

Discrete-time: Markov process/Kraus maps

Continuous-time: stochastic/Lindblad master equations Stabilization of "Schrödinger cats" by reservoir engineering

The principle

Discrete-time example: the LKB photon box

Continuous-time examples and Fokker-Planck equations Conclusion

Appendix

Design of a strict control Lyapunov function State estimation and stability of quantum filtering Schrödinger cats and Wigner functions

Reservoir engineering stabilization: complements Books on open quantum systems

3 / 52

The first experimental realization of a

1

Stabilization of a quantum state with exactlynphoton(s) (n=0,1,2,3, . . .).

Experiment:C. Sayrin et. al., Nature 477, 73-77, September 2011.

Theory:I. Dotsenko et al., Physical Review A, 80: 013805-013813, 2009.

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

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

1Courtesy of I. Dotsenko. Sampling period 80µs.

4 / 52

Three quantum features emphasized by the LKB photon box

d

dt|ψi=−_~ⁱH|ψi, d

dtρ=−_~ⁱ[H, ρ], H=H₀+uH₁ 2. Origin of dissipation: collapse of the wave packetinduced by the

measure of observableOwith spectral decompositionP

µλ_µPµ:

I measure outcomeµwith proba.pµ =hψ|Pµ|ψi= Tr(ρPµ) depending on|ψi,ρjust before the measurement

I measure back-action if outcomeµ:

|ψi 7→ |ψi⁺= Pµ|ψi

phψ|Pµ|ψi, ρ7→ρ+= PµρPµ

Tr(ρPµ) 3. Tensor product for the description of composite systems(S,M):

I Hilbert spaceH=HS⊗ HM

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.

5 / 52

The composite system

I SystemS

corresponds to a quantized mode in

HS = ( _∞

X

n=0

ψ_n|ni

(ψ_n)^∞_n=0∈l²(C) )

,

where

represents the Fock state associated to exactly

photons inside the cavity

I MeterM

is associated to atoms :

, each atom admits two energy levels and is described by a wave function

with

1; atoms leaving

are all in state

I

When atom comes out

of the composite system atom/field is

|ΨiB =|ψi ⊗ |gi.

6 / 52

S: quantum harmonic oscillator

I Hilbert space:

HS ={P

n≥0ψ_n|ni, (ψ_n)_n≥0∈l²(C)}.

I Quantum state space:

D={ρ∈ L(H^S), ρ^† =ρ, Tr(ρ) =1, ρ≥0}.

I Operators and commutations:

a|ni=√

n|n−1i,a^†|ni=√

n+1|n+1i; N=a^†a,N|ni=n|ni;

[a,a^†] =I,af(N) =f(N+I)a;

Dα=e^{αa†−α†a}. a=X +iP= ^√1

2 x+_∂x^∂

, [X,P] =ıI/2.

I Hamiltonian:H_S/~=ω_ca^†a+uc(a+a^†).

(associated classical dynamics:

dx

dt =ω_cp, ^dp_dt =−ω_cx−√ 2uc).

I Coherent state of amplitude α∈C:|αi=P

n≥0

e^{−|α|2/2√αn}

n!

|ni;|αi ≡_π^11/4e^ı

√2x=αe^−(x−

√ 2<α)2 2

a|αi=α|αi,Dα|0i=|αi.

|0

|1

|2 ω

|n

ω

u

... ...

7 / 52

M: 2-level system, i.e. a qubit

I Hilbert space:

H^M =C²={cg|gi+ce|ei, cg,ce∈C}.

I Quantum state space:

D={ρ∈ L(H^M), ρ^† =ρ, Tr(ρ) =1, ρ≥0}.

I Operators and commutations:

σ_-=|gihe|,σ₊=σ_-^†=|eihg| σ_x=σ_-+σ₊=|gihe|+|eihg|; σ_y=iσ_-−iσ₊=i|gihe| −i|eihg|; σ_z =σ₊σ_-−σ_-σ₊=|eihe| − |gihg|; σ_x²=I,σ_xσ_y =iσ_z,[σ_x,σ_y] =2iσ_z, . . .

I Hamiltonian:H_M/~=ω_qσ_z/2+u_qσ_x.

I Bloch sphere representation:

D=n

1

2 I+xσx+yσ_y+zσ_z (x,y,z)∈R³, x²+y²+z²≤1o

| g

| e ω

u

8 / 52

The Markov model (1)

C

B

D R

R

I When atom comes outB:|ΨiB =|ψi ⊗ |gi.

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

|Ψi^R2 =U_SM |ψi ⊗ |gi

= Mg|ψi

⊗ |gi+ Me|ψi

⊗ |ei whereU_SM is the total unitary transformation (Schrödinger propagator) defining the linear measurement operatorsMgand MeonHS. SinceU_SM is unitary,M^†_gMg+M^†_eMe=I.

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The Markov model (2)

C

B

D R

R

The unitary propagatorU_SM is derived from Jaynes-Cummings HamiltonianH_SMin the interaction frame. Two kind of qubit/cavity Halmitonians:

resonant,H_SM/~=i Ω(vt)/2

a^†⊗σ_-−a⊗σ₊ , dispersive,HSM/~= Ω²(vt)/(2δ)

N⊗σ_z, whereΩ(x) = Ω₀e^−x

2

w2,x =vt withv atom velocity,Ω₀vacuum Rabi pulsation,wradial mode-width and whereδ=ω_q−ω_c is the detuning between qubit pulsationω_qand cavity pulsationω_c (|δ| Ω₀).

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The Markov model (3)

Just before the measurement inD, the atom/field state is:

M_g|ψi ⊗ |gi+M_e|ψi ⊗ |ei

Denote byµ∈ {g,e}the measurement outcome in detectorD: with probabilitypµ=hψ|M^†_µMµ|ψiwe getµ. Just after the measurement outcomeµ,the state becomes separable:

|Ψi^D=^√1_p

µ (Mµ|ψi)⊗ |µi=

Mµ

√hψ|M^†µMµ|ψi|ψi

⊗ |µi.

Markov process(density matrix formulationρ∼ |ψihψ|)

ρ+=







M^g(ρ) = ^MgρM

† g

Tr(MgρM^†_g), with probabilityp_g= Tr

M_gρM^†_g

; Me(ρ) = ^MeρM†e

Tr(MeρM^†e), with probabilityp_e= Tr

M_eρM^†_e . Kraus map:

E

11 / 52

A controlled Markov process (input

)

Stateρ: the density operator of the photon(s) trapped in the cavity.

Outputy: quantum projective measure of the probe atom.

Theideal modelreads

ρ_k+1=



















DukMgρ_kM^†_gD^†_uk Tr

M_gρ_kM^†_g yk =gwith probabilitypg,k = Tr

M_gρ_kM^†_g

DukMeρ_kM^†_eD^†_uk Tr

Meρ_kM^†_e y_k =ewith probabilityp_e,k = Tr

M_eρ_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 operators in the linear dispersive case M_g=cos_φ

0N+φR

2

andM_e=sin_φ

0N+φR

2

:M^†_gM_g+M^†_eM_e=I withN=a^†a=diag(0,1,2, . . .)the photon number operator.

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u =

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

ρ_k+1=



















cos

φ₀N+φR

2

ρ_kcos

φ₀N+φR

2

Tr

cos²

φ₀N+φR

2

ρ_k

with prob. Tr cos²_φ

0N+φR

2

ρ_k

sin

φ₀N+φR

2

ρ_ksin

φ₀N+φR

2

Tr

sin²

φ₀N+φR

2

ρ_k

with prob. Tr sin²_φ

0N+φR

2

ρ_k

Steady state:any Fock stateρ=|n¯ihn¯|(n¯∈N) is a steady-state (no other steady state when(φ_R, φ₀, π)areQ-independent)

Martingales:for any real functiong,Vg(ρ) = Tr(g(N)ρ)is a martingale:

E

Convergence to a Fock state when(φ_R, φ₀, π)areQ-independent:

V(ρ) =−¹2

P

nhn|ρ|ni²is a super-martingale with

E

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

For a realization starting fromρ₀, the probability to converge towards the Fock state|¯nihn¯|is equal to Tr(|n¯ihn¯|ρ₀) =h¯n|ρ₀|n¯i. _{13 / 52}

Structure of the stabilizing quantum feedback scheme

I Goal: stabilization of the steady-state|n¯ihn¯|(controller set-point).

I At each time stepk:

1. ready_k the measurement outcome for probe atomk. 2. update the quantum state estimationρ^est_k−1toρ^est_k fromy_k 3. computeuk as a function ofρ^est_k (state feedback).

4. apply the micro-wave pulse of amplitudeuk. Anobserver/controllerstructure:

1. real-time state estimationbased on asymptotic observer: here quantum filteringtechniques;

2. state feedbackstabilization towards a stationary regime: here control Lyapunovtechniques based on open-loop martingales

Tr(g(N)ρ).

It takes into accountimperfections,delays(5 sampling) and cavity decoherence.

In finite dimension (truncation ton^maxphotons), all the mathematical details and convergence proof are given in the Automatica 2013

paper 14 / 52

LKB photon-box: Markov process with detection errors (1)

I With pure stateρ=|ψihψ|, we have ρ₊=|ψ₊ihψ₊|= 1

Tr

MµρM^†_µMµρM^†_µ

when the atom collapses inµ=g,ewith proba. Tr

MµρM^†_µ .

I Detection error rates: P(y =e/µ=g) =η_g∈[0,1]the probability of erroneous assignation toewhen the atom collapses ing;P(y =g/µ=e) =η_e∈[0,1](given by the contrast of the Ramsey fringes).

Bayes law: expectationρ+of|ψ+ihψ+|knowingρand the imperfect detectiony.

ρ₊=











(1−ηg)MgρM^†_g+η_eMeρM^†_e

Tr(^(1−ηg)MgρM^†g+ηeMeρM^†e)^ify=g, prob. Tr

(1−ηg)MgρM^†_g+ηeMeρM^†_e

;

η_gMgρM^†_g+(1−ηe)MeρM^†_e

Tr(^ηgMgρM^†_g+(1−ηe)MeρM^†_e)^ify=e, prob. Tr

ηgMgρM^†_g+ (1−ηe)MeρM^†_e .

ρ+does not remain pure: the quantum stateρ+becomes a mixed

state;|ψ+ibecomes physically irrelevant (not numerically). _{15 / 52}

LKB photon-box: Markov process with detection errors (2)

ρ₊=











(1−ηg)MgρM^†_g+η_eMeρM^†_e

Tr(^(1−ηg)MgρM^†g+ηeMeρM^†e), with prob. Tr

(1−ηg)MgρM^†_g+ηeMeρM^†_e

;

η_gMgρM^†_g+(1−ηe)MeρM^†_e

Tr(^ηgMgρM^†_g+(1−ηe)MeρM^†_e) with prob. Tr

ηgMgρM^†_g+ (1−ηe)MeρM^†_e .

Key point:

Tr

(1−η_g)M_gρM^†_g+η_eM_eρM^†_e

and Tr

η_gM_gρM^†_g+ (1−η_e)M_eρM^†_e are the probabilities to detecty =g ande, knowingρ.

Generalization:with(ηµ⁰,µ)a left stochastic matrixηµ⁰,µ≥0 and P

µ⁰ηµ⁰,µ=1, we have ρ₊=

P

µηµ⁰,µMµρM^†_µ Tr

P

µηµ⁰,µMµρM^†_µ when we detecty =µ⁰. The probability to detecty =µ⁰knowingρis Tr

P

µη_µ⁰_,µMµρM^†_µ .

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Continuous/discrete-time Stochastic Master Equation (SME)

ρ_k+1=

Pm

µ=1η_µ0,µMµρ_kM^†_µ

Tr(^Pmµ=1η_µ0,µMµρ_kM^†µ), with proba. pµ⁰(ρ_k) =Pm

µ=1ηµ⁰,µTr

Mµρ_kM^†_µ associated toKraus maps(ensemble average, quantum channel)

E

µ

Mµρ_kM^†_µ with X

µ

M^†_µMµ=I

Continuous-time modelsarestochastic differential systems dρ_t = −_~ⁱ[H, ρt] +X

ν

Lνρ_tL^†_ν−¹₂(L^†_νLνρ_t+ρ_tL^†_νLν)

! dt

+X

ν

√ην

Lνρ_t+ρ_tL^†_ν− Tr

(Lν+L^†_ν)ρ_t ρ_t

dWν,t

driven byWiener processdWν,t =dyν,t−√ην Tr

(Lν+L^†_ν)ρ_t dt with measuresy_ν,t, detection efficienciesη_ν ∈[0,1]and

Lindblad-Kossakowskimaster equations (ην ≡0):

d

dtρ=−_~ⁱ[H, ρ] + +X

ν

Lνρ_tL^†_ν−¹₂(L^†_νLνρ_t+ρ_tL^†_νLν)

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Continuous/discrete-time diffusive SME

With a single imperfect measuredyt =√η Tr

(L+L^†)ρ_t

dt+dWt

and detection efficiencyη∈[0,1], the quantum stateρ_t is usually mixed and obeys to

dρ_t =

−_~ⁱ[H, ρ_t] +Lρ_tL^†−¹₂(L^†Lρ_t+ρ_tL^†L) dt +√η

Lρ_t+ρ_tL^†− Tr

(L+L^†)ρ_t ρ_t

dW_t

driven by the Wiener processdWt

WithIt¯o rules, it can be written as the following "discrete-time" Markov model

ρ_t+dt= Mdytρ_tM^†_dyt + (1−η)LρtL^†dt Tr

Mdytρ_tM^†_dyt + (1−η)LρtL^†dt withM_dyt =I+

−_~ⁱH−¹2

L^†L

dt+√ηdy_tL.

18 / 52

A key physical example in circuit QED

Superconducting qubit dispersively coupled to a cavity traversed by a microwave signal (input/output theory).

The back-action on the qubit state of a single measurement of both output fieldquadraturesIt

andQt is described by a simple SME for the qubit density operator.

dρ_t = [u^∗σ_-−uσ₊, ρ_t] +γ_t(σ_zρσ_z−ρ) dt +p

ηγ_t/2 σ_zρ_t+ρ_tσ_z−2 Tr(σ_zρ_t)ρ_t

dW_t^I+ıp

ηγ_t/2[σz, ρ_t]dW_t^Q withI_t andQ_t given bydI_t =p

ηγ_t/2 Tr(2σ_zρ_t)dt+dW_t^I and

dQ_t =dW_t^Q, whereγ_t ≥0 is related by the read-outpulse shapeand η∈[0,1]is the detection efficiency.

3M. Hatridge et al. Quantum Back-Action of an Individual Variable-Strength Measurement. Science, 2013, 339, 178-181.

19 / 52

Watt regulator: classical analogue of quantum coherent feedback.

From WikiPedia

The first variations of speedδωand governor angleδθobey to

d

dtδω=−aδθ d²

dt²δθ=−Λd

dtδθ−Ω²(δθ−bδω) with (a,b,Λ,Ω) positive parame- ters.

Third order system d³

dt³δω+ Λd²

dt²δω+ Ω²d

dtδω+abΩ²δω=0.

Characteristic polynomialP(s) =s³+ Λs²+ Ω²s+abΩ²with roots having negative real parts iffΛ>ab:governor damping must be strong enough to ensure asymptotic stability.

Key issues:asymptotic stability and convergence rates.

4J.C. Maxwell: On governors. Proc. of the Royal Society, No.100, 1868.

20 / 52

Reservoir Engineering and coherent feedback

System Reservoir

Engineered interaction

dissipation

κ

H

H

H

H =H_res+H_int+H_syst

if

exponentially on a time scale of

1/κ then . . . .

5See, e.g., the lectures of H. Mabuchi delivered at the "Ecole de physique des Houches", July 2011.

21 / 52

Reservoir Engineering and coherent feedback

System Reservoir

Engineered interaction

dissipation

κ

H

H

H

γ

H =H_res+H_int+H_syst

. . . .

then

1

5See, e.g., the lectures of H. Mabuchi delivered at the "Ecole de physique des Houches", July 2011.

21 / 52

Reservoir stabilizing "Schrödinger cats"

-α

2.

Cavity mode (system) atom (reservoir)

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⊗I_M+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.

6A. Sarlette et al: Stabilization of Nonclassical States of the Radiation Field in a Cavity by Reservoir Engineering. Physical Review Letters, Volume 107, Issue 1, 2011.

22 / 52

Composite interaction with

ge

time t t_r/2

-t_r/2

-T/2 T/2

δ(t)δ₀

-δ₀ 0

0

top cavity mirror

bottom cavity mirror Ω(vt)

U =Uoff-resonant-1

U =X(ξ_N)Z(φ_N),

23 / 52

Composite interaction with

ge

time t t_r/2

-t_r/2

-T/2 T/2

δ(t)δ₀

-δ₀ 0

0

top cavity mirror

bottom cavity mirror Ω(vt)

U =U_resonantUoff-resonant-1

U =Y(θ_N^r) X(ξ_N)Z(φ_N),

23 / 52

Composite interaction with

ge

time t t_r/2

-t_r/2

-T/2 T/2

δ(t)δ₀

-δ₀ 0

0

top cavity mirror

bottom cavity mirror Ω(vt)

U =Uoff-resonant-2U_resonantUoff-resonant-1

U =Z(−φ_N)X(ξ_N) Y(θ^r_N) X(ξ_N)Z(φ_N),

23 / 52

Composite interaction with

ge

time t 0 t_r/2

-t_r/2

-T/2 T/2

δ(t)δ₀

-δ₀ 0

top cavity mirror

bottom cavity mirror Ω(vt)

U ≈e^−iφKerrN2U_resonante^iφKerrN2

23 / 52

Convergence of

iterates towards

-α

2

ρ^Kerr_k+1=K^Kerr(ρ^Kerr_k ) =M^Kerr_g ρ^Kerr_k (M^Kerr_g )^†+M^Kerr_e ρ^Kerr_k (M^Kerr_e )^†. withM^Kerr_g =cos(^u₂)cos(θN/2) +sin(^u₂)^{sin(θ√N/2)}

N a^† and M^Kerr_e =sin(^u₂)cos(θN+1/2)−cos(^u₂)a^{sin(θ√N/2)}

N .

Assume|u| ≤π/2,θ₀=0,θ_n∈]0, π[forn>0 and limn7→+∞θ_n=π/2, then (Zaki Leghtas, PhD thesis (2012))

I exists aunique common eigen-state|ψ^KerriofM^Kerr_g andM^Kerr_e : ρ^Kerr_∞ =|ψ^Kerrihψ^Kerr|fixed point ofK^Kerr.

I if, moreovern7→θ_nis increasing, limk7→+∞ρ^Kerr_k =ρ^Kerr_∞. For well chosen experimental parameters,ρ^Kerr_∞ ≈ |α_∞ihα_∞|and h^Kerr_N ≈πN²/2. Sincee^−iπ2N2|α_∞i= ^e−iπ/4√

2 |α_∞i+i|-α∞i :

k7→+∞lim ρ_k = ¹₂

|α_∞i+i|-α∞i

hα_∞|+ih-α∞|

6

= ¹₂|α_∞ihα_∞|+¹₂|-α∞ih-α∞|.

24 / 52

Wigner function

for different values of the density operator

W^ρ:C3ξ→ ²_π

Tr

e^iπND_−ξρD_ξ

∈[−

2/π, 2/π]

Re(ξ)

Im(ξ)

−0.6

−0.4

−0.2 0 0.2 0.4 Fock state |n=0> Fock state |n=3> Coherent state |α=1.8> 0.6

Coherent state |-α> Statistical mixture of

|-α> and |α> Cat state |-α>+|α>

-3-3

0 0

3 3

25 / 52

An approximation by a continuous-time model

Cavity mode (system) atom (reservoir)

R1

R2

Box of atoms

Aim:

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

DC field:

(controls atom frequency) ENS experiment

In the Kerr frameρ=e^−iπ/2N2ρ^Kerre^iπ/2N2:

d

dtρ^Kerr=u[a^†−a, ρ^Kerr] +κ aρ^Kerra^†−(Nρ^Kerr+ρ^KerrN)/2 Identical to the Lindbald master equation of adamped harmonic oscillator(κ >0) driven by a coherent input field of amplitudeu.

Simulations: convergence from vacuum inidealandrealisticcases.

7A. Sarlette et al: Stabilization of nonclassical states of one and two-mode radiation fields by reservoir engineering. Phys. Rev. A 86, 012114 (2012).

26 / 52

Convergence of the quantum damped harmonic oscillator

d

dtρ=u[a^†−a, ρ] +κ aρa^†−(Nρ+ρN)/2 converge exponentially towards|α∞ihα∞|withα∞=2u/κ.

Elementary proof:under the unitary change of frame ρ=e^{(α∞a†−α∞a)} ξe^{−(α∞a†−α∞a)} the new density operatorξis governed by

d

dtξ=κ aξa^†−(Nξ+ξN)/2

; its energyE = Tr(Nξ) = Tr a^†aξ

converges exponentially to 0 since it obeys to_dt^dE =−κE; thusξconverges exponentially to|0ih0|. Computation only based on commutation relations:

[a,a^†] =1, af(N) =f(N+I)a, e^{−(αa†+α∗a)}ae^{(αa†−α∗a)}=a+α.

27 / 52

Quantum Fokker-Planck equation: damped harmonic oscillator

d

dtρ=u[a^†−a, ρ] +κ aρa^†−(Nρ+ρN)/2 ρcan be represented by itsWigner functionW^ρdefined by

C3ξ=x +ip7→W^ρ(ξ) =_π² Tr

e^iπNe^{−ξa†+ξ∗a}ρe^{ξa†−ξ∗a} With the correspondences

∂

∂ξ = ¹₂ ∂

∂x −i ∂

∂p

, ∂

∂ξ^∗ =¹₂ ∂

∂x +i ∂

∂p

W^ρa=

ξ−¹2

∂

∂ξ^∗

W^ρ, W^aρ =

ξ+¹₂ ∂

∂ξ^∗

W^ρ W^ρa† =

ξ^∗+¹₂ ∂

∂ξ

W^ρ, W^a†ρ=

ξ^∗−¹2

∂

∂ξ

W^ρ we get the followingPDEforW^ρ(α∞=2u/κ):

∂W^ρ

∂t = ^κ₂ ∂

∂x

(x−α_∞)W^ρ + ∂

∂p

pW^ρ

+¹₄∆W^ρ

converging toward theGaussianW^ρ∞(x,p) =_π²e^{−2(x−α∞)2−2p2}.

28 / 52

Reservoir with the cavity relaxation (1/κ

photon life-time)

Cavity mode (system) atom (reservoir)

R1

R2

Box of atoms

Aim:

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

DC field:

(controls atom frequency) ENS experiment

In the Kerr representation frame

:

d dtρ^Kerr=

reservoir relaxation

z }| {

u[a^†−a, ρ^Kerr] +κ aρ^Kerra^†−(Nρ^Kerr+ρ^KerrN)/2 +κc e^iπNaρ^Kerra^†e^−iπN−(Nρ^Kerr+ρ^KerrN)/2

| {z }

cavity decoherence

.

8A. Sarlette et al: Stabilization of nonclassical states of one and two-mode radiation fields by reservoir engineering. Phys. Rev. A 86, 012114 (2012).

29 / 52

The steady-state for

0

The steady stateρ^Kerr_∞ in the Kerr frame

0=u[a^†−a, ρ^Kerr_∞ ] +κ aρ^Kerr_∞a^†−(Nρ^Kerr_∞ +ρ^Kerr_∞N)/2

+κ_c e^iπNaρ^Kerr_∞a^†e^−iπN−(Nρ^Kerr_∞ +ρ^Kerr_∞N)/2 is unique

ρ^Kerr_∞ = Z α^c_∞

−α^c_∞

µ(x)|xihx|dx.

The positive weight functionµ(Glauber-ShudarshanPdistribution) is given by

µ(x) =µ₀

((α^c_∞)²−x²)^(αc∞)2e^x2rc

α^c_∞−x ,

withr_c =2κ_c/(κ+κ_c)andα^c_∞=2u/(κ+κ_c). The normalization factorµ₀>0 ensures thatRα^c_∞

−α^c_∞µ(x)dx =1.

Conjecture: global (exponential) convergence towardsρ^Kerr_∞ ofρ^Kerr(t) ast 7→+∞.

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Robustness of the reservoir stabilizing the two-leg cat.

SinceW^{eiπNρKerre−iπN}(ξ) =W^ρKerr(−ξ) the master Lindblad equation

d dtρ^Kerr=

reservoir relaxation

z }| {

u[a^†−a, ρ^Kerr] +κ aρ^Kerra^†−(Nρ^Kerr+ρ^KerrN)/2 +κ_c ae^iπNρ^Kerre^−iπNa^†−(Nρ^Kerr+ρ^KerrN)/2

| {z }

cavity decoherence

.

yields to the followingnon localdiffusion PDE (quantum Fokker-Planck equation):

∂W^ρKerr

∂t (x,p)

=^κ+κ₂ ^c ∂

∂x

(x−α∞)W^ρKerr + ∂

∂p

pW^ρKerr

+¹₄∆W^ρKerr

(x,p)

+κc (x²+p²+¹₂) W^ρKerr (−x,−p)

−W^ρKerr (x,p)

!

+₁₆¹ ∆W^ρKerr (−x,−p)

−∆W^ρKerr (x,p)

!!

−κc





x 2





∂W^ρKerr

∂x (−x,−p)

+ ∂W^ρKerr

∂x (x,p)



+^p₂





∂W^ρKerr

∂p (−x,−p)

+ ∂W^ρKerr

∂p (x,p)









Convergence towardsW^ρKerr∞ (x,p) =Rα^c_∞

−α^c_∞ 2µ(α)

π e^{−2(x−α)2−2p2}dα remains to be proved.

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Quantum information processing with cat-qubits

It is possible withcircuit QEDto design an open quantum system governed by

d

dtρ=u[(a²)^†−a², ρ] +κ

a²ρ(a²)^†−((a²)^†a²ρ+ρ(a²)^†a²)/2

whereais replaced bya². The supports of all solutionsρ(t)converge to thedecoherence free spacespanned by the even and odd

cat-state;

|C_α⁺_∞i ∝ |α∞i+|-α∞i, |C_α⁻_∞i ∝ |α∞i − |-α∞iwithα∞=p 2u/κ.

The corresponding PDE forW^ρ is oforder 4inx andp.

A similar systemwhereais replaced now witha⁴could be very interesting for quantum information processing where the logical qubit is encoded in the planes spanned by even and odd cat-states:

|C⁺_α∞i,|C_i⁺_α

∞i ,

|C_α⁻_∞i,|C⁻_iα

∞i .withα∞=p⁴ 2u/κ.

The corresponding PDE forW^ρ is oforder 8inx andp.

9M. Mirrahimi et al: Dynamically protected cat-qubits: a new paradigm for universal quantum computation,arXiv:1312.2017v1, 2014.

32 / 52

Conclusion: convergence issues of open-quantum systems

ρ_k+1=K(ρ_k) =X

ν

Mνρ_kM_ν^† with X

ν

M_ν^†Mν =I Continuous-time model (Lindbald, Fokker-Planck eq.):

d

dtρ=−_~ⁱ[H, ρ] +X

ν

LνρL^†_ν−(L^†_νLνρ+ρL^†_νLν)/2

, Stability induces by contraction for a lot of metrics (nuclear norm

Tr(|ρ−σ|), fidelity Tr p√ρσ√ρ

, see the work of D. Petz).

Open issuesmotivated byrobustquantum information processing:

1. characterization of theΩ-limit support ofρ:decoherence free spaces are affine spaces where the dynamics are of Schrödinger types; they can be reduced to a point (pointer-state);

2. Estimation of convergence rate and robustness.

3. Reservoir engineering:design of realisticMν andLν to achieve rapid convergence towards prescribed affine spaces (protection against decoherence).

33 / 52

Collaborators and scientific environment

I Former PhDs and PostDocs:Hadis Amini, Hector Bessa Silveira, Zaki Leghtas, Alain Sarlette, Ram Somaraju.

I LKB Physicists: Michel Brune, Igor Dotsenko, Sébastien Gleyzes, Serge Haroche, Jean-Michel Raimond,Bruno Peaudecerf, Clément Sayrin, Xingxing Zhou.

I Quantic project (INRIA/ENS/MINES):Benjamin Huard, François Mallet, Mazyar Mirrahimi,Landry Bretheau, Philippe Campagne, Joachim Cohen, Emmanuel Flurin, Ananda Roy, Pierre Six.

I Mathematicians: Karine Beauchard, Jean-Michel Coron, Thomas Chambrion, Bernard Bonnard, Ugo Boscain, Sylvain Ervedoza, Stéphane Gaubert, Andrea Grigoriu, Claude Le Bris, Yvon Maday, Vahagn Nersesyan, Clément Pellégrini, Paulo Sergio Pereira da Silva, Jean-Pierre Puel, Lionel Rosier, Julien Salomon, Rodolphe Sepulchre, Mario Sigalotti, Gabriel Turinici.

I Centre Automatique et Systèmes:François Chaplais, Florent Di Méglio, Jean Lévine, Philippe Martin, Nicolas Petit, Laurent Praly.

I And also: Lectures at Collège de France, ANR projects CQUID

and EMAQS, UPS-COFECUB, . . . _{34 / 52}

Quantum state feedback to stabilize the set-point

The Lyapunov feedback scheme is based on astrict control Lyapunov function:

V(ρ) =X

n

−hn|ρ|ni²+σ_nhn|ρ|ni

where >0 is small enough and

σ_n=











1 4+P^¯n

ν=1 1

ν −ν¹², ifn=0;

P^¯n ν=n+1

1

ν −ν¹², ifn∈[1,¯n−1];

0, ifn= ¯n;

Pn ν=¯n+11

ν +_ν¹₂, ifn∈[¯n+1,+∞] Feedback law:u=f(ρ) =:Argmin

υ∈[−¯u,¯u]

V

Dυ

MgρM^†_g+MeρM^†_e D^†_υ

.

Achieveglobal stabilizationsince the decrease isstrict

∀ρ6=|n¯ihn¯|, V

D_f(ρ)

MgρM^†_g+MeρM^†_e D^†_f(ρ)

<V

ρ .

35 / 52

The control Lyapunov function used for experiment.

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) =P

n −hn|ρ|ni²+σnhn|ρ|ni

for

3.

σ_n∼log(n): key issue to avoid trajectories escaping ton= +∞.

36 / 52